{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v00 --supports--> v02\n- v05 --refutes--> v06\n- v06 --refutes--> v01\n- v06 --supports--> v00\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7c2bb375a1813ae5ba62571c3ef9593cc9dd77b016e024fbcb85cdf60928bf2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v05 --supports--> v06\n- v03 --refutes--> v04\n- v04 --supports--> v01\n- v04 --supports--> v05\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":"f2b72d435b6aba9893353f7e020700c93de84019eb93e92a0e276194258e0202","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v04 --refutes--> v01\n- v01 --refutes--> v02\n- v05 --supports--> v03\n- v01 --supports--> v00\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d122827782e656dcd17a69b3387b9c689d007c151904f081029b0f74b224085e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v06 --supports--> v05\n- v02 --supports--> v00\n- v06 --refutes--> v03\n- v04 --supports--> v06\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0ec0d39b052b20a4b31812f7805cbc02a716b21ea72374c0df83b5bd22b3dc3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v00 --refutes--> v01\n- v06 --refutes--> v05\n- v04 --supports--> v00\n- v05 --refutes--> v04\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f047f72eac7fa34f9e6434d02a22a66c32ab190ab457b89ceef9e193ec8a014","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v00 --supports--> v02\n- v00 --refutes--> v03\n- v02 --supports--> v05\n- v04 --supports--> v00\n- v06 --refutes--> 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":"fc07fb7cff0a7ebb2fdf6bb2e25ebf3bd22d86a3ef96ff4c1ba651b223db0d77","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v01 --supports--> v05\n- v03 --refutes--> v04\n- v00 --refutes--> v03\n- v04 --supports--> v01\n- v01 --supports--> v06\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":"3886e40c9fef0135a00310110307db94f4374955c50cbbe48683af96955e6fb8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v01 --refutes--> v00\n- v02 --supports--> v05\n- v03 --refutes--> v04\n- v02 --refutes--> v06\n- v00 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"46b0ab1d10336d9e8b9aec083b7213849ffdb3b137cdbb13520198c9d80c7957","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v06\n- v02 --supports--> v00\n- v05 --supports--> v01\n- v03 --refutes--> v02\n- v04 --refutes--> v05\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"be065447d9bba6a0b88f84e4646766a2b7ba5b19d20c3e3d51da18d2e2920db1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v05 --refutes--> v06\n- v02 --refutes--> v01\n- v02 --supports--> v00\n- v04 --refutes--> v02\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":"d86fdfe21ece566606799023ab4fdfc4b8068fe147668b138481e320b3293b21","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v05 --supports--> v00\n- v02 --supports--> v06\n- v02 --refutes--> v01\n- v02 --supports--> v03\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b4154cb61e5e77607678663ec6383189e76f9b29ea367ba37f99b0bbb02a61a3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v02 --refutes--> v06\n- v06 --supports--> v03\n- v06 --supports--> v01\n- v05 --refutes--> v00\n- v02 --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":"5a1666bfbe89886a44720df9966032a65ce74393e021e273a3e39894403fe70f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v04 --refutes--> v02\n- v00 --refutes--> v03\n- v00 --supports--> v05\n- v03 --supports--> v06\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":"1e1f507fb0847ac700de314cd40000b68f8c6946c87880001513c489ca0c0c12","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v00 --supports--> v03\n- v02 --refutes--> v06\n- v01 --supports--> v00\n- v00 --supports--> v04\n- v03 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7649c975f4d9f5ccce8e863e7d94b15f4acef04c30c633561ec62a8e07b85cfe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v00 --supports--> v06\n- v04 --refutes--> v01\n- v02 --supports--> v03\n- v01 --refutes--> v02\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd721b7afd69d038d0d7a5dc7713f429651079c9b719757eedddedde561dcafb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v02 --supports--> v01\n- v05 --refutes--> v03\n- v02 --supports--> v06\n- v04 --refutes--> v05\n- v02 --supports--> v00\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":"0f7432e9fd25e7ad34c65b74358e4351e9889ba81b3a002b92f9b21925081454","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v01 --refutes--> v06\n- v01 --refutes--> v05\n- v01 --supports--> v02\n- v03 --supports--> v01\n- v04 --refutes--> v00\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba5d41179c61c590db5b90da49d95d12b646faa94949f8fcb2db78d8c8417bad","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --supports--> v06\n- v01 --supports--> v05\n- v03 --refutes--> v01\n- v00 --refutes--> v04\n- v00 --supports--> v02\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":"7969e7e3275fade58a19d2b5d962ede2c4831e4aa2e917753d30e70af601e9f3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v05 --refutes--> v02\n- v05 --refutes--> v03\n- v05 --refutes--> v00\n- v06 --refutes--> v01\n- v00 --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":"d41296d694080a9a2a3484342eec4955ac06fa327ca01d4be59cc8dd218d64b9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v01 --refutes--> v06\n- v01 --refutes--> v03\n- v05 --supports--> v02\n- v00 --supports--> v05\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d189faf6696f665b1325ab175b7d608ca0ce959d67941ea9884338daf098362d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v05\n- v04 --refutes--> v03\n- v05 --supports--> v06\n- v05 --supports--> v02\n- v05 --supports--> v00\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":"50b5be83b9c03498ec7e7e282c42c7a14b160a0db95bd589459b6564db0bae2e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v04 --refutes--> v02\n- v00 --refutes--> v04\n- v05 --refutes--> v00\n- v06 --refutes--> v05\n- v00 --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":"71b8d70fe0c389d3b49cf2eddbaa537e26ed1a3713109afa4f706151dd8500ae","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v06 --refutes--> v00\n- v00 --refutes--> v02\n- v03 --supports--> v04\n- v05 --supports--> v03\n- v00 --refutes--> v05\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":"48bd452aa5db544c451b5b24a7bc6b99ea04740b123288065a4d17af32387fa1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v02 --supports--> v04\n- v05 --refutes--> v00\n- v02 --supports--> v03\n- v05 --refutes--> v02\n- v03 --supports--> v06\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":"53a539c473dd70a3eae4a5cc1aad162ae6fc703603aa6a62d665f353dd74e583","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v03 --refutes--> v00\n- v05 --supports--> v04\n- v05 --refutes--> v01\n- v06 --refutes--> v03\n- v00 --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":"df98e4be738c11e03dccc3ef11795dc68bef68d5edd2e5cc912884c8dbeea6de","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v06 --supports--> v00\n- v02 --supports--> v01\n- v00 --refutes--> v05\n- v00 --supports--> v03\n- v01 --supports--> v06\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":"6e7b17ba4a1785bf8c2788d9730c52ea198ed9df7588e1b56ab7fe2cf5c07e29","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v02 --refutes--> v00\n- v06 --supports--> v02\n- v03 --refutes--> v01\n- v01 --refutes--> v05\n- v01 --supports--> v04\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":"6200ec3a7daa63d0982d1483e9e29361614225f94bb3531feeda0ef324fef062","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v00 --supports--> v05\n- v05 --refutes--> v04\n- v03 --supports--> v02\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":"a8612578a254a03242d2068805767e700e4262c412173ecb6b716b96cea83e3e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v02 --supports--> v05\n- v02 --refutes--> v04\n- v03 --refutes--> v01\n- v05 --supports--> v03\n- v01 --refutes--> v00\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":"ccc79fade50265889309491b992ca59ea82e69c4111b792b032b4c82222e35d4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v05 --supports--> v04\n- v02 --supports--> v06\n- v00 --refutes--> v01\n- v03 --supports--> v00\n- v02 --supports--> v03\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":"64526f1d41e30460668b81d538aa78ab81bf84d773e2afb74948c2cffb538df9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v06 --supports--> v01\n- v04 --refutes--> v05\n- v02 --supports--> v04\n- v05 --supports--> v06\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":"d9ba5a14718fe5944cadf754dd95e11aca1eecccca04d97182d08bdbce5f590d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v04\n- v04 --supports--> v02\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":"d7277f9bbcdcc49af0755bfa2037c11c1e7b34c06fe0af38d168b9205ff3c6b7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v05 --refutes--> v03\n- v01 --supports--> v00\n- v02 --refutes--> v04\n- v01 --refutes--> v02\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cd5734a099256bc21fe4d6b200aa7121f44a025a28aeac571540ae0b8b5d204e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v02 --supports--> v06\n- v03 --refutes--> v02\n- v00 --refutes--> v03\n- v01 --supports--> v00\n- v01 --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":"128b1212a2fd99501c0209b872e4665f756f4abf742359ae48d513e6a61e0ab4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v03 --refutes--> v04\n- v04 --refutes--> v01\n- v02 --refutes--> v06\n- v04 --supports--> v05\n- v02 --supports--> v03\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":"78c647628f6eb2008dac42bd07b77395df8fb9e56c6bfbee197fa5f2f3e42f69","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v00 --supports--> v04\n- v04 --refutes--> v02\n- v01 --supports--> v00\n- v00 --refutes--> v06\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb4de8cbd0a167b6657c3664cfa19de93d7ae82ed22769b61c89bb397b0764ac","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v06 --refutes--> v00\n- v05 --supports--> v04\n- v06 --supports--> v01\n- v04 --refutes--> v03\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2d4087b2828426758bbd0b2265ca225132075730773cb31f5f777ccb9c7b36e4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v06 --refutes--> v01\n- v05 --refutes--> v00\n- v06 --refutes--> v04\n- v04 --supports--> v05\n- v05 --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":"7c7226fcb4b87ada174fe8da790e681f4e3ab5a159342ec9d3d466d08a30eb24","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v06 --refutes--> v03\n- v01 --supports--> v02\n- v06 --supports--> v05\n- v00 --supports--> v01\n- v03 --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":"e0caf87e5329bfd9c0e0d90d3fb32726edcc10e429d3f00c57d912aa6e871aa1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v06 --refutes--> v01\n- v06 --refutes--> v04\n- v02 --refutes--> v06\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":"ec0799ec7c6e0aa84ca39e85ca9bc9b88fc2397cf45f9b8f054650274a9719a3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v01 --refutes--> v05\n- v03 --refutes--> v06\n- v01 --supports--> v00\n- v02 --supports--> v03\n- v03 --supports--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ef6135cf5a9b613dab07054e22ca7dca2ec2b142636edb5c5d40f50f53157d28","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --refutes--> v04\n- v02 --refutes--> v00\n- v00 --refutes--> v01\n- v03 --refutes--> v02\n- v06 --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":"6338dc3c6f91ac99e03ebc53a4912269795a973e430c2be0c2079ae351e2f7b5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v00\n- v00 --supports--> v06\n- v04 --refutes--> v02\n- v06 --supports--> v03\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":"f5288b452fb3d4795438b62567eef83911bd4025f6757ebaf0664d3edadc2935","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v05 --refutes--> v04\n- v04 --supports--> v00\n- v02 --supports--> v05\n- v06 --supports--> v02\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":"dc8b96f67031f58117a15963fa24f69553d52f6ff197385efc39190273993b3a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v05 --refutes--> v03\n- v01 --refutes--> v02\n- v02 --refutes--> v00\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":"69e48e1c42cef9489047f1b112258d3ce1c6098f69e09703f7d5be8f524b5d79","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v02 --supports--> v04\n- v06 --supports--> v01\n- v03 --supports--> v05\n- v02 --refutes--> v00\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d46fb0a27f2e0f859759e698fa1b6e8c29cfd5aed4ca1fe4f7ae4b180f17727","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v00 --supports--> v02\n- v05 --refutes--> v04\n- v01 --supports--> v06\n- v05 --refutes--> v01\n- v01 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2858585374ca6705086ba5519d47b3907d1fb933a567ee2813f07b373fd680b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v04 --supports--> v06\n- v04 --supports--> v01\n- v00 --supports--> v02\n- v02 --supports--> v04\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":"dd966d120e91b03232587a34ecbb977e65ec67a7ec10d8017111f08b4f59f12b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v02 --refutes--> v03\n- v02 --refutes--> v00\n- v06 --refutes--> v02\n- v01 --supports--> v04\n- v00 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"81270a8d91c691a6d26dd038c661362ebb8cf270940b71eb1c38c4aaec47626e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v00 --refutes--> v01\n- v04 --supports--> v02\n- v03 --refutes--> v04\n- v04 --supports--> v06\n- v05 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9d97dcdfa2b6a16e808739d0a934749737679b11193dbf1d2dd3bebb5f6ab0da","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v01 --refutes--> v02\n- v05 --refutes--> v01\n- v04 --refutes--> v06\n- v06 --supports--> v00\n- v06 --refutes--> 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":"3ee6d39f5a8b9b467a433b72437f280a21d100f304dae633264f3cf77f3eb14f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v01 --supports--> v05\n- v01 --supports--> v04\n- v02 --refutes--> v06\n- v02 --refutes--> v01\n- v03 --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":"0e2a57ba6f7e78cdbdb9c9db10224c8b8e42fb32ecb154f8c6d0c7754c453b6c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v03 --supports--> v04\n- v02 --refutes--> v06\n- v06 --supports--> v03\n- v06 --refutes--> v05\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0caa6f3b7066939e5ce82958d154bf68aea3c79007cfe890cc441c5be361c1fb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v04 --supports--> v03\n- v00 --refutes--> v02\n- v04 --supports--> v05\n- v02 --refutes--> v01\n- v03 --supports--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bd38f109fc63f3f18429ef2f2c133ca8d2236b97bc1cfeb5577dfc0465da76b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v01\n- v03 --refutes--> v04\n- v00 --supports--> v03\n- v04 --supports--> v02\n- v02 --supports--> v06\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":"ca305d0efe7109a39f5d428b4665edaa8ef53545ffc4ef3db9eff3ce8da10640","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v00 --refutes--> v01\n- v04 --refutes--> v03\n- v04 --supports--> v00\n- v04 --supports--> v05\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":"9abaf123ac87a84e837ab5767a12cc4de79729748fac7fe34d95e89a30ae47d4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v05 --refutes--> v01\n- v00 --refutes--> v06\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":"cb8df573c0ae8bacb19e679c2fc5feb24177e0243dfcaae53f81e1fb42bbef69","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v02 --supports--> v05\n- v05 --refutes--> v06\n- v00 --refutes--> v02\n- v06 --refutes--> v04\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e72084a40bbdf61d88e10adb08b5d6c6841ab13560f959b34b8abfba2c19cc9f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v01 --supports--> v00\n- v04 --refutes--> v06\n- v04 --refutes--> v02\n- v00 --refutes--> v05\n- v05 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cc8035463dda6f2d66910b3d40562d81930d9d085059c4802701b60a7c798834","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v04 --supports--> v03\n- v02 --supports--> v00\n- v05 --refutes--> v02\n- v04 --refutes--> v06\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ba981369622a32e3dcc87e8a851b4227967a9f7dc54f525638f657d8a979c74a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v00 --refutes--> v04\n- v04 --refutes--> v01\n- v00 --supports--> v06\n- v03 --supports--> v05\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b48909496a0f8cd8e5c8e9b0911c365bcb45bc2d9bfd74d622933fa75fec9c64","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v04 --supports--> v06\n- v02 --supports--> v00\n- v06 --supports--> v03\n- v03 --refutes--> v02\n- v02 --refutes--> v05\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f50d625aa7c0a3fc47d55f63f63639a5f34b1c80697e0c472b81c2c87b43c5b9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v00 --supports--> v02\n- v00 --supports--> v06\n- v04 --refutes--> v03\n- v01 --supports--> v00\n- v03 --supports--> 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":"05470211ef160fa7529df27df372498a991903c720306ea128ce5ccd67d6c058","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --refutes--> v02\n- v06 --refutes--> v00\n- v02 --supports--> v03\n- v05 --supports--> v04\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f3061a4703ae74880082e018c889a22d6a4bc5b949d550691fd49fe7d283fa0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v03 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v05\n- v06 --supports--> v03\n- v04 --supports--> v00\n- v05 --refutes--> v07\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":"45555419635cd95dd13dc91e2d48663c550984d05f9dfebd38122f58dea702d8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v01 --refutes--> v03\n- v00 --supports--> v05\n- v05 --supports--> v01\n- v02 --supports--> v07\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":"7a446d7d320e8a2dc807cb5aebf058cabb3ba4ac6ebcb12967828e9b331b15f8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v04 --refutes--> v01\n- v00 --refutes--> v02\n- v05 --refutes--> v03\n- v02 --supports--> v05\n- v01 --refutes--> v00\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7b4594c32d450a725c80f112c4efa52bf4ae17e41a1bbd23ffe7470d8ca1f7c7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v07 --refutes--> v06\n- v03 --refutes--> v00\n- v00 --supports--> v04\n- v02 --supports--> v03\n- v01 --supports--> v02\n- v01 --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":"6ba683e5f7c3c67625d2ccb1edd4e3ac190cbb360dcc953ab2c3a1e7f27ef94c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v03 --supports--> v05\n- v06 --supports--> v04\n- v07 --refutes--> v00\n- v06 --refutes--> v01\n- v03 --supports--> v06\n- v02 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b7895b8196316b45646ff6942ec648cc7c7853d94a01165dca28c1a4e4a1c88a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v00 --refutes--> v06\n- v04 --supports--> v02\n- v03 --supports--> v04\n- v06 --refutes--> v01\n- v02 --refutes--> v00\n- v03 --supports--> v07\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":"9640c847326b6559cc2636049d8e4d1b98e58b9f53e31beef8680eee85c28bd8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v05 --supports--> v02\n- v01 --supports--> v04\n- v01 --supports--> v03\n- v00 --supports--> v05\n- v00 --supports--> v06\n- v04 --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":"c0bdb6e349d5f776b0ab36afe4b37d4f917215b1cb9b5a6ddd0ab69f088d2463","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v02 --supports--> v05\n- v07 --supports--> v02\n- v01 --supports--> v03\n- v03 --supports--> v06\n- v02 --refutes--> v00\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30f2c0792fd0d2d7343c8d4064a29008879b3330cf544c555bee10eeb697661c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v07 --supports--> v04\n- v00 --refutes--> v03\n- v02 --supports--> v05\n- v01 --supports--> v06\n- v00 --refutes--> v02\n- v06 --supports--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fd4307e7af8997b5f04a1cb05ae4f121073eed7eca8ab062eda4dc77ea5b91a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v07 --supports--> v02\n- v04 --refutes--> v01\n- v02 --refutes--> v04\n- v06 --refutes--> v03\n- v04 --supports--> v05\n- v01 --refutes--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69c1a423600368b9383ea891cf9df0c001bdd678cd14bc9b795dee218401adc5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v00 --supports--> v06\n- v05 --supports--> v07\n- v02 --supports--> v05\n- v06 --refutes--> v04\n- v07 --refutes--> v00\n- v02 --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":"c0ac3e0b9777f0c8cc12a67e9531f6d52b25895f2153660bd8147170f807d743","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v00 --refutes--> v05\n- v03 --refutes--> v07\n- v02 --refutes--> v06\n- v06 --supports--> v04\n- v02 --supports--> v01\n- v07 --supports--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6fe8e5b1b966c62902bb849162f5785d3d7ab6279a9328972b7cfa5ccae32fe7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v01\n- v00 --supports--> v04\n- v01 --supports--> v00\n- v04 --supports--> v03\n- v05 --refutes--> v07\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":"85ad1de0e2f7c5632866f1d39ee6e6e0d371c9c77f07f3631bd68949eb280987","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v00 --supports--> v04\n- v04 --supports--> v02\n- v06 --refutes--> v01\n- v02 --refutes--> v07\n- v03 --supports--> v00\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97912050355b1ac09e8e1cbfa1547a0147cdfc645caaa5fa4289e1345461d5fc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v04 --supports--> v00\n- v01 --supports--> v02\n- v04 --refutes--> v05\n- v01 --refutes--> v04\n- v06 --refutes--> v03\n- v05 --supports--> v07\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":"0a846b0ac0cc20f96ea00f81724060c17da14e0db00f52812efe763f38b1cefe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v05\n- v07 --refutes--> v02\n- v04 --supports--> v01\n- v00 --supports--> v07\n- v03 --supports--> v00\n- v04 --refutes--> v06\n- v01 --supports--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c0d0d97de640223a8bb3d7914d6e2b38d355e01dff8a9d7dfba5dc08f1c8896f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v00 --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v05 --refutes--> v03\n- v02 --supports--> v05\n- v03 --supports--> v07\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":"583833cc738f688595cdc22650c7706925418009ffe0296d132ce6675c88e067","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v00 --supports--> v07\n- v04 --supports--> v06\n- v00 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v04\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35479aa50b0feac742a79ab6c385562fc25ab656a7c89ec95cc4289f7e8336e5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v03 --supports--> v01\n- v05 --supports--> v00\n- v01 --refutes--> v02\n- v03 --refutes--> v06\n- v07 --refutes--> v03\n- v02 --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":"66d23b7d9b300eae0b7be0dc82827b731930105551b10eacc22344da2876b0d8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v06 --refutes--> v04\n- v07 --supports--> v03\n- v04 --supports--> v01\n- v06 --supports--> v05\n- v00 --refutes--> v02\n- v07 --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":"bb441d7a1a0720f4ef8ca0d72bd5bd9537078faaa4a4223ec5f01f3fc1c634f3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v02\n- v01 --supports--> v04\n- v04 --refutes--> v00\n- v03 --supports--> v05\n- v01 --refutes--> v06\n- v07 --refutes--> v03\n- v04 --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":"e360c981f3032434eda61ac820d5409330b43fd1a3c9f492ec3982f373164708","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v07 --refutes--> v01\n- v02 --supports--> v03\n- v04 --supports--> v05\n- v03 --supports--> v06\n- v04 --refutes--> v02\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":"ccd6f8a30b07e863c548a147ce12ed7bde5bfb8ee0ebb683bab859833664094d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v06\n- v04 --supports--> v02\n- v06 --refutes--> v01\n- v06 --supports--> v00\n- v06 --supports--> v03\n- v02 --refutes--> v05\n- v07 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"288c0dc4685576358f67f75af17f78ffdedc323e0ad57f7e627853fa7a283e0c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v00 --refutes--> v05\n- v05 --supports--> v03\n- v00 --supports--> v06\n- v07 --refutes--> v04\n- v04 --refutes--> v00\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":"443f2278ba4f3290573c490f2cbfad77668503bda863f407fb389d3f0b5b399d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v07 --supports--> v03\n- v00 --supports--> v04\n- v05 --supports--> v00\n- v07 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --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":"2b6021d41fa7b1892f8f1bec832f313193233b14e029d13b397911c26a732dae","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v02 --supports--> v04\n- v01 --supports--> v07\n- v02 --refutes--> v06\n- v00 --refutes--> v03\n- v03 --supports--> v02\n- v01 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63dc83ccfc2cf3df4391ae474f009439b85ce279e4a0f9dc2687b3f26529ef4e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v03 --refutes--> v05\n- v07 --refutes--> v03\n- v06 --refutes--> v01\n- v06 --refutes--> v00\n- v05 --refutes--> v02\n- v02 --supports--> 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":"7ac04045ed6c308c3eeebf3f179764da65618278de0a1f4bceaab7a35aeb5644","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v00 --supports--> v01\n- v04 --refutes--> v02\n- v00 --supports--> v04\n- v05 --supports--> v00\n- v02 --refutes--> v07\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13bb3e0627cf7bf1a3d664b3e44593e04e1c31dc659967cf6ea2ca01ae166714","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v04 --refutes--> v02\n- v04 --refutes--> v00\n- v07 --refutes--> v03\n- v00 --supports--> v05\n- v04 --refutes--> v01\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a30e40874acdb72629c2c3ffe9620ea91a11f7772f1f8c382aea3380de4d4df9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v07 --refutes--> v04\n- v01 --refutes--> v00\n- v07 --supports--> v06\n- v02 --supports--> v07\n- v07 --supports--> v01\n- v05 --refutes--> v03\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2001a3f7538dd18a23929e9d41ac21651fc514d0e66ef4c0b9e7fdea0d226189","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v05\n- v02 --refutes--> v03\n- v02 --supports--> v04\n- v04 --supports--> v01\n- v04 --supports--> v06\n- v05 --supports--> v07\n- v06 --refutes--> v00\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":"23249367e7c5a45e705004119b13c332c3764005786281f446ac1e26fbedab7b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v05 --supports--> v03\n- v01 --refutes--> v07\n- v07 --supports--> v02\n- v04 --supports--> v01\n- v01 --supports--> v00\n- v05 --supports--> 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":"02b15bf0fe0b7b4d97755cd154aa712656bcf363b6beb2ce308c7a83a1826051","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v00 --refutes--> v07\n- v02 --supports--> v00\n- v03 --supports--> v06\n- v05 --refutes--> v03\n- v06 --refutes--> v04\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ecf879c6a48cfe84dae120202d8823a277bef05deb3b26caa2fbfbd1707d2b31","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v06 --supports--> v05\n- v04 --supports--> v02\n- v00 --refutes--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v01\n- v03 --refutes--> v07\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":"b369311082e691ab1933f2e1d0d3658ed1c38d9fc96225b179a2cc8a66e28454","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v01 --refutes--> v04\n- v06 --supports--> v01\n- v05 --refutes--> v07\n- v01 --refutes--> v05\n- v06 --refutes--> v00\n- v03 --supports--> v02\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":"082290baf9240387f4de5763ac8b675eda0e008332bc2d1cf6592febf594e66f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v04 --refutes--> v02\n- v04 --refutes--> v06\n- v02 --supports--> v03\n- v00 --refutes--> v01\n- v03 --refutes--> v07\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1885337fb120a3c73733e2fa880f1db14c02389953073a43ecd9b0c5068fa93a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v05 --refutes--> v03\n- v07 --supports--> v04\n- v07 --supports--> v00\n- v02 --supports--> v06\n- v03 --supports--> v02\n- v04 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"df645f3113b441ab60c61487c7e6c596c02f082937fd8b18190193c98c0380fd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v01\n- v06 --refutes--> v07\n- v06 --refutes--> v05\n- v04 --supports--> v06\n- v00 --refutes--> v03\n- v05 --refutes--> v02\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e26d2b94b8c678e1fe98c7ef0f7e71fc6ffeb295ae965db0842d1fd8cf5d3a6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v03 --refutes--> v00\n- v02 --supports--> v06\n- v00 --supports--> v01\n- v04 --supports--> v03\n- v02 --supports--> v05\n- v01 --supports--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e7e521073d3b732c503f09c67fab44d1ace78dca8bea820de852e8196240512","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v06 --refutes--> v07\n- v03 --refutes--> v02\n- v06 --supports--> v05\n- v04 --supports--> v06\n- v01 --refutes--> v04\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"92ec8d78d8f7937a714884edbddc65965f8f3af278d8c80484788986596216d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v02 --refutes--> v00\n- v01 --refutes--> v02\n- v07 --refutes--> v06\n- v03 --refutes--> v07\n- v02 --refutes--> v03\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"82533ba3857fdcbd342153d5797a1277969c33d66a3a782ddc272c262db1a0f2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v04 --refutes--> v05\n- v00 --refutes--> v02\n- v00 --supports--> v03\n- v01 --refutes--> v06\n- v05 --supports--> v00\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"197bc90ff3ca9d1c85ddc77ffe4a755d88b6ae185e13a080c151fb68ff999fff","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v02 --refutes--> v07\n- v00 --refutes--> v01\n- v01 --supports--> v02\n- v03 --refutes--> v05\n- v05 --refutes--> v06\n- v05 --refutes--> v00\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":"14a550ab0a72e465b451856faa317126ec387615ba3b05962f475d15af96b424","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v07 --refutes--> v02\n- v07 --supports--> v01\n- v00 --supports--> v07\n- v04 --refutes--> v06\n- v00 --supports--> v03\n- v05 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e2ec7b8bb3fe75fb6bf9cb7d5d94efc97cb549873a42d342ad09a9191131a1c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v04 --supports--> v00\n- v02 --supports--> v04\n- v00 --refutes--> v07\n- v00 --refutes--> v03\n- v07 --refutes--> v05\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a88b3a2d0ba26dacaec2ea3aee440e28345952a13a94d65d24ed5dfc994ab715","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v06 --supports--> v02\n- v00 --supports--> v05\n- v03 --refutes--> v01\n- v02 --refutes--> v03\n- v05 --supports--> v06\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":"baf9340a4801e39968305109d95f45619c75bc96a0647df3e0a596b3d546968f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v01 --supports--> v06\n- v00 --supports--> v05\n- v01 --refutes--> v02\n- v00 --refutes--> v01\n- v07 --supports--> v03\n- v02 --supports--> v07\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":"74dae6306aa486bf3f6890879946ed0d0f15557a9df4353db7f7b5e96cdd1129","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v07\n- v05 --supports--> v02\n- v00 --refutes--> v06\n- v02 --refutes--> v00\n- v00 --refutes--> v01\n- v04 --refutes--> v05\n- v03 --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"199e253912260ed32fb6ec9c36bf3d6ae943ad3507aadaebd64bda956ca66fd3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v04\n- v05 --supports--> v02\n- v03 --refutes--> v01\n- v01 --supports--> v06\n- v02 --refutes--> v07\n- v01 --refutes--> v00\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c8685006fbba47b697abf9618ceecc48b94e9a34d2b9a9da5f732144397265d2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v05 --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v07\n- v00 --supports--> v01\n- v01 --supports--> v02\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":"ea3b32b4ce241d368297b520c6694963d7f5b84667f160b79280773ee905b57f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v05\n- v00 --refutes--> v02\n- v06 --refutes--> v03\n- v05 --supports--> v01\n- v05 --supports--> v04\n- v07 --refutes--> v00\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d5c5f440e736b21ba0de54f03b76321df0ff37025325116b634b5f5089b917b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v04\n- v04 --supports--> v03\n- v01 --refutes--> v06\n- v04 --supports--> v05\n- v06 --refutes--> v00\n- v03 --supports--> v01\n- v01 --supports--> v07\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":"07b92dfa17c3e0aa853329549d607dcf7d9fd48010cfd1a1947ab2d76f554e39","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v05 --supports--> v00\n- v06 --supports--> v02\n- v05 --refutes--> v07\n- v04 --supports--> v03\n- v07 --supports--> v01\n- v01 --supports--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5fcfea8f9a68e03bcc3e6d9dedfa9f5eca30b416c8ba62f80a3118bd34fdf81d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v07\n- v04 --refutes--> v01\n- v02 --supports--> v03\n- v07 --refutes--> v06\n- v02 --supports--> v05\n- v05 --refutes--> v04\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":"612216163d4b8178b718ee8905f47edc23fc8eba47e6ccfd5595b0d9c58fbd22","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v03\n- v02 --refutes--> v05\n- v04 --supports--> v00\n- v05 --refutes--> v07\n- v03 --supports--> v01\n- v05 --supports--> v06\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36f90ee4e9426721fdbeea051d00d6bd08a42b6fe07505cbeea0a9bf8fe509d5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v04\n- v01 --supports--> v00\n- v05 --supports--> v02\n- v04 --supports--> v05\n- v02 --refutes--> v01\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":"f3f0cb354b868e51240393ec030e59f5c36bcb8730549feba23140ddf2e5d062","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v07 --refutes--> v05\n- v07 --supports--> v04\n- v00 --refutes--> v02\n- v02 --supports--> v01\n- v06 --refutes--> v07\n- v01 --refutes--> v06\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":"de383224df2e32d91b738e0f6586d1e0318f099c764188250f586e5236806e4d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v03\n- v02 --refutes--> v07\n- v05 --supports--> v06\n- v07 --supports--> v05\n- v01 --supports--> v00\n- v02 --refutes--> v04\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6832bdab48aef6607fa6c5b7d3a750d9359d7205d71dbb351a4e886679dc560","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v02 --refutes--> v05\n- v04 --supports--> v00\n- v03 --supports--> v07\n- v03 --refutes--> v04\n- v01 --refutes--> v02\n- v04 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c996c584cbf1776efecd78dbb306a07a8bee16d14e658d2c9020dc63c9ecd510","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v06 --supports--> v05\n- v01 --supports--> v04\n- v05 --refutes--> v00\n- v05 --refutes--> v01\n- v03 --refutes--> v02\n- v02 --refutes--> v06\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":"5aa93343ac35f3c1ea10b4d3eea2621a7ab87ede314c323c40cc960888c319c6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v03 --supports--> v02\n- v06 --supports--> v05\n- v05 --refutes--> v00\n- v07 --refutes--> v06\n- v00 --supports--> v04\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":"1d3a2b33f817373156d51c7733cb72710bf3fbf41ef0992806148868b19acc7b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v06 --supports--> v07\n- v07 --refutes--> v01\n- v06 --refutes--> v03\n- v02 --refutes--> v05\n- v05 --supports--> v04\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"590736b0bdb8d40060ce6fd17649a1281a0aa923fc4be3dfa440bd038a8538fc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v01 --supports--> v00\n- v01 --refutes--> v04\n- v02 --refutes--> v05\n- v06 --refutes--> v07\n- v06 --supports--> v02\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d044270d7fb5de0c1dd3519254d73ece8f04f84808b002ba2444af91f67507a8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v00 --refutes--> v03\n- v03 --supports--> v07\n- v05 --supports--> v02\n- v05 --supports--> v01\n- v02 --refutes--> v06\n- v06 --supports--> v04\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7ef56d4525d0980628bff5c5970a6975e3a13467cd18317803da9144c44459b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v01 --supports--> v08\n- v08 --refutes--> v05\n- v08 --refutes--> v03\n- v08 --supports--> v04\n- v08 --supports--> v00\n- v02 --refutes--> v07\n- v07 --refutes--> v01\n- v09 --supports--> v02\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cab6a5e372a8eb03ca47e9237568589c8f60e8fb1ce4ab9b48849f1e2140f3ea","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v04 --supports--> v00\n- v01 --supports--> v08\n- v01 --supports--> v06\n- v00 --supports--> v02\n- v00 --supports--> v03\n- v05 --supports--> v01\n- v07 --refutes--> v09\n- v01 --refutes--> v04\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1498e0a70c73c0f071aaec53c37669d4ec33fbbf01d50615f45fd41409aafb0f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v07 --refutes--> v09\n- v05 --refutes--> v06\n- v00 --refutes--> v05\n- v07 --supports--> v08\n- v06 --refutes--> v04\n- v09 --supports--> v03\n- v09 --supports--> v01\n- v01 --refutes--> v00\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":"cf4779f2a3f2bb5c27308dd4c808a7aeb98ae604a5f9a29690693b4b0f35031a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v05 --refutes--> v09\n- v07 --supports--> v03\n- v06 --refutes--> v02\n- v03 --refutes--> v04\n- v04 --refutes--> v08\n- v05 --supports--> v06\n- v03 --refutes--> v00\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":"7ffd98a021498180079c74e18abb327fd003760547c1b0c278c23d6f068b189d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v03 --refutes--> v07\n- v02 --refutes--> v08\n- v08 --supports--> v09\n- v05 --refutes--> v06\n- v07 --refutes--> v02\n- v03 --supports--> v00\n- v03 --refutes--> v04\n- v06 --refutes--> v03\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":"33a2fab31ece82a33a5de4ff41ca7ac7aed05a5a9bac773ae48d51cc721d6dce","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v06 --refutes--> v08\n- v04 --refutes--> v00\n- v01 --refutes--> v05\n- v01 --refutes--> v02\n- v06 --refutes--> v03\n- v05 --refutes--> v04\n- v09 --supports--> v06\n- v04 --refutes--> v07\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe859235b444c0a61eb879d7c7b8bc1f9f02edf74a23c6ed16a40a87d1d03f10","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v09 --supports--> v04\n- v05 --supports--> v09\n- v08 --supports--> v01\n- v08 --supports--> v00\n- v04 --supports--> v08\n- v01 --supports--> v07\n- v04 --supports--> v06\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d70318e49c0e8aa2f32181fc4c867e6d4b7eafa199a64bb36009ad7decaed20","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v01 --refutes--> v07\n- v05 --supports--> v02\n- v06 --refutes--> v03\n- v05 --supports--> v06\n- v06 --supports--> v08\n- v07 --refutes--> v04\n- v00 --supports--> v01\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"317f4f5dc6ca9d609d9aed3b8d96c3d93ae6195013f3872b10cfb7b463612e55","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v00 --refutes--> v05\n- v06 --refutes--> v09\n- v00 --supports--> v01\n- v02 --refutes--> v04\n- v01 --supports--> v08\n- v04 --refutes--> v07\n- v08 --refutes--> v02\n- v04 --refutes--> v03\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":"0cea26644a669fc60aafbda8d3219f491c82d1dc14ab79999ab49174a5efd487","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v04 --refutes--> v09\n- v04 --supports--> v03\n- v01 --refutes--> v04\n- v08 --refutes--> v05\n- v02 --supports--> v06\n- v05 --supports--> v01\n- v04 --supports--> v07\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8ecf402338fe656709116ba8ed85987a0117a0413cf9f4aaaf39bb23dd979f0e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v01 --refutes--> v04\n- v06 --refutes--> v09\n- v04 --supports--> v03\n- v03 --supports--> v02\n- v05 --supports--> v07\n- v05 --refutes--> v00\n- v07 --refutes--> v08\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23a1201c96338ecc28c8e7beaa6f19a983dc4ac3fb9679b778ed6ccb5b526876","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v03 --supports--> v01\n- v09 --refutes--> v06\n- v00 --supports--> v09\n- v09 --refutes--> v07\n- v02 --supports--> v04\n- v05 --supports--> v08\n- v09 --supports--> v02\n- v08 --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":"077d23fe5821c76acf0c5779d518eb56880edee48c6c6d3891b4f13690e97701","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v05 --supports--> v09\n- v06 --supports--> v00\n- v05 --supports--> v07\n- v06 --supports--> v01\n- v08 --supports--> v06\n- v04 --refutes--> v03\n- v06 --supports--> v02\n- v02 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fbc536cd127b6d4a21824de72118366e9eaaf15e499a5884e6672a9d6404e222","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v09 --supports--> v05\n- v06 --supports--> v00\n- v05 --refutes--> v06\n- v09 --supports--> v07\n- v01 --refutes--> v09\n- v00 --refutes--> v03\n- v09 --refutes--> v02\n- v09 --refutes--> 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":"1f7173aa4888978d5e06b8d909fbde03871047eca8fc145ba0817e0e99c431d4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v08\n- v00 --refutes--> v05\n- v05 --refutes--> v04\n- v09 --refutes--> v03\n- v01 --refutes--> v07\n- v07 --supports--> v06\n- v09 --refutes--> v02\n- v05 --refutes--> v09\n- v06 --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":"99718af956d2e580ad0f36a27828ffbe575422f149e6967b44129feb7cffa01f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v05 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v08\n- v06 --refutes--> v03\n- v09 --refutes--> v07\n- v00 --refutes--> v09\n- v05 --supports--> v01\n- v08 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"17e233b2d03e99017321cf0340ae745ddafc98984c4cc35ae8c98ccaa7f32c0b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v05 --supports--> v02\n- v06 --refutes--> v07\n- v03 --refutes--> v01\n- v04 --supports--> v06\n- v06 --refutes--> v09\n- v02 --supports--> v03\n- v00 --supports--> v05\n- v00 --supports--> v08\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":"ab0b4238f6e29ee9e01c0002a2f0328ed15b7df690de1d9edb48f553de2b9610","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v04 --refutes--> v06\n- v01 --refutes--> v05\n- v07 --refutes--> v03\n- v05 --supports--> v09\n- v07 --refutes--> v08\n- v07 --refutes--> v00\n- v02 --refutes--> v01\n- v09 --refutes--> v04\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":"d12e9efb494f01128202f9293bf18644fc534f97c5af2481501a688edb572ccf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v00 --supports--> v01\n- v08 --supports--> v06\n- v01 --refutes--> v08\n- v00 --refutes--> v07\n- v00 --refutes--> v03\n- v01 --refutes--> v09\n- v05 --supports--> v04\n- v06 --refutes--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"edc66726be91b906a53a7419d5af4ee369ac3f3fe2e7401b6eafadb80946be74","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v00\n- v07 --refutes--> v03\n- v04 --supports--> v05\n- v02 --supports--> v08\n- v02 --refutes--> v01\n- v02 --refutes--> v06\n- v00 --supports--> v09\n- v01 --refutes--> v04\n- v09 --refutes--> v07\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"469309abfce6f17c8d3ef2396fd9f00b5cabfbbf22104bdea8331833e415fac4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v07 --supports--> v08\n- v03 --supports--> v07\n- v06 --refutes--> v09\n- v02 --supports--> v06\n- v04 --supports--> v02\n- v05 --supports--> v04\n- v04 --refutes--> v01\n- v06 --supports--> v03\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":"a9fb234efd4cf3defcaa83bc64ad324d24bd09998646deeafaceae1f48903336","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v06 --supports--> v00\n- v05 --refutes--> v02\n- v04 --refutes--> v01\n- v07 --supports--> v03\n- v05 --refutes--> v09\n- v04 --supports--> v06\n- v05 --supports--> v04\n- v00 --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":"afc3799b7a02dc845c8cc9afb63d888134ec8bab69a02f6ca3636298dc58d265","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v01\n- v02 --supports--> v09\n- v04 --refutes--> v05\n- v03 --refutes--> v06\n- v00 --refutes--> v03\n- v07 --refutes--> v00\n- v04 --supports--> v02\n- v00 --supports--> v08\n- v09 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"77b9a8ee5905c0eadb5ecbf18c181917abef369930f10b6ffd022e14b520080e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v01 --supports--> v08\n- v02 --supports--> v01\n- v04 --supports--> v02\n- v08 --refutes--> v06\n- v01 --supports--> v03\n- v05 --refutes--> v09\n- v02 --supports--> v07\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9adfa7f712b03a4b42f01bd2e436275ff737c504292447dbcb2397172f725a3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v03 --refutes--> v07\n- v05 --supports--> v06\n- v05 --refutes--> v00\n- v05 --supports--> v01\n- v02 --refutes--> v04\n- v08 --supports--> v03\n- v07 --supports--> v02\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2c2054834f0c240de228dcf53d193b27aeac9904fcbd284f28d72b750bb5f768","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v02 --refutes--> v09\n- v08 --supports--> v03\n- v02 --refutes--> v04\n- v00 --supports--> v06\n- v07 --refutes--> v01\n- v03 --supports--> v05\n- v00 --supports--> v02\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5c5099b5d9e1081dc1e40c498fc7815540dd2b30ba1e236397b0458245800b3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v01\n- v03 --refutes--> v04\n- v09 --refutes--> v07\n- v09 --refutes--> v03\n- v00 --refutes--> v09\n- v08 --supports--> v00\n- v09 --refutes--> v02\n- v09 --supports--> v05\n- v06 --refutes--> v08\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":"db317d4f6b7be2f9523f01815fb09486280ec9842bbf7be856fe1604bc821a1f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v04 --refutes--> v05\n- v03 --supports--> v04\n- v05 --refutes--> v02\n- v01 --refutes--> v07\n- v08 --refutes--> v00\n- v03 --refutes--> v06\n- v07 --supports--> v08\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4347d9b611cc932597b670d5427d727491e492b2ec181a1f9d6abaed22765421","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v02\n- v00 --supports--> v06\n- v04 --supports--> v05\n- v01 --supports--> v07\n- v01 --refutes--> v03\n- v06 --supports--> v09\n- v02 --supports--> v01\n- v08 --supports--> v04\n- v01 --supports--> v08\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":"b22e66315ba615d50660db7263e29355543d2cf9f7afba80972137b4d563653f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v02 --refutes--> v05\n- v01 --supports--> v07\n- v03 --supports--> v04\n- v02 --refutes--> v08\n- v07 --refutes--> v09\n- v08 --refutes--> v03\n- v03 --supports--> v01\n- v09 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0e3a7a97784a8e753a681ceb3579a88139d75cdc1ac3e7afde3c44b33f8f1bd1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v04 --supports--> v05\n- v01 --refutes--> v06\n- v02 --refutes--> v03\n- v03 --refutes--> v08\n- v00 --supports--> v04\n- v04 --supports--> v07\n- v05 --supports--> v01\n- v00 --supports--> v09\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":"9d7a4ef175b41c3c4d4de1e9981ccebb4e75aa5050114b2ade3e7d958db60319","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v08 --supports--> v01\n- v04 --supports--> v00\n- v03 --refutes--> v08\n- v02 --supports--> v05\n- v05 --supports--> v03\n- v08 --refutes--> v06\n- v04 --supports--> v07\n- v05 --refutes--> v09\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":"850ac46d25c5b2c4f4a42c06854c4e6ebe1bd866900c4d41fb67e5d365bc41dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v02\n- v09 --refutes--> v04\n- v09 --refutes--> v07\n- v06 --refutes--> v09\n- v03 --refutes--> v08\n- v02 --supports--> v06\n- v03 --refutes--> v05\n- v06 --supports--> v01\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6279b6c4b6fa573c9925e405fb7b25c80fb92cde744fef768721fa396d6b8606","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v05 --supports--> v09\n- v01 --supports--> v06\n- v07 --refutes--> v04\n- v06 --refutes--> v08\n- v07 --refutes--> v02\n- v05 --supports--> v07\n- v00 --supports--> v01\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bcc2aa5b18e0f7255f0bdfa7a24f21bca468b778d28fdcef38abca2250d2afe0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v08 --supports--> v05\n- v03 --supports--> v00\n- v02 --refutes--> v07\n- v00 --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v04\n- v03 --refutes--> v06\n- v05 --supports--> v03\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":"2e817697e3d61409d96631858c31fbbfa272966557aa15c3eb5cb8784e6b6e3a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v07 --supports--> v02\n- v03 --supports--> v09\n- v07 --supports--> v06\n- v04 --refutes--> v03\n- v01 --refutes--> v04\n- v07 --refutes--> v00\n- v08 --supports--> v07\n- v05 --refutes--> 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":"1b5c9b1fabdce408832f6568d63ccbd508aa6e8a8c64711c34772699b6887a6d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v02 --refutes--> v03\n- v06 --supports--> v00\n- v06 --supports--> v07\n- v03 --refutes--> v01\n- v07 --refutes--> v02\n- v03 --supports--> v08\n- v01 --supports--> v05\n- v01 --refutes--> v04\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":"419607a2ba455e9ced272ddea1bb0f158f6692793cd467e4a5c6e1b78787c93a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v02 --supports--> v06\n- v09 --refutes--> v05\n- v07 --refutes--> v03\n- v03 --supports--> v02\n- v01 --supports--> v04\n- v09 --refutes--> v08\n- v08 --refutes--> v00\n- v02 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28c5fac682c1b5e2b76524bb97908ccd1e6cd28a02d9464b4cec4ffdc9fb62c8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v08 --supports--> v06\n- v05 --supports--> v03\n- v07 --supports--> v09\n- v04 --supports--> v07\n- v03 --refutes--> v01\n- v08 --supports--> v05\n- v04 --supports--> v00\n- v07 --supports--> v08\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":"f181272375a9cb432d68ba32277c6a22c7e118962f4e4522e3e065698c5652d4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v09\n- v07 --supports--> v04\n- v06 --refutes--> v01\n- v06 --supports--> v02\n- v08 --supports--> v03\n- v06 --supports--> v05\n- v07 --supports--> v08\n- v03 --supports--> v06\n- v00 --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":"9bea442765da5b7e0747a6e8610aab4ae746b28b8a854fa8d24830294623d5f4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v02 --supports--> v07\n- v01 --refutes--> v00\n- v02 --refutes--> v06\n- v03 --refutes--> v01\n- v01 --refutes--> v02\n- v09 --supports--> v05\n- v01 --refutes--> v08\n- v04 --supports--> v09\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":"d2e19be136ff2fc64222f5ca546e6dfa4cd5f5d6af523d3555c0e62825037c58","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v04\n- v03 --refutes--> v07\n- v07 --refutes--> v06\n- v07 --refutes--> v09\n- v08 --supports--> v03\n- v06 --refutes--> v01\n- v02 --supports--> v08\n- v03 --supports--> v05\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a7d43cb07179bb686b3e544a27ce36452688d406163b0d7a11fecb965438ee4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v03 --refutes--> v08\n- v06 --supports--> v04\n- v03 --refutes--> v07\n- v07 --supports--> v02\n- v06 --refutes--> v03\n- v01 --refutes--> v09\n- v02 --supports--> v01\n- v09 --supports--> v00\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":"b0d776a3cd04bb663a324274c8d00050457813a34c490aca9ae8c9aa05a5bf6f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v03 --refutes--> v09\n- v03 --supports--> v07\n- v00 --supports--> v02\n- v05 --refutes--> v08\n- v08 --refutes--> v03\n- v00 --refutes--> v06\n- v07 --refutes--> v01\n- v01 --refutes--> v00\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fe626957a2a30fa11c41a070717d98ab6b3893208fdd910eed535347b2d2b376","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v08 --supports--> v05\n- v05 --supports--> v09\n- v00 --supports--> v08\n- v07 --supports--> v03\n- v08 --refutes--> v02\n- v03 --refutes--> v00\n- v08 --refutes--> v01\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc1f4f94c09f14a9fdcddac26862b2763b0c2ce76eaa56c4347a711f5fe68e2a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v06 --supports--> v00\n- v04 --refutes--> v09\n- v00 --supports--> v08\n- v04 --supports--> v03\n- v03 --supports--> v05\n- v01 --refutes--> v06\n- v01 --refutes--> v02\n- v07 --refutes--> v04\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":"51445d3c05ae0cf6093f65397e8bb40934f75b7e3bbdad737933d545e4f936d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v00\n- v07 --supports--> v01\n- v07 --supports--> v05\n- v03 --supports--> v08\n- v00 --refutes--> v02\n- v01 --refutes--> v04\n- v08 --supports--> v06\n- v00 --supports--> v07\n- v09 --supports--> v03\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":"e1ca1f17a2b5757fb5a5a28754b8b24952141cb100832da1b9b50b3cbcf7cc72","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v08 --supports--> v04\n- v08 --supports--> v03\n- v03 --refutes--> v07\n- v06 --refutes--> v00\n- v00 --supports--> v09\n- v03 --refutes--> v06\n- v02 --refutes--> v05\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1d97e5d346519ae2d7cf07dc68ebb3246a3512da47b10dbfc2e77df13b743501","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v03\n- v00 --supports--> v05\n- v08 --refutes--> v00\n- v07 --refutes--> v01\n- v01 --refutes--> v08\n- v05 --supports--> v04\n- v05 --refutes--> v02\n- v06 --refutes--> v07\n- v01 --refutes--> v09\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":"0972bc56f30e31783bac4e4be17cc18dd28c822d495b86360d1ae980b7dcbad0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v00 --refutes--> v01\n- v07 --supports--> v00\n- v02 --refutes--> v06\n- v09 --supports--> v07\n- v04 --supports--> v08\n- v04 --supports--> v02\n- v00 --supports--> v04\n- v04 --refutes--> v03\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b018d4bda6dd0da024a7acede2391d07880e8cb030c17fa6e64f988e2cb2a8b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v07 --refutes--> v04\n- v09 --refutes--> v06\n- v07 --refutes--> v03\n- v08 --refutes--> v01\n- v03 --supports--> v08\n- v01 --refutes--> v02\n- v06 --refutes--> v07\n- v07 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6127695fcce6ec2bfc92371cf35d517285bbe23edba93d141c3e8134988acfe0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v00 --supports--> v08\n- v02 --refutes--> v06\n- v03 --supports--> v00\n- v08 --supports--> v02\n- v07 --supports--> v09\n- v09 --refutes--> v05\n- v02 --supports--> v01\n- v02 --supports--> v04\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":"eb023a691c3c05371e3e6b7ed1bf3b9767ce07dc4a70928837b6c280e4c6b9c8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v09 --supports--> v06\n- v09 --refutes--> v03\n- v01 --supports--> v05\n- v06 --supports--> v07\n- v09 --refutes--> v00\n- v07 --refutes--> v04\n- v02 --supports--> v09\n- v06 --refutes--> v08\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":"df0adcf32d82c52f20b14f2cf4005b2dee0031a2f0c265e8051be089b8bf0fcb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v02 --supports--> v03\n- v02 --supports--> v09\n- v01 --supports--> v07\n- v05 --supports--> v08\n- v08 --refutes--> v04\n- v08 --supports--> v02\n- v07 --supports--> v00\n- v01 --supports--> v06\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":"3282b23ed06e0f10b93320d659fe016b3ee42f86fc458de67b9d2ae9007d98e4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v08 --supports--> v06\n- v07 --supports--> v09\n- v08 --refutes--> v02\n- v01 --refutes--> v00\n- v09 --refutes--> v03\n- v07 --supports--> v05\n- v09 --supports--> v08\n- v04 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa1523cde4a6c1245c2b67dfe35d8db44b009529523f9b661fc1e53e698e5142","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v01\n- v04 --supports--> v00\n- v09 --refutes--> v02\n- v02 --supports--> v08\n- v04 --refutes--> v06\n- v01 --supports--> v03\n- v02 --supports--> v04\n- v00 --supports--> v05\n- v01 --supports--> v07\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":"903ba7ea874b6b67ec3878a2618db2a9df5d507fbdd95ae355ccc7b43618102d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v08 --refutes--> v02\n- v04 --refutes--> v09\n- v01 --supports--> v05\n- v06 --supports--> v07\n- v08 --refutes--> v00\n- v05 --supports--> v06\n- v06 --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":"d90cc1535918747013538e75c7b0ce75cc09da98388c3b6fd06871a227031897","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v07 --supports--> v04\n- v08 --refutes--> v09\n- v09 --refutes--> v07\n- v04 --supports--> v02\n- v01 --supports--> v00\n- v00 --supports--> v08\n- v07 --refutes--> v06\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bed14852361c77007bd0c14b099504e6257ed32388181a3c9e8ffad643c46ec9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v06 --refutes--> v03\n- v09 --refutes--> v05\n- v07 --supports--> v00\n- v00 --refutes--> v08\n- v09 --refutes--> v06\n- v01 --refutes--> v02\n- v03 --refutes--> v01\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e166b00ba9bca9ea3da81f27b437ca3238770899e48ad9d43f8c2b4b8a39e0ca","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v00 --refutes--> v07\n- v09 --refutes--> v08\n- v01 --supports--> v06\n- v04 --refutes--> v05\n- v02 --refutes--> v00\n- v01 --refutes--> v03\n- v00 --refutes--> v04\n- v08 --supports--> v02\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":"e5fe69c4771f67f97902aca9402c57bb6eca9fef8408236a68f3e7c0e9ae4763","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v00 --refutes--> v02\n- v07 --refutes--> v08\n- v08 --supports--> v09\n- v01 --supports--> v07\n- v06 --supports--> v00\n- v00 --refutes--> v03\n- v04 --refutes--> v01\n- v00 --supports--> v04\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":"605ac979c79306d62c26f695b5a23687a0096cfd2db3660eaf5f5642b00d0167","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v05 --refutes--> v03\n- v08 --refutes--> v04\n- v07 --refutes--> v01\n- v00 --supports--> v02\n- v00 --refutes--> v06\n- v09 --refutes--> v07\n- v05 --supports--> v00\n- v04 --refutes--> v09\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":"19b9221a75d3c68384891689dd73cc15aaf96a48cc19f3a0d65f9bf7e25e3524","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v03 --supports--> v09\n- v09 --refutes--> v01\n- v07 --refutes--> v03\n- v06 --supports--> v07\n- v05 --supports--> v06\n- v05 --refutes--> v00\n- v01 --refutes--> v04\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":"5b4ab0e20e4d6788c3d2c8a195dacee6ce17e1580793ca12fcc5a7c8d4a8b7a2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v08 --supports--> v09\n- v03 --refutes--> v04\n- v03 --supports--> v06\n- v00 --refutes--> v08\n- v05 --refutes--> v07\n- v09 --refutes--> v01\n- v06 --supports--> v02\n- v06 --refutes--> v00\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":"483d3525de01bd7cf26bfbb270926538bfe63c616642f56eb7a2fd51547f4e9c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v08 --refutes--> v0a\n- v05 --refutes--> v03\n- v06 --supports--> v04\n- v06 --supports--> v02\n- v08 --refutes--> v01\n- v04 --refutes--> v08\n- v04 --refutes--> v07\n- v09 --refutes--> v05\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ff4978faa17d67f36b5cc9fb61698be372d18f4e74636e1ac9e05c6139e2244a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v03\n- v05 --supports--> v00\n- v05 --refutes--> v06\n- v05 --refutes--> v0a\n- v08 --supports--> v04\n- v09 --refutes--> v05\n- v03 --refutes--> v02\n- v0a --refutes--> v07\n- v07 --refutes--> v08\n- v02 --supports--> v01\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":"b802fbe69d839697f4bce5a7a726ba394bb6c8de1a14730a2db1cea2ce9576a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0a\n- v06 --refutes--> v02\n- v07 --supports--> v03\n- v06 --supports--> v07\n- v07 --refutes--> v05\n- v04 --supports--> v06\n- v05 --refutes--> v09\n- v08 --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v04\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":"a65dbd302fe655803713f23eef4cfec0719c7e0d4245f47a33e1547774073745","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v00\n- v05 --refutes--> v01\n- v09 --supports--> v08\n- v01 --refutes--> v04\n- v03 --supports--> v02\n- v0a --supports--> v03\n- v04 --supports--> v0a\n- v09 --refutes--> v05\n- v03 --supports--> v06\n- v02 --supports--> v07\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":"2125828f4a9f4f6214081ec0d1bde369392ceea36f1236299229578a697d2030","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v05 --supports--> v03\n- v01 --supports--> v04\n- v09 --refutes--> v08\n- v08 --refutes--> v0a\n- v00 --refutes--> v07\n- v04 --refutes--> v00\n- v08 --refutes--> v06\n- v07 --refutes--> v05\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"69dbc9353543fee84d1e9b184273d5009e0177547fb079e9eccd92b48dd221f4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v02 --refutes--> v08\n- v0a --refutes--> v07\n- v08 --refutes--> v0a\n- v09 --supports--> v06\n- v01 --refutes--> v03\n- v09 --refutes--> v00\n- v07 --refutes--> v09\n- v06 --supports--> v01\n- v08 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb13bac3ab09d907419a97159707240e4ce98d77c2a701c7b6095e1522f1923f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v03 --supports--> v04\n- v04 --refutes--> v07\n- v08 --supports--> v05\n- v01 --refutes--> v08\n- v04 --refutes--> v06\n- v02 --supports--> v01\n- v05 --supports--> v0a\n- v09 --refutes--> v02\n- v03 --supports--> v00\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7002a17faefc4b5c97b2ac8416e91c3748243c6b847c99cc996e2eeb52d27674","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v09 --supports--> v03\n- v02 --supports--> v09\n- v0a --supports--> v08\n- v06 --refutes--> v0a\n- v08 --refutes--> v07\n- v0a --supports--> v00\n- v08 --supports--> v01\n- v03 --refutes--> v06\n- v07 --refutes--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19534c01d19b157f864908310d1060e5d9d962701660ed2171bfb3c86ba97a10","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v08 --supports--> v02\n- v06 --refutes--> v07\n- v08 --supports--> v03\n- v05 --refutes--> v09\n- v01 --supports--> v05\n- v05 --supports--> v0a\n- v03 --supports--> v04\n- v03 --supports--> v00\n- v04 --supports--> v06\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e9694ff0134b5c3c82beca8d20d6967685e8f7dcf29ebd0bcef93bc61ce0d926","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v05 --refutes--> v07\n- v04 --refutes--> v02\n- v08 --refutes--> v05\n- v04 --supports--> v01\n- v0a --supports--> v08\n- v09 --refutes--> v03\n- v04 --refutes--> v00\n- v07 --supports--> v09\n- v00 --supports--> v0a\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":"6a9112301be6f2274246416a877554c462546de8f6d5d2fbf5e4fffc68e28e06","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v02 --refutes--> v00\n- v03 --refutes--> v07\n- v09 --supports--> v0a\n- v08 --supports--> v02\n- v00 --supports--> v09\n- v03 --supports--> v08\n- v04 --supports--> v01\n- v03 --supports--> v06\n- v01 --refutes--> v03\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":"7763e81d0d35fda4cc1a845bb287f302529add853850f7056aa12ad2026c9bd3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v04 --refutes--> v02\n- v01 --supports--> v04\n- v07 --supports--> v03\n- v01 --refutes--> v00\n- v09 --refutes--> v07\n- v03 --supports--> v0a\n- v04 --refutes--> v09\n- v06 --refutes--> v01\n- v0a --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f6a844ed435e55641e460768682af9b784f9847cbcf5cc8c96d2dd111567a4c6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v00\n- v0a --refutes--> v06\n- v09 --refutes--> v05\n- v08 --supports--> v01\n- v06 --supports--> v09\n- v08 --supports--> v07\n- v04 --supports--> v0a\n- v03 --supports--> v04\n- v03 --supports--> v02\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f1c383f7936e5873092531588ed5abd3ae2f6a649a2092ea7966f173c74cf4b6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v08\n- v09 --supports--> v0a\n- v02 --supports--> v09\n- v0a --refutes--> v05\n- v01 --refutes--> v00\n- v03 --supports--> v01\n- v01 --supports--> v06\n- v05 --supports--> v03\n- v04 --refutes--> v02\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3bc3364627663851c75335d41f760336cd35fdf94cf8c31a28cc4be37b6bc10e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v0a --supports--> v08\n- v09 --refutes--> v0a\n- v00 --refutes--> v09\n- v08 --supports--> v05\n- v02 --refutes--> v04\n- v02 --supports--> v03\n- v01 --refutes--> v00\n- v08 --refutes--> v07\n- v06 --supports--> 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":"3fb13f5b36169a2ea6a69f76f9e00c3086b16739823a513dec4585da580b8e3e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v08 --refutes--> v0a\n- v08 --refutes--> v03\n- v09 --supports--> v04\n- v00 --supports--> v07\n- v0a --supports--> v01\n- v01 --refutes--> v05\n- v09 --supports--> v06\n- v04 --supports--> v00\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec767a22121f8b16c2fd7e3d30247c7090512db0246077b5f586e84dbfd48668","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v03 --refutes--> v00\n- v07 --supports--> v03\n- v04 --supports--> v07\n- v0a --supports--> v09\n- v05 --supports--> v04\n- v02 --refutes--> v06\n- v0a --supports--> v01\n- v03 --supports--> v08\n- v09 --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":"50cd40e866b5f2d3fd6bef685877dc97a8d4e0a0766e3d77b2d30eeb30eb1ca4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0a\n- v0a --supports--> v01\n- v03 --supports--> v04\n- v01 --supports--> v02\n- v05 --supports--> v03\n- v00 --supports--> v06\n- v03 --supports--> v00\n- v06 --supports--> v08\n- v01 --supports--> v07\n- v01 --supports--> v09\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":"ed44636a5d602a72694a509b9e6a41cb06c035109326b17f7cc82242929e8572","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v00\n- v00 --supports--> v08\n- v07 --refutes--> v04\n- v00 --supports--> v01\n- v04 --supports--> v03\n- v05 --refutes--> v0a\n- v04 --supports--> v06\n- v00 --refutes--> v09\n- v09 --supports--> v07\n- v06 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3bc374572f05b396ed44be60894c020fe6b68edc026851b41e05612719c0e34","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v04 --refutes--> v07\n- v06 --refutes--> v0a\n- v06 --refutes--> v01\n- v08 --refutes--> v02\n- v00 --refutes--> v08\n- v03 --refutes--> v05\n- v06 --refutes--> v03\n- v08 --refutes--> v04\n- v09 --refutes--> v06\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":"faf1231189ec67f7b634de1d69e39d85203cc244da837ee34e8213ab389da7e9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v05 --refutes--> v09\n- v04 --supports--> v02\n- v08 --refutes--> v06\n- v06 --supports--> v0a\n- v06 --refutes--> v03\n- v06 --supports--> v05\n- v00 --supports--> v04\n- v02 --refutes--> v08\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":"afb1c0eac839bd1df784bf5c9c3e3cdcd0728812290e3834674f58aa536000bf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v02 --supports--> v00\n- v00 --refutes--> v05\n- v01 --supports--> v04\n- v03 --supports--> v0a\n- v07 --refutes--> v02\n- v02 --supports--> v08\n- v00 --refutes--> v09\n- v0a --supports--> v07\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16d8336b235645bbc0195696d190b62d7957e77de5950a89c5db75c33268bf02","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v00 --supports--> v04\n- v06 --supports--> v08\n- v07 --refutes--> v01\n- v08 --supports--> v02\n- v00 --refutes--> v07\n- v02 --refutes--> v0a\n- v03 --refutes--> v05\n- v06 --supports--> v09\n- v07 --refutes--> v06\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":"aa34345ece5fd4a6ba28cec65b2b85e02f09a9e50bf4df7b133169861d65470a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v05\n- v08 --supports--> v02\n- v07 --supports--> v09\n- v06 --refutes--> v08\n- v05 --supports--> v04\n- v04 --supports--> v07\n- v09 --supports--> v0a\n- v09 --supports--> v01\n- v09 --supports--> v03\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"28abe1e12ac4fff9376dde5b5ddbb078dee890f513b1eedec0d7627923e11c9f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v00 --supports--> v09\n- v08 --refutes--> v01\n- v02 --refutes--> v07\n- v00 --refutes--> v05\n- v07 --supports--> v06\n- v01 --refutes--> v03\n- v06 --supports--> v00\n- v01 --supports--> v0a\n- v09 --supports--> v08\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6ac86d88243753980f3358e349f937d465e4310f9d3153ad9edb6c35bf02d91a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v05 --supports--> v09\n- v0a --supports--> v01\n- v06 --supports--> v04\n- v01 --supports--> v07\n- v03 --supports--> v00\n- v05 --supports--> v03\n- v04 --refutes--> v0a\n- v03 --refutes--> v02\n- v07 --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":"c67f1dc7ff94e56ee3db87f5403c94503f36388a40f6baa06d7ff25e5f42c0e9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v00\n- v00 --refutes--> v04\n- v01 --supports--> v05\n- v04 --refutes--> v09\n- v03 --refutes--> v0a\n- v04 --supports--> v03\n- v02 --supports--> v06\n- v0a --refutes--> v02\n- v03 --refutes--> v08\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73fde410881a2f916b545d29af7ad03e7d38ca7a9c6bb7e2619cf65fd60b81ad","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v03 --supports--> v08\n- v05 --supports--> v01\n- v03 --supports--> v04\n- v06 --supports--> v09\n- v07 --refutes--> v00\n- v00 --refutes--> v06\n- v01 --refutes--> v03\n- v09 --refutes--> v05\n- v01 --refutes--> v02\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":"b0681ac6446c8891a9f55b252d77407864b93fdca4cb1a47931818b0948c7a4a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v07 --supports--> v0a\n- v00 --supports--> v04\n- v09 --supports--> v03\n- v01 --supports--> v06\n- v0a --refutes--> v02\n- v01 --supports--> v07\n- v08 --supports--> v01\n- v0a --supports--> v05\n- v09 --refutes--> v08\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":"a3063818cb6d1f6240ed52fd0b55cf2c2ccfb09ba25288af2dbc79f3d7c5b888","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v02\n- v06 --supports--> v0a\n- v01 --supports--> v00\n- v05 --supports--> v04\n- v09 --refutes--> v01\n- v02 --supports--> v06\n- v09 --supports--> v08\n- v02 --refutes--> v07\n- v04 --refutes--> v09\n- v06 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97035e3eafafd1c3b6118e4cbf1ccd8683752cda24ad9ef466a6c4138f7f5107","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v09\n- v02 --supports--> v00\n- v06 --refutes--> v03\n- v00 --refutes--> v07\n- v02 --refutes--> v0a\n- v09 --refutes--> v06\n- v01 --refutes--> v05\n- v00 --refutes--> v04\n- v03 --refutes--> v08\n- v07 --refutes--> v01\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":"eefc96c2101ecf360ed3641384ee5200a156d1fce3afb84ae5d64d16b82de6a7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v09 --refutes--> v02\n- v09 --supports--> v07\n- v08 --supports--> v03\n- v01 --supports--> v04\n- v04 --supports--> v06\n- v09 --supports--> v05\n- v03 --supports--> v0a\n- v07 --supports--> v01\n- v06 --refutes--> v08\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":"fea1c1bf4ae660f66045322957ece08ada566a164899ba3ec8c191d29dfde5a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v09 --refutes--> v06\n- v02 --refutes--> v04\n- v02 --refutes--> v05\n- v08 --refutes--> v07\n- v03 --refutes--> v0a\n- v00 --supports--> v03\n- v00 --supports--> v01\n- v02 --supports--> v08\n- v07 --refutes--> v00\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":"86729f841cc28f0bca48edb3a732c4d4a4aee59d8cb914996ab196ec26dfa48c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v04\n- v04 --supports--> v03\n- v02 --refutes--> v08\n- v06 --supports--> v01\n- v08 --supports--> v05\n- v04 --refutes--> v06\n- v09 --refutes--> v0a\n- v01 --supports--> v07\n- v08 --supports--> v00\n- v07 --supports--> v09\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":"da578c1fd8759144e567fc0ac1fbd6423635117bbae3dc03cd625410a4879779","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v00 --refutes--> v07\n- v00 --supports--> v05\n- v08 --refutes--> v04\n- v00 --supports--> v06\n- v0a --supports--> v08\n- v08 --refutes--> v02\n- v01 --refutes--> v00\n- v06 --refutes--> v09\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3e32b9d54d9ed1a72ca3d83ad2b615b00d5ccb8c676e5e8fa71ea7d9a4520a7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v02 --refutes--> v0a\n- v04 --supports--> v05\n- v09 --supports--> v08\n- v09 --refutes--> v07\n- v00 --supports--> v03\n- v08 --supports--> v06\n- v03 --supports--> v09\n- v02 --refutes--> v00\n- v08 --supports--> 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":"a5fcc4f0f97f558178bf110f3cdc572cdb656ec82112a3a9a80d9fe8870acc45","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v08 --supports--> v0a\n- v07 --refutes--> v01\n- v07 --refutes--> v03\n- v03 --refutes--> v08\n- v04 --refutes--> v06\n- v02 --supports--> v00\n- v09 --refutes--> v04\n- v0a --refutes--> v02\n- v02 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7770ec34174e2b594ac3c3834653b9aa06449c583d0923df80efbfdb7c9526d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v07\n- v03 --refutes--> v09\n- v04 --supports--> v01\n- v06 --supports--> v02\n- v0a --supports--> v08\n- v09 --supports--> v06\n- v04 --refutes--> v03\n- v08 --supports--> v04\n- v06 --supports--> v05\n- v00 --supports--> v0a\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":"f54b390b940607e7937256d3218438e30e64b1bcf95c5adde79c352845f9b4e1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v07 --refutes--> v01\n- v02 --supports--> v09\n- v07 --refutes--> v00\n- v06 --supports--> v02\n- v00 --supports--> v0a\n- v05 --refutes--> v03\n- v02 --supports--> v07\n- v08 --supports--> v06\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":"47524c042e68cdc8cf50361b126d862b55b041999cae97c11c4b479add3ef44a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v03 --refutes--> v04\n- v07 --refutes--> v08\n- v09 --refutes--> v00\n- v07 --refutes--> v06\n- v05 --refutes--> v07\n- v01 --supports--> v02\n- v04 --supports--> v05\n- v04 --refutes--> v0a\n- v02 --refutes--> v03\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":"06a69f0b84279e4400820665e1a735169a20ca0894512ca80e7db7bbbb1565dd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v06 --supports--> v02\n- v00 --refutes--> v0a\n- v05 --supports--> v09\n- v05 --supports--> v07\n- v08 --supports--> v03\n- v0a --refutes--> v05\n- v04 --refutes--> v00\n- v05 --supports--> v06\n- v06 --refutes--> v01\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3e13087f47d2d20a5020175912bb2b7a42d53e6c455818a9f80f06630d9fd02","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v0a --supports--> v04\n- v09 --supports--> v07\n- v09 --supports--> v00\n- v05 --supports--> v09\n- v07 --supports--> v0a\n- v05 --refutes--> v06\n- v0a --refutes--> v08\n- v03 --supports--> v01\n- v01 --supports--> v05\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":"60d8352e3049a19437d0a3b08497c0d5fd0eab5aa6c383059effc565dd2ff823","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0a\n- v01 --supports--> v05\n- v06 --refutes--> v01\n- v02 --refutes--> v06\n- v02 --refutes--> v04\n- v07 --refutes--> v09\n- v05 --refutes--> v07\n- v09 --supports--> v08\n- v00 --refutes--> v02\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":"5d05a5fdf358d88ab7923b461204042420be06f6a0caabfe8193d7d4b40527bf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v03\n- v06 --refutes--> v0a\n- v04 --supports--> v06\n- v04 --supports--> v05\n- v01 --refutes--> v04\n- v03 --supports--> v09\n- v06 --refutes--> v08\n- v09 --refutes--> v07\n- v08 --refutes--> v00\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8b4ad11970519204ceb8f205bd10a5bb6ab422ba4f6be579c5fb654949edb290","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v04 --refutes--> v08\n- v00 --supports--> v07\n- v03 --refutes--> v00\n- v08 --refutes--> v09\n- v03 --supports--> v0a\n- v03 --refutes--> v05\n- v02 --refutes--> v04\n- v01 --supports--> v02\n- v00 --refutes--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"059499252f7151fb0207de5de7db416d19b70a7c368179d69f1bce54bd610847","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v0a --refutes--> v07\n- v03 --supports--> v06\n- v08 --supports--> v01\n- v05 --refutes--> v00\n- v05 --refutes--> v09\n- v06 --supports--> v05\n- v03 --supports--> v04\n- v07 --supports--> v03\n- v01 --refutes--> v0a\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":"e9dfb86dad06d7d3028a1d4522ad304c087503955014bc5da9ae2e33aac979c6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v07\n- v07 --supports--> v05\n- v08 --refutes--> v01\n- v06 --refutes--> v09\n- v05 --refutes--> v0a\n- v01 --refutes--> v03\n- v03 --supports--> v06\n- v0a --supports--> v02\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":"54e09ada8683a3ebe8f6d590a0feb3afb14c4dd015c6f95e8c821bc6f9b3dfac","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v01\n- v00 --supports--> v04\n- v06 --supports--> v09\n- v03 --refutes--> v08\n- v06 --supports--> v0a\n- v04 --supports--> v07\n- v05 --supports--> v06\n- v01 --refutes--> v00\n- v01 --refutes--> v02\n- v03 --supports--> v05\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":"eee4812e700bdc2baff82058cd679560dd6f767126314c288d84a5585d9287f1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v04 --refutes--> v00\n- v07 --supports--> v06\n- v05 --refutes--> v0a\n- v06 --supports--> v05\n- v03 --supports--> v02\n- v05 --supports--> v03\n- v01 --refutes--> v07\n- v00 --supports--> v01\n- v01 --refutes--> v09\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"31166f20e45e87b8e89af1d066d8bf3184fb183ac55dcc59c1631e448a393ebe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v08 --supports--> v06\n- v0a --refutes--> v04\n- v09 --supports--> v08\n- v08 --supports--> v02\n- v05 --supports--> v0a\n- v07 --refutes--> v03\n- v02 --refutes--> v05\n- v09 --supports--> v01\n- v09 --supports--> v00\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":"f348aa0034eadcedb1c81a031908791c8158a3ea95cfe5a18f25896245c44081","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v07 --supports--> v04\n- v01 --supports--> v09\n- v06 --supports--> v07\n- v03 --refutes--> v05\n- v09 --supports--> v02\n- v07 --supports--> v0a\n- v04 --supports--> v03\n- v04 --refutes--> v00\n- v06 --refutes--> v08\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":"6447aa1c5acbd16f4f27764107581a3632aaa658d3fd8f384ae50c45eddbfcf7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v05 --supports--> v02\n- v03 --refutes--> v09\n- v07 --refutes--> v00\n- v05 --refutes--> v0a\n- v07 --supports--> v05\n- v09 --supports--> v07\n- v06 --refutes--> v01\n- v01 --supports--> v08\n- v06 --supports--> v04\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":"bfaee5c0cfb88a3df4f5754e06a98987ad392fc8aea121ae7550fddfd18ecd12","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v07 --supports--> v08\n- v02 --refutes--> v06\n- v06 --supports--> v04\n- v0a --refutes--> v03\n- v01 --refutes--> v0a\n- v00 --supports--> v05\n- v07 --supports--> v02\n- v04 --supports--> v01\n- v0a --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2096c746fb7c7739b96ea5b17421f5c070bf8e08c1c61f1599dd72125c324e07","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v08\n- v09 --supports--> v0a\n- v06 --refutes--> v07\n- v09 --supports--> v01\n- v04 --refutes--> v03\n- v04 --refutes--> v06\n- v05 --refutes--> v00\n- v09 --refutes--> v05\n- v00 --supports--> v04\n- v07 --refutes--> v02\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8331a103c36aae128619ac6b7cb4d1454c47e7a2dfd100eade294d7270aa4721","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v03 --supports--> v01\n- v07 --supports--> v03\n- v04 --refutes--> v05\n- v0a --supports--> v00\n- v00 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v06\n- v05 --refutes--> v0a\n- v04 --supports--> v09\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":"8507a1bed5f9db6f5991be16c680f7ab815ac54f9d44dc7b5a05963046cba0c5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v0a --supports--> v00\n- v06 --refutes--> v05\n- v05 --supports--> v09\n- v03 --supports--> v08\n- v00 --refutes--> v03\n- v0a --refutes--> v04\n- v01 --refutes--> v06\n- v09 --supports--> v0a\n- v01 --refutes--> v02\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a4ffe9681b998a0a63e08246e7460d874e5beed635aab0496fc5a168770d935","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0a\n- v02 --supports--> v08\n- v01 --refutes--> v05\n- v0a --supports--> v00\n- v0a --supports--> v06\n- v08 --supports--> v09\n- v09 --supports--> v03\n- v06 --refutes--> v02\n- v0a --refutes--> v04\n- v06 --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":"0918da57d60f5c8cbe826cd5f5e3bfff79aca1f92b2eece09e677bf933a32d25","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0a\n- v03 --refutes--> v09\n- v01 --refutes--> v07\n- v04 --refutes--> v05\n- v09 --refutes--> v06\n- v07 --refutes--> v04\n- v09 --refutes--> v00\n- v0a --refutes--> v01\n- v01 --supports--> v08\n- v04 --refutes--> v02\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02cdf87fc9ca706146c50e9241844222339d3d779326ee77707d638a9db7d2a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v07 --supports--> v06\n- v09 --supports--> v04\n- v03 --refutes--> v08\n- v02 --supports--> v0a\n- v08 --supports--> v07\n- v03 --refutes--> v05\n- v00 --supports--> v09\n- v02 --supports--> v01\n- v04 --refutes--> v02\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":"802672a3075c399d34a4a966896d169d3e845c9d1342577f84aa5a51329ed0f5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v08 --supports--> v05\n- v09 --supports--> v0a\n- v03 --refutes--> v06\n- v06 --refutes--> v09\n- v08 --refutes--> v01\n- v09 --refutes--> v00\n- v07 --supports--> v02\n- v05 --supports--> v04\n- v05 --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":"95e8fdb1fa7335a92a630b41b20d40fe7079baf6ba6ec685bec738e976d1ed6f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v04 --refutes--> v0a\n- v09 --refutes--> v04\n- v00 --supports--> v09\n- v05 --supports--> v06\n- v05 --supports--> v07\n- v05 --supports--> v03\n- v08 --refutes--> v05\n- v00 --supports--> v02\n- v0a --supports--> v01\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":"b35b31ffdcee396cbb3e23ab8f5bfd4865c4267fc965ef97629c29124bdd25df","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v00 --supports--> v07\n- v06 --supports--> v00\n- v04 --refutes--> v08\n- v00 --refutes--> v0a\n- v0a --supports--> v05\n- v04 --refutes--> v03\n- v09 --supports--> v06\n- v06 --supports--> v01\n- v08 --supports--> v09\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":"7b14749fc9644ef5e3b44151489cf66bcd243444fd72d7948060bb8c9e0093d3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v07 --supports--> v0a\n- v02 --supports--> v00\n- v04 --refutes--> v07\n- v07 --refutes--> v03\n- v01 --refutes--> v06\n- v06 --supports--> v09\n- v02 --supports--> v08\n- v09 --refutes--> v04\n- v07 --supports--> v05\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":"d79d5d9b03a0a480389a87f1dc063fc8d1e8df65f95b18a79037de7c86d1c072","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v08 --supports--> v03\n- v03 --supports--> v09\n- v02 --refutes--> v07\n- v07 --supports--> v04\n- v05 --supports--> v01\n- v09 --supports--> v06\n- v06 --supports--> v02\n- v06 --supports--> v00\n- v06 --supports--> v0a\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":"154c903778f7a5e9d0be5df1895707f38317ef0e39a75a02d9e6ac7bddf1e439","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v05 --refutes--> v0b\n- v07 --refutes--> v03\n- v05 --refutes--> v06\n- v02 --refutes--> v09\n- v0a --refutes--> v05\n- v0a --refutes--> v08\n- v00 --supports--> v07\n- v0a --refutes--> v01\n- v0c --supports--> v02\n- v09 --refutes--> v04\n- v06 --supports--> v0c\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":"d5cc2bb97b0b53b867e8dc14e7728aa84d3e17d10382d4c4bceb2093ba71b784","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v0b --refutes--> v01\n- v08 --refutes--> v0a\n- v05 --supports--> v04\n- v0a --supports--> v0b\n- v07 --refutes--> v08\n- v05 --supports--> v00\n- v0b --supports--> v02\n- v06 --supports--> v05\n- v05 --refutes--> v07\n- v09 --refutes--> v06\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c63c4dc594b3a77155bc20153eba6fcc04a7ee232fcdcda374c00dcadae18380","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v06 --refutes--> v05\n- v07 --supports--> v0b\n- v08 --refutes--> v04\n- v0a --supports--> v02\n- v09 --refutes--> v06\n- v04 --refutes--> v00\n- v07 --refutes--> v01\n- v05 --refutes--> v03\n- v04 --supports--> v07\n- v08 --supports--> v0c\n- v05 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ffc80851e437557127dbebbe105d7f6c135ba6ca30373a508b8a82a9d1bd8317","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v0a --supports--> v04\n- v09 --supports--> v0c\n- v0a --refutes--> v00\n- v02 --supports--> v0b\n- v0c --supports--> v01\n- v0c --refutes--> v05\n- v06 --refutes--> v08\n- v02 --refutes--> v03\n- v03 --supports--> v09\n- v08 --supports--> v02\n- v0c --refutes--> v07\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":"b2b6ea764951af5fd5b46df2b8f6e106bd1aa991688e2d94702d0e888b64bde9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v00 --refutes--> v03\n- v02 --supports--> v00\n- v0b --supports--> v06\n- v04 --supports--> v05\n- v03 --refutes--> v0b\n- v0c --supports--> v0a\n- v0b --refutes--> v08\n- v04 --refutes--> v09\n- v02 --supports--> v01\n- v08 --supports--> v0c\n- v02 --supports--> v07\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":"638b50f88f286f405e59624d8ce155fae097b68587af3a6be26c7fd59d727103","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v04\n- v04 --refutes--> v00\n- v04 --supports--> v03\n- v0c --refutes--> v01\n- v0c --refutes--> v0b\n- v0a --supports--> v09\n- v06 --refutes--> v07\n- v08 --supports--> v05\n- v05 --supports--> v0a\n- v00 --supports--> v02\n- v00 --refutes--> v0c\n- v07 --refutes--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"890e7256d6d57715e47bf872bf1a5c94dcbb8f07faff3335a01047cab37b4ac5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v0b --refutes--> v06\n- v09 --refutes--> v03\n- v07 --refutes--> v09\n- v0a --supports--> v0c\n- v07 --refutes--> v01\n- v06 --refutes--> v00\n- v00 --refutes--> v02\n- v0a --refutes--> v08\n- v03 --supports--> v04\n- v0c --refutes--> v07\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":"87d12d0ba921905a924a6ca1262851626d9fb3566c33a492444edc0e401950bc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v0a --refutes--> v03\n- v02 --refutes--> v05\n- v06 --supports--> v09\n- v06 --refutes--> v0b\n- v03 --supports--> v07\n- v09 --refutes--> v08\n- v09 --supports--> v00\n- v05 --supports--> v06\n- v0c --supports--> v04\n- v09 --supports--> v0a\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c24f2156a2b91dc5166cab7d4d9d062952a45f5a10320aafce19028a42f8be98","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v00\n- v03 --supports--> v0a\n- v09 --supports--> v0b\n- v00 --refutes--> v02\n- v07 --refutes--> v05\n- v0b --refutes--> v04\n- v03 --refutes--> v09\n- v08 --refutes--> v07\n- v0c --refutes--> v06\n- v00 --refutes--> v08\n- v07 --supports--> v03\n- v00 --refutes--> v01\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":"397ab10cb786bebc5fd7e0062b6160e9f55984698085633576faa6f94ac3d12a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v04\n- v0c --supports--> v0a\n- v01 --supports--> v06\n- v02 --refutes--> v00\n- v05 --refutes--> v0c\n- v09 --supports--> v02\n- v06 --supports--> v0b\n- v0c --refutes--> v07\n- v0c --refutes--> v01\n- v00 --refutes--> v08\n- v02 --supports--> v03\n- v04 --refutes--> v09\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":"115dfc3cecf4f0ab9fcbb71cc92907975fe40bf800a47707c788b19af5853fb9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v05 --refutes--> v0a\n- v04 --refutes--> v06\n- v03 --refutes--> v05\n- v00 --refutes--> v02\n- v0a --refutes--> v09\n- v05 --supports--> v01\n- v03 --refutes--> v0c\n- v02 --supports--> v04\n- v0b --refutes--> v03\n- v03 --supports--> v08\n- v0a --supports--> v07\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":"555e25dc8a930bb3eadb2aec4d12f62d40bcd05406d3fbeefb2e16a5776ab84d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v04\n- v06 --supports--> v02\n- v03 --refutes--> v00\n- v0a --supports--> v07\n- v00 --supports--> v08\n- v05 --supports--> v0b\n- v02 --supports--> v09\n- v09 --supports--> v05\n- v02 --supports--> v0c\n- v00 --supports--> v0a\n- v07 --supports--> v01\n- v07 --supports--> v06\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":"5ae26694a71807290080b3c57a33f858eba89330b190d6c6cd240ad24e68ac71","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v0b --supports--> v09\n- v04 --refutes--> v0a\n- v0c --supports--> v07\n- v08 --refutes--> v04\n- v00 --refutes--> v02\n- v09 --supports--> v08\n- v06 --refutes--> v00\n- v0a --refutes--> v0c\n- v0a --supports--> v05\n- v09 --supports--> v01\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ceb32bb058154db12aa35c64a4209f40b3e40fd01f24e2edc92252b0d5a0cde5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v01\n- v07 --refutes--> v04\n- v03 --supports--> v0a\n- v09 --supports--> v07\n- v06 --supports--> v0b\n- v0a --supports--> v00\n- v03 --supports--> v02\n- v04 --supports--> v05\n- v01 --refutes--> v08\n- v08 --refutes--> v09\n- v06 --supports--> v0c\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad6dc146b34aa92940eafa20c7d5cb618132b74c69abe5f07e356584d9a9b45e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v02\n- v0b --refutes--> v00\n- v09 --refutes--> v04\n- v09 --supports--> v03\n- v05 --refutes--> v0c\n- v0b --supports--> v07\n- v08 --supports--> v09\n- v00 --refutes--> v01\n- v09 --refutes--> v0a\n- v06 --refutes--> v0b\n- v03 --refutes--> v06\n- v01 --supports--> v05\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":"6cb54193e458f363b871a0337cedf579f7de407c342c1d17d9bbd9b166a35f44","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v02 --refutes--> v09\n- v07 --supports--> v01\n- v01 --refutes--> v03\n- v06 --refutes--> v00\n- v03 --refutes--> v0c\n- v09 --supports--> v05\n- v0a --refutes--> v02\n- v03 --supports--> v06\n- v00 --refutes--> v0b\n- v02 --supports--> v08\n- v00 --refutes--> v0a\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":"ec20fedde7b605bd9687ad67a4b8aee1d169c734e6b8b6e08501c843cc4ea5f3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v09\n- v0a --refutes--> v06\n- v07 --refutes--> v01\n- v06 --refutes--> v0c\n- v0c --refutes--> v00\n- v01 --refutes--> v04\n- v00 --refutes--> v0b\n- v03 --refutes--> v07\n- v0b --supports--> v05\n- v01 --supports--> v08\n- v00 --supports--> v02\n- v0b --supports--> v03\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":"472435cca6b3d20ebea1c2217ba59cd64ccf7548e49ea741cff9a1571593ef0f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v00\n- v01 --supports--> v07\n- v02 --supports--> v05\n- v0a --supports--> v0c\n- v03 --refutes--> v08\n- v08 --refutes--> v0b\n- v0a --refutes--> v09\n- v09 --supports--> v04\n- v07 --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v06\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5462e50a2ded8f796293329b508baf39e456b34769881cd0f464467e3d6c533","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v00 --supports--> v05\n- v07 --supports--> v0c\n- v05 --supports--> v03\n- v01 --supports--> v00\n- v02 --supports--> v08\n- v06 --supports--> v04\n- v02 --refutes--> v06\n- v04 --supports--> v01\n- v0c --supports--> v0b\n- v05 --supports--> v07\n- v02 --refutes--> v0a\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":"da05e11bc512448d3b22ba3cb49280fbe6e893817c1ac998ec08682a7eb252c2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v01 --refutes--> v08\n- v04 --refutes--> v01\n- v0b --refutes--> v0a\n- v0b --supports--> v05\n- v03 --supports--> v07\n- v05 --refutes--> v03\n- v0b --supports--> v06\n- v09 --supports--> v04\n- v05 --refutes--> v02\n- v08 --refutes--> v0b\n- v03 --supports--> v00\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":"6fb0dffa2fcbf3063640904ba15abe2827bece738318b72998ae3afd8c3a3cd9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v0a --refutes--> v08\n- v02 --refutes--> v00\n- v03 --supports--> v0b\n- v0b --supports--> v01\n- v06 --supports--> v05\n- v09 --supports--> v03\n- v01 --refutes--> v02\n- v06 --supports--> v0c\n- v0c --supports--> v04\n- v0c --supports--> v09\n- v09 --supports--> v07\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":"548f6f21d99fd257b9018abf75f3221ffe153b8ca56ed9be253e52a1ae5eb6ac","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v08\n- v08 --refutes--> v06\n- v0c --refutes--> v00\n- v09 --supports--> v05\n- v01 --refutes--> v0c\n- v06 --refutes--> v09\n- v05 --supports--> v0a\n- v0a --refutes--> v04\n- v0a --supports--> v01\n- v09 --refutes--> v03\n- v0c --refutes--> v02\n- v06 --refutes--> v0b\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":"237d2afd3bda6d104bd146730b7a4a4d5bbc4c250069a08d7c8e3b9b891adab2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v0a --supports--> v0b\n- v09 --refutes--> v05\n- v06 --supports--> v00\n- v06 --refutes--> v01\n- v08 --supports--> v0a\n- v0c --supports--> v07\n- v07 --refutes--> v06\n- v01 --supports--> v09\n- v0a --supports--> v04\n- v08 --supports--> v02\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4c1b88f65bfeadeb6d0136c70f63a985ab90f701d27d9a76aa86c3b6f9e596d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v04\n- v05 --supports--> v03\n- v06 --supports--> v07\n- v0c --refutes--> v08\n- v02 --supports--> v06\n- v07 --supports--> v0c\n- v05 --supports--> v0a\n- v0a --supports--> v02\n- v00 --supports--> v01\n- v01 --supports--> v05\n- v00 --refutes--> v0b\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":"100127ade4c1089504f06f2abde30a8018cb1687a9e4863b75b97a46ff942d20","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v0c --refutes--> v00\n- v06 --supports--> v01\n- v08 --refutes--> v09\n- v0b --supports--> v08\n- v08 --refutes--> v05\n- v05 --supports--> v0a\n- v0c --refutes--> v02\n- v01 --supports--> v04\n- v04 --supports--> v03\n- v0c --supports--> v07\n- v04 --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":"f433bbb8e3e697dacf1a2ddd7d804fcbddd881cefabd3b84cdfce22093f85e1a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v01\n- v02 --supports--> v03\n- v00 --refutes--> v04\n- v09 --refutes--> v0c\n- v02 --supports--> v0a\n- v03 --supports--> v0b\n- v08 --supports--> v05\n- v07 --supports--> v09\n- v05 --refutes--> v07\n- v00 --refutes--> v08\n- v09 --refutes--> v02\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd8f52017933779b8c17e35d20ac70fc90e9081bcce31d02cdb8b6b3c49f696c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v02 --refutes--> v0c\n- v00 --refutes--> v09\n- v0a --supports--> v08\n- v0b --refutes--> v03\n- v0b --supports--> v06\n- v07 --supports--> v05\n- v08 --supports--> v0b\n- v07 --supports--> v0a\n- v00 --supports--> v04\n- v0b --refutes--> v00\n- v09 --refutes--> v01\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8859a0784d8f003060a0e4d339a63bb2f3c0c6bc613e168118b2b3b911502064","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v0a --refutes--> v02\n- v0a --refutes--> v01\n- v09 --refutes--> v07\n- v07 --supports--> v00\n- v09 --supports--> v03\n- v06 --refutes--> v09\n- v07 --supports--> v0b\n- v0a --supports--> v05\n- v00 --supports--> v04\n- v08 --refutes--> v06\n- v0c --refutes--> v08\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":"c4b56cde58ad41c542a6295977b30d4e1968f9c2f76ebc7533f21977a86b3be2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0b\n- v07 --refutes--> v0a\n- v0a --supports--> v09\n- v0b --refutes--> v0c\n- v08 --refutes--> v03\n- v0a --refutes--> v08\n- v08 --refutes--> v00\n- v08 --supports--> v02\n- v0c --supports--> v04\n- v04 --refutes--> v05\n- v04 --supports--> v01\n- v01 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c95c20c376037646895c9e1407976cd535db3e585196f5a7e51519d6c2dbb888","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v05\n- v05 --refutes--> v09\n- v02 --refutes--> v0b\n- v02 --supports--> v01\n- v0c --supports--> v02\n- v03 --supports--> v07\n- v00 --supports--> v08\n- v00 --supports--> v03\n- v02 --supports--> v04\n- v00 --refutes--> v06\n- v04 --refutes--> v0a\n- v09 --refutes--> v0c\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":"2f4d1d9b69c98fc08850eb7b5c58fa7f1b5d8226503be3ff726d95d874dd7def","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v0a --refutes--> v08\n- v0a --refutes--> v0b\n- v01 --supports--> v06\n- v04 --refutes--> v01\n- v06 --refutes--> v02\n- v07 --refutes--> v0a\n- v05 --refutes--> v0c\n- v0a --refutes--> v03\n- v0b --supports--> v05\n- v0c --refutes--> v00\n- v06 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3fbcdaebe23b6f765cae04d981c64facc1bc74ccb649a237bb394019a4952b6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v01 --supports--> v03\n- v04 --refutes--> v09\n- v0a --refutes--> v08\n- v01 --refutes--> v0c\n- v0b --supports--> v05\n- v08 --refutes--> v06\n- v06 --refutes--> v01\n- v08 --supports--> v00\n- v07 --refutes--> v02\n- v07 --supports--> v0b\n- v01 --refutes--> v04\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":"e49e175fec90fb5e4bcaae6eb5673e8907e1491af550c5ed965cecf6155ef018","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v05 --refutes--> v0b\n- v03 --refutes--> v06\n- v0b --supports--> v00\n- v0a --refutes--> v05\n- v0b --supports--> v09\n- v0c --supports--> v03\n- v01 --refutes--> v02\n- v0b --refutes--> v07\n- v01 --supports--> v0a\n- v05 --supports--> v08\n- v00 --supports--> v04\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":"62b5ae56ebb3a9aa6110036cadac767f0aa9fe9f73426abd084f4a34ba57ad91","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0a\n- v09 --supports--> v0c\n- v02 --refutes--> v09\n- v0b --supports--> v00\n- v0b --supports--> v06\n- v05 --refutes--> v04\n- v00 --refutes--> v01\n- v0a --refutes--> v0b\n- v0a --supports--> v07\n- v0c --refutes--> v03\n- v09 --supports--> v08\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4443125502980f1d55d215ef98fa77d8eb136721d0e290b0c2ed2d5e097c985","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v02 --supports--> v0b\n- v01 --refutes--> v00\n- v0a --refutes--> v09\n- v03 --refutes--> v04\n- v0a --refutes--> v0c\n- v0b --supports--> v05\n- v05 --supports--> v01\n- v00 --supports--> v07\n- v00 --supports--> v03\n- v02 --refutes--> v08\n- v06 --supports--> v0a\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e12dd3dc005dc890c7f633966988275e52095f0bbc19345e25992fd76049566","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v01\n- v0a --refutes--> v0b\n- v02 --refutes--> v09\n- v0c --refutes--> v08\n- v05 --supports--> v07\n- v06 --refutes--> v0a\n- v00 --supports--> v02\n- v01 --supports--> v04\n- v0a --supports--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v03\n- v04 --supports--> v0c\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":"5d848506c93cdfa4cfb7de23ebfc76bc6edd0812a03ca757c49dd841da2224b4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v09 --refutes--> v02\n- v01 --refutes--> v05\n- v09 --supports--> v0b\n- v0a --supports--> v00\n- v06 --refutes--> v07\n- v07 --supports--> v0a\n- v05 --supports--> v08\n- v07 --supports--> v0c\n- v0b --refutes--> v06\n- v0a --refutes--> v03\n- v05 --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":"7fa05002847bdb5514b96cf2c70d52b536694b3f98a7e864c4d65a0a1a4b0ed8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0b\n- v01 --refutes--> v0a\n- v06 --supports--> v08\n- v0b --refutes--> v02\n- v08 --supports--> v0c\n- v0b --refutes--> v04\n- v0b --supports--> v00\n- v09 --refutes--> v03\n- v00 --refutes--> v01\n- v0a --refutes--> v09\n- v01 --supports--> v07\n- v09 --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":"a61aa16e6c1b9ca4a9d325dd671df0bc7b90c922c2d1642d52a06da5d2016874","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v08 --supports--> v02\n- v04 --refutes--> v09\n- v04 --supports--> v03\n- v0b --refutes--> v01\n- v00 --supports--> v07\n- v05 --refutes--> v0a\n- v05 --refutes--> v04\n- v06 --supports--> v08\n- v00 --supports--> v06\n- v0c --supports--> v05\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"64d212ed6ad027f6e0280be14b58a86b367699fa289fb3b1fdec8771d287048a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v05 --refutes--> v02\n- v08 --supports--> v03\n- v04 --refutes--> v05\n- v02 --refutes--> v08\n- v00 --refutes--> v04\n- v08 --supports--> v01\n- v0a --supports--> v09\n- v0a --refutes--> v00\n- v01 --supports--> v06\n- v02 --refutes--> v07\n- v08 --supports--> v0b\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":"0684e56ef95b66ecdfa6ad4283af9a54a335d96d7c2c73c77d3ffdf30fe19a70","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v0b --refutes--> v02\n- v05 --supports--> v06\n- v02 --refutes--> v07\n- v05 --refutes--> v09\n- v00 --supports--> v0c\n- v05 --refutes--> v0a\n- v08 --refutes--> v05\n- v07 --supports--> v01\n- v01 --refutes--> v00\n- v01 --refutes--> v04\n- v09 --refutes--> v03\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":"33bbf5f295a2b43f314eb9e73fe4b08954896440eea64049865c20abcb4071d6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v0a --refutes--> v03\n- v08 --refutes--> v06\n- v06 --supports--> v09\n- v0a --refutes--> v0c\n- v08 --refutes--> v07\n- v08 --supports--> v04\n- v0b --supports--> v01\n- v05 --supports--> v0b\n- v09 --refutes--> v02\n- v01 --refutes--> v08\n- v03 --refutes--> v05\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"afc4d01dc1687dd1d503ea3d7046ceff0125858995689838b6674764ffa9c42e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v09\n- v07 --refutes--> v05\n- v04 --refutes--> v02\n- v06 --supports--> v0c\n- v01 --supports--> v06\n- v03 --refutes--> v00\n- v0c --refutes--> v0b\n- v08 --supports--> v04\n- v0c --supports--> v07\n- v04 --supports--> v01\n- v07 --refutes--> v0a\n- v07 --refutes--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"259f42297af30ee888ea81f50c0e6a2f470f113e3c59d87deec0c253d7c010c3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v07 --refutes--> v0c\n- v0a --supports--> v09\n- v03 --supports--> v01\n- v07 --supports--> v02\n- v0c --supports--> v05\n- v05 --supports--> v06\n- v06 --supports--> v0b\n- v09 --supports--> v03\n- v06 --supports--> v08\n- v06 --refutes--> v0a\n- v03 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b23838e054b89195fe8ef355e3b9b4ae263bb7729c537861f9b8a1950dd338ac","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v05\n- v00 --refutes--> v07\n- v08 --supports--> v04\n- v00 --supports--> v0c\n- v06 --supports--> v02\n- v06 --supports--> v0a\n- v09 --supports--> v08\n- v0b --supports--> v06\n- v03 --supports--> v00\n- v08 --refutes--> v01\n- v01 --refutes--> v03\n- v07 --supports--> v0b\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":"4e63c999d65e4037c3d1df98efd8063eef168ce7afe8b42a3e941f230dfd0165","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v02 --supports--> v0b\n- v0b --supports--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v07\n- v01 --supports--> v0a\n- v08 --supports--> v03\n- v0a --supports--> v09\n- v06 --supports--> v01\n- v0c --refutes--> v02\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":"56d9d0eb65a8b47096e0ad810c41ee631f4e1ec3001ab2afb35ce20274af94fe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v07 --supports--> v0a\n- v0b --supports--> v08\n- v04 --refutes--> v0b\n- v03 --refutes--> v05\n- v02 --refutes--> v03\n- v0c --supports--> v02\n- v00 --refutes--> v09\n- v07 --supports--> v01\n- v0a --supports--> v00\n- v00 --refutes--> v06\n- v03 --supports--> v07\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"618cd9fb0ef36ce8b51c892e29776241133c5d1ae2768b5d6bebe6fb30d157dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v0a --supports--> v08\n- v04 --supports--> v01\n- v06 --refutes--> v02\n- v09 --supports--> v07\n- v06 --refutes--> v00\n- v01 --supports--> v0b\n- v09 --supports--> v0a\n- v03 --supports--> v04\n- v0c --refutes--> v06\n- v01 --supports--> v05\n- v05 --refutes--> 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":"ac0578e1c797c7228ea88025b51490acecd5a569beeadc2a6988fba19d15b298","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0c\n- v07 --refutes--> v05\n- v0a --supports--> v03\n- v04 --refutes--> v06\n- v03 --refutes--> v0b\n- v00 --refutes--> v09\n- v07 --supports--> v08\n- v09 --supports--> v07\n- v03 --supports--> v04\n- v08 --refutes--> v0a\n- v01 --supports--> v00\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9ea0dd9a6749d65b8b23b15d23514a91bccacfbcbdcd53f88e52f99767caeb3d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v0a --refutes--> v08\n- v00 --supports--> v0b\n- v08 --refutes--> v02\n- v07 --supports--> v01\n- v07 --refutes--> v05\n- v0b --refutes--> v0c\n- v03 --refutes--> v0a\n- v08 --supports--> v00\n- v03 --supports--> v09\n- v0a --refutes--> v04\n- v06 --refutes--> v07\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":"88d1a9918ff0fb30085943b38382a301ef5fe3540ba62babbb66944180752677","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v02 --supports--> v03\n- v06 --refutes--> v02\n- v0c --refutes--> v07\n- v08 --supports--> v04\n- v0c --refutes--> v00\n- v05 --refutes--> v0c\n- v04 --refutes--> v01\n- v07 --refutes--> v08\n- v07 --supports--> v09\n- v07 --supports--> v0a\n- v02 --supports--> v05\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":"1628e17ccb14cd956d31dbe9a4675eb546bf42cf27580fb29bea9c27ac20a592","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v03\n- v0c --refutes--> v01\n- v07 --supports--> v0c\n- v0c --supports--> v06\n- v01 --supports--> v04\n- v00 --supports--> v02\n- v03 --refutes--> v08\n- v0a --refutes--> v07\n- v08 --supports--> v00\n- v00 --refutes--> v0a\n- v08 --supports--> v0b\n- v0a --refutes--> v09\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0103f79f23335fa98ca000fd72f63f31c3b2724c2415acb8b47da84ce0c92650","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v08 --refutes--> v07\n- v06 --refutes--> v05\n- v01 --supports--> v0b\n- v01 --supports--> v04\n- v07 --supports--> v0c\n- v05 --refutes--> v01\n- v07 --supports--> v0a\n- v01 --supports--> v09\n- v03 --supports--> v00\n- v08 --supports--> v02\n- v0a --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":"076788685105eb7ed235063fe09c1eca3758c08d6c0cbe31c3630910b4254ba5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0c\n- v0b --supports--> v05\n- v05 --refutes--> v04\n- v03 --supports--> v09\n- v05 --refutes--> v00\n- v04 --supports--> v07\n- v03 --supports--> v06\n- v0c --supports--> v01\n- v01 --supports--> v08\n- v04 --refutes--> v03\n- v08 --refutes--> v0a\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":"21737850c7f9c7e9a65e75ea0468b1ec4523f3a8a85078b65fab801500861286","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v03\n- v0c --refutes--> v09\n- v05 --supports--> v06\n- v05 --supports--> v01\n- v01 --supports--> v04\n- v03 --supports--> v07\n- v01 --supports--> v0a\n- v07 --supports--> v08\n- v08 --refutes--> v05\n- v04 --refutes--> v02\n- v03 --refutes--> v0b\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":"d85e26ab2e2ac28693aaff4f01cb9eda6fbefa9cfc343a950c0640deb54f786e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v08\n- v02 --supports--> v0b\n- v02 --refutes--> v01\n- v0c --supports--> v03\n- v01 --supports--> v06\n- v0c --refutes--> v00\n- v06 --refutes--> v07\n- v08 --refutes--> v0a\n- v05 --supports--> v02\n- v07 --supports--> v0c\n- v09 --supports--> v05\n- v02 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97c9f31b43a0f7040b8e99aee639b6d3871915d38478cb27705e7b7b858843c3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0a\n- v01 --supports--> v03\n- v0b --supports--> v05\n- v08 --supports--> v07\n- v07 --refutes--> v01\n- v04 --refutes--> v09\n- v04 --refutes--> v00\n- v09 --supports--> v06\n- v03 --refutes--> v0c\n- v0c --refutes--> v02\n- v0b --refutes--> v04\n- v09 --refutes--> v08\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":"c97aa24141ff8b8b1b181015b8e084a0a4aba8def9259e89f4a7fd2903d66937","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v09\n- v00 --refutes--> v07\n- v0b --supports--> v08\n- v01 --supports--> v02\n- v05 --refutes--> v00\n- v0a --supports--> v0b\n- v03 --refutes--> v0a\n- v03 --supports--> v06\n- v08 --refutes--> v01\n- v09 --supports--> v04\n- v00 --refutes--> v03\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":"24907ebf297e2d9e855b11f5705a9122238e9c71ee6659f22c4d5d7ffa9c5500","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0b\n- v08 --refutes--> v0a\n- v02 --supports--> v05\n- v09 --supports--> v07\n- v07 --supports--> v01\n- v04 --refutes--> v03\n- v00 --supports--> v0c\n- v04 --refutes--> v09\n- v02 --refutes--> v08\n- v0c --supports--> v02\n- v01 --supports--> v00\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":"e3f4e12308477169492b785dde9f92dac56bd24027d8fca85e7e09a0319f71aa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0a\n- v0b --supports--> v00\n- v08 --supports--> v07\n- v06 --supports--> v02\n- v08 --refutes--> v04\n- v01 --refutes--> v06\n- v08 --refutes--> v09\n- v09 --supports--> v0c\n- v00 --refutes--> v01\n- v02 --supports--> v03\n- v03 --supports--> v05\n- v09 --refutes--> v0b\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":"9b0cc077ef0bd4f0a82a383174f5be559e976e2d012988193a8c723605811919","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v09\n- v05 --refutes--> v03\n- v0b --refutes--> v0c\n- v09 --supports--> v06\n- v09 --refutes--> v01\n- v07 --refutes--> v0a\n- v0c --supports--> v07\n- v0c --refutes--> v04\n- v09 --supports--> v02\n- v03 --refutes--> v00\n- v01 --supports--> v05\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":"1f46a914bd828cccfd36fbffbdc81dd9fa47fb585a8c4dfd5def7a8865aca00e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v09 --refutes--> v00\n- v09 --refutes--> v0a\n- v01 --supports--> v04\n- v0a --refutes--> v07\n- v02 --supports--> v08\n- v05 --refutes--> v0b\n- v04 --supports--> v06\n- v02 --supports--> v09\n- v0b --supports--> v02\n- v0a --supports--> v01\n- v05 --refutes--> v03\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":"62b7163589f0a8397328ebf9d67fc930ff1a09c215b7fc6bde09ca03b54cc8cb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v0c --refutes--> v01\n- v06 --supports--> v0c\n- v02 --supports--> v00\n- v03 --refutes--> v07\n- v00 --supports--> v05\n- v00 --supports--> v09\n- v07 --refutes--> v0b\n- v00 --supports--> v04\n- v07 --supports--> v06\n- v04 --refutes--> v0a\n- v04 --supports--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b6ebaeeeb5aa7bd453875a9da989e48534ed645a18c3fccb5abd0a2f88a28b7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v01\n- v01 --refutes--> v03\n- v01 --refutes--> v0c\n- v0c --refutes--> v05\n- v07 --supports--> v04\n- v05 --supports--> v09\n- v07 --refutes--> v06\n- v0b --refutes--> v07\n- v06 --supports--> v08\n- v09 --refutes--> v0b\n- v06 --supports--> v02\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":"53540d0a5a83e7e705aac42800137d5e5d9f95a48690c0203a912f3e074b4f3b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v02\n- v07 --refutes--> v01\n- v08 --refutes--> v0c\n- v08 --refutes--> v04\n- v0d --refutes--> v05\n- v0c --refutes--> v03\n- v0c --supports--> v0d\n- v02 --refutes--> v0b\n- v09 --supports--> v07\n- v05 --supports--> v06\n- v02 --refutes--> v00\n- v01 --refutes--> v08\n- v0b --supports--> v0a\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":"6aa6e3a2b9a1a8deaccd59b313437c46fc922856299dfc228d72e59d0f1b71d2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v03\n- v0d --refutes--> v08\n- v09 --refutes--> v0b\n- v04 --refutes--> v01\n- v02 --supports--> v07\n- v0b --refutes--> v00\n- v08 --refutes--> v02\n- v07 --refutes--> v04\n- v0c --supports--> v05\n- v0c --supports--> v0d\n- v0d --supports--> v0a\n- v03 --supports--> v0c\n- v0b --refutes--> v06\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":"058dd14170895f0635b7844f81cf1f174a4b778b3a0acc1f79e85cc12e189f6f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v01 --supports--> v00\n- v02 --refutes--> v08\n- v0a --supports--> v07\n- v04 --supports--> v02\n- v0b --refutes--> v0a\n- v07 --supports--> v09\n- v07 --supports--> v03\n- v06 --refutes--> v0d\n- v0d --supports--> v01\n- v02 --refutes--> v0c\n- v02 --refutes--> v0b\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":"89bdad2b489d5ed56259a9d2cbf77763cba8ba343e39618ea701c5d1ca795dd9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v09 --refutes--> v05\n- v0d --supports--> v02\n- v07 --supports--> v09\n- v0d --supports--> v0b\n- v05 --refutes--> v04\n- v0c --refutes--> v06\n- v0c --supports--> v07\n- v0b --supports--> v01\n- v0c --supports--> v03\n- v08 --refutes--> v0d\n- v04 --supports--> v00\n- v0a --supports--> v0c\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":"45746222d36d2df10a9caf70f999af1bc191b385e64a67a74d291661c46de558","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v01 --supports--> v02\n- v05 --refutes--> v0b\n- v06 --refutes--> v00\n- v07 --supports--> v03\n- v06 --refutes--> v05\n- v04 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v0d\n- v09 --refutes--> v06\n- v03 --refutes--> v04\n- v02 --refutes--> v08\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":"f03652fff1039ece0de77bfec612b75f966c3a9a63cb516a5a4c9daa6a25682f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v04\n- v03 --supports--> v09\n- v07 --refutes--> v01\n- v07 --refutes--> v0b\n- v08 --refutes--> v0c\n- v08 --supports--> v06\n- v04 --supports--> v03\n- v00 --supports--> v07\n- v0a --refutes--> v08\n- v01 --refutes--> v0a\n- v08 --supports--> v05\n- v09 --supports--> v00\n- v03 --refutes--> v02\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":"72202a6c8ac995e64ce8e32b05790bcfbff2a85e182d94d27e904f3065210647","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v07 --supports--> v06\n- v0b --supports--> v04\n- v01 --refutes--> v09\n- v09 --supports--> v02\n- v02 --supports--> v03\n- v01 --supports--> v0c\n- v08 --refutes--> v0b\n- v09 --refutes--> v00\n- v05 --supports--> v0a\n- v0b --supports--> v05\n- v06 --supports--> v08\n- v03 --refutes--> v07\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":"beeb2ce8983a9e9260bb6f0066187dbe2dc350579e16f4e46a8eed7e2233e2fe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0c\n- v0a --refutes--> v03\n- v0d --refutes--> v05\n- v09 --refutes--> v08\n- v0c --refutes--> v07\n- v00 --supports--> v0a\n- v07 --supports--> v0d\n- v01 --supports--> v02\n- v00 --supports--> v0b\n- v0a --refutes--> v01\n- v0d --refutes--> v09\n- v02 --refutes--> v04\n- v0a --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb31f54cce6334230349db7ea43da985fd48618da32a073e423240cb3022e786","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v00 --supports--> v03\n- v04 --supports--> v0a\n- v02 --refutes--> v0b\n- v07 --refutes--> v05\n- v0c --supports--> v06\n- v0d --refutes--> v01\n- v04 --supports--> v08\n- v03 --supports--> v0c\n- v05 --supports--> v02\n- v04 --refutes--> v0d\n- v04 --refutes--> v09\n- v0b --refutes--> v00\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":"4f8f340880a0bd0d710e27bcd472a46c2cfe95cdeb0b4742c073f090bf802bbd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v06 --refutes--> v09\n- v00 --supports--> v06\n- v0c --supports--> v03\n- v0d --refutes--> v0b\n- v0a --supports--> v02\n- v08 --supports--> v01\n- v02 --refutes--> v07\n- v0d --refutes--> v05\n- v0c --refutes--> v0a\n- v01 --refutes--> v0c\n- v06 --refutes--> v04\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de2e8ce1d67efd3ad74382b26ea91de7bc2dec9baddcb48e97dfa4a8fc37f5c6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v07\n- v02 --supports--> v0c\n- v07 --refutes--> v0a\n- v00 --refutes--> v06\n- v08 --supports--> v09\n- v01 --refutes--> v00\n- v0b --supports--> v03\n- v02 --supports--> v0d\n- v03 --supports--> v01\n- v00 --supports--> v08\n- v07 --refutes--> v04\n- v03 --supports--> v05\n- v0a --refutes--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2197b32a800466266883e35159147f29db671d692b41b7a80cee0a4b5c37a72e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0b\n- v0c --refutes--> v02\n- v0a --supports--> v03\n- v0a --supports--> v07\n- v09 --refutes--> v0d\n- v04 --supports--> v0a\n- v09 --supports--> v05\n- v07 --refutes--> v09\n- v00 --supports--> v04\n- v0a --refutes--> v01\n- v06 --supports--> v00\n- v0b --refutes--> v06\n- v04 --supports--> v08\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":"d7bb139a9f705c2571e71bad42af374c24ea6ad52dac256b51c35f8b6221af3d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v0c --refutes--> v0d\n- v08 --refutes--> v00\n- v0d --supports--> v04\n- v0d --supports--> v02\n- v0b --supports--> v05\n- v06 --supports--> v08\n- v07 --supports--> v0b\n- v03 --refutes--> v06\n- v01 --supports--> v03\n- v0a --supports--> v09\n- v0a --refutes--> v07\n- v0b --refutes--> v01\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":"89c2a5225af5020ef20e25bc893763f84241469e9e0f23945e3a9c9157e04256","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v07\n- v00 --refutes--> v06\n- v01 --supports--> v0d\n- v0c --refutes--> v05\n- v06 --refutes--> v0b\n- v0a --supports--> v08\n- v05 --supports--> v03\n- v0b --supports--> v01\n- v0a --supports--> v02\n- v06 --refutes--> v04\n- v05 --refutes--> v09\n- v0a --supports--> v00\n- v09 --supports--> v0a\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":"6cde6ae592ffe1e03bceada1efaa88808c5da75e260d2faebd575ea11265adfe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0a\n- v01 --refutes--> v05\n- v0d --supports--> v03\n- v06 --refutes--> v0b\n- v04 --refutes--> v02\n- v0c --refutes--> v06\n- v05 --refutes--> v04\n- v0c --supports--> v08\n- v0a --refutes--> v01\n- v00 --supports--> v0d\n- v06 --supports--> v00\n- v02 --supports--> v07\n- v0a --refutes--> 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":"eb555106865a873443bd339ebecc396b0509cdfc56af5f40ef600d20a75a39bb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0b\n- v06 --supports--> v0a\n- v0a --supports--> v04\n- v05 --refutes--> v07\n- v0c --refutes--> v05\n- v07 --supports--> v08\n- v03 --supports--> v02\n- v0a --refutes--> v00\n- v09 --refutes--> v01\n- v0d --supports--> v0c\n- v03 --refutes--> v06\n- v01 --supports--> v03\n- v07 --refutes--> v09\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e483ac8a3f03b20a35eb92b7da0caa54aa682456257edf036314a32aaf851d0b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v07\n- v07 --refutes--> v0b\n- v03 --refutes--> v08\n- v04 --supports--> v09\n- v00 --refutes--> v0c\n- v05 --refutes--> v03\n- v09 --supports--> v01\n- v0c --supports--> v05\n- v0b --refutes--> v0d\n- v03 --refutes--> v04\n- v0b --supports--> v00\n- v07 --refutes--> v06\n- v09 --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3fbd4a0323a1111eeac732bedef1cac70712265b39f42dd619fdd2a62b0eb31","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v09 --refutes--> v04\n- v01 --supports--> v02\n- v0b --supports--> v0d\n- v02 --supports--> v09\n- v05 --refutes--> v08\n- v09 --refutes--> v05\n- v06 --refutes--> v01\n- v0b --supports--> v0a\n- v05 --supports--> v03\n- v0d --supports--> v06\n- v07 --supports--> v00\n- v00 --supports--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"966225ae1df0ddaa5df57126aa2456422476fc9ab7a016b558c8565f1caf6a44","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v09\n- v0b --supports--> v07\n- v01 --refutes--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v0d\n- v03 --refutes--> v0a\n- v01 --supports--> v04\n- v05 --supports--> v0c\n- v08 --supports--> v00\n- v08 --refutes--> v06\n- v0c --supports--> v01\n- v06 --refutes--> v02\n- v0a --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":"e9429c70db957f1de4b8b80fc63a57f6c7e329f424e2a51dec50206eecd374a7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0a\n- v07 --refutes--> v06\n- v0a --refutes--> v03\n- v02 --refutes--> v07\n- v07 --supports--> v08\n- v03 --supports--> v09\n- v0a --supports--> v0b\n- v08 --refutes--> v01\n- v09 --supports--> v02\n- v04 --refutes--> v0c\n- v03 --supports--> v0d\n- v05 --refutes--> v04\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61e02a6f51ca5882864174e5e109a6e00bd4c68d2ca15395e9c33856cb3505a8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v08\n- v00 --supports--> v01\n- v09 --refutes--> v0c\n- v01 --refutes--> v02\n- v08 --refutes--> v07\n- v08 --supports--> v0b\n- v04 --refutes--> v09\n- v09 --supports--> v0a\n- v01 --refutes--> v03\n- v08 --refutes--> v05\n- v02 --supports--> v06\n- v0d --refutes--> v00\n- v07 --supports--> v04\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":"7f240efd74e85ec20447a2f3aaed93cbafe8c30f46b9d91e9b9a1fa4d8f1fb74","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v01 --refutes--> v0d\n- v07 --refutes--> v0a\n- v07 --supports--> v02\n- v0d --refutes--> v08\n- v05 --supports--> v00\n- v05 --supports--> v04\n- v03 --supports--> v01\n- v06 --refutes--> v03\n- v01 --supports--> v09\n- v04 --supports--> v0b\n- v07 --refutes--> v0c\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b3c79badce8b3d9a399e21f03c507adb2a92a13ce554a45c6b3b885323b7db84","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v0d --supports--> v0c\n- v08 --refutes--> v09\n- v04 --refutes--> v02\n- v03 --refutes--> v0b\n- v03 --supports--> v05\n- v06 --refutes--> v0a\n- v0c --refutes--> v08\n- v03 --refutes--> v07\n- v06 --refutes--> v03\n- v02 --supports--> v0d\n- v09 --refutes--> v00\n- v05 --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":"5960e2cfcf5ef316801be62ee8828e97514eb051cc3dcdd7b4fa47d7fd4705f3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0c\n- v0b --supports--> v00\n- v04 --supports--> v03\n- v05 --supports--> v0a\n- v06 --supports--> v08\n- v05 --refutes--> v07\n- v00 --supports--> v02\n- v03 --refutes--> v09\n- v03 --refutes--> v05\n- v02 --supports--> v06\n- v06 --supports--> v0d\n- v01 --supports--> v0b\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8877f8345f157c1f68eb825cb958218dc2665eca84b9d6f0cb293b81b3ac8f40","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v03\n- v01 --refutes--> v0d\n- v0a --supports--> v04\n- v09 --refutes--> v01\n- v04 --supports--> v05\n- v06 --supports--> v0a\n- v07 --refutes--> v0b\n- v07 --refutes--> v02\n- v05 --supports--> v00\n- v01 --refutes--> v07\n- v08 --supports--> v09\n- v08 --refutes--> v0c\n- v02 --refutes--> v06\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":"7d6f6c168fa73e4a8a04e7a30c6600328957f93c7fb5e117fb2ece62cbc0d38e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0d\n- v05 --supports--> v09\n- v08 --refutes--> v07\n- v05 --supports--> v01\n- v0b --refutes--> v00\n- v0c --supports--> v0a\n- v01 --supports--> v04\n- v08 --refutes--> v05\n- v0a --supports--> v06\n- v00 --supports--> v02\n- v0d --supports--> v0b\n- v01 --supports--> v0c\n- v05 --refutes--> v03\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":"a01dfc57cfac40a2ec9f12237e0ae4b6e9fe482112b50e564b7edc283701b355","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0b\n- v0c --supports--> v00\n- v0c --supports--> v06\n- v02 --refutes--> v05\n- v0b --supports--> v01\n- v03 --supports--> v08\n- v0a --supports--> v03\n- v03 --refutes--> v07\n- v09 --refutes--> v02\n- v07 --supports--> v09\n- v01 --supports--> v04\n- v04 --refutes--> v0c\n- v0a --refutes--> v0d\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e0c38ea3fa831d313bab3cfb21dc96fdcc522bc9d0fe2a75bf0ecb5d589ceee0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v07 --refutes--> v0b\n- v0b --refutes--> v08\n- v09 --supports--> v07\n- v0a --refutes--> v03\n- v01 --refutes--> v06\n- v0a --refutes--> v02\n- v03 --refutes--> v0c\n- v0a --refutes--> v0d\n- v00 --supports--> v0a\n- v0b --supports--> v04\n- v07 --supports--> v05\n- v04 --supports--> v00\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":"db227d0ba6cbaf8a6f5d6bea613d823fcc091ee80d611bb9882c850490f50744","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v09\n- v0c --refutes--> v00\n- v0d --supports--> v03\n- v03 --supports--> v06\n- v08 --supports--> v07\n- v08 --refutes--> v01\n- v02 --refutes--> v0c\n- v01 --supports--> v0b\n- v04 --refutes--> v0d\n- v05 --refutes--> v02\n- v03 --refutes--> v05\n- v0a --supports--> v08\n- v0c --supports--> v0a\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":"f6ce82e684a5ebcfcd6f1dcd239b087db17b8bb0ccd78c3606269c6bf1dd93e4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v01\n- v02 --supports--> v07\n- v0d --refutes--> v00\n- v09 --refutes--> v0c\n- v0a --supports--> v06\n- v02 --refutes--> v03\n- v00 --supports--> v09\n- v03 --supports--> v0d\n- v0b --refutes--> v04\n- v00 --refutes--> v08\n- v0a --refutes--> v0b\n- v04 --refutes--> v02\n- v0c --refutes--> v05\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":"ad40b299d6df79a1be899f57849bc05976e9a0db7d9657e7d84cd937a42658b3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v00 --supports--> v02\n- v09 --supports--> v05\n- v06 --supports--> v09\n- v01 --refutes--> v00\n- v09 --refutes--> v04\n- v0d --refutes--> v0b\n- v0c --refutes--> v03\n- v02 --refutes--> v0d\n- v09 --refutes--> v07\n- v0c --supports--> v06\n- v01 --supports--> v0a\n- v08 --refutes--> v01\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":"85a9d448bdf641d0616e3135b381c17a1e978d1e2a52fe8505f09adf2fe94c12","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v01\n- v0b --supports--> v06\n- v0b --supports--> v0d\n- v09 --supports--> v0c\n- v04 --supports--> v00\n- v0c --supports--> v05\n- v0a --supports--> v0b\n- v02 --refutes--> v0a\n- v0a --refutes--> v08\n- v04 --supports--> v07\n- v06 --supports--> v03\n- v04 --supports--> v09\n- v03 --supports--> v04\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b24d9ecbdb659a2268c3b72fc30cc8d6c6fd7d4bd8035cf152a7c38b07c265aa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0b\n- v00 --supports--> v0c\n- v06 --supports--> v0d\n- v09 --refutes--> v00\n- v04 --refutes--> v07\n- v0c --supports--> v06\n- v01 --supports--> v08\n- v0a --refutes--> v09\n- v06 --supports--> v03\n- v04 --refutes--> v02\n- v06 --refutes--> v05\n- v08 --supports--> v0a\n- v02 --refutes--> v01\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":"b51489b6c0d4e4ce92745dad25db10e41b2819535c51b7907c35bce1918ba9f8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v02 --refutes--> v00\n- v00 --supports--> v0b\n- v04 --supports--> v03\n- v0d --supports--> v02\n- v06 --refutes--> v05\n- v03 --supports--> v0c\n- v06 --refutes--> v09\n- v02 --supports--> v01\n- v08 --supports--> v0d\n- v0b --supports--> v04\n- v06 --refutes--> v0a\n- v0c --refutes--> v06\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":"e9b11566811691ce688733edb6757ace8ccf109d44b18d29a8077d54f22de828","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v03\n- v02 --supports--> v06\n- v0c --supports--> v08\n- v01 --refutes--> v07\n- v0b --supports--> v0d\n- v05 --supports--> v0b\n- v03 --refutes--> v0c\n- v09 --refutes--> v04\n- v07 --supports--> v05\n- v0b --supports--> v02\n- v0c --supports--> v00\n- v02 --refutes--> v0a\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0392986fd04953cf5de859790ea5060aef589ee4fd5095c9d6ceb99dfdcb179b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v02\n- v04 --supports--> v09\n- v0b --supports--> v0a\n- v05 --supports--> v07\n- v01 --supports--> v0b\n- v09 --supports--> v0c\n- v08 --supports--> v01\n- v05 --refutes--> v00\n- v0b --supports--> v0d\n- v0c --refutes--> v05\n- v01 --refutes--> v03\n- v00 --refutes--> v08\n- v06 --refutes--> v04\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":"4680a752ed9800b02cb4ebeb790425702d0cd3c5be09fd7413e3d87378f8b2cf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v06 --refutes--> v08\n- v0b --supports--> v0a\n- v0d --refutes--> v06\n- v04 --refutes--> v01\n- v04 --refutes--> v05\n- v02 --supports--> v0d\n- v03 --refutes--> v0b\n- v08 --supports--> v09\n- v0c --refutes--> v02\n- v08 --supports--> v03\n- v04 --refutes--> v0c\n- v0b --supports--> v07\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":"0f2dbd89eeed73475fbc3f8570c0683029657842dde0cce877a1c6899dda2deb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v00 --supports--> v08\n- v0b --supports--> v03\n- v07 --supports--> v09\n- v02 --refutes--> v01\n- v00 --refutes--> v0a\n- v08 --refutes--> v0c\n- v03 --refutes--> v02\n- v08 --refutes--> v0d\n- v01 --supports--> v04\n- v09 --supports--> v0b\n- v0c --supports--> v07\n- v07 --refutes--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0ee7de540319664c5a790d88bd011397c1ad4b381f016d4648e977454bfe43ee","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v00\n- v00 --refutes--> v08\n- v0c --supports--> v07\n- v09 --supports--> v01\n- v03 --supports--> v0a\n- v03 --supports--> v04\n- v0b --refutes--> v05\n- v03 --refutes--> v09\n- v00 --refutes--> v02\n- v02 --refutes--> v03\n- v0b --supports--> v06\n- v01 --supports--> v0b\n- v07 --refutes--> 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":"059dc4e6906f9202ae0c9261c95831f47f50f17fb333fd1fd1281bc94ba046a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v09\n- v01 --refutes--> v0a\n- v0b --supports--> v05\n- v07 --refutes--> v0b\n- v06 --supports--> v02\n- v0a --supports--> v03\n- v08 --supports--> v06\n- v05 --refutes--> v0d\n- v03 --refutes--> v08\n- v05 --supports--> v01\n- v02 --supports--> v0c\n- v02 --refutes--> v00\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":"33d1e4a08040f9b88b12979108ab4a03bf5c40b73eff7fb24b3e5eab0ff434d1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v08 --supports--> v0b\n- v08 --refutes--> v01\n- v0a --refutes--> v0c\n- v09 --refutes--> v02\n- v04 --supports--> v03\n- v03 --refutes--> v09\n- v06 --supports--> v04\n- v0a --supports--> v05\n- v09 --refutes--> v0d\n- v07 --supports--> v06\n- v01 --supports--> v00\n- v01 --supports--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bd6d2b05e265ee707d0c7958d65c3217e349205ee4c7b267017b20bbcdcfdb26","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v05 --refutes--> v0a\n- v0b --supports--> v08\n- v0b --refutes--> v0c\n- v04 --supports--> v0d\n- v03 --supports--> v05\n- v0d --refutes--> v0b\n- v09 --refutes--> v02\n- v0c --supports--> v03\n- v05 --refutes--> v06\n- v0a --refutes--> v07\n- v0c --supports--> v01\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":"f175aa4b32ff8107bcd30d65129e18950cc8669bcbd5b4b461f73bcdf4ad4618","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v0c --refutes--> v0d\n- v02 --supports--> v06\n- v0b --refutes--> v03\n- v0d --refutes--> v05\n- v0a --supports--> v09\n- v04 --refutes--> v02\n- v05 --supports--> v0b\n- v06 --refutes--> v00\n- v02 --refutes--> v08\n- v02 --supports--> v07\n- v09 --refutes--> v0c\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bfe356fa71d52a48dbcdf56b56833170ce373eba39c76c5d980e12ce311ad18","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v09\n- v06 --supports--> v03\n- v07 --refutes--> v06\n- v0b --refutes--> v07\n- v06 --supports--> v0c\n- v08 --supports--> v04\n- v02 --supports--> v0d\n- v09 --supports--> v05\n- v0d --refutes--> v00\n- v00 --refutes--> v08\n- v00 --refutes--> v0a\n- v02 --refutes--> v01\n- v08 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7544cfcbb4fb6929d4be1b2648cff92fd5b6c89a6fc1f36e90abdd23ae39b426","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v08 --supports--> v03\n- v02 --refutes--> v09\n- v0a --refutes--> v0b\n- v08 --refutes--> v07\n- v05 --refutes--> v06\n- v05 --refutes--> v0c\n- v08 --supports--> v0a\n- v01 --refutes--> v04\n- v01 --refutes--> v02\n- v06 --supports--> v00\n- v0b --supports--> v0d\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"817c0ccf6b765c2e4015b0123ac1950f79ccdbd8f2bc5aca8db94a05d8c5ac58","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v01\n- v01 --refutes--> v0b\n- v00 --refutes--> v06\n- v06 --supports--> v04\n- v08 --supports--> v0d\n- v04 --supports--> v07\n- v0a --refutes--> v05\n- v0b --refutes--> v08\n- v00 --supports--> v0c\n- v00 --refutes--> v09\n- v0d --refutes--> v0a\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":"65c519d1e4673bec1971a61a7e8d9fd5f9a436d879e54311656c5e7be4c2c5dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v01\n- v0b --refutes--> v00\n- v0a --supports--> v09\n- v07 --supports--> v08\n- v08 --supports--> v06\n- v02 --refutes--> v03\n- v03 --supports--> v0c\n- v04 --refutes--> v0a\n- v02 --refutes--> v05\n- v0c --refutes--> v04\n- v09 --refutes--> v07\n- v0a --refutes--> v0d\n- v08 --supports--> v0b\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":"1b06529e42b1e22be9240858c3e9e7e9eb4344ed43ecffd660ce24fad657bc16","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0d\n- v09 --supports--> v05\n- v02 --refutes--> v03\n- v05 --refutes--> v08\n- v07 --supports--> v0b\n- v07 --refutes--> v0a\n- v00 --refutes--> v0c\n- v03 --supports--> v01\n- v02 --supports--> v04\n- v06 --supports--> v00\n- v0c --refutes--> v09\n- v05 --supports--> v07\n- v0d --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":"69b44c292bbd3837d830c9bbab4d6d5125b0c21399a0aeea2b3bc035e0dae90e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v03 --refutes--> v01\n- v04 --supports--> v0b\n- v01 --supports--> v07\n- v05 --refutes--> v02\n- v08 --refutes--> v09\n- v07 --refutes--> v0d\n- v08 --supports--> v04\n- v0b --refutes--> v06\n- v0b --refutes--> v03\n- v02 --refutes--> v0c\n- v01 --supports--> v0a\n- v0c --supports--> v08\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19815248a77cc76480834726e11a7e39fc9f968c2ec739da80e31cefae877d69","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v0a --refutes--> v02\n- v04 --supports--> v08\n- v0a --supports--> v09\n- v01 --supports--> v0a\n- v02 --supports--> v06\n- v08 --supports--> v0c\n- v0c --supports--> v01\n- v00 --supports--> v07\n- v00 --supports--> v03\n- v05 --supports--> v04\n- v0c --refutes--> v0b\n- v00 --refutes--> v0d\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f76e0bc2aab801547d70aba1397fd1c86cdb54a70d062ac7737b4c55c5ecbe14","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v0d --supports--> v05\n- v0b --supports--> v06\n- v0b --supports--> v0d\n- v02 --refutes--> v07\n- v00 --supports--> v01\n- v0c --supports--> v04\n- v07 --supports--> v0b\n- v08 --refutes--> v09\n- v00 --supports--> v03\n- v04 --refutes--> v02\n- v05 --refutes--> v00\n- v05 --supports--> v0a\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":"f3b7d946da53750fb955f420b3423bf20314f91311f7ebdd45c2e3089e4f9bb7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v09 --supports--> v06\n- v04 --refutes--> v09\n- v03 --supports--> v07\n- v07 --supports--> v02\n- v0b --refutes--> v01\n- v04 --refutes--> v08\n- v07 --refutes--> v04\n- v0c --refutes--> v05\n- v00 --refutes--> v0c\n- v0c --refutes--> v0d\n- v04 --supports--> v0a\n- v01 --refutes--> v00\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":"521819d71a416a4ae4b04762c7eff2c6e6baef5c7082988906ffc99d5c862d02","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v03\n- v02 --supports--> v09\n- v01 --refutes--> v0b\n- v0a --supports--> v0d\n- v0d --refutes--> v05\n- v00 --supports--> v06\n- v09 --refutes--> v0a\n- v08 --refutes--> v01\n- v08 --refutes--> v0c\n- v09 --refutes--> v07\n- v0d --refutes--> v04\n- v0b --refutes--> v02\n- v04 --refutes--> v00\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":"b414e343092ab75ab2ff7f15cf7f6f60f695189bc804c172b09ccb2944b56ad9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v02\n- v01 --supports--> v00\n- v00 --refutes--> v03\n- v04 --supports--> v01\n- v05 --refutes--> v0b\n- v09 --supports--> v0a\n- v06 --refutes--> v05\n- v01 --refutes--> v07\n- v0a --refutes--> v0d\n- v00 --supports--> v0c\n- v0d --refutes--> v08\n- v05 --refutes--> v04\n- v08 --supports--> v06\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"092076a2ba353acdc83bf27de375d62eefaa1ed3b003f909f3c36c422c307c03","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v03 --supports--> v01\n- v03 --supports--> v0c\n- v00 --supports--> v02\n- v02 --refutes--> v03\n- v09 --supports--> v05\n- v0c --supports--> v07\n- v08 --refutes--> v0a\n- v0b --supports--> v08\n- v0c --supports--> v0d\n- v0a --refutes--> v09\n- v07 --refutes--> v0b\n- v02 --refutes--> v06\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":"d1c45de4727e30cdee7d22a847a059fc3ece7ce39f1c8ec111961e7f8282d897","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v03\n- v03 --supports--> v07\n- v03 --refutes--> v01\n- v0c --supports--> v02\n- v07 --supports--> v09\n- v0b --supports--> v00\n- v0a --refutes--> v05\n- v08 --supports--> v0b\n- v07 --supports--> v0d\n- v02 --refutes--> v04\n- v04 --supports--> v0a\n- v02 --refutes--> v06\n- v0d --refutes--> 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":"b2758646956b27198167420a1e83a6e05e94302e23660a3b07612a1d9bf76a17","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v00 --refutes--> v03\n- v05 --supports--> v00\n- v02 --supports--> v06\n- v01 --supports--> v07\n- v07 --supports--> v09\n- v03 --supports--> v0d\n- v09 --supports--> v08\n- v02 --refutes--> v0a\n- v0d --supports--> v02\n- v02 --supports--> v01\n- v07 --supports--> v04\n- v0c --refutes--> v05\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":"52966915a0191488bf08151fc61fa559a4dcd5bf9f534eafe66c202819f7c2bf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v01 --supports--> v0b\n- v02 --refutes--> v0a\n- v05 --supports--> v0d\n- v01 --supports--> v06\n- v0b --supports--> v02\n- v07 --refutes--> v01\n- v0c --refutes--> v07\n- v03 --supports--> v05\n- v0a --supports--> v03\n- v07 --refutes--> v04\n- v0c --refutes--> v09\n- v00 --refutes--> v0c\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":"9f727e2872f635e59b9b85f59e56dfa0784aa312febb872b30bc9b1412090eb7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v09\n- v01 --supports--> v0c\n- v02 --refutes--> v07\n- v09 --refutes--> v04\n- v09 --refutes--> v05\n- v00 --supports--> v02\n- v0a --supports--> v06\n- v0b --supports--> v08\n- v0b --supports--> v00\n- v01 --supports--> v03\n- v02 --supports--> v0a\n- v03 --refutes--> v0b\n- v06 --refutes--> v0d\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":"df75fc3937539a4d683c9483bbfc264537d2d974c55f7bf69b5bd5c8a8d8d420","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v01 --supports--> v05\n- v0a --refutes--> v00\n- v0b --refutes--> v0d\n- v05 --supports--> v08\n- v0c --supports--> v09\n- v0b --refutes--> v07\n- v04 --refutes--> v02\n- v05 --supports--> v06\n- v00 --refutes--> v03\n- v00 --refutes--> v0b\n- v06 --supports--> v0c\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4106301b41f32bf77ddd8654b027ea002420558139bed8b8f6a948032a845a4c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v0b --supports--> v04\n- v03 --supports--> v00\n- v03 --refutes--> v0d\n- v0c --refutes--> v08\n- v01 --refutes--> v0c\n- v08 --supports--> v07\n- v05 --refutes--> v0b\n- v09 --refutes--> v03\n- v0b --refutes--> v0a\n- v05 --refutes--> v06\n- v0a --supports--> v09\n- v03 --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9c0bc8dcac4d6e587164b40f05d9d08c04db2113d7c051f1dd1f177b3c9c838b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v08 --supports--> v03\n- v09 --supports--> v0a\n- v09 --refutes--> v01\n- v06 --supports--> v09\n- v0a --supports--> v0c\n- v0c --supports--> v08\n- v03 --supports--> v0d\n- v07 --refutes--> v04\n- v02 --supports--> v07\n- v0c --refutes--> v00\n- v05 --supports--> v02\n- v0a --supports--> 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":"7d82f817d69234fc9e57032b20c7e3ce1e32767dc41e81e5c6130ffe32929656","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v02\n- v02 --supports--> v0a\n- v03 --refutes--> v0c\n- v01 --refutes--> v04\n- v0b --refutes--> v05\n- v06 --refutes--> v0d\n- v01 --refutes--> v00\n- v0a --supports--> v06\n- v06 --refutes--> v09\n- v01 --refutes--> v08\n- v0d --refutes--> v01\n- v00 --refutes--> v0b\n- v02 --refutes--> v07\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":"2e7fcc3e59f1da203959c021fc910247afc37870445fdc361707881f6c61d321","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v02 --refutes--> v00\n- v00 --refutes--> v03\n- v00 --supports--> v08\n- v0c --refutes--> v0a\n- v0a --refutes--> v02\n- v01 --supports--> v07\n- v08 --refutes--> v05\n- v05 --supports--> v01\n- v07 --supports--> v04\n- v07 --refutes--> v0b\n- v00 --supports--> v09\n- v04 --refutes--> v0d\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":"d17e0c5c08a645f0f3ac786994a487e9291e3e1d0eddc64f2499a50b3236eccc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v0d --refutes--> v00\n- v0f --refutes--> v05\n- v00 --refutes--> v08\n- v0b --refutes--> v0f\n- v08 --refutes--> v07\n- v0c --refutes--> v0d\n- v0a --supports--> v0b\n- v0c --refutes--> v06\n- v03 --supports--> v09\n- v0f --refutes--> v02\n- v08 --supports--> v04\n- v05 --refutes--> v0c\n- v07 --refutes--> v03\n- v0d --refutes--> v0e\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":"77e3d9da898268aee8a909916e2db6d40aab7fbaaf532acb91f955102930ed06","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0c\n- v0b --refutes--> v0e\n- v0f --supports--> v0b\n- v0e --supports--> v05\n- v03 --supports--> v07\n- v03 --refutes--> v0a\n- v05 --supports--> v03\n- v03 --refutes--> v09\n- v04 --refutes--> v0f\n- v0c --supports--> v01\n- v0f --supports--> v00\n- v0f --supports--> v06\n- v0c --refutes--> v04\n- v0a --refutes--> v0d\n- v02 --refutes--> v08\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":"5facdbe79668c536ceb920830dc96264e267089ef28e1d9cea46c0f0669cc224","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v0d\n- v07 --refutes--> v0c\n- v09 --refutes--> v07\n- v03 --supports--> v04\n- v07 --refutes--> v01\n- v0b --refutes--> v0f\n- v0d --refutes--> v05\n- v06 --refutes--> v09\n- v0f --supports--> v00\n- v03 --refutes--> v0b\n- v0c --supports--> v0a\n- v0a --supports--> v02\n- v05 --supports--> v03\n- v0a --refutes--> v08\n- v03 --refutes--> v0e\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":"911c2fc96e8945ce24fdc246519b6bc540324b86bb74367f565de3ebb128a87a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v05 --refutes--> v01\n- v08 --refutes--> v03\n- v04 --refutes--> v0b\n- v0b --refutes--> v0f\n- v06 --refutes--> v08\n- v0f --refutes--> v0d\n- v00 --supports--> v0e\n- v01 --refutes--> v06\n- v03 --supports--> v00\n- v0f --refutes--> v09\n- v0b --supports--> v0c\n- v00 --supports--> v04\n- v06 --supports--> v0a\n- v02 --refutes--> v05\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":"4acc6e3da5deaba3bd5bf57c939a78fc2513fb298e4c93e46db6278c1dd175b2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v01 --refutes--> v09\n- v06 --refutes--> v0f\n- v02 --refutes--> v01\n- v03 --supports--> v00\n- v0a --supports--> v07\n- v09 --refutes--> v03\n- v04 --supports--> v0c\n- v04 --refutes--> v08\n- v0f --refutes--> v0a\n- v03 --refutes--> v0d\n- v0a --supports--> v04\n- v0c --refutes--> v02\n- v00 --supports--> v0b\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":"6e637b614476639b76273cb26dc8303fe0ac4e9200aed6cf78acb311b4f2292c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0c\n- v05 --supports--> v0d\n- v0f --refutes--> v07\n- v0e --refutes--> v01\n- v01 --refutes--> v05\n- v0e --refutes--> v00\n- v0a --refutes--> v06\n- v0f --refutes--> v09\n- v06 --supports--> v04\n- v0c --refutes--> v0e\n- v04 --supports--> v0b\n- v01 --refutes--> v03\n- v05 --supports--> v02\n- v02 --refutes--> v0f\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"931d064cdb1da05ab5f97111bd51cf69a9fdd442dc31800e48277c02fc70fc14","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v00\n- v09 --refutes--> v05\n- v0b --refutes--> v01\n- v08 --refutes--> v03\n- v02 --refutes--> v06\n- v02 --refutes--> v0e\n- v0e --supports--> v09\n- v03 --supports--> v0f\n- v0c --supports--> v02\n- v04 --refutes--> v0c\n- v0f --supports--> v0b\n- v04 --supports--> v07\n- v0a --refutes--> v04\n- v0b --refutes--> v0a\n- v0a --supports--> v0d\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":"eac1078b2a1b631391145398843536c2846384b59b1c96799dbbeba324a50c6d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v0e --refutes--> v0f\n- v09 --refutes--> v0e\n- v0b --supports--> v08\n- v0c --refutes--> v0d\n- v06 --supports--> v05\n- v00 --supports--> v06\n- v0b --refutes--> v01\n- v08 --supports--> v02\n- v0d --supports--> v04\n- v09 --supports--> v07\n- v05 --refutes--> v0c\n- v0b --supports--> v0a\n- v0f --refutes--> v00\n- v0d --supports--> v0b\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"756155168c79f9fdf435eafa972521862c4024956ebbe729c4c6c6ade7fefd28","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v07\n- v00 --supports--> v0c\n- v00 --refutes--> v02\n- v04 --refutes--> v00\n- v06 --supports--> v0f\n- v0a --refutes--> v09\n- v08 --refutes--> v0d\n- v09 --supports--> v05\n- v04 --supports--> v03\n- v05 --supports--> v0e\n- v07 --refutes--> v0a\n- v0d --refutes--> v01\n- v00 --supports--> v06\n- v0d --supports--> v0b\n- v05 --refutes--> v08\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":"43b8485c5dcf98ed47965ba5762a28d8630e595b7bfd141ed512fa15cc7226d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v09\n- v06 --supports--> v0c\n- v07 --refutes--> v03\n- v01 --refutes--> v0d\n- v03 --refutes--> v0e\n- v0d --supports--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v06\n- v02 --refutes--> v01\n- v09 --refutes--> v02\n- v0b --supports--> v05\n- v03 --supports--> v0f\n- v0a --refutes--> v04\n- v02 --refutes--> v08\n- v0a --supports--> v07\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":"52d9a9c78af82ffd11f816de1ecb18d375c3ee6f8d16ca79756983bb5e9d8ff6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v0b --refutes--> v02\n- v0d --supports--> v04\n- v01 --supports--> v08\n- v0f --supports--> v09\n- v04 --supports--> v00\n- v03 --supports--> v0a\n- v05 --supports--> v0b\n- v03 --supports--> v05\n- v0c --supports--> v01\n- v05 --refutes--> v0e\n- v09 --refutes--> v07\n- v08 --refutes--> v0f\n- v09 --supports--> v03\n- v0b --supports--> v0d\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":"8db990af48452c2574de477fa21f8bd69d8f86d6ae626930f674d9e7e9d23643","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0d\n- v03 --refutes--> v07\n- v0a --refutes--> v0e\n- v0e --supports--> v0f\n- v03 --supports--> v0a\n- v01 --supports--> v09\n- v09 --supports--> v0b\n- v0d --refutes--> v0c\n- v08 --supports--> v04\n- v05 --refutes--> v08\n- v0d --refutes--> v06\n- v0d --refutes--> v02\n- v0c --refutes--> v01\n- v05 --supports--> v00\n- v0b --refutes--> v05\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2322ba7ffad2444856e2d6df304b21b76e072117216d5e884f21460075ef9888","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0d\n- v07 --supports--> v06\n- v0b --supports--> v0c\n- v06 --refutes--> v00\n- v0d --refutes--> v0b\n- v0b --refutes--> v0a\n- v06 --refutes--> v09\n- v04 --refutes--> v01\n- v06 --supports--> v03\n- v08 --refutes--> v07\n- v0c --refutes--> v08\n- v06 --refutes--> v05\n- v00 --supports--> v04\n- v01 --refutes--> v02\n- v07 --supports--> v0e\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":"b4529ca9a1d1eb8c73ba0c6441502dcad47108951f42c4bcc8ab774a6084596d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v01 --refutes--> v00\n- v0b --supports--> v07\n- v00 --refutes--> v05\n- v0e --refutes--> v0d\n- v04 --supports--> v09\n- v0f --supports--> v0c\n- v00 --supports--> v02\n- v0d --refutes--> v0b\n- v0b --supports--> v01\n- v0c --refutes--> v04\n- v0f --supports--> v0a\n- v03 --supports--> v0f\n- v04 --refutes--> v06\n- v03 --supports--> v08\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":"96bc3c992e87a086ee260b4171b117b022fbe3a3e4e5e901a6a24d74a12770a4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v06 --supports--> v08\n- v07 --refutes--> v04\n- v0f --supports--> v03\n- v0b --supports--> v09\n- v0d --refutes--> v05\n- v00 --supports--> v01\n- v0f --supports--> v0d\n- v0c --refutes--> v00\n- v08 --supports--> v0c\n- v05 --refutes--> v07\n- v04 --refutes--> v02\n- v00 --supports--> v0b\n- v00 --supports--> v0e\n- v04 --refutes--> v06\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":"80501c33d1a27ef0536d82acaa9d57e3a044aef9bb3e4dc669da4fcaa0aac444","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v08 --refutes--> v02\n- v0f --supports--> v06\n- v0a --supports--> v09\n- v0a --refutes--> v0b\n- v04 --supports--> v05\n- v09 --supports--> v0e\n- v04 --refutes--> v0a\n- v03 --supports--> v01\n- v09 --supports--> v0d\n- v06 --refutes--> v00\n- v03 --supports--> v04\n- v02 --refutes--> v03\n- v00 --refutes--> v0c\n- v0c --refutes--> v08\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":"d2bc9471f248233c3b9fff1f31f23f884a977e13355429fd32719229a453fd5a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0e\n- v09 --refutes--> v0a\n- v0f --supports--> v07\n- v0e --refutes--> v05\n- v0e --supports--> v0b\n- v09 --supports--> v03\n- v02 --supports--> v0c\n- v09 --supports--> v0d\n- v0c --supports--> v08\n- v00 --supports--> v0f\n- v08 --supports--> v04\n- v0b --supports--> v00\n- v08 --refutes--> v06\n- v01 --supports--> v02\n- v0f --refutes--> v09\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":"99205b010b5583ecfc0b9ca220c4cff2d17d2a48c3b11185beacb022d7ee9337","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0d\n- v08 --refutes--> v04\n- v0f --supports--> v03\n- v0a --supports--> v07\n- v08 --refutes--> v0e\n- v00 --supports--> v0b\n- v09 --supports--> v0a\n- v07 --supports--> v02\n- v0d --refutes--> v0c\n- v08 --refutes--> v05\n- v0d --supports--> v08\n- v0a --refutes--> v06\n- v07 --refutes--> v00\n- v01 --supports--> v09\n- v03 --supports--> v01\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":"b1930598a3a5ff5d562fd28892ff4a060d5cc929cc8ba8e6a51acc63edb9c01f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v06\n- v0d --supports--> v03\n- v0c --refutes--> v04\n- v01 --refutes--> v0d\n- v0b --refutes--> v0e\n- v0c --supports--> v07\n- v05 --supports--> v0c\n- v0c --supports--> v0a\n- v0e --refutes--> v09\n- v0e --refutes--> v0f\n- v0d --refutes--> v08\n- v0a --refutes--> v0b\n- v03 --refutes--> v05\n- v09 --supports--> v02\n- v00 --supports--> 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":"c430355e45fb6eacdf10c4ccaebe6d96b5eed9ba12ab50cb24d0e3372740688b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0a\n- v02 --refutes--> v0e\n- v04 --refutes--> v06\n- v07 --refutes--> v0f\n- v01 --supports--> v08\n- v0b --refutes--> v0d\n- v09 --refutes--> v02\n- v03 --supports--> v07\n- v0e --refutes--> v0b\n- v0d --supports--> v01\n- v0d --refutes--> v00\n- v06 --supports--> v05\n- v06 --supports--> v09\n- v06 --refutes--> v0c\n- v07 --refutes--> v04\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":"349f380cb8203447691f89c6b799941611f08c37c2531347f012ad7a3934861f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v06\n- v0b --supports--> v09\n- v09 --refutes--> v02\n- v04 --supports--> v01\n- v04 --supports--> v08\n- v06 --supports--> v05\n- v0f --refutes--> v0c\n- v0c --supports--> v07\n- v0c --supports--> v0d\n- v0a --refutes--> v0f\n- v09 --refutes--> v04\n- v0d --supports--> v0b\n- v06 --refutes--> v0e\n- v09 --supports--> v00\n- v03 --refutes--> v0a\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":"53e9f90d1ec1077fad8031cdf5d038c28f12e1f05044d705b6a441c197b3e879","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v09\n- v0f --refutes--> v0b\n- v0a --refutes--> v02\n- v0d --refutes--> v0c\n- v04 --supports--> v0e\n- v0d --supports--> v00\n- v09 --refutes--> v05\n- v0e --supports--> v07\n- v0b --refutes--> v03\n- v04 --refutes--> v01\n- v0a --supports--> v04\n- v0f --refutes--> v08\n- v07 --supports--> v0f\n- v03 --refutes--> v0d\n- v07 --supports--> v06\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":"c1b705079557f0470fb7b6cf6f67502de176141a24a607654ae6d8c11c2c7a6d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v03 --supports--> v07\n- v07 --supports--> v09\n- v0b --supports--> v00\n- v08 --refutes--> v06\n- v0c --refutes--> v0e\n- v07 --refutes--> v05\n- v03 --supports--> v02\n- v07 --refutes--> v01\n- v06 --supports--> v0c\n- v0d --refutes--> v0b\n- v0a --refutes--> v04\n- v0f --supports--> v0d\n- v0a --supports--> v03\n- v0e --supports--> v0a\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":"ea50b9c93bfae282af970080b4524f4d7f63d56d1e5c92e46a5591f383c9cdf5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v04\n- v04 --supports--> v09\n- v07 --supports--> v0b\n- v04 --supports--> v02\n- v0a --supports--> v08\n- v07 --refutes--> v0c\n- v04 --refutes--> v07\n- v01 --refutes--> v05\n- v06 --supports--> v0a\n- v0d --refutes--> v0f\n- v0d --supports--> v01\n- v0b --refutes--> v06\n- v08 --supports--> v0e\n- v0e --supports--> v03\n- v08 --refutes--> v00\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":"94f5473aaf2b2bac107291a7343c30f5f8ecff5dc2bb2f7d6b816c50f5b1536b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v08\n- v05 --refutes--> v04\n- v05 --supports--> v0c\n- v06 --refutes--> v07\n- v04 --supports--> v0e\n- v03 --refutes--> v01\n- v0b --supports--> v00\n- v02 --supports--> v0a\n- v00 --supports--> v0f\n- v06 --refutes--> v0d\n- v0d --refutes--> v0b\n- v0a --supports--> v09\n- v03 --supports--> v05\n- v0a --supports--> v03\n- v0e --refutes--> v06\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":"a54a1f2c3c6b2804916c3067c2c968054399e35f2224e76130ade3082c5931f6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v03\n- v07 --supports--> v0b\n- v07 --supports--> v05\n- v09 --refutes--> v04\n- v06 --supports--> v00\n- v0e --refutes--> v07\n- v04 --supports--> v0e\n- v06 --refutes--> v0a\n- v0f --refutes--> v0d\n- v0d --supports--> v02\n- v00 --supports--> v01\n- v05 --supports--> v0f\n- v0f --refutes--> v08\n- v00 --refutes--> v09\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5ba0ff2e9bb7e0442315259444124c72d229b29d534675b976edabeb80e6559","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v05 --supports--> v0a\n- v02 --supports--> v0d\n- v0a --refutes--> v0b\n- v03 --supports--> v08\n- v09 --refutes--> v0e\n- v09 --refutes--> v06\n- v0b --refutes--> v07\n- v0c --supports--> v09\n- v0b --refutes--> v0f\n- v03 --refutes--> v01\n- v01 --supports--> v05\n- v02 --supports--> v00\n- v07 --supports--> v02\n- v06 --refutes--> v03\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":"ba9b34a9fc511aa67795f72dac1f832a6743c9815406d70d5a4ca9aae3b69ed0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0b\n- v02 --refutes--> v08\n- v04 --refutes--> v05\n- v0d --supports--> v07\n- v05 --refutes--> v0e\n- v0b --supports--> v04\n- v0d --refutes--> v00\n- v08 --supports--> v09\n- v01 --supports--> v0f\n- v08 --supports--> v03\n- v08 --supports--> v0a\n- v0e --supports--> v06\n- v03 --supports--> v0c\n- v06 --supports--> v02\n- v01 --refutes--> v0d\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":"b7ef4cf9e3762599bdd6cb07b84d6ead52863df00f605dcc14978258a9a3bfd9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0e\n- v0f --refutes--> v06\n- v0e --refutes--> v05\n- v08 --refutes--> v0a\n- v02 --supports--> v08\n- v05 --refutes--> v02\n- v0a --refutes--> v09\n- v0d --refutes--> v0c\n- v0f --supports--> v00\n- v0a --refutes--> v01\n- v0d --supports--> v0b\n- v01 --refutes--> v0d\n- v07 --refutes--> v03\n- v0e --refutes--> v04\n- v0b --supports--> v0f\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":"cf95a52c71738ef31e7ab104864ebe70b96bb6a4c60427baeaba5be848a876ad","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v00\n- v08 --supports--> v01\n- v02 --supports--> v0e\n- v01 --refutes--> v0a\n- v05 --refutes--> v06\n- v0e --supports--> v0d\n- v00 --refutes--> v03\n- v01 --refutes--> v09\n- v07 --supports--> v05\n- v05 --supports--> v0f\n- v0d --supports--> v07\n- v0e --refutes--> v04\n- v07 --refutes--> v0c\n- v05 --refutes--> v0b\n- v03 --supports--> v02\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":"91c275f33f19ec4c17f7be95d36ce565adc64bbacde4daa1246f58922bdd9f0b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v09\n- v03 --supports--> v0c\n- v02 --supports--> v08\n- v07 --refutes--> v05\n- v06 --refutes--> v0b\n- v03 --refutes--> v00\n- v03 --supports--> v0d\n- v05 --supports--> v0e\n- v0f --refutes--> v02\n- v0a --refutes--> v04\n- v0b --refutes--> v0a\n- v0f --refutes--> v01\n- v0e --refutes--> v03\n- v02 --refutes--> v06\n- v04 --supports--> v07\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":"720e2767b0bf28c4b8ade7daa3a70296a81e91fc60f88167ed6e079306ce7777","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v0b --refutes--> v0d\n- v0a --refutes--> v0c\n- v0c --supports--> v08\n- v07 --refutes--> v01\n- v0e --refutes--> v0a\n- v0a --supports--> v00\n- v08 --supports--> v02\n- v03 --supports--> v0f\n- v0d --supports--> v05\n- v03 --refutes--> v0e\n- v0b --refutes--> v04\n- v0a --supports--> v09\n- v01 --supports--> v03\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":"5a198d90c3e5206841649636308a61f4be01357a39bc2add3b01c419593bfd4d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v06 --supports--> v04\n- v00 --refutes--> v09\n- v05 --supports--> v08\n- v03 --supports--> v0e\n- v00 --supports--> v03\n- v03 --supports--> v0f\n- v08 --refutes--> v00\n- v04 --supports--> v0b\n- v00 --refutes--> v07\n- v08 --refutes--> v0a\n- v06 --supports--> v02\n- v0f --supports--> v0c\n- v0d --supports--> v01\n- v0c --refutes--> v0d\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":"c4b48df795c8fb37a707972a59910c0eb86681032736679810dc508a410be62c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v0f --supports--> v05\n- v0c --supports--> v08\n- v03 --supports--> v07\n- v08 --supports--> v0f\n- v08 --refutes--> v06\n- v0e --refutes--> v0d\n- v0f --supports--> v09\n- v00 --refutes--> v0b\n- v00 --refutes--> v01\n- v09 --refutes--> v0a\n- v00 --supports--> v02\n- v07 --refutes--> v0e\n- v0f --refutes--> v04\n- v0b --refutes--> v0c\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":"bda3b1c5ab401d7c44aac5b845eaf859bafd7de4bf0a251f84ece88155b83b41","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v05\n- v0d --supports--> v08\n- v0a --supports--> v0b\n- v0c --refutes--> v0d\n- v0b --refutes--> v03\n- v09 --supports--> v0e\n- v0f --refutes--> v09\n- v09 --supports--> v07\n- v0b --supports--> v0f\n- v0d --supports--> v00\n- v01 --refutes--> v0a\n- v09 --refutes--> v02\n- v05 --supports--> v01\n- v05 --refutes--> v06\n- v07 --refutes--> v04\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":"2d71cc5343724e98fe802634ad55c29eaeef81ac75313f4d656b65eaea0731db","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v09 --refutes--> v0e\n- v0f --refutes--> v04\n- v0b --refutes--> v0a\n- v07 --refutes--> v0b\n- v02 --supports--> v0f\n- v06 --supports--> v09\n- v09 --refutes--> v07\n- v0f --supports--> v08\n- v07 --refutes--> v01\n- v08 --supports--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v06\n- v02 --refutes--> v00\n- v05 --refutes--> v0c\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":"9e65c60db3136b6aff1e697846661bbea29e94b37919874007654a20bef96b20","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v00\n- v0d --refutes--> v07\n- v05 --refutes--> v0a\n- v0f --refutes--> v06\n- v01 --supports--> v0e\n- v05 --refutes--> v08\n- v0f --supports--> v0d\n- v04 --refutes--> v05\n- v08 --supports--> v02\n- v02 --supports--> v0c\n- v02 --supports--> v09\n- v09 --refutes--> v0f\n- v07 --refutes--> v03\n- v08 --refutes--> v0b\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e193f7ebb87febc82ec970ef78221261d2e162454c91153ebc536cbbd8a588a6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v08\n- v06 --supports--> v07\n- v0e --refutes--> v0c\n- v0b --refutes--> v0e\n- v0c --refutes--> v09\n- v08 --refutes--> v00\n- v03 --refutes--> v05\n- v07 --supports--> v02\n- v05 --supports--> v0b\n- v08 --supports--> v04\n- v02 --refutes--> v0a\n- v07 --supports--> v0d\n- v0c --refutes--> v06\n- v0a --supports--> v01\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":"d3760dfc79cd85fb315b53e3db4e1beb9f6d6cf7d401a661a7277b8a2b5bcbf6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v02\n- v08 --refutes--> v09\n- v04 --refutes--> v06\n- v0d --supports--> v0a\n- v05 --refutes--> v0c\n- v01 --refutes--> v0f\n- v09 --supports--> v0b\n- v0d --supports--> v00\n- v07 --supports--> v0e\n- v01 --refutes--> v03\n- v0e --supports--> v04\n- v03 --refutes--> v05\n- v09 --supports--> v01\n- v04 --supports--> v08\n- v0d --supports--> v07\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":"062da083e1cfc8036d25fa731b57c00fec0abb18d6287b3d74b0d127d4a94352","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v0e --refutes--> v04\n- v07 --refutes--> v0e\n- v05 --supports--> v08\n- v06 --supports--> v07\n- v0e --refutes--> v0d\n- v00 --supports--> v0f\n- v0f --supports--> v06\n- v0f --supports--> v0a\n- v01 --supports--> v03\n- v08 --refutes--> v09\n- v04 --refutes--> v01\n- v07 --supports--> v0c\n- v03 --refutes--> v0b\n- v08 --supports--> v00\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a46f896a3314c08715369b464f9a483e050e9b842784a3099c59a95e507cd73c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v02\n- v0e --supports--> v0f\n- v03 --refutes--> v0c\n- v02 --refutes--> v05\n- v04 --supports--> v03\n- v00 --refutes--> v09\n- v0f --refutes--> v04\n- v0d --supports--> v07\n- v01 --refutes--> v06\n- v02 --supports--> v0b\n- v0c --supports--> v0d\n- v09 --refutes--> v08\n- v02 --refutes--> v0a\n- v09 --supports--> v0e\n- v0d --supports--> v01\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14587210344a91987d82454752d3defc37e17c1e033b3ec8074ebfbae6a47c73","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v08 --refutes--> v09\n- v0a --refutes--> v0c\n- v03 --supports--> v02\n- v05 --refutes--> v00\n- v0a --refutes--> v0f\n- v0a --refutes--> v0e\n- v0b --refutes--> v0d\n- v06 --supports--> v0b\n- v08 --supports--> v07\n- v07 --supports--> v05\n- v00 --refutes--> v03\n- v02 --refutes--> v01\n- v01 --refutes--> v04\n- v01 --refutes--> v06\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":"b87539fd9f92bfaba5267e2910d765a00d8574fd7e9804c94c9f85dc573f871d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0e\n- v0a --refutes--> v0c\n- v0a --supports--> v04\n- v05 --supports--> v0a\n- v01 --refutes--> v02\n- v06 --refutes--> v00\n- v05 --refutes--> v0d\n- v09 --refutes--> v03\n- v01 --supports--> v09\n- v04 --refutes--> v07\n- v08 --refutes--> v0b\n- v00 --supports--> v05\n- v06 --supports--> v0f\n- v0b --refutes--> v01\n- v03 --supports--> 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":"5d17bd67aab1224d702e38ed26f46a26ed1d44e6891eb3a3672cbbd76d128a03","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v04\n- v00 --supports--> v0d\n- v09 --supports--> v0b\n- v01 --supports--> v0e\n- v0a --refutes--> v0f\n- v00 --refutes--> v0a\n- v07 --refutes--> v00\n- v0f --refutes--> v08\n- v09 --refutes--> v07\n- v06 --refutes--> v05\n- v00 --refutes--> v02\n- v05 --supports--> v09\n- v03 --refutes--> v01\n- v00 --refutes--> v0c\n- v0e --supports--> v06\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2cd9f9b80baceb2ee9abb4509a4a743f700b463db6af1ea627612d94f5db7a1e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v00\n- v0b --supports--> v09\n- v01 --supports--> v08\n- v0e --supports--> v0a\n- v07 --refutes--> v0c\n- v08 --refutes--> v0f\n- v0c --refutes--> v0d\n- v04 --refutes--> v03\n- v09 --supports--> v01\n- v0e --supports--> v02\n- v01 --supports--> v06\n- v08 --supports--> v04\n- v04 --supports--> v05\n- v0a --refutes--> v07\n- v05 --refutes--> v0e\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":"427b2866e54c048acab0d3aa66234944da226d8193e5e8cfc7a795c93f9fc4fd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v06\n- v0a --refutes--> v0b\n- v0e --refutes--> v01\n- v0e --refutes--> v03\n- v04 --refutes--> v05\n- v05 --refutes--> v07\n- v04 --refutes--> v08\n- v09 --refutes--> v04\n- v0e --refutes--> v0c\n- v00 --refutes--> v0d\n- v06 --refutes--> v00\n- v03 --supports--> v09\n- v0d --refutes--> v02\n- v00 --refutes--> v0f\n- v0d --refutes--> v0a\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":"be8abb215c9c35f0414c88dfa6642378c4c262862894a58f7b97f5bd42688e6a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v03 --supports--> v0b\n- v09 --supports--> v0c\n- v04 --refutes--> v06\n- v06 --supports--> v0f\n- v0c --refutes--> v00\n- v0b --refutes--> v0d\n- v0a --supports--> v03\n- v0f --supports--> v0e\n- v09 --supports--> v02\n- v0f --supports--> v08\n- v04 --supports--> v07\n- v0c --supports--> v01\n- v0e --supports--> v0a\n- v0f --supports--> v05\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":"7a2e66a3ede8f19e2f5d7cbae379a87fbe1dc77a67b7267cb883ef4431ee79fa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0c\n- v08 --supports--> v0f\n- v07 --refutes--> v03\n- v0f --refutes--> v01\n- v02 --supports--> v00\n- v0d --refutes--> v04\n- v04 --refutes--> v08\n- v06 --refutes--> v0e\n- v03 --refutes--> v06\n- v00 --supports--> v07\n- v03 --supports--> v0b\n- v04 --refutes--> v09\n- v06 --refutes--> v0a\n- v0f --refutes--> v05\n- v0e --supports--> v0d\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35b4ac6b0aa31f49b13c98bf2bb227d38bc9174e332ac5393fdacf1aab5593bd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v07\n- v00 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --refutes--> v06\n- v0f --supports--> v02\n- v01 --supports--> v0f\n- v08 --supports--> v05\n- v01 --refutes--> v0e\n- v08 --refutes--> v0c\n- v06 --supports--> v01\n- v05 --supports--> v00\n- v01 --supports--> v04\n- v05 --supports--> v0a\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":"69c8ef2b087963181723c537f5d0a0d94c1736d10c546f4f0a83c7b7739296c8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0d\n- v0c --refutes--> v08\n- v0a --supports--> v03\n- v0c --supports--> v0f\n- v07 --supports--> v00\n- v09 --refutes--> v0b\n- v06 --supports--> v07\n- v0b --supports--> v05\n- v02 --refutes--> v09\n- v0a --supports--> v0e\n- v03 --refutes--> v0c\n- v00 --refutes--> v02\n- v09 --refutes--> v04\n- v02 --refutes--> v01\n- v05 --supports--> v0a\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":"b90c88dff7dfae37d47846a20166402966e29f2994babc7ad4430d0f53dbf10b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v03\n- v0f --refutes--> v02\n- v07 --supports--> v0f\n- v0f --supports--> v0a\n- v08 --supports--> v00\n- v06 --supports--> v0e\n- v08 --refutes--> v07\n- v08 --refutes--> v01\n- v04 --refutes--> v09\n- v05 --refutes--> v0c\n- v0e --refutes--> v0b\n- v03 --supports--> v04\n- v09 --refutes--> v06\n- v0e --refutes--> v05\n- v08 --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78cbc0521fbe803a2a7cb1d449c95f504be985a2bacc9fbad2d3044e5bccd0a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v0e --refutes--> v00\n- v0a --supports--> v01\n- v0a --supports--> v09\n- v0b --refutes--> v0d\n- v0d --refutes--> v02\n- v08 --refutes--> v0a\n- v0e --supports--> v03\n- v0d --refutes--> v05\n- v01 --refutes--> v0b\n- v07 --supports--> v0e\n- v0b --refutes--> v0f\n- v04 --refutes--> v08\n- v0e --supports--> v06\n- v06 --supports--> v04\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":"8d5296a8cb6e4cebeb31d3656af34e118fd8baf231fdcb04394a5b3017d8a5a0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v03\n- v0e --refutes--> v07\n- v07 --supports--> v0f\n- v06 --supports--> v0e\n- v0d --refutes--> v0c\n- v08 --refutes--> v02\n- v08 --refutes--> v04\n- v05 --supports--> v06\n- v0d --supports--> v08\n- v0f --supports--> v0b\n- v01 --refutes--> v05\n- v0f --supports--> v0d\n- v0e --refutes--> v00\n- v0a --refutes--> v09\n- v04 --supports--> v0a\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":"9b038c2a7ae03c7958ffe1dd035acbfe68634e73eb643521a249f3a791c6e189","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v04\n- v03 --supports--> v05\n- v04 --supports--> v0c\n- v0d --supports--> v02\n- v0b --refutes--> v0a\n- v04 --supports--> v00\n- v05 --refutes--> v01\n- v0c --supports--> v06\n- v0b --supports--> v0d\n- v02 --supports--> v0e\n- v06 --refutes--> v03\n- v0e --refutes--> v09\n- v0d --refutes--> v08\n- v05 --refutes--> v0b\n- v0e --supports--> v07\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":"9fd9a45db73f24cf44db2022f0cb75dd233b88084a9d5bca1a6735fe977e3dc5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v05\n- v0d --refutes--> v0e\n- v09 --supports--> v0f\n- v07 --refutes--> v04\n- v09 --refutes--> v00\n- v0c --refutes--> v07\n- v0d --supports--> v01\n- v00 --refutes--> v03\n- v00 --supports--> v06\n- v0a --supports--> v09\n- v0e --supports--> v0b\n- v08 --supports--> v0d\n- v05 --supports--> v0c\n- v03 --refutes--> v08\n- v0a --supports--> 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":"18e2aa7592ef046d0841fb2d8ebe62acbdde170053d89a714962cc68f360b2bd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v02\n- v01 --refutes--> v08\n- v05 --refutes--> v06\n- v07 --supports--> v04\n- v0b --supports--> v09\n- v0a --supports--> v00\n- v09 --supports--> v01\n- v0a --supports--> v0b\n- v01 --refutes--> v0d\n- v07 --refutes--> v0a\n- v0f --supports--> v05\n- v0b --supports--> v0e\n- v0f --refutes--> v03\n- v06 --supports--> v0c\n- v08 --refutes--> v0f\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":"7892c5b7cebb142980fabcec64fa0d6287569ec967222432ca1f485d59160682","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0b\n- v00 --supports--> v0f\n- v0d --supports--> v01\n- v05 --refutes--> v08\n- v03 --supports--> v0d\n- v05 --refutes--> v0a\n- v0b --refutes--> v09\n- v0d --supports--> v07\n- v00 --refutes--> v0c\n- v09 --refutes--> v04\n- v09 --refutes--> v02\n- v0c --refutes--> v03\n- v06 --refutes--> v05\n- v09 --supports--> v0e\n- v02 --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aad303ee7d89ac64706b8de5baf2cf0f12a4d0ebcc5f6831b9f080b525417aca","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v0c\n- v03 --supports--> v0a\n- v0d --refutes--> v05\n- v00 --refutes--> v02\n- v02 --supports--> v03\n- v0a --supports--> v04\n- v04 --supports--> v0f\n- v01 --refutes--> v0b\n- v01 --supports--> v00\n- v0c --supports--> v01\n- v00 --refutes--> v0e\n- v04 --supports--> v09\n- v08 --supports--> v0d\n- v09 --supports--> v07\n- v00 --supports--> 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":"d764ff83c151ce2eebd0fd1bae34323eee49ce8a3489f627a619d155025fe7ee","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0f\n- v05 --refutes--> v0e\n- v04 --refutes--> v01\n- v0b --supports--> v09\n- v01 --supports--> v0d\n- v05 --supports--> v0c\n- v0a --supports--> v0b\n- v07 --supports--> v02\n- v07 --refutes--> v08\n- v09 --supports--> v07\n- v04 --supports--> v06\n- v05 --supports--> v03\n- v07 --supports--> v00\n- v0f --supports--> v05\n- v0d --refutes--> v0a\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":"9adcf69a09c4e66a0abab9ed803ed7ddd77918d9e53a87127fc037136afd29bf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v04 --refutes--> v0a\n- v03 --supports--> v01\n- v07 --supports--> v0c\n- v0d --refutes--> v0b\n- v09 --supports--> v03\n- v02 --supports--> v08\n- v00 --supports--> v0d\n- v09 --refutes--> v06\n- v08 --supports--> v05\n- v00 --supports--> v0f\n- v0a --supports--> v07\n- v08 --refutes--> v00\n- v01 --refutes--> v02\n- v08 --supports--> v0e\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":"c9c10ac5df63fb66968a3b25854c42167fb04b2aea1fa0c04aa971b8093d9edf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v03\n- v0c --supports--> v04\n- v02 --refutes--> v0f\n- v09 --supports--> v0d\n- v0e --refutes--> v0a\n- v02 --supports--> v0b\n- v07 --refutes--> v02\n- v0e --refutes--> v0c\n- v0c --refutes--> v06\n- v01 --supports--> v00\n- v08 --supports--> v01\n- v03 --refutes--> v08\n- v01 --refutes--> v0e\n- v0d --refutes--> v07\n- v0c --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"841bd1c9e0366a3d05a55adc704129fa010e85cb74cfe3b8eaa1abb5ea6e6a0e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0f\n- v0c --supports--> v03\n- v07 --supports--> v0e\n- v0f --supports--> v0c\n- v0c --supports--> v02\n- v0d --supports--> v05\n- v0f --refutes--> v06\n- v00 --supports--> v0a\n- v00 --supports--> v01\n- v05 --refutes--> v08\n- v0c --refutes--> v09\n- v0c --refutes--> v04\n- v0e --refutes--> v00\n- v0a --refutes--> v0d\n- v04 --supports--> v0b\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":"d1c33c79486d5034c8dabcf4d83e81b95240d8fbd3dc92be5dd6b1cd21f10a37","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0f\n- v0c --supports--> v04\n- v0a --supports--> v09\n- v0d --supports--> v05\n- v03 --refutes--> v0c\n- v0f --refutes--> v0e\n- v0b --refutes--> v03\n- v01 --refutes--> v0a\n- v0d --supports--> v08\n- v0f --refutes--> v0b\n- v0c --supports--> v02\n- v0c --refutes--> v06\n- v0c --refutes--> v00\n- v09 --refutes--> v0d\n- v08 --supports--> v07\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5a3d1b101a5cf77748c533e233e8a45dd546367a3ce1cb15119236232940b67f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v0f\n- v01 --refutes--> v02\n- v04 --supports--> v05\n- v0a --supports--> v0b\n- v0c --supports--> v0a\n- v00 --supports--> v03\n- v00 --refutes--> v0d\n- v01 --supports--> v07\n- v0d --supports--> v09\n- v05 --refutes--> v0c\n- v0d --supports--> v06\n- v01 --supports--> v08\n- v0c --refutes--> v0e\n- v0b --supports--> v01\n- v09 --refutes--> v04\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":"b5a8da92ca35140a050d4e2578a0d01e51f6549de0829fceab1da68bd7889fc8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0c\n- v07 --refutes--> v06\n- v0e --refutes--> v03\n- v01 --refutes--> v0d\n- v09 --supports--> v0a\n- v10 --supports--> v07\n- v06 --refutes--> v01\n- v0e --refutes--> v08\n- v0c --refutes--> v04\n- v00 --supports--> v05\n- v02 --supports--> v0e\n- v10 --refutes--> v0f\n- v01 --supports--> v00\n- v03 --supports--> v10\n- v0a --supports--> v02\n- v0e --supports--> v0b\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":"efcbf33f464b3bf812082028faa2fd31f5919885df8c10ae2e798011748e3d6a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v00\n- v06 --refutes--> v04\n- v0d --refutes--> v0c\n- v09 --refutes--> v03\n- v06 --refutes--> v0b\n- v06 --refutes--> v0e\n- v02 --supports--> v0a\n- v0d --supports--> v05\n- v0d --refutes--> v08\n- v0f --refutes--> v01\n- v0a --supports--> v10\n- v05 --supports--> v06\n- v00 --refutes--> v0f\n- v0f --refutes--> v07\n- v10 --refutes--> v09\n- v07 --refutes--> v02\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":"3f274a9b423059d74fe438acec939e0b63af64660f4af7a9dfc0e9a48873a149","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v01\n- v06 --refutes--> v0b\n- v0e --supports--> v10\n- v05 --refutes--> v09\n- v0a --refutes--> v05\n- v0f --refutes--> v06\n- v00 --supports--> v0a\n- v0e --supports--> v02\n- v02 --supports--> v04\n- v0c --refutes--> v00\n- v06 --refutes--> v0d\n- v0b --supports--> v0c\n- v02 --supports--> v07\n- v04 --refutes--> v0f\n- v09 --refutes--> v08\n- v00 --supports--> v03\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":"7c062acb353fbb8ffefc8ee9e5a4cf8140100b5cc159477ac82f3202ca170d8e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v03\n- v08 --refutes--> v0f\n- v0c --supports--> v00\n- v02 --refutes--> v09\n- v05 --supports--> v0b\n- v05 --supports--> v0c\n- v04 --refutes--> v0a\n- v00 --refutes--> v06\n- v06 --supports--> v0d\n- v02 --supports--> v05\n- v07 --supports--> v01\n- v03 --supports--> v04\n- v03 --supports--> v10\n- v04 --supports--> v02\n- v0d --supports--> v07\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":"23ee22028b1b35cc966208d72ab6e321305f85525f8432b657f695bd44e3a191","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v08\n- v05 --supports--> v06\n- v01 --supports--> v0c\n- v00 --supports--> v02\n- v01 --supports--> v03\n- v0a --supports--> v10\n- v0b --supports--> v07\n- v05 --refutes--> v0e\n- v0d --supports--> v01\n- v07 --refutes--> v04\n- v08 --refutes--> v0a\n- v03 --refutes--> v05\n- v10 --refutes--> v0d\n- v03 --refutes--> v09\n- v0b --supports--> v0f\n- v04 --supports--> v00\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":"38272849e1a29b6b33c5caf6b522cb2afb8e92e8c9bdab480a13474fbbfcee3c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v01\n- v0c --supports--> v0d\n- v0b --refutes--> v03\n- v03 --refutes--> v09\n- v00 --supports--> v08\n- v0a --refutes--> v0e\n- v10 --refutes--> v0b\n- v0a --supports--> v05\n- v0c --refutes--> v07\n- v03 --supports--> v06\n- v0c --supports--> v02\n- v0f --refutes--> v00\n- v02 --refutes--> v0f\n- v0a --refutes--> v04\n- v08 --refutes--> v0a\n- v01 --refutes--> v10\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":"e5d28d1600e119d94c65e26f614f7499593917db17676b66dacd83c0c15b0e94","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v06\n- v04 --refutes--> v0c\n- v10 --supports--> v0e\n- v05 --refutes--> v02\n- v05 --supports--> v00\n- v06 --refutes--> v05\n- v0e --refutes--> v01\n- v0a --supports--> v10\n- v0c --refutes--> v07\n- v03 --refutes--> v0d\n- v10 --refutes--> v08\n- v0d --refutes--> v04\n- v07 --refutes--> v0f\n- v0c --supports--> v09\n- v0e --supports--> v03\n- v0c --refutes--> v0b\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"877ef7757f9a3591563d320a3486c55974b61a4a942ecc87f78baead94389cc2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v07\n- v0d --supports--> v06\n- v0b --supports--> v00\n- v08 --supports--> v05\n- v0e --refutes--> v10\n- v06 --refutes--> v0e\n- v0f --refutes--> v01\n- v03 --refutes--> v02\n- v0a --refutes--> v03\n- v0c --refutes--> v0f\n- v0c --supports--> v09\n- v05 --supports--> v0c\n- v03 --refutes--> v08\n- v0f --refutes--> v04\n- v00 --supports--> v0d\n- v0e --supports--> v0a\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9dbf9658bbc159f2b28e7812ecaab9e3bf7be551da83f5d7e46f8909c83b697a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0f\n- v03 --supports--> v00\n- v09 --refutes--> v08\n- v05 --refutes--> v10\n- v0c --supports--> v05\n- v0a --refutes--> v04\n- v09 --supports--> v02\n- v0b --refutes--> v09\n- v0a --refutes--> v03\n- v0e --supports--> v0c\n- v01 --refutes--> v0b\n- v0f --supports--> v0a\n- v06 --supports--> v01\n- v0a --refutes--> v0d\n- v0f --supports--> v07\n- v03 --supports--> v06\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":"9b94c90a3eb035121cde0901b386cf806ada2de73e4e8febb20f8fcf14313869","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v0f --refutes--> v0c\n- v04 --refutes--> v00\n- v0e --refutes--> v05\n- v08 --refutes--> v03\n- v0b --refutes--> v0a\n- v0e --refutes--> v01\n- v10 --refutes--> v09\n- v0a --refutes--> v0d\n- v0a --refutes--> v0e\n- v0c --supports--> v06\n- v0e --supports--> v0f\n- v09 --refutes--> v0b\n- v03 --refutes--> v10\n- v02 --refutes--> v04\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":"868318ffae6ae77de9226280702bca3627781415a55ceb778d9b042f4bb107b6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0d\n- v00 --refutes--> v04\n- v08 --refutes--> v01\n- v02 --supports--> v03\n- v0a --supports--> v09\n- v0d --refutes--> v07\n- v05 --supports--> v0a\n- v0c --supports--> v10\n- v09 --refutes--> v08\n- v0e --refutes--> v05\n- v00 --refutes--> v06\n- v07 --refutes--> v00\n- v0d --supports--> v0b\n- v07 --refutes--> v0f\n- v00 --refutes--> v0e\n- v01 --refutes--> v02\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":"2cd3be212493c74b91d2ef2411ccfd4a185315a439cc177e7a6973757a430794","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v08\n- v04 --supports--> v09\n- v0e --supports--> v0b\n- v0c --supports--> v10\n- v0f --supports--> v03\n- v04 --refutes--> v0c\n- v0d --supports--> v04\n- v0e --supports--> v05\n- v03 --refutes--> v0a\n- v02 --supports--> v00\n- v10 --refutes--> v07\n- v06 --supports--> v0d\n- v05 --refutes--> v06\n- v03 --refutes--> v0e\n- v0f --refutes--> v01\n- v07 --refutes--> v02\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":"01bbfdb3bd62ba4635ec5eb346a8082d34288ee47fabb74bbb6f7b03befa54c4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v04 --supports--> v0f\n- v09 --refutes--> v0b\n- v0b --refutes--> v06\n- v0a --supports--> v0d\n- v0f --refutes--> v08\n- v0b --supports--> v00\n- v00 --refutes--> v0e\n- v03 --refutes--> v02\n- v0a --supports--> v0c\n- v0a --refutes--> v05\n- v03 --refutes--> v0a\n- v10 --refutes--> v04\n- v0e --supports--> v01\n- v0c --refutes--> v10\n- v08 --supports--> v09\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":"37a9f596372a293f4081626be005b47abfa0fdaca223c0d51bf23589eb3e1819","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v09\n- v0e --refutes--> v0c\n- v0e --supports--> v03\n- v0b --refutes--> v0f\n- v07 --refutes--> v02\n- v0e --supports--> v05\n- v0e --supports--> v06\n- v07 --refutes--> v0b\n- v03 --refutes--> v07\n- v0b --refutes--> v01\n- v00 --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v10\n- v10 --refutes--> v04\n- v09 --supports--> v0e\n- v0a --supports--> v00\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc9aa20ba8a4fd0b6f66ec532b567f1fb4401cab37cc77a483cd01f1d1f8ba1a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v06\n- v02 --refutes--> v03\n- v0c --refutes--> v0f\n- v08 --supports--> v09\n- v03 --refutes--> v01\n- v0e --refutes--> v0b\n- v05 --supports--> v10\n- v05 --refutes--> v0e\n- v01 --supports--> v0d\n- v00 --supports--> v02\n- v02 --supports--> v0a\n- v0e --supports--> v04\n- v04 --supports--> v07\n- v09 --refutes--> v0c\n- v07 --refutes--> v08\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":"2ea2096354589b8567ddff1c423405bce2c495f7f3f8b1fb28c12914e69d31a2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0c\n- v01 --refutes--> v00\n- v03 --supports--> v0b\n- v0e --supports--> v0f\n- v0e --refutes--> v05\n- v0b --supports--> v04\n- v09 --refutes--> v07\n- v07 --supports--> v0d\n- v05 --refutes--> v0a\n- v09 --refutes--> v02\n- v04 --supports--> v10\n- v00 --supports--> v03\n- v09 --refutes--> v06\n- v0a --refutes--> v01\n- v0d --refutes--> v0e\n- v05 --supports--> v08\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":"fcd6637899dc507722bec398195ddde02f89a59435b124b5f9280e49e1410b2d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v05\n- v0c --refutes--> v07\n- v02 --supports--> v09\n- v08 --refutes--> v01\n- v01 --refutes--> v0e\n- v09 --refutes--> v03\n- v01 --refutes--> v0c\n- v03 --refutes--> v0b\n- v03 --supports--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v06\n- v10 --supports--> v0f\n- v07 --supports--> v10\n- v0f --supports--> v04\n- v07 --supports--> v00\n- v04 --refutes--> v0a\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":"4566d2266db89ce89f563cb218365f54f27d05e0e875f53f1cc6f5f3699ae3f2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v10\n- v04 --supports--> v0c\n- v0f --refutes--> v06\n- v08 --supports--> v04\n- v04 --supports--> v01\n- v05 --supports--> v0d\n- v0e --supports--> v07\n- v04 --refutes--> v00\n- v0c --supports--> v05\n- v09 --supports--> v0a\n- v06 --supports--> v0b\n- v0a --supports--> v0f\n- v0d --supports--> v02\n- v02 --refutes--> v09\n- v02 --refutes--> v03\n- v06 --refutes--> v0e\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":"0bf86a92f5ffac2c2fcddb73d802dbfc431b44358c18e9a36149074b9e500674","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v10\n- v07 --supports--> v01\n- v05 --supports--> v0d\n- v00 --refutes--> v09\n- v03 --refutes--> v00\n- v05 --refutes--> v07\n- v0a --supports--> v08\n- v08 --supports--> v0e\n- v01 --refutes--> v02\n- v06 --refutes--> v0c\n- v0b --refutes--> v05\n- v0e --refutes--> v03\n- v10 --supports--> v0a\n- v01 --supports--> v04\n- v0e --refutes--> v0f\n- v06 --supports--> v0b\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":"ed6b1fe5fc029f3ae19dd3a4ceb9165cd3c0d82968d471387673614cb97c1c73","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v04\n- v0f --refutes--> v07\n- v06 --supports--> v0e\n- v10 --supports--> v00\n- v06 --refutes--> v0c\n- v0b --supports--> v05\n- v03 --supports--> v0d\n- v0a --supports--> v09\n- v09 --refutes--> v06\n- v03 --refutes--> v0a\n- v0f --supports--> v03\n- v0b --refutes--> v02\n- v02 --refutes--> v0f\n- v01 --refutes--> v0b\n- v08 --refutes--> v10\n- v10 --refutes--> v01\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":"eae7073bed25324d5f640693ea8d7115a194ba142d34089594efe750d6398759","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v06\n- v03 --refutes--> v0e\n- v0d --refutes--> v0b\n- v02 --supports--> v03\n- v01 --refutes--> v09\n- v0d --supports--> v05\n- v0c --refutes--> v0a\n- v07 --refutes--> v08\n- v01 --refutes--> v02\n- v10 --supports--> v04\n- v05 --supports--> v01\n- v00 --refutes--> v0d\n- v0f --supports--> v0c\n- v0c --supports--> v10\n- v07 --supports--> v00\n- v04 --refutes--> v07\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":"78f89a81504ed2ccdec1c5985d74ef3d452821c1a1b04f9b55cf4987dc43612a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0b\n- v01 --refutes--> v10\n- v02 --refutes--> v01\n- v09 --supports--> v04\n- v04 --supports--> v0e\n- v07 --refutes--> v0d\n- v0c --supports--> v03\n- v03 --refutes--> v09\n- v07 --refutes--> v0c\n- v08 --supports--> v0f\n- v01 --supports--> v07\n- v06 --refutes--> v0a\n- v00 --supports--> v06\n- v0e --refutes--> v05\n- v0e --supports--> v00\n- v06 --supports--> v08\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":"d4ce011cdf890688bb7bd0df69c7014e5030910978377f1601fd9135c938303d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0a\n- v06 --refutes--> v02\n- v08 --refutes--> v03\n- v02 --supports--> v0f\n- v08 --refutes--> v0c\n- v00 --refutes--> v08\n- v09 --refutes--> v10\n- v0c --supports--> v04\n- v0f --refutes--> v0d\n- v0b --refutes--> v06\n- v02 --supports--> v0e\n- v0c --supports--> v01\n- v10 --supports--> v0b\n- v07 --refutes--> v09\n- v0d --refutes--> v00\n- v00 --supports--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1f0c42c4b0268db8a578c896ce3b20021f1349133dda5e217953f491817fafc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v06 --supports--> v02\n- v04 --refutes--> v0f\n- v0b --supports--> v10\n- v05 --supports--> v08\n- v0b --supports--> v01\n- v0d --supports--> v03\n- v0e --refutes--> v00\n- v0f --refutes--> v0e\n- v02 --supports--> v04\n- v03 --refutes--> v07\n- v08 --supports--> v0b\n- v09 --supports--> v0c\n- v0b --refutes--> v0d\n- v03 --supports--> v09\n- v09 --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":"a5a529091f0878c23a6796999cf002d96c9620391d3553925ff03342e4224319","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v03\n- v03 --supports--> v02\n- v04 --refutes--> v07\n- v0e --supports--> v04\n- v05 --supports--> v0a\n- v00 --supports--> v0f\n- v05 --supports--> v0e\n- v00 --refutes--> v08\n- v00 --supports--> v0b\n- v10 --supports--> v05\n- v07 --supports--> v0c\n- v07 --supports--> v00\n- v03 --supports--> v01\n- v08 --refutes--> v09\n- v06 --refutes--> v10\n- v09 --supports--> v0d\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":"160a2e968f7118974097cd3aa78019e960edb0f4f05084e48103639fdadaebfd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v02\n- v0f --supports--> v05\n- v0c --refutes--> v01\n- v07 --refutes--> v06\n- v06 --supports--> v0b\n- v0d --refutes--> v03\n- v0b --refutes--> v09\n- v08 --refutes--> v0d\n- v10 --supports--> v0e\n- v0d --supports--> v07\n- v07 --refutes--> v04\n- v0c --supports--> v08\n- v0a --refutes--> v0c\n- v05 --refutes--> v10\n- v09 --supports--> v0f\n- v05 --supports--> v00\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":"5e5b0c314910232d2ec5774224800cfb739242381e5c701f52652fac53e71461","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0e\n- v0a --supports--> v08\n- v06 --refutes--> v03\n- v08 --refutes--> v01\n- v03 --supports--> v04\n- v04 --refutes--> v0f\n- v00 --refutes--> v02\n- v0b --refutes--> v06\n- v10 --supports--> v0b\n- v08 --refutes--> v09\n- v02 --refutes--> v0d\n- v10 --supports--> v0c\n- v05 --refutes--> v10\n- v07 --supports--> v05\n- v02 --supports--> v07\n- v08 --supports--> v00\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":"8cf8ed71272d10a8bf56b99a85c87a421b68806ee18275d7388945a6bc894a52","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v00\n- v0b --supports--> v07\n- v0a --supports--> v04\n- v04 --supports--> v03\n- v0b --supports--> v05\n- v09 --refutes--> v0e\n- v04 --refutes--> v08\n- v08 --refutes--> v10\n- v0b --supports--> v0d\n- v10 --refutes--> v06\n- v06 --refutes--> v02\n- v05 --refutes--> v09\n- v04 --refutes--> v0f\n- v0c --refutes--> v0b\n- v02 --refutes--> v01\n- v02 --supports--> v0c\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8bdb2a1f1287cb19e0e6110e98b68e0c3fe01e225b40d84629bf7ee218c4ea43","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0e\n- v0c --supports--> v00\n- v04 --supports--> v10\n- v0b --supports--> v02\n- v10 --refutes--> v09\n- v08 --supports--> v0b\n- v0b --refutes--> v0c\n- v0a --refutes--> v07\n- v0d --refutes--> v08\n- v0f --refutes--> v0a\n- v00 --supports--> v06\n- v05 --refutes--> v0d\n- v10 --refutes--> v0f\n- v0e --refutes--> v04\n- v0a --refutes--> v03\n- v00 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"857c3081f0c673dafc95d0585bdf825dafbaff79142aa1922f5e0b20afd1fa65","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v0a --refutes--> v10\n- v10 --supports--> v08\n- v03 --supports--> v01\n- v06 --supports--> v0c\n- v04 --supports--> v06\n- v0e --refutes--> v0f\n- v03 --supports--> v02\n- v06 --supports--> v03\n- v03 --supports--> v0a\n- v0e --supports--> v0b\n- v07 --refutes--> v0e\n- v03 --refutes--> v09\n- v08 --supports--> v0d\n- v0f --supports--> v05\n- v00 --refutes--> v07\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":"c2ec880a1484be1bcd77d3a524e358727537f651b4543d73485797c3fdd831fc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v07\n- v07 --supports--> v10\n- v0f --refutes--> v02\n- v0d --supports--> v0f\n- v04 --supports--> v08\n- v10 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v00\n- v0d --refutes--> v05\n- v08 --supports--> v0d\n- v01 --refutes--> v0e\n- v04 --supports--> v0a\n- v01 --supports--> v09\n- v09 --supports--> v04\n- v06 --supports--> v0c\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":"35e27f94ba273cc03bb5d811e93232915d56668ae535d6afef5c61b7379e037c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0b\n- v03 --refutes--> v08\n- v01 --refutes--> v04\n- v06 --supports--> v0f\n- v04 --supports--> v0c\n- v0e --supports--> v07\n- v08 --supports--> v00\n- v04 --supports--> v0d\n- v10 --supports--> v05\n- v04 --refutes--> v02\n- v08 --refutes--> v0e\n- v05 --refutes--> v06\n- v02 --supports--> v0a\n- v02 --refutes--> v09\n- v07 --supports--> v01\n- v0f --supports--> v03\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":"99ebd1bd97a79ef756d2a39737dec8d7019b06f12e685911de45a78a13a8a32d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0b\n- v06 --refutes--> v0c\n- v00 --refutes--> v09\n- v10 --supports--> v0e\n- v04 --refutes--> v00\n- v08 --refutes--> v03\n- v03 --supports--> v02\n- v09 --refutes--> v06\n- v08 --supports--> v01\n- v0d --refutes--> v04\n- v0c --refutes--> v0a\n- v0d --refutes--> v07\n- v0a --supports--> v08\n- v09 --refutes--> v05\n- v0b --supports--> v10\n- v0c --refutes--> v0f\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":"ddc39f95d44c3d1384d7957f380fccbcb13cb904cffd5dbcffd8271ed17cb8f0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v05\n- v0a --refutes--> v0f\n- v0f --refutes--> v0c\n- v07 --refutes--> v03\n- v0c --refutes--> v01\n- v06 --refutes--> v0e\n- v0e --refutes--> v08\n- v07 --supports--> v04\n- v0e --supports--> v10\n- v09 --supports--> v07\n- v10 --refutes--> v02\n- v10 --supports--> v0a\n- v10 --supports--> v0b\n- v01 --supports--> v09\n- v00 --supports--> v0d\n- v0d --refutes--> v06\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8d7bf85c28716abe39d17420f8de33bfc27ab56f92426ca011038f8c40732c48","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v06\n- v10 --refutes--> v0c\n- v0d --refutes--> v09\n- v0f --supports--> v02\n- v05 --supports--> v0e\n- v0f --refutes--> v01\n- v0b --supports--> v0f\n- v07 --supports--> v08\n- v06 --supports--> v0a\n- v00 --refutes--> v07\n- v07 --refutes--> v05\n- v0e --supports--> v0b\n- v0c --refutes--> v00\n- v06 --supports--> v0d\n- v06 --refutes--> v03\n- v05 --refutes--> v04\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":"12b49b1563d0c4d2ce150f297eaec2f89c0002b6d0c4f89f0252f278076aacd2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0e\n- v02 --supports--> v05\n- v05 --refutes--> v03\n- v06 --refutes--> v02\n- v09 --supports--> v10\n- v0f --supports--> v09\n- v05 --supports--> v0c\n- v0c --refutes--> v0a\n- v08 --refutes--> v0f\n- v10 --supports--> v07\n- v04 --refutes--> v0d\n- v0d --refutes--> v08\n- v06 --refutes--> v01\n- v08 --refutes--> v00\n- v0e --supports--> v04\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":"bd98001a736454738d89dbd214ead1d6aef8b92a27a648aebbfd4ff40a5a7009","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v09\n- v06 --refutes--> v0b\n- v09 --supports--> v0e\n- v01 --refutes--> v05\n- v04 --refutes--> v08\n- v0c --supports--> v04\n- v02 --refutes--> v07\n- v0d --refutes--> v0a\n- v08 --supports--> v03\n- v08 --refutes--> v10\n- v05 --supports--> v0f\n- v06 --refutes--> v0d\n- v0d --supports--> v02\n- v0e --supports--> v01\n- v08 --refutes--> v00\n- v0f --refutes--> v06\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":"fd1872359a1fbe48cec428f9345f1c261f10c4811d753c4955daddc50bad70b2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0c\n- v0f --supports--> v02\n- v09 --refutes--> v0f\n- v0b --refutes--> v03\n- v0f --refutes--> v07\n- v0f --refutes--> v0a\n- v01 --refutes--> v06\n- v08 --supports--> v00\n- v04 --supports--> v0d\n- v0c --supports--> v01\n- v0f --refutes--> v05\n- v00 --supports--> v09\n- v04 --refutes--> v08\n- v06 --refutes--> v0b\n- v03 --refutes--> v04\n- v03 --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":"d1eb9fedfcb39d83693f1ed4f70b5ac4c322eb23363f9316c7c229b28affce08","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v07 --refutes--> v10\n- v07 --supports--> v08\n- v02 --supports--> v00\n- v08 --supports--> v03\n- v00 --supports--> v0b\n- v0b --refutes--> v0c\n- v01 --supports--> v07\n- v05 --supports--> v01\n- v06 --refutes--> v0a\n- v0b --supports--> v0d\n- v04 --refutes--> v06\n- v09 --refutes--> v0f\n- v05 --supports--> v0e\n- v08 --refutes--> v04\n- v0a --supports--> v09\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a85e72a836450cf4752b2ed3c252ffb69474fdcb9726026e9a3bfbb7f87a21b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0d\n- v0f --refutes--> v0b\n- v01 --refutes--> v06\n- v09 --refutes--> v0e\n- v10 --supports--> v02\n- v08 --refutes--> v07\n- v10 --refutes--> v05\n- v07 --supports--> v10\n- v0a --refutes--> v08\n- v09 --refutes--> v0f\n- v01 --supports--> v0c\n- v00 --refutes--> v09\n- v0b --supports--> v0a\n- v09 --refutes--> v04\n- v02 --refutes--> v03\n- v0c --refutes--> 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":"86cb6c404088403f4d3fee091a8240d4e9ee390ff7d67707c128ecec0d713d15","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0a\n- v0d --refutes--> v02\n- v09 --supports--> v03\n- v0d --supports--> v0c\n- v01 --supports--> v08\n- v0d --supports--> v07\n- v00 --refutes--> v0f\n- v0b --supports--> v04\n- v02 --supports--> v05\n- v03 --refutes--> v01\n- v08 --refutes--> v0d\n- v0f --supports--> v0b\n- v0d --refutes--> v0e\n- v0a --refutes--> v09\n- v02 --supports--> v10\n- v0b --refutes--> v06\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":"aaa8c0345f574640bb5c93bcba2544ee3fe240a04a19f26ffacd58468b998ac1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v10\n- v07 --refutes--> v01\n- v03 --refutes--> v08\n- v04 --supports--> v0a\n- v09 --refutes--> v07\n- v06 --supports--> v05\n- v0a --refutes--> v09\n- v00 --supports--> v04\n- v05 --supports--> v0e\n- v09 --refutes--> v0b\n- v08 --refutes--> v06\n- v03 --supports--> v0c\n- v01 --supports--> v03\n- v03 --supports--> v0f\n- v09 --refutes--> v0d\n- v0e --supports--> v02\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cc8e31295ec94ab0046e675e5eaad0308373e44693e16fbd783e79833e059b1d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0c\n- v0e --refutes--> v06\n- v0b --refutes--> v03\n- v04 --supports--> v0a\n- v10 --supports--> v01\n- v03 --supports--> v0f\n- v04 --supports--> v10\n- v05 --refutes--> v0e\n- v08 --supports--> v0b\n- v09 --refutes--> v02\n- v0b --supports--> v0d\n- v0b --supports--> v07\n- v0f --supports--> v05\n- v06 --supports--> v00\n- v06 --refutes--> v04\n- v02 --refutes--> v08\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f41f6d31bacf3063ceb4641509cbeed988dea1a3e579a58672bd6d5b295a02bc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v07\n- v0a --supports--> v01\n- v0a --refutes--> v09\n- v02 --supports--> v04\n- v07 --refutes--> v0e\n- v00 --refutes--> v0c\n- v04 --refutes--> v0f\n- v0e --supports--> v0a\n- v00 --refutes--> v05\n- v05 --refutes--> v0d\n- v09 --refutes--> v10\n- v07 --supports--> v03\n- v05 --refutes--> v02\n- v09 --supports--> v06\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":"62a74f4140bf2b2c857c7bcacf4166356ff15c614bd7ae6b0b9c10cd8d0d1090","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v0e\n- v05 --supports--> v0b\n- v0a --refutes--> v01\n- v10 --supports--> v04\n- v0f --refutes--> v09\n- v0d --refutes--> v02\n- v0f --supports--> v03\n- v03 --refutes--> v10\n- v05 --supports--> v06\n- v08 --supports--> v0a\n- v10 --refutes--> v07\n- v0c --refutes--> v05\n- v04 --refutes--> v08\n- v0b --supports--> v00\n- v0d --refutes--> v0c\n- v00 --supports--> v0f\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":"8bd618590af2e3dbb969e4f6b740e87f2d863d8340774d928f0bfcba6abc16e7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0e\n- v09 --supports--> v05\n- v0e --supports--> v02\n- v00 --refutes--> v06\n- v01 --supports--> v10\n- v07 --supports--> v0a\n- v0d --supports--> v0f\n- v0f --refutes--> v00\n- v02 --refutes--> v07\n- v08 --supports--> v09\n- v07 --refutes--> v01\n- v0c --refutes--> v04\n- v08 --supports--> v0b\n- v08 --supports--> v03\n- v05 --refutes--> v0c\n- v01 --refutes--> v0d\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f086fd83b09e352121da9ac1ddffcd4585bafdabc49524070ebb240c4526c700","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v09\n- v01 --refutes--> v02\n- v09 --supports--> v00\n- v07 --refutes--> v0a\n- v0f --supports--> v0e\n- v10 --supports--> v03\n- v0a --refutes--> v04\n- v01 --refutes--> v0f\n- v0e --supports--> v06\n- v00 --refutes--> v07\n- v10 --refutes--> v08\n- v04 --refutes--> v10\n- v0f --supports--> v0c\n- v08 --supports--> v01\n- v0a --refutes--> v0b\n- v07 --supports--> v0d\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":"656888cd220ab58f7fec2cd81ebaf1817905ffcd0be63e7ada136365404a5988","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v06\n- v0a --supports--> v07\n- v00 --refutes--> v10\n- v0e --supports--> v05\n- v07 --refutes--> v0e\n- v03 --supports--> v08\n- v0b --refutes--> v00\n- v09 --refutes--> v0d\n- v0e --refutes--> v0c\n- v09 --supports--> v0b\n- v02 --refutes--> v03\n- v05 --supports--> v04\n- v01 --refutes--> v09\n- v03 --supports--> v0f\n- v10 --supports--> v0a\n- v05 --refutes--> v02\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":"5f5a4504bad193f1fff518454238c65a520f1f71ffec653ea73ac9231759cf64","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v02\n- v04 --refutes--> v0e\n- v09 --supports--> v0b\n- v03 --supports--> v0d\n- v08 --refutes--> v03\n- v09 --supports--> v0a\n- v05 --refutes--> v07\n- v0e --supports--> v01\n- v0b --supports--> v00\n- v0f --refutes--> v09\n- v0f --refutes--> v0c\n- v0d --supports--> v10\n- v0d --supports--> v0f\n- v00 --supports--> v05\n- v07 --refutes--> v04\n- v05 --supports--> v06\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":"f0cf71b1378eccc47628030f3a0af3dab917d779f023762d76604a29720b5ecb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0d\n- v09 --supports--> v0f\n- v00 --supports--> v0b\n- v01 --refutes--> v07\n- v0b --supports--> v04\n- v0b --refutes--> v0c\n- v08 --refutes--> v02\n- v0d --supports--> v09\n- v01 --supports--> v00\n- v0f --refutes--> v06\n- v0d --refutes--> v05\n- v0e --supports--> v0a\n- v0f --supports--> v0e\n- v0a --supports--> v10\n- v0a --supports--> v01\n- v03 --refutes--> v08\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":"70e8e4737124149a85f0ae3c35d2b22e12304b843b625f1c1f52244eeaa2e482","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v04 --refutes--> v01\n- v0e --refutes--> v04\n- v10 --refutes--> v09\n- v0c --supports--> v0e\n- v06 --refutes--> v05\n- v01 --supports--> v10\n- v10 --refutes--> v08\n- v08 --refutes--> v0b\n- v02 --supports--> v0f\n- v08 --refutes--> v07\n- v06 --supports--> v03\n- v0b --refutes--> v0d\n- v02 --supports--> v0c\n- v01 --supports--> v0a\n- v00 --supports--> v06\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"00255e30c9a17e46887f63809cc2a52e3b5eaa4af4345ef931ac85ad0c51dab4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v06\n- v06 --supports--> v08\n- v04 --refutes--> v0d\n- v09 --supports--> v04\n- v09 --supports--> v01\n- v10 --refutes--> v09\n- v0d --supports--> v05\n- v0b --supports--> v0a\n- v05 --refutes--> v07\n- v0c --refutes--> v0b\n- v07 --supports--> v0c\n- v0d --supports--> v00\n- v0a --supports--> v0e\n- v0c --refutes--> v02\n- v07 --supports--> v0f\n- v0a --refutes--> v03\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":"36dc4c4983a88cc233809a21ce702939ad960a7a3663027da74067d2b7a4a187","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0a\n- v0b --refutes--> v04\n- v0e --supports--> v00\n- v04 --refutes--> v08\n- v06 --supports--> v07\n- v0e --supports--> v0d\n- v0a --supports--> v03\n- v0c --supports--> v06\n- v09 --refutes--> v0b\n- v0a --supports--> v10\n- v0e --supports--> v0f\n- v07 --refutes--> v01\n- v0a --supports--> v02\n- v10 --refutes--> v0e\n- v0d --refutes--> v0c\n- v04 --supports--> v05\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c4e8f5792e1c6d2eb6d37564049530df1000e2adff02ff040228bae021d1ae7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v04\n- v0b --refutes--> v06\n- v01 --refutes--> v0a\n- v02 --supports--> v0c\n- v05 --supports--> v09\n- v0e --supports--> v10\n- v08 --supports--> v02\n- v0d --supports--> v01\n- v0f --supports--> v05\n- v0e --refutes--> v00\n- v0a --refutes--> v0f\n- v0e --supports--> v0b\n- v04 --supports--> v08\n- v0c --refutes--> v0e\n- v0d --supports--> v07\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":"bdca8409e9c7c8a234d8d8585da3ba9975954f4a0a5804048df81f43e03122f1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0f\n- v0e --refutes--> v06\n- v0d --supports--> v04\n- v03 --refutes--> v0d\n- v05 --supports--> v02\n- v0a --supports--> v0b\n- v0d --supports--> v10\n- v03 --refutes--> v0c\n- v05 --supports--> v0e\n- v09 --supports--> v05\n- v04 --supports--> v07\n- v0f --refutes--> v03\n- v00 --refutes--> v01\n- v06 --supports--> v00\n- v0f --refutes--> v08\n- v0a --refutes--> v09\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":"60fedd9541c3049fa88813b73e5285ec8cdd5f9abdd1696e804d3d7fa2ad6f9a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v03 --supports--> v0e\n- v0c --supports--> v0b\n- v0f --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v06\n- v00 --refutes--> v02\n- v10 --refutes--> v0d\n- v0d --supports--> v04\n- v05 --supports--> v0f\n- v06 --supports--> v10\n- v01 --supports--> v08\n- v00 --supports--> v05\n- v04 --refutes--> v0c\n- v0c --supports--> v09\n- v00 --supports--> v0a\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":"f7ef0129b08b5b45764648d996aea8e94e34a41d078a9305339b00ce9d244a74","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0c\n- v07 --refutes--> v0d\n- v06 --refutes--> v0e\n- v00 --supports--> v0a\n- v05 --supports--> v09\n- v01 --refutes--> v00\n- v05 --refutes--> v04\n- v00 --refutes--> v05\n- v0b --supports--> v07\n- v08 --refutes--> v0f\n- v0d --refutes--> v10\n- v10 --refutes--> v01\n- v08 --refutes--> v0b\n- v07 --refutes--> v02\n- v0e --supports--> v08\n- v03 --refutes--> v06\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0bcd1848bc33ffd27ff87e59a8d68162fe8f8f7a52d1dac741adf4bcdd74973b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v06 --refutes--> v0b\n- v04 --refutes--> v05\n- v07 --supports--> v02\n- v0a --refutes--> v08\n- v08 --supports--> v10\n- v09 --refutes--> v07\n- v08 --supports--> v00\n- v02 --supports--> v04\n- v02 --refutes--> v0f\n- v10 --supports--> v01\n- v0e --supports--> v0c\n- v04 --refutes--> v03\n- v0d --supports--> v06\n- v06 --supports--> v0a\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e294cefe61407620e46d7f5ff9e8acfe44edbea0b48aa50bfb28faa776efb80b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v00\n- v0b --supports--> v07\n- v08 --supports--> v0b\n- v06 --supports--> v0d\n- v0c --refutes--> v02\n- v06 --refutes--> v04\n- v0d --supports--> v10\n- v10 --refutes--> v0a\n- v0b --supports--> v09\n- v0f --refutes--> v03\n- v02 --supports--> v05\n- v0f --refutes--> v01\n- v09 --refutes--> v06\n- v10 --refutes--> v0e\n- v02 --supports--> v0f\n- v0e --supports--> v0c\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":"df4c2e33da63e837575aa7cffacc2b2024bb9912d39c47d5e572a750ece60d47","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v07 --supports--> v00\n- v09 --refutes--> v05\n- v0d --refutes--> v06\n- v0f --supports--> v04\n- v06 --supports--> v08\n- v08 --refutes--> v07\n- v0a --supports--> v03\n- v00 --refutes--> v0a\n- v08 --refutes--> v10\n- v04 --refutes--> v0e\n- v0a --refutes--> v01\n- v06 --refutes--> v0b\n- v0e --supports--> v0c\n- v09 --refutes--> v0f\n- v08 --supports--> v02\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":"169dbaff4b4f4ef5017b382be9dc63b86478dbabe4ee960216bb1e0273bd7c6f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v0a --refutes--> v07\n- v08 --refutes--> v10\n- v08 --supports--> v0b\n- v07 --refutes--> v06\n- v0e --supports--> v0f\n- v04 --refutes--> v03\n- v0d --supports--> v0a\n- v02 --supports--> v05\n- v04 --supports--> v0c\n- v10 --refutes--> v0d\n- v02 --refutes--> v01\n- v0f --refutes--> v02\n- v05 --supports--> v08\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":"f42be1aa380cd3fbbfb22b16a65756dcb35fdc3ed588531c3a720cf221675468","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v0f --refutes--> v06\n- v06 --supports--> v08\n- v0c --refutes--> v01\n- v0b --supports--> v0f\n- v0b --supports--> v0e\n- v01 --refutes--> v05\n- v0d --refutes--> v10\n- v05 --supports--> v0a\n- v10 --supports--> v0c\n- v00 --refutes--> v07\n- v04 --supports--> v0d\n- v0b --refutes--> v09\n- v07 --refutes--> v04\n- v06 --refutes--> v00\n- v00 --refutes--> v03\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":"23538019379129d387058f4dfad3b75e8a16878cd16848d255d438d4df0fc338","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v02\n- v0d --supports--> v00\n- v0e --supports--> v0f\n- v06 --supports--> v0c\n- v03 --supports--> v10\n- v0c --refutes--> v04\n- v00 --supports--> v03\n- v05 --supports--> v06\n- v0b --refutes--> v01\n- v08 --refutes--> v0e\n- v0c --supports--> v0b\n- v08 --refutes--> v07\n- v06 --refutes--> v09\n- v0b --supports--> v0d\n- v0f --supports--> v0a\n- v02 --supports--> v08\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":"939391d86e749b4eb52fe7fe67063c74a4b23b0077203e941127a4ed80b73722","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0e\n- v0d --refutes--> v0c\n- v0f --supports--> v0b\n- v01 --supports--> v08\n- v0d --supports--> v06\n- v00 --supports--> v10\n- v01 --refutes--> v00\n- v04 --refutes--> v05\n- v02 --supports--> v0f\n- v0a --refutes--> v09\n- v07 --refutes--> v04\n- v10 --refutes--> v02\n- v0f --supports--> v07\n- v05 --refutes--> v03\n- v06 --refutes--> v0a\n- v05 --supports--> v0d\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0676bf0907897fc9ff4f2f98665bed9bf7f61e37621c5d0e2c782b5afd8c79aa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0e\n- v03 --supports--> v0a\n- v0e --supports--> v07\n- v0f --refutes--> v0b\n- v0b --refutes--> v08\n- v0b --supports--> v02\n- v04 --supports--> v12\n- v00 --supports--> v10\n- v10 --refutes--> v0d\n- v02 --supports--> v09\n- v07 --supports--> v04\n- v0c --refutes--> v03\n- v10 --supports--> v0c\n- v01 --supports--> v0f\n- v02 --refutes--> v06\n- v04 --refutes--> v01\n- v03 --refutes--> v05\n- v0c --refutes--> v11\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":"72ad9aaae134b69bdda9e4be8181db871191420d326f72c11ba11b48207a3956","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0f\n- v0c --refutes--> v01\n- v0d --refutes--> v0e\n- v0e --refutes--> v0b\n- v03 --supports--> v07\n- v0b --supports--> v0a\n- v0c --refutes--> v00\n- v12 --supports--> v08\n- v10 --supports--> v12\n- v0f --supports--> v0c\n- v10 --supports--> v02\n- v07 --supports--> v09\n- v0e --supports--> v11\n- v0f --refutes--> v04\n- v00 --refutes--> v05\n- v05 --supports--> v10\n- v0b --refutes--> v03\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":"2f6120792f3e6c8ca0f972363b537fb8fb9b8ce436ff1270c53474921b9dd1d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0c\n- v0a --refutes--> v00\n- v0a --refutes--> v01\n- v09 --refutes--> v03\n- v11 --refutes--> v0b\n- v0f --supports--> v04\n- v0f --refutes--> v0d\n- v08 --supports--> v11\n- v05 --refutes--> v10\n- v06 --refutes--> v09\n- v11 --refutes--> v0a\n- v0d --supports--> v08\n- v02 --supports--> v12\n- v01 --refutes--> v02\n- v09 --refutes--> v0e\n- v09 --refutes--> v05\n- v05 --supports--> v07\n- v0c --refutes--> v06\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":"c0b7298f32c5862c6f3fc64c37663811078edd203c5d85f9b9ff65b31fcf0b77","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v01\n- v0a --supports--> v04\n- v00 --supports--> v0f\n- v06 --refutes--> v05\n- v0f --refutes--> v12\n- v09 --supports--> v0e\n- v11 --refutes--> v02\n- v0d --supports--> v09\n- v04 --refutes--> v11\n- v12 --supports--> v0a\n- v0d --supports--> v07\n- v0c --supports--> v03\n- v11 --refutes--> v0d\n- v08 --refutes--> v06\n- v05 --refutes--> v0c\n- v0c --supports--> v00\n- v12 --refutes--> v0b\n- v09 --supports--> v10\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":"371f08f5a95763d11a1838d1de25ce830a73a4aa6939b31821fc3f7238a2023e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v04\n- v01 --supports--> v09\n- v0b --refutes--> v02\n- v0b --supports--> v12\n- v08 --refutes--> v11\n- v0d --refutes--> v00\n- v06 --supports--> v03\n- v01 --supports--> v05\n- v03 --refutes--> v0b\n- v03 --refutes--> v07\n- v0f --supports--> v0d\n- v10 --refutes--> v06\n- v11 --supports--> v0f\n- v09 --supports--> v0c\n- v09 --supports--> v08\n- v00 --supports--> v0e\n- v11 --refutes--> v0a\n- v12 --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":"de61ce1dfdb4738a140dcbbf3e6aaca91f2bfc4f35ec4dcecc336e7c94d7a498","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v04\n- v02 --supports--> v01\n- v10 --supports--> v02\n- v03 --refutes--> v06\n- v10 --supports--> v05\n- v02 --refutes--> v0d\n- v0f --refutes--> v08\n- v03 --refutes--> v0b\n- v00 --supports--> v0c\n- v0c --supports--> v0e\n- v11 --supports--> v03\n- v11 --supports--> v09\n- v0e --refutes--> v0a\n- v03 --supports--> v12\n- v0a --refutes--> v0f\n- v12 --supports--> v07\n- v0d --supports--> v00\n- v07 --supports--> v10\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3809f165ea9dceb1a6787eace2564be929607ffd7a7f8d1ce4212fbebc5ca70b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v01\n- v12 --supports--> v0d\n- v02 --supports--> v0f\n- v12 --supports--> v0a\n- v04 --supports--> v0e\n- v0a --supports--> v04\n- v03 --refutes--> v0b\n- v00 --refutes--> v09\n- v0f --supports--> v03\n- v10 --refutes--> v00\n- v09 --supports--> v11\n- v03 --refutes--> v08\n- v07 --refutes--> v05\n- v09 --refutes--> v02\n- v07 --supports--> v12\n- v0e --refutes--> v10\n- v03 --refutes--> v06\n- v07 --refutes--> v0c\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dfc489513915afafef4f5b63fed4bb718a8207e4f1dce4fe0dc13c93e689cf26","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v01\n- v0c --supports--> v02\n- v05 --refutes--> v00\n- v0c --refutes--> v05\n- v0f --supports--> v06\n- v06 --refutes--> v11\n- v03 --supports--> v10\n- v00 --supports--> v09\n- v0c --refutes--> v0e\n- v11 --refutes--> v12\n- v01 --supports--> v03\n- v03 --refutes--> v0a\n- v00 --refutes--> v07\n- v0d --supports--> v0b\n- v03 --supports--> v0f\n- v09 --supports--> v08\n- v0b --supports--> v0c\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":"f81f6dfc044681050346008a6104d2075855a0a8468d65b0c606b6b6b03e0926","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v08\n- v0e --supports--> v06\n- v0a --refutes--> v03\n- v03 --refutes--> v07\n- v03 --supports--> v0d\n- v03 --refutes--> v04\n- v08 --supports--> v12\n- v07 --refutes--> v09\n- v01 --supports--> v10\n- v10 --refutes--> v0f\n- v0b --refutes--> v0c\n- v00 --supports--> v0e\n- v02 --refutes--> v05\n- v02 --supports--> v00\n- v10 --refutes--> v0b\n- v0b --supports--> v02\n- v06 --supports--> v11\n- v12 --refutes--> v01\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":"60f9550bc969b4eec8d53ef6cf1d4d3c089f552e3852406cd9f3e8910f5e5994","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v06\n- v08 --supports--> v0d\n- v06 --supports--> v10\n- v07 --supports--> v0b\n- v05 --refutes--> v01\n- v06 --supports--> v11\n- v11 --supports--> v0e\n- v0d --supports--> v0a\n- v0f --supports--> v04\n- v01 --refutes--> v12\n- v09 --refutes--> v07\n- v0b --refutes--> v00\n- v0e --refutes--> v09\n- v0d --refutes--> v02\n- v01 --refutes--> v0f\n- v05 --refutes--> v03\n- v0b --supports--> v08\n- v0b --refutes--> v0c\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":"1d6a73a67b90aaeaa35f7797430912adb94c22202cbf9ce5ebf0393e6d5d308f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0c\n- v04 --refutes--> v12\n- v07 --supports--> v04\n- v0c --supports--> v08\n- v08 --refutes--> v0e\n- v03 --supports--> v0a\n- v08 --supports--> v05\n- v0f --supports--> v11\n- v0a --supports--> v02\n- v05 --refutes--> v06\n- v0b --refutes--> v10\n- v09 --supports--> v07\n- v04 --refutes--> v03\n- v07 --refutes--> v0d\n- v04 --refutes--> v00\n- v05 --refutes--> v0f\n- v0b --refutes--> v01\n- v01 --supports--> v09\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":"4bbc6aa3ed60f3ecab32dfda54e126669d625d96631f5b403724e6780f2d0514","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v06\n- v02 --refutes--> v0b\n- v00 --refutes--> v0f\n- v0e --refutes--> v12\n- v05 --refutes--> v07\n- v0e --supports--> v02\n- v03 --supports--> v08\n- v01 --refutes--> v0e\n- v0b --refutes--> v0a\n- v0c --supports--> v0d\n- v10 --supports--> v00\n- v03 --supports--> v04\n- v11 --supports--> v03\n- v03 --refutes--> v10\n- v06 --refutes--> v11\n- v00 --refutes--> v05\n- v0c --supports--> v01\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":"436d6e79cff73d33558a45e6d22440e049a86ad868febedd31412283433d7451","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v05\n- v0e --supports--> v0b\n- v08 --refutes--> v0d\n- v08 --supports--> v03\n- v08 --refutes--> v0f\n- v04 --refutes--> v01\n- v0d --refutes--> v09\n- v12 --supports--> v02\n- v05 --refutes--> v07\n- v0b --refutes--> v0a\n- v05 --refutes--> v12\n- v09 --refutes--> v0c\n- v12 --refutes--> v0e\n- v0a --refutes--> v11\n- v11 --refutes--> v04\n- v05 --refutes--> v06\n- v01 --refutes--> v08\n- v0e --refutes--> v10\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":"773229b0959a2b6159309b1f89a424e971fb52b316927174a15bd5e15426d17b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v11\n- v07 --refutes--> v12\n- v0f --supports--> v07\n- v01 --supports--> v10\n- v12 --supports--> v09\n- v03 --refutes--> v06\n- v10 --refutes--> v04\n- v0f --supports--> v0b\n- v05 --refutes--> v03\n- v08 --supports--> v05\n- v12 --supports--> v08\n- v00 --supports--> v0d\n- v06 --refutes--> v0a\n- v08 --supports--> v0e\n- v04 --supports--> v00\n- v05 --supports--> v02\n- v06 --refutes--> v01\n- v0d --refutes--> v0c\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":"ec921392599d8f2dcd8b4b443e36a10a36ed9f1e0d09443cb4d601c5b56d6ccd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v11\n- v05 --refutes--> v00\n- v0c --supports--> v02\n- v01 --refutes--> v04\n- v07 --refutes--> v05\n- v0f --refutes--> v10\n- v03 --refutes--> v01\n- v10 --supports--> v0c\n- v12 --refutes--> v08\n- v11 --refutes--> v0f\n- v0c --supports--> v09\n- v02 --refutes--> v06\n- v01 --refutes--> v0a\n- v06 --refutes--> v0e\n- v0a --refutes--> v12\n- v06 --supports--> v07\n- v06 --supports--> v0d\n- v01 --supports--> v0b\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":"a1bcefb5226176a143fcf19fcba886f77ae33d606ab73dfca3078cc7c38a26c4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v05\n- v08 --refutes--> v01\n- v11 --supports--> v09\n- v09 --refutes--> v0f\n- v08 --supports--> v0d\n- v12 --refutes--> v04\n- v00 --supports--> v12\n- v0d --refutes--> v0a\n- v06 --refutes--> v11\n- v0f --supports--> v02\n- v02 --supports--> v0c\n- v08 --refutes--> v0e\n- v0d --refutes--> v10\n- v03 --refutes--> v08\n- v07 --supports--> v06\n- v10 --refutes--> v07\n- v12 --supports--> v03\n- v0d --refutes--> v0b\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":"c481dd073b8253bb6ccd6af1c07d57d9d7e11c9a7592e40e3bb438597fb420d5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v11\n- v09 --refutes--> v12\n- v04 --refutes--> v09\n- v10 --refutes--> v04\n- v07 --supports--> v0f\n- v0f --supports--> v0a\n- v05 --supports--> v07\n- v0e --refutes--> v08\n- v01 --refutes--> v0e\n- v09 --refutes--> v06\n- v09 --supports--> v02\n- v10 --refutes--> v0c\n- v08 --refutes--> v00\n- v06 --supports--> v0d\n- v0d --supports--> v01\n- v00 --supports--> v05\n- v04 --supports--> v0b\n- v07 --supports--> v03\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":"4cd9b3ad34b382523524adcee25428d06e3f71f32929d318f337ef2e49c4cd56","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0b\n- v05 --supports--> v06\n- v05 --supports--> v02\n- v00 --refutes--> v04\n- v0f --refutes--> v09\n- v0e --refutes--> v07\n- v01 --refutes--> v05\n- v03 --refutes--> v11\n- v09 --supports--> v00\n- v00 --supports--> v12\n- v08 --refutes--> v0f\n- v03 --refutes--> v08\n- v0d --supports--> v01\n- v07 --supports--> v03\n- v02 --supports--> v10\n- v00 --refutes--> v0d\n- v00 --refutes--> v0a\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":"6759042a5b467ed2fac8e95dde081fc8c804539fa66fa0154eb568a1e05235f7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0f\n- v0c --supports--> v01\n- v04 --supports--> v09\n- v0c --refutes--> v07\n- v09 --supports--> v05\n- v08 --refutes--> v10\n- v0b --supports--> v04\n- v09 --refutes--> v03\n- v0a --refutes--> v12\n- v11 --refutes--> v06\n- v03 --refutes--> v02\n- v0a --supports--> v08\n- v03 --supports--> v00\n- v07 --supports--> v0d\n- v0e --supports--> v0c\n- v07 --refutes--> v0a\n- v10 --supports--> v11\n- v00 --supports--> v0e\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":"9a79c0cdc3f3d2a02cf9a9ca33297f9b2b7443d3122da158ed877992d6a9786a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0a\n- v04 --refutes--> v0d\n- v0c --refutes--> v08\n- v05 --supports--> v03\n- v04 --supports--> v07\n- v0a --supports--> v00\n- v08 --supports--> v06\n- v0a --refutes--> v01\n- v03 --supports--> v0e\n- v0c --refutes--> v11\n- v06 --supports--> v04\n- v0e --refutes--> v02\n- v03 --refutes--> v0f\n- v01 --refutes--> v05\n- v10 --supports--> v0b\n- v10 --supports--> v12\n- v07 --supports--> v10\n- v0e --supports--> v09\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":"2f4431d6138e20a401e7e459adc7a2836c55c2f15d81f90b32437cf842e81786","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v09\n- v08 --refutes--> v06\n- v09 --supports--> v07\n- v00 --refutes--> v03\n- v09 --supports--> v0f\n- v02 --supports--> v0c\n- v07 --refutes--> v0e\n- v0e --supports--> v0d\n- v11 --refutes--> v12\n- v02 --refutes--> v10\n- v03 --supports--> v02\n- v0e --refutes--> v08\n- v04 --supports--> v11\n- v02 --supports--> v04\n- v0b --supports--> v05\n- v06 --refutes--> v00\n- v02 --refutes--> v0a\n- v07 --refutes--> v01\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":"51bd7fe25f1d7838fcceda147b0642cd128b99564542c6da5a826046738270bc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v07 --supports--> v03\n- v10 --supports--> v0d\n- v0f --refutes--> v01\n- v00 --refutes--> v08\n- v08 --refutes--> v0e\n- v03 --refutes--> v02\n- v0a --supports--> v05\n- v08 --supports--> v0a\n- v12 --refutes--> v0b\n- v04 --refutes--> v12\n- v09 --refutes--> v07\n- v0a --supports--> v04\n- v10 --supports--> v00\n- v02 --supports--> v10\n- v08 --refutes--> v11\n- v12 --supports--> v0f\n- v01 --supports--> v0c\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":"af1c3153b24200ab081f760f85c06d741ee17e38daa9b381b37f7a9cabbfabb8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v10\n- v0f --supports--> v0e\n- v03 --refutes--> v05\n- v0f --supports--> v01\n- v0a --refutes--> v12\n- v00 --supports--> v09\n- v02 --refutes--> v03\n- v05 --refutes--> v0c\n- v12 --supports--> v11\n- v0a --refutes--> v04\n- v0c --supports--> v0f\n- v0d --refutes--> v0b\n- v11 --supports--> v08\n- v12 --supports--> v07\n- v09 --refutes--> v02\n- v11 --supports--> v06\n- v0d --supports--> v00\n- v10 --refutes--> v0d\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":"61a6952b11efad8e774484d6e422fd3e1ea566a6c38b58134790173c7ddb9715","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0b\n- v05 --supports--> v0f\n- v03 --refutes--> v0e\n- v0f --supports--> v12\n- v09 --supports--> v08\n- v0c --supports--> v01\n- v0c --supports--> v11\n- v01 --refutes--> v0a\n- v08 --refutes--> v05\n- v0f --supports--> v02\n- v0a --supports--> v04\n- v07 --supports--> v0c\n- v09 --refutes--> v0d\n- v0a --refutes--> v00\n- v08 --refutes--> v06\n- v0a --refutes--> v10\n- v04 --refutes--> v09\n- v02 --refutes--> v03\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":"598ad654c58216b8f2e0a783b581d8691ae5bcb33d137f8492232a6b9b9a9398","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v01\n- v0e --refutes--> v0f\n- v09 --refutes--> v0b\n- v11 --refutes--> v07\n- v0a --supports--> v05\n- v0a --refutes--> v08\n- v0f --supports--> v03\n- v0b --supports--> v0e\n- v12 --supports--> v09\n- v02 --supports--> v00\n- v02 --refutes--> v04\n- v0a --supports--> v0c\n- v0f --refutes--> v06\n- v04 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0f --supports--> v02\n- v0f --refutes--> v10\n- v07 --refutes--> v12\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":"174023d7c7b044cf0e05d8dfbceaa2d388bf67c10ad51488a9fe666281a83f26","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v0e\n- v11 --refutes--> v08\n- v0f --refutes--> v06\n- v0c --refutes--> v00\n- v0e --refutes--> v11\n- v03 --supports--> v01\n- v03 --refutes--> v0d\n- v00 --refutes--> v02\n- v02 --refutes--> v10\n- v12 --supports--> v03\n- v0a --supports--> v04\n- v08 --refutes--> v0c\n- v11 --supports--> v05\n- v0a --supports--> v07\n- v03 --supports--> v09\n- v10 --refutes--> v12\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":"51df21a5918248ad7a7e7703c41741752c8411be67e858ee65cbaffd82dbc407","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v03\n- v0a --supports--> v00\n- v0e --refutes--> v02\n- v0a --refutes--> v10\n- v08 --refutes--> v11\n- v02 --refutes--> v05\n- v06 --supports--> v0a\n- v0d --supports--> v0f\n- v00 --supports--> v0c\n- v0a --refutes--> v04\n- v08 --supports--> v0e\n- v0f --refutes--> v07\n- v0f --refutes--> v09\n- v08 --refutes--> v0b\n- v09 --refutes--> v06\n- v0c --supports--> v01\n- v00 --supports--> v12\n- v03 --supports--> v0d\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":"ffeba1e85536b64faebff2d0f97d72b4ee9180459cf1c72ad34bee077351070f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v08\n- v01 --refutes--> v0b\n- v04 --supports--> v00\n- v03 --supports--> v02\n- v04 --refutes--> v05\n- v10 --supports--> v01\n- v04 --supports--> v0e\n- v06 --supports--> v07\n- v11 --supports--> v0d\n- v05 --refutes--> v03\n- v10 --supports--> v0c\n- v11 --refutes--> v0a\n- v0b --refutes--> v12\n- v0d --refutes--> v09\n- v0b --refutes--> v04\n- v08 --refutes--> v11\n- v06 --supports--> v0f\n- v09 --refutes--> v06\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":"c87f0d9adc002239164d1ed6fb0146f70d21385ee7a21ae87645d74c1338dad8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v04\n- v05 --supports--> v0f\n- v0d --supports--> v0b\n- v0f --refutes--> v0d\n- v02 --refutes--> v07\n- v03 --supports--> v00\n- v04 --supports--> v09\n- v00 --refutes--> v08\n- v10 --refutes--> v12\n- v0f --refutes--> v06\n- v0c --supports--> v10\n- v02 --supports--> v03\n- v0a --supports--> v05\n- v02 --supports--> v11\n- v0e --supports--> v02\n- v04 --refutes--> v0a\n- v0d --supports--> v01\n- v08 --supports--> v0c\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":"286a49e3ccb29c9475706b605b2924f10d60b7c46c15db8aff7dcd418b4e40aa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v04\n- v0a --refutes--> v0f\n- v02 --supports--> v0a\n- v07 --supports--> v00\n- v0e --refutes--> v09\n- v0f --refutes--> v08\n- v0e --refutes--> v02\n- v03 --refutes--> v0d\n- v08 --supports--> v12\n- v02 --refutes--> v0b\n- v02 --refutes--> v06\n- v07 --supports--> v01\n- v0d --refutes--> v11\n- v01 --supports--> v0c\n- v05 --supports--> v0e\n- v10 --refutes--> v05\n- v01 --supports--> v10\n- v11 --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":"cd058cca5269234f2f7a61d88a279a2966fc405a07f98f8d37331d3e10a63593","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v06 --refutes--> v02\n- v03 --supports--> v0f\n- v12 --refutes--> v0d\n- v09 --supports--> v0e\n- v07 --refutes--> v0a\n- v00 --refutes--> v0c\n- v02 --supports--> v05\n- v02 --supports--> v04\n- v04 --supports--> v12\n- v0f --refutes--> v08\n- v12 --refutes--> v07\n- v08 --refutes--> v06\n- v0c --refutes--> v10\n- v08 --supports--> v0b\n- v10 --refutes--> v01\n- v09 --refutes--> v03\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":"0ced12a8cefe7d0c04acf21f583c652afb11ad8e3ab4ebad0fe15956dafb4d07","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v08\n- v12 --refutes--> v09\n- v09 --supports--> v00\n- v05 --supports--> v01\n- v0e --supports--> v10\n- v02 --supports--> v0e\n- v0e --supports--> v04\n- v10 --refutes--> v0d\n- v05 --supports--> v0a\n- v0c --refutes--> v03\n- v07 --refutes--> v02\n- v09 --refutes--> v07\n- v0f --refutes--> v11\n- v0d --refutes--> v0f\n- v03 --supports--> v12\n- v01 --refutes--> v0c\n- v01 --supports--> v0b\n- v0c --supports--> v06\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":"fe47e0b8c97efb305837b94950a050cf7680ec451278b633fa2952cc37e680ec","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v05\n- v00 --refutes--> v01\n- v06 --supports--> v0e\n- v0a --supports--> v07\n- v0d --refutes--> v08\n- v05 --refutes--> v0d\n- v12 --refutes--> v03\n- v06 --refutes--> v0c\n- v0a --supports--> v11\n- v07 --supports--> v12\n- v08 --supports--> v0a\n- v08 --refutes--> v09\n- v02 --refutes--> v0f\n- v06 --refutes--> v04\n- v0f --refutes--> v10\n- v05 --refutes--> v0b\n- v00 --refutes--> v02\n- v03 --supports--> v00\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":"3be7a5dec2b69a036c7bb66bb96da57aff32a11aa6ec10515a81d3e5945f40af","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v06\n- v0b --refutes--> v03\n- v00 --refutes--> v04\n- v02 --refutes--> v0d\n- v00 --refutes--> v0f\n- v04 --refutes--> v12\n- v0c --refutes--> v0a\n- v12 --refutes--> v0e\n- v0a --supports--> v07\n- v10 --refutes--> v0c\n- v11 --supports--> v09\n- v0b --supports--> v02\n- v07 --supports--> v11\n- v07 --supports--> v05\n- v09 --supports--> v0b\n- v09 --refutes--> v08\n- v0d --supports--> v00\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":"72c3ee042bc86627f810b7036248d30f25160f0423abb8aa87fe63e559e8b902","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v0c\n- v0c --refutes--> v11\n- v02 --refutes--> v0b\n- v0b --refutes--> v07\n- v00 --refutes--> v02\n- v04 --refutes--> v0e\n- v0a --supports--> v0f\n- v0c --refutes--> v08\n- v02 --refutes--> v0d\n- v01 --refutes--> v00\n- v11 --refutes--> v04\n- v01 --supports--> v10\n- v09 --supports--> v05\n- v0e --supports--> v09\n- v03 --refutes--> v0a\n- v07 --supports--> v03\n- v07 --supports--> v06\n- v09 --refutes--> v12\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":"8a4ee0c4229f0ea1aa7ab420c9cec39af8b61c2c1bda2245b52e1d00a7ac46e9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v0e --refutes--> v06\n- v07 --refutes--> v0a\n- v07 --supports--> v0b\n- v02 --refutes--> v01\n- v09 --refutes--> v12\n- v0b --supports--> v03\n- v02 --supports--> v0d\n- v05 --refutes--> v08\n- v01 --supports--> v00\n- v0e --refutes--> v07\n- v05 --supports--> v04\n- v01 --supports--> v0c\n- v09 --supports--> v05\n- v0c --supports--> v0f\n- v10 --supports--> v02\n- v0f --refutes--> v11\n- v03 --refutes--> v10\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":"ba8d13ddac6131192a8af8e03799d4563e845e6a88282ccac3f8d4b61c55d8c4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v08\n- v10 --refutes--> v04\n- v02 --refutes--> v01\n- v10 --refutes--> v11\n- v02 --refutes--> v12\n- v0a --refutes--> v0d\n- v06 --refutes--> v0f\n- v12 --refutes--> v10\n- v00 --supports--> v0c\n- v12 --refutes--> v09\n- v06 --supports--> v00\n- v0d --supports--> v06\n- v0a --supports--> v0b\n- v07 --refutes--> v05\n- v08 --refutes--> v07\n- v05 --refutes--> v02\n- v04 --refutes--> v0e\n- v00 --supports--> v03\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":"e746755322ab212cf915d73b6547a7a924043c7dc44211d86b8eca81518ae60b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0b\n- v05 --refutes--> v02\n- v05 --supports--> v0d\n- v0e --supports--> v08\n- v03 --refutes--> v10\n- v03 --supports--> v0f\n- v02 --refutes--> v03\n- v00 --supports--> v0a\n- v04 --supports--> v05\n- v11 --supports--> v0e\n- v01 --refutes--> v09\n- v09 --refutes--> v00\n- v03 --refutes--> v07\n- v00 --supports--> v04\n- v10 --refutes--> v11\n- v04 --supports--> v12\n- v08 --supports--> v0c\n- v06 --supports--> v01\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9e6f95f6f699c6fc31adde8c8c5567f7cf4c194db52b9fba29923246753a269e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v03\n- v11 --refutes--> v10\n- v00 --supports--> v04\n- v00 --refutes--> v08\n- v07 --supports--> v01\n- v00 --refutes--> v0a\n- v0d --refutes--> v07\n- v10 --refutes--> v06\n- v08 --refutes--> v11\n- v0c --refutes--> v0e\n- v01 --refutes--> v09\n- v02 --refutes--> v0c\n- v0b --refutes--> v02\n- v10 --refutes--> v12\n- v07 --refutes--> v0f\n- v0c --refutes--> v05\n- v05 --supports--> v00\n- v09 --supports--> v0b\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":"62330954719cec768aa7d30c8e50c46851cfe5e72f3b6b690fbca56f1e2fa99a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v0e --supports--> v12\n- v03 --refutes--> v0e\n- v0f --supports--> v00\n- v06 --supports--> v03\n- v03 --supports--> v0a\n- v0c --refutes--> v02\n- v09 --supports--> v0f\n- v05 --refutes--> v06\n- v01 --supports--> v0b\n- v12 --refutes--> v11\n- v0b --supports--> v08\n- v12 --refutes--> v07\n- v02 --refutes--> v10\n- v04 --refutes--> v0c\n- v02 --refutes--> v09\n- v12 --refutes--> v0d\n- v0f --supports--> v01\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":"e4ce60bf66b82255df3141ca4f5d2a0e87ab4329f31f2ab6eb29de34b2e63d7f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v01\n- v10 --supports--> v0f\n- v07 --refutes--> v0d\n- v0d --supports--> v04\n- v00 --supports--> v07\n- v0a --supports--> v02\n- v0f --supports--> v09\n- v0e --refutes--> v10\n- v04 --refutes--> v0c\n- v0f --supports--> v11\n- v03 --refutes--> v0e\n- v0c --supports--> v0b\n- v04 --refutes--> v05\n- v02 --supports--> v00\n- v09 --refutes--> v0a\n- v03 --refutes--> v12\n- v0c --supports--> v06\n- v04 --refutes--> v08\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":"826260414efb40f43c38db3b9898651a083b137f327a834c1da5bc0193e770fa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0d\n- v0d --supports--> v08\n- v11 --supports--> v07\n- v12 --refutes--> v06\n- v00 --refutes--> v12\n- v0b --refutes--> v10\n- v11 --supports--> v05\n- v12 --supports--> v04\n- v01 --supports--> v0f\n- v0b --refutes--> v02\n- v0a --refutes--> v00\n- v0f --supports--> v11\n- v04 --supports--> v0b\n- v05 --supports--> v0c\n- v12 --supports--> v09\n- v03 --refutes--> v0a\n- v05 --refutes--> v03\n- v0a --refutes--> v0e\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":"525caeec79e3511ba97d329705ef5ef3a196fe9ae765bdbe45bc3c38824f26c9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v02 --refutes--> v03\n- v0e --refutes--> v0b\n- v01 --refutes--> v0e\n- v00 --refutes--> v05\n- v0f --refutes--> v08\n- v0a --refutes--> v11\n- v10 --refutes--> v0a\n- v05 --refutes--> v0f\n- v0a --supports--> v0c\n- v08 --refutes--> v04\n- v0f --refutes--> v07\n- v00 --supports--> v0d\n- v0b --supports--> v02\n- v08 --refutes--> v10\n- v0a --supports--> v09\n- v03 --refutes--> v12\n- v0c --supports--> v01\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":"2eb2dd752a47fe7e67a6aca95bcaabe4355fc745e73009eca118e9b89e2c653a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v0b\n- v08 --refutes--> v0f\n- v12 --supports--> v08\n- v05 --supports--> v07\n- v10 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v05\n- v0e --refutes--> v00\n- v11 --supports--> v06\n- v0e --refutes--> v03\n- v04 --refutes--> v0d\n- v00 --supports--> v04\n- v05 --supports--> v12\n- v0e --supports--> v0c\n- v0d --refutes--> v10\n- v06 --supports--> v0e\n- v11 --refutes--> v02\n- v05 --supports--> v01\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":"000f35cc7b406c1aed940c9cabe7deef6f4fe5f23fa4a67e687995fa1ab0ab20","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0d\n- v10 --refutes--> v0a\n- v05 --supports--> v09\n- v0b --refutes--> v12\n- v0b --supports--> v03\n- v04 --supports--> v0c\n- v0f --refutes--> v08\n- v0d --supports--> v07\n- v01 --refutes--> v05\n- v10 --refutes--> v11\n- v08 --refutes--> v00\n- v10 --supports--> v04\n- v03 --refutes--> v10\n- v0c --refutes--> v0e\n- v00 --refutes--> v0b\n- v0c --supports--> v02\n- v0c --refutes--> v01\n- v0d --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"87f8da6e9369d0a6f458925cf846bc8e38bf689a36f2b2c532a7dfc43861120b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v05 --refutes--> v12\n- v0f --supports--> v00\n- v05 --refutes--> v0e\n- v06 --supports--> v0c\n- v06 --supports--> v01\n- v0c --refutes--> v03\n- v05 --refutes--> v11\n- v00 --refutes--> v07\n- v07 --supports--> v06\n- v0c --refutes--> v02\n- v0d --refutes--> v04\n- v10 --refutes--> v05\n- v09 --refutes--> v10\n- v08 --refutes--> v09\n- v02 --supports--> v0d\n- v0e --supports--> v0f\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":"7001fcd7d0ac6aefaf3351dd19ce87d3c286756b36b93fe62ee6c1e4acd238b2","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v10\n- v0e --supports--> v06\n- v05 --supports--> v12\n- v04 --refutes--> v03\n- v0f --supports--> v0d\n- v10 --supports--> v01\n- v04 --refutes--> v05\n- v0c --refutes--> v0a\n- v05 --supports--> v0e\n- v02 --supports--> v0c\n- v01 --supports--> v02\n- v0a --refutes--> v00\n- v0e --refutes--> v0f\n- v05 --supports--> v0b\n- v0d --refutes--> v07\n- v0d --supports--> v11\n- v0c --supports--> v09\n- v00 --supports--> v04\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":"800012bcc295f0ae139ce0148e8b2378357ad42b1ac35af626172b5cfa88e866","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v04\n- v0f --refutes--> v0b\n- v04 --supports--> v07\n- v12 --refutes--> v0d\n- v00 --supports--> v03\n- v12 --refutes--> v01\n- v00 --refutes--> v05\n- v10 --supports--> v0c\n- v05 --supports--> v10\n- v11 --supports--> v09\n- v07 --supports--> v12\n- v0d --supports--> v00\n- v10 --refutes--> v11\n- v0d --supports--> v0a\n- v12 --refutes--> v08\n- v06 --refutes--> v02\n- v10 --refutes--> v0e\n- v09 --refutes--> v06\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":"a21d295dce64f9ce895d0f826e6f64cac16ceadd60e67ebf8d8bf3e2f25461a9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v07\n- v0b --refutes--> v0a\n- v11 --supports--> v02\n- v07 --supports--> v0c\n- v0c --supports--> v12\n- v0a --refutes--> v09\n- v11 --supports--> v04\n- v0c --refutes--> v0d\n- v10 --refutes--> v00\n- v04 --refutes--> v0e\n- v01 --refutes--> v06\n- v04 --refutes--> v08\n- v10 --refutes--> v01\n- v01 --supports--> v0b\n- v09 --supports--> v03\n- v01 --supports--> v0f\n- v12 --refutes--> v05\n- v05 --supports--> v10\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":"ee2fb6eb16e356a043acdb7fdb8c3c76f53c158d88368da241831e6abf143cd1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v00\n- v0f --refutes--> v03\n- v03 --supports--> v10\n- v0a --supports--> v0c\n- v04 --supports--> v0b\n- v06 --supports--> v0f\n- v0a --supports--> v05\n- v00 --supports--> v09\n- v05 --supports--> v12\n- v03 --supports--> v04\n- v0b --refutes--> v0a\n- v0f --refutes--> v08\n- v04 --refutes--> v01\n- v07 --refutes--> v0d\n- v05 --refutes--> v07\n- v02 --refutes--> v0e\n- v09 --supports--> v02\n- v11 --supports--> v06\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":"08aa18ef5314f69e24c17e2af8f72f45c9d91a4020ddd59ccaf7ac7d85eca062","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v08 --supports--> v0a\n- v12 --supports--> v11\n- v05 --refutes--> v08\n- v0f --supports--> v06\n- v05 --refutes--> v03\n- v02 --refutes--> v09\n- v01 --supports--> v0d\n- v0a --refutes--> v0f\n- v00 --refutes--> v0c\n- v00 --refutes--> v12\n- v05 --refutes--> v07\n- v02 --supports--> v05\n- v0e --supports--> v02\n- v06 --supports--> v0b\n- v0a --supports--> v04\n- v0d --supports--> v10\n- v11 --refutes--> v0e\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":"f072fc1f1eff7a202dc32627d5e4f4fc0f4125192fddbbbabfdd90ff3bf73b51","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v06\n- v04 --refutes--> v0d\n- v09 --refutes--> v0b\n- v0f --refutes--> v0e\n- v08 --supports--> v05\n- v06 --supports--> v0f\n- v0b --supports--> v12\n- v03 --supports--> v08\n- v05 --supports--> v04\n- v0b --supports--> v00\n- v0f --refutes--> v11\n- v03 --refutes--> v07\n- v0c --supports--> v09\n- v0e --refutes--> v01\n- v05 --refutes--> v10\n- v03 --refutes--> v02\n- v0d --supports--> v0c\n- v0e --supports--> v0a\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":"6871cf7f57678c8cf23d70eaec295ac7d493b324fab4908f04b2a2e52047799e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v04\n- v06 --supports--> v05\n- v09 --supports--> v0f\n- v04 --supports--> v11\n- v06 --refutes--> v00\n- v0c --supports--> v09\n- v0e --supports--> v02\n- v0c --refutes--> v03\n- v11 --refutes--> v0d\n- v07 --refutes--> v12\n- v0a --refutes--> v07\n- v05 --supports--> v10\n- v09 --supports--> v0a\n- v0d --refutes--> v01\n- v01 --refutes--> v06\n- v05 --refutes--> v0c\n- v04 --supports--> v0b\n- v07 --refutes--> v08\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":"742ecc6b4ec593828d5722510d45fcb5a0e5916bcc8f482ecebb890c2cac72ef","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v0a\n- v0c --refutes--> v00\n- v0d --refutes--> v03\n- v0b --supports--> v07\n- v03 --supports--> v05\n- v03 --refutes--> v12\n- v08 --supports--> v06\n- v03 --supports--> v0c\n- v0a --supports--> v09\n- v0d --refutes--> v0e\n- v04 --supports--> v0b\n- v10 --supports--> v0d\n- v0a --refutes--> v01\n- v0b --supports--> v11\n- v01 --refutes--> v08\n- v0c --supports--> v0f\n- v06 --refutes--> v02\n- v00 --refutes--> v04\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":"43654b976c4be54758415ebd0c3c7d1c82304b29ca95c19658d24d9ab82c7d5a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0d\n- v11 --refutes--> v03\n- v0a --refutes--> v08\n- v10 --supports--> v0a\n- v07 --supports--> v05\n- v09 --supports--> v00\n- v0d --refutes--> v01\n- v0d --refutes--> v06\n- v12 --refutes--> v0c\n- v0a --supports--> v11\n- v0f --refutes--> v09\n- v0b --refutes--> v07\n- v11 --refutes--> v12\n- v00 --supports--> v10\n- v0f --refutes--> v02\n- v0c --refutes--> v0b\n- v07 --refutes--> v04\n- v0a --refutes--> v0e\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":"70fc01a74702a1b52227422ca235711c02ab10391ef71d43d7f0533e73ab544f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v06\n- v08 --refutes--> v03\n- v0f --supports--> v08\n- v0b --supports--> v09\n- v09 --refutes--> v12\n- v00 --supports--> v01\n- v01 --supports--> v0a\n- v05 --refutes--> v10\n- v10 --supports--> v0f\n- v12 --refutes--> v0e\n- v00 --refutes--> v0d\n- v01 --supports--> v07\n- v11 --refutes--> v02\n- v08 --refutes--> v11\n- v06 --supports--> v00\n- v09 --refutes--> v04\n- v07 --supports--> v0b\n- v12 --refutes--> v0c\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":"2098bb16c56f563b8478e55a37d7671d0912d37aa9903c68dde836fc8e3e8851","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v09\n- v0a --refutes--> v0e\n- v0e --refutes--> v03\n- v07 --supports--> v0c\n- v09 --refutes--> v05\n- v08 --refutes--> v0a\n- v08 --refutes--> v11\n- v07 --supports--> v0d\n- v00 --refutes--> v12\n- v0f --supports--> v08\n- v0a --supports--> v06\n- v02 --refutes--> v0b\n- v01 --supports--> v0f\n- v02 --refutes--> v00\n- v03 --refutes--> v07\n- v01 --supports--> v10\n- v0c --supports--> v02\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":"f69b7775bc0bbbdb8a080f99ceeeb72591859ac8965784209cd4b959ee89fe0d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v09\n- v06 --refutes--> v10\n- v12 --supports--> v03\n- v0e --supports--> v12\n- v03 --refutes--> v0a\n- v03 --supports--> v02\n- v0c --refutes--> v0f\n- v0d --refutes--> v0c\n- v0d --supports--> v11\n- v0d --supports--> v05\n- v10 --refutes--> v0b\n- v04 --refutes--> v08\n- v0c --refutes--> v01\n- v02 --refutes--> v06\n- v09 --refutes--> v0e\n- v01 --supports--> v04\n- v12 --supports--> v07\n- v0c --supports--> v00\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0df942eb1296c7fcc8a5951e5e84aff75710f64ac3cbc42673ec19945cebec6b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0e\n- v09 --refutes--> v02\n- v11 --refutes--> v06\n- v0f --supports--> v11\n- v00 --supports--> v07\n- v0e --supports--> v0a\n- v03 --supports--> v08\n- v0c --refutes--> v00\n- v00 --supports--> v04\n- v02 --refutes--> v0c\n- v11 --refutes--> v0d\n- v0e --refutes--> v01\n- v0f --supports--> v0b\n- v03 --supports--> v09\n- v0e --supports--> v05\n- v07 --supports--> v0f\n- v0a --supports--> v03\n- v06 --supports--> v12\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":"96b3dd3b6a1761fa3c2f46239692904f14643b60afb92c52b185d6c2824d34bd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v10\n- v0b --refutes--> v0e\n- v0b --supports--> v07\n- v08 --refutes--> v05\n- v0c --supports--> v0f\n- v06 --supports--> v01\n- v09 --supports--> v0b\n- v04 --supports--> v11\n- v07 --supports--> v12\n- v0f --refutes--> v00\n- v09 --refutes--> v0d\n- v00 --refutes--> v0a\n- v08 --refutes--> v02\n- v06 --supports--> v09\n- v03 --supports--> v08\n- v05 --supports--> v04\n- v00 --refutes--> v03\n- v12 --refutes--> v0c\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":"bb74fea2d73278f22a3636d3c3405a768fefcb8555ad62b63dc624ca1fa84b6c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v11\n- v01 --refutes--> v07\n- v03 --refutes--> v04\n- v02 --supports--> v12\n- v0a --refutes--> v0c\n- v0a --supports--> v09\n- v0e --refutes--> v0b\n- v08 --supports--> v0e\n- v08 --supports--> v0d\n- v0f --refutes--> v02\n- v09 --supports--> v03\n- v01 --supports--> v0a\n- v12 --refutes--> v10\n- v05 --supports--> v0f\n- v11 --refutes--> v01\n- v0f --supports--> v00\n- v03 --supports--> v06\n- v12 --supports--> v08\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":"817b95eebef735e13ccfb72f422ead2bab58a1f1a497405ac95a78b9ebf60938","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0a\n- v01 --supports--> v07\n- v0c --refutes--> v0f\n- v09 --refutes--> v01\n- v0f --supports--> v11\n- v10 --supports--> v04\n- v05 --supports--> v0c\n- v0d --supports--> v06\n- v11 --supports--> v03\n- v0d --supports--> v12\n- v0b --refutes--> v02\n- v01 --refutes--> v0b\n- v0e --supports--> v10\n- v09 --refutes--> v00\n- v02 --refutes--> v05\n- v10 --refutes--> v0d\n- v08 --refutes--> v0e\n- v0d --supports--> v09\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":"0e6d9c4f83e3046ddc6737a118d07adb481bb22f4c52abb38a7c28f407317d77","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v03\n- v04 --refutes--> v0b\n- v0b --refutes--> v0c\n- v07 --refutes--> v0a\n- v02 --refutes--> v00\n- v0f --refutes--> v09\n- v0f --refutes--> v04\n- v0a --refutes--> v02\n- v0b --supports--> v0e\n- v02 --supports--> v08\n- v03 --supports--> v07\n- v01 --supports--> v05\n- v0f --refutes--> v11\n- v00 --refutes--> v0d\n- v0b --refutes--> v01\n- v12 --refutes--> v0f\n- v05 --refutes--> v06\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":"41c5221c175a1a7abbdc76515c218946467793423e0a3b184a52ab9db19ecfd0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v02\n- v10 --refutes--> v12\n- v04 --refutes--> v09\n- v0e --supports--> v0d\n- v02 --supports--> v05\n- v10 --supports--> v0f\n- v02 --supports--> v10\n- v09 --supports--> v06\n- v08 --supports--> v03\n- v11 --refutes--> v08\n- v08 --supports--> v0a\n- v06 --refutes--> v0c\n- v0e --refutes--> v07\n- v00 --refutes--> v04\n- v0b --supports--> v11\n- v0c --refutes--> v01\n- v0c --supports--> v0b\n- v0f --supports--> v00\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":"816b0a8dca68d551166d184e8c1b1164979989175f00dbbadca37dd601cb775c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v05\n- v10 --refutes--> v0c\n- v12 --supports--> v13\n- v09 --refutes--> v12\n- v07 --supports--> v0a\n- v07 --supports--> v0d\n- v03 --refutes--> v0f\n- v13 --refutes--> v0b\n- v08 --refutes--> v02\n- v03 --refutes--> v07\n- v04 --refutes--> v0e\n- v02 --supports--> v04\n- v00 --refutes--> v08\n- v0b --supports--> v10\n- v12 --refutes--> v01\n- v0c --supports--> v03\n- v11 --refutes--> v00\n- v00 --refutes--> v06\n- v04 --refutes--> v09\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":"2a0c61b98249b6c0d4086ad67edec50cca1a3beb58c7038658b14d4f94716eeb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v11\n- v08 --supports--> v00\n- v06 --refutes--> v0c\n- v08 --refutes--> v13\n- v10 --refutes--> v04\n- v04 --supports--> v07\n- v09 --supports--> v01\n- v0f --supports--> v05\n- v07 --refutes--> v06\n- v0b --supports--> v0d\n- v05 --supports--> v12\n- v09 --refutes--> v0a\n- v03 --supports--> v0f\n- v04 --supports--> v0e\n- v13 --refutes--> v10\n- v0c --supports--> v09\n- v11 --supports--> v03\n- v0f --supports--> v02\n- v09 --refutes--> v0b\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":"7c248e11034d9d075085329455b12fb515025171421f311a23d025b15b0fcec6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0f\n- v13 --supports--> v10\n- v12 --supports--> v05\n- v0e --supports--> v0c\n- v01 --supports--> v03\n- v0f --refutes--> v01\n- v06 --refutes--> v0a\n- v08 --refutes--> v0e\n- v11 --supports--> v04\n- v00 --supports--> v11\n- v03 --supports--> v07\n- v03 --refutes--> v09\n- v11 --supports--> v0b\n- v02 --refutes--> v13\n- v06 --refutes--> v00\n- v0e --supports--> v0d\n- v0b --refutes--> v12\n- v03 --supports--> v06\n- v05 --supports--> v02\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":"cdc8faf08858fd5cee4e7bbef813f5bb98a97ddf2091d34a00e901e3212bd184","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v04 --refutes--> v01\n- v09 --refutes--> v10\n- v04 --supports--> v0d\n- v08 --supports--> v02\n- v0e --supports--> v08\n- v04 --refutes--> v0b\n- v10 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0a --supports--> v13\n- v02 --refutes--> v11\n- v00 --supports--> v07\n- v03 --refutes--> v04\n- v0e --refutes--> v12\n- v05 --supports--> v00\n- v13 --supports--> v03\n- v13 --refutes--> v06\n- v0f --supports--> v05\n- v00 --supports--> v0a\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":"bf93b44714ad703be64cf0fe166d9a15300cf9e21b30c6905d4d95794653fcb8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0b\n- v0b --refutes--> v02\n- v04 --supports--> v01\n- v06 --supports--> v04\n- v0e --refutes--> v0c\n- v00 --supports--> v0f\n- v0f --supports--> v07\n- v0a --supports--> v03\n- v11 --refutes--> v06\n- v06 --refutes--> v13\n- v12 --refutes--> v11\n- v0e --refutes--> v0a\n- v04 --supports--> v00\n- v0d --supports--> v0e\n- v02 --supports--> v12\n- v00 --refutes--> v08\n- v07 --supports--> v0d\n- v0d --supports--> v05\n- v0e --refutes--> v09\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":"452c4de69c20980a96525be2c43a6159faeda18d902742eb67cf58828cab0b49","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0f\n- v13 --supports--> v10\n- v09 --refutes--> v0a\n- v0a --refutes--> v08\n- v00 --refutes--> v0e\n- v04 --supports--> v05\n- v00 --refutes--> v04\n- v07 --refutes--> v02\n- v05 --supports--> v0d\n- v11 --refutes--> v12\n- v05 --refutes--> v13\n- v11 --supports--> v0c\n- v0f --supports--> v03\n- v0b --supports--> v00\n- v0f --supports--> v06\n- v06 --refutes--> v07\n- v0a --supports--> v11\n- v06 --refutes--> v01\n- v02 --supports--> v09\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":"6b7be5decb3b02de9ed286522ba7deb525ca54374c03ee5ac637bccffa891d63","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0a\n- v02 --refutes--> v01\n- v09 --refutes--> v06\n- v01 --supports--> v07\n- v00 --supports--> v0c\n- v10 --refutes--> v11\n- v0f --refutes--> v0b\n- v06 --refutes--> v10\n- v06 --refutes--> v08\n- v07 --supports--> v09\n- v0e --refutes--> v04\n- v0e --supports--> v13\n- v11 --supports--> v0d\n- v0d --supports--> v03\n- v0b --refutes--> v12\n- v0c --refutes--> v02\n- v02 --supports--> v05\n- v0b --refutes--> v0e\n- v0e --supports--> v00\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":"7b22a3553c1bf351797e3e9933136ed0521cf955e5a446c630b6631a75421747","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v02\n- v0d --supports--> v13\n- v11 --supports--> v0c\n- v04 --refutes--> v01\n- v13 --supports--> v05\n- v09 --refutes--> v00\n- v07 --supports--> v0f\n- v0b --refutes--> v11\n- v00 --refutes--> v0e\n- v0a --supports--> v0b\n- v11 --refutes--> v10\n- v12 --supports--> v07\n- v04 --supports--> v06\n- v0f --supports--> v0d\n- v03 --supports--> v0a\n- v10 --supports--> v12\n- v04 --refutes--> v03\n- v13 --refutes--> v09\n- v00 --supports--> v08\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":"1c0124c9566c10f392acfce244298f9555d13946764e31b4b1ba4369c31c98a6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v06\n- v0d --supports--> v11\n- v0b --supports--> v0d\n- v04 --refutes--> v08\n- v08 --refutes--> v02\n- v0f --supports--> v10\n- v11 --supports--> v0e\n- v0a --refutes--> v07\n- v07 --refutes--> v00\n- v09 --refutes--> v0c\n- v0f --refutes--> v01\n- v00 --supports--> v0f\n- v08 --supports--> v03\n- v09 --refutes--> v12\n- v0f --refutes--> v09\n- v02 --supports--> v0b\n- v04 --refutes--> v05\n- v00 --supports--> v13\n- v06 --refutes--> v04\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":"79f366d83a64e3c5ef77ffde3e5092c5ef62dc72aaef4605e0dd2b0a08ff5d6b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0a\n- v02 --refutes--> v10\n- v06 --refutes--> v07\n- v03 --refutes--> v04\n- v13 --supports--> v08\n- v09 --supports--> v02\n- v08 --supports--> v0c\n- v12 --supports--> v0f\n- v10 --refutes--> v0d\n- v09 --supports--> v0e\n- v11 --supports--> v13\n- v0d --supports--> v00\n- v0a --supports--> v03\n- v08 --refutes--> v0b\n- v0d --refutes--> v01\n- v03 --supports--> v11\n- v05 --refutes--> v06\n- v01 --supports--> v05\n- v08 --supports--> v09\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":"c72bb50f4d36ab9b3e70bf44f3d5cd7aab543cf2cf339e7c3bc0570b45bcfde4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v00\n- v01 --refutes--> v0c\n- v0c --refutes--> v13\n- v10 --refutes--> v0b\n- v04 --refutes--> v12\n- v06 --refutes--> v03\n- v10 --supports--> v0e\n- v0f --supports--> v09\n- v09 --supports--> v06\n- v13 --refutes--> v10\n- v01 --supports--> v11\n- v10 --supports--> v05\n- v00 --refutes--> v02\n- v08 --supports--> v04\n- v12 --refutes--> v0d\n- v00 --refutes--> v0f\n- v05 --refutes--> v07\n- v04 --refutes--> v0a\n- v05 --supports--> v08\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0649b4a486f58f3b74245a6ef89e90b23ddc545d31ccdcb77c5b601b104ba9a1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v03\n- v01 --supports--> v05\n- v0d --refutes--> v09\n- v10 --supports--> v06\n- v08 --supports--> v12\n- v01 --supports--> v11\n- v08 --refutes--> v0d\n- v13 --supports--> v07\n- v0c --supports--> v10\n- v0a --refutes--> v13\n- v09 --refutes--> v04\n- v0f --supports--> v0b\n- v04 --supports--> v01\n- v06 --refutes--> v08\n- v0f --supports--> v0c\n- v0e --refutes--> v0f\n- v06 --supports--> v02\n- v0a --refutes--> v00\n- v07 --refutes--> v0e\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":"e56c3564ae4baca7a3e5197aa68ae7c4cde01bcbe96dba03ac4b9e5689f488ad","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v13\n- v02 --refutes--> v0c\n- v06 --supports--> v03\n- v0a --supports--> v05\n- v10 --supports--> v0d\n- v06 --refutes--> v07\n- v0c --refutes--> v06\n- v02 --supports--> v0f\n- v00 --supports--> v11\n- v06 --refutes--> v12\n- v0e --refutes--> v02\n- v11 --refutes--> v0e\n- v01 --supports--> v04\n- v09 --refutes--> v00\n- v04 --refutes--> v08\n- v0d --refutes--> v01\n- v07 --refutes--> v10\n- v0a --supports--> v09\n- v04 --refutes--> v0b\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2741b7a29f00a2a830cff03caa5629f9ea60dc0009651b6b0cef95114e991117","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0e\n- v0b --supports--> v0f\n- v11 --supports--> v08\n- v0e --refutes--> v0c\n- v07 --refutes--> v00\n- v0e --supports--> v09\n- v06 --supports--> v11\n- v10 --supports--> v05\n- v0b --refutes--> v06\n- v0c --refutes--> v01\n- v08 --refutes--> v0a\n- v07 --refutes--> v13\n- v06 --supports--> v02\n- v05 --supports--> v07\n- v0c --refutes--> v10\n- v12 --refutes--> v04\n- v00 --supports--> v0d\n- v08 --supports--> v03\n- v0a --refutes--> v12\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":"ce3c603219134965a57c3223e910dad89f088654ef373bd8dc64076671387fb5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v08\n- v12 --refutes--> v02\n- v05 --refutes--> v10\n- v06 --refutes--> v04\n- v01 --refutes--> v09\n- v0d --refutes--> v12\n- v0c --refutes--> v0d\n- v09 --supports--> v0e\n- v0f --refutes--> v11\n- v07 --refutes--> v00\n- v06 --refutes--> v0f\n- v09 --supports--> v05\n- v08 --refutes--> v13\n- v00 --refutes--> v03\n- v08 --supports--> v07\n- v0f --refutes--> v0b\n- v10 --refutes--> v0a\n- v11 --refutes--> v01\n- v10 --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db86c94fd3d95d41bebc85c0f97e1bf272cc0540ddbdf6e32cd05f2df6a52e46","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v12\n- v08 --supports--> v07\n- v09 --supports--> v00\n- v07 --supports--> v0e\n- v00 --refutes--> v01\n- v00 --supports--> v0c\n- v04 --refutes--> v02\n- v0d --supports--> v0f\n- v08 --refutes--> v11\n- v01 --refutes--> v06\n- v09 --refutes--> v10\n- v06 --refutes--> v04\n- v0e --supports--> v03\n- v02 --supports--> v0d\n- v07 --refutes--> v05\n- v0f --refutes--> v08\n- v12 --refutes--> v0a\n- v00 --supports--> v0b\n- v13 --supports--> v09\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":"ef51f57b6ebbb79094b811cbf71ff3c59eac2f1c1bac7a63ae071180e7400675","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v00\n- v03 --refutes--> v0e\n- v0f --refutes--> v0b\n- v07 --supports--> v0f\n- v0b --refutes--> v04\n- v04 --refutes--> v09\n- v06 --supports--> v11\n- v10 --supports--> v0a\n- v0b --refutes--> v05\n- v01 --refutes--> v13\n- v02 --supports--> v03\n- v11 --refutes--> v07\n- v13 --supports--> v02\n- v11 --supports--> v08\n- v03 --refutes--> v06\n- v01 --refutes--> v12\n- v0a --supports--> v0d\n- v09 --supports--> v10\n- v0f --supports--> v0c\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":"abfbf191a2b3258e3d7a410f305493091567c240bb13324032295debd0ae3973","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v04\n- v12 --refutes--> v10\n- v0b --supports--> v0d\n- v0d --refutes--> v02\n- v08 --refutes--> v12\n- v13 --refutes--> v05\n- v10 --refutes--> v0e\n- v09 --refutes--> v11\n- v04 --supports--> v07\n- v10 --supports--> v0b\n- v0f --supports--> v01\n- v02 --supports--> v06\n- v10 --supports--> v0a\n- v06 --supports--> v09\n- v01 --refutes--> v00\n- v01 --refutes--> v08\n- v10 --supports--> v03\n- v13 --supports--> v0f\n- v09 --supports--> v0c\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":"bca90e2418ce2b2efa3e4b10fd9eef896516a0b6499a764e7c33c6c8bc468e01","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v07\n- v01 --refutes--> v11\n- v02 --supports--> v09\n- v08 --supports--> v06\n- v0f --refutes--> v04\n- v13 --refutes--> v10\n- v0f --refutes--> v12\n- v06 --refutes--> v0f\n- v08 --supports--> v0e\n- v11 --supports--> v02\n- v0c --supports--> v05\n- v0d --supports--> v0c\n- v05 --refutes--> v08\n- v03 --supports--> v0b\n- v02 --supports--> v13\n- v03 --refutes--> v01\n- v10 --supports--> v0d\n- v05 --refutes--> v00\n- v04 --refutes--> v0a\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aed1e4e542ede8294e4f0f8bb7273caca4364a8500a2e0bf7b3030c4598f20b6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v02\n- v13 --supports--> v00\n- v0d --refutes--> v04\n- v0c --refutes--> v0e\n- v11 --refutes--> v13\n- v0f --supports--> v0a\n- v0a --supports--> v12\n- v00 --refutes--> v06\n- v01 --supports--> v03\n- v05 --supports--> v0f\n- v01 --refutes--> v0c\n- v0c --supports--> v0b\n- v13 --supports--> v07\n- v04 --refutes--> v09\n- v02 --refutes--> v01\n- v0a --refutes--> v10\n- v0c --refutes--> v05\n- v12 --refutes--> v11\n- v11 --refutes--> v08\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dbbf91172650c6592323b68ae50dc9bd1d4542daf2f3d573ddf7182b175f67ff","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0d\n- v12 --refutes--> v0a\n- v10 --supports--> v13\n- v04 --refutes--> v0e\n- v0c --supports--> v08\n- v02 --refutes--> v03\n- v10 --refutes--> v05\n- v00 --supports--> v10\n- v10 --supports--> v07\n- v12 --refutes--> v0f\n- v06 --refutes--> v00\n- v07 --refutes--> v02\n- v0e --supports--> v09\n- v01 --supports--> v0c\n- v10 --refutes--> v0b\n- v0a --refutes--> v06\n- v09 --supports--> v12\n- v12 --supports--> v11\n- v08 --refutes--> v04\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"89bf2dec556c14ad6d3ec6fac6f3ed025d02cab134117f1a34cb2990ada438a5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v11\n- v0e --refutes--> v06\n- v09 --refutes--> v0a\n- v03 --refutes--> v0d\n- v07 --refutes--> v05\n- v03 --refutes--> v00\n- v0b --refutes--> v04\n- v0c --supports--> v09\n- v04 --supports--> v0f\n- v12 --refutes--> v13\n- v0d --refutes--> v07\n- v06 --supports--> v0c\n- v07 --supports--> v10\n- v0a --refutes--> v03\n- v05 --refutes--> v0b\n- v0d --refutes--> v08\n- v0f --refutes--> v12\n- v0c --supports--> v02\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":"5a7308a145db5d9b593e49a5448f7c18620efd00479c8b68a5faed78b2067c72","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0e\n- v0b --supports--> v0c\n- v0b --refutes--> v0a\n- v09 --refutes--> v03\n- v05 --refutes--> v12\n- v0e --refutes--> v01\n- v05 --supports--> v08\n- v11 --supports--> v13\n- v11 --refutes--> v10\n- v01 --refutes--> v04\n- v00 --refutes--> v0f\n- v03 --refutes--> v06\n- v02 --refutes--> v0d\n- v12 --refutes--> v07\n- v0a --supports--> v09\n- v12 --supports--> v11\n- v13 --supports--> v02\n- v00 --supports--> v0b\n- v03 --supports--> v05\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5c346971f7230fabc81b23ec6b58d01dc66b56e1d12e722d69da9044b5b6b6c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v08\n- v07 --refutes--> v0a\n- v0b --refutes--> v12\n- v0f --supports--> v07\n- v03 --refutes--> v0e\n- v01 --supports--> v0b\n- v07 --refutes--> v0d\n- v13 --supports--> v11\n- v11 --refutes--> v06\n- v0b --refutes--> v03\n- v09 --supports--> v13\n- v13 --refutes--> v0c\n- v06 --refutes--> v04\n- v01 --refutes--> v10\n- v08 --refutes--> v05\n- v05 --supports--> v02\n- v13 --refutes--> v00\n- v05 --supports--> v09\n- v04 --refutes--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"23404535a0f3b0dea90ae8a47159aceaf1e0b69a2e2f61f18915d37c96520e40","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v04\n- v0f --supports--> v05\n- v0f --supports--> v01\n- v09 --refutes--> v02\n- v11 --supports--> v07\n- v10 --supports--> v09\n- v04 --supports--> v13\n- v13 --supports--> v06\n- v0d --supports--> v0e\n- v07 --refutes--> v03\n- v07 --supports--> v10\n- v02 --refutes--> v0b\n- v02 --refutes--> v0d\n- v06 --supports--> v0a\n- v12 --supports--> v0f\n- v04 --refutes--> v0c\n- v0a --refutes--> v12\n- v11 --refutes--> v00\n- v0a --refutes--> v08\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":"6635d9c397863067446d2027a5bf893b4b9340530df121bc316546680ff50daa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v0c\n- v06 --supports--> v0b\n- v05 --supports--> v0f\n- v0e --supports--> v06\n- v05 --refutes--> v13\n- v0a --refutes--> v08\n- v04 --supports--> v0a\n- v07 --supports--> v11\n- v06 --refutes--> v09\n- v09 --refutes--> v12\n- v0a --supports--> v01\n- v00 --supports--> v10\n- v03 --supports--> v00\n- v09 --refutes--> v02\n- v02 --supports--> v04\n- v08 --supports--> v05\n- v10 --refutes--> v0e\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":"bf3d5caba58c97b59cafc4d6772ca364b716df4bef9ceaa1b75a164b69721aca","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v02\n- v0d --refutes--> v0f\n- v05 --supports--> v12\n- v0c --refutes--> v00\n- v07 --supports--> v03\n- v0d --refutes--> v06\n- v10 --supports--> v11\n- v0a --refutes--> v13\n- v00 --supports--> v10\n- v01 --refutes--> v0d\n- v02 --supports--> v0a\n- v00 --supports--> v08\n- v05 --refutes--> v09\n- v12 --refutes--> v0c\n- v0f --refutes--> v07\n- v07 --supports--> v0b\n- v01 --supports--> v0e\n- v01 --refutes--> v04\n- v0b --refutes--> v05\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":"d9b5936c7e40abe170a87c1f2d592ad976e2c1cd9bd56105ab5fc0395aea2d07","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v0e\n- v01 --refutes--> v07\n- v01 --supports--> v02\n- v0d --refutes--> v01\n- v12 --refutes--> v10\n- v0c --refutes--> v09\n- v10 --supports--> v0c\n- v06 --refutes--> v0f\n- v02 --refutes--> v04\n- v0a --refutes--> v06\n- v02 --supports--> v0a\n- v09 --refutes--> v11\n- v0f --supports--> v12\n- v00 --supports--> v0d\n- v11 --refutes--> v0b\n- v0a --refutes--> v03\n- v02 --supports--> v08\n- v09 --supports--> v05\n- v09 --supports--> v13\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":"36c628be00b2a86712bf7d859106bd0312a58b6eb670e057a37417873e3b8159","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v08\n- v10 --refutes--> v03\n- v11 --supports--> v00\n- v0b --supports--> v07\n- v03 --supports--> v12\n- v09 --supports--> v0a\n- v00 --refutes--> v02\n- v09 --refutes--> v05\n- v11 --supports--> v0c\n- v03 --refutes--> v01\n- v0d --refutes--> v04\n- v06 --refutes--> v13\n- v04 --supports--> v0e\n- v12 --refutes--> v0d\n- v0a --refutes--> v06\n- v04 --refutes--> v09\n- v03 --supports--> v0f\n- v07 --refutes--> v11\n- v02 --supports--> v10\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":"8b3c6b9f765b1c493048094decbc93dd0fdad2d63b8488bbaf6408961e30aaa4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v08 --refutes--> v0b\n- v0d --supports--> v0c\n- v0e --supports--> v08\n- v0e --refutes--> v00\n- v0d --refutes--> v02\n- v06 --supports--> v0f\n- v0b --supports--> v03\n- v06 --refutes--> v0e\n- v0c --refutes--> v13\n- v06 --supports--> v05\n- v0e --refutes--> v01\n- v12 --supports--> v11\n- v04 --refutes--> v06\n- v03 --supports--> v10\n- v10 --refutes--> v0d\n- v09 --refutes--> v04\n- v07 --refutes--> v12\n- v0a --supports--> v07\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":"9c50ee1f4936ead6e98956ff71463418ddaa0a27f826104aa7429e1404089373","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v0f\n- v09 --supports--> v01\n- v05 --supports--> v02\n- v07 --supports--> v05\n- v0f --supports--> v0b\n- v05 --refutes--> v06\n- v11 --refutes--> v10\n- v08 --supports--> v12\n- v0b --refutes--> v0d\n- v0e --supports--> v13\n- v00 --refutes--> v0a\n- v02 --supports--> v0c\n- v00 --refutes--> v11\n- v01 --refutes--> v07\n- v0b --refutes--> v0e\n- v11 --supports--> v03\n- v02 --refutes--> v08\n- v10 --refutes--> v04\n- v13 --supports--> v09\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":"d57487d732a537ff24ef73aa08857b3a86c0c9317ce32d37a49572fadf1ab2db","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v03\n- v05 --refutes--> v07\n- v03 --refutes--> v09\n- v10 --refutes--> v0b\n- v0c --refutes--> v10\n- v12 --supports--> v0a\n- v0d --supports--> v13\n- v0f --refutes--> v01\n- v0f --supports--> v0c\n- v05 --supports--> v04\n- v11 --supports--> v02\n- v03 --refutes--> v00\n- v0a --supports--> v0d\n- v03 --supports--> v06\n- v0b --refutes--> v05\n- v02 --supports--> v08\n- v04 --refutes--> v12\n- v0d --supports--> v0e\n- v0d --refutes--> v11\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":"e13da21adac62508027c9400458f94be53f26893ff50b6561bd0a668230048dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v05\n- v07 --supports--> v12\n- v06 --supports--> v07\n- v0a --refutes--> v09\n- v0f --supports--> v03\n- v02 --refutes--> v0e\n- v11 --supports--> v0f\n- v06 --refutes--> v08\n- v0e --refutes--> v0d\n- v05 --refutes--> v0c\n- v0c --supports--> v04\n- v11 --refutes--> v0b\n- v02 --supports--> v00\n- v12 --refutes--> v0a\n- v0d --refutes--> v06\n- v03 --supports--> v02\n- v12 --supports--> v10\n- v0e --refutes--> v13\n- v06 --supports--> v01\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":"3cd6dc9d449fa208cd554694d7c1b50ad48667f6718aadb9aee2126638f45c9e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v10\n- v06 --refutes--> v0b\n- v0d --supports--> v0e\n- v11 --refutes--> v0a\n- v10 --supports--> v0c\n- v0f --refutes--> v12\n- v13 --supports--> v08\n- v0a --supports--> v03\n- v13 --refutes--> v0d\n- v12 --supports--> v06\n- v06 --supports--> v13\n- v10 --refutes--> v11\n- v02 --supports--> v0f\n- v0f --refutes--> v04\n- v0e --refutes--> v05\n- v02 --refutes--> v00\n- v01 --refutes--> v09\n- v07 --refutes--> v01\n- v0a --supports--> v02\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":"33187c3d57a9271fe8ff72ef630ee865c90f402c093ca6be6dea512f42579eaa","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0e\n- v12 --refutes--> v11\n- v00 --refutes--> v12\n- v05 --supports--> v04\n- v13 --refutes--> v0b\n- v05 --supports--> v09\n- v0e --supports--> v01\n- v0a --refutes--> v0d\n- v06 --refutes--> v00\n- v07 --supports--> v13\n- v0d --refutes--> v05\n- v03 --refutes--> v0c\n- v0f --supports--> v0a\n- v03 --supports--> v06\n- v13 --refutes--> v02\n- v01 --supports--> v03\n- v13 --refutes--> v0f\n- v0e --refutes--> v08\n- v0e --supports--> v10\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":"104f47e6c665c8e84e268e5c0e98484ba0de54acf9b3e1f3d24de2c9e2183de3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0b\n- v05 --supports--> v06\n- v0f --refutes--> v13\n- v09 --supports--> v05\n- v10 --refutes--> v01\n- v10 --refutes--> v0a\n- v05 --refutes--> v0c\n- v0d --supports--> v00\n- v0e --refutes--> v09\n- v00 --supports--> v02\n- v11 --refutes--> v12\n- v02 --refutes--> v11\n- v07 --supports--> v0d\n- v10 --supports--> v07\n- v0f --refutes--> v08\n- v09 --refutes--> v04\n- v0c --refutes--> v0f\n- v07 --refutes--> v03\n- v12 --refutes--> v0e\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":"cc19faa1d57355f5c6a62b7c2e05fd5828f781b0f3ebd6642b8654d4423ce88e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v08\n- v13 --supports--> v10\n- v0b --refutes--> v05\n- v09 --refutes--> v02\n- v10 --supports--> v07\n- v06 --supports--> v0a\n- v08 --refutes--> v12\n- v0b --refutes--> v11\n- v08 --supports--> v03\n- v01 --refutes--> v0f\n- v12 --supports--> v0c\n- v09 --supports--> v13\n- v11 --refutes--> v09\n- v0d --refutes--> v04\n- v09 --refutes--> v0e\n- v0c --refutes--> v0b\n- v0a --refutes--> v01\n- v01 --supports--> v0d\n- v0b --refutes--> v00\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36e6171bd5c2a8c619246bc4547462fe13a50ae92c373972289815868860bf6e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v04 --supports--> v0a\n- v09 --refutes--> v06\n- v00 --supports--> v0d\n- v06 --refutes--> v00\n- v05 --refutes--> v09\n- v0c --refutes--> v10\n- v03 --refutes--> v11\n- v01 --supports--> v12\n- v05 --supports--> v0b\n- v05 --refutes--> v0f\n- v0a --supports--> v02\n- v11 --refutes--> v04\n- v0d --supports--> v0e\n- v00 --supports--> v07\n- v0e --supports--> v01\n- v12 --supports--> v0c\n- v11 --refutes--> v13\n- v0a --refutes--> v05\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":"bddcbe5ebbac31a444dfde07415178b93abf716bbf008db50ceafb43e866aa80","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0f\n- v05 --supports--> v0e\n- v13 --supports--> v0c\n- v13 --refutes--> v03\n- v08 --supports--> v12\n- v04 --supports--> v00\n- v04 --supports--> v09\n- v08 --refutes--> v07\n- v02 --supports--> v0b\n- v0f --refutes--> v11\n- v0a --supports--> v01\n- v0d --supports--> v0a\n- v11 --supports--> v04\n- v09 --supports--> v05\n- v0c --refutes--> v08\n- v0f --supports--> v10\n- v06 --supports--> v02\n- v05 --refutes--> v0d\n- v01 --supports--> v13\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":"62ada3309f7998174143b9fe5e2420e1f2ed7f477fc9c5153ab26a44e9923a22","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v09\n- v01 --supports--> v0b\n- v0c --refutes--> v13\n- v05 --supports--> v0e\n- v08 --supports--> v01\n- v0d --refutes--> v06\n- v02 --supports--> v10\n- v0e --supports--> v11\n- v0f --supports--> v02\n- v10 --refutes--> v08\n- v0a --refutes--> v0d\n- v0a --supports--> v03\n- v13 --supports--> v12\n- v08 --supports--> v07\n- v0d --refutes--> v00\n- v12 --refutes--> v0a\n- v01 --refutes--> v05\n- v0a --supports--> v04\n- v00 --refutes--> v0f\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":"858db6cfe5363e48915947e3481685cffb69a923c711af390f9653ef2e2a85e4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0b\n- v07 --refutes--> v10\n- v09 --supports--> v13\n- v06 --refutes--> v12\n- v0d --supports--> v07\n- v0b --supports--> v08\n- v04 --supports--> v01\n- v08 --refutes--> v06\n- v0e --supports--> v04\n- v07 --refutes--> v02\n- v01 --refutes--> v03\n- v05 --supports--> v0f\n- v12 --refutes--> v11\n- v0a --supports--> v0d\n- v0f --supports--> v09\n- v07 --supports--> v05\n- v12 --supports--> v0e\n- v0f --refutes--> v0c\n- v01 --supports--> v00\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3f23dab7b67fe4c5d9813dcbd3628bb280686525d39709aa5c95d46a6f385163","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v11\n- v08 --supports--> v12\n- v06 --refutes--> v10\n- v07 --supports--> v08\n- v0b --refutes--> v0d\n- v08 --supports--> v0f\n- v0f --refutes--> v05\n- v06 --supports--> v0a\n- v13 --refutes--> v01\n- v05 --refutes--> v04\n- v13 --supports--> v02\n- v0d --supports--> v07\n- v11 --refutes--> v0e\n- v02 --supports--> v00\n- v00 --supports--> v03\n- v0a --refutes--> v0b\n- v08 --refutes--> v0c\n- v03 --supports--> v06\n- v06 --refutes--> v09\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":"e8b4230549f86c7310eb70da9a54b0909df782187f25715d8c2961d58a64b452","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v11\n- v03 --refutes--> v10\n- v03 --refutes--> v04\n- v0d --supports--> v0e\n- v10 --refutes--> v12\n- v11 --supports--> v0d\n- v02 --supports--> v00\n- v08 --refutes--> v09\n- v07 --refutes--> v08\n- v01 --supports--> v0f\n- v08 --refutes--> v0a\n- v0f --refutes--> v13\n- v0c --refutes--> v01\n- v09 --refutes--> v0c\n- v02 --supports--> v06\n- v08 --supports--> v0b\n- v0e --refutes--> v03\n- v0f --supports--> v02\n- v05 --refutes--> v07\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":"407b9c535b21ccd8432cd34a94b09b3e6616b8ed12506aa1f1509d2c7d0d72db","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v08\n- v11 --refutes--> v04\n- v02 --refutes--> v11\n- v13 --supports--> v0d\n- v0f --supports--> v0e\n- v0f --supports--> v0b\n- v04 --refutes--> v03\n- v02 --supports--> v05\n- v0c --supports--> v0f\n- v10 --refutes--> v02\n- v03 --refutes--> v00\n- v00 --refutes--> v12\n- v10 --refutes--> v0a\n- v06 --refutes--> v07\n- v00 --refutes--> v13\n- v07 --supports--> v01\n- v13 --refutes--> v09\n- v0e --refutes--> v06\n- v01 --refutes--> v10\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":"fc28a5f6be66146d60844e2da5f1dfc8626da696aa14cdb6ddd94204ff278247","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v06\n- v07 --refutes--> v12\n- v0b --supports--> v10\n- v0a --refutes--> v0e\n- v07 --refutes--> v03\n- v06 --supports--> v00\n- v0b --supports--> v08\n- v11 --refutes--> v0b\n- v05 --refutes--> v07\n- v11 --refutes--> v04\n- v13 --refutes--> v0d\n- v0e --refutes--> v11\n- v08 --refutes--> v02\n- v0c --refutes--> v13\n- v0e --refutes--> v01\n- v00 --refutes--> v0c\n- v02 --refutes--> v05\n- v13 --refutes--> v0a\n- v06 --refutes--> v09\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":"91d037bf5bb1b46a4ee7a8643ac172dfcbd6819d240296845679cc80e1cae8af","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0a\n- v04 --refutes--> v0d\n- v08 --supports--> v09\n- v00 --supports--> v02\n- v13 --refutes--> v00\n- v12 --supports--> v10\n- v02 --refutes--> v0c\n- v0c --supports--> v06\n- v07 --supports--> v0f\n- v0e --supports--> v07\n- v0b --supports--> v12\n- v00 --refutes--> v11\n- v0f --supports--> v13\n- v0c --refutes--> v08\n- v01 --supports--> v0b\n- v0a --supports--> v01\n- v13 --supports--> v03\n- v08 --supports--> v05\n- v12 --refutes--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78bbce181860f3a265f24884cfd7e2a66aa5c68f6e23ebec5b1c4eab60641bd9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v10\n- v11 --refutes--> v01\n- v07 --refutes--> v11\n- v10 --supports--> v0a\n- v10 --supports--> v0e\n- v12 --supports--> v0c\n- v04 --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v0f\n- v03 --supports--> v13\n- v02 --refutes--> v00\n- v08 --refutes--> v02\n- v09 --refutes--> v12\n- v0c --refutes--> v0d\n- v10 --refutes--> v08\n- v01 --refutes--> v05\n- v05 --refutes--> v06\n- v02 --refutes--> v04\n- v05 --supports--> v0b\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":"35aa60c819189900927e2a9b23e8dc71e70050aecec5b421af3fa5fcb93a8149","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v01\n- v07 --supports--> v03\n- v04 --refutes--> v10\n- v0a --supports--> v0b\n- v10 --supports--> v08\n- v0d --supports--> v0e\n- v08 --refutes--> v09\n- v12 --refutes--> v07\n- v05 --refutes--> v04\n- v07 --supports--> v00\n- v0b --refutes--> v0c\n- v0f --refutes--> v12\n- v0c --supports--> v0d\n- v07 --refutes--> v06\n- v01 --refutes--> v0a\n- v0b --refutes--> v02\n- v01 --supports--> v13\n- v04 --refutes--> v11\n- v08 --refutes--> v0f\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":"e68ccca196b5a7665cd3a5c679454455f4d9f7398194eacf7cbe49efc4403bfb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0e\n- v04 --refutes--> v0b\n- v0f --supports--> v04\n- v0f --supports--> v08\n- v0d --supports--> v11\n- v0c --supports--> v00\n- v0e --supports--> v10\n- v0f --supports--> v07\n- v13 --refutes--> v0f\n- v00 --refutes--> v01\n- v01 --refutes--> v12\n- v00 --supports--> v03\n- v10 --supports--> v0a\n- v02 --supports--> v13\n- v01 --refutes--> v09\n- v06 --supports--> v02\n- v0b --supports--> v05\n- v10 --refutes--> v0c\n- v09 --refutes--> v06\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":"9d4dbf4cf95a6220a006f89a058a5e4af2a41e7a03f11a1247af1b6e4fe5e905","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v10\n- v03 --refutes--> v08\n- v11 --refutes--> v0f\n- v11 --supports--> v0d\n- v12 --supports--> v00\n- v0b --refutes--> v13\n- v02 --supports--> v05\n- v04 --refutes--> v0e\n- v0a --supports--> v12\n- v0e --refutes--> v11\n- v02 --refutes--> v0b\n- v12 --supports--> v06\n- v04 --supports--> v01\n- v0e --refutes--> v09\n- v06 --supports--> v07\n- v10 --refutes--> v04\n- v0d --refutes--> v0c\n- v0c --refutes--> v03\n- v06 --supports--> v02\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":"d4c70e3821806c8ec4c5b99f7a9ec7eeb3d2bc1c026c65863903b6b76035d31d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v05\n- v10 --supports--> v03\n- v0f --supports--> v0d\n- v03 --refutes--> v0f\n- v02 --supports--> v0a\n- v06 --refutes--> v13\n- v13 --refutes--> v11\n- v0d --refutes--> v09\n- v07 --supports--> v08\n- v10 --refutes--> v0c\n- v0d --supports--> v00\n- v00 --supports--> v0b\n- v02 --supports--> v01\n- v06 --refutes--> v0e\n- v13 --refutes--> v04\n- v07 --supports--> v06\n- v05 --refutes--> v07\n- v11 --supports--> v02\n- v12 --refutes--> v10\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":"0f8872ffcca48485426b0f3c3ba0aa7874ac3c841f68bc6c897dec7e0f586ea6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v11\n- v04 --supports--> v06\n- v05 --supports--> v0d\n- v0a --refutes--> v08\n- v0c --supports--> v04\n- v02 --refutes--> v0b\n- v09 --supports--> v03\n- v06 --supports--> v09\n- v10 --supports--> v02\n- v00 --refutes--> v10\n- v01 --supports--> v0a\n- v07 --refutes--> v05\n- v12 --supports--> v0c\n- v07 --refutes--> v0e\n- v0d --supports--> v0f\n- v0a --supports--> v00\n- v04 --supports--> v13\n- v09 --supports--> v01\n- v02 --refutes--> v07\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":"9ffeecc2c5d43821a2fc5dfb266e46fc4899c07423cfd70a6cbc64de343c7d5c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v09\n- v0e --supports--> v0a\n- v0f --refutes--> v07\n- v05 --refutes--> v00\n- v0b --supports--> v03\n- v0c --supports--> v13\n- v0f --refutes--> v01\n- v12 --refutes--> v0c\n- v04 --refutes--> v10\n- v13 --refutes--> v11\n- v13 --refutes--> v05\n- v02 --supports--> v0f\n- v04 --supports--> v0b\n- v07 --refutes--> v08\n- v02 --supports--> v0d\n- v07 --supports--> v06\n- v08 --supports--> v0e\n- v03 --supports--> v12\n- v00 --refutes--> v02\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":"ab264c36a43ad9a3d3e2585865d09a63edeec38eb5a88a90061589961858eea6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v12\n- v0e --refutes--> v00\n- v11 --supports--> v09\n- v05 --refutes--> v10\n- v0e --refutes--> v04\n- v13 --refutes--> v05\n- v0e --refutes--> v02\n- v07 --refutes--> v0f\n- v13 --supports--> v01\n- v10 --supports--> v0d\n- v07 --supports--> v08\n- v0c --refutes--> v0b\n- v0a --refutes--> v13\n- v11 --supports--> v03\n- v02 --supports--> v0c\n- v12 --supports--> v07\n- v09 --refutes--> v0e\n- v08 --supports--> v06\n- v08 --supports--> v0a\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":"6dc6b50e4a567fee4883403117c78c3d1b364cec6fd3daed4795a11aa532fa84","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v00\n- v09 --refutes--> v12\n- v0b --supports--> v0e\n- v00 --supports--> v0a\n- v0b --refutes--> v10\n- v0a --refutes--> v05\n- v0a --refutes--> v03\n- v00 --supports--> v0f\n- v0c --supports--> v11\n- v00 --refutes--> v0d\n- v11 --supports--> v13\n- v09 --refutes--> v0c\n- v0e --supports--> v01\n- v06 --refutes--> v09\n- v13 --refutes--> v08\n- v09 --refutes--> v02\n- v04 --refutes--> v07\n- v08 --refutes--> v0b\n- v05 --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"844159af33c2fef308028a9b245f578c36362e7cafcc066af2d5f42e91e44657","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0e\n- v12 --refutes--> v06\n- v01 --supports--> v0c\n- v12 --refutes--> v08\n- v07 --supports--> v0b\n- v01 --supports--> v0d\n- v08 --refutes--> v10\n- v11 --refutes--> v07\n- v00 --refutes--> v0a\n- v02 --refutes--> v11\n- v13 --supports--> v02\n- v0c --refutes--> v0f\n- v0b --supports--> v00\n- v09 --refutes--> v04\n- v00 --refutes--> v01\n- v0f --refutes--> v03\n- v09 --refutes--> v12\n- v0c --refutes--> v05\n- v10 --supports--> v13\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":"13a8edc81ad8c5a13da41db84b9aebdc83e2bb16aa56c4baa174f109537aef7f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v09\n- v0d --refutes--> v07\n- v11 --supports--> v00\n- v00 --supports--> v05\n- v0f --refutes--> v06\n- v09 --supports--> v0e\n- v06 --supports--> v10\n- v06 --refutes--> v0a\n- v06 --refutes--> v12\n- v11 --supports--> v08\n- v09 --refutes--> v0b\n- v01 --supports--> v13\n- v0b --refutes--> v02\n- v0d --supports--> v01\n- v0c --refutes--> v0f\n- v0b --supports--> v03\n- v02 --refutes--> v04\n- v04 --supports--> v0c\n- v12 --refutes--> v11\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":"dc2b6212b77b3322f6c9df62492a082e4e0fe8bb930c1808597437ac07830c38","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v11\n- v02 --refutes--> v0d\n- v0f --refutes--> v0c\n- v09 --supports--> v0e\n- v0a --supports--> v00\n- v13 --refutes--> v05\n- v0e --supports--> v0a\n- v04 --refutes--> v08\n- v0c --refutes--> v06\n- v12 --refutes--> v0b\n- v07 --supports--> v13\n- v0f --supports--> v01\n- v0d --refutes--> v04\n- v0c --refutes--> v03\n- v0b --supports--> v0f\n- v05 --refutes--> v12\n- v0f --refutes--> v10\n- v0a --supports--> v07\n- v04 --supports--> v09\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":"84c86ab85bbded66e020ad7b1b4ffd62dc26edd9f9e6044b91f2a7745dd19ddb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v0f\n- v00 --refutes--> v03\n- v13 --refutes--> v04\n- v05 --refutes--> v0b\n- v0e --refutes--> v12\n- v0e --supports--> v11\n- v0a --refutes--> v01\n- v02 --supports--> v10\n- v0a --refutes--> v06\n- v10 --refutes--> v07\n- v07 --supports--> v0c\n- v05 --supports--> v0a\n- v09 --refutes--> v08\n- v08 --supports--> v0d\n- v08 --refutes--> v0e\n- v11 --refutes--> v02\n- v03 --supports--> v13\n- v06 --supports--> v00\n- v07 --supports--> v05\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":"a6a052c3a71e04ed5851beeab1af17cc565ccfd7b4c4ebb45eed040eaccebe6f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v01\n- v11 --supports--> v03\n- v13 --supports--> v08\n- v0f --supports--> v0a\n- v10 --refutes--> v0b\n- v10 --supports--> v00\n- v09 --refutes--> v0c\n- v00 --supports--> v11\n- v02 --supports--> v0f\n- v01 --refutes--> v06\n- v10 --supports--> v12\n- v03 --supports--> v05\n- v0c --refutes--> v13\n- v0a --refutes--> v10\n- v0e --supports--> v09\n- v05 --supports--> v0e\n- v11 --supports--> v07\n- v0f --refutes--> v0d\n- v0e --supports--> v04\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a7fa0cee802477ff100a0dbd43068998a21dc5ede0c685d4fc86043581d3ab53","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0e\n- v0c --refutes--> v04\n- v04 --refutes--> v0a\n- v09 --refutes--> v11\n- v0a --refutes--> v0d\n- v11 --refutes--> v05\n- v00 --refutes--> v10\n- v07 --refutes--> v0b\n- v0f --supports--> v00\n- v01 --refutes--> v03\n- v01 --refutes--> v09\n- v06 --refutes--> v12\n- v11 --refutes--> v0c\n- v10 --supports--> v02\n- v04 --supports--> v13\n- v0d --refutes--> v06\n- v12 --refutes--> v07\n- v06 --refutes--> v08\n- v07 --supports--> v0f\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"41200a4f263870983d80814133feb4431a04b94e3488957ad5e10ad03b91d4ec","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v01 --supports--> v0f\n- v0c --supports--> v11\n- v05 --refutes--> v0e\n- v00 --supports--> v07\n- v0e --refutes--> v0c\n- v10 --supports--> v00\n- v04 --refutes--> v03\n- v0b --supports--> v10\n- v11 --supports--> v0a\n- v07 --refutes--> v12\n- v11 --supports--> v06\n- v0e --refutes--> v13\n- v09 --refutes--> v04\n- v10 --supports--> v0d\n- v03 --refutes--> v01\n- v07 --refutes--> v05\n- v03 --refutes--> v02\n- v09 --supports--> v08\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":"2b5034de9291aa8c513b70e5297446681ae7f85fea946b3dae6732fb7e7ad953","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0c\n- v11 --refutes--> v10\n- v0e --supports--> v11\n- v13 --refutes--> v0f\n- v13 --supports--> v09\n- v10 --refutes--> v12\n- v04 --supports--> v0b\n- v13 --supports--> v00\n- v0c --refutes--> v0d\n- v02 --supports--> v08\n- v0a --refutes--> v05\n- v0d --refutes--> v0a\n- v06 --refutes--> v13\n- v08 --supports--> v0e\n- v0c --refutes--> v01\n- v12 --supports--> v06\n- v04 --refutes--> v02\n- v0a --refutes--> v04\n- v12 --supports--> v07\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eb06612865b38d9eb42868652906026e4208f8669fa69d97e04af20e3a68d38d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0d\n- v07 --refutes--> v0a\n- v06 --refutes--> v04\n- v01 --refutes--> v0f\n- v02 --supports--> v11\n- v03 --refutes--> v05\n- v11 --refutes--> v0c\n- v0d --supports--> v10\n- v10 --supports--> v0b\n- v06 --supports--> v0e\n- v11 --supports--> v00\n- v10 --supports--> v01\n- v09 --supports--> v13\n- v09 --refutes--> v02\n- v05 --refutes--> v07\n- v0f --refutes--> v12\n- v08 --refutes--> v09\n- v0f --refutes--> v03\n- v07 --supports--> v06\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":"a16a23850722a681f0981bead1119c04dd98b7abc35c5482b54ba00531bc28f8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v08\n- v12 --refutes--> v0b\n- v14 --supports--> v11\n- v09 --refutes--> v13\n- v0a --supports--> v01\n- v0c --refutes--> v0d\n- v07 --refutes--> v04\n- v09 --supports--> v07\n- v0a --refutes--> v09\n- v15 --refutes--> v14\n- v0b --supports--> v00\n- v06 --refutes--> v15\n- v04 --refutes--> v06\n- v0d --refutes--> v0e\n- v00 --refutes--> v10\n- v06 --supports--> v03\n- v00 --refutes--> v0c\n- v0b --supports--> v02\n- v11 --supports--> v12\n- v0f --refutes--> v0a\n- v0d --supports--> v05\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":"265aaf73e56495008133846ca4e076c3b243ff75e3abeeecf36c8680e0512d4e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v02\n- v00 --refutes--> v11\n- v04 --supports--> v06\n- v07 --refutes--> v0c\n- v02 --refutes--> v00\n- v13 --supports--> v07\n- v0d --supports--> v05\n- v14 --supports--> v12\n- v03 --supports--> v14\n- v11 --supports--> v0b\n- v02 --refutes--> v0e\n- v09 --refutes--> v0d\n- v08 --supports--> v10\n- v0a --supports--> v08\n- v14 --supports--> v09\n- v08 --refutes--> v03\n- v06 --refutes--> v13\n- v08 --supports--> v01\n- v0d --refutes--> v04\n- v03 --supports--> v0f\n- v0b --refutes--> v15\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":"d31888d2723875d7f29ad283ba375a7eea3988230edf4312c45a430c4195f6a6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v12\n- v01 --supports--> v02\n- v0e --supports--> v00\n- v0b --supports--> v01\n- v03 --supports--> v0a\n- v01 --refutes--> v08\n- v07 --supports--> v03\n- v00 --supports--> v14\n- v04 --refutes--> v06\n- v0d --refutes--> v0c\n- v09 --supports--> v04\n- v04 --refutes--> v0d\n- v0f --refutes--> v15\n- v0f --supports--> v0e\n- v03 --refutes--> v09\n- v01 --supports--> v11\n- v05 --supports--> v13\n- v10 --refutes--> v07\n- v00 --supports--> v10\n- v0c --supports--> v0b\n- v13 --refutes--> v0f\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":"08dfaea1fb7fb80bfbcbe30824f09ffd59312e1edea35cdceb968a3a4577a0e1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v08\n- v0f --refutes--> v12\n- v08 --refutes--> v07\n- v10 --refutes--> v0f\n- v01 --supports--> v00\n- v0f --supports--> v03\n- v11 --refutes--> v0d\n- v0e --supports--> v14\n- v06 --refutes--> v0b\n- v13 --refutes--> v0e\n- v0e --supports--> v05\n- v08 --supports--> v01\n- v05 --supports--> v10\n- v15 --refutes--> v11\n- v10 --refutes--> v0c\n- v09 --refutes--> v15\n- v0b --refutes--> v13\n- v12 --refutes--> v04\n- v01 --refutes--> v06\n- v01 --refutes--> v02\n- v0a --supports--> v09\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":"61c39a4931519b882f99c77ec93dd97effd43a9a72f0c56290272f79dfd7eb37","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v03\n- v08 --refutes--> v06\n- v05 --supports--> v09\n- v00 --supports--> v0f\n- v0d --supports--> v08\n- v0f --supports--> v0a\n- v08 --refutes--> v0b\n- v12 --refutes--> v07\n- v13 --supports--> v0c\n- v09 --supports--> v0e\n- v15 --refutes--> v14\n- v0e --supports--> v11\n- v00 --supports--> v04\n- v12 --supports--> v10\n- v10 --supports--> v05\n- v09 --supports--> v01\n- v08 --supports--> v13\n- v0c --refutes--> v15\n- v0a --refutes--> v12\n- v11 --supports--> v0d\n- v0c --supports--> v02\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b516b4785c736a8379ec36dcb90560caba9b4a01bdd6cec1349705185c1a020","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0b\n- v09 --refutes--> v04\n- v13 --supports--> v08\n- v05 --supports--> v06\n- v09 --refutes--> v07\n- v02 --supports--> v0f\n- v0d --refutes--> v02\n- v07 --supports--> v0a\n- v0e --supports--> v12\n- v06 --supports--> v14\n- v06 --refutes--> v0c\n- v12 --refutes--> v13\n- v0c --refutes--> v00\n- v09 --refutes--> v10\n- v02 --refutes--> v0e\n- v0a --supports--> v03\n- v03 --refutes--> v0d\n- v08 --supports--> v05\n- v13 --supports--> v15\n- v0d --refutes--> v01\n- v11 --supports--> v09\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":"2229133eee753b99bb307d5b89c2096b35dcd2178a898da5dadce455171dba30","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0d\n- v0f --supports--> v13\n- v13 --refutes--> v14\n- v15 --refutes--> v05\n- v13 --supports--> v0a\n- v0b --refutes--> v09\n- v02 --supports--> v01\n- v05 --refutes--> v06\n- v13 --refutes--> v15\n- v03 --refutes--> v0b\n- v01 --refutes--> v0c\n- v08 --supports--> v11\n- v08 --refutes--> v12\n- v0c --refutes--> v07\n- v07 --refutes--> v0e\n- v00 --refutes--> v02\n- v04 --refutes--> v0f\n- v0e --supports--> v08\n- v08 --refutes--> v04\n- v0c --supports--> v10\n- v05 --refutes--> v03\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":"21d43e398beb174dcd22a73482490c37e03c969374b2ec499182385bcf20f346","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v04\n- v02 --supports--> v0c\n- v03 --refutes--> v06\n- v10 --supports--> v13\n- v10 --refutes--> v0f\n- v06 --supports--> v02\n- v04 --refutes--> v03\n- v12 --refutes--> v0a\n- v09 --refutes--> v11\n- v14 --supports--> v10\n- v0d --refutes--> v05\n- v10 --refutes--> v0d\n- v05 --refutes--> v0e\n- v0e --refutes--> v12\n- v10 --refutes--> v01\n- v0a --supports--> v09\n- v04 --refutes--> v0b\n- v00 --supports--> v14\n- v04 --refutes--> v07\n- v08 --refutes--> v00\n- v03 --supports--> v15\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":"c0858196922cc79d9ba7cae7f09a8268e277d2aab9d904a7d1848ab01890e243","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0a\n- v08 --refutes--> v05\n- v05 --refutes--> v07\n- v0c --refutes--> v14\n- v0a --refutes--> v03\n- v0b --refutes--> v01\n- v07 --refutes--> v00\n- v13 --supports--> v0c\n- v03 --supports--> v02\n- v12 --supports--> v0b\n- v0c --supports--> v15\n- v04 --supports--> v12\n- v01 --supports--> v0d\n- v00 --refutes--> v04\n- v13 --supports--> v11\n- v05 --supports--> v09\n- v12 --refutes--> v06\n- v01 --supports--> v10\n- v0c --supports--> v0e\n- v0f --refutes--> v13\n- v03 --supports--> v08\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":"0100a31171334e5f52a17a17ee31c5eccdcb51d946c31619ea9074024bd9b1fe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v05\n- v14 --supports--> v11\n- v0b --supports--> v15\n- v04 --refutes--> v00\n- v0f --supports--> v0e\n- v00 --refutes--> v14\n- v07 --refutes--> v12\n- v00 --refutes--> v06\n- v08 --refutes--> v04\n- v11 --refutes--> v0f\n- v0c --supports--> v03\n- v0c --supports--> v0a\n- v12 --refutes--> v0b\n- v10 --supports--> v13\n- v02 --supports--> v0d\n- v0e --supports--> v02\n- v13 --refutes--> v0c\n- v02 --refutes--> v07\n- v10 --refutes--> v01\n- v00 --supports--> v09\n- v0b --supports--> v10\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"427b520ac356705a14b31eb326b671785e8c0ce6ed93cb821a58b1e5d73bf60c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v11\n- v15 --refutes--> v05\n- v10 --supports--> v12\n- v0f --supports--> v0a\n- v10 --supports--> v13\n- v13 --refutes--> v08\n- v08 --supports--> v03\n- v06 --supports--> v07\n- v0b --supports--> v00\n- v0f --supports--> v04\n- v09 --refutes--> v0f\n- v08 --refutes--> v14\n- v0e --supports--> v0d\n- v01 --supports--> v15\n- v09 --refutes--> v02\n- v05 --refutes--> v10\n- v06 --supports--> v01\n- v0d --refutes--> v09\n- v0c --supports--> v0b\n- v08 --refutes--> v0e\n- v0a --refutes--> v0c\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":"4311729ac14f857f01446955b0bbebb53c544c030eb8a9f6fa31ba47861dd72d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v14\n- v0d --refutes--> v15\n- v0d --supports--> v08\n- v0c --refutes--> v03\n- v14 --refutes--> v0c\n- v0b --refutes--> v0a\n- v12 --refutes--> v09\n- v04 --supports--> v0b\n- v0a --supports--> v13\n- v05 --supports--> v04\n- v0b --refutes--> v02\n- v0d --supports--> v01\n- v15 --supports--> v07\n- v11 --supports--> v0d\n- v09 --refutes--> v0e\n- v0c --supports--> v00\n- v10 --supports--> v12\n- v09 --supports--> v0f\n- v04 --refutes--> v06\n- v0a --refutes--> v10\n- v07 --refutes--> v05\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":"61c4b0612a7f757e21227155affd0d9061b4fc1f311cdbff279d489352a77786","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v07\n- v0d --refutes--> v0c\n- v01 --refutes--> v04\n- v05 --supports--> v13\n- v14 --supports--> v11\n- v0a --supports--> v0d\n- v07 --supports--> v00\n- v07 --refutes--> v15\n- v0d --refutes--> v03\n- v12 --supports--> v01\n- v00 --refutes--> v10\n- v03 --refutes--> v14\n- v0d --supports--> v02\n- v08 --refutes--> v12\n- v00 --refutes--> v0e\n- v06 --supports--> v05\n- v0a --supports--> v0b\n- v07 --supports--> v09\n- v0d --supports--> v0f\n- v10 --supports--> v06\n- v13 --refutes--> v0a\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":"92c56ebf807b7ff0dff78fa2aa895e6e6a3d74bbd79d86f63c5f79cba2d881c6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v0c\n- v12 --supports--> v05\n- v08 --refutes--> v0d\n- v12 --refutes--> v0b\n- v01 --supports--> v04\n- v06 --refutes--> v0a\n- v0b --refutes--> v08\n- v15 --supports--> v07\n- v09 --refutes--> v15\n- v01 --refutes--> v12\n- v0d --supports--> v00\n- v08 --supports--> v0f\n- v08 --supports--> v03\n- v00 --supports--> v14\n- v0a --supports--> v09\n- v13 --supports--> v02\n- v10 --refutes--> v06\n- v0c --refutes--> v10\n- v10 --supports--> v11\n- v12 --supports--> v0e\n- v15 --refutes--> v01\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":"04ae86f6335feafac2ba663215e3a9b7a9b452c913553359524062322ae9677c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v09\n- v00 --refutes--> v14\n- v14 --supports--> v02\n- v12 --refutes--> v0c\n- v04 --refutes--> v0a\n- v13 --refutes--> v00\n- v0d --refutes--> v03\n- v0f --supports--> v05\n- v01 --refutes--> v15\n- v11 --supports--> v07\n- v01 --supports--> v08\n- v12 --refutes--> v11\n- v03 --refutes--> v04\n- v14 --supports--> v10\n- v0a --refutes--> v06\n- v12 --supports--> v0b\n- v09 --supports--> v0e\n- v0e --supports--> v13\n- v0a --supports--> v01\n- v14 --refutes--> v0d\n- v05 --supports--> v12\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":"c3f7ebaf663e430663fcad05279de2605be71341b6b8b43701735c0c55f845da","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v07\n- v05 --refutes--> v03\n- v01 --refutes--> v08\n- v08 --refutes--> v0c\n- v11 --refutes--> v0d\n- v04 --refutes--> v09\n- v03 --refutes--> v14\n- v01 --supports--> v02\n- v0a --refutes--> v05\n- v07 --refutes--> v0b\n- v06 --refutes--> v12\n- v11 --refutes--> v13\n- v14 --refutes--> v00\n- v10 --supports--> v01\n- v06 --supports--> v0e\n- v09 --refutes--> v0a\n- v0f --supports--> v06\n- v08 --refutes--> v11\n- v04 --supports--> v15\n- v14 --refutes--> v0f\n- v06 --refutes--> v10\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73904252b6d27bb758fe5364d5891a5ecb848958c9776c971221706a0456cff1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v07\n- v05 --refutes--> v0e\n- v12 --supports--> v00\n- v0c --refutes--> v03\n- v0b --refutes--> v12\n- v06 --refutes--> v0b\n- v00 --supports--> v13\n- v11 --supports--> v0a\n- v03 --supports--> v15\n- v01 --refutes--> v0c\n- v0e --supports--> v10\n- v0e --refutes--> v02\n- v02 --refutes--> v04\n- v03 --supports--> v08\n- v11 --refutes--> v05\n- v02 --supports--> v14\n- v04 --refutes--> v06\n- v0c --supports--> v0f\n- v09 --refutes--> v11\n- v11 --refutes--> v0d\n- v13 --supports--> v01\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9fdb027c7125f3a1971122a29e26b03cf65be2a731d0035013489bcd9811cc4e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0a\n- v02 --supports--> v05\n- v02 --refutes--> v08\n- v0d --supports--> v01\n- v06 --supports--> v00\n- v0a --supports--> v0c\n- v08 --supports--> v09\n- v0e --refutes--> v14\n- v13 --supports--> v12\n- v01 --refutes--> v13\n- v13 --refutes--> v11\n- v03 --refutes--> v0d\n- v10 --refutes--> v06\n- v0d --supports--> v15\n- v11 --refutes--> v10\n- v0f --refutes--> v02\n- v14 --refutes--> v0f\n- v08 --supports--> v04\n- v10 --refutes--> v0b\n- v0e --supports--> v07\n- v04 --refutes--> v03\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":"06849f30a40d86ab17d02645bf863db2af6710beca3b62b6360c806f8878fc09","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v02\n- v00 --supports--> v0c\n- v01 --refutes--> v12\n- v08 --refutes--> v04\n- v02 --refutes--> v0f\n- v14 --supports--> v07\n- v10 --supports--> v0b\n- v14 --refutes--> v09\n- v15 --supports--> v14\n- v0d --supports--> v01\n- v05 --supports--> v10\n- v06 --supports--> v0d\n- v10 --refutes--> v00\n- v06 --supports--> v13\n- v15 --refutes--> v11\n- v0e --supports--> v06\n- v08 --refutes--> v0e\n- v03 --supports--> v05\n- v14 --supports--> v08\n- v02 --supports--> v0a\n- v0f --refutes--> v03\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":"bbd34e805d04ce6b45287b453591377310faf18bb28c486bc1a8d0d427c507c3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v13\n- v0f --supports--> v0c\n- v05 --refutes--> v08\n- v06 --supports--> v03\n- v05 --supports--> v09\n- v13 --refutes--> v11\n- v09 --supports--> v0b\n- v10 --supports--> v0e\n- v15 --supports--> v0d\n- v09 --refutes--> v12\n- v03 --supports--> v10\n- v15 --refutes--> v07\n- v04 --refutes--> v05\n- v01 --refutes--> v0a\n- v01 --refutes--> v06\n- v06 --supports--> v14\n- v15 --supports--> v02\n- v13 --supports--> v00\n- v00 --supports--> v01\n- v0e --refutes--> v0f\n- v0c --supports--> v15\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":"f09fecc7d173193db20c46b4e871d194f64235e801662274c43e260403d88f2e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v12\n- v05 --refutes--> v09\n- v05 --supports--> v14\n- v03 --supports--> v06\n- v0c --refutes--> v11\n- v13 --refutes--> v05\n- v0f --supports--> v00\n- v10 --supports--> v0b\n- v14 --refutes--> v10\n- v06 --refutes--> v02\n- v05 --supports--> v04\n- v06 --refutes--> v07\n- v02 --supports--> v15\n- v0e --refutes--> v0c\n- v0b --supports--> v0a\n- v02 --refutes--> v13\n- v00 --refutes--> v03\n- v06 --supports--> v0d\n- v0a --refutes--> v01\n- v11 --refutes--> v0f\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":"28ade3b4e1024ca9e20c59bc615337ec656ff411c6c32a0bbff1fceff2aafb0f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0c --refutes--> v0d\n- v11 --refutes--> v09\n- v0e --refutes--> v11\n- v07 --supports--> v0a\n- v06 --supports--> v07\n- v13 --supports--> v02\n- v0a --refutes--> v03\n- v04 --refutes--> v0e\n- v10 --refutes--> v01\n- v02 --refutes--> v04\n- v13 --supports--> v0b\n- v0d --supports--> v14\n- v09 --supports--> v06\n- v0d --supports--> v0f\n- v0e --refutes--> v08\n- v14 --supports--> v10\n- v02 --supports--> v12\n- v0a --supports--> v0c\n- v01 --refutes--> v00\n- v09 --refutes--> v15\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":"7000b88b76e71297e59918d501beda55ccde15acfe5be269f058782817a0fffc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v01\n- v0c --refutes--> v11\n- v08 --refutes--> v07\n- v02 --refutes--> v15\n- v07 --supports--> v0e\n- v11 --refutes--> v10\n- v15 --supports--> v06\n- v01 --supports--> v0c\n- v0c --refutes--> v0a\n- v11 --refutes--> v0f\n- v10 --supports--> v08\n- v05 --refutes--> v0b\n- v09 --refutes--> v0d\n- v14 --refutes--> v02\n- v00 --refutes--> v09\n- v0b --refutes--> v14\n- v07 --supports--> v13\n- v01 --supports--> v03\n- v01 --refutes--> v04\n- v0e --supports--> v00\n- v10 --supports--> v12\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":"d1d931693e416207570148211045545465894563fe2279254e8683a5409b79f7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v11\n- v10 --refutes--> v05\n- v0e --supports--> v00\n- v00 --refutes--> v0b\n- v12 --supports--> v14\n- v02 --supports--> v0e\n- v02 --supports--> v03\n- v01 --supports--> v0a\n- v10 --supports--> v09\n- v02 --supports--> v13\n- v0a --supports--> v0c\n- v15 --refutes--> v10\n- v0a --supports--> v07\n- v0f --refutes--> v08\n- v11 --supports--> v12\n- v0d --refutes--> v06\n- v0a --supports--> v02\n- v10 --supports--> v01\n- v0b --refutes--> v0d\n- v04 --refutes--> v15\n- v0d --refutes--> v0f\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":"4ec7f039fc141e0d917b1d5c11c1b74097d449e15a08d4213e071fbf1584ae66","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v11\n- v0d --refutes--> v13\n- v0b --supports--> v0c\n- v12 --supports--> v03\n- v06 --supports--> v0e\n- v06 --refutes--> v07\n- v0d --supports--> v00\n- v10 --supports--> v08\n- v08 --refutes--> v05\n- v0d --supports--> v14\n- v0f --refutes--> v01\n- v0f --supports--> v0d\n- v09 --supports--> v0b\n- v0e --refutes--> v02\n- v0c --refutes--> v06\n- v03 --supports--> v09\n- v03 --refutes--> v15\n- v0c --refutes--> v0a\n- v02 --refutes--> v0f\n- v0b --refutes--> v04\n- v11 --supports--> v10\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":"0139fec1b1158fa800b76ade43aeefb055a98a8ec3ea40d6bc9d26a2c775f29d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v13 --refutes--> v12\n- v14 --refutes--> v0a\n- v03 --supports--> v01\n- v10 --supports--> v15\n- v02 --supports--> v0d\n- v12 --refutes--> v0e\n- v08 --refutes--> v06\n- v12 --refutes--> v03\n- v11 --refutes--> v00\n- v00 --supports--> v0c\n- v14 --supports--> v0f\n- v02 --refutes--> v14\n- v0a --refutes--> v13\n- v15 --supports--> v0b\n- v0c --refutes--> v10\n- v00 --supports--> v07\n- v0a --supports--> v05\n- v06 --refutes--> v02\n- v0b --refutes--> v08\n- v0c --refutes--> v09\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":"cbd519cd6a891d4555e2534ef6dc86b1212de78759b43e1455996fc40cd38343","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --refutes--> v06\n- v0b --supports--> v15\n- v07 --refutes--> v0d\n- v10 --supports--> v12\n- v0c --refutes--> v10\n- v04 --supports--> v02\n- v0f --supports--> v09\n- v0e --supports--> v11\n- v01 --refutes--> v0e\n- v13 --refutes--> v0f\n- v0f --supports--> v07\n- v0f --refutes--> v00\n- v01 --supports--> v05\n- v03 --refutes--> v0b\n- v0d --refutes--> v14\n- v15 --refutes--> v13\n- v12 --supports--> v01\n- v11 --refutes--> v04\n- v12 --refutes--> v0a\n- v0d --supports--> v08\n- v0d --refutes--> v0c\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":"a27d6ac052fb1c59e4b86cf97ba3005df4ade6eea4b375eb69cca9680e6a079b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v15\n- v0b --supports--> v01\n- v07 --refutes--> v00\n- v00 --supports--> v09\n- v04 --refutes--> v07\n- v0b --refutes--> v0e\n- v09 --refutes--> v12\n- v05 --refutes--> v0d\n- v0d --refutes--> v02\n- v0f --supports--> v05\n- v0f --supports--> v11\n- v06 --supports--> v0c\n- v03 --refutes--> v0b\n- v10 --refutes--> v08\n- v09 --refutes--> v0f\n- v0b --supports--> v13\n- v01 --supports--> v14\n- v06 --refutes--> v04\n- v08 --refutes--> v03\n- v0f --supports--> v0a\n- v02 --refutes--> v10\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":"cec68f6a4111686a8c410281b57968f200f42cddb224043e69a30dbbc9825486","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v13\n- v13 --refutes--> v07\n- v14 --refutes--> v04\n- v01 --supports--> v15\n- v0d --refutes--> v0a\n- v14 --supports--> v01\n- v08 --refutes--> v14\n- v00 --refutes--> v0f\n- v05 --supports--> v12\n- v0a --supports--> v09\n- v11 --refutes--> v00\n- v06 --refutes--> v05\n- v0c --refutes--> v06\n- v05 --refutes--> v10\n- v01 --refutes--> v0c\n- v09 --supports--> v0e\n- v00 --refutes--> v0d\n- v10 --supports--> v03\n- v07 --refutes--> v11\n- v0d --supports--> v0b\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":"46044ff36f802fc93edbbafca40c9b975deb1f532429eaad84570670a74249f1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v14\n- v01 --supports--> v0e\n- v09 --supports--> v10\n- v13 --refutes--> v01\n- v09 --refutes--> v0d\n- v15 --supports--> v0a\n- v15 --supports--> v0c\n- v0f --refutes--> v09\n- v0b --refutes--> v07\n- v09 --refutes--> v08\n- v07 --refutes--> v15\n- v12 --refutes--> v0b\n- v11 --supports--> v0f\n- v09 --refutes--> v06\n- v04 --supports--> v11\n- v0e --supports--> v04\n- v05 --supports--> v12\n- v15 --refutes--> v00\n- v0d --refutes--> v05\n- v12 --refutes--> v02\n- v0b --refutes--> v03\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e594b0da416b0999f44864f558636e197aa1aa633c8a0904e6c91aba564b0f8e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v00\n- v01 --supports--> v0e\n- v00 --refutes--> v13\n- v07 --refutes--> v15\n- v13 --supports--> v0f\n- v15 --supports--> v14\n- v06 --supports--> v04\n- v02 --refutes--> v0b\n- v0f --supports--> v11\n- v01 --refutes--> v0a\n- v14 --supports--> v06\n- v02 --refutes--> v10\n- v12 --refutes--> v0d\n- v12 --refutes--> v0c\n- v03 --supports--> v01\n- v0e --refutes--> v07\n- v01 --refutes--> v09\n- v01 --supports--> v05\n- v03 --refutes--> v08\n- v04 --refutes--> v02\n- v10 --supports--> v12\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":"c5bec05a79c3329a6b8f00b9c044c60e9425a0aa96dff2b647726df4b88057d6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v0c\n- v0e --supports--> v0d\n- v05 --refutes--> v0e\n- v08 --refutes--> v06\n- v15 --supports--> v01\n- v0a --refutes--> v09\n- v0b --supports--> v04\n- v02 --refutes--> v10\n- v0d --supports--> v15\n- v15 --refutes--> v02\n- v06 --supports--> v00\n- v09 --supports--> v08\n- v00 --refutes--> v12\n- v0e --refutes--> v14\n- v00 --refutes--> v0b\n- v09 --refutes--> v13\n- v07 --refutes--> v05\n- v10 --supports--> v0a\n- v00 --supports--> v03\n- v04 --supports--> v11\n- v09 --refutes--> v0f\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":"81aa36a8b02801f74cd9aed97778473f11e7a1e4814a3dfaf31a26077158b4ab","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v01 --supports--> v00\n- v0b --refutes--> v0c\n- v13 --supports--> v11\n- v00 --refutes--> v0b\n- v12 --supports--> v0e\n- v0f --supports--> v0a\n- v06 --supports--> v15\n- v12 --supports--> v14\n- v03 --supports--> v04\n- v13 --supports--> v01\n- v0e --refutes--> v10\n- v06 --refutes--> v05\n- v08 --refutes--> v09\n- v12 --refutes--> v02\n- v0b --refutes--> v0d\n- v10 --refutes--> v08\n- v05 --refutes--> v13\n- v0f --refutes--> v07\n- v0d --supports--> v03\n- v03 --refutes--> v0f\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":"5a8b01552b1732c228eae9fea184e4964f8ecf72c5eb2bcdb5c6f9288bdf51b6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v06\n- v02 --supports--> v11\n- v01 --refutes--> v04\n- v15 --refutes--> v0c\n- v0c --refutes--> v0a\n- v02 --supports--> v0e\n- v15 --supports--> v13\n- v06 --supports--> v08\n- v00 --supports--> v02\n- v14 --supports--> v03\n- v0a --refutes--> v0b\n- v11 --refutes--> v14\n- v11 --refutes--> v0d\n- v0f --supports--> v10\n- v0f --supports--> v07\n- v04 --supports--> v12\n- v03 --supports--> v0f\n- v04 --supports--> v15\n- v0a --supports--> v09\n- v00 --supports--> v05\n- v08 --refutes--> v00\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":"f07503c4b6cc512d532d9efe665e5286647fa247662da36da84e29bc8b62f8a0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v04\n- v0d --refutes--> v01\n- v02 --refutes--> v13\n- v06 --supports--> v0d\n- v06 --refutes--> v09\n- v01 --supports--> v0e\n- v11 --refutes--> v12\n- v0a --supports--> v02\n- v05 --refutes--> v0a\n- v02 --supports--> v15\n- v15 --refutes--> v03\n- v0f --supports--> v00\n- v00 --supports--> v06\n- v02 --refutes--> v0c\n- v0e --refutes--> v11\n- v11 --supports--> v0b\n- v15 --refutes--> v07\n- v0b --refutes--> v14\n- v14 --refutes--> v10\n- v08 --supports--> v0f\n- v14 --supports--> v05\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d3c1de7490a354bb63e71aedebed9bad778d239c8bf79df7527f915d8e9b141f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0f\n- v05 --refutes--> v06\n- v11 --supports--> v03\n- v10 --refutes--> v00\n- v13 --refutes--> v05\n- v03 --refutes--> v13\n- v0a --supports--> v0c\n- v04 --refutes--> v0e\n- v0e --supports--> v12\n- v13 --refutes--> v07\n- v10 --supports--> v11\n- v0b --refutes--> v0d\n- v01 --refutes--> v02\n- v05 --supports--> v04\n- v14 --supports--> v01\n- v0c --supports--> v10\n- v01 --refutes--> v08\n- v12 --supports--> v14\n- v05 --supports--> v15\n- v0d --refutes--> v0a\n- v01 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"532b85188c73b4f6f16f478ffd78598c06f6238e3c4e010f851efb6a82169912","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v13\n- v0c --refutes--> v05\n- v03 --refutes--> v14\n- v08 --supports--> v04\n- v11 --supports--> v0e\n- v14 --supports--> v15\n- v02 --refutes--> v0d\n- v0b --supports--> v12\n- v01 --refutes--> v07\n- v0b --refutes--> v10\n- v08 --refutes--> v06\n- v0e --supports--> v0a\n- v03 --supports--> v00\n- v0c --refutes--> v11\n- v0d --refutes--> v09\n- v15 --supports--> v01\n- v0d --refutes--> v0f\n- v12 --supports--> v0c\n- v08 --supports--> v02\n- v0f --supports--> v03\n- v07 --refutes--> v0b\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":"1eb8b442c644901bfd350805cf1bc2e033ccbb36c86c8cc45f29f36dc6e681ee","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v15\n- v0f --supports--> v11\n- v11 --supports--> v02\n- v02 --supports--> v12\n- v05 --refutes--> v04\n- v00 --supports--> v13\n- v12 --supports--> v05\n- v02 --refutes--> v10\n- v04 --supports--> v09\n- v0c --supports--> v0b\n- v09 --supports--> v00\n- v15 --supports--> v14\n- v0c --refutes--> v0d\n- v00 --supports--> v0c\n- v07 --supports--> v0f\n- v03 --supports--> v07\n- v03 --refutes--> v0e\n- v15 --supports--> v01\n- v09 --supports--> v0a\n- v11 --refutes--> v06\n- v01 --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6b845f9a71d272b1cb3efae6db4b41c9fce48f47f4d59b7e73e3caed305d894","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v13\n- v08 --refutes--> v05\n- v07 --refutes--> v0b\n- v13 --supports--> v12\n- v0b --refutes--> v02\n- v0c --refutes--> v00\n- v09 --supports--> v0c\n- v0a --refutes--> v08\n- v13 --refutes--> v14\n- v01 --supports--> v0e\n- v01 --refutes--> v06\n- v03 --supports--> v01\n- v06 --supports--> v15\n- v06 --refutes--> v09\n- v10 --refutes--> v03\n- v03 --refutes--> v0d\n- v04 --refutes--> v07\n- v00 --refutes--> v0a\n- v08 --supports--> v0f\n- v08 --refutes--> v11\n- v11 --supports--> v04\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"af56c32880d160baead3287444c948d9c8ac4e88af6194745fe81db3c31c015e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v00\n- v03 --supports--> v01\n- v0d --refutes--> v14\n- v06 --supports--> v12\n- v02 --supports--> v05\n- v08 --refutes--> v07\n- v03 --supports--> v15\n- v10 --supports--> v04\n- v08 --supports--> v0d\n- v12 --supports--> v0a\n- v0c --supports--> v10\n- v0b --supports--> v13\n- v11 --supports--> v09\n- v09 --refutes--> v02\n- v04 --refutes--> v03\n- v0b --refutes--> v06\n- v0d --supports--> v0b\n- v12 --supports--> v0f\n- v0a --refutes--> v0e\n- v02 --supports--> v0c\n- v0e --refutes--> v11\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":"82166d065d840b364c4a8c9c752998287e05eb8b04bf549a06725c97f17e515c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v14\n- v15 --supports--> v11\n- v06 --supports--> v0e\n- v0a --refutes--> v06\n- v12 --refutes--> v08\n- v00 --refutes--> v01\n- v13 --refutes--> v0c\n- v02 --supports--> v07\n- v03 --supports--> v15\n- v13 --refutes--> v09\n- v01 --supports--> v02\n- v12 --supports--> v0b\n- v05 --supports--> v04\n- v06 --supports--> v03\n- v13 --refutes--> v0d\n- v15 --refutes--> v0f\n- v11 --supports--> v05\n- v0c --supports--> v00\n- v14 --supports--> v13\n- v05 --refutes--> v10\n- v07 --supports--> v12\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":"5db0d47ec8d197642fbcd0631d5039bd2222b471b261c142f515757be0612b5c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0b\n- v05 --supports--> v0d\n- v10 --refutes--> v02\n- v03 --refutes--> v09\n- v12 --supports--> v11\n- v03 --supports--> v07\n- v14 --refutes--> v13\n- v03 --refutes--> v0f\n- v07 --supports--> v15\n- v07 --refutes--> v12\n- v0d --supports--> v03\n- v02 --refutes--> v14\n- v13 --supports--> v04\n- v13 --refutes--> v0e\n- v03 --refutes--> v0a\n- v0e --supports--> v00\n- v0b --supports--> v08\n- v14 --refutes--> v0c\n- v05 --supports--> v01\n- v08 --refutes--> v05\n- v11 --supports--> v06\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":"6c1b40ca615727fa3b5afeea6da4545ee13357646b9894bbbf2bfd5111f5c2b9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v08\n- v09 --supports--> v01\n- v0a --supports--> v0e\n- v01 --supports--> v06\n- v15 --refutes--> v0f\n- v03 --supports--> v10\n- v11 --refutes--> v02\n- v11 --refutes--> v12\n- v00 --supports--> v0a\n- v05 --supports--> v07\n- v12 --refutes--> v13\n- v14 --refutes--> v05\n- v10 --supports--> v14\n- v05 --supports--> v0c\n- v0f --supports--> v0b\n- v05 --supports--> v00\n- v05 --supports--> v0d\n- v13 --supports--> v04\n- v13 --supports--> v09\n- v0f --refutes--> v03\n- v06 --refutes--> v15\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":"f4f0a93f02dccd5fa7906252b34ff62b156f55601d4523a9af307ea54f4fc1f4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v15\n- v0c --supports--> v13\n- v05 --refutes--> v04\n- v0e --refutes--> v08\n- v12 --refutes--> v07\n- v0a --refutes--> v0f\n- v15 --supports--> v02\n- v0c --supports--> v0e\n- v01 --supports--> v14\n- v05 --refutes--> v12\n- v0b --refutes--> v00\n- v06 --supports--> v0a\n- v11 --refutes--> v10\n- v12 --supports--> v0d\n- v01 --refutes--> v0c\n- v02 --supports--> v06\n- v02 --refutes--> v09\n- v08 --refutes--> v11\n- v08 --supports--> v03\n- v11 --supports--> v05\n- v00 --refutes--> v01\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":"d583f096d58c4af69fbf0c00b11742ca19b2c841ef0ddde68d0b69f763e69f2d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v15\n- v10 --refutes--> v07\n- v02 --refutes--> v10\n- v07 --supports--> v13\n- v0b --supports--> v04\n- v0d --supports--> v0a\n- v11 --supports--> v0c\n- v05 --refutes--> v12\n- v00 --refutes--> v01\n- v0f --supports--> v09\n- v06 --refutes--> v11\n- v13 --refutes--> v05\n- v05 --refutes--> v08\n- v0c --supports--> v00\n- v0b --refutes--> v06\n- v0a --refutes--> v02\n- v0f --refutes--> v0b\n- v11 --refutes--> v03\n- v06 --refutes--> v0e\n- v01 --refutes--> v0d\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":"1ec7aedf3bcad11092eee5df886de36a5d9b3658ca0f75db495aac08752ee453","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v02\n- v07 --refutes--> v12\n- v0e --refutes--> v0a\n- v14 --supports--> v00\n- v02 --supports--> v09\n- v0d --refutes--> v0b\n- v05 --refutes--> v07\n- v11 --supports--> v14\n- v00 --supports--> v0d\n- v05 --supports--> v15\n- v0a --supports--> v08\n- v12 --refutes--> v03\n- v13 --refutes--> v0f\n- v13 --supports--> v0c\n- v09 --refutes--> v10\n- v0a --refutes--> v11\n- v05 --refutes--> v06\n- v09 --supports--> v05\n- v0b --refutes--> v01\n- v13 --refutes--> v04\n- v12 --refutes--> v13\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":"16ce35ec44fe378566401d1b5a51ba99d2c2caf1d81d66b8dbce0b3aa08b4fe7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v00\n- v05 --refutes--> v0f\n- v07 --refutes--> v06\n- v0d --supports--> v05\n- v06 --supports--> v0a\n- v03 --supports--> v0d\n- v13 --refutes--> v0e\n- v14 --refutes--> v15\n- v0b --refutes--> v03\n- v15 --supports--> v10\n- v13 --supports--> v02\n- v11 --refutes--> v04\n- v06 --refutes--> v01\n- v11 --refutes--> v0b\n- v12 --refutes--> v07\n- v15 --refutes--> v13\n- v0f --supports--> v14\n- v0c --refutes--> v09\n- v00 --refutes--> v11\n- v12 --refutes--> v08\n- v0c --refutes--> v12\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":"cfc97459d142ab8b2c7ee46b1b4ff595eded583dde125b5c72a5dfe032878a02","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v07\n- v0b --supports--> v13\n- v0d --refutes--> v14\n- v14 --refutes--> v15\n- v14 --refutes--> v09\n- v0c --refutes--> v08\n- v02 --supports--> v0c\n- v07 --refutes--> v0b\n- v05 --refutes--> v00\n- v12 --refutes--> v0f\n- v15 --supports--> v03\n- v13 --refutes--> v12\n- v05 --refutes--> v0d\n- v0c --supports--> v0e\n- v0b --supports--> v01\n- v10 --refutes--> v04\n- v14 --refutes--> v11\n- v04 --supports--> v02\n- v12 --supports--> v10\n- v0a --supports--> v06\n- v06 --supports--> v05\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"214c820a7d8811c89cf7292498f535f2310d0a137cf0c684b002c2204663bd29","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0f\n- v14 --refutes--> v08\n- v01 --supports--> v05\n- v13 --refutes--> v04\n- v13 --supports--> v07\n- v0f --refutes--> v0c\n- v0c --refutes--> v12\n- v01 --supports--> v10\n- v03 --refutes--> v0b\n- v02 --supports--> v00\n- v11 --refutes--> v02\n- v0c --refutes--> v09\n- v08 --supports--> v13\n- v13 --supports--> v06\n- v0d --supports--> v14\n- v01 --supports--> v0a\n- v05 --supports--> v11\n- v07 --supports--> v0e\n- v11 --refutes--> v15\n- v0e --refutes--> v01\n- v03 --supports--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7adb157974a4bab038febc6dcd934fabd45c1000237d82fceb1223849dd47ded","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0b\n- v0f --refutes--> v09\n- v02 --refutes--> v0f\n- v14 --supports--> v0c\n- v0c --refutes--> v0d\n- v08 --refutes--> v11\n- v0f --supports--> v15\n- v0f --refutes--> v07\n- v0c --refutes--> v12\n- v04 --refutes--> v08\n- v0c --supports--> v13\n- v01 --supports--> v0e\n- v06 --supports--> v00\n- v0a --refutes--> v02\n- v07 --supports--> v06\n- v11 --refutes--> v05\n- v03 --refutes--> v14\n- v11 --supports--> v0a\n- v12 --refutes--> v04\n- v06 --supports--> v01\n- v04 --refutes--> v10\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c2038adadae8d1cf721c5062e9287ef0f02082961066269dcb77f10e77c489e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v12\n- v09 --supports--> v06\n- v15 --supports--> v09\n- v0a --refutes--> v07\n- v05 --refutes--> v00\n- v0e --supports--> v01\n- v07 --supports--> v0f\n- v15 --refutes--> v08\n- v13 --supports--> v0d\n- v14 --refutes--> v04\n- v15 --refutes--> v0b\n- v12 --refutes--> v13\n- v04 --supports--> v0e\n- v10 --supports--> v02\n- v02 --supports--> v15\n- v0d --supports--> v14\n- v0e --supports--> v05\n- v10 --supports--> v0c\n- v10 --refutes--> v11\n- v09 --refutes--> v03\n- v05 --refutes--> v0a\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":"04ea4c4d2ed7c07dd9d314780087913c6f46b02458ad824e43b46b188a2c0a1d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0a --supports--> v01\n- v0c --refutes--> v00\n- v11 --supports--> v02\n- v02 --supports--> v15\n- v10 --refutes--> v0f\n- v0f --refutes--> v13\n- v08 --supports--> v06\n- v02 --supports--> v09\n- v03 --supports--> v0e\n- v0e --supports--> v05\n- v05 --supports--> v04\n- v07 --supports--> v12\n- v0e --supports--> v14\n- v0a --refutes--> v03\n- v02 --refutes--> v08\n- v06 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v10\n- v04 --refutes--> v11\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":"ae47425e1a8daf3c26f34834e78951b6c64d045adee68a449b70701bcba66d40","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v0f\n- v04 --refutes--> v0d\n- v00 --refutes--> v04\n- v12 --refutes--> v0a\n- v08 --supports--> v01\n- v06 --refutes--> v0b\n- v02 --supports--> v05\n- v0d --refutes--> v11\n- v09 --supports--> v07\n- v0b --supports--> v13\n- v14 --refutes--> v06\n- v0b --supports--> v0e\n- v0e --supports--> v02\n- v07 --refutes--> v08\n- v00 --supports--> v10\n- v05 --refutes--> v12\n- v14 --supports--> v15\n- v08 --refutes--> v14\n- v0a --supports--> v00\n- v00 --refutes--> v0c\n- v14 --supports--> v03\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":"439f5940cba6fbb67334961a66c20815101bbf1ef8812922acd2f3cdf1cc5f98","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v06\n- v00 --supports--> v13\n- v01 --supports--> v00\n- v06 --refutes--> v10\n- v0c --refutes--> v05\n- v0d --refutes--> v0a\n- v03 --supports--> v0b\n- v0d --refutes--> v02\n- v04 --supports--> v0e\n- v07 --refutes--> v0d\n- v0f --supports--> v0c\n- v0e --supports--> v15\n- v03 --refutes--> v04\n- v11 --supports--> v01\n- v12 --supports--> v11\n- v0c --supports--> v07\n- v01 --refutes--> v08\n- v06 --refutes--> v03\n- v15 --supports--> v09\n- v12 --refutes--> v14\n- v15 --supports--> v12\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":"e21ec00508db2b5bf6c8e08064aadcac59dea1072aece5b1ba6c03e3bf901e34","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v10\n- v07 --refutes--> v03\n- v15 --refutes--> v07\n- v06 --supports--> v02\n- v0f --supports--> v0c\n- v03 --supports--> v06\n- v11 --refutes--> v0f\n- v14 --refutes--> v05\n- v0c --supports--> v0d\n- v0f --refutes--> v04\n- v06 --refutes--> v00\n- v0f --refutes--> v0a\n- v0e --supports--> v13\n- v15 --refutes--> v09\n- v14 --refutes--> v12\n- v01 --refutes--> v08\n- v0c --refutes--> v14\n- v10 --refutes--> v01\n- v13 --refutes--> v0b\n- v01 --supports--> v0e\n- v00 --supports--> v11\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":"beccc1cf5a7d325232c9c043582a257470f0bca813b8a23fad8a4d2bd88051d9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v02\n- v15 --supports--> v05\n- v0c --refutes--> v0e\n- v08 --refutes--> v09\n- v0d --supports--> v06\n- v09 --supports--> v14\n- v03 --supports--> v13\n- v10 --supports--> v0d\n- v11 --supports--> v12\n- v0f --supports--> v0c\n- v10 --refutes--> v07\n- v03 --refutes--> v0a\n- v09 --supports--> v00\n- v08 --supports--> v04\n- v09 --supports--> v01\n- v0e --refutes--> v15\n- v0a --supports--> v0f\n- v0d --refutes--> v0b\n- v05 --supports--> v10\n- v02 --supports--> v08\n- v09 --refutes--> v11\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":"31c5b8400a9ded050c5f06a06dc345b3d9eb48167a997c3a584e430216dce73c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v13\n- v02 --refutes--> v0e\n- v0c --refutes--> v04\n- v01 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --refutes--> v07\n- v08 --supports--> v00\n- v13 --supports--> v0d\n- v14 --refutes--> v0f\n- v08 --refutes--> v09\n- v14 --refutes--> v12\n- v0a --refutes--> v06\n- v06 --refutes--> v0b\n- v0e --supports--> v15\n- v0c --supports--> v0a\n- v09 --refutes--> v11\n- v14 --supports--> v01\n- v05 --supports--> v10\n- v15 --supports--> v14\n- v13 --supports--> v03\n- v11 --supports--> v0c\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7ad4ee6b73a46b3bae05d6574cac859ee18b103eeb355ac62e613b1acce58543","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v0f\n- v02 --refutes--> v06\n- v07 --supports--> v14\n- v15 --supports--> v13\n- v05 --supports--> v02\n- v04 --supports--> v0e\n- v13 --supports--> v10\n- v06 --supports--> v07\n- v0b --refutes--> v0c\n- v02 --supports--> v03\n- v0c --supports--> v0d\n- v12 --supports--> v11\n- v08 --supports--> v15\n- v0e --supports--> v0b\n- v11 --refutes--> v09\n- v12 --supports--> v01\n- v11 --refutes--> v08\n- v0e --supports--> v00\n- v12 --refutes--> v0a\n- v14 --refutes--> v04\n- v0c --supports--> v12\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":"17cbc9564471a1c76e0439de5dbfc6666a69a4d29dc5ed1ac9a4c79a196eade9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v04 --refutes--> v14\n- v05 --refutes--> v10\n- v05 --supports--> v02\n- v14 --supports--> v07\n- v0f --refutes--> v0c\n- v07 --supports--> v08\n- v00 --refutes--> v0f\n- v09 --refutes--> v0a\n- v07 --supports--> v09\n- v12 --supports--> v04\n- v0c --refutes--> v01\n- v0e --refutes--> v00\n- v0c --refutes--> v15\n- v02 --refutes--> v13\n- v01 --supports--> v06\n- v07 --refutes--> v03\n- v09 --refutes--> v0e\n- v11 --supports--> v05\n- v0b --supports--> v11\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":"9d7095e62988cbdfa6a7bcc573a526a4d9c719e2425350e2d9278e187a88e5f1","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v0b\n- v0a --refutes--> v03\n- v15 --supports--> v0f\n- v06 --supports--> v09\n- v07 --supports--> v00\n- v14 --refutes--> v02\n- v06 --refutes--> v0a\n- v01 --supports--> v04\n- v11 --refutes--> v01\n- v03 --refutes--> v10\n- v0c --supports--> v15\n- v12 --supports--> v14\n- v07 --supports--> v13\n- v10 --refutes--> v11\n- v08 --refutes--> v12\n- v00 --refutes--> v0c\n- v0a --refutes--> v0d\n- v04 --supports--> v08\n- v04 --supports--> v0e\n- v14 --supports--> v07\n- v10 --refutes--> v05\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":"3d465ccb9f3df5e53c5bb1aa70385c6b2ad622fc051cd13dcc94f2f8269faf1c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0c\n- v04 --refutes--> v14\n- v06 --refutes--> v09\n- v06 --refutes--> v12\n- v12 --refutes--> v0f\n- v03 --refutes--> v10\n- v15 --supports--> v13\n- v0a --refutes--> v07\n- v13 --refutes--> v0d\n- v10 --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v06\n- v13 --refutes--> v0e\n- v12 --supports--> v0a\n- v0c --refutes--> v11\n- v13 --refutes--> v05\n- v11 --refutes--> v0b\n- v11 --refutes--> v04\n- v07 --refutes--> v00\n- v0c --refutes--> v08\n- v05 --refutes--> v03\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":"41303ab858ebd3d4ae8de33799235bfee94522f1bb70fa2fcbf080aeec02237c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v08\n- v03 --supports--> v12\n- v0c --supports--> v10\n- v12 --refutes--> v0b\n- v0f --supports--> v03\n- v05 --refutes--> v07\n- v10 --supports--> v05\n- v02 --supports--> v0e\n- v07 --refutes--> v14\n- v06 --refutes--> v0c\n- v0d --refutes--> v02\n- v09 --supports--> v0d\n- v14 --refutes--> v0f\n- v09 --refutes--> v04\n- v02 --refutes--> v01\n- v03 --refutes--> v13\n- v10 --refutes--> v00\n- v01 --refutes--> v06\n- v07 --refutes--> v15\n- v0b --refutes--> v0a\n- v05 --refutes--> v11\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":"f489891b70831b62320b83e6f83632a6c385a09a50b91e8844ebab492e292762","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v10\n- v01 --supports--> v06\n- v0f --supports--> v07\n- v00 --supports--> v09\n- v0f --refutes--> v02\n- v01 --refutes--> v11\n- v04 --supports--> v15\n- v08 --refutes--> v0e\n- v0a --supports--> v00\n- v01 --supports--> v0a\n- v09 --refutes--> v03\n- v05 --refutes--> v14\n- v0f --supports--> v0b\n- v10 --refutes--> v0f\n- v0c --supports--> v04\n- v04 --supports--> v12\n- v13 --refutes--> v05\n- v01 --refutes--> v0d\n- v0f --refutes--> v01\n- v03 --supports--> v08\n- v14 --supports--> v0c\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"35bf0b417fe03046fcd046b1ba63bed9426ee609dd02bcd2159ed6c8101bfad3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v09\n- v08 --refutes--> v03\n- v04 --supports--> v0a\n- v11 --refutes--> v10\n- v0f --refutes--> v05\n- v01 --supports--> v15\n- v0c --refutes--> v0f\n- v0e --refutes--> v02\n- v01 --refutes--> v14\n- v14 --refutes--> v0e\n- v0c --supports--> v07\n- v08 --supports--> v00\n- v0e --refutes--> v06\n- v03 --refutes--> v0d\n- v06 --supports--> v13\n- v09 --supports--> v01\n- v03 --refutes--> v11\n- v0e --refutes--> v12\n- v0a --supports--> v0c\n- v0a --refutes--> v0b\n- v05 --supports--> v08\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":"3935bce97e5e557d0df6b8a79203f717f6d0b469e764c619c5a8a3ad7df0a7af","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v10\n- v15 --refutes--> v09\n- v11 --refutes--> v13\n- v13 --supports--> v15\n- v12 --supports--> v0b\n- v01 --refutes--> v0c\n- v0b --refutes--> v11\n- v0a --refutes--> v12\n- v0c --refutes--> v08\n- v0a --supports--> v00\n- v07 --supports--> v0d\n- v0e --refutes--> v07\n- v02 --supports--> v04\n- v02 --supports--> v03\n- v07 --refutes--> v0f\n- v14 --refutes--> v0a\n- v08 --supports--> v14\n- v12 --refutes--> v16\n- v02 --refutes--> v06\n- v10 --refutes--> v05\n- v05 --supports--> v0e\n- v15 --supports--> v02\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":"c495398d91a9dbf3f7f36da01ebc2157abd98e8ea0df5e61bb4ff60a3ee59af6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v08\n- v14 --refutes--> v12\n- v13 --refutes--> v07\n- v0c --supports--> v0a\n- v01 --refutes--> v0b\n- v09 --supports--> v0e\n- v16 --supports--> v0f\n- v0b --refutes--> v0c\n- v00 --supports--> v10\n- v02 --refutes--> v03\n- v00 --refutes--> v04\n- v08 --supports--> v16\n- v0c --refutes--> v15\n- v0a --refutes--> v13\n- v08 --refutes--> v05\n- v12 --refutes--> v01\n- v0e --supports--> v14\n- v13 --refutes--> v00\n- v16 --refutes--> v11\n- v10 --supports--> v02\n- v10 --refutes--> v0d\n- v12 --supports--> 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":"918baedaf906517f131c764d573acc79e53ba76fd5397f636f0c13790c212461","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0a\n- v04 --refutes--> v13\n- v13 --refutes--> v0d\n- v01 --supports--> v15\n- v0f --supports--> v12\n- v15 --refutes--> v0c\n- v06 --refutes--> v08\n- v0f --supports--> v00\n- v14 --supports--> v0f\n- v07 --refutes--> v09\n- v12 --supports--> v01\n- v0f --supports--> v05\n- v08 --supports--> v10\n- v12 --supports--> v0e\n- v0a --supports--> v03\n- v14 --supports--> v0b\n- v02 --supports--> v07\n- v0a --supports--> v06\n- v09 --refutes--> v14\n- v14 --refutes--> v11\n- v07 --refutes--> v16\n- v0d --refutes--> v02\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":"ac1666ed7785985ac477230a8b365d665828bb5418685b1ce438a243768d3381","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v09\n- v10 --supports--> v15\n- v0b --supports--> v02\n- v15 --supports--> v08\n- v14 --refutes--> v0f\n- v08 --supports--> v13\n- v16 --refutes--> v0b\n- v09 --refutes--> v12\n- v05 --supports--> v06\n- v06 --supports--> v11\n- v13 --refutes--> v04\n- v14 --supports--> v0d\n- v0f --refutes--> v10\n- v06 --supports--> v01\n- v07 --refutes--> v05\n- v01 --supports--> v0c\n- v07 --refutes--> v0a\n- v01 --supports--> v0e\n- v10 --refutes--> v00\n- v02 --refutes--> v14\n- v13 --supports--> v07\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":"2d1d15c315d6f90771f88952b70b776aadd302fb12759cc86fa86fb513fffa27","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v0e\n- v0e --supports--> v0a\n- v0d --supports--> v09\n- v0b --refutes--> v15\n- v0a --refutes--> v04\n- v16 --refutes--> v00\n- v02 --refutes--> v11\n- v02 --refutes--> v0f\n- v09 --refutes--> v0b\n- v13 --supports--> v16\n- v14 --refutes--> v10\n- v03 --refutes--> v0c\n- v14 --supports--> v03\n- v13 --refutes--> v01\n- v15 --supports--> v07\n- v03 --refutes--> v06\n- v04 --refutes--> v0d\n- v07 --refutes--> v12\n- v03 --refutes--> v05\n- v0e --supports--> v08\n- v07 --supports--> v02\n- v11 --refutes--> v14\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":"fc449294955b047c1dd6b57a42af15641d3e44c4602888287538af7c97469bdf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0d\n- v0c --refutes--> v12\n- v0b --supports--> v11\n- v09 --supports--> v08\n- v06 --supports--> v09\n- v04 --supports--> v0f\n- v08 --refutes--> v04\n- v14 --refutes--> v0b\n- v16 --supports--> v0e\n- v07 --refutes--> v15\n- v16 --refutes--> v14\n- v05 --supports--> v03\n- v11 --supports--> v0c\n- v05 --supports--> v06\n- v00 --supports--> v0a\n- v13 --refutes--> v16\n- v11 --supports--> v02\n- v0a --refutes--> v01\n- v07 --refutes--> v00\n- v04 --supports--> v13\n- v12 --supports--> v07\n- v0a --refutes--> v10\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":"e326c6f12dfb95e8ada708f8bcd6251771ebb4663b14554b09159f8b2a0645cc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v0c --refutes--> v08\n- v14 --supports--> v05\n- v0b --supports--> v06\n- v06 --supports--> v09\n- v03 --supports--> v07\n- v08 --refutes--> v12\n- v0d --supports--> v0b\n- v05 --refutes--> v10\n- v07 --refutes--> v01\n- v10 --refutes--> v16\n- v0a --supports--> v03\n- v0e --refutes--> v0c\n- v11 --supports--> v14\n- v07 --refutes--> v0f\n- v08 --supports--> v04\n- v03 --refutes--> v13\n- v00 --refutes--> v0a\n- v03 --refutes--> v15\n- v07 --refutes--> v0e\n- v03 --refutes--> v02\n- v10 --supports--> v00\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":"52b77e7b3a3043cb03c9ec1281c5101d78d19d728df356427c6874102cdaf3f5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v16\n- v0b --refutes--> v0c\n- v12 --refutes--> v13\n- v13 --refutes--> v07\n- v10 --refutes--> v00\n- v0a --supports--> v08\n- v04 --supports--> v03\n- v03 --supports--> v0e\n- v15 --supports--> v0d\n- v13 --refutes--> v0a\n- v00 --refutes--> v14\n- v09 --supports--> v05\n- v14 --refutes--> v01\n- v0d --refutes--> v04\n- v08 --refutes--> v06\n- v0e --refutes--> v12\n- v15 --refutes--> v0f\n- v16 --supports--> v0b\n- v0b --supports--> v11\n- v0b --refutes--> v10\n- v09 --supports--> v02\n- v09 --refutes--> v15\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":"070305ebbef779f31b37e68f422c652dfdd9cfb55fa67b1f835cb0fff2bfc26b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0c\n- v13 --supports--> v12\n- v0e --supports--> v06\n- v15 --refutes--> v02\n- v0e --supports--> v09\n- v0c --supports--> v0a\n- v02 --supports--> v05\n- v12 --supports--> v00\n- v00 --supports--> v04\n- v04 --supports--> v0f\n- v03 --supports--> v0b\n- v01 --supports--> v13\n- v06 --refutes--> v11\n- v11 --supports--> v01\n- v02 --refutes--> v03\n- v00 --refutes--> v0d\n- v03 --supports--> v0e\n- v12 --supports--> v14\n- v07 --supports--> v08\n- v0a --supports--> v15\n- v0a --supports--> v10\n- v16 --supports--> v07\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":"e4e0ddcb74059b983b594a7045353fca0e4f5d6cdd7445993547a184f8e3086a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0d\n- v01 --refutes--> v03\n- v10 --supports--> v00\n- v12 --supports--> v07\n- v07 --supports--> v11\n- v09 --refutes--> v0f\n- v02 --supports--> v09\n- v16 --refutes--> v13\n- v09 --refutes--> v0a\n- v10 --supports--> v15\n- v0d --refutes--> v02\n- v16 --refutes--> v12\n- v0c --refutes--> v14\n- v0b --refutes--> v0c\n- v0e --supports--> v08\n- v05 --refutes--> v0b\n- v00 --refutes--> v01\n- v0e --supports--> v16\n- v07 --refutes--> v04\n- v01 --refutes--> v06\n- v0c --supports--> v10\n- v0f --supports--> v0e\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":"e057407f4a57d5c71635c4b3b45095620e1d70a5b19c88e821abb8009500234d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v03\n- v16 --refutes--> v0d\n- v0e --refutes--> v0f\n- v06 --refutes--> v10\n- v07 --supports--> v13\n- v01 --supports--> v11\n- v16 --refutes--> v05\n- v0d --supports--> v00\n- v01 --supports--> v02\n- v00 --refutes--> v0e\n- v0b --supports--> v0a\n- v10 --refutes--> v01\n- v09 --supports--> v0b\n- v08 --refutes--> v0c\n- v12 --refutes--> v08\n- v0c --supports--> v06\n- v07 --supports--> v09\n- v0a --refutes--> v16\n- v0c --refutes--> v04\n- v01 --refutes--> v14\n- v0a --supports--> v15\n- v0f --refutes--> v12\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":"26adc634ecc9f7ef57a5d1c018f6a2d044212ecda7ec04e62d9ab7e08e53903b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v14\n- v05 --refutes--> v16\n- v0d --supports--> v0c\n- v09 --refutes--> v01\n- v16 --refutes--> v07\n- v07 --refutes--> v03\n- v02 --supports--> v04\n- v01 --refutes--> v11\n- v04 --refutes--> v0e\n- v14 --refutes--> v10\n- v11 --refutes--> v12\n- v11 --refutes--> v00\n- v0c --refutes--> v0b\n- v0e --supports--> v05\n- v05 --supports--> v0a\n- v0e --supports--> v15\n- v09 --supports--> v13\n- v05 --refutes--> v06\n- v10 --refutes--> v09\n- v0c --refutes--> v08\n- v0d --refutes--> v0f\n- v00 --refutes--> v0d\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":"dce7a34546631250a89bde466f2d3b784074d0ea3331098a6ff3db87c5845f75","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v06\n- v08 --supports--> v14\n- v14 --supports--> v0d\n- v16 --refutes--> v02\n- v11 --supports--> v15\n- v15 --supports--> v03\n- v13 --refutes--> v05\n- v05 --supports--> v04\n- v0e --refutes--> v13\n- v01 --supports--> v11\n- v11 --refutes--> v00\n- v03 --refutes--> v16\n- v04 --supports--> v12\n- v07 --refutes--> v01\n- v09 --refutes--> v08\n- v04 --refutes--> v07\n- v13 --refutes--> v0f\n- v11 --refutes--> v10\n- v0d --refutes--> v0e\n- v01 --refutes--> v0b\n- v0a --supports--> v09\n- v11 --supports--> 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":"25ad0da5d7578a6dc71bb3434ac4cba40f2fd9f5a4fea3aa1bfb72d60ab8ceb9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v15\n- v08 --supports--> v12\n- v14 --refutes--> v13\n- v02 --refutes--> v00\n- v0f --supports--> v0c\n- v0e --supports--> v03\n- v00 --refutes--> v0f\n- v0b --refutes--> v07\n- v13 --refutes--> v01\n- v14 --refutes--> v09\n- v05 --supports--> v0d\n- v16 --refutes--> v11\n- v03 --supports--> v10\n- v03 --refutes--> v04\n- v0f --refutes--> v14\n- v0d --refutes--> v06\n- v07 --refutes--> v08\n- v05 --refutes--> v0a\n- v06 --supports--> v0b\n- v08 --refutes--> v16\n- v11 --supports--> v0e\n- v13 --refutes--> v05\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f05f573f61c261e66eefee978c002fe301e529e8cc4330f886e0720d7fb3aeb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0e\n- v04 --supports--> v15\n- v11 --refutes--> v09\n- v13 --refutes--> v06\n- v05 --refutes--> v08\n- v11 --supports--> v0d\n- v00 --refutes--> v10\n- v0c --supports--> v13\n- v13 --refutes--> v07\n- v0e --supports--> v0a\n- v06 --refutes--> v02\n- v00 --supports--> v03\n- v0b --refutes--> v11\n- v14 --refutes--> v0c\n- v00 --supports--> v05\n- v05 --refutes--> v0b\n- v02 --refutes--> v16\n- v0a --refutes--> v0f\n- v05 --supports--> v01\n- v02 --supports--> v00\n- v12 --supports--> v14\n- v0d --refutes--> v04\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":"c5f40067e97cb75743dba601a7a49ffa82b5784521672ca8ba2514435762f263","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v02\n- v02 --refutes--> v09\n- v00 --supports--> v05\n- v03 --refutes--> v14\n- v03 --supports--> v07\n- v0b --refutes--> v0d\n- v15 --refutes--> v0c\n- v08 --supports--> v06\n- v03 --refutes--> v01\n- v13 --supports--> v16\n- v15 --supports--> v04\n- v0c --refutes--> v11\n- v06 --supports--> v15\n- v03 --refutes--> v0a\n- v0f --supports--> v0e\n- v10 --supports--> v00\n- v12 --refutes--> v0f\n- v0e --supports--> v03\n- v00 --supports--> v12\n- v02 --refutes--> v10\n- v13 --supports--> v08\n- v0a --supports--> v13\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":"76e304a4caba07554dbfd890e9943dbaaf39eaeec9c8c20b2997037385e5d5af","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v15\n- v0d --refutes--> v14\n- v16 --supports--> v12\n- v0a --supports--> v0e\n- v07 --refutes--> v0d\n- v12 --refutes--> v10\n- v00 --refutes--> v01\n- v02 --supports--> v07\n- v11 --supports--> v13\n- v0f --refutes--> v09\n- v02 --refutes--> v08\n- v12 --supports--> v0a\n- v0d --supports--> v11\n- v0f --supports--> v00\n- v01 --refutes--> v04\n- v0e --supports--> v02\n- v02 --supports--> v06\n- v13 --refutes--> v05\n- v0e --refutes--> v03\n- v0c --supports--> v0f\n- v04 --refutes--> v0b\n- v0b --supports--> v16\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":"62df5f8ef9f8690a358a39691b544a142ea04ce376f19561cfbf72bd74798b4d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v09 --supports--> v08\n- v07 --refutes--> v09\n- v13 --refutes--> v0c\n- v06 --supports--> v12\n- v00 --refutes--> v01\n- v15 --supports--> v13\n- v0e --supports--> v00\n- v0e --refutes--> v14\n- v07 --refutes--> v11\n- v0d --supports--> v15\n- v0e --supports--> v04\n- v0c --supports--> v07\n- v02 --refutes--> v0e\n- v08 --refutes--> v05\n- v12 --supports--> v0f\n- v00 --supports--> v0d\n- v0a --refutes--> v02\n- v0f --supports--> v0b\n- v0d --refutes--> v16\n- v0b --supports--> v0a\n- v02 --refutes--> v10\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":"ba0ac500e4c37c0af488a656db900e6cc880145c6bebbc4083f66f75521b70cc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v16\n- v04 --refutes--> v0f\n- v01 --refutes--> v09\n- v0c --refutes--> v08\n- v16 --refutes--> v0c\n- v14 --supports--> v0e\n- v12 --refutes--> v15\n- v0d --supports--> v07\n- v10 --refutes--> v00\n- v13 --refutes--> v0d\n- v08 --supports--> v06\n- v05 --supports--> v03\n- v05 --supports--> v11\n- v15 --supports--> v04\n- v05 --refutes--> v12\n- v13 --refutes--> v0b\n- v0a --refutes--> v01\n- v0d --supports--> v10\n- v10 --supports--> v05\n- v0e --supports--> v0a\n- v04 --supports--> v14\n- v08 --supports--> v02\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":"1e661fcccfc682942337b307dadf2b2c7f7fa4793eee643e96b435f4abede079","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v07\n- v0d --supports--> v09\n- v12 --supports--> v0d\n- v0a --supports--> v0b\n- v0f --refutes--> v13\n- v0e --refutes--> v06\n- v13 --refutes--> v12\n- v0b --refutes--> v00\n- v03 --supports--> v08\n- v16 --supports--> v10\n- v0e --refutes--> v14\n- v0d --supports--> v15\n- v15 --refutes--> v04\n- v05 --supports--> v0e\n- v01 --supports--> v05\n- v15 --refutes--> v11\n- v0a --supports--> v02\n- v16 --supports--> v0a\n- v14 --supports--> v0f\n- v11 --refutes--> v16\n- v08 --refutes--> v01\n- v05 --supports--> v0c\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":"8bfb18cab03231fb0ec4124f05f208ad9231a9ed5525f2b1158347acdef9b9f7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v14\n- v03 --refutes--> v08\n- v0c --supports--> v0a\n- v0e --supports--> v11\n- v07 --supports--> v06\n- v07 --supports--> v0c\n- v14 --supports--> v00\n- v07 --supports--> v02\n- v0d --refutes--> v15\n- v09 --supports--> v04\n- v08 --refutes--> v13\n- v16 --supports--> v01\n- v13 --refutes--> v12\n- v11 --supports--> v09\n- v15 --supports--> v0e\n- v10 --supports--> v0f\n- v14 --supports--> v0d\n- v0c --supports--> v05\n- v13 --supports--> v16\n- v09 --refutes--> v10\n- v0a --supports--> v03\n- v0e --supports--> v0b\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"60a9aa87df016f04b03a81f1a7962008bb198a8473a160a4890cd6d05bf0ab49","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0d\n- v10 --supports--> v16\n- v00 --refutes--> v0c\n- v02 --supports--> v0e\n- v14 --supports--> v10\n- v13 --refutes--> v06\n- v12 --refutes--> v0b\n- v01 --refutes--> v05\n- v00 --refutes--> v11\n- v10 --supports--> v13\n- v0b --refutes--> v09\n- v04 --supports--> v01\n- v09 --refutes--> v15\n- v08 --refutes--> v02\n- v0a --supports--> v12\n- v0c --supports--> v07\n- v09 --refutes--> v0f\n- v0f --refutes--> v14\n- v01 --supports--> v00\n- v08 --supports--> v03\n- v0e --supports--> v0a\n- v13 --supports--> v04\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea1987dd3349c6c3ba1ad49e47cef52c56834b721e61bfad2733bb8c4081a1dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v08\n- v0b --refutes--> v03\n- v04 --refutes--> v0f\n- v05 --supports--> v06\n- v12 --supports--> v02\n- v11 --refutes--> v13\n- v0b --supports--> v01\n- v15 --supports--> v0d\n- v00 --refutes--> v15\n- v0d --supports--> v0e\n- v05 --supports--> v09\n- v12 --refutes--> v0a\n- v16 --supports--> v0c\n- v04 --refutes--> v14\n- v06 --supports--> v07\n- v0a --supports--> v16\n- v04 --supports--> v00\n- v03 --supports--> v11\n- v07 --refutes--> v12\n- v13 --supports--> v10\n- v13 --supports--> v04\n- v0c --supports--> v0b\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":"98e0a39df0c0cba7ff2f66bef4ab357ef5c8989cb19aa1be3613546e40192426","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v16\n- v10 --supports--> v01\n- v06 --refutes--> v0d\n- v0e --refutes--> v06\n- v07 --refutes--> v0e\n- v10 --refutes--> v13\n- v14 --supports--> v10\n- v15 --refutes--> v0b\n- v05 --refutes--> v00\n- v03 --supports--> v15\n- v01 --refutes--> v02\n- v0b --refutes--> v09\n- v0b --refutes--> v11\n- v14 --refutes--> v08\n- v11 --supports--> v0f\n- v07 --supports--> v04\n- v11 --refutes--> v0a\n- v0b --refutes--> v0c\n- v12 --refutes--> v14\n- v00 --supports--> v07\n- v0d --supports--> v03\n- v02 --supports--> v05\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":"fbc9b7f22c76c5e395fa1410392daec3283f457b1048a0efbcb08c8a183186b5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v05\n- v06 --refutes--> v08\n- v00 --refutes--> v0e\n- v07 --supports--> v01\n- v0a --refutes--> v0f\n- v14 --supports--> v0d\n- v15 --refutes--> v02\n- v11 --supports--> v03\n- v07 --refutes--> v13\n- v06 --supports--> v07\n- v04 --supports--> v16\n- v0e --refutes--> v06\n- v01 --refutes--> v11\n- v05 --supports--> v04\n- v10 --refutes--> v15\n- v0a --supports--> v14\n- v01 --supports--> v09\n- v02 --supports--> v0a\n- v03 --supports--> v10\n- v0d --refutes--> v0c\n- v07 --refutes--> v12\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":"3dc3ff2d9a6d30d50a3b99bfbf60b8adeab1904d839047990f5e9e58b44a4c05","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v08\n- v11 --refutes--> v06\n- v0d --refutes--> v05\n- v0c --supports--> v09\n- v00 --supports--> v10\n- v05 --supports--> v0e\n- v16 --supports--> v14\n- v07 --supports--> v0b\n- v13 --supports--> v01\n- v12 --refutes--> v13\n- v04 --refutes--> v03\n- v04 --refutes--> v0a\n- v12 --refutes--> v02\n- v15 --refutes--> v0d\n- v14 --refutes--> v0c\n- v12 --supports--> v0f\n- v0e --supports--> v07\n- v00 --refutes--> v16\n- v09 --supports--> v15\n- v07 --supports--> v12\n- v11 --supports--> v00\n- v03 --refutes--> v11\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":"834c7b337afc54548d7a06d127bc4802099054708f1c550469390b641f6ead2b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0c\n- v0c --refutes--> v07\n- v0a --supports--> v01\n- v06 --supports--> v09\n- v02 --supports--> v15\n- v14 --supports--> v0a\n- v04 --refutes--> v11\n- v0f --refutes--> v16\n- v02 --refutes--> v0b\n- v11 --supports--> v06\n- v0e --refutes--> v14\n- v0c --supports--> v04\n- v09 --supports--> v0e\n- v12 --refutes--> v0f\n- v15 --refutes--> v12\n- v11 --refutes--> v08\n- v15 --supports--> v03\n- v05 --supports--> v00\n- v16 --supports--> v05\n- v0d --supports--> v10\n- v0d --refutes--> v02\n- v14 --supports--> v13\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":"e5025990ccb661f06d79811e3825d1fa6cc6707c97d4d30c9928318e0644baa9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v14\n- v10 --refutes--> v00\n- v0c --supports--> v06\n- v03 --refutes--> v0b\n- v0b --supports--> v12\n- v15 --refutes--> v07\n- v14 --refutes--> v0f\n- v05 --refutes--> v13\n- v0b --refutes--> v15\n- v02 --supports--> v0d\n- v11 --refutes--> v09\n- v11 --supports--> v0a\n- v14 --supports--> v04\n- v07 --supports--> v11\n- v11 --supports--> v02\n- v16 --refutes--> v03\n- v0c --supports--> v08\n- v0c --refutes--> v10\n- v0d --refutes--> v05\n- v01 --refutes--> v0c\n- v03 --supports--> v0e\n- v04 --refutes--> v16\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"924b05730784461e3914c73a4244cd5fcba33704d72bd09d0bf895d3091da2ef","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v12\n- v0d --supports--> v01\n- v10 --supports--> v11\n- v04 --refutes--> v03\n- v03 --supports--> v0c\n- v16 --refutes--> v06\n- v09 --refutes--> v07\n- v07 --supports--> v15\n- v07 --refutes--> v02\n- v0e --refutes--> v0a\n- v10 --supports--> v08\n- v12 --supports--> v05\n- v05 --refutes--> v0e\n- v0c --supports--> v09\n- v05 --refutes--> v14\n- v01 --supports--> v13\n- v06 --supports--> v00\n- v06 --supports--> v04\n- v13 --refutes--> v0f\n- v0e --refutes--> v0d\n- v0f --refutes--> v16\n- v06 --refutes--> v0b\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":"558be4cd9df72378e3393588cb883db258e5c3a318855bedb71c531b6ad1d646","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v04\n- v15 --refutes--> v0d\n- v06 --supports--> v09\n- v0a --refutes--> v08\n- v03 --refutes--> v00\n- v0d --supports--> v07\n- v0f --supports--> v15\n- v08 --refutes--> v12\n- v13 --supports--> v16\n- v08 --supports--> v03\n- v06 --refutes--> v10\n- v0f --supports--> v0e\n- v0b --refutes--> v0c\n- v06 --supports--> v11\n- v05 --supports--> v0a\n- v01 --refutes--> v0f\n- v0b --supports--> v05\n- v14 --refutes--> v06\n- v04 --refutes--> v0b\n- v03 --refutes--> v14\n- v06 --supports--> v02\n- v07 --refutes--> v13\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":"7186e5e68c5cbe34d321274c961870c06f5ef6cd89b4ad9425133108906a4882","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v05\n- v14 --supports--> v11\n- v12 --supports--> v16\n- v10 --supports--> v08\n- v08 --supports--> v07\n- v00 --supports--> v06\n- v11 --refutes--> v10\n- v02 --refutes--> v0c\n- v10 --supports--> v15\n- v07 --refutes--> v00\n- v16 --refutes--> v0f\n- v06 --supports--> v0a\n- v04 --refutes--> v0b\n- v02 --refutes--> v0e\n- v0e --supports--> v03\n- v08 --supports--> v01\n- v04 --refutes--> v02\n- v13 --supports--> v14\n- v03 --refutes--> v12\n- v0f --supports--> v13\n- v0f --supports--> v09\n- v06 --supports--> v0d\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":"d27c8907801986dd359cc982b051e82f316b9fd696c8fbb16add0d74395efaec","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v0b\n- v03 --supports--> v14\n- v0d --refutes--> v08\n- v08 --refutes--> v00\n- v15 --refutes--> v05\n- v0f --refutes--> v09\n- v03 --supports--> v0e\n- v02 --supports--> v0a\n- v06 --refutes--> v16\n- v03 --supports--> v11\n- v11 --supports--> v12\n- v09 --refutes--> v07\n- v00 --refutes--> v06\n- v11 --refutes--> v13\n- v13 --supports--> v01\n- v09 --refutes--> v0d\n- v0f --supports--> v0c\n- v01 --supports--> v04\n- v05 --supports--> v03\n- v16 --supports--> v15\n- v04 --refutes--> v02\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":"533d6a2bb6193a0db756a086a3a6db0efb62dbb4f682b0c827543c878ea645d7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v03\n- v02 --refutes--> v07\n- v0b --supports--> v0a\n- v05 --supports--> v0b\n- v13 --refutes--> v14\n- v16 --refutes--> v0e\n- v01 --supports--> v10\n- v01 --supports--> v06\n- v13 --refutes--> v16\n- v13 --supports--> v0d\n- v03 --supports--> v01\n- v0b --supports--> v13\n- v11 --refutes--> v04\n- v09 --refutes--> v08\n- v00 --supports--> v15\n- v08 --supports--> v05\n- v00 --refutes--> v11\n- v03 --supports--> v0f\n- v0c --supports--> v02\n- v02 --refutes--> v12\n- v04 --refutes--> v0c\n- v06 --supports--> v00\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":"5a8f6c24b467cb2236682139a9008bb2bbd814a051081fddd374370570bd79fe","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v01\n- v0a --refutes--> v10\n- v01 --supports--> v08\n- v01 --refutes--> v0b\n- v07 --supports--> v14\n- v06 --refutes--> v0d\n- v0d --refutes--> v07\n- v0f --supports--> v09\n- v13 --refutes--> v16\n- v0d --supports--> v15\n- v00 --refutes--> v11\n- v0a --refutes--> v04\n- v05 --supports--> v0f\n- v14 --supports--> v05\n- v0e --supports--> v0c\n- v0c --supports--> v12\n- v09 --refutes--> v00\n- v0e --supports--> v02\n- v0a --supports--> v06\n- v13 --supports--> v0a\n- v0b --refutes--> v0e\n- v06 --supports--> v03\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":"9106bd2e02ce8db155b57c74396a40358d10fda3db3cdb6ee2a8062437a254ec","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v0a\n- v13 --supports--> v04\n- v16 --refutes--> v13\n- v0b --supports--> v07\n- v09 --supports--> v0e\n- v0f --refutes--> v08\n- v12 --supports--> v06\n- v0d --refutes--> v03\n- v0a --refutes--> v0f\n- v13 --supports--> v14\n- v0c --refutes--> v02\n- v09 --supports--> v0d\n- v05 --supports--> v00\n- v00 --refutes--> v10\n- v03 --refutes--> v01\n- v16 --supports--> v11\n- v14 --refutes--> v05\n- v0b --refutes--> v15\n- v03 --supports--> v0b\n- v06 --refutes--> v16\n- v0f --refutes--> v12\n- v10 --refutes--> v09\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"71a2882b056f36ca3a636d88e0156e9ce0057988585d4259195b27362e2226c7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0a\n- v16 --refutes--> v06\n- v09 --refutes--> v0d\n- v09 --supports--> v0c\n- v10 --supports--> v0f\n- v06 --supports--> v13\n- v0c --supports--> v01\n- v04 --refutes--> v14\n- v05 --supports--> v09\n- v14 --supports--> v0b\n- v03 --supports--> v05\n- v06 --supports--> v00\n- v10 --supports--> v11\n- v0e --supports--> v16\n- v14 --supports--> v07\n- v02 --supports--> v08\n- v15 --refutes--> v02\n- v13 --supports--> v15\n- v07 --refutes--> v03\n- v0c --supports--> v0e\n- v07 --refutes--> v12\n- v0f --supports--> v04\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":"dd81ba3bfbdc716c0ccf16614057d62de210a484b96a04c8e079dcfeb79b5866","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v15\n- v01 --refutes--> v0e\n- v0f --supports--> v07\n- v0c --supports--> v13\n- v16 --refutes--> v03\n- v12 --refutes--> v14\n- v05 --supports--> v12\n- v16 --supports--> v0a\n- v0b --supports--> v16\n- v0e --refutes--> v0f\n- v02 --refutes--> v00\n- v06 --refutes--> v11\n- v11 --refutes--> v0b\n- v0c --supports--> v01\n- v0a --supports--> v10\n- v10 --supports--> v05\n- v02 --refutes--> v08\n- v0f --supports--> v09\n- v00 --supports--> v06\n- v02 --supports--> v04\n- v0a --refutes--> v0d\n- v07 --supports--> v02\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":"b44f6c5a7f6749d05c846202d51dfdaab79b9de5d756b64ab1e3514156fc29de","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0b\n- v0a --supports--> v12\n- v05 --supports--> v11\n- v0f --supports--> v09\n- v0f --refutes--> v03\n- v15 --supports--> v10\n- v01 --supports--> v08\n- v04 --refutes--> v05\n- v11 --supports--> v15\n- v0b --supports--> v07\n- v12 --refutes--> v0f\n- v0a --refutes--> v16\n- v08 --supports--> v00\n- v14 --refutes--> v0e\n- v07 --refutes--> v14\n- v0e --refutes--> v06\n- v15 --supports--> v0d\n- v08 --refutes--> v02\n- v00 --refutes--> v0a\n- v13 --refutes--> v01\n- v0e --supports--> v13\n- v14 --supports--> v0c\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":"23bb640ed946a2aac2500e000ba9ab1b1912d2335d57babe03bc34c388146adf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v06\n- v08 --refutes--> v0c\n- v0d --supports--> v0f\n- v09 --supports--> v16\n- v12 --refutes--> v03\n- v01 --supports--> v09\n- v16 --supports--> v0d\n- v08 --supports--> v05\n- v0e --refutes--> v07\n- v14 --supports--> v13\n- v01 --supports--> v11\n- v13 --refutes--> v00\n- v0a --supports--> v0e\n- v0d --refutes--> v0b\n- v0b --supports--> v10\n- v0b --refutes--> v12\n- v05 --supports--> v14\n- v15 --supports--> v08\n- v07 --refutes--> v01\n- v03 --refutes--> v04\n- v03 --refutes--> v15\n- v05 --supports--> v02\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":"2f6d97839e4c41f01400a6ceca1d0f892b14a70224eab1eaac2adc18ca1b5223","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v0c --supports--> v12\n- v0e --supports--> v0a\n- v05 --refutes--> v0f\n- v08 --refutes--> v07\n- v0f --supports--> v16\n- v0f --supports--> v13\n- v05 --refutes--> v00\n- v03 --refutes--> v0e\n- v08 --supports--> v03\n- v13 --refutes--> v04\n- v01 --supports--> v06\n- v04 --refutes--> v02\n- v02 --supports--> v08\n- v08 --supports--> v15\n- v0f --refutes--> v11\n- v0a --refutes--> v14\n- v0c --supports--> v0b\n- v14 --supports--> v0d\n- v09 --refutes--> v0c\n- v0b --supports--> v10\n- v0b --supports--> v05\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":"21979c0e59034961feb604748ad8a698f4971e17ce65cff1d25e351f2d644194","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v04\n- v07 --refutes--> v03\n- v03 --supports--> v05\n- v06 --refutes--> v09\n- v09 --refutes--> v0f\n- v08 --refutes--> v15\n- v0c --refutes--> v01\n- v08 --supports--> v0a\n- v06 --refutes--> v02\n- v03 --supports--> v08\n- v15 --supports--> v0d\n- v01 --refutes--> v10\n- v0c --refutes--> v11\n- v04 --refutes--> v0c\n- v07 --refutes--> v0e\n- v14 --refutes--> v13\n- v0f --refutes--> v0b\n- v00 --supports--> v12\n- v10 --supports--> v14\n- v13 --refutes--> v07\n- v08 --refutes--> v16\n- v00 --supports--> v06\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":"6d7d3b0b2d9ba3ec2a994bd7c5e18a61fdf4fddd9f56c949d432756c585aa683","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v13\n- v01 --supports--> v05\n- v09 --supports--> v03\n- v0f --refutes--> v0c\n- v12 --refutes--> v0e\n- v03 --supports--> v07\n- v08 --supports--> v00\n- v0a --supports--> v08\n- v0e --supports--> v10\n- v0e --refutes--> v15\n- v00 --supports--> v12\n- v15 --refutes--> v0b\n- v16 --refutes--> v04\n- v0c --refutes--> v11\n- v09 --supports--> v02\n- v16 --supports--> v0d\n- v05 --supports--> v06\n- v09 --refutes--> v14\n- v0c --supports--> v16\n- v03 --supports--> v0a\n- v0b --supports--> v0f\n- v0d --refutes--> v01\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":"56dbd285b86ff30896c064a351f9f5cee871e47d362f3e3a40cc16ebb9564e58","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v13\n- v14 --refutes--> v08\n- v01 --refutes--> v03\n- v13 --refutes--> v09\n- v09 --refutes--> v00\n- v04 --supports--> v15\n- v0f --supports--> v14\n- v0e --refutes--> v0b\n- v09 --supports--> v12\n- v08 --refutes--> v04\n- v07 --supports--> v10\n- v04 --refutes--> v0d\n- v0c --refutes--> v05\n- v10 --refutes--> v01\n- v12 --refutes--> v16\n- v0b --refutes--> v0c\n- v04 --supports--> v0a\n- v12 --refutes--> v07\n- v10 --refutes--> v11\n- v03 --refutes--> v0f\n- v06 --refutes--> v0e\n- v03 --refutes--> v02\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1930ff479b42eb050af4e9bb3a7634cf986660a6d2239a97ef785775ef81904b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v10\n- v0c --refutes--> v04\n- v10 --refutes--> v08\n- v00 --refutes--> v0a\n- v12 --supports--> v05\n- v14 --refutes--> v16\n- v0f --refutes--> v11\n- v0b --supports--> v07\n- v11 --supports--> v03\n- v00 --supports--> v02\n- v01 --refutes--> v0f\n- v0c --refutes--> v0d\n- v09 --supports--> v01\n- v08 --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v09\n- v03 --refutes--> v00\n- v0c --supports--> v14\n- v03 --supports--> v15\n- v11 --refutes--> v13\n- v16 --supports--> v12\n- v0c --refutes--> v06\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":"6228384ec272097aeda352680eaba798f19d9b8711154d526c9487dd0156db27","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0f\n- v14 --supports--> v06\n- v15 --supports--> v01\n- v0b --supports--> v03\n- v01 --refutes--> v08\n- v04 --refutes--> v15\n- v00 --supports--> v09\n- v07 --supports--> v0a\n- v03 --supports--> v05\n- v03 --supports--> v0d\n- v00 --supports--> v04\n- v06 --refutes--> v10\n- v06 --refutes--> v16\n- v0a --supports--> v00\n- v16 --supports--> v0e\n- v01 --supports--> v13\n- v0e --supports--> v07\n- v02 --refutes--> v0c\n- v01 --supports--> v02\n- v02 --refutes--> v12\n- v11 --refutes--> v14\n- v0c --supports--> v0b\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":"21fa1d2ffc7f864d7b20286669be6ac0b270cd0340983bee5ab929a46ebeab6b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v01\n- v09 --refutes--> v10\n- v14 --supports--> v02\n- v15 --supports--> v07\n- v0a --refutes--> v0f\n- v0e --supports--> v0d\n- v0e --supports--> v00\n- v14 --supports--> v04\n- v03 --supports--> v13\n- v16 --refutes--> v03\n- v10 --refutes--> v05\n- v01 --refutes--> v0a\n- v0a --refutes--> v0b\n- v04 --refutes--> v15\n- v0e --refutes--> v14\n- v16 --refutes--> v11\n- v13 --supports--> v08\n- v0b --refutes--> v06\n- v16 --refutes--> v0c\n- v07 --supports--> v09\n- v06 --supports--> v0e\n- v04 --supports--> v12\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":"1d0f508dd791f801b967ee25f0db7f5b9ed8986508d6613f455ee2f241fd3852","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v12\n- v0a --refutes--> v00\n- v07 --refutes--> v14\n- v06 --refutes--> v03\n- v0a --refutes--> v0d\n- v08 --supports--> v02\n- v16 --refutes--> v0b\n- v00 --supports--> v13\n- v16 --supports--> v0a\n- v04 --supports--> v08\n- v0f --supports--> v09\n- v01 --supports--> v11\n- v07 --supports--> v0f\n- v10 --refutes--> v0c\n- v05 --supports--> v01\n- v02 --refutes--> v15\n- v03 --supports--> v05\n- v0c --refutes--> v06\n- v0f --supports--> v0e\n- v00 --supports--> v04\n- v01 --refutes--> v07\n- v0e --refutes--> v16\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":"b7003521b36b85a49fceacfd3492055ae23ac4afd21d4eae2afc27aa2db96eac","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v11\n- v09 --refutes--> v14\n- v14 --refutes--> v0a\n- v00 --supports--> v04\n- v05 --refutes--> v09\n- v10 --supports--> v15\n- v0d --refutes--> v16\n- v0a --supports--> v12\n- v11 --supports--> v00\n- v04 --refutes--> v0b\n- v0b --refutes--> v0e\n- v0d --refutes--> v03\n- v01 --refutes--> v0d\n- v02 --refutes--> v10\n- v0b --supports--> v0c\n- v0f --supports--> v01\n- v10 --supports--> v06\n- v0a --supports--> v08\n- v0b --refutes--> v02\n- v15 --supports--> v05\n- v0a --supports--> v0f\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":"c3ff6718f2ca4c63e2f23111b1ea82b541858efe974005374b614db883cf4408","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v10\n- v09 --refutes--> v15\n- v11 --supports--> v04\n- v07 --refutes--> v16\n- v12 --supports--> v08\n- v0b --supports--> v00\n- v0e --supports--> v07\n- v14 --supports--> v13\n- v0d --refutes--> v06\n- v01 --refutes--> v14\n- v03 --supports--> v0b\n- v0c --supports--> v12\n- v13 --refutes--> v03\n- v09 --supports--> v0d\n- v16 --supports--> v11\n- v09 --refutes--> v02\n- v12 --refutes--> v0a\n- v12 --refutes--> v05\n- v06 --supports--> v01\n- v07 --refutes--> v0f\n- v11 --refutes--> v09\n- v00 --supports--> v0c\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":"09ca8aa87db584420439abc883c37081d8bd50a163f4ce756dd91af2944c0e88","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v04\n- v16 --refutes--> v02\n- v0d --refutes--> v11\n- v15 --refutes--> v06\n- v02 --supports--> v01\n- v00 --supports--> v09\n- v0e --refutes--> v16\n- v07 --refutes--> v12\n- v14 --refutes--> v10\n- v01 --refutes--> v15\n- v0d --refutes--> v0e\n- v06 --supports--> v14\n- v0a --supports--> v00\n- v07 --supports--> v13\n- v13 --supports--> v05\n- v08 --refutes--> v0c\n- v02 --supports--> v0b\n- v14 --refutes--> v07\n- v0c --refutes--> v0a\n- v09 --refutes--> v0d\n- v16 --supports--> v0f\n- v0d --supports--> v03\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"755df222ba468bda254011804eb14b8b3329f7f5d15b702cf24350349dfc270f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v14\n- v10 --supports--> v04\n- v00 --refutes--> v0b\n- v0c --refutes--> v0e\n- v0f --supports--> v10\n- v01 --supports--> v0f\n- v13 --refutes--> v01\n- v05 --refutes--> v03\n- v09 --refutes--> v13\n- v10 --refutes--> v00\n- v13 --refutes--> v08\n- v07 --supports--> v11\n- v02 --refutes--> v12\n- v09 --refutes--> v06\n- v02 --supports--> v0c\n- v14 --refutes--> v02\n- v0c --refutes--> v0a\n- v0a --refutes--> v05\n- v03 --supports--> v07\n- v0b --supports--> v16\n- v0b --refutes--> v0d\n- v05 --refutes--> v15\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":"09c6a923dd80aa94d2dbac78e93c3ba552c295cdf3176b0f9916bfa66061bf69","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0e\n- v10 --supports--> v04\n- v00 --supports--> v0b\n- v04 --supports--> v00\n- v16 --supports--> v12\n- v0e --refutes--> v10\n- v07 --supports--> v11\n- v09 --supports--> v16\n- v11 --refutes--> v0f\n- v01 --refutes--> v05\n- v08 --supports--> v03\n- v16 --supports--> v0d\n- v11 --refutes--> v09\n- v05 --refutes--> v08\n- v16 --refutes--> v0c\n- v02 --supports--> v07\n- v0b --supports--> v02\n- v03 --supports--> v14\n- v16 --supports--> v06\n- v02 --refutes--> v0a\n- v09 --refutes--> v15\n- v0d --supports--> v13\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":"346b900b5f03194866741872cff5733f131ef618bab5fcb3b0e91cc84f881eec","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0a\n- v0d --refutes--> v10\n- v0c --refutes--> v15\n- v04 --supports--> v02\n- v08 --refutes--> v00\n- v11 --refutes--> v13\n- v13 --supports--> v0c\n- v08 --supports--> v0f\n- v00 --supports--> v05\n- v01 --supports--> v09\n- v15 --supports--> v07\n- v0a --supports--> v01\n- v04 --supports--> v06\n- v0d --supports--> v03\n- v09 --refutes--> v11\n- v04 --supports--> v0b\n- v08 --refutes--> v14\n- v03 --refutes--> v12\n- v0d --supports--> v16\n- v0e --supports--> v0d\n- v06 --refutes--> v0e\n- v07 --supports--> v08\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e76755cea64c7afc55e552e85cadb66505b74d46de0c3b7cd7a677b7e8e70165","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v05\n- v14 --supports--> v13\n- v05 --supports--> v09\n- v0f --refutes--> v0b\n- v0c --supports--> v14\n- v16 --refutes--> v0e\n- v10 --supports--> v0f\n- v0d --refutes--> v0c\n- v0f --refutes--> v12\n- v04 --supports--> v01\n- v14 --refutes--> v11\n- v16 --refutes--> v0d\n- v03 --refutes--> v08\n- v03 --refutes--> v0a\n- v00 --supports--> v15\n- v09 --supports--> v16\n- v0f --supports--> v00\n- v00 --refutes--> v03\n- v14 --refutes--> v02\n- v0a --supports--> v06\n- v06 --refutes--> v04\n- v02 --refutes--> v10\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":"5bc3d36119629361271bf45c37a4f85ea28042b5e0ccd362c56ea70b4f8c257a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0a\n- v0e --refutes--> v03\n- v0d --refutes--> v06\n- v04 --refutes--> v0d\n- v07 --supports--> v11\n- v00 --refutes--> v01\n- v12 --supports--> v10\n- v0c --supports--> v0f\n- v02 --supports--> v13\n- v0d --supports--> v07\n- v13 --supports--> v12\n- v0c --refutes--> v02\n- v0b --refutes--> v08\n- v01 --refutes--> v09\n- v00 --supports--> v05\n- v14 --supports--> v0c\n- v07 --refutes--> v0e\n- v02 --refutes--> v15\n- v08 --refutes--> v14\n- v03 --supports--> v16\n- v09 --supports--> v0b\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b74376959a987c0cd516e7d8a9868e52d59ca3d71130ba308784ae3e1dd3e8cf","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v15\n- v0d --refutes--> v09\n- v07 --refutes--> v16\n- v01 --supports--> v0a\n- v0a --supports--> v07\n- v00 --refutes--> v04\n- v0b --supports--> v02\n- v0e --refutes--> v0d\n- v15 --supports--> v08\n- v02 --supports--> v0e\n- v0f --refutes--> v0c\n- v12 --refutes--> v05\n- v08 --refutes--> v01\n- v08 --supports--> v14\n- v03 --refutes--> v00\n- v0d --refutes--> v12\n- v0c --refutes--> v11\n- v07 --refutes--> v03\n- v01 --supports--> v10\n- v11 --refutes--> v0b\n- v11 --supports--> v13\n- v0c --supports--> v06\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":"30292e1402add34655ea572fc4fdc3803b091c7a424ad5bda36fc62f176099e4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v12\n- v08 --refutes--> v0b\n- v15 --refutes--> v10\n- v11 --refutes--> v09\n- v11 --supports--> v08\n- v04 --supports--> v00\n- v0d --supports--> v0f\n- v10 --refutes--> v04\n- v01 --supports--> v03\n- v01 --supports--> v0d\n- v14 --refutes--> v13\n- v05 --refutes--> v0a\n- v0a --supports--> v01\n- v12 --supports--> v11\n- v16 --supports--> v0c\n- v14 --refutes--> v06\n- v16 --refutes--> v15\n- v13 --supports--> v02\n- v0d --supports--> v0e\n- v04 --supports--> v14\n- v04 --refutes--> v07\n- v0b --refutes--> v05\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":"70ae0cf79ff8249de4b466cd2ab02d825ae3ac8ec3708e35824c557eef67bef4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v04\n- v16 --refutes--> v06\n- v0e --supports--> v05\n- v04 --supports--> v0d\n- v08 --refutes--> v16\n- v07 --refutes--> v11\n- v13 --refutes--> v15\n- v13 --refutes--> v0b\n- v0a --refutes--> v08\n- v14 --supports--> v01\n- v00 --supports--> v0a\n- v14 --supports--> v09\n- v0b --refutes--> v07\n- v06 --refutes--> v02\n- v0c --refutes--> v12\n- v0d --refutes--> v14\n- v13 --supports--> v03\n- v05 --refutes--> v0c\n- v14 --supports--> v0e\n- v0c --supports--> v13\n- v11 --supports--> v0f\n- v08 --refutes--> v10\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":"03cc37fd92f3c75dda7fab550ed2daa810a6c84e694380643efb2debbcc15492","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v05\n- v0f --supports--> v02\n- v0d --refutes--> v07\n- v08 --refutes--> v15\n- v0b --supports--> v10\n- v04 --supports--> v14\n- v09 --supports--> v0a\n- v11 --supports--> v12\n- v0e --supports--> v09\n- v02 --refutes--> v04\n- v05 --refutes--> v00\n- v15 --supports--> v01\n- v15 --refutes--> v16\n- v0b --refutes--> v08\n- v00 --supports--> v11\n- v09 --refutes--> v0b\n- v0d --refutes--> v06\n- v15 --refutes--> v13\n- v0c --supports--> v03\n- v12 --supports--> v0d\n- v03 --supports--> v0e\n- v04 --supports--> v0c\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5189219b8a9e2366ca966e6e9f065eb65ca841d6e36140a8bd292a0cddd7a8b5","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v10\n- v07 --supports--> v05\n- v16 --supports--> v0b\n- v0d --refutes--> v0a\n- v12 --refutes--> v06\n- v0f --supports--> v04\n- v01 --refutes--> v02\n- v13 --refutes--> v03\n- v08 --refutes--> v16\n- v07 --refutes--> v13\n- v11 --supports--> v0f\n- v00 --supports--> v08\n- v11 --refutes--> v14\n- v13 --refutes--> v11\n- v02 --supports--> v0d\n- v0e --refutes--> v15\n- v0b --supports--> v0c\n- v04 --refutes--> v12\n- v12 --refutes--> v0e\n- v0b --supports--> v01\n- v09 --refutes--> v07\n- v0e --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":"598eb299155a0f32eb221f0ef259fac16cb8795c8001d2ceedb88b43cdc7b286","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v04\n- v11 --supports--> v05\n- v14 --supports--> v01\n- v0c --supports--> v07\n- v02 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v09\n- v08 --refutes--> v00\n- v11 --refutes--> v12\n- v06 --supports--> v13\n- v05 --supports--> v0d\n- v0a --refutes--> v06\n- v10 --refutes--> v0a\n- v0f --refutes--> v0b\n- v06 --supports--> v02\n- v04 --supports--> v0c\n- v03 --refutes--> v15\n- v0b --supports--> v08\n- v03 --supports--> v14\n- v07 --refutes--> v03\n- v02 --supports--> v16\n- v05 --refutes--> v10\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":"ba90cf2bc8c51c10d272cb21086b8a81c836a4d8c54bf0e75eeee91286c20df6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v13\n- v12 --supports--> v05\n- v07 --supports--> v06\n- v05 --refutes--> v00\n- v11 --refutes--> v03\n- v11 --supports--> v0d\n- v09 --refutes--> v0a\n- v03 --supports--> v0b\n- v06 --refutes--> v0f\n- v00 --refutes--> v09\n- v13 --refutes--> v01\n- v07 --refutes--> v08\n- v05 --supports--> v04\n- v02 --refutes--> v16\n- v0f --supports--> v12\n- v09 --refutes--> v02\n- v10 --supports--> v14\n- v15 --refutes--> v10\n- v03 --refutes--> v0e\n- v16 --refutes--> v15\n- v0e --supports--> v07\n- v13 --supports--> v0c\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":"7e91f786366f2731cb423af9680dd626ec51a0a6d6a3d7b844f97c9ce7135313","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v11\n- v03 --refutes--> v04\n- v0d --refutes--> v09\n- v15 --supports--> v13\n- v13 --refutes--> v01\n- v0d --refutes--> v06\n- v0d --supports--> v16\n- v0e --refutes--> v14\n- v0a --refutes--> v15\n- v12 --refutes--> v07\n- v13 --refutes--> v10\n- v02 --refutes--> v0a\n- v07 --refutes--> v0d\n- v0c --supports--> v08\n- v0f --supports--> v0e\n- v16 --supports--> v03\n- v15 --refutes--> v05\n- v14 --supports--> v00\n- v07 --supports--> v0b\n- v04 --refutes--> v0c\n- v00 --refutes--> v02\n- v0c --refutes--> v0f\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":"a671f8d1e5895d90c779a58759988fae1bfc6578ddf82400fa18677a02a5b3d4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v05\n- v09 --refutes--> v01\n- v08 --refutes--> v12\n- v02 --refutes--> v07\n- v05 --supports--> v09\n- v15 --refutes--> v13\n- v14 --supports--> v10\n- v0e --refutes--> v0a\n- v11 --refutes--> v04\n- v0d --refutes--> v16\n- v06 --refutes--> v0b\n- v06 --refutes--> v03\n- v03 --refutes--> v14\n- v01 --supports--> v15\n- v03 --refutes--> v0f\n- v11 --refutes--> v0e\n- v11 --refutes--> v00\n- v08 --supports--> v06\n- v14 --refutes--> v0c\n- v15 --refutes--> v02\n- v16 --supports--> v11\n- v07 --supports--> v08\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":"6894f4686a8d1422d1bc8e3355cab678b7e606c471f3396556bc017bcf9e5eb7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v03\n- v08 --refutes--> v09\n- v09 --refutes--> v0c\n- v11 --refutes--> v01\n- v04 --supports--> v06\n- v0e --supports--> v14\n- v0a --refutes--> v08\n- v02 --refutes--> v10\n- v07 --refutes--> v0e\n- v0f --supports--> v18\n- v18 --supports--> v05\n- v0c --supports--> v11\n- v0e --refutes--> v04\n- v04 --supports--> v0d\n- v11 --refutes--> v0b\n- v03 --refutes--> v07\n- v01 --refutes--> v17\n- v02 --supports--> v16\n- v18 --supports--> v13\n- v02 --refutes--> v12\n- v01 --supports--> v02\n- v0e --supports--> v00\n- v05 --refutes--> v15\n- v06 --refutes--> v0f\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":"51c7caa41baefd7a4cb4d78b98cbc6f980f1fcf414dda702e56c264c19c0ccab","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v16\n- v01 --refutes--> v0d\n- v13 --supports--> v0f\n- v0d --refutes--> v0c\n- v0b --supports--> v04\n- v00 --refutes--> v14\n- v07 --refutes--> v09\n- v0e --refutes--> v10\n- v0d --refutes--> v0e\n- v16 --supports--> v12\n- v06 --supports--> v02\n- v04 --supports--> v06\n- v17 --refutes--> v11\n- v18 --supports--> v07\n- v10 --refutes--> v00\n- v06 --refutes--> v0a\n- v09 --refutes--> v15\n- v0e --supports--> v03\n- v0f --supports--> v0b\n- v09 --refutes--> v05\n- v00 --supports--> v13\n- v09 --supports--> v01\n- v10 --refutes--> v08\n- v16 --refutes--> v18\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":"59c375cbae9f8162f645e080059e7bb639bb15df78db4ea8802769514abc3847","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v00\n- v0c --supports--> v12\n- v0d --refutes--> v16\n- v0f --supports--> v0e\n- v0f --supports--> v05\n- v17 --supports--> v11\n- v05 --supports--> v0d\n- v09 --supports--> v17\n- v06 --refutes--> v01\n- v14 --supports--> v0b\n- v15 --refutes--> v04\n- v15 --refutes--> v08\n- v08 --refutes--> v06\n- v16 --refutes--> v13\n- v00 --refutes--> v0f\n- v17 --refutes--> v0a\n- v16 --supports--> v07\n- v02 --refutes--> v10\n- v02 --supports--> v0c\n- v11 --supports--> v15\n- v06 --refutes--> v14\n- v14 --refutes--> v03\n- v08 --refutes--> v18\n- v12 --supports--> v09\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":"b68625adaa9863399f1ffe9d7c54f9f01e90145d297f583f542c1bf65f0ade07","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v0f\n- v06 --supports--> v14\n- v0d --refutes--> v02\n- v03 --refutes--> v07\n- v0c --supports--> v0d\n- v05 --supports--> v16\n- v09 --supports--> v00\n- v18 --supports--> v05\n- v05 --refutes--> v08\n- v03 --refutes--> v13\n- v08 --supports--> v06\n- v00 --supports--> v15\n- v04 --refutes--> v01\n- v12 --supports--> v0b\n- v08 --supports--> v17\n- v01 --refutes--> v0a\n- v0b --refutes--> v03\n- v01 --refutes--> v18\n- v18 --refutes--> v10\n- v03 --refutes--> v04\n- v0d --refutes--> v11\n- v00 --supports--> v12\n- v0f --supports--> v0c\n- v0e --supports--> v09\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":"8f57235f0fea7bb8767bc89ee657fa49f5ce4be1e02bac2a13788ffe257ff374","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v02\n- v14 --refutes--> v07\n- v01 --supports--> v0a\n- v00 --supports--> v05\n- v08 --refutes--> v0e\n- v01 --refutes--> v08\n- v0e --supports--> v0f\n- v06 --refutes--> v03\n- v0d --refutes--> v17\n- v0b --supports--> v10\n- v0e --refutes--> v12\n- v07 --refutes--> v0d\n- v12 --supports--> v09\n- v01 --refutes--> v18\n- v03 --refutes--> v00\n- v05 --refutes--> v15\n- v04 --refutes--> v16\n- v0d --refutes--> v04\n- v05 --refutes--> v11\n- v16 --supports--> v0b\n- v09 --supports--> v13\n- v0b --supports--> v01\n- v00 --refutes--> v0c\n- v13 --supports--> v06\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":"014ad9caab36369977021070a7b4c605478deac4c6684ca68805785001e659a7","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v0e\n- v0d --refutes--> v0c\n- v02 --refutes--> v15\n- v01 --refutes--> v14\n- v0a --supports--> v10\n- v0a --refutes--> v05\n- v06 --refutes--> v12\n- v15 --supports--> v08\n- v14 --refutes--> v16\n- v0d --refutes--> v0b\n- v07 --refutes--> v09\n- v12 --supports--> v0a\n- v13 --refutes--> v11\n- v11 --refutes--> v01\n- v0f --refutes--> v00\n- v06 --supports--> v18\n- v03 --supports--> v06\n- v07 --supports--> v04\n- v0b --supports--> v17\n- v0a --supports--> v0f\n- v09 --supports--> v03\n- v0f --refutes--> v02\n- v15 --refutes--> v0d\n- v17 --refutes--> v13\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":"77d134122fc5cd8a6893d6a809788261e4072f2604311de6025a6168479e6303","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v16\n- v04 --refutes--> v10\n- v08 --supports--> v05\n- v00 --refutes--> v0f\n- v17 --refutes--> v02\n- v06 --supports--> v15\n- v0d --refutes--> v13\n- v14 --refutes--> v0a\n- v11 --refutes--> v08\n- v05 --supports--> v12\n- v15 --supports--> v00\n- v12 --supports--> v0b\n- v11 --supports--> v18\n- v02 --refutes--> v03\n- v0a --supports--> v11\n- v0b --refutes--> v06\n- v06 --refutes--> v07\n- v04 --supports--> v0c\n- v11 --refutes--> v0e\n- v12 --supports--> v01\n- v0c --supports--> v17\n- v0c --refutes--> v09\n- v0d --supports--> v04\n- v16 --refutes--> v0d\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":"af733df264a54920f4f2dce98b5095b04f9c609d9f0a9162507da458a0cea181","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v14\n- v01 --supports--> v04\n- v07 --supports--> v0a\n- v0a --refutes--> v09\n- v18 --supports--> v07\n- v00 --supports--> v12\n- v14 --supports--> v0e\n- v0e --supports--> v05\n- v0b --refutes--> v10\n- v09 --refutes--> v0f\n- v16 --refutes--> v0b\n- v01 --refutes--> v17\n- v06 --supports--> v08\n- v10 --refutes--> v06\n- v0d --supports--> v15\n- v08 --refutes--> v02\n- v04 --refutes--> v03\n- v0d --refutes--> v01\n- v0e --refutes--> v00\n- v09 --supports--> v11\n- v0b --supports--> v0c\n- v08 --refutes--> v18\n- v11 --refutes--> v0d\n- v00 --refutes--> v13\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":"9430b479e6f69c09a7e1332bfd18b3b6ab8e9fdbb5c6706399914a7ead1e87b3","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v0b --refutes--> v17\n- v01 --refutes--> v16\n- v07 --supports--> v18\n- v14 --supports--> v11\n- v0c --refutes--> v08\n- v09 --supports--> v0b\n- v13 --supports--> v06\n- v03 --refutes--> v12\n- v07 --supports--> v0d\n- v0f --supports--> v14\n- v0a --refutes--> v00\n- v0b --refutes--> v0a\n- v0c --refutes--> v02\n- v13 --refutes--> v15\n- v0e --supports--> v01\n- v07 --supports--> v0c\n- v12 --refutes--> v09\n- v0c --supports--> v04\n- v14 --refutes--> v0e\n- v16 --supports--> v13\n- v05 --refutes--> v07\n- v08 --supports--> v03\n- v0a --refutes--> v10\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":"a4001daea2044fce4582026fddd8ab0650e1c8bbb19adb40ec3558766e8680b9","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v09\n- v02 --supports--> v13\n- v09 --supports--> v07\n- v04 --supports--> v01\n- v00 --refutes--> v18\n- v11 --supports--> v17\n- v06 --supports--> v11\n- v07 --supports--> v0e\n- v10 --supports--> v15\n- v09 --supports--> v16\n- v0f --refutes--> v0a\n- v0a --supports--> v02\n- v15 --refutes--> v03\n- v0c --supports--> v0b\n- v05 --supports--> v06\n- v11 --refutes--> v08\n- v04 --supports--> v10\n- v18 --supports--> v12\n- v15 --refutes--> v14\n- v02 --supports--> v04\n- v07 --refutes--> v05\n- v09 --refutes--> v0d\n- v0c --supports--> v00\n- v14 --supports--> v0c\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":"8e5f7e3c92734c5ddfc6eb400f4ab67e399bdb750a6432398ade25934ea40ecb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v07\n- v0e --supports--> v02\n- v0b --refutes--> v0f\n- v11 --refutes--> v06\n- v0b --refutes--> v01\n- v17 --refutes--> v14\n- v00 --refutes--> v0e\n- v17 --supports--> v16\n- v06 --refutes--> v04\n- v06 --supports--> v12\n- v17 --refutes--> v00\n- v0d --supports--> v0c\n- v0c --supports--> v10\n- v07 --refutes--> v15\n- v07 --supports--> v11\n- v05 --supports--> v17\n- v12 --supports--> v09\n- v09 --refutes--> v0b\n- v18 --refutes--> v08\n- v10 --supports--> v05\n- v18 --supports--> v03\n- v18 --refutes--> v13\n- v13 --refutes--> v0a\n- v0f --supports--> v18\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":"c7e44bd21fc5fc6ab90bce02c182d58ee6168af8028bd0864bb471dceee40454","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v0d\n- v02 --supports--> v11\n- v07 --supports--> v17\n- v15 --supports--> v0e\n- v08 --supports--> v12\n- v03 --supports--> v0c\n- v16 --refutes--> v14\n- v11 --refutes--> v10\n- v03 --refutes--> v13\n- v16 --supports--> v15\n- v11 --supports--> v05\n- v12 --supports--> v09\n- v17 --supports--> v02\n- v15 --supports--> v06\n- v01 --refutes--> v0a\n- v0e --refutes--> v01\n- v0c --refutes--> v08\n- v02 --refutes--> v18\n- v07 --refutes--> v00\n- v12 --refutes--> v07\n- v0c --supports--> v04\n- v0a --refutes--> v0b\n- v0b --refutes--> v0f\n- v0f --supports--> v03\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":"70f7ac2f90c3707ee64885fd420d5912d1a01a92cd2cacc34fbd135ca496140c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v06\n- v12 --supports--> v04\n- v09 --refutes--> v18\n- v0e --supports--> v02\n- v14 --supports--> v07\n- v12 --supports--> v10\n- v01 --refutes--> v03\n- v02 --refutes--> v01\n- v0c --supports--> v0d\n- v0a --supports--> v16\n- v05 --refutes--> v12\n- v16 --refutes--> v09\n- v07 --refutes--> v05\n- v06 --refutes--> v14\n- v01 --refutes--> v0b\n- v10 --refutes--> v08\n- v07 --refutes--> v17\n- v0b --refutes--> v00\n- v06 --supports--> v13\n- v10 --supports--> v0a\n- v0c --refutes--> v0f\n- v0a --supports--> v15\n- v0c --refutes--> v0e\n- v00 --refutes--> v11\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":"5da65292113ad1278057ed3b5146eed0a76e36874c9e0f731ce59d3a105803ba","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0f\n- v15 --refutes--> v00\n- v03 --refutes--> v10\n- v11 --refutes--> v17\n- v14 --supports--> v0a\n- v18 --supports--> v12\n- v0b --refutes--> v0c\n- v05 --supports--> v06\n- v07 --refutes--> v18\n- v0d --refutes--> v0e\n- v00 --refutes--> v08\n- v0d --refutes--> v0b\n- v01 --supports--> v15\n- v11 --supports--> v05\n- v00 --supports--> v16\n- v09 --supports--> v04\n- v14 --supports--> v11\n- v12 --supports--> v03\n- v00 --refutes--> v13\n- v08 --supports--> v07\n- v05 --refutes--> v09\n- v09 --supports--> v01\n- v0f --refutes--> v14\n- v0b --supports--> v02\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":"9deebddcf19042bee2868f0c0ad849965b8c621621fb64419e6a1312985d6fff","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v13\n- v18 --supports--> v00\n- v0b --supports--> v02\n- v08 --refutes--> v09\n- v08 --refutes--> v06\n- v12 --supports--> v0d\n- v08 --refutes--> v01\n- v0c --refutes--> v07\n- v02 --refutes--> v0f\n- v14 --supports--> v0a\n- v03 --supports--> v04\n- v0a --refutes--> v11\n- v17 --supports--> v0e\n- v0e --supports--> v0b\n- v00 --supports--> v03\n- v06 --refutes--> v17\n- v0f --supports--> v0c\n- v07 --refutes--> v15\n- v04 --supports--> v16\n- v07 --supports--> v12\n- v12 --refutes--> v10\n- v04 --refutes--> v08\n- v0f --refutes--> v05\n- v12 --refutes--> v14\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":"5c3accd7bafb12af19307e8b5fd8263f2ff3544ec96ad45dc2dd2bac259c63a8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v15\n- v0d --refutes--> v0a\n- v0e --refutes--> v07\n- v16 --supports--> v11\n- v09 --supports--> v03\n- v08 --refutes--> v04\n- v18 --refutes--> v17\n- v17 --supports--> v0b\n- v09 --supports--> v10\n- v01 --supports--> v08\n- v08 --refutes--> v05\n- v04 --supports--> v14\n- v16 --supports--> v12\n- v10 --refutes--> v02\n- v0a --refutes--> v00\n- v11 --supports--> v0d\n- v14 --supports--> v13\n- v0b --supports--> v09\n- v10 --supports--> v16\n- v07 --supports--> v01\n- v0f --supports--> v18\n- v07 --supports--> v06\n- v14 --supports--> v0f\n- v0d --refutes--> v0c\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":"ac488089940e882d5e1557bd0ba4f2560bc2227454ec396100663e75ecd9356d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v07\n- v0a --refutes--> v13\n- v17 --refutes--> v15\n- v06 --refutes--> v17\n- v07 --supports--> v02\n- v11 --supports--> v10\n- v09 --refutes--> v14\n- v10 --refutes--> v0d\n- v03 --refutes--> v00\n- v0f --refutes--> v04\n- v0d --supports--> v0e\n- v05 --refutes--> v11\n- v09 --refutes--> v18\n- v16 --supports--> v0b\n- v12 --refutes--> v09\n- v05 --refutes--> v0c\n- v0f --supports--> v06\n- v09 --refutes--> v05\n- v0a --refutes--> v03\n- v03 --supports--> v12\n- v02 --refutes--> v01\n- v01 --refutes--> v0f\n- v11 --supports--> v08\n- v15 --refutes--> v16\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":"542cd874e7f1773c71630b27c1a50a8d14e2f656f971cfc3be7f84b56c2e7df8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v01\n- v15 --supports--> v17\n- v06 --supports--> v0d\n- v05 --supports--> v00\n- v17 --refutes--> v09\n- v01 --refutes--> v04\n- v0a --refutes--> v10\n- v16 --supports--> v0e\n- v00 --refutes--> v12\n- v12 --refutes--> v18\n- v12 --supports--> v13\n- v17 --supports--> v0f\n- v13 --supports--> v15\n- v04 --refutes--> v0a\n- v16 --refutes--> v07\n- v0c --refutes--> v11\n- v10 --supports--> v02\n- v0b --refutes--> v05\n- v16 --supports--> v03\n- v06 --refutes--> v14\n- v0c --supports--> v0b\n- v09 --refutes--> v16\n- v06 --supports--> v0c\n- v13 --supports--> v08\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"432198dfd4c03cf09e4b2b545a97f8f93140dbcdf9d9e49b08f893e0c7dff54e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v18\n- v0f --supports--> v13\n- v04 --refutes--> v0b\n- v10 --refutes--> v11\n- v14 --refutes--> v01\n- v05 --supports--> v09\n- v07 --supports--> v0f\n- v13 --supports--> v06\n- v0b --refutes--> v10\n- v10 --supports--> v05\n- v0c --supports--> v15\n- v08 --refutes--> v16\n- v13 --supports--> v12\n- v02 --refutes--> v14\n- v01 --refutes--> v04\n- v08 --supports--> v0d\n- v02 --supports--> v17\n- v0c --supports--> v0e\n- v12 --supports--> v00\n- v0e --refutes--> v0a\n- v16 --refutes--> v0c\n- v12 --refutes--> v02\n- v05 --supports--> v08\n- v01 --refutes--> v03\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":"f451abfaf71f10a312972896b6dfe10268c1cbacee17b5c0eb3791efe0e9e979","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v13\n- v09 --refutes--> v0b\n- v0c --refutes--> v0e\n- v08 --supports--> v12\n- v09 --refutes--> v05\n- v0e --supports--> v16\n- v0a --supports--> v18\n- v01 --refutes--> v0d\n- v10 --refutes--> v14\n- v14 --supports--> v08\n- v13 --refutes--> v15\n- v09 --refutes--> v06\n- v08 --supports--> v17\n- v0e --supports--> v03\n- v02 --refutes--> v0c\n- v0e --refutes--> v00\n- v07 --refutes--> v0f\n- v18 --supports--> v10\n- v03 --refutes--> v01\n- v08 --refutes--> v02\n- v0d --supports--> v07\n- v06 --refutes--> v11\n- v15 --refutes--> v09\n- v0c --supports--> v04\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":"69ee2e74b9f371b5ddeae88a089f63b4b9375b62751686006ba88520c812a2d8","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v00\n- v03 --refutes--> v02\n- v0a --supports--> v09\n- v09 --refutes--> v0c\n- v06 --supports--> v10\n- v10 --supports--> v16\n- v05 --supports--> v01\n- v0f --refutes--> v06\n- v10 --refutes--> v15\n- v02 --supports--> v07\n- v16 --refutes--> v17\n- v04 --refutes--> v0a\n- v02 --supports--> v0d\n- v0c --refutes--> v12\n- v01 --supports--> v0b\n- v01 --refutes--> v13\n- v18 --refutes--> v03\n- v01 --supports--> v18\n- v02 --refutes--> v08\n- v04 --supports--> v11\n- v0f --refutes--> v0e\n- v00 --supports--> v04\n- v17 --refutes--> v05\n- v02 --supports--> v14\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":"452a3a1c9f30ecae0388bacc0fa42ad84e1ab088a37a393dbf50449ee8968c84","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v13\n- v05 --refutes--> v12\n- v14 --supports--> v01\n- v10 --refutes--> v0b\n- v09 --refutes--> v04\n- v00 --supports--> v08\n- v03 --refutes--> v18\n- v05 --supports--> v0e\n- v10 --refutes--> v03\n- v17 --refutes--> v05\n- v0c --refutes--> v0f\n- v17 --supports--> v07\n- v00 --supports--> v11\n- v18 --refutes--> v06\n- v02 --supports--> v09\n- v08 --supports--> v0a\n- v06 --supports--> v02\n- v0f --supports--> v14\n- v16 --refutes--> v0d\n- v0e --supports--> v16\n- v0a --refutes--> v10\n- v16 --refutes--> v0c\n- v09 --supports--> v17\n- v17 --refutes--> v15\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":"2029ffed357a5d827ad72715a07209037d026e6de516f5e4ea7c4c22b80b66c4","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v15\n- v05 --refutes--> v13\n- v06 --supports--> v00\n- v09 --refutes--> v11\n- v13 --refutes--> v03\n- v01 --refutes--> v0b\n- v00 --refutes--> v0c\n- v0f --supports--> v07\n- v18 --supports--> v06\n- v11 --supports--> v08\n- v16 --refutes--> v0f\n- v0c --supports--> v09\n- v12 --refutes--> v10\n- v10 --supports--> v16\n- v07 --supports--> v01\n- v11 --refutes--> v0d\n- v06 --refutes--> v04\n- v13 --refutes--> v0e\n- v13 --supports--> v17\n- v0e --refutes--> v12\n- v0c --refutes--> v02\n- v16 --refutes--> v0a\n- v08 --refutes--> v05\n- v00 --supports--> v14\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":"be61f5c164c4578bd8dff7122afd3589d1707f12f78f65f4293be0440650ae94","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v07\n- v09 --supports--> v04\n- v0d --refutes--> v0a\n- v0e --refutes--> v00\n- v0b --refutes--> v13\n- v16 --refutes--> v03\n- v0f --supports--> v16\n- v0a --supports--> v0c\n- v00 --supports--> v08\n- v09 --supports--> v15\n- v07 --refutes--> v14\n- v17 --refutes--> v0f\n- v0b --refutes--> v17\n- v0a --refutes--> v02\n- v06 --refutes--> v12\n- v09 --supports--> v0e\n- v02 --supports--> v05\n- v03 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v01\n- v07 --refutes--> v09\n- v16 --refutes--> v10\n- v08 --refutes--> v11\n- v11 --refutes--> v0b\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":"0085b16cc7c50d3db769f28c9acb459dbfde4821996d1c1b7a1a8623efbc77f6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v16\n- v0b --refutes--> v0c\n- v02 --supports--> v0b\n- v0a --supports--> v07\n- v00 --refutes--> v02\n- v05 --refutes--> v0a\n- v0c --supports--> v15\n- v0a --supports--> v03\n- v15 --supports--> v08\n- v02 --supports--> v17\n- v18 --supports--> v12\n- v0a --supports--> v0e\n- v06 --refutes--> v00\n- v11 --supports--> v10\n- v10 --refutes--> v09\n- v03 --refutes--> v13\n- v05 --supports--> v04\n- v0d --supports--> v14\n- v14 --supports--> v06\n- v03 --refutes--> v0d\n- v08 --refutes--> v18\n- v03 --supports--> v0f\n- v01 --refutes--> v11\n- v18 --supports--> v01\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d0231b8bc545e8a4ee755e69f0eb0e7ec4dfd22ded9e6a3f2412de09677f9a6","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --supports--> v0d\n- v07 --refutes--> v06\n- v16 --refutes--> v00\n- v0a --supports--> v03\n- v13 --refutes--> v02\n- v10 --supports--> v07\n- v02 --refutes--> v05\n- v0e --supports--> v01\n- v06 --refutes--> v0f\n- v17 --supports--> v0b\n- v06 --refutes--> v0c\n- v04 --refutes--> v17\n- v03 --refutes--> v14\n- v10 --supports--> v11\n- v08 --supports--> v18\n- v03 --refutes--> v09\n- v0a --refutes--> v15\n- v01 --refutes--> v16\n- v03 --refutes--> v13\n- v07 --supports--> v12\n- v05 --refutes--> v04\n- v0b --supports--> v0e\n- v06 --refutes--> v08\n- v18 --supports--> v0a\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":"02d4485d28a778dcf88f897bf9210aac8eddca495fa6c063e995af3e3c8d231e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v06\n- v16 --refutes--> v04\n- v16 --refutes--> v02\n- v05 --refutes--> v03\n- v0a --supports--> v00\n- v12 --refutes--> v15\n- v13 --refutes--> v0c\n- v15 --refutes--> v07\n- v14 --refutes--> v10\n- v10 --supports--> v0f\n- v0e --supports--> v12\n- v03 --supports--> v08\n- v04 --refutes--> v05\n- v04 --supports--> v0d\n- v0a --refutes--> v16\n- v0f --refutes--> v0e\n- v10 --refutes--> v01\n- v14 --supports--> v18\n- v11 --refutes--> v13\n- v13 --supports--> v17\n- v0c --supports--> v0a\n- v08 --refutes--> v14\n- v14 --refutes--> v0b\n- v09 --supports--> v11\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":"69c64eada0ca21a510b23a44b2125b1e76b3ed7370e0fcea7179844bd05c0e0f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v06\n- v0b --supports--> v14\n- v0e --refutes--> v01\n- v00 --supports--> v08\n- v13 --supports--> v17\n- v03 --supports--> v0c\n- v05 --refutes--> v10\n- v05 --refutes--> v03\n- v14 --supports--> v09\n- v16 --supports--> v02\n- v00 --supports--> v0a\n- v06 --refutes--> v12\n- v0e --refutes--> v0b\n- v08 --refutes--> v0d\n- v14 --supports--> v18\n- v04 --refutes--> v00\n- v05 --refutes--> v11\n- v17 --supports--> v05\n- v07 --supports--> v0f\n- v0c --refutes--> v07\n- v07 --refutes--> v04\n- v05 --supports--> v15\n- v12 --supports--> v0e\n- v14 --supports--> v13\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":"315261637b51bb0ce8de1adc6deee474feecadb7eb7480e00781a1e8b6da67cd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v05\n- v16 --refutes--> v17\n- v10 --supports--> v12\n- v0e --supports--> v11\n- v0c --refutes--> v10\n- v05 --refutes--> v14\n- v03 --refutes--> v06\n- v0e --refutes--> v09\n- v17 --refutes--> v07\n- v16 --supports--> v0d\n- v05 --supports--> v0b\n- v17 --supports--> v15\n- v04 --refutes--> v0f\n- v04 --refutes--> v16\n- v00 --refutes--> v04\n- v08 --refutes--> v0a\n- v07 --supports--> v03\n- v04 --supports--> v13\n- v11 --supports--> v08\n- v18 --supports--> v00\n- v02 --refutes--> v0c\n- v08 --refutes--> v01\n- v0a --supports--> v02\n- v14 --refutes--> v18\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":"c42ad4b0e5554a4f334283de99076d8dba6e63ae810cb1f4d7931f0bbcb5e200","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v15\n- v14 --refutes--> v18\n- v09 --supports--> v02\n- v01 --refutes--> v04\n- v0b --refutes--> v0f\n- v02 --refutes--> v0b\n- v11 --refutes--> v09\n- v08 --refutes--> v0d\n- v0e --refutes--> v01\n- v05 --refutes--> v0c\n- v12 --supports--> v08\n- v15 --supports--> v06\n- v16 --refutes--> v13\n- v16 --supports--> v0a\n- v18 --supports--> v16\n- v0a --refutes--> v03\n- v0c --refutes--> v11\n- v05 --refutes--> v07\n- v05 --refutes--> v10\n- v01 --supports--> v12\n- v01 --refutes--> v17\n- v0b --refutes--> v14\n- v06 --supports--> v05\n- v06 --supports--> v00\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":"700e25657a0bc86bb9a6ab254a07990280adc0f973285aea2bb34b818c77e76f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v05\n- v09 --supports--> v13\n- v14 --supports--> v0e\n- v0e --refutes--> v15\n- v04 --supports--> v0b\n- v0a --supports--> v02\n- v16 --supports--> v14\n- v01 --supports--> v17\n- v05 --refutes--> v01\n- v0d --refutes--> v00\n- v04 --refutes--> v0a\n- v0c --refutes--> v07\n- v17 --refutes--> v08\n- v10 --refutes--> v0c\n- v18 --supports--> v10\n- v04 --supports--> v06\n- v18 --refutes--> v11\n- v07 --refutes--> v09\n- v02 --supports--> v16\n- v08 --supports--> v0d\n- v17 --supports--> v12\n- v09 --refutes--> v0f\n- v05 --refutes--> v03\n- v09 --supports--> v04\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":"c03948058b22a0c6fa61a94dd3106aab09536a77c8fe17305eddd013efa94991","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --refutes--> v02\n- v02 --refutes--> v06\n- v14 --refutes--> v07\n- v05 --refutes--> v0a\n- v01 --refutes--> v15\n- v17 --supports--> v13\n- v0d --refutes--> v00\n- v12 --refutes--> v05\n- v0a --refutes--> v16\n- v12 --refutes--> v04\n- v15 --refutes--> v0f\n- v0e --refutes--> v14\n- v17 --supports--> v18\n- v06 --refutes--> v01\n- v0f --refutes--> v0c\n- v14 --supports--> v12\n- v15 --refutes--> v03\n- v06 --supports--> v11\n- v0c --supports--> v17\n- v09 --refutes--> v0e\n- v0e --supports--> v10\n- v12 --supports--> v0b\n- v18 --supports--> v0d\n- v0f --supports--> v08\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c0c6770f53040ed5bb2a9e0c739ca38267e9bcc42aebbe9a227e1eb38148d0d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v17\n- v16 --supports--> v15\n- v10 --refutes--> v11\n- v06 --supports--> v16\n- v0f --refutes--> v0d\n- v08 --supports--> v03\n- v04 --refutes--> v08\n- v12 --supports--> v06\n- v02 --supports--> v00\n- v03 --refutes--> v14\n- v00 --supports--> v18\n- v0e --refutes--> v0f\n- v18 --supports--> v09\n- v0d --supports--> v0c\n- v10 --refutes--> v0b\n- v15 --supports--> v04\n- v18 --refutes--> v01\n- v00 --supports--> v0a\n- v0c --supports--> v02\n- v08 --supports--> v05\n- v03 --refutes--> v10\n- v09 --refutes--> v12\n- v18 --refutes--> v13\n- v00 --supports--> v07\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":"1e5e74c40326360940b65b3265632f0a5ba703c4b7ea8c8174cf411c00a95d61","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v14\n- v12 --refutes--> v0e\n- v02 --refutes--> v05\n- v12 --refutes--> v0c\n- v0e --supports--> v08\n- v03 --supports--> v11\n- v0a --supports--> v16\n- v10 --refutes--> v0f\n- v07 --refutes--> v03\n- v17 --refutes--> v0b\n- v03 --refutes--> v09\n- v13 --supports--> v0d\n- v11 --refutes--> v10\n- v07 --refutes--> v06\n- v15 --refutes--> v13\n- v05 --refutes--> v12\n- v10 --supports--> v02\n- v12 --refutes--> v01\n- v16 --supports--> v07\n- v11 --refutes--> v18\n- v0d --refutes--> v17\n- v13 --supports--> v00\n- v04 --refutes--> v15\n- v08 --supports--> v04\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":"60b72dcf876748011e32603c2bc127b91bec6f48c9790812210bca6db6aff2df","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v09\n- v05 --refutes--> v13\n- v18 --refutes--> v12\n- v0a --refutes--> v14\n- v05 --supports--> v16\n- v0c --refutes--> v06\n- v0d --supports--> v11\n- v0b --refutes--> v08\n- v04 --refutes--> v0f\n- v14 --refutes--> v0b\n- v0f --refutes--> v10\n- v17 --refutes--> v0a\n- v10 --refutes--> v03\n- v0a --refutes--> v00\n- v10 --refutes--> v15\n- v08 --supports--> v05\n- v04 --supports--> v02\n- v07 --refutes--> v0d\n- v0c --refutes--> v0e\n- v01 --supports--> v18\n- v10 --refutes--> v07\n- v05 --supports--> v0c\n- v12 --supports--> v04\n- v06 --supports--> v01\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":"2eab04d3a17613e612271d01fe8a2ed3f0e494e57ce5af7d0954823cd026f0e0","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v03\n- v05 --supports--> v17\n- v17 --supports--> v0e\n- v0f --supports--> v0c\n- v03 --supports--> v15\n- v02 --supports--> v08\n- v17 --refutes--> v11\n- v0f --supports--> v04\n- v05 --refutes--> v10\n- v08 --refutes--> v09\n- v07 --supports--> v14\n- v17 --refutes--> v0a\n- v0f --refutes--> v0d\n- v14 --supports--> v0f\n- v11 --supports--> v06\n- v09 --refutes--> v16\n- v0f --supports--> v13\n- v12 --supports--> v00\n- v12 --supports--> v05\n- v0b --refutes--> v18\n- v01 --refutes--> v07\n- v06 --refutes--> v0b\n- v04 --refutes--> v12\n- v0b --supports--> v02\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":"7666784b458d72e1ba25a70376f014eb8073325551f451387ad93ef70625b8cd","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0b --supports--> v17\n- v15 --refutes--> v04\n- v04 --refutes--> v05\n- v10 --refutes--> v0b\n- v05 --supports--> v18\n- v10 --supports--> v06\n- v08 --supports--> v0c\n- v07 --refutes--> v09\n- v07 --refutes--> v13\n- v14 --refutes--> v08\n- v0f --refutes--> v0a\n- v0b --supports--> v02\n- v01 --refutes--> v0f\n- v00 --refutes--> v10\n- v17 --refutes--> v12\n- v16 --supports--> v01\n- v0a --supports--> v07\n- v04 --supports--> v03\n- v0c --refutes--> v0e\n- v15 --refutes--> v11\n- v13 --supports--> v00\n- v05 --supports--> v16\n- v17 --supports--> v14\n- v04 --refutes--> v0d\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":"01d9498b13ff6a8d8625133541fb2ecd1b3f9eae2aa1a2a45fd3052b9584d673","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v00\n- v02 --refutes--> v11\n- v03 --supports--> v0d\n- v02 --supports--> v12\n- v06 --refutes--> v05\n- v0f --supports--> v0b\n- v00 --refutes--> v03\n- v0f --refutes--> v16\n- v14 --refutes--> v0c\n- v10 --supports--> v0e\n- v16 --refutes--> v07\n- v06 --refutes--> v04\n- v17 --supports--> v14\n- v08 --supports--> v17\n- v04 --supports--> v0a\n- v04 --refutes--> v15\n- v12 --refutes--> v0f\n- v0e --refutes--> v02\n- v03 --supports--> v06\n- v0c --refutes--> v18\n- v04 --supports--> v13\n- v15 --supports--> v10\n- v03 --supports--> v09\n- v0f --supports--> v01\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":"d0cc00a70d77442be8e157cdbdc7050dd70b29ed216351d20d4bfaa038911e71","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v12\n- v06 --refutes--> v00\n- v12 --supports--> v0f\n- v10 --supports--> v09\n- v03 --refutes--> v0d\n- v15 --supports--> v17\n- v0e --supports--> v0a\n- v15 --supports--> v0c\n- v13 --refutes--> v18\n- v13 --supports--> v0b\n- v0e --supports--> v05\n- v08 --refutes--> v16\n- v13 --refutes--> v04\n- v14 --supports--> v13\n- v0c --supports--> v07\n- v02 --supports--> v06\n- v0c --supports--> v11\n- v06 --supports--> v14\n- v0e --refutes--> v15\n- v0d --refutes--> v01\n- v01 --supports--> v0e\n- v11 --refutes--> v08\n- v09 --refutes--> v02\n- v04 --refutes--> v03\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":"40fd971757e5c41a390c7358aa41140eb4cf83ca9a26468c83199945075b7905","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v16\n- v0b --supports--> v17\n- v12 --refutes--> v07\n- v05 --refutes--> v15\n- v05 --supports--> v0c\n- v13 --supports--> v0d\n- v08 --supports--> v03\n- v10 --refutes--> v05\n- v10 --refutes--> v0a\n- v07 --supports--> v11\n- v09 --supports--> v08\n- v14 --refutes--> v06\n- v11 --supports--> v02\n- v0f --refutes--> v00\n- v14 --refutes--> v0b\n- v17 --refutes--> v09\n- v0b --supports--> v18\n- v0e --supports--> v14\n- v0f --refutes--> v0e\n- v0f --supports--> v04\n- v08 --supports--> v12\n- v02 --refutes--> v13\n- v0d --supports--> v01\n- v15 --refutes--> v0f\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":"4ca08d36a5e0608f47ea46b40f75f45621a73814fb033ee56f59f9b5323875ad","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v02\n- v0a --refutes--> v18\n- v11 --refutes--> v00\n- v12 --refutes--> v13\n- v0a --supports--> v07\n- v17 --supports--> v05\n- v14 --supports--> v01\n- v14 --supports--> v0d\n- v10 --supports--> v0c\n- v01 --supports--> v0f\n- v13 --refutes--> v06\n- v0c --refutes--> v09\n- v0c --supports--> v16\n- v17 --supports--> v15\n- v03 --supports--> v14\n- v0e --refutes--> v12\n- v13 --supports--> v04\n- v0f --supports--> v10\n- v02 --refutes--> v0a\n- v16 --supports--> v0e\n- v07 --supports--> v03\n- v11 --supports--> v17\n- v0f --supports--> v0b\n- v0c --supports--> v08\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":"c2bc0071fc8d9c3235b1f812f507ab30cfca5064c3dfc1887618ed0d7791b5ed","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v00\n- v14 --supports--> v17\n- v13 --supports--> v08\n- v13 --supports--> v11\n- v15 --supports--> v10\n- v05 --refutes--> v02\n- v01 --supports--> v0b\n- v16 --supports--> v0a\n- v01 --supports--> v0f\n- v15 --refutes--> v0c\n- v0a --refutes--> v0e\n- v0a --supports--> v04\n- v0b --supports--> v05\n- v0d --refutes--> v01\n- v02 --supports--> v15\n- v08 --refutes--> v12\n- v14 --supports--> v16\n- v10 --refutes--> v14\n- v00 --refutes--> v06\n- v0d --refutes--> v03\n- v08 --refutes--> v0d\n- v07 --refutes--> v18\n- v0a --refutes--> v07\n- v06 --refutes--> v09\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":"9f50ca6f20188db3e03dbbbe91dbe2ebf41a0445fedcc5e2383659e6e717222a","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v0b\n- v12 --supports--> v0d\n- v16 --supports--> v04\n- v16 --supports--> v07\n- v0e --supports--> v0a\n- v0d --refutes--> v16\n- v16 --supports--> v13\n- v05 --refutes--> v0e\n- v09 --supports--> v14\n- v00 --supports--> v06\n- v03 --refutes--> v01\n- v03 --supports--> v05\n- v0e --refutes--> v10\n- v17 --supports--> v00\n- v17 --refutes--> v11\n- v07 --refutes--> v0c\n- v02 --refutes--> v03\n- v0b --supports--> v08\n- v08 --supports--> v0f\n- v0c --supports--> v17\n- v07 --refutes--> v15\n- v0c --refutes--> v18\n- v08 --supports--> v09\n- v14 --supports--> v02\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":"34b2e5dccc5501bb5af0cd593457fdd891717e471b492e1211203c497a4f0d45","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v07\n- v07 --refutes--> v04\n- v13 --refutes--> v11\n- v14 --supports--> v00\n- v10 --supports--> v08\n- v11 --refutes--> v01\n- v05 --refutes--> v0a\n- v10 --supports--> v0e\n- v03 --supports--> v05\n- v0b --refutes--> v0c\n- v04 --refutes--> v06\n- v12 --refutes--> v14\n- v02 --refutes--> v03\n- v08 --refutes--> v02\n- v05 --supports--> v0b\n- v04 --refutes--> v18\n- v0b --refutes--> v15\n- v07 --refutes--> v0f\n- v16 --refutes--> v12\n- v04 --supports--> v13\n- v0c --refutes--> v16\n- v01 --supports--> v10\n- v01 --refutes--> v0d\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":"74f19457bb91403ae352fcf7f92318f15e2e9ec387d47185f992161705a42925","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v0a\n- v05 --refutes--> v18\n- v0b --refutes--> v09\n- v09 --refutes--> v05\n- v15 --supports--> v12\n- v13 --supports--> v14\n- v0d --refutes--> v04\n- v09 --supports--> v11\n- v0c --refutes--> v13\n- v0e --refutes--> v06\n- v10 --refutes--> v03\n- v07 --supports--> v0b\n- v00 --refutes--> v15\n- v17 --refutes--> v10\n- v10 --refutes--> v0e\n- v14 --refutes--> v0d\n- v17 --refutes--> v02\n- v05 --refutes--> v0f\n- v12 --refutes--> v0c\n- v0d --refutes--> v17\n- v01 --supports--> v00\n- v0d --refutes--> v08\n- v18 --refutes--> v16\n- v0e --refutes--> v07\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":"5a55bea8662a40a63e88e3a1ab363f1f1dace11ad9860db9214f60ccaf315169","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v10\n- v0c --supports--> v07\n- v0d --supports--> v0e\n- v12 --refutes--> v00\n- v02 --supports--> v16\n- v00 --refutes--> v03\n- v05 --refutes--> v0c\n- v05 --supports--> v04\n- v11 --supports--> v15\n- v05 --refutes--> v01\n- v07 --refutes--> v0f\n- v14 --supports--> v17\n- v15 --refutes--> v08\n- v0d --refutes--> v11\n- v16 --refutes--> v0d\n- v03 --supports--> v09\n- v07 --supports--> v18\n- v16 --refutes--> v0b\n- v10 --supports--> v14\n- v09 --supports--> v0a\n- v12 --refutes--> v13\n- v15 --supports--> v06\n- v14 --refutes--> v05\n- v0f --supports--> v02\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":"015ba92c4914f702706a02a7d74e1ae804037369e151674d19498c4110c8ae0f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v03\n- v0f --refutes--> v0e\n- v10 --supports--> v12\n- v16 --supports--> v15\n- v0f --supports--> v09\n- v08 --refutes--> v0c\n- v0a --supports--> v01\n- v02 --refutes--> v07\n- v0b --refutes--> v14\n- v16 --supports--> v05\n- v03 --refutes--> v02\n- v15 --refutes--> v0f\n- v0a --supports--> v04\n- v06 --refutes--> v13\n- v01 --supports--> v08\n- v0d --supports--> v16\n- v0f --refutes--> v17\n- v0c --supports--> v0b\n- v17 --refutes--> v10\n- v07 --supports--> v0a\n- v10 --refutes--> v18\n- v14 --supports--> v0d\n- v14 --supports--> v11\n- v16 --supports--> v00\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":"037f6bdaccd447e54f80c86b57d5d5085944195a0caee63fb198be57ea935914","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v15\n- v0f --refutes--> v0d\n- v0a --supports--> v10\n- v16 --supports--> v14\n- v01 --refutes--> v0b\n- v0d --refutes--> v16\n- v17 --refutes--> v13\n- v11 --refutes--> v0a\n- v13 --supports--> v0f\n- v00 --supports--> v12\n- v12 --supports--> v03\n- v18 --refutes--> v08\n- v08 --supports--> v00\n- v04 --refutes--> v17\n- v01 --supports--> v0e\n- v03 --supports--> v0c\n- v13 --refutes--> v07\n- v0c --refutes--> v05\n- v11 --supports--> v02\n- v0a --supports--> v01\n- v0c --supports--> v11\n- v16 --supports--> v18\n- v03 --supports--> v09\n- v0b --refutes--> v06\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":"00c16dc1a37fe46a56afe9002c216be3fc02dc46395554d841537712b2105508","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --supports--> v07\n- v08 --refutes--> v02\n- v03 --refutes--> v0b\n- v01 --refutes--> v11\n- v14 --refutes--> v15\n- v01 --supports--> v17\n- v0b --supports--> v10\n- v08 --refutes--> v01\n- v0d --supports--> v06\n- v01 --refutes--> v0a\n- v0d --refutes--> v00\n- v07 --supports--> v05\n- v13 --supports--> v16\n- v04 --refutes--> v03\n- v16 --refutes--> v0f\n- v00 --supports--> v08\n- v0e --supports--> v12\n- v11 --supports--> v18\n- v05 --supports--> v04\n- v13 --supports--> v0c\n- v10 --supports--> v13\n- v18 --refutes--> v0e\n- v16 --supports--> v0d\n- v03 --refutes--> v09\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":"19c529f224f1971db019138a1cd3749870a48ecb25667682c6e66fa9fd818e31","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v09\n- v15 --supports--> v18\n- v18 --refutes--> v0c\n- v07 --supports--> v05\n- v10 --refutes--> v0b\n- v01 --supports--> v03\n- v12 --refutes--> v02\n- v00 --supports--> v0f\n- v06 --supports--> v14\n- v08 --refutes--> v0a\n- v08 --supports--> v12\n- v01 --refutes--> v08\n- v14 --supports--> v15\n- v04 --supports--> v00\n- v16 --supports--> v06\n- v0f --supports--> v01\n- v11 --refutes--> v13\n- v0f --refutes--> v17\n- v13 --refutes--> v16\n- v07 --refutes--> v10\n- v0c --supports--> v04\n- v10 --refutes--> v11\n- v12 --supports--> v0e\n- v04 --refutes--> v0d\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":"ff3b361e7efec5106ed895ea6ca811b981cbc75f82bb210815964c9d1383c628","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v06\n- v15 --supports--> v03\n- v12 --supports--> v00\n- v0b --supports--> v17\n- v14 --supports--> v09\n- v00 --supports--> v08\n- v08 --supports--> v11\n- v05 --supports--> v12\n- v0c --supports--> v14\n- v06 --supports--> v01\n- v07 --supports--> v0f\n- v07 --refutes--> v16\n- v02 --supports--> v0b\n- v0b --supports--> v18\n- v15 --refutes--> v0a\n- v04 --refutes--> v02\n- v15 --supports--> v10\n- v09 --supports--> v04\n- v00 --refutes--> v0d\n- v06 --refutes--> v05\n- v07 --supports--> v0c\n- v18 --refutes--> v15\n- v07 --refutes--> v13\n- v11 --supports--> v0e\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":"12438634dc1dc9418e2f80e39af4fac0a302d4840a96e83d11ee5c3b54e20e5f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v09\n- v15 --refutes--> v17\n- v05 --refutes--> v0b\n- v09 --refutes--> v05\n- v0f --refutes--> v18\n- v06 --refutes--> v01\n- v01 --supports--> v0c\n- v10 --supports--> v13\n- v09 --supports--> v14\n- v11 --refutes--> v08\n- v03 --refutes--> v02\n- v02 --supports--> v12\n- v17 --supports--> v10\n- v18 --supports--> v15\n- v08 --refutes--> v16\n- v18 --supports--> v0e\n- v08 --refutes--> v06\n- v12 --supports--> v0f\n- v01 --refutes--> v00\n- v10 --refutes--> v11\n- v0b --supports--> v03\n- v01 --refutes--> v04\n- v08 --refutes--> v0a\n- v15 --supports--> v0d\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":"a9cae796cb57ac3737a609530a701be30293e291ecb018d8196341a64f345e8b","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v15\n- v09 --refutes--> v08\n- v00 --refutes--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v09\n- v0e --refutes--> v0b\n- v08 --supports--> v06\n- v10 --refutes--> v0f\n- v15 --supports--> v11\n- v16 --refutes--> v10\n- v08 --supports--> v13\n- v11 --supports--> v01\n- v0b --refutes--> v07\n- v0c --refutes--> v17\n- v01 --refutes--> v0e\n- v0e --refutes--> v0a\n- v13 --refutes--> v0c\n- v03 --refutes--> v12\n- v17 --supports--> v16\n- v10 --supports--> v18\n- v16 --supports--> v05\n- v11 --refutes--> v0d\n- v10 --refutes--> v02\n- v10 --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":"618358811627601428572ed449b0e2ba855d2ade068536a03a6e02771f38963d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --supports--> v15\n- v10 --refutes--> v0c\n- v13 --refutes--> v17\n- v02 --supports--> v00\n- v0d --supports--> v18\n- v16 --refutes--> v0d\n- v15 --refutes--> v0b\n- v01 --supports--> v07\n- v09 --refutes--> v02\n- v07 --refutes--> v11\n- v0a --supports--> v09\n- v02 --supports--> v10\n- v04 --supports--> v0a\n- v0b --refutes--> v05\n- v07 --refutes--> v04\n- v02 --supports--> v06\n- v07 --supports--> v0e\n- v0f --refutes--> v13\n- v0d --supports--> v08\n- v0b --refutes--> v16\n- v0c --refutes--> v03\n- v16 --refutes--> v12\n- v0a --supports--> v14\n- v08 --supports--> v01\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2afc5a900d0d6cad511504e6ebfadb6509cb7ee730a4a96944c6d6d3c4f2b36d","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v14\n- v17 --refutes--> v04\n- v0f --refutes--> v00\n- v04 --refutes--> v0d\n- v17 --refutes--> v01\n- v15 --refutes--> v10\n- v15 --refutes--> v07\n- v0c --refutes--> v03\n- v00 --refutes--> v0b\n- v09 --supports--> v16\n- v11 --refutes--> v02\n- v08 --refutes--> v0e\n- v13 --supports--> v0c\n- v07 --refutes--> v08\n- v07 --supports--> v18\n- v04 --supports--> v09\n- v14 --supports--> v12\n- v0e --refutes--> v11\n- v11 --supports--> v17\n- v09 --refutes--> v0a\n- v13 --supports--> v05\n- v12 --refutes--> v13\n- v04 --supports--> v06\n- v0b --supports--> v15\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":"2db6428a3840e7bea0b3bfb3b37588b5308bd7eba4884b6f582e0e12d8fb723c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v17\n- v0b --supports--> v0d\n- v14 --supports--> v0c\n- v0f --refutes--> v18\n- v10 --supports--> v03\n- v16 --refutes--> v0f\n- v18 --supports--> v13\n- v18 --supports--> v0b\n- v17 --refutes--> v04\n- v0a --refutes--> v02\n- v05 --refutes--> v16\n- v17 --refutes--> v00\n- v05 --supports--> v01\n- v05 --refutes--> v08\n- v11 --refutes--> v06\n- v0a --supports--> v15\n- v15 --refutes--> v14\n- v06 --supports--> v05\n- v0e --refutes--> v11\n- v0f --refutes--> v09\n- v17 --refutes--> v12\n- v07 --refutes--> v10\n- v14 --refutes--> v0e\n- v03 --refutes--> v0a\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"36c7cdddacbfd858a892c86bf7fd1367c4ad57220ccb75d3c2a63739432e5d85","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v06\n- v14 --supports--> v0e\n- v12 --supports--> v16\n- v03 --refutes--> v04\n- v10 --supports--> v14\n- v07 --supports--> v10\n- v11 --refutes--> v05\n- v06 --refutes--> v08\n- v03 --refutes--> v0b\n- v14 --supports--> v01\n- v0e --supports--> v09\n- v0f --refutes--> v03\n- v03 --refutes--> v00\n- v05 --refutes--> v0d\n- v04 --refutes--> v07\n- v09 --supports--> v0a\n- v09 --supports--> v0c\n- v05 --refutes--> v17\n- v08 --refutes--> v15\n- v18 --supports--> v12\n- v0d --refutes--> v0f\n- v11 --refutes--> v13\n- v12 --supports--> v02\n- v15 --supports--> v11\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":"9654eb607d1c5007fb1c7c2f50dc9570b70f8efc74168129f40d4c89d6f0413c","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0c\n- v02 --supports--> v14\n- v10 --refutes--> v13\n- v0f --supports--> v01\n- v06 --supports--> v0d\n- v03 --supports--> v18\n- v07 --refutes--> v0e\n- v00 --supports--> v10\n- v0e --refutes--> v06\n- v08 --supports--> v02\n- v07 --refutes--> v09\n- v0d --supports--> v0b\n- v0a --supports--> v17\n- v0b --supports--> v16\n- v0b --supports--> v05\n- v04 --refutes--> v07\n- v0a --refutes--> v04\n- v0f --supports--> v11\n- v0f --refutes--> v12\n- v11 --refutes--> v08\n- v10 --refutes--> v03\n- v16 --refutes--> v00\n- v02 --refutes--> v15\n- v18 --supports--> v0f\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":"689d8fde8d033c1f2e0c438247b8ad238ce297216fe4ebb8eb898afb317e4d17","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v17\n- v06 --refutes--> v16\n- v0c --refutes--> v05\n- v06 --refutes--> v15\n- v16 --refutes--> v02\n- v07 --supports--> v0c\n- v00 --refutes--> v0f\n- v0e --supports--> v11\n- v18 --supports--> v0e\n- v16 --refutes--> v12\n- v13 --supports--> v01\n- v0f --supports--> v0a\n- v0e --supports--> v08\n- v04 --refutes--> v00\n- v10 --supports--> v07\n- v0f --refutes--> v09\n- v01 --refutes--> v04\n- v11 --refutes--> v10\n- v05 --refutes--> v0b\n- v07 --supports--> v0d\n- v0a --refutes--> v06\n- v13 --supports--> v14\n- v0b --supports--> v13\n- v04 --refutes--> v03\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":"abee975273f1fbc8fc4a9d6cccfb1e978e282d512fb5ec2c7ea20c3c3501357e","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0f\n- v05 --supports--> v02\n- v11 --supports--> v13\n- v0c --refutes--> v06\n- v03 --refutes--> v0d\n- v17 --supports--> v01\n- v06 --refutes--> v04\n- v17 --refutes--> v09\n- v0e --supports--> v0a\n- v06 --refutes--> v08\n- v18 --supports--> v05\n- v04 --supports--> v07\n- v03 --refutes--> v15\n- v0a --refutes--> v00\n- v0d --supports--> v0e\n- v17 --supports--> v14\n- v17 --supports--> v16\n- v13 --refutes--> v0c\n- v0b --supports--> v17\n- v09 --refutes--> v18\n- v11 --refutes--> v10\n- v12 --refutes--> v0b\n- v05 --refutes--> v03\n- v07 --refutes--> v12\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":"51d8074163a3ecceff94a79f7311fdddf2eb1b0cecda394ab6bb3540b3738634","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --refutes--> v00\n- v18 --refutes--> v09\n- v07 --supports--> v04\n- v05 --refutes--> v03\n- v11 --supports--> v12\n- v06 --supports--> v07\n- v13 --supports--> v0d\n- v00 --supports--> v17\n- v16 --refutes--> v11\n- v08 --refutes--> v02\n- v0d --supports--> v06\n- v0e --refutes--> v08\n- v16 --refutes--> v0f\n- v09 --refutes--> v0e\n- v0e --refutes--> v0c\n- v04 --supports--> v05\n- v00 --refutes--> v16\n- v02 --supports--> v0a\n- v01 --refutes--> v10\n- v0e --supports--> v14\n- v06 --supports--> v15\n- v01 --refutes--> v0b\n- v10 --supports--> v13\n- v05 --supports--> v18\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":"1c9022b2d2b5b8c408a86b6893a9d9546fb59543f80c213ec57898c913da69cb","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0a\n- v0c --supports--> v00\n- v03 --refutes--> v0d\n- v17 --refutes--> v09\n- v0f --refutes--> v01\n- v07 --refutes--> v17\n- v0c --supports--> v05\n- v02 --supports--> v0b\n- v13 --supports--> v02\n- v17 --supports--> v03\n- v10 --refutes--> v14\n- v06 --supports--> v08\n- v06 --refutes--> v16\n- v03 --supports--> v0f\n- v00 --supports--> v07\n- v10 --refutes--> v0e\n- v0b --supports--> v11\n- v02 --supports--> v04\n- v0a --supports--> v13\n- v11 --supports--> v10\n- v16 --refutes--> v0c\n- v0f --supports--> v15\n- v0f --supports--> v18\n- v14 --refutes--> v06\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":"848ab97aec9eeae9b02a785ca5ec97ea2b6a9e36fdc10cc4803714792546e4dc","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v06\n- v10 --supports--> v08\n- v0b --refutes--> v12\n- v06 --supports--> v10\n- v01 --refutes--> v17\n- v10 --refutes--> v00\n- v17 --refutes--> v0d\n- v00 --supports--> v03\n- v0b --supports--> v0c\n- v0c --supports--> v13\n- v0e --supports--> v0b\n- v04 --supports--> v0e\n- v03 --refutes--> v0a\n- v13 --supports--> v09\n- v18 --supports--> v02\n- v09 --refutes--> v01\n- v03 --refutes--> v04\n- v13 --supports--> v14\n- v11 --refutes--> v0f\n- v01 --supports--> v05\n- v0d --supports--> v18\n- v00 --supports--> v07\n- v0e --refutes--> v16\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":"56abebf2c9c1045a002fc579ddc0c40957d44ae40e0ff9314bc2fed05c42f59f","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":"graph"}
{"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":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v00\n- v0b --supports--> v07\n- v05 --supports--> v10\n- v11 --supports--> v0b\n- v03 --supports--> v01\n- v00 --supports--> v0a\n- v0d --refutes--> v15\n- v07 --supports--> v17\n- v0d --refutes--> v16\n- v04 --refutes--> v18\n- v15 --supports--> v0c\n- v05 --refutes--> v02\n- v0e --supports--> v12\n- v12 --refutes--> v11\n- v09 --refutes--> v0f\n- v02 --supports--> v0d\n- v01 --refutes--> v09\n- v03 --supports--> v13\n- v0a --refutes--> v0e\n- v07 --supports--> v14\n- v07 --supports--> v06\n- v0f --refutes--> v05\n- v07 --supports--> v04\n- v03 --supports--> v08\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":"352429dddf297765b127ccca97bf7ed029ad3c7b33ad639ff34528a69b977402","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":"graph"}