{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-605122684df404408d82","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v1d --supports--> v14\n- v19 --refutes--> v1d\n- v1b --supports--> v00\n- v1d --supports--> v04\n- v1b --refutes--> v0f\n- v08 --refutes--> v06\n- v09 --refutes--> v11\n- v1b --supports--> v13\n- v06 --refutes--> v1a\n- v04 --refutes--> v15\n- v01 --refutes--> v1e\n- v15 --supports--> v05\n- v16 --supports--> v0b\n- v11 --refutes--> v18\n- v01 --refutes--> v07\n- v0d --refutes--> v16\n- v11 --refutes--> v0a\n- v0c --supports--> v03\n- v17 --supports--> v0d\n- v0a --refutes--> v12\n- v12 --refutes--> v17\n- v1e --supports--> v1b\n- v16 --supports--> v1c\n- v00 --refutes--> v0e\n- v0c --supports--> v08\n- v05 --refutes--> v0c\n- v16 --supports--> v02\n- v16 --refutes--> v01\n- v08 --supports--> v10\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"97b2cfdac286ba65133553a43b46091ef3b2956d950dd96823b060ed2646db2a","schema":"livegraph.boolean-propagation.v1","spec_hash":"605122684df404408d82b86301c120bde0728fc7c6b1613ad3104a57dc609a6e","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-7d4ecb8cd2b804648ae8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v0e --supports--> v02\n- v02 --supports--> v0b\n- v1b --supports--> v01\n- v1d --supports--> v03\n- v15 --supports--> v1e\n- v1c --refutes--> v0c\n- v07 --refutes--> v1d\n- v00 --supports--> v12\n- v18 --supports--> v16\n- v1d --supports--> v1b\n- v11 --supports--> v17\n- v0d --supports--> v07\n- v0b --refutes--> v05\n- v1a --supports--> v00\n- v09 --supports--> v1c\n- v16 --supports--> v11\n- v16 --refutes--> v10\n- v0e --supports--> v13\n- v02 --refutes--> v0a\n- v01 --refutes--> v08\n- v11 --supports--> v0d\n- v19 --supports--> v18\n- v08 --supports--> v06\n- v19 --refutes--> v0f\n- v06 --supports--> v15\n- v0b --supports--> v09\n- v15 --supports--> v0e\n- v00 --refutes--> v14\n- v14 --refutes--> v04\n\n[VERIFIED ROOT]\nv1a = 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":"faad18e0660f1253e5e86c51c8fd5991ddc67a441eef25702cf212461e7d0c22","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d4ecb8cd2b804648ae8189d65a28464404f32413f82dd2145c83ff8c3ce5f6c","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-69c302a2db25d81c331e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v19\n- v1b --supports--> v1c\n- v08 --refutes--> v1d\n- v1a --supports--> v05\n- v06 --refutes--> v04\n- v1e --supports--> v12\n- v00 --refutes--> v10\n- v05 --supports--> v01\n- v10 --supports--> v09\n- v15 --refutes--> v0f\n- v01 --refutes--> v0b\n- v15 --refutes--> v06\n- v10 --refutes--> v14\n- v02 --refutes--> v15\n- v1e --supports--> v0e\n- v11 --refutes--> v00\n- v1a --refutes--> v16\n- v0e --supports--> v18\n- v09 --refutes--> v1e\n- v1c --supports--> v11\n- v0e --refutes--> v1a\n- v04 --supports--> v1b\n- v15 --supports--> v0d\n- v02 --refutes--> v07\n- v02 --supports--> v0c\n- v04 --supports--> v17\n- v13 --supports--> v02\n- v19 --supports--> v13\n- v0b --supports--> v0a\n- v0b --refutes--> v08\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"91c57238b0a37f1422b185a0d0c0ea76a29125584ab99942780383c053c4ba38","schema":"livegraph.boolean-propagation.v1","spec_hash":"69c302a2db25d81c331e6c2dfc5dc77865a26693314371d724ef9aa5a3fa7c99","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-96c7562560a9f5b48eae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1b\n- v06 --refutes--> v0b\n- v07 --supports--> v00\n- v09 --refutes--> v14\n- v10 --refutes--> v12\n- v17 --supports--> v18\n- v14 --refutes--> v05\n- v1e --supports--> v19\n- v19 --refutes--> v07\n- v19 --refutes--> v13\n- v0d --supports--> v16\n- v05 --supports--> v0d\n- v01 --supports--> v1d\n- v1e --supports--> v1a\n- v04 --supports--> v10\n- v01 --supports--> v0e\n- v0c --supports--> v03\n- v0d --supports--> v01\n- v09 --supports--> v11\n- v18 --supports--> v02\n- v19 --refutes--> v0a\n- v0b --supports--> v04\n- v15 --supports--> v1c\n- v12 --refutes--> v09\n- v05 --supports--> v08\n- v02 --refutes--> v15\n- v07 --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v06\n- v1c --supports--> v1e\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6618b02fa3bc833c1d61a810731907d08c68de4edee5e4c5906eab46b06dc31","schema":"livegraph.boolean-propagation.v1","spec_hash":"96c7562560a9f5b48eae9736dfe26b48930b0bbd4eaefee2d35a6925a58994ca","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-a0c31eb28dd7628d95fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v06 --supports--> v02\n- v0b --refutes--> v17\n- v12 --supports--> v18\n- v1d --supports--> v13\n- v0a --supports--> v1b\n- v04 --refutes--> v0b\n- v10 --refutes--> v0f\n- v1b --supports--> v01\n- v1a --supports--> v10\n- v09 --supports--> v04\n- v0d --refutes--> v12\n- v00 --refutes--> v05\n- v1a --refutes--> v0e\n- v14 --refutes--> v06\n- v1d --refutes--> v14\n- v1a --supports--> v16\n- v19 --supports--> v0c\n- v06 --supports--> v1a\n- v0c --supports--> v00\n- v0d --refutes--> v1e\n- v05 --refutes--> v0d\n- v17 --refutes--> v1d\n- v17 --refutes--> v11\n- v1a --supports--> v07\n- v0f --refutes--> v19\n- v04 --refutes--> v1c\n- v18 --refutes--> v03\n- v0b --supports--> v08\n- v01 --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1529efe8e9456f22473feccb8a0af4f0e8662f59fea9a8d88526f9cfd67a51de","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0c31eb28dd7628d95fed7b5e1d0801c8a19083895aa26bd59c90e92a54101d1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-69e1780dca3bfc18381a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v0b\n- v08 --refutes--> v1c\n- v1d --refutes--> v10\n- v0e --refutes--> v0f\n- v18 --supports--> v1d\n- v01 --refutes--> v0e\n- v10 --refutes--> v1b\n- v03 --refutes--> v00\n- v19 --supports--> v06\n- v00 --supports--> v09\n- v00 --refutes--> v13\n- v18 --refutes--> v17\n- v0e --supports--> v0c\n- v11 --supports--> v16\n- v04 --supports--> v18\n- v1b --refutes--> v03\n- v1c --supports--> v04\n- v1a --refutes--> v14\n- v09 --refutes--> v07\n- v11 --supports--> v08\n- v10 --refutes--> v15\n- v0a --supports--> v1a\n- v0b --refutes--> v01\n- v07 --refutes--> v0d\n- v19 --refutes--> v1e\n- v0d --supports--> v0a\n- v0c --refutes--> v11\n- v03 --refutes--> v12\n- v07 --refutes--> v05\n- v1a --supports--> v02\n\n[VERIFIED ROOT]\nv19 = 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":"fe1717783227035734df70709bb46bc92827f5853adaa0c4616d726bdd96646b","schema":"livegraph.boolean-propagation.v1","spec_hash":"69e1780dca3bfc18381aee6bcc51f4b4dd6767ac4155c76b36054a88adab82d6","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-26c9b21f2e18abcaf005","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v17\n- v1c --supports--> v05\n- v0c --supports--> v1b\n- v01 --supports--> v10\n- v11 --supports--> v15\n- v18 --refutes--> v0d\n- v04 --refutes--> v0b\n- v06 --supports--> v0e\n- v06 --supports--> v16\n- v1b --supports--> v00\n- v1a --supports--> v04\n- v01 --supports--> v0a\n- v13 --refutes--> v03\n- v04 --supports--> v1e\n- v17 --refutes--> v01\n- v07 --supports--> v1c\n- v13 --supports--> v0c\n- v1d --refutes--> v13\n- v1c --supports--> v02\n- v0c --supports--> v09\n- v1b --supports--> v0f\n- v0f --refutes--> v1a\n- v14 --refutes--> v19\n- v0e --refutes--> v18\n- v0a --supports--> v14\n- v07 --refutes--> v12\n- v11 --refutes--> v07\n- v0d --supports--> v1d\n- v17 --supports--> v08\n- v02 --refutes--> v06\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aa4ca1146c9222a722a8825f14f6ced8b8a6bdfe8473fd13d6628bd27f802080","schema":"livegraph.boolean-propagation.v1","spec_hash":"26c9b21f2e18abcaf005104b8e82a517f14dfb72686ac0cf6d51bb1e3bb1b56e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-a8a6f512057103623af8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v10\n- v0f --refutes--> v03\n- v0c --refutes--> v0b\n- v12 --supports--> v07\n- v0d --refutes--> v0a\n- v07 --refutes--> v09\n- v15 --supports--> v0d\n- v18 --supports--> v12\n- v0c --supports--> v1d\n- v0f --supports--> v0e\n- v11 --supports--> v1b\n- v16 --refutes--> v1e\n- v1d --supports--> v1a\n- v18 --supports--> v1c\n- v1d --supports--> v01\n- v16 --supports--> v04\n- v09 --supports--> v0c\n- v12 --refutes--> v05\n- v0e --supports--> v15\n- v01 --supports--> v0f\n- v10 --refutes--> v06\n- v02 --supports--> v16\n- v0e --refutes--> v19\n- v02 --supports--> v17\n- v13 --supports--> v02\n- v0d --refutes--> v13\n- v14 --refutes--> v11\n- v01 --supports--> v08\n- v1b --supports--> v00\n- v06 --refutes--> v18\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83c5bacfc11e5a4d183adb5153b5734adb407ddbab101a8614a97c3b30e7c3c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"a8a6f512057103623af8e6581c5f3407b25d0992ce9e37272d665db0631875d3","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-a324ebf3ae6023ccda67","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v14 --refutes--> v06\n- v09 --refutes--> v0f\n- v19 --supports--> v0a\n- v03 --supports--> v16\n- v10 --supports--> v00\n- v0d --refutes--> v04\n- v0b --supports--> v11\n- v0e --supports--> v13\n- v1c --refutes--> v1e\n- v0e --refutes--> v19\n- v1c --refutes--> v0d\n- v14 --refutes--> v05\n- v0a --refutes--> v0c\n- v10 --supports--> v02\n- v05 --refutes--> v1d\n- v11 --supports--> v12\n- v0f --supports--> v10\n- v0d --supports--> v18\n- v03 --refutes--> v08\n- v0a --supports--> v17\n- v15 --supports--> v1b\n- v12 --supports--> v09\n- v1b --refutes--> v0b\n- v10 --supports--> v07\n- v15 --supports--> v1a\n- v17 --supports--> v1c\n- v04 --refutes--> v03\n- v17 --supports--> v01\n- v07 --supports--> v0e\n- v1d --supports--> v15\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0c4af5945e46ad003484a6c8ff9b8b9a8996ebcc852e91165f327848acce283e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a324ebf3ae6023ccda6703d1880afa3b21b9ea26e50e7a835e0386bfe88f84b8","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-e4f4f71c343c2f597287","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v0f\n- v19 --refutes--> v07\n- v0e --supports--> v06\n- v0c --supports--> v14\n- v00 --refutes--> v17\n- v10 --supports--> v13\n- v08 --refutes--> v02\n- v17 --refutes--> v1c\n- v03 --refutes--> v16\n- v09 --refutes--> v03\n- v1e --refutes--> v04\n- v1a --refutes--> v0b\n- v16 --supports--> v11\n- v13 --refutes--> v19\n- v12 --supports--> v05\n- v19 --refutes--> v09\n- v19 --supports--> v0d\n- v04 --supports--> v15\n- v12 --supports--> v08\n- v12 --refutes--> v01\n- v14 --refutes--> v0e\n- v11 --supports--> v1e\n- v1b --refutes--> v18\n- v17 --refutes--> v1b\n- v17 --supports--> v0a\n- v06 --supports--> v1a\n- v15 --refutes--> v0c\n- v1e --supports--> v1d\n- v1a --refutes--> v00\n- v08 --supports--> v10\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30c25043382025be19649c0c8f6e160eb44e31c695bd0aec55f884f7012339cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4f4f71c343c2f59728733c4d0907d7298782153cac79ec356c76a2232311744","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-4e405e74392de0a5d8b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --refutes--> v07\n- v10 --refutes--> v08\n- v07 --supports--> v14\n- v03 --supports--> v17\n- v01 --supports--> v1d\n- v09 --supports--> v0b\n- v12 --refutes--> v15\n- v0b --refutes--> v0e\n- v1b --supports--> v0d\n- v16 --refutes--> v10\n- v1a --refutes--> v1e\n- v0a --refutes--> v09\n- v14 --refutes--> v19\n- v04 --supports--> v03\n- v16 --supports--> v1c\n- v0f --supports--> v01\n- v1b --refutes--> v0a\n- v15 --supports--> v0f\n- v17 --supports--> v1a\n- v05 --supports--> v18\n- v13 --refutes--> v12\n- v0b --refutes--> v04\n- v1a --supports--> v16\n- v19 --supports--> v1b\n- v09 --refutes--> v00\n- v1a --supports--> v11\n- v13 --refutes--> v06\n- v12 --refutes--> v02\n- v0f --supports--> v0c\n- v18 --refutes--> v13\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":"ec2e9d082652c49b137bc3c897ce32270c720784238012ebd195761ce1dbfaf0","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e405e74392de0a5d8b2611b9272caddd0beddf735b5550c1df8718a66a83b87","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"}],\"hop\":20,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-b98b4000e18b9cea3a3c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v18\n- v17 --supports--> v09\n- v0d --supports--> v06\n- v0d --refutes--> v07\n- v10 --refutes--> v11\n- v05 --supports--> v01\n- v07 --refutes--> v16\n- v05 --refutes--> v0c\n- v0b --supports--> v0e\n- v08 --refutes--> v0d\n- v0d --supports--> v00\n- v16 --supports--> v1a\n- v14 --supports--> v05\n- v0a --supports--> v17\n- v12 --supports--> v14\n- v0a --refutes--> v19\n- v0d --refutes--> v1b\n- v13 --supports--> v15\n- v0b --supports--> v1e\n- v1c --supports--> v0b\n- v1c --refutes--> v04\n- v1a --supports--> v12\n- v1d --supports--> v1c\n- v0f --supports--> v08\n- v07 --supports--> v03\n- v11 --supports--> v1d\n- v0e --refutes--> v13\n- v17 --refutes--> v02\n- v18 --refutes--> v0a\n- v02 --supports--> v10\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d68d0b34e138b7c973f63fde822ed342a9fd3e6f0eb933fbc72f9db3de7ac3be","schema":"livegraph.boolean-propagation.v1","spec_hash":"b98b4000e18b9cea3a3c46eb935d3e0fc028b9cd8635207bdc2ce752b591b010","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-b79ca33439a3b8c303d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v05\n- v03 --refutes--> v1d\n- v1c --supports--> v0e\n- v07 --refutes--> v11\n- v0b --supports--> v07\n- v06 --supports--> v04\n- v08 --refutes--> v0b\n- v05 --refutes--> v17\n- v17 --supports--> v1b\n- v00 --refutes--> v0c\n- v1d --supports--> v06\n- v01 --refutes--> v1e\n- v1c --supports--> v01\n- v19 --refutes--> v16\n- v07 --supports--> v09\n- v1e --supports--> v19\n- v03 --refutes--> v0a\n- v1e --supports--> v14\n- v11 --refutes--> v15\n- v12 --supports--> v1a\n- v10 --refutes--> v12\n- v1a --supports--> v03\n- v12 --refutes--> v02\n- v17 --supports--> v18\n- v03 --supports--> v0d\n- v16 --supports--> v08\n- v11 --refutes--> v0f\n- v1b --refutes--> v10\n- v1d --supports--> v13\n- v06 --supports--> v1c\n\n[VERIFIED ROOT]\nv00 = 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":"883cbdd2516a7209ac4d8d66ee03f4c87ecd316272d552d212a42b35f3aa2e41","schema":"livegraph.boolean-propagation.v1","spec_hash":"b79ca33439a3b8c303d97a3d739385839ac5c87ee7cbe2339992615556724f57","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-288463bf941b6bcef8fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v08\n- v0b --supports--> v1a\n- v1b --supports--> v1e\n- v04 --refutes--> v06\n- v1c --supports--> v11\n- v1e --refutes--> v0a\n- v1e --supports--> v16\n- v1a --refutes--> v1c\n- v18 --supports--> v1d\n- v08 --refutes--> v14\n- v00 --supports--> v1b\n- v13 --refutes--> v04\n- v0d --supports--> v0c\n- v10 --refutes--> v17\n- v0b --supports--> v15\n- v16 --refutes--> v05\n- v08 --refutes--> v07\n- v11 --refutes--> v10\n- v12 --refutes--> v0b\n- v18 --refutes--> v0e\n- v07 --refutes--> v18\n- v18 --refutes--> v19\n- v1d --supports--> v0f\n- v18 --refutes--> v01\n- v04 --supports--> v02\n- v16 --supports--> v09\n- v02 --refutes--> v12\n- v0c --refutes--> v13\n- v12 --supports--> v03\n- v1d --refutes--> v00\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":"7e65b9cb9fa88e2c5fa477a99b05cc56ec8bac3c5a09de73891bb81b798aebc7","schema":"livegraph.boolean-propagation.v1","spec_hash":"288463bf941b6bcef8fa17e326558e174543bf3e94b33a069c9b2daf909928a4","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-dfe6a20f2bf6617d0870","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v15\n- v1b --refutes--> v10\n- v14 --supports--> v03\n- v12 --refutes--> v1b\n- v19 --supports--> v18\n- v1b --supports--> v11\n- v1b --supports--> v16\n- v15 --supports--> v0d\n- v1c --supports--> v09\n- v02 --refutes--> v06\n- v11 --supports--> v05\n- v15 --supports--> v08\n- v0e --refutes--> v01\n- v02 --refutes--> v12\n- v17 --supports--> v1a\n- v0e --supports--> v0b\n- v0f --refutes--> v1d\n- v0e --supports--> v0f\n- v1a --supports--> v14\n- v09 --refutes--> v02\n- v00 --refutes--> v0c\n- v1e --supports--> v0e\n- v0f --refutes--> v04\n- v0d --supports--> v1e\n- v0e --refutes--> v13\n- v0a --refutes--> v00\n- v07 --supports--> v17\n- v05 --supports--> v07\n- v14 --supports--> v19\n- v0c --refutes--> v1c\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"feb4f164a882b780263832ee52139c8fc89dae8c60513ca53fac3f578e392887","schema":"livegraph.boolean-propagation.v1","spec_hash":"dfe6a20f2bf6617d08701ca88d8d26296ae6ab212fa981e86eeddb97b9ce2b02","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-89c6d38616f54586ceed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v0c\n- v03 --refutes--> v0e\n- v1c --refutes--> v19\n- v15 --supports--> v08\n- v1b --supports--> v16\n- v1c --refutes--> v15\n- v12 --refutes--> v1a\n- v09 --refutes--> v1b\n- v0c --refutes--> v00\n- v03 --supports--> v05\n- v07 --refutes--> v04\n- v06 --refutes--> v11\n- v00 --refutes--> v1c\n- v12 --supports--> v0d\n- v14 --supports--> v18\n- v0c --supports--> v0b\n- v17 --supports--> v07\n- v0e --refutes--> v10\n- v08 --refutes--> v0f\n- v13 --refutes--> v17\n- v12 --supports--> v01\n- v18 --supports--> v0a\n- v14 --refutes--> v02\n- v04 --refutes--> v09\n- v1b --supports--> v14\n- v06 --supports--> v03\n- v08 --supports--> v06\n- v0a --refutes--> v12\n- v0d --supports--> v1e\n- v08 --refutes--> v1d\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e50a188eea8f4d3102ddd0709ab123cbb27f3ff445e86b1602847feb2493f1bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"89c6d38616f54586ceed08b8a0799b1d210f1c3acf075dd617d7e21ce3536283","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-20fe8840ed5964400e1a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v06\n- v19 --refutes--> v0f\n- v03 --supports--> v1d\n- v0c --supports--> v1a\n- v15 --refutes--> v17\n- v0c --refutes--> v1b\n- v15 --supports--> v12\n- v1b --supports--> v1c\n- v04 --refutes--> v01\n- v03 --supports--> v19\n- v02 --supports--> v04\n- v0b --supports--> v1e\n- v05 --supports--> v03\n- v04 --refutes--> v0a\n- v16 --supports--> v15\n- v0f --supports--> v07\n- v06 --supports--> v05\n- v13 --supports--> v00\n- v0c --refutes--> v18\n- v0a --supports--> v0e\n- v07 --refutes--> v11\n- v00 --supports--> v0c\n- v08 --supports--> v02\n- v00 --supports--> v0d\n- v11 --supports--> v08\n- v0a --refutes--> v14\n- v0c --supports--> v10\n- v1e --supports--> v16\n- v14 --supports--> v13\n- v14 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f69b584b00717e96ba1cd34070acc7d2f566e47037aac867b5948c95616e38f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"20fe8840ed5964400e1ad5473ab8a43c4f88c4d3338eecbcf8857f3426f6e5d2","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-cceaa3beb5a7db549187","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --supports--> v0d\n- v0f --refutes--> v03\n- v17 --supports--> v12\n- v07 --supports--> v05\n- v0d --refutes--> v0e\n- v0e --supports--> v08\n- v03 --supports--> v0b\n- v0b --refutes--> v13\n- v01 --refutes--> v0f\n- v07 --supports--> v02\n- v0b --refutes--> v15\n- v1c --refutes--> v18\n- v04 --supports--> v00\n- v0e --refutes--> v19\n- v17 --refutes--> v06\n- v03 --supports--> v14\n- v11 --refutes--> v0c\n- v05 --refutes--> v1c\n- v00 --refutes--> v11\n- v0f --refutes--> v1d\n- v11 --supports--> v09\n- v11 --supports--> v01\n- v18 --refutes--> v04\n- v08 --refutes--> v10\n- v0f --supports--> v1e\n- v15 --refutes--> v16\n- v1b --supports--> v17\n- v1a --supports--> v07\n- v18 --refutes--> v0a\n- v16 --refutes--> v1b\n\n[VERIFIED ROOT]\nv1a = 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":"f3cff079d473f647bce8765f015b7dc5735a73667343e924fce09d82bf276222","schema":"livegraph.boolean-propagation.v1","spec_hash":"cceaa3beb5a7db54918741253f683e4375773c43983b5d864b02a9e8993878ce","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-6a099a39822067d0625d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v0c\n- v0c --supports--> v04\n- v0a --supports--> v02\n- v15 --refutes--> v1d\n- v15 --refutes--> v10\n- v04 --supports--> v12\n- v15 --refutes--> v11\n- v17 --refutes--> v05\n- v08 --supports--> v06\n- v1c --supports--> v1e\n- v14 --supports--> v13\n- v1e --supports--> v03\n- v0b --refutes--> v16\n- v02 --refutes--> v1a\n- v07 --supports--> v01\n- v17 --refutes--> v0f\n- v10 --refutes--> v18\n- v13 --supports--> v15\n- v18 --refutes--> v0d\n- v01 --refutes--> v17\n- v16 --refutes--> v1c\n- v01 --supports--> v00\n- v07 --refutes--> v1b\n- v1a --supports--> v0b\n- v08 --refutes--> v19\n- v08 --supports--> v0a\n- v17 --supports--> v09\n- v12 --supports--> v14\n- v1e --supports--> v07\n- v15 --refutes--> v0e\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d5c82679eaaca3c4954c3d59d06580e95227b21573f31b98ed030e9a625e470f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a099a39822067d0625d70f0a698d8b945d0d30a9c23eb3162db1b73a4115d6d","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-a2821beff4a94ca48150","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v0a\n- v11 --supports--> v1a\n- v02 --refutes--> v19\n- v05 --refutes--> v11\n- v17 --supports--> v04\n- v0c --refutes--> v16\n- v02 --supports--> v14\n- v1a --refutes--> v0e\n- v19 --supports--> v01\n- v0f --supports--> v1e\n- v0e --supports--> v02\n- v02 --refutes--> v1c\n- v0e --supports--> v10\n- v06 --refutes--> v05\n- v03 --supports--> v15\n- v13 --supports--> v09\n- v13 --refutes--> v03\n- v1d --refutes--> v06\n- v03 --refutes--> v18\n- v0b --refutes--> v17\n- v1e --supports--> v12\n- v1e --refutes--> v0c\n- v0c --supports--> v1d\n- v0b --refutes--> v00\n- v0a --supports--> v0f\n- v18 --refutes--> v0b\n- v07 --refutes--> v13\n- v01 --supports--> v07\n- v17 --refutes--> v08\n- v1e --supports--> v0d\n\n[VERIFIED ROOT]\nv1b = 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":"196e379c811d80ce6cea57852dcfc59eb7e93aaafb5672b9ffbe0f9b7b44129a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2821beff4a94ca48150553a7861901a5221d3964e314153e6edb78e468007bd","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-9d9ca7db48c5c3536fad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v17 --refutes--> v16\n- v12 --refutes--> v0c\n- v11 --refutes--> v1a\n- v1a --supports--> v0f\n- v08 --refutes--> v09\n- v17 --supports--> v0b\n- v05 --refutes--> v1d\n- v15 --supports--> v04\n- v09 --refutes--> v02\n- v1d --refutes--> v1b\n- v13 --refutes--> v15\n- v0c --refutes--> v0d\n- v06 --supports--> v18\n- v1c --supports--> v1e\n- v04 --refutes--> v12\n- v02 --supports--> v19\n- v11 --supports--> v07\n- v04 --supports--> v10\n- v1d --supports--> v06\n- v16 --refutes--> v1c\n- v03 --supports--> v11\n- v1e --supports--> v13\n- v1e --supports--> v00\n- v1e --supports--> v0a\n- v1d --refutes--> v0e\n- v19 --refutes--> v17\n- v03 --refutes--> v01\n- v18 --supports--> v08\n- v1a --supports--> v05\n\n[VERIFIED ROOT]\nv03 = 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":"a15912c770831a1228056552a5190bacf3678e7bebb8f3927e374db56497a03f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9d9ca7db48c5c3536fade51e27ac46b66265dae7a0783950474d7aca742d36cc","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-f058fefb7b105f74ab05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v1b\n- v01 --refutes--> v1a\n- v16 --refutes--> v14\n- v11 --supports--> v0c\n- v0d --supports--> v15\n- v1a --refutes--> v19\n- v19 --supports--> v17\n- v02 --refutes--> v04\n- v15 --refutes--> v11\n- v1a --refutes--> v0a\n- v16 --supports--> v0f\n- v1e --refutes--> v03\n- v10 --refutes--> v01\n- v07 --supports--> v12\n- v07 --supports--> v1d\n- v01 --refutes--> v0b\n- v1b --refutes--> v05\n- v11 --supports--> v09\n- v07 --supports--> v08\n- v0e --refutes--> v0d\n- v18 --refutes--> v0e\n- v0c --supports--> v02\n- v1e --supports--> v13\n- v06 --supports--> v18\n- v19 --refutes--> v1e\n- v04 --refutes--> v16\n- v12 --supports--> v06\n- v05 --supports--> v1c\n- v11 --refutes--> v00\n- v1c --refutes--> v07\n\n[VERIFIED ROOT]\nv10 = 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":"481d7918b6e37cb1ee7955ea44314c1ad1e57f3c5e4a1263ba89337f9a43151c","schema":"livegraph.boolean-propagation.v1","spec_hash":"f058fefb7b105f74ab05b14cf13d9c5bf2a52dafe84a2ac523b40e2e69fed281","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-94ebcbbfacb2ea7d85bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v14 --supports--> v04\n- v1a --supports--> v0f\n- v14 --supports--> v02\n- v01 --supports--> v18\n- v03 --supports--> v19\n- v07 --supports--> v0e\n- v01 --supports--> v17\n- v05 --refutes--> v1b\n- v0a --refutes--> v0b\n- v18 --supports--> v0c\n- v03 --refutes--> v15\n- v08 --refutes--> v1e\n- v19 --supports--> v14\n- v0d --supports--> v03\n- v12 --supports--> v0a\n- v0b --supports--> v08\n- v01 --supports--> v1c\n- v1e --supports--> v09\n- v1e --refutes--> v13\n- v07 --refutes--> v06\n- v07 --supports--> v00\n- v06 --supports--> v1a\n- v02 --refutes--> v07\n- v0a --refutes--> v16\n- v1e --refutes--> v01\n- v08 --supports--> v11\n- v1d --supports--> v10\n- v0f --refutes--> v12\n- v0c --supports--> v1d\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c7ee930a3884ba7602fa2596a14ddcffe616378fb16f81e54fb27bd745a3aa9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"94ebcbbfacb2ea7d85bd061f93f34b81168afd5be10f43c482d61055ea91d0ec","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-eba766e9f8d1ed4c8f3b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v16\n- v0a --refutes--> v00\n- v07 --refutes--> v0c\n- v15 --supports--> v1d\n- v13 --refutes--> v17\n- v07 --refutes--> v1a\n- v09 --refutes--> v10\n- v10 --supports--> v15\n- v06 --supports--> v12\n- v0c --supports--> v13\n- v11 --refutes--> v09\n- v0e --refutes--> v05\n- v05 --supports--> v06\n- v17 --supports--> v0f\n- v06 --refutes--> v14\n- v0d --refutes--> v0a\n- v12 --refutes--> v1e\n- v0b --refutes--> v07\n- v18 --supports--> v0d\n- v0b --refutes--> v1c\n- v0b --refutes--> v19\n- v15 --refutes--> v0e\n- v16 --refutes--> v01\n- v08 --supports--> v0b\n- v0f --refutes--> v04\n- v0a --supports--> v02\n- v02 --supports--> v11\n- v16 --refutes--> v08\n- v15 --supports--> v1b\n- v11 --supports--> v03\n\n[VERIFIED ROOT]\nv18 = 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":"ef7f578ddc782e0f0f3488678343efbdbaf3001ee27d953ac30740a5ad9209fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"eba766e9f8d1ed4c8f3bd47845e3f760f25d25ca1ca02316d8f5584a77e7e715","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-c59f0bdad23682a046d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --supports--> v0a\n- v05 --refutes--> v01\n- v0d --supports--> v13\n- v12 --refutes--> v0b\n- v03 --supports--> v04\n- v03 --supports--> v17\n- v15 --supports--> v18\n- v08 --supports--> v1b\n- v1c --refutes--> v05\n- v08 --supports--> v1a\n- v07 --supports--> v03\n- v1b --refutes--> v1e\n- v0e --supports--> v08\n- v1b --supports--> v1c\n- v07 --supports--> v16\n- v07 --refutes--> v02\n- v19 --refutes--> v00\n- v12 --supports--> v11\n- v06 --supports--> v0c\n- v13 --supports--> v10\n- v05 --refutes--> v12\n- v11 --supports--> v1d\n- v0f --supports--> v14\n- v09 --supports--> v0e\n- v1d --supports--> v15\n- v00 --supports--> v0f\n- v13 --supports--> v07\n- v18 --refutes--> v0d\n- v14 --refutes--> v06\n- v06 --supports--> v09\n\n[VERIFIED ROOT]\nv19 = 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":"a2f228f980748c648e40b79cfbb1ea70cb2bcb373a3f4ee993b6e7b0d2af27ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"c59f0bdad23682a046d9fbde7a39c8b0f068febd55a36287e9c228815b50e48d","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-a918ed313a762b0e28d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v05\n- v1b --supports--> v10\n- v0d --supports--> v1d\n- v0e --supports--> v14\n- v1b --refutes--> v1e\n- v03 --supports--> v06\n- v1c --refutes--> v02\n- v15 --refutes--> v13\n- v04 --refutes--> v11\n- v16 --supports--> v1b\n- v11 --supports--> v15\n- v0c --refutes--> v07\n- v07 --supports--> v08\n- v11 --refutes--> v0f\n- v1b --refutes--> v12\n- v19 --refutes--> v0d\n- v03 --supports--> v1c\n- v02 --supports--> v0c\n- v1e --refutes--> v17\n- v0b --supports--> v16\n- v14 --refutes--> v19\n- v08 --supports--> v0e\n- v1c --supports--> v09\n- v16 --refutes--> v18\n- v0a --supports--> v04\n- v07 --supports--> v1a\n- v0d --refutes--> v00\n- v15 --supports--> v03\n- v01 --refutes--> v0b\n- v17 --refutes--> v0a\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":"b04c62013b205e897f25d53e87dd94299d61ba84a147158f33500ff7978fae07","schema":"livegraph.boolean-propagation.v1","spec_hash":"a918ed313a762b0e28d809edaf9fd5c798040f051540342933da39c8dfdc74ec","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-a98c0e30a7dd7edd368e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v14 --refutes--> v06\n- v0c --supports--> v12\n- v15 --supports--> v19\n- v12 --supports--> v0d\n- v12 --refutes--> v0b\n- v11 --refutes--> v14\n- v0e --supports--> v05\n- v0c --refutes--> v16\n- v0c --refutes--> v18\n- v07 --refutes--> v1d\n- v03 --supports--> v07\n- v14 --refutes--> v13\n- v0b --supports--> v1e\n- v17 --supports--> v0f\n- v0e --refutes--> v15\n- v02 --supports--> v01\n- v13 --refutes--> v0e\n- v02 --supports--> v1b\n- v08 --supports--> v09\n- v10 --refutes--> v17\n- v0a --refutes--> v0c\n- v19 --supports--> v02\n- v04 --supports--> v0a\n- v0b --supports--> v1a\n- v1c --refutes--> v10\n- v0b --supports--> v00\n- v17 --refutes--> v08\n- v00 --supports--> v1c\n- v09 --refutes--> v03\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98fb3ae1f8cbbcf6bba048af5b9228c6a6f6e13ee61b9d729fcdf2c84716dfc1","schema":"livegraph.boolean-propagation.v1","spec_hash":"a98c0e30a7dd7edd368e0b4a7e8b647c61608519b22a877641f158787b6da48f","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-d2d962281025505cb7e7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v0a\n- v1b --supports--> v04\n- v08 --refutes--> v1e\n- v13 --refutes--> v07\n- v06 --refutes--> v11\n- v03 --refutes--> v0b\n- v05 --refutes--> v0f\n- v03 --supports--> v08\n- v15 --refutes--> v09\n- v00 --refutes--> v05\n- v16 --refutes--> v01\n- v1e --supports--> v1d\n- v1e --supports--> v14\n- v1b --refutes--> v17\n- v1e --refutes--> v0d\n- v0c --supports--> v15\n- v09 --refutes--> v10\n- v18 --supports--> v02\n- v1c --refutes--> v00\n- v18 --supports--> v1a\n- v16 --refutes--> v03\n- v07 --supports--> v0c\n- v11 --supports--> v0e\n- v12 --refutes--> v18\n- v1e --supports--> v19\n- v11 --supports--> v12\n- v1d --supports--> v06\n- v1b --refutes--> v16\n- v0f --supports--> v13\n- v10 --supports--> v1b\n\n[VERIFIED ROOT]\nv1c = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"adad3663a701be35bd4a6d611a89356f82833cf8a73eb5953c06f457ad80715a","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2d962281025505cb7e72319b0847bd85f77afd7ba9e3af1495481f5f085e793","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-7a89cdf8411b6c52cf56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1d\n- v1d --refutes--> v16\n- v03 --supports--> v05\n- v0c --supports--> v0f\n- v01 --refutes--> v06\n- v1c --supports--> v1b\n- v1e --refutes--> v0e\n- v06 --supports--> v09\n- v00 --refutes--> v11\n- v0a --refutes--> v01\n- v08 --supports--> v0c\n- v05 --supports--> v19\n- v05 --supports--> v00\n- v0b --refutes--> v03\n- v00 --supports--> v14\n- v1e --refutes--> v15\n- v1d --supports--> v18\n- v10 --supports--> v08\n- v14 --refutes--> v10\n- v1b --supports--> v0d\n- v18 --supports--> v1a\n- v15 --supports--> v04\n- v1a --supports--> v0b\n- v15 --supports--> v17\n- v0c --supports--> v1c\n- v17 --refutes--> v0a\n- v00 --refutes--> v02\n- v12 --supports--> v1e\n- v10 --supports--> v13\n- v1e --supports--> v07\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a795e39a6da758a17891b3b5fc180a22c7fe05ad3e822e9f8772517ad626525","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a89cdf8411b6c52cf5622d1ec634d51d210b3b1ce71a7136e2b39c421de1a1f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-4bd6b9e65c11dfc6eacd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --refutes--> v10\n- v19 --supports--> v0f\n- v17 --supports--> v16\n- v17 --refutes--> v11\n- v05 --supports--> v00\n- v12 --supports--> v01\n- v1d --refutes--> v18\n- v10 --supports--> v09\n- v13 --refutes--> v1c\n- v19 --supports--> v05\n- v0b --refutes--> v13\n- v00 --supports--> v14\n- v01 --refutes--> v0c\n- v11 --refutes--> v1e\n- v05 --supports--> v0e\n- v07 --refutes--> v1b\n- v13 --supports--> v1a\n- v1a --refutes--> v03\n- v06 --supports--> v0b\n- v12 --supports--> v04\n- v00 --supports--> v08\n- v07 --supports--> v12\n- v12 --refutes--> v0a\n- v0d --refutes--> v06\n- v0d --supports--> v02\n- v18 --supports--> v19\n- v08 --supports--> v07\n- v0c --supports--> v0d\n- v1e --supports--> v15\n- v1e --refutes--> v1d\n\n[VERIFIED ROOT]\nv17 = 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":"0311c35a6424e666d227d6a558b828f75cbe0b90eaf0e7e3689f6be72b2fcef6","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bd6b9e65c11dfc6eacd3d642e94eba97f0e3de866955bcf700cf290ec99f3f9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-95af5f550ce6ce83ac75","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v00\n- v0f --supports--> v07\n- v07 --supports--> v14\n- v04 --supports--> v01\n- v19 --refutes--> v10\n- v03 --supports--> v18\n- v0d --refutes--> v19\n- v03 --refutes--> v0a\n- v11 --supports--> v13\n- v06 --refutes--> v1c\n- v00 --refutes--> v09\n- v17 --supports--> v08\n- v08 --refutes--> v05\n- v11 --refutes--> v03\n- v17 --supports--> v02\n- v18 --refutes--> v1e\n- v0b --supports--> v16\n- v10 --supports--> v0e\n- v0b --supports--> v0d\n- v10 --refutes--> v12\n- v05 --refutes--> v06\n- v00 --supports--> v1d\n- v00 --refutes--> v0f\n- v0c --refutes--> v0b\n- v1a --refutes--> v11\n- v04 --supports--> v1b\n- v1b --refutes--> v15\n- v06 --refutes--> v1a\n- v15 --refutes--> v0c\n- v14 --refutes--> v17\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0b91ea87050104c64cb7c254063014d39b39111604cea631b17ebf1a6ecb75d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"95af5f550ce6ce83ac75e17a0f853464f0f6eb8aa9ba106f7e543b640d38f1e8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1b\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-5a0d21ebc50c48423c65","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v09\n- v1b --refutes--> v14\n- v13 --supports--> v06\n- v00 --refutes--> v0a\n- v07 --refutes--> v1a\n- v0f --refutes--> v0d\n- v15 --supports--> v04\n- v06 --supports--> v10\n- v1a --supports--> v13\n- v1b --refutes--> v0f\n- v1c --supports--> v07\n- v1b --refutes--> v02\n- v12 --refutes--> v0e\n- v04 --supports--> v01\n- v16 --refutes--> v08\n- v04 --refutes--> v05\n- v0b --refutes--> v15\n- v18 --refutes--> v1c\n- v0e --refutes--> v11\n- v19 --refutes--> v18\n- v15 --supports--> v03\n- v16 --refutes--> v19\n- v04 --supports--> v1e\n- v10 --refutes--> v00\n- v1e --supports--> v12\n- v07 --refutes--> v1d\n- v00 --refutes--> v0b\n- v11 --supports--> v1b\n- v17 --refutes--> v16\n- v0f --supports--> v0c\n\n[VERIFIED ROOT]\nv17 = 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":"45709584fa082d8b6ab11ddd342a4bfeb3072079bed637e0f868952f2db72dbf","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a0d21ebc50c48423c6587adaf31c77ca048bf83389b0c7f8617351d508743a8","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-6ba3bf10989430431b37","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v19\n- v0a --refutes--> v0b\n- v06 --refutes--> v02\n- v1e --refutes--> v18\n- v1c --refutes--> v06\n- v09 --supports--> v07\n- v07 --supports--> v16\n- v16 --refutes--> v1e\n- v1c --refutes--> v11\n- v09 --refutes--> v12\n- v0e --supports--> v1c\n- v0e --supports--> v15\n- v14 --refutes--> v1b\n- v1c --refutes--> v01\n- v18 --supports--> v0e\n- v10 --supports--> v13\n- v02 --supports--> v00\n- v04 --supports--> v03\n- v04 --refutes--> v1a\n- v1a --supports--> v08\n- v1e --refutes--> v17\n- v0c --supports--> v1d\n- v1d --supports--> v0f\n- v05 --refutes--> v10\n- v14 --refutes--> v05\n- v00 --supports--> v0c\n- v0f --refutes--> v0a\n- v10 --supports--> v04\n- v04 --supports--> v0d\n- v08 --supports--> v09\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b542dbc1dbcb54d2ad65e45199cc2a2d4ac42218c6fdd6ba27d753ae62a4b038","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ba3bf10989430431b3713fd68e1ad5c022e8ae41a47f89bc2a30d4e89a2cf0a","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-c10ff93b4daa99e2ef3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v02 --refutes--> v08\n- v0c --refutes--> v02\n- v18 --supports--> v0d\n- v19 --supports--> v1b\n- v12 --supports--> v10\n- v0a --refutes--> v04\n- v12 --supports--> v16\n- v03 --supports--> v11\n- v03 --supports--> v19\n- v07 --supports--> v09\n- v0a --supports--> v0b\n- v08 --supports--> v1a\n- v1b --refutes--> v1e\n- v19 --refutes--> v01\n- v08 --supports--> v07\n- v1c --refutes--> v12\n- v10 --supports--> v14\n- v10 --refutes--> v05\n- v0d --refutes--> v1d\n- v0b --supports--> v0c\n- v0e --supports--> v0a\n- v1b --refutes--> v15\n- v1b --refutes--> v0f\n- v0c --refutes--> v13\n- v0f --supports--> v00\n- v00 --supports--> v0e\n- v14 --supports--> v18\n- v09 --supports--> v1c\n- v06 --refutes--> v03\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eea7f75c5750ce69a06e4b15b2a8e07d8f0a23ab21759f8575fe42034a3824d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"c10ff93b4daa99e2ef3d9677ac98eceb4f440b710ae728b792412a77ff799572","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-9ab1a3e7a5762ed8296f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v14\n- v11 --supports--> v08\n- v05 --refutes--> v17\n- v08 --refutes--> v0d\n- v0c --supports--> v16\n- v00 --refutes--> v04\n- v17 --refutes--> v1c\n- v03 --supports--> v1a\n- v13 --supports--> v0b\n- v0e --refutes--> v15\n- v10 --refutes--> v02\n- v1e --refutes--> v03\n- v02 --supports--> v0c\n- v00 --refutes--> v1b\n- v0d --supports--> v05\n- v0b --supports--> v01\n- v0f --refutes--> v06\n- v03 --supports--> v0f\n- v00 --supports--> v07\n- v0a --supports--> v11\n- v06 --refutes--> v00\n- v17 --supports--> v0e\n- v0a --supports--> v18\n- v0b --supports--> v19\n- v19 --refutes--> v1e\n- v09 --refutes--> v13\n- v0e --supports--> v10\n- v08 --refutes--> v1d\n- v0c --refutes--> v12\n- v16 --supports--> v09\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e44b57fe76ceace35fa612aafa55b190c99ec8def927013c596e0b868f9d2c38","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ab1a3e7a5762ed8296fddd71aaa43e60e515046ce14c0ab116a96a776c46ad6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-082f084f12e093d14d14","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1c\n- v1c --supports--> v05\n- v12 --refutes--> v1e\n- v1a --supports--> v0e\n- v07 --refutes--> v12\n- v10 --supports--> v19\n- v0f --supports--> v03\n- v18 --refutes--> v0b\n- v01 --supports--> v04\n- v0e --refutes--> v1b\n- v13 --supports--> v07\n- v0e --supports--> v08\n- v18 --refutes--> v0a\n- v0c --refutes--> v11\n- v1a --supports--> v17\n- v12 --refutes--> v02\n- v03 --supports--> v16\n- v1a --refutes--> v09\n- v08 --refutes--> v10\n- v15 --refutes--> v13\n- v05 --refutes--> v15\n- v1d --refutes--> v1a\n- v03 --refutes--> v0d\n- v04 --refutes--> v18\n- v0d --supports--> v0c\n- v18 --supports--> v0f\n- v13 --refutes--> v14\n- v1e --supports--> v1d\n- v0f --refutes--> v00\n- v0f --supports--> v06\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1465b40540f57e92f07329953d2cfc5e87cdfd8c0a53db9dda94880302ddeda7","schema":"livegraph.boolean-propagation.v1","spec_hash":"082f084f12e093d14d14a82b4bc954380bcd57d5af5983a5f46b81b639bc673f","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-20107f3bcb295f8d1241","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v18 --refutes--> v14\n- v11 --refutes--> v1d\n- v15 --supports--> v03\n- v01 --refutes--> v1e\n- v1b --supports--> v09\n- v11 --supports--> v0b\n- v06 --refutes--> v0f\n- v15 --supports--> v0d\n- v00 --refutes--> v11\n- v0d --refutes--> v05\n- v1a --supports--> v08\n- v06 --supports--> v02\n- v1d --refutes--> v17\n- v18 --refutes--> v10\n- v18 --supports--> v16\n- v02 --refutes--> v00\n- v09 --refutes--> v0e\n- v05 --supports--> v1c\n- v1c --refutes--> v0c\n- v07 --supports--> v12\n- v01 --refutes--> v18\n- v14 --supports--> v15\n- v0e --refutes--> v01\n- v1d --supports--> v19\n- v07 --supports--> v1a\n- v0e --refutes--> v13\n- v1c --refutes--> v0a\n- v17 --supports--> v07\n- v04 --refutes--> v06\n- v0c --refutes--> v04\n\n[VERIFIED ROOT]\nv1b = 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":"ba5c6f1e1802ffbaae1bcd9f4c7cd62b9ca54c3cd33f8b52b89c4267360ee589","schema":"livegraph.boolean-propagation.v1","spec_hash":"20107f3bcb295f8d12418c9492a3ccf8e031cce37bb1c234012ad5d3fd3f7269","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-3bdfe600153908ba3ad3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --supports--> v0b\n- v04 --refutes--> v09\n- v09 --refutes--> v14\n- v1c --refutes--> v01\n- v14 --supports--> v0c\n- v06 --refutes--> v00\n- v01 --supports--> v17\n- v12 --refutes--> v1a\n- v02 --supports--> v05\n- v0f --supports--> v08\n- v1d --supports--> v10\n- v10 --refutes--> v03\n- v14 --supports--> v1d\n- v12 --refutes--> v02\n- v09 --supports--> v13\n- v00 --supports--> v11\n- v0f --supports--> v1e\n- v02 --supports--> v0a\n- v1c --refutes--> v18\n- v01 --refutes--> v12\n- v16 --supports--> v07\n- v05 --refutes--> v0f\n- v19 --supports--> v0d\n- v15 --refutes--> v0e\n- v16 --supports--> v19\n- v03 --supports--> v1b\n- v19 --supports--> v15\n- v11 --supports--> v1c\n- v08 --refutes--> v04\n- v1b --supports--> v16\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2030fc8948b0b488ba8850d1779ad96e317a10de6974d02912f9ee1cc3f6ab9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bdfe600153908ba3ad3cd55cdc165faa809ae1188d07c5d84de34c3575507a7","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-beb4af9a5eb075bd6dc6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v0d --supports--> v1b\n- v15 --refutes--> v0d\n- v11 --supports--> v08\n- v00 --supports--> v16\n- v1e --supports--> v1c\n- v0b --supports--> v1d\n- v0e --refutes--> v14\n- v1a --refutes--> v01\n- v18 --refutes--> v11\n- v0c --refutes--> v19\n- v0f --refutes--> v0b\n- v1d --refutes--> v1a\n- v15 --supports--> v12\n- v03 --refutes--> v02\n- v00 --supports--> v1e\n- v10 --refutes--> v18\n- v1e --supports--> v03\n- v0a --supports--> v0e\n- v07 --supports--> v06\n- v17 --supports--> v10\n- v18 --supports--> v04\n- v09 --refutes--> v0f\n- v19 --refutes--> v15\n- v0d --supports--> v09\n- v1a --supports--> v07\n- v0e --refutes--> v0c\n- v11 --supports--> v00\n- v06 --supports--> v17\n- v15 --supports--> v13\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":"e9658e357c080e211f82a5fea52cdeb5390f4cb4ee901d5654a594c74c0ad805","schema":"livegraph.boolean-propagation.v1","spec_hash":"beb4af9a5eb075bd6dc60b113e8b4aa8cbf96de35f30500a5747c2138cb39848","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-d0cc9c397ee1bae97ecb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v1e\n- v1c --supports--> v0c\n- v03 --refutes--> v07\n- v0a --supports--> v09\n- v10 --supports--> v0e\n- v11 --supports--> v18\n- v03 --supports--> v0f\n- v15 --refutes--> v0d\n- v05 --refutes--> v11\n- v09 --refutes--> v10\n- v1e --supports--> v03\n- v09 --supports--> v0b\n- v06 --supports--> v01\n- v10 --refutes--> v12\n- v01 --supports--> v05\n- v02 --supports--> v14\n- v06 --refutes--> v13\n- v1c --refutes--> v08\n- v02 --supports--> v16\n- v16 --supports--> v17\n- v1e --supports--> v1b\n- v17 --supports--> v1a\n- v0d --refutes--> v0a\n- v17 --supports--> v06\n- v1d --supports--> v04\n- v11 --refutes--> v1d\n- v0f --refutes--> v1c\n- v0c --refutes--> v02\n- v04 --refutes--> v15\n- v16 --supports--> v00\n\n[VERIFIED ROOT]\nv19 = 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":"faaba9604e450b30a3224af4e70127c0feabaa9d39f8d8e22528c757c3ce3520","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0cc9c397ee1bae97ecb53fc57d3677d0c1975dec762e245cee5d079533e1ff7","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-2d182cb1d7eed6dee8b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v12 --refutes--> v05\n- v06 --supports--> v1c\n- v0c --supports--> v16\n- v13 --supports--> v0c\n- v0f --supports--> v09\n- v1a --refutes--> v11\n- v1d --refutes--> v03\n- v0d --supports--> v17\n- v16 --supports--> v06\n- v0f --supports--> v0b\n- v19 --supports--> v02\n- v00 --refutes--> v07\n- v09 --supports--> v13\n- v03 --refutes--> v0d\n- v00 --supports--> v04\n- v06 --supports--> v18\n- v1e --supports--> v08\n- v0e --supports--> v00\n- v0c --refutes--> v10\n- v13 --supports--> v14\n- v12 --refutes--> v0f\n- v01 --supports--> v19\n- v1b --refutes--> v0e\n- v1a --supports--> v12\n- v18 --refutes--> v1b\n- v02 --refutes--> v1a\n- v07 --refutes--> v0a\n- v07 --refutes--> v1d\n- v16 --supports--> v15\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":"d52f2b83a8ac07a49224473d94143baaa262b4401eb6238ff7b74d61a18fdc44","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d182cb1d7eed6dee8b07d42cff6893528dac0345ba95f1951c42781a3b5bcab","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-00eb6ef646acd5169ddc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v19 --supports--> v04\n- v1a --supports--> v10\n- v09 --supports--> v07\n- v07 --supports--> v1e\n- v1e --refutes--> v05\n- v19 --supports--> v15\n- v1b --supports--> v13\n- v1e --supports--> v1a\n- v1d --supports--> v11\n- v06 --refutes--> v12\n- v13 --refutes--> v18\n- v08 --supports--> v06\n- v16 --refutes--> v17\n- v12 --supports--> v01\n- v0e --supports--> v1b\n- v12 --supports--> v0a\n- v1e --supports--> v14\n- v18 --supports--> v16\n- v15 --refutes--> v09\n- v0c --refutes--> v19\n- v0b --supports--> v08\n- v1a --refutes--> v0b\n- v03 --refutes--> v1d\n- v11 --supports--> v0e\n- v07 --supports--> v1c\n- v18 --refutes--> v02\n- v12 --refutes--> v00\n- v17 --refutes--> v0c\n- v15 --supports--> v0d\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c695e50cabb84faea15200b4b278788e398e1c9c136da407e24dceab28325bb9","schema":"livegraph.boolean-propagation.v1","spec_hash":"00eb6ef646acd5169ddc74714074cb562fdda8a7576fb60d696e56169ae0bed3","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-511de59c1d076e1fcaa0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1c\n- v14 --refutes--> v0c\n- v16 --refutes--> v18\n- v05 --supports--> v0a\n- v12 --refutes--> v08\n- v1b --supports--> v09\n- v07 --supports--> v06\n- v1e --refutes--> v1b\n- v0e --refutes--> v00\n- v1a --supports--> v1d\n- v03 --refutes--> v0b\n- v14 --refutes--> v0f\n- v11 --refutes--> v04\n- v0c --supports--> v15\n- v1d --refutes--> v01\n- v16 --refutes--> v14\n- v02 --supports--> v12\n- v1b --supports--> v17\n- v12 --refutes--> v10\n- v16 --supports--> v0d\n- v00 --supports--> v02\n- v19 --supports--> v16\n- v17 --supports--> v07\n- v11 --refutes--> v1e\n- v00 --supports--> v13\n- v1c --refutes--> v11\n- v0b --supports--> v1a\n- v05 --refutes--> v0e\n- v1d --refutes--> v19\n- v07 --supports--> v03\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a86885c5e12115bcd88bca6bef36f2f44280d50e3728aaf1fe8bed155481502","schema":"livegraph.boolean-propagation.v1","spec_hash":"511de59c1d076e1fcaa0b28aa9af9dafc7d2e25a5196f1dc74b67c65c1e51f3f","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-289e16a8b12d191e4dfd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1c --refutes--> v1a\n- v1a --refutes--> v1b\n- v07 --refutes--> v11\n- v1d --refutes--> v18\n- v18 --supports--> v10\n- v13 --supports--> v1d\n- v00 --refutes--> v1e\n- v0d --refutes--> v02\n- v0e --supports--> v0b\n- v02 --supports--> v05\n- v12 --refutes--> v01\n- v01 --supports--> v09\n- v10 --refutes--> v08\n- v0e --supports--> v0a\n- v10 --refutes--> v16\n- v04 --refutes--> v12\n- v00 --supports--> v19\n- v15 --supports--> v17\n- v0f --supports--> v0d\n- v02 --refutes--> v0e\n- v08 --supports--> v14\n- v17 --supports--> v07\n- v1a --refutes--> v04\n- v01 --supports--> v0f\n- v13 --refutes--> v03\n- v04 --supports--> v0c\n- v14 --refutes--> v15\n- v11 --supports--> v1c\n- v00 --supports--> v13\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":"69e39cc37c6e6cba2c9d2f7b685f26fd2b672864ee166dccafeb85e8259dadef","schema":"livegraph.boolean-propagation.v1","spec_hash":"289e16a8b12d191e4dfd1aedd43c4f89f205513aa6ff849a685e4f9ab723ad40","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-16098fd317c2b42fc7b3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v12\n- v13 --refutes--> v0e\n- v00 --supports--> v03\n- v0e --supports--> v08\n- v07 --refutes--> v02\n- v0d --refutes--> v18\n- v00 --supports--> v05\n- v0c --refutes--> v1a\n- v1d --refutes--> v1c\n- v18 --supports--> v09\n- v08 --refutes--> v0c\n- v05 --refutes--> v1e\n- v17 --supports--> v07\n- v1b --refutes--> v16\n- v19 --supports--> v13\n- v1d --refutes--> v06\n- v1d --refutes--> v0b\n- v19 --supports--> v10\n- v07 --supports--> v1d\n- v04 --supports--> v0a\n- v1d --supports--> v11\n- v18 --refutes--> v15\n- v16 --refutes--> v0d\n- v09 --refutes--> v01\n- v09 --supports--> v0f\n- v1d --supports--> v04\n- v0f --refutes--> v14\n- v14 --supports--> v19\n- v12 --refutes--> v1b\n- v04 --supports--> v00\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"21b69f4abf646878e64bd703361cce8c0c929adeef5b8eaa0fee8989cdb8809c","schema":"livegraph.boolean-propagation.v1","spec_hash":"16098fd317c2b42fc7b3006d4f37ecc6e28edb6c960d2f496ac215246c824bf7","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-0054cb5f3299793c00f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v10\n- v1c --refutes--> v18\n- v1c --refutes--> v04\n- v00 --refutes--> v0a\n- v1d --supports--> v16\n- v0b --refutes--> v0e\n- v02 --refutes--> v1d\n- v18 --supports--> v19\n- v02 --refutes--> v05\n- v1a --refutes--> v14\n- v16 --refutes--> v08\n- v07 --supports--> v0d\n- v00 --supports--> v0f\n- v03 --supports--> v1b\n- v1a --supports--> v11\n- v15 --refutes--> v02\n- v06 --supports--> v09\n- v0b --refutes--> v06\n- v12 --supports--> v1e\n- v06 --supports--> v1a\n- v01 --supports--> v07\n- v0a --refutes--> v15\n- v0b --supports--> v13\n- v14 --supports--> v03\n- v1b --refutes--> v1c\n- v07 --supports--> v12\n- v1e --refutes--> v0c\n- v0c --refutes--> v00\n- v19 --supports--> v01\n- v1e --supports--> v17\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv08\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ecae05c2ca17f4535375d1ce34eaeaa675bd06d7509deb707877b71b7f7c7bb8","schema":"livegraph.boolean-propagation.v1","spec_hash":"0054cb5f3299793c00f151b453e22e0159aebc59eb5d4d7153ac3d2200219f55","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-25bc796865304384283e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v06\n- v0c --refutes--> v1a\n- v10 --refutes--> v08\n- v0f --supports--> v13\n- v0c --refutes--> v18\n- v13 --refutes--> v11\n- v05 --refutes--> v17\n- v10 --refutes--> v01\n- v17 --refutes--> v02\n- v10 --supports--> v0c\n- v1a --supports--> v1e\n- v1d --supports--> v10\n- v0a --supports--> v1d\n- v12 --refutes--> v1b\n- v11 --supports--> v05\n- v1e --supports--> v19\n- v07 --supports--> v1c\n- v00 --supports--> v0f\n- v04 --supports--> v16\n- v02 --supports--> v04\n- v0b --supports--> v12\n- v0c --supports--> v14\n- v13 --refutes--> v03\n- v0b --supports--> v15\n- v1b --refutes--> v00\n- v0f --refutes--> v0e\n- v17 --refutes--> v09\n- v1c --refutes--> v0a\n- v19 --refutes--> v0b\n- v11 --refutes--> v0d\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":"56bba51929e14daaf1724c65ce0a4eb2b0a8fdd3cd401530afa3ee9bd47cbb7c","schema":"livegraph.boolean-propagation.v1","spec_hash":"25bc796865304384283eefebda4e8aefa4458a0cb372aef7937be2da240d86d7","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-e73c56683c41b4c735a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v08\n- v18 --refutes--> v01\n- v08 --supports--> v17\n- v15 --refutes--> v07\n- v00 --refutes--> v0d\n- v11 --supports--> v05\n- v0f --refutes--> v10\n- v06 --supports--> v0b\n- v1a --supports--> v0f\n- v1c --supports--> v03\n- v1e --supports--> v1a\n- v17 --refutes--> v1e\n- v12 --refutes--> v15\n- v0f --refutes--> v13\n- v19 --refutes--> v0e\n- v03 --supports--> v04\n- v14 --refutes--> v06\n- v0a --refutes--> v1d\n- v18 --refutes--> v11\n- v00 --supports--> v18\n- v05 --supports--> v12\n- v02 --refutes--> v1c\n- v13 --refutes--> v02\n- v0c --refutes--> v19\n- v0a --supports--> v14\n- v1c --supports--> v09\n- v06 --supports--> v0c\n- v19 --refutes--> v00\n- v1c --supports--> v1b\n- v12 --supports--> v16\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":"040ebd86d0a8e12d21f28c6590c5102fe54b2efa2ecb79d81867e3e8edc97d4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"e73c56683c41b4c735a4f214db7d20f0319f078e762288950bd6913440c92195","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-b4794cb679810b815db1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v00\n- v1c --refutes--> v02\n- v1b --refutes--> v0f\n- v19 --supports--> v14\n- v05 --refutes--> v13\n- v1b --supports--> v09\n- v05 --refutes--> v15\n- v00 --refutes--> v1d\n- v06 --refutes--> v07\n- v1e --refutes--> v1c\n- v10 --supports--> v17\n- v06 --supports--> v10\n- v0e --refutes--> v16\n- v1a --supports--> v06\n- v0a --refutes--> v03\n- v10 --refutes--> v18\n- v0b --supports--> v19\n- v1d --supports--> v0a\n- v14 --refutes--> v0e\n- v0e --refutes--> v01\n- v0c --refutes--> v1a\n- v0c --refutes--> v04\n- v01 --refutes--> v1b\n- v02 --supports--> v05\n- v01 --refutes--> v0d\n- v01 --refutes--> v11\n- v0a --supports--> v12\n- v03 --supports--> v0b\n- v17 --supports--> v1e\n- v0a --refutes--> v08\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fafc67eecd4ee67db5f291fd6b559e06784d99b69f509320ec6919f7bd2191e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4794cb679810b815db111d2a69c6cfab2d6c1f896187f33fdffd4a9955e2536","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-cdd499febf49e10bdd43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v13\n- v14 --refutes--> v11\n- v05 --supports--> v17\n- v0c --supports--> v06\n- v01 --supports--> v1d\n- v0d --supports--> v00\n- v0e --supports--> v0f\n- v1e --supports--> v15\n- v1a --refutes--> v14\n- v13 --refutes--> v18\n- v12 --supports--> v08\n- v09 --refutes--> v1e\n- v13 --supports--> v1b\n- v10 --supports--> v04\n- v04 --supports--> v19\n- v00 --supports--> v09\n- v18 --refutes--> v03\n- v18 --supports--> v12\n- v10 --refutes--> v16\n- v14 --refutes--> v1c\n- v1d --supports--> v05\n- v17 --refutes--> v10\n- v1e --supports--> v0c\n- v1c --supports--> v0e\n- v04 --refutes--> v07\n- v0e --refutes--> v0d\n- v04 --refutes--> v1a\n- v06 --supports--> v0b\n- v12 --refutes--> v02\n- v06 --supports--> v0a\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":"d682144f2b789d064023b65f5bf6198ac19bc12e00918981ca4e751e4cdfd4bd","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdd499febf49e10bdd43d7940921ca8cecfcda88888762edf65aaf521f40cf45","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-e68527a0bde1e5cdf0e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v13 --supports--> v06\n- v18 --refutes--> v03\n- v19 --refutes--> v0e\n- v1c --supports--> v02\n- v0b --refutes--> v17\n- v1d --refutes--> v13\n- v1a --supports--> v12\n- v0d --supports--> v1c\n- v01 --supports--> v0a\n- v04 --refutes--> v0d\n- v15 --refutes--> v0f\n- v15 --supports--> v1b\n- v13 --supports--> v1a\n- v0f --refutes--> v18\n- v0b --refutes--> v09\n- v0d --supports--> v00\n- v0a --supports--> v1d\n- v09 --refutes--> v19\n- v11 --supports--> v0b\n- v14 --supports--> v0c\n- v02 --supports--> v10\n- v11 --supports--> v16\n- v11 --supports--> v05\n- v02 --supports--> v14\n- v14 --refutes--> v01\n- v03 --supports--> v07\n- v03 --supports--> v08\n- v0f --supports--> v1e\n- v12 --refutes--> v11\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":"6ed4c0882305fbc6ce5db5c7962334053d752b20e10b0d09fdaa911907702374","schema":"livegraph.boolean-propagation.v1","spec_hash":"e68527a0bde1e5cdf0e5eee4d2c86c24816e02435f4bec0bc91849dd1868d443","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-7cd09c2110b9196153f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v08 --supports--> v15\n- v07 --refutes--> v0d\n- v00 --supports--> v0a\n- v05 --refutes--> v06\n- v03 --refutes--> v07\n- v19 --refutes--> v18\n- v06 --supports--> v0c\n- v06 --supports--> v17\n- v05 --supports--> v12\n- v14 --supports--> v0b\n- v1c --refutes--> v1b\n- v02 --refutes--> v16\n- v09 --refutes--> v0f\n- v09 --refutes--> v0e\n- v1b --refutes--> v19\n- v07 --supports--> v04\n- v1a --refutes--> v03\n- v05 --supports--> v11\n- v0e --supports--> v00\n- v18 --supports--> v08\n- v08 --supports--> v14\n- v13 --supports--> v1e\n- v04 --supports--> v05\n- v0b --supports--> v09\n- v00 --supports--> v1a\n- v1a --refutes--> v1d\n- v02 --refutes--> v13\n- v10 --supports--> v1c\n- v02 --supports--> v01\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3c8ae3cd81de0a23fc797445b144b3b3cee6c47bd6e1bb74a33896384a1fc8cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cd09c2110b9196153f69b928d66aaff1d56a5a4496b98f6b9dcf1e47c8e1620","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-735849f58db2bba6cffe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v14\n- v00 --refutes--> v04\n- v09 --refutes--> v15\n- v18 --refutes--> v0c\n- v18 --supports--> v0d\n- v01 --refutes--> v07\n- v1c --refutes--> v03\n- v16 --refutes--> v1e\n- v0a --supports--> v08\n- v19 --supports--> v0a\n- v1c --refutes--> v11\n- v17 --refutes--> v16\n- v04 --supports--> v19\n- v12 --supports--> v0f\n- v09 --supports--> v18\n- v1a --supports--> v02\n- v1d --refutes--> v10\n- v0d --supports--> v00\n- v0d --supports--> v06\n- v0f --supports--> v09\n- v08 --supports--> v05\n- v1e --refutes--> v1a\n- v16 --refutes--> v0e\n- v18 --supports--> v13\n- v0b --supports--> v1b\n- v08 --refutes--> v1c\n- v0b --supports--> v17\n- v02 --supports--> v01\n- v1d --refutes--> v0b\n- v07 --refutes--> v12\n\n[VERIFIED ROOT]\nv1d = 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":"113b91715996586df0aed6dd5149d4f6c92f23c15563388ae10d99119b24226e","schema":"livegraph.boolean-propagation.v1","spec_hash":"735849f58db2bba6cffed17eef1571c5139645eea698ba50bdcf58b6dfd2dc9a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-877c366da31dfb08ae3a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --supports--> v08\n- v0b --refutes--> v1a\n- v0a --refutes--> v09\n- v19 --supports--> v1c\n- v0a --refutes--> v14\n- v16 --refutes--> v1e\n- v1a --refutes--> v12\n- v14 --supports--> v11\n- v1c --supports--> v03\n- v06 --supports--> v19\n- v04 --refutes--> v0f\n- v03 --supports--> v16\n- v1a --supports--> v0e\n- v00 --refutes--> v02\n- v19 --refutes--> v1d\n- v14 --supports--> v1b\n- v10 --supports--> v18\n- v00 --supports--> v0d\n- v01 --supports--> v10\n- v0c --supports--> v05\n- v11 --refutes--> v06\n- v1a --refutes--> v17\n- v1e --refutes--> v0c\n- v0f --supports--> v00\n- v00 --supports--> v0b\n- v13 --refutes--> v04\n- v11 --refutes--> v07\n- v05 --refutes--> v01\n- v11 --supports--> v15\n- v12 --refutes--> v0a\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv18\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d99cf9675d6b13ec50c2b28542d0de03c501616a500fa9f2d8594bc081b3988b","schema":"livegraph.boolean-propagation.v1","spec_hash":"877c366da31dfb08ae3a8da9b3f76190cb6d9b2c74c5247ed99b35cbad53182a","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-745d7ad07ce6355338fe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v0f\n- v14 --supports--> v01\n- v06 --refutes--> v0d\n- v00 --refutes--> v0a\n- v15 --refutes--> v17\n- v1a --supports--> v0e\n- v15 --supports--> v19\n- v0e --supports--> v18\n- v03 --refutes--> v15\n- v0c --refutes--> v04\n- v07 --supports--> v1b\n- v14 --refutes--> v16\n- v1c --refutes--> v02\n- v11 --refutes--> v0c\n- v10 --refutes--> v07\n- v13 --refutes--> v00\n- v0c --supports--> v05\n- v09 --supports--> v1a\n- v09 --refutes--> v12\n- v07 --refutes--> v0b\n- v15 --refutes--> v1d\n- v1e --supports--> v11\n- v08 --supports--> v03\n- v02 --refutes--> v08\n- v00 --refutes--> v14\n- v06 --supports--> v09\n- v07 --supports--> v1c\n- v01 --supports--> v1e\n- v18 --refutes--> v10\n- v1d --refutes--> v13\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":"d0a358f93ada60ba83fdd9305f49194643c3bae9f11c0ed25e352a8ab830a93c","schema":"livegraph.boolean-propagation.v1","spec_hash":"745d7ad07ce6355338fe3579c9d572e3cb4e595c8a0b476a97aa14b5c7a30c67","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-25077ce7dacba4c93e3c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v1b --refutes--> v1c\n- v10 --refutes--> v15\n- v0a --supports--> v0c\n- v19 --supports--> v1e\n- v02 --refutes--> v17\n- v11 --supports--> v02\n- v0e --supports--> v0f\n- v1c --refutes--> v0d\n- v01 --refutes--> v07\n- v17 --supports--> v19\n- v1d --refutes--> v14\n- v01 --supports--> v08\n- v00 --refutes--> v18\n- v01 --refutes--> v04\n- v17 --supports--> v1a\n- v17 --refutes--> v13\n- v15 --supports--> v06\n- v19 --supports--> v16\n- v00 --supports--> v05\n- v19 --refutes--> v01\n- v04 --refutes--> v09\n- v06 --supports--> v0e\n- v12 --refutes--> v1d\n- v1c --refutes--> v03\n- v03 --refutes--> v0a\n- v18 --supports--> v12\n- v1c --refutes--> v0b\n- v0f --supports--> v1b\n- v04 --supports--> v10\n\n[VERIFIED ROOT]\nv11 = 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":"2cacd3c902da6192c3353559d0f2924db18e0204987e8868a48ff2ea3ba81ff4","schema":"livegraph.boolean-propagation.v1","spec_hash":"25077ce7dacba4c93e3c131d3bf5e7112e047ae53ba4b1d9334e4d74d5996fac","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-75fc6312f8ed1acef03e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --supports--> v07\n- v06 --refutes--> v13\n- v04 --supports--> v0a\n- v0e --refutes--> v01\n- v05 --refutes--> v11\n- v1e --refutes--> v0c\n- v08 --refutes--> v05\n- v16 --supports--> v1e\n- v06 --refutes--> v10\n- v05 --refutes--> v1b\n- v1c --refutes--> v19\n- v18 --refutes--> v02\n- v0f --refutes--> v00\n- v12 --refutes--> v1c\n- v1a --supports--> v16\n- v02 --refutes--> v09\n- v17 --supports--> v06\n- v11 --refutes--> v03\n- v03 --refutes--> v17\n- v16 --supports--> v14\n- v07 --refutes--> v0d\n- v05 --refutes--> v1d\n- v0f --supports--> v0b\n- v1c --refutes--> v0e\n- v09 --refutes--> v04\n- v1a --supports--> v15\n- v0e --supports--> v08\n- v1e --refutes--> v12\n- v0d --supports--> v1a\n- v06 --refutes--> v0f\n\n[VERIFIED ROOT]\nv18 = 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":"e6829e5421dc3c006a0016b1859684b44afe947e2d2ba9d022afec857f2eca2b","schema":"livegraph.boolean-propagation.v1","spec_hash":"75fc6312f8ed1acef03e9bf4cff7ca774f9dd19dda552250c90f7891ae7ce40e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-b50a8cc5350b331b396b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v06 --refutes--> v1c\n- v10 --supports--> v06\n- v04 --supports--> v03\n- v02 --supports--> v1b\n- v0c --supports--> v11\n- v02 --refutes--> v17\n- v19 --refutes--> v1e\n- v02 --supports--> v1a\n- v09 --supports--> v1d\n- v1e --refutes--> v0e\n- v00 --supports--> v0f\n- v0a --supports--> v08\n- v19 --refutes--> v0b\n- v1e --refutes--> v0a\n- v05 --refutes--> v19\n- v1a --refutes--> v05\n- v0c --supports--> v12\n- v00 --refutes--> v04\n- v08 --refutes--> v14\n- v19 --supports--> v18\n- v11 --refutes--> v0d\n- v1c --refutes--> v07\n- v07 --supports--> v09\n- v16 --supports--> v10\n- v07 --refutes--> v15\n- v13 --refutes--> v16\n- v01 --supports--> v02\n- v04 --refutes--> v0c\n- v1d --supports--> v01\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5833656be7105d6dc390179e3909f35c80e651804d6b4c6cd078dd4037c1e0c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"b50a8cc5350b331b396b3832bc558bb4b7aa4ba858250d7184ddbc3a23a37e69","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-6f7095ea7f16c74f039c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v04\n- v13 --refutes--> v01\n- v1b --supports--> v1e\n- v00 --refutes--> v19\n- v1a --supports--> v10\n- v09 --refutes--> v0a\n- v0a --supports--> v16\n- v03 --refutes--> v00\n- v19 --refutes--> v0e\n- v08 --refutes--> v03\n- v18 --refutes--> v12\n- v17 --supports--> v0b\n- v09 --supports--> v1d\n- v0c --supports--> v05\n- v1b --supports--> v13\n- v01 --refutes--> v07\n- v15 --supports--> v18\n- v12 --refutes--> v1b\n- v13 --supports--> v14\n- v01 --supports--> v1c\n- v16 --supports--> v11\n- v11 --refutes--> v1a\n- v02 --supports--> v09\n- v00 --supports--> v06\n- v1c --supports--> v0c\n- v1a --supports--> v15\n- v17 --supports--> v02\n- v12 --refutes--> v0d\n- v0c --refutes--> v08\n- v0f --supports--> v17\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":"2f4c974b6e69ba32dd4592468edf6042691af555e207e2ae943055d23ca3466e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6f7095ea7f16c74f039c1aea4d0ee26da1bcbc387d2085ae4b3f044378ee3e98","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-9c5c699490090f7d1c9c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v05 --supports--> v0a\n- v08 --supports--> v09\n- v0b --supports--> v0f\n- v02 --refutes--> v07\n- v09 --refutes--> v0d\n- v1e --refutes--> v15\n- v0f --supports--> v1d\n- v05 --supports--> v13\n- v0d --supports--> v1a\n- v04 --supports--> v1c\n- v0a --refutes--> v08\n- v0f --supports--> v02\n- v15 --refutes--> v05\n- v06 --supports--> v0c\n- v01 --refutes--> v19\n- v0a --supports--> v11\n- v14 --refutes--> v18\n- v01 --supports--> v16\n- v14 --refutes--> v1e\n- v17 --refutes--> v01\n- v0c --supports--> v10\n- v0c --supports--> v12\n- v03 --refutes--> v06\n- v05 --supports--> v00\n- v07 --refutes--> v14\n- v0d --refutes--> v17\n- v16 --refutes--> v03\n- v1c --refutes--> v0b\n- v08 --supports--> v0e\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5575da5a40280927796787a96a0d71e536e16c07df41b3a8728cc2590b31ecae","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c5c699490090f7d1c9cbce0743728b139ccbc60cf61382c43b0589e5cc8ffdf","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-f24c2d4759604869681c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v0b --refutes--> v01\n- v0c --refutes--> v17\n- v0e --refutes--> v14\n- v0e --supports--> v03\n- v15 --refutes--> v1c\n- v13 --supports--> v18\n- v1a --supports--> v16\n- v11 --supports--> v00\n- v0a --refutes--> v0b\n- v0f --supports--> v0e\n- v08 --refutes--> v11\n- v07 --refutes--> v1d\n- v1c --supports--> v0f\n- v14 --supports--> v06\n- v00 --refutes--> v10\n- v1d --supports--> v15\n- v08 --supports--> v09\n- v04 --supports--> v13\n- v18 --refutes--> v05\n- v0b --supports--> v08\n- v00 --supports--> v0c\n- v0f --supports--> v1b\n- v04 --supports--> v12\n- v17 --supports--> v02\n- v17 --supports--> v07\n- v06 --refutes--> v04\n- v07 --supports--> v0d\n- v18 --refutes--> v19\n- v1e --refutes--> v1a\n\n[VERIFIED ROOT]\nv0a = 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":"024b77d2ee2e44b10d6253e3fd7a27ca934fed56810ffa5137501d58fc953f2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f24c2d4759604869681cb6586ce81100f36e5b18940746fb88688c8b8e0f4a53","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-3bab775c41ec09504f6b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --refutes--> v1a\n- v01 --supports--> v12\n- v18 --refutes--> v1c\n- v17 --refutes--> v06\n- v14 --supports--> v16\n- v04 --refutes--> v14\n- v13 --refutes--> v0a\n- v09 --refutes--> v13\n- v0d --refutes--> v1b\n- v1a --supports--> v0b\n- v04 --supports--> v10\n- v1e --refutes--> v02\n- v12 --supports--> v04\n- v05 --supports--> v00\n- v01 --refutes--> v11\n- v15 --supports--> v0d\n- v04 --refutes--> v07\n- v1d --refutes--> v18\n- v00 --supports--> v0e\n- v14 --refutes--> v09\n- v00 --refutes--> v17\n- v06 --supports--> v01\n- v13 --supports--> v1e\n- v1e --refutes--> v0f\n- v1c --refutes--> v05\n- v0c --refutes--> v15\n- v0d --refutes--> v08\n- v0f --supports--> v0c\n- v1a --refutes--> v19\n- v1a --refutes--> v03\n\n[VERIFIED ROOT]\nv1d = 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":"cbd194fac9e37455ab09f1ba22f7f52a20f062e52f0407a6c5b280c8d08cd161","schema":"livegraph.boolean-propagation.v1","spec_hash":"3bab775c41ec09504f6bf67348d0f7c5b47b341d623364a226cfa8237d75b555","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-36deaf2248f9d278be1e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v02 --refutes--> v1d\n- v02 --refutes--> v1a\n- v17 --refutes--> v12\n- v01 --supports--> v16\n- v15 --supports--> v19\n- v0e --supports--> v07\n- v05 --refutes--> v04\n- v09 --supports--> v17\n- v19 --supports--> v0a\n- v12 --refutes--> v1b\n- v18 --supports--> v02\n- v0d --refutes--> v01\n- v16 --supports--> v08\n- v08 --refutes--> v13\n- v09 --supports--> v0c\n- v0f --refutes--> v00\n- v1c --supports--> v0d\n- v03 --refutes--> v09\n- v14 --supports--> v15\n- v1a --refutes--> v1c\n- v12 --refutes--> v1e\n- v04 --supports--> v18\n- v07 --supports--> v05\n- v0d --supports--> v11\n- v01 --supports--> v06\n- v19 --refutes--> v0e\n- v18 --supports--> v0b\n- v0f --supports--> v03\n- v1a --supports--> v10\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad691494003cdbef9a172b4cde59fd20a2f93221cef30156e2bf7c01c22841fd","schema":"livegraph.boolean-propagation.v1","spec_hash":"36deaf2248f9d278be1ed93acad35f4bc80562018884f8f13131ce3f6e988f3f","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-de9ad3458fce366ce56a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v10\n- v06 --refutes--> v0f\n- v17 --refutes--> v04\n- v04 --supports--> v0a\n- v18 --refutes--> v08\n- v19 --supports--> v11\n- v1c --refutes--> v0c\n- v0f --supports--> v1e\n- v13 --refutes--> v03\n- v1d --supports--> v13\n- v13 --supports--> v0b\n- v1c --supports--> v15\n- v0f --refutes--> v12\n- v0f --supports--> v02\n- v1a --refutes--> v1d\n- v16 --supports--> v1a\n- v0e --supports--> v1c\n- v01 --supports--> v05\n- v01 --refutes--> v19\n- v0f --refutes--> v07\n- v04 --refutes--> v0e\n- v0f --supports--> v0d\n- v15 --supports--> v16\n- v08 --refutes--> v06\n- v12 --supports--> v09\n- v11 --supports--> v17\n- v00 --supports--> v1b\n- v12 --refutes--> v14\n- v1b --refutes--> v01\n- v14 --supports--> v00\n\n[VERIFIED ROOT]\nv18 = 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":"524d8e73f7a14b30c4cacb2f97569392d33ca2b436c1dffad4c93906ccbdea64","schema":"livegraph.boolean-propagation.v1","spec_hash":"de9ad3458fce366ce56a678c142d8db859303d093dce1a6e187e0a67db141336","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-21e769c55292b26cb5bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0d\n- v1a --supports--> v01\n- v0f --supports--> v18\n- v14 --supports--> v13\n- v09 --supports--> v0a\n- v16 --supports--> v19\n- v02 --supports--> v07\n- v1d --refutes--> v12\n- v0b --supports--> v1a\n- v10 --supports--> v0b\n- v1d --refutes--> v03\n- v1a --refutes--> v06\n- v13 --supports--> v04\n- v07 --refutes--> v11\n- v1e --refutes--> v16\n- v0a --refutes--> v1b\n- v00 --supports--> v1c\n- v0f --refutes--> v09\n- v1e --supports--> v17\n- v11 --supports--> v0f\n- v0a --refutes--> v1e\n- v12 --refutes--> v14\n- v19 --supports--> v1d\n- v09 --refutes--> v08\n- v0a --supports--> v0c\n- v19 --refutes--> v15\n- v06 --supports--> v02\n- v0e --supports--> v05\n- v0e --supports--> v00\n- v0d --supports--> v0e\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dd000ceb39ac685ed9d58ce1f19a723edd3ec568e53f3a05ef98c6caac80c173","schema":"livegraph.boolean-propagation.v1","spec_hash":"21e769c55292b26cb5bb132a58dbb26fd53310c2a33e5571912021e85c003e15","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-dd30a3ad862e67153b96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0b\n- v11 --supports--> v0c\n- v16 --supports--> v04\n- v19 --supports--> v1b\n- v10 --supports--> v18\n- v03 --supports--> v01\n- v06 --supports--> v15\n- v1c --refutes--> v13\n- v01 --refutes--> v11\n- v00 --supports--> v07\n- v17 --supports--> v03\n- v12 --refutes--> v17\n- v1c --refutes--> v1e\n- v0e --supports--> v14\n- v0e --refutes--> v1c\n- v02 --supports--> v06\n- v0c --supports--> v0d\n- v0e --supports--> v09\n- v16 --refutes--> v12\n- v18 --refutes--> v02\n- v0f --refutes--> v19\n- v1c --refutes--> v16\n- v19 --refutes--> v1a\n- v03 --supports--> v05\n- v07 --refutes--> v08\n- v0d --refutes--> v0f\n- v08 --supports--> v1d\n- v1d --refutes--> v0e\n- v19 --refutes--> v0a\n- v06 --refutes--> v00\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6d3f0f89f13783528dd608335199e798fe62ee5eee99ebe2b4dd25056c685ce0","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd30a3ad862e67153b96c11d57436902485dc59417d304936354a19d3a80adff","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-c364b775e822aae79663","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v0b\n- v0f --supports--> v05\n- v1c --supports--> v12\n- v06 --refutes--> v19\n- v1a --refutes--> v1b\n- v0c --supports--> v14\n- v0e --refutes--> v15\n- v02 --refutes--> v0a\n- v0d --supports--> v0e\n- v12 --supports--> v0c\n- v1a --refutes--> v03\n- v17 --refutes--> v07\n- v02 --refutes--> v01\n- v1a --refutes--> v10\n- v0a --refutes--> v1e\n- v11 --refutes--> v0f\n- v1a --refutes--> v18\n- v0b --supports--> v11\n- v11 --supports--> v00\n- v06 --supports--> v13\n- v08 --refutes--> v1d\n- v1e --supports--> v06\n- v06 --supports--> v09\n- v07 --supports--> v02\n- v10 --supports--> v17\n- v12 --refutes--> v04\n- v0d --refutes--> v16\n- v05 --supports--> v1c\n- v1d --supports--> v1a\n- v0c --supports--> v0d\n\n[VERIFIED ROOT]\nv08 = 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":"1f4dc5a7cd7708ce9357f64189b7380bd09d6e244900c7f00426f1751b43f600","schema":"livegraph.boolean-propagation.v1","spec_hash":"c364b775e822aae796632dffbe60b0eef4d202e3440644bb90b11784b0ede232","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-a87fdf779a6a7d886784","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v03 --supports--> v01\n- v08 --refutes--> v1a\n- v1b --refutes--> v0d\n- v18 --refutes--> v16\n- v07 --supports--> v15\n- v15 --refutes--> v02\n- v19 --supports--> v17\n- v0a --refutes--> v18\n- v0a --refutes--> v04\n- v0c --supports--> v0e\n- v16 --refutes--> v0f\n- v12 --supports--> v07\n- v17 --supports--> v11\n- v16 --supports--> v1c\n- v15 --supports--> v06\n- v12 --refutes--> v1d\n- v00 --refutes--> v12\n- v17 --refutes--> v13\n- v1e --supports--> v05\n- v1b --supports--> v1e\n- v10 --refutes--> v09\n- v02 --refutes--> v10\n- v0c --supports--> v00\n- v09 --refutes--> v14\n- v16 --supports--> v03\n- v05 --supports--> v19\n- v14 --supports--> v1b\n- v01 --supports--> v0c\n- v17 --supports--> v0b\n\n[VERIFIED ROOT]\nv08 = 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":"f7652e6212d1dc29e39821ff25e16d6aeff7e1ade52947738e9a69f8b3c522bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"a87fdf779a6a7d886784050a89a60aa28445efe3122d85a6eb11804676a893bc","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-22768dc66cf6fe3fc794","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v13\n- v0b --supports--> v1e\n- v0e --supports--> v11\n- v07 --refutes--> v05\n- v17 --supports--> v10\n- v08 --supports--> v1b\n- v06 --refutes--> v0e\n- v12 --supports--> v0a\n- v16 --refutes--> v1a\n- v1c --refutes--> v01\n- v1c --supports--> v16\n- v15 --supports--> v0d\n- v0f --supports--> v15\n- v02 --refutes--> v08\n- v12 --refutes--> v1d\n- v18 --supports--> v02\n- v1b --supports--> v0b\n- v05 --refutes--> v09\n- v0b --refutes--> v06\n- v1d --supports--> v19\n- v04 --refutes--> v14\n- v10 --refutes--> v1c\n- v17 --refutes--> v03\n- v0b --refutes--> v0c\n- v0e --refutes--> v07\n- v16 --refutes--> v0f\n- v0d --supports--> v00\n- v14 --supports--> v17\n- v1d --supports--> v04\n- v05 --refutes--> v12\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":"5b0703438a53519046144cd68f37ee6e84433d02298965c7dd70746f324227c2","schema":"livegraph.boolean-propagation.v1","spec_hash":"22768dc66cf6fe3fc794286d7d25114afbcbb48c6e49df29156212ffcd1bc090","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-0654c68e4b021cd4d37a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --supports--> v02\n- v12 --refutes--> v04\n- v04 --supports--> v13\n- v09 --supports--> v0c\n- v17 --supports--> v1a\n- v1a --supports--> v12\n- v18 --supports--> v01\n- v11 --refutes--> v0d\n- v14 --refutes--> v0f\n- v07 --supports--> v0e\n- v18 --supports--> v0a\n- v04 --refutes--> v11\n- v07 --supports--> v15\n- v06 --supports--> v10\n- v03 --refutes--> v1c\n- v0b --supports--> v07\n- v1d --refutes--> v08\n- v0d --supports--> v1d\n- v1b --supports--> v00\n- v08 --supports--> v06\n- v14 --supports--> v17\n- v1e --refutes--> v1b\n- v06 --supports--> v1e\n- v1d --supports--> v05\n- v02 --refutes--> v14\n- v18 --supports--> v16\n- v18 --supports--> v19\n- v03 --refutes--> v09\n- v00 --refutes--> v18\n- v16 --refutes--> v0b\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"905a32447724469b721a53188157241ef7dd5fd4e069a31522f67da7e973e699","schema":"livegraph.boolean-propagation.v1","spec_hash":"0654c68e4b021cd4d37ac6bf354e793933e42707cb7700558bbf0e8739017e02","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-dd61827acba18e51dfc5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0c\n- v1c --supports--> v05\n- v08 --supports--> v15\n- v09 --supports--> v1d\n- v09 --refutes--> v02\n- v09 --refutes--> v0b\n- v12 --supports--> v1e\n- v07 --supports--> v10\n- v0c --supports--> v01\n- v00 --refutes--> v0a\n- v0e --supports--> v14\n- v1a --refutes--> v04\n- v17 --supports--> v08\n- v0a --supports--> v0e\n- v1d --supports--> v1c\n- v14 --supports--> v0f\n- v0a --supports--> v16\n- v05 --refutes--> v0d\n- v0d --refutes--> v03\n- v0c --supports--> v17\n- v13 --supports--> v00\n- v11 --supports--> v13\n- v0c --refutes--> v18\n- v17 --supports--> v06\n- v07 --supports--> v09\n- v04 --supports--> v07\n- v1d --refutes--> v1b\n- v1e --refutes--> v1a\n- v0c --supports--> v19\n- v08 --refutes--> v12\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":"3d74fc162f7cdba1bfc6807611bd7046a37bb8719f23be811bb7d6c727ca5390","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd61827acba18e51dfc5d70b1df30ef7aa44aa943d254a35a208ea3600777d8a","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-c7c995181a768ef7418f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v0a --refutes--> v03\n- v15 --refutes--> v12\n- v14 --refutes--> v15\n- v0e --refutes--> v14\n- v01 --supports--> v04\n- v14 --refutes--> v10\n- v07 --supports--> v0f\n- v1c --refutes--> v1d\n- v1a --refutes--> v17\n- v0a --refutes--> v0d\n- v1b --supports--> v13\n- v11 --supports--> v07\n- v0f --refutes--> v02\n- v0d --supports--> v0c\n- v00 --refutes--> v0b\n- v08 --refutes--> v11\n- v01 --refutes--> v16\n- v0d --supports--> v18\n- v19 --supports--> v1c\n- v00 --refutes--> v01\n- v16 --supports--> v1e\n- v1e --refutes--> v1a\n- v07 --supports--> v05\n- v1d --supports--> v1b\n- v1a --refutes--> v08\n- v13 --supports--> v0e\n- v18 --supports--> v00\n- v1d --supports--> v09\n- v1a --refutes--> v06\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a6a3d6edeb0c340c64570ec18d96dc8dbfb7b7b9ee42e0ee9d1211c958f23164","schema":"livegraph.boolean-propagation.v1","spec_hash":"c7c995181a768ef7418fb2deec56126a3b088d078595a245f806fc56846123a6","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-5022c592f036a9d1b606","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1b --supports--> v0d\n- v0c --refutes--> v06\n- v0a --refutes--> v11\n- v1d --refutes--> v0c\n- v00 --supports--> v05\n- v05 --refutes--> v08\n- v0e --refutes--> v0b\n- v0a --supports--> v0e\n- v01 --refutes--> v07\n- v03 --supports--> v1c\n- v1e --refutes--> v12\n- v0d --supports--> v0f\n- v06 --supports--> v1a\n- v14 --supports--> v02\n- v1c --refutes--> v10\n- v06 --supports--> v13\n- v1c --refutes--> v19\n- v14 --supports--> v1b\n- v15 --refutes--> v18\n- v03 --supports--> v17\n- v19 --supports--> v09\n- v1a --refutes--> v00\n- v12 --supports--> v1d\n- v19 --supports--> v01\n- v0f --supports--> v03\n- v0e --refutes--> v04\n- v15 --supports--> v0a\n- v0b --supports--> v1e\n- v06 --refutes--> v16\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3824be11dcb051a0170458e1974e2f01b5bc803f2868ffe8d5e4ba3b2fc3d448","schema":"livegraph.boolean-propagation.v1","spec_hash":"5022c592f036a9d1b606b98e92c5c0d92e0570f747c2f3af0a5d6eef176d86a0","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-da732b7f1a4868d8e554","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v17\n- v14 --refutes--> v0b\n- v04 --supports--> v16\n- v0b --refutes--> v13\n- v12 --refutes--> v00\n- v02 --refutes--> v04\n- v11 --supports--> v10\n- v0d --refutes--> v0e\n- v1b --supports--> v14\n- v15 --refutes--> v0a\n- v0c --refutes--> v02\n- v03 --refutes--> v11\n- v0e --refutes--> v1e\n- v16 --refutes--> v0d\n- v06 --supports--> v1d\n- v1c --supports--> v05\n- v00 --supports--> v06\n- v11 --refutes--> v12\n- v05 --refutes--> v03\n- v08 --supports--> v1a\n- v1d --refutes--> v1b\n- v08 --supports--> v15\n- v15 --supports--> v18\n- v0e --supports--> v0f\n- v03 --refutes--> v01\n- v1d --refutes--> v19\n- v00 --refutes--> v07\n- v15 --refutes--> v0c\n- v1e --supports--> v1c\n- v08 --supports--> v09\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"29204ed4e290d719a3ccb287ea352b0fa285ffff7bd32fafbaeb3cd965a11916","schema":"livegraph.boolean-propagation.v1","spec_hash":"da732b7f1a4868d8e554be17655dfaa8c6a7b4b04364083986a36d255ce2d893","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-1280eb3eee827b8d2d79","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v05\n- v15 --supports--> v01\n- v1d --supports--> v15\n- v19 --supports--> v0a\n- v1c --refutes--> v0f\n- v0b --supports--> v02\n- v12 --refutes--> v1b\n- v11 --refutes--> v1d\n- v0b --refutes--> v16\n- v04 --refutes--> v09\n- v02 --refutes--> v17\n- v03 --refutes--> v0c\n- v1e --supports--> v18\n- v17 --supports--> v1e\n- v15 --supports--> v08\n- v0c --refutes--> v19\n- v10 --refutes--> v12\n- v09 --supports--> v03\n- v12 --supports--> v0b\n- v0e --supports--> v04\n- v01 --refutes--> v13\n- v0f --refutes--> v14\n- v18 --supports--> v07\n- v15 --refutes--> v0d\n- v19 --supports--> v11\n- v01 --supports--> v10\n- v09 --supports--> v1a\n- v07 --refutes--> v1c\n- v09 --refutes--> v06\n- v17 --supports--> v00\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5a74f0b4348acaa6441eebd090db46d79f77503ad62e2167e065a1481ccb3ab","schema":"livegraph.boolean-propagation.v1","spec_hash":"1280eb3eee827b8d2d79dcf268895768f6fbcc4b667bffe5099f021676cefb16","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-8c03b3fdce9e45696fde","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v16 --refutes--> v12\n- v1a --refutes--> v18\n- v0d --refutes--> v0f\n- v06 --refutes--> v0a\n- v17 --supports--> v0d\n- v04 --supports--> v09\n- v13 --refutes--> v11\n- v0a --refutes--> v1d\n- v0c --refutes--> v16\n- v00 --supports--> v15\n- v0b --refutes--> v17\n- v0e --refutes--> v1e\n- v09 --refutes--> v01\n- v00 --refutes--> v13\n- v09 --refutes--> v1a\n- v1d --refutes--> v04\n- v08 --refutes--> v14\n- v1c --supports--> v06\n- v14 --refutes--> v03\n- v0a --refutes--> v19\n- v17 --supports--> v10\n- v0e --supports--> v0c\n- v12 --refutes--> v08\n- v07 --supports--> v0e\n- v0a --supports--> v05\n- v11 --refutes--> v07\n- v17 --supports--> v1b\n- v1a --refutes--> v00\n- v14 --supports--> v0b\n\n[VERIFIED ROOT]\nv1c = 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":"a644b9ad6bcc0f7dc58c447415d91e2ab2d8fe59e600e610d81d8651dcd4bfc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c03b3fdce9e45696fdec07d21ece44a0112fbc0c386dbaa7a450b092c322adf","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-b5be341d3166d0a909a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v16 --supports--> v14\n- v1c --refutes--> v17\n- v0f --refutes--> v09\n- v0b --supports--> v16\n- v16 --refutes--> v18\n- v19 --refutes--> v0d\n- v15 --refutes--> v0a\n- v11 --supports--> v08\n- v1b --supports--> v0f\n- v06 --supports--> v11\n- v19 --refutes--> v12\n- v0c --supports--> v1d\n- v05 --supports--> v0b\n- v04 --refutes--> v00\n- v01 --supports--> v13\n- v03 --refutes--> v0c\n- v15 --refutes--> v10\n- v0b --supports--> v1a\n- v17 --supports--> v04\n- v01 --supports--> v1c\n- v0e --supports--> v06\n- v0f --supports--> v19\n- v0b --supports--> v02\n- v00 --supports--> v1e\n- v1d --refutes--> v01\n- v16 --supports--> v0e\n- v0a --supports--> v03\n- v08 --refutes--> v1b\n- v0e --supports--> v07\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5857e5525fd9a5748a13ca2e4fb4057697cdaf3a0b4d9093a28ce43a88e4bbe7","schema":"livegraph.boolean-propagation.v1","spec_hash":"b5be341d3166d0a909a762f3276622ac115b0154f314ae85abd03a5a9b41d21a","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-d7b0f9c023adbe609305","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v14 --refutes--> v09\n- v14 --refutes--> v12\n- v0a --supports--> v10\n- v15 --supports--> v0b\n- v01 --refutes--> v19\n- v16 --supports--> v1e\n- v15 --refutes--> v0c\n- v0f --supports--> v05\n- v13 --refutes--> v1a\n- v0a --refutes--> v11\n- v19 --supports--> v02\n- v13 --refutes--> v06\n- v0d --supports--> v08\n- v08 --refutes--> v16\n- v05 --supports--> v18\n- v0a --supports--> v00\n- v07 --refutes--> v0a\n- v02 --refutes--> v1d\n- v0a --supports--> v17\n- v0d --supports--> v03\n- v1d --refutes--> v1c\n- v1e --refutes--> v01\n- v06 --supports--> v0f\n- v07 --supports--> v0e\n- v0c --refutes--> v07\n- v18 --supports--> v15\n- v19 --refutes--> v04\n- v11 --refutes--> v0d\n- v1c --supports--> v14\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":"8832c987e89fc078efe96caaa0965fc58b277c9bfa96327ca305366319507d3a","schema":"livegraph.boolean-propagation.v1","spec_hash":"d7b0f9c023adbe6093054f67a144311a559c5e728087fde2f890fd0678c322cf","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-cf06d63b5b9ffa07b064","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v0f --refutes--> v1b\n- v0d --supports--> v0b\n- v01 --refutes--> v19\n- v17 --supports--> v05\n- v1b --supports--> v10\n- v19 --refutes--> v1e\n- v16 --supports--> v11\n- v18 --refutes--> v0d\n- v19 --refutes--> v13\n- v1a --refutes--> v18\n- v0e --supports--> v0c\n- v1d --refutes--> v16\n- v05 --supports--> v07\n- v0a --supports--> v08\n- v15 --supports--> v01\n- v01 --supports--> v02\n- v10 --refutes--> v12\n- v00 --supports--> v1c\n- v0a --supports--> v17\n- v01 --supports--> v04\n- v15 --supports--> v14\n- v10 --supports--> v1d\n- v16 --supports--> v0a\n- v1a --supports--> v09\n- v19 --refutes--> v06\n- v1c --supports--> v0e\n- v06 --supports--> v1a\n- v07 --supports--> v15\n- v0b --refutes--> v00\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":"524ae9df4ff960bae280bb875a737b9c02ee7858ea398de6e423be3a82d56938","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf06d63b5b9ffa07b064b5046ec87655eb7b2ae8edd3abe518485583b89b03f1","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-6c7b3fb1cbd737f5af96","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v19\n- v0d --refutes--> v00\n- v11 --refutes--> v1c\n- v07 --supports--> v03\n- v1b --refutes--> v0b\n- v02 --supports--> v09\n- v0c --supports--> v0a\n- v18 --refutes--> v15\n- v18 --refutes--> v05\n- v1d --supports--> v16\n- v1e --supports--> v18\n- v0c --refutes--> v07\n- v1c --supports--> v12\n- v0e --refutes--> v04\n- v17 --supports--> v1a\n- v0b --supports--> v08\n- v0c --supports--> v14\n- v08 --refutes--> v01\n- v16 --refutes--> v0c\n- v06 --refutes--> v1b\n- v1c --supports--> v02\n- v02 --supports--> v0d\n- v19 --refutes--> v1d\n- v0f --refutes--> v11\n- v08 --refutes--> v13\n- v0e --supports--> v0f\n- v00 --refutes--> v17\n- v08 --refutes--> v1e\n- v17 --refutes--> v06\n- v0f --refutes--> v10\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":"bf1fb0ada7f362e765da5f6f614cc6ddd82ac7770c5efc7040d00b086ab9203f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c7b3fb1cbd737f5af96b9dfa64389849035ee0a42104cfeeea06908b80aca43","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-a84621c83e3baca0c0bc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1d\n- v05 --supports--> v09\n- v10 --supports--> v0b\n- v0f --supports--> v11\n- v17 --refutes--> v0a\n- v06 --supports--> v00\n- v0e --refutes--> v02\n- v0e --supports--> v17\n- v13 --supports--> v0d\n- v01 --supports--> v1a\n- v0e --refutes--> v03\n- v0b --supports--> v07\n- v0b --refutes--> v14\n- v01 --refutes--> v0c\n- v1e --supports--> v06\n- v09 --refutes--> v0f\n- v11 --refutes--> v15\n- v19 --supports--> v10\n- v0c --supports--> v18\n- v10 --refutes--> v04\n- v15 --supports--> v19\n- v08 --refutes--> v13\n- v0c --refutes--> v1b\n- v00 --refutes--> v08\n- v08 --refutes--> v1c\n- v0a --supports--> v1e\n- v14 --supports--> v01\n- v0c --refutes--> v12\n- v16 --supports--> v05\n- v0c --refutes--> v0e\n\n[VERIFIED ROOT]\nv16 = 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":"8c8e881c7deea2b38331902a5070d67ca9bdfd88815d991f02f8a8478b9584e7","schema":"livegraph.boolean-propagation.v1","spec_hash":"a84621c83e3baca0c0bc43da42f1f08d7c79f9133368f76a1088a5b3c0b70c16","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-1c2a31781009db26d040","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v17\n- v12 --refutes--> v1d\n- v1e --refutes--> v14\n- v12 --supports--> v09\n- v0a --supports--> v15\n- v0e --supports--> v07\n- v08 --supports--> v03\n- v1b --supports--> v1c\n- v08 --supports--> v11\n- v0f --refutes--> v01\n- v08 --refutes--> v1a\n- v1b --supports--> v12\n- v19 --refutes--> v02\n- v05 --refutes--> v13\n- v06 --refutes--> v0f\n- v07 --refutes--> v10\n- v09 --supports--> v06\n- v1e --supports--> v19\n- v0c --refutes--> v04\n- v02 --supports--> v0e\n- v19 --supports--> v16\n- v05 --supports--> v08\n- v15 --supports--> v1e\n- v10 --supports--> v0c\n- v18 --refutes--> v1b\n- v07 --supports--> v00\n- v05 --supports--> v0d\n- v03 --supports--> v0a\n- v01 --refutes--> v05\n- v0b --refutes--> v18\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"928f03ad7dba67457388fa1e4f0bc84609124df47badfe8e6224ae1074756c04","schema":"livegraph.boolean-propagation.v1","spec_hash":"1c2a31781009db26d0409463f7b20888448ee0d8199619d2f5945af72ddc29d2","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-9f42a24afd72860f7565","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v18\n- v02 --refutes--> v11\n- v0c --supports--> v15\n- v1e --refutes--> v12\n- v0b --refutes--> v0c\n- v0c --supports--> v07\n- v1b --refutes--> v13\n- v15 --refutes--> v1d\n- v1d --refutes--> v01\n- v0f --refutes--> v0e\n- v01 --refutes--> v08\n- v01 --refutes--> v06\n- v13 --refutes--> v03\n- v06 --supports--> v02\n- v18 --supports--> v1a\n- v0a --supports--> v1e\n- v1e --refutes--> v10\n- v1d --refutes--> v1c\n- v18 --refutes--> v05\n- v04 --refutes--> v0b\n- v0b --refutes--> v19\n- v0f --supports--> v14\n- v02 --supports--> v17\n- v00 --supports--> v09\n- v12 --refutes--> v0f\n- v06 --supports--> v16\n- v17 --supports--> v1b\n- v13 --supports--> v00\n- v0e --supports--> v0d\n- v09 --refutes--> v0a\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c147c5b3546aeedde6cb4c8f2f2bd65b2252340d1981a1e3dddfaecb21270c47","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f42a24afd72860f75650d9eb597b985e44e5e235be1330ea342433fc767065e","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-67f28bb02cfc5230273d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v11 --supports--> v00\n- v02 --refutes--> v01\n- v0e --refutes--> v13\n- v01 --supports--> v0b\n- v06 --refutes--> v0f\n- v1d --supports--> v1c\n- v1b --refutes--> v11\n- v07 --supports--> v10\n- v0f --supports--> v1e\n- v00 --refutes--> v0c\n- v04 --supports--> v17\n- v15 --supports--> v04\n- v12 --refutes--> v1b\n- v0d --refutes--> v16\n- v10 --refutes--> v02\n- v1e --refutes--> v08\n- v01 --refutes--> v15\n- v01 --refutes--> v0a\n- v0d --refutes--> v1d\n- v1e --refutes--> v18\n- v1d --refutes--> v05\n- v0e --supports--> v12\n- v15 --refutes--> v1a\n- v09 --supports--> v06\n- v08 --refutes--> v0d\n- v17 --refutes--> v0e\n- v17 --refutes--> v14\n- v0c --refutes--> v09\n- v11 --supports--> v03\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e8cfa7c83a4e685bc6707384d5f9ae0aa1689a5e668cdc8d732b498b10c50dfd","schema":"livegraph.boolean-propagation.v1","spec_hash":"67f28bb02cfc5230273d127886548d6ba7a3722ab0450fbc749153e53bf31c9b","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-0a79c797521f7e7aafcf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1d\n- v03 --refutes--> v1c\n- v0e --refutes--> v10\n- v1d --supports--> v05\n- v1a --supports--> v0e\n- v1b --refutes--> v00\n- v18 --refutes--> v01\n- v11 --supports--> v0c\n- v1d --refutes--> v11\n- v03 --supports--> v02\n- v0f --supports--> v0b\n- v0a --refutes--> v04\n- v15 --refutes--> v1b\n- v06 --refutes--> v18\n- v10 --refutes--> v0d\n- v02 --refutes--> v1e\n- v0a --supports--> v07\n- v09 --supports--> v0a\n- v11 --supports--> v17\n- v11 --refutes--> v13\n- v11 --refutes--> v16\n- v0f --supports--> v09\n- v11 --refutes--> v15\n- v14 --supports--> v06\n- v01 --refutes--> v12\n- v1b --supports--> v19\n- v00 --refutes--> v14\n- v12 --refutes--> v03\n- v07 --refutes--> v08\n- v10 --refutes--> v0f\n\n[VERIFIED ROOT]\nv1a = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae1d197ee09cb90edacc0f3525cee88fb88f25caa42da0dd2fdb9747ecce9614","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a79c797521f7e7aafcf389f56cabdaad80f3469e09e21956d9c536d50ac14dd","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-ad573729edb6e8d08ce6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v18 --supports--> v0f\n- v14 --supports--> v0e\n- v09 --refutes--> v13\n- v05 --refutes--> v1b\n- v17 --refutes--> v0d\n- v11 --supports--> v09\n- v05 --supports--> v0c\n- v05 --supports--> v08\n- v0a --supports--> v17\n- v08 --supports--> v03\n- v10 --supports--> v05\n- v19 --refutes--> v11\n- v11 --refutes--> v01\n- v08 --supports--> v02\n- v02 --refutes--> v1a\n- v16 --supports--> v10\n- v1e --refutes--> v19\n- v00 --supports--> v14\n- v1c --supports--> v0b\n- v0f --supports--> v0a\n- v04 --refutes--> v07\n- v14 --refutes--> v04\n- v17 --refutes--> v16\n- v1a --refutes--> v1c\n- v19 --refutes--> v15\n- v0b --supports--> v12\n- v18 --refutes--> v1d\n- v09 --refutes--> v00\n- v12 --supports--> v06\n\n[VERIFIED ROOT]\nv18 = 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":"f1cdba1b626ad1fb9bddbe4a8d57c57d427dd7fa78d1e2fd5d1ea27ea40edbcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"ad573729edb6e8d08ce652e0aa08bb9ff5a0cb2fd004faf1a3921ff4246cf08e","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-4a6410569701c70c0e73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v09\n- v12 --supports--> v18\n- v18 --refutes--> v08\n- v04 --supports--> v1a\n- v00 --refutes--> v0c\n- v09 --supports--> v12\n- v0c --supports--> v14\n- v18 --supports--> v17\n- v06 --supports--> v03\n- v11 --supports--> v0b\n- v1b --supports--> v01\n- v1a --supports--> v06\n- v19 --refutes--> v04\n- v0b --supports--> v0a\n- v0f --supports--> v1c\n- v1b --refutes--> v10\n- v05 --supports--> v02\n- v1d --supports--> v1b\n- v0a --refutes--> v0e\n- v0a --supports--> v1e\n- v05 --supports--> v00\n- v18 --supports--> v16\n- v0f --refutes--> v0d\n- v1e --refutes--> v19\n- v1c --supports--> v13\n- v0c --refutes--> v0f\n- v1e --supports--> v15\n- v13 --supports--> v11\n- v06 --supports--> v07\n- v07 --supports--> v1d\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":"3a46c56d216b31a5a598375f8f4f98f37a58c7bd795f9b6872e6c46a67d4f39d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a6410569701c70c0e730cc332e1b007b87088cdaa895714ad5fa8230e3bf30e","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-3153315d15489128b7ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v0d --refutes--> v16\n- v0a --refutes--> v17\n- v02 --supports--> v1a\n- v1e --refutes--> v07\n- v07 --supports--> v12\n- v17 --supports--> v0c\n- v04 --refutes--> v00\n- v11 --supports--> v06\n- v04 --supports--> v03\n- v03 --supports--> v01\n- v1a --refutes--> v08\n- v1b --supports--> v0f\n- v0d --supports--> v15\n- v0f --supports--> v05\n- v1c --supports--> v18\n- v06 --refutes--> v13\n- v0c --refutes--> v19\n- v08 --supports--> v1c\n- v18 --supports--> v0a\n- v03 --refutes--> v0b\n- v0d --supports--> v1e\n- v03 --supports--> v1d\n- v12 --refutes--> v14\n- v19 --refutes--> v0d\n- v12 --refutes--> v1b\n- v0b --refutes--> v02\n- v18 --supports--> v09\n- v1e --refutes--> v10\n- v07 --refutes--> v0e\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"48cf7d05c1a7a2b294b6ab6a7b0ebf3c7b9381fc3f46d87d62828b7eb1c16e4f","schema":"livegraph.boolean-propagation.v1","spec_hash":"3153315d15489128b7ad26050366528403bbddb84796aa696e18f04cfe595668","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-dbf38548caf339b60c0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v13\n- v1a --supports--> v0e\n- v11 --refutes--> v06\n- v1e --supports--> v0c\n- v0a --refutes--> v11\n- v12 --supports--> v14\n- v03 --refutes--> v1a\n- v09 --refutes--> v15\n- v17 --supports--> v10\n- v12 --refutes--> v0a\n- v0e --refutes--> v1c\n- v1b --supports--> v12\n- v02 --refutes--> v0d\n- v1c --refutes--> v04\n- v12 --refutes--> v00\n- v1c --supports--> v09\n- v0e --refutes--> v1d\n- v05 --supports--> v16\n- v11 --refutes--> v1e\n- v17 --refutes--> v0f\n- v08 --supports--> v18\n- v0d --refutes--> v08\n- v15 --refutes--> v05\n- v1a --refutes--> v19\n- v18 --supports--> v17\n- v0c --refutes--> v03\n- v01 --refutes--> v02\n- v03 --refutes--> v07\n- v10 --refutes--> v1b\n- v05 --refutes--> v0b\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":"74e76de057238699e273fb34baf68caff4f28f95f276ef109d198bc945ea0b9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbf38548caf339b60c0cc7488cf3cbc428e4ee37fad4b6c40184e70cf8c06001","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-09f80b46ad7050fec665","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v03\n- v06 --refutes--> v05\n- v07 --refutes--> v1e\n- v05 --supports--> v08\n- v05 --refutes--> v0c\n- v19 --supports--> v00\n- v0c --supports--> v13\n- v15 --refutes--> v09\n- v0c --refutes--> v0a\n- v0b --refutes--> v18\n- v11 --supports--> v1b\n- v06 --refutes--> v1c\n- v1b --refutes--> v02\n- v19 --refutes--> v10\n- v0f --refutes--> v15\n- v04 --supports--> v01\n- v1e --supports--> v0d\n- v06 --supports--> v17\n- v0a --refutes--> v0e\n- v0d --refutes--> v1d\n- v18 --refutes--> v07\n- v01 --refutes--> v06\n- v1b --refutes--> v16\n- v19 --supports--> v14\n- v0e --supports--> v0b\n- v1d --supports--> v19\n- v1e --supports--> v1a\n- v0f --supports--> v12\n- v02 --supports--> v04\n- v00 --refutes--> v0f\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":"c588ee1cda9288d796017007a0d5b418daa03d7ebd9fe79eadeaafb2ae049246","schema":"livegraph.boolean-propagation.v1","spec_hash":"09f80b46ad7050fec665a2b5f44d7ab5621eef7963b20ff9fe3a541cffb18786","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-ffaa81b117ffc71dd6e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --supports--> v0d\n- v04 --supports--> v14\n- v15 --refutes--> v1a\n- v1d --supports--> v0b\n- v10 --supports--> v08\n- v05 --refutes--> v1b\n- v08 --supports--> v1c\n- v07 --supports--> v0e\n- v02 --supports--> v1e\n- v06 --refutes--> v1d\n- v08 --refutes--> v18\n- v1a --supports--> v00\n- v05 --supports--> v06\n- v03 --supports--> v10\n- v11 --refutes--> v02\n- v06 --supports--> v01\n- v0f --refutes--> v13\n- v0a --supports--> v05\n- v06 --refutes--> v0c\n- v12 --refutes--> v0a\n- v06 --supports--> v16\n- v15 --refutes--> v09\n- v0f --supports--> v19\n- v13 --supports--> v15\n- v00 --supports--> v03\n- v0e --refutes--> v0f\n- v04 --supports--> v12\n- v18 --refutes--> v04\n- v0b --supports--> v11\n- v1d --refutes--> v17\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"400223637ec8c8100992060f371a5d957952079fdd6f9b9086df757933ee52d3","schema":"livegraph.boolean-propagation.v1","spec_hash":"ffaa81b117ffc71dd6e571a9955c75f959c0e2d92f7ca00f2cb750ae3e8443b6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-c72399864bd77fee96da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v07\n- v18 --supports--> v1e\n- v0e --supports--> v08\n- v12 --supports--> v16\n- v00 --supports--> v0d\n- v0b --supports--> v1c\n- v0e --supports--> v04\n- v11 --refutes--> v0e\n- v17 --refutes--> v09\n- v12 --refutes--> v06\n- v04 --supports--> v0c\n- v0c --supports--> v12\n- v12 --supports--> v01\n- v19 --supports--> v02\n- v15 --refutes--> v1b\n- v15 --supports--> v11\n- v09 --supports--> v14\n- v03 --supports--> v05\n- v03 --refutes--> v0a\n- v1a --refutes--> v13\n- v19 --refutes--> v1d\n- v1a --refutes--> v0f\n- v16 --refutes--> v19\n- v09 --supports--> v10\n- v18 --supports--> v17\n- v10 --refutes--> v00\n- v02 --supports--> v0b\n- v0f --refutes--> v03\n- v0d --refutes--> v15\n- v0a --supports--> v18\n\n[VERIFIED ROOT]\nv1a = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"32f5c50ad3273f40526c83d678d03739461650094d6e3b235269c8a7f43631e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c72399864bd77fee96dac396106cd16a3718ed20f1be96c3af0b44ac0aa77b56","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"}],\"hop\":20,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-ae186a49d6c087e64f15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --refutes--> v0f\n- v07 --refutes--> v0b\n- v10 --refutes--> v08\n- v07 --refutes--> v1d\n- v09 --refutes--> v0c\n- v14 --supports--> v16\n- v0e --refutes--> v17\n- v13 --supports--> v12\n- v0a --supports--> v00\n- v1d --refutes--> v15\n- v03 --supports--> v19\n- v02 --refutes--> v04\n- v14 --supports--> v06\n- v17 --supports--> v13\n- v04 --refutes--> v0e\n- v1e --supports--> v1a\n- v10 --supports--> v07\n- v15 --supports--> v14\n- v1d --refutes--> v1b\n- v10 --refutes--> v11\n- v1e --refutes--> v01\n- v16 --refutes--> v1c\n- v15 --refutes--> v0d\n- v00 --supports--> v1e\n- v1c --refutes--> v05\n- v09 --refutes--> v0a\n- v1e --refutes--> v18\n- v1c --refutes--> v09\n- v19 --supports--> v02\n- v01 --supports--> 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":"d136b4779dc2ea53371ecbc95860f8228e1976f35fd56126d222190a50922d16","schema":"livegraph.boolean-propagation.v1","spec_hash":"ae186a49d6c087e64f15d4d5d96a2baaa27bd8b412ac44a06491c5c594ad7425","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-5ce6c554f25ed7e0335c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v16\n- v09 --supports--> v1a\n- v07 --refutes--> v05\n- v0c --refutes--> v06\n- v10 --supports--> v07\n- v1c --supports--> v15\n- v08 --refutes--> v04\n- v13 --refutes--> v03\n- v19 --refutes--> v0a\n- v06 --supports--> v0b\n- v01 --refutes--> v14\n- v0e --supports--> v0d\n- v15 --supports--> v0e\n- v08 --supports--> v18\n- v18 --refutes--> v11\n- v07 --refutes--> v08\n- v0b --supports--> v00\n- v19 --supports--> v09\n- v06 --supports--> v1b\n- v15 --supports--> v1d\n- v1c --refutes--> v0f\n- v00 --refutes--> v10\n- v1a --supports--> v0c\n- v14 --refutes--> v12\n- v03 --refutes--> v19\n- v0d --supports--> v02\n- v15 --supports--> v1e\n- v02 --refutes--> v01\n- v14 --refutes--> v13\n- v19 --supports--> v17\n\n[VERIFIED ROOT]\nv1c = 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":"9f949afd14f3e6fb99904db63cac4a8a5a09dd5870305e4bd950c974a1d1d92a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ce6c554f25ed7e0335c527e33a99e61b02f64b32469924cd243810b5d992a88","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-1f7e0ff5656342673604","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v0b\n- v0e --refutes--> v16\n- v17 --supports--> v12\n- v01 --supports--> v0d\n- v01 --refutes--> v00\n- v1a --supports--> v18\n- v18 --refutes--> v1c\n- v0b --supports--> v1d\n- v09 --refutes--> v03\n- v1d --supports--> v06\n- v10 --supports--> v15\n- v14 --refutes--> v04\n- v07 --refutes--> v19\n- v07 --supports--> v08\n- v14 --refutes--> v1e\n- v1c --supports--> v11\n- v17 --refutes--> v1a\n- v00 --supports--> v10\n- v0c --supports--> v01\n- v07 --refutes--> v0f\n- v03 --supports--> v0e\n- v19 --supports--> v05\n- v04 --supports--> v0c\n- v13 --refutes--> v17\n- v05 --supports--> v14\n- v04 --supports--> v0a\n- v16 --refutes--> v07\n- v10 --refutes--> v13\n- v00 --supports--> v1b\n- v16 --supports--> v02\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"37af7f5ad080da885218d17c9ef10b34cd9c74601524066b24b072d2f54a8adb","schema":"livegraph.boolean-propagation.v1","spec_hash":"1f7e0ff565634267360406fe427ea6452e2331ceaebca5f7adc440c1bac974d8","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-a19b8ee4b6ea02a44376","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1d\n- v00 --refutes--> v0f\n- v1b --supports--> v13\n- v12 --refutes--> v16\n- v1c --refutes--> v01\n- v17 --supports--> v10\n- v1e --supports--> v07\n- v1b --supports--> v1a\n- v10 --supports--> v12\n- v0d --supports--> v1e\n- v03 --refutes--> v0e\n- v13 --refutes--> v02\n- v0c --supports--> v0d\n- v01 --refutes--> v14\n- v01 --supports--> v18\n- v07 --supports--> v00\n- v0f --refutes--> v1c\n- v01 --supports--> v17\n- v00 --supports--> v0b\n- v1d --refutes--> v19\n- v1d --supports--> v04\n- v02 --refutes--> v0a\n- v19 --refutes--> v08\n- v0a --refutes--> v0c\n- v0e --refutes--> v05\n- v12 --supports--> v03\n- v12 --supports--> v06\n- v1d --refutes--> v11\n- v08 --supports--> v09\n- v0e --refutes--> v15\n\n[VERIFIED ROOT]\nv1b = 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":"b40bce65f3533e506f7de566eb450b60ea2a729aa888d69e30e2513c96e4fe06","schema":"livegraph.boolean-propagation.v1","spec_hash":"a19b8ee4b6ea02a44376b70b065d640679840547e83f7e93013e5a948ea1c38c","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-0528ded769ad5134eff2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1c\n- v1e --supports--> v06\n- v02 --refutes--> v03\n- v15 --refutes--> v1e\n- v0a --refutes--> v00\n- v05 --refutes--> v01\n- v1d --supports--> v08\n- v07 --refutes--> v10\n- v00 --supports--> v0d\n- v19 --refutes--> v0a\n- v08 --refutes--> v0f\n- v14 --supports--> v17\n- v04 --refutes--> v0e\n- v08 --supports--> v12\n- v12 --refutes--> v13\n- v03 --refutes--> v1d\n- v00 --refutes--> v05\n- v12 --refutes--> v0c\n- v10 --refutes--> v11\n- v06 --refutes--> v19\n- v1b --supports--> v1a\n- v08 --refutes--> v16\n- v1e --supports--> v0b\n- v05 --supports--> v1b\n- v14 --supports--> v18\n- v14 --refutes--> v04\n- v0c --refutes--> v15\n- v1a --refutes--> v14\n- v0e --supports--> v07\n- v0c --supports--> v09\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9a7cbc655f3688d252bfbcb491878b86a6a9cb526cd26b85a25fabc1b51100a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"0528ded769ad5134eff2b3ad5df0e74ae465de6388f41812ca6442ac862adb44","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-cb5b6cb23eafd188c4c6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v07 --refutes--> v1d\n- v05 --supports--> v1a\n- v16 --supports--> v08\n- v0e --refutes--> v1e\n- v12 --supports--> v1b\n- v12 --supports--> v11\n- v09 --supports--> v05\n- v0a --supports--> v06\n- v10 --refutes--> v15\n- v0b --supports--> v13\n- v10 --refutes--> v0f\n- v01 --supports--> v12\n- v00 --supports--> v09\n- v17 --supports--> v04\n- v0e --refutes--> v07\n- v1a --refutes--> v16\n- v0b --supports--> v0d\n- v02 --refutes--> v1c\n- v12 --refutes--> v14\n- v07 --supports--> v0c\n- v1c --supports--> v0a\n- v1b --refutes--> v00\n- v0d --refutes--> v17\n- v07 --supports--> v18\n- v08 --refutes--> v03\n- v03 --supports--> v02\n- v15 --supports--> v0e\n- v17 --supports--> v01\n- v1c --supports--> v19\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8972541144df1504d3463369a1815181eebe60e30296de44903901915bdf03cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"cb5b6cb23eafd188c4c6e34be29441822729f776c95ea62cbf3e7a0d9b277607","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-7bfe7c1aa2d6c85be647","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v17 --refutes--> v0c\n- v1b --supports--> v06\n- v0a --supports--> v1d\n- v1b --refutes--> v16\n- v07 --supports--> v01\n- v04 --supports--> v1e\n- v15 --supports--> v1c\n- v10 --refutes--> v04\n- v1a --refutes--> v10\n- v15 --supports--> v08\n- v0f --refutes--> v17\n- v0c --refutes--> v13\n- v12 --supports--> v18\n- v1b --refutes--> v03\n- v0d --supports--> v07\n- v08 --refutes--> v0f\n- v06 --refutes--> v12\n- v0a --supports--> v0b\n- v11 --refutes--> v00\n- v12 --supports--> v02\n- v15 --supports--> v14\n- v1a --refutes--> v05\n- v19 --supports--> v11\n- v18 --refutes--> v0d\n- v01 --refutes--> v1a\n- v1e --refutes--> v15\n- v0b --supports--> v19\n- v0f --refutes--> v0e\n- v19 --refutes--> v09\n\n[VERIFIED ROOT]\nv1b = 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":"367af64cd7a1b4dbcc1ceb121073c636fbfd6e5cfaaaceffe870e3b5b1b8162a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7bfe7c1aa2d6c85be64794073ae310aea98a943dc10375aa02721d622367931a","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-4bb775c1c4c98004cb4d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1b --refutes--> v11\n- v15 --refutes--> v04\n- v08 --refutes--> v19\n- v04 --supports--> v0b\n- v0f --supports--> v1a\n- v19 --refutes--> v0e\n- v12 --supports--> v10\n- v1c --refutes--> v16\n- v10 --supports--> v15\n- v0e --supports--> v1c\n- v1c --supports--> v0a\n- v08 --refutes--> v1e\n- v0f --supports--> v09\n- v03 --refutes--> v06\n- v09 --refutes--> v1b\n- v14 --supports--> v0d\n- v04 --supports--> v0f\n- v01 --refutes--> v14\n- v05 --refutes--> v12\n- v0d --refutes--> v18\n- v03 --supports--> v05\n- v1c --supports--> v07\n- v0c --supports--> v1d\n- v15 --refutes--> v00\n- v02 --supports--> v17\n- v07 --supports--> v0c\n- v08 --supports--> v13\n- v02 --supports--> v03\n- v1d --refutes--> v01\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":"bba8a3f776b9986ab1b606fabacc84e4ba2055376997b3b1eaa79c5a624edb92","schema":"livegraph.boolean-propagation.v1","spec_hash":"4bb775c1c4c98004cb4d809b5eaa0b2589ce63d05dbacae5e6f1e3742e6c2ac7","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-eb8420be275d7217f321","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v01 --supports--> v1c\n- v05 --supports--> v14\n- v17 --refutes--> v0d\n- v01 --supports--> v05\n- v17 --refutes--> v15\n- v11 --refutes--> v1d\n- v07 --supports--> v0c\n- v0f --supports--> v0a\n- v1a --refutes--> v03\n- v0f --supports--> v08\n- v00 --refutes--> v1e\n- v0e --refutes--> v17\n- v03 --refutes--> v0f\n- v1d --supports--> v01\n- v09 --refutes--> v0e\n- v12 --refutes--> v04\n- v0f --refutes--> v0b\n- v06 --supports--> v13\n- v16 --supports--> v11\n- v11 --supports--> v10\n- v17 --refutes--> v02\n- v00 --refutes--> v07\n- v07 --refutes--> v09\n- v0d --refutes--> v16\n- v0b --refutes--> v19\n- v18 --supports--> v1a\n- v1a --refutes--> v1b\n- v19 --refutes--> v00\n- v04 --supports--> v18\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":"5ca46a5c06b528b4c9ee0af5fad8e652b2ea258ef2144a8faef2b909d030c735","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb8420be275d7217f321d42ca0f14dc2de94434fed3912c9ff99291876e93914","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-8d37d137cdb8b1066bce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1d --refutes--> v14\n- v05 --refutes--> v15\n- v0d --refutes--> v00\n- v01 --refutes--> v05\n- v11 --refutes--> v04\n- v11 --refutes--> v19\n- v17 --refutes--> v10\n- v0d --supports--> v17\n- v0a --refutes--> v0d\n- v1d --supports--> v18\n- v01 --supports--> v02\n- v0c --refutes--> v13\n- v10 --supports--> v03\n- v08 --refutes--> v0f\n- v1b --refutes--> v1e\n- v0e --supports--> v09\n- v1b --refutes--> v06\n- v03 --refutes--> v12\n- v1c --refutes--> v1d\n- v0e --supports--> v1a\n- v15 --supports--> v08\n- v1b --supports--> v11\n- v12 --refutes--> v0c\n- v07 --refutes--> v01\n- v0f --refutes--> v0b\n- v1d --refutes--> v0e\n- v11 --supports--> v07\n- v0b --supports--> v16\n- v13 --supports--> v1c\n\n[VERIFIED ROOT]\nv1b = 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":"3adba11bbd5d14b5b840c3cdb8bacf54d76ba06e6fd899dc860b181939a7433b","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d37d137cdb8b1066bcea96a502535dd9ed8d3f4502788a3bcffca69f107bf57","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-eb7c005efefcd25e8c1c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v0c\n- v07 --refutes--> v03\n- v07 --refutes--> v1c\n- v0d --supports--> v09\n- v15 --refutes--> v0e\n- v1a --refutes--> v18\n- v11 --supports--> v1b\n- v00 --supports--> v0b\n- v1d --supports--> v16\n- v16 --supports--> v12\n- v02 --supports--> v0a\n- v18 --supports--> v11\n- v14 --supports--> v10\n- v18 --supports--> v08\n- v19 --refutes--> v15\n- v0b --supports--> v1d\n- v1e --refutes--> v01\n- v10 --refutes--> v02\n- v1c --refutes--> v05\n- v15 --supports--> v0d\n- v0a --refutes--> v1e\n- v05 --refutes--> v13\n- v16 --supports--> v19\n- v05 --supports--> v0f\n- v0f --supports--> v1a\n- v07 --refutes--> v04\n- v10 --supports--> v06\n- v1b --supports--> v14\n- v0d --supports--> v07\n- v02 --refutes--> v17\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":"183742e379c73511beab5dc92bba9804e8d6addf676c14b92b469392adcb9ef9","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb7c005efefcd25e8c1c40bfd614645d853f956314f435644a3f17bfcd121484","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-c957a23b1903c7428e55","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v03\n- v14 --supports--> v16\n- v11 --supports--> v10\n- v1b --supports--> v04\n- v13 --supports--> v0b\n- v1c --supports--> v0f\n- v1d --supports--> v0c\n- v0c --supports--> v12\n- v01 --supports--> v06\n- v0f --refutes--> v0d\n- v16 --supports--> v18\n- v1a --refutes--> v07\n- v18 --refutes--> v08\n- v19 --supports--> v15\n- v18 --supports--> v1e\n- v05 --refutes--> v1a\n- v1a --refutes--> v0e\n- v08 --supports--> v02\n- v01 --supports--> v0a\n- v0e --refutes--> v13\n- v04 --refutes--> v19\n- v06 --supports--> v1d\n- v11 --refutes--> v05\n- v15 --refutes--> v11\n- v1a --supports--> v09\n- v0d --supports--> v01\n- v13 --supports--> v17\n- v13 --refutes--> v14\n- v0d --supports--> v00\n- v0c --supports--> v1b\n\n[VERIFIED ROOT]\nv1c = 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":"e1d3677532988709931ee130a3a05f43c12f99e3e68b975c3a5d84aa7a26eb02","schema":"livegraph.boolean-propagation.v1","spec_hash":"c957a23b1903c7428e554a4206828357e2ffa4d48491901fa923d7f50b2a2b3d","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-59f83237caf678c76e93","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v09\n- v15 --refutes--> v17\n- v12 --supports--> v19\n- v05 --refutes--> v15\n- v08 --supports--> v02\n- v17 --refutes--> v1e\n- v1b --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v1c\n- v0f --supports--> v16\n- v0d --refutes--> v01\n- v1c --refutes--> v04\n- v1d --supports--> v0f\n- v16 --supports--> v12\n- v0e --refutes--> v00\n- v13 --supports--> v0b\n- v0d --supports--> v1b\n- v1c --supports--> v0e\n- v12 --refutes--> v0a\n- v05 --supports--> v1a\n- v11 --refutes--> v1d\n- v18 --refutes--> v10\n- v18 --refutes--> v0c\n- v14 --refutes--> v0d\n- v0a --refutes--> v14\n- v00 --supports--> v18\n- v18 --supports--> v03\n- v15 --refutes--> v07\n- v17 --supports--> v13\n- v0b --refutes--> v11\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":"121512a025d92ecc2a871d5d9a9a82990039721bf244af4071818e7f41e89265","schema":"livegraph.boolean-propagation.v1","spec_hash":"59f83237caf678c76e932616a382b8907ea1f2332915e14363ed25169bc3a627","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-7934196974dc36968c0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v0b\n- v1c --refutes--> v13\n- v0b --supports--> v09\n- v1d --supports--> v11\n- v09 --supports--> v07\n- v1c --supports--> v1b\n- v05 --supports--> v1e\n- v1d --supports--> v05\n- v00 --supports--> v02\n- v0a --refutes--> v1a\n- v18 --refutes--> v15\n- v1a --refutes--> v0c\n- v12 --refutes--> v06\n- v13 --refutes--> v10\n- v15 --supports--> v0a\n- v16 --refutes--> v03\n- v09 --supports--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v1c\n- v02 --supports--> v04\n- v0d --supports--> v16\n- v05 --refutes--> v01\n- v0c --refutes--> v14\n- v14 --supports--> v0d\n- v13 --refutes--> v18\n- v03 --supports--> v0f\n- v0a --refutes--> v17\n- v02 --refutes--> v1d\n- v14 --refutes--> v0e\n- v18 --supports--> v19\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0f0930edc1f565a6670680c8c3768d4e5954334c5ceb9d1dbe19f0df6232518e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7934196974dc36968c0bbcfa2f1aa199fe2e934688b705d1a681150d2a8f7cee","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-46363f6bbaaef420a1af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v0e --refutes--> v16\n- v16 --refutes--> v0b\n- v02 --refutes--> v0d\n- v0f --supports--> v00\n- v14 --refutes--> v1b\n- v18 --refutes--> v03\n- v15 --refutes--> v02\n- v07 --supports--> v01\n- v1e --refutes--> v10\n- v09 --supports--> v1d\n- v1e --refutes--> v15\n- v01 --supports--> v04\n- v00 --refutes--> v0a\n- v1e --supports--> v12\n- v11 --supports--> v0f\n- v09 --refutes--> v11\n- v1e --refutes--> v06\n- v05 --supports--> v07\n- v18 --supports--> v13\n- v18 --refutes--> v17\n- v08 --supports--> v05\n- v1b --supports--> v08\n- v0d --refutes--> v1c\n- v1b --refutes--> v1a\n- v1c --supports--> v14\n- v18 --refutes--> v1e\n- v04 --refutes--> v0e\n- v0b --refutes--> v09\n- v1e --refutes--> v0c\n\n[VERIFIED ROOT]\nv18 = 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":"e0110fb76613574211271dfa8ccfc0ddb26e61c7298d705a17d1e410cc84ced6","schema":"livegraph.boolean-propagation.v1","spec_hash":"46363f6bbaaef420a1af7162993d646b9371c3020578e06dc340aa61caf9ec2b","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-41cb14803d122ba10663","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v09\n- v09 --supports--> v05\n- v12 --supports--> v0e\n- v14 --supports--> v15\n- v14 --refutes--> v18\n- v00 --refutes--> v0c\n- v1b --supports--> v17\n- v09 --refutes--> v11\n- v00 --supports--> v1b\n- v0e --refutes--> v0f\n- v1d --refutes--> v0b\n- v10 --supports--> v06\n- v12 --refutes--> v19\n- v07 --refutes--> v0a\n- v17 --refutes--> v03\n- v10 --refutes--> v1a\n- v02 --supports--> v16\n- v06 --refutes--> v1e\n- v03 --refutes--> v10\n- v06 --refutes--> v08\n- v15 --refutes--> v04\n- v0a --refutes--> v14\n- v11 --refutes--> v1c\n- v02 --refutes--> v12\n- v04 --supports--> v01\n- v1c --refutes--> v02\n- v08 --refutes--> v07\n- v00 --refutes--> v0d\n- v1d --refutes--> v13\n- v0f --refutes--> v1d\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9170c340c1ff9443a61c8fa6a3b1e780cc5752e7adee43ecb2642237b6eacbf0","schema":"livegraph.boolean-propagation.v1","spec_hash":"41cb14803d122ba106630d054eb5939dbba7e5cc8afdebd33b5b24a72ff3f811","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-fcfba904ee7ddd5c6657","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0e\n- v15 --refutes--> v00\n- v18 --refutes--> v0b\n- v09 --supports--> v08\n- v14 --supports--> v1d\n- v19 --refutes--> v09\n- v16 --refutes--> v17\n- v09 --refutes--> v18\n- v09 --supports--> v1c\n- v1d --supports--> v16\n- v03 --supports--> v0f\n- v0c --supports--> v10\n- v12 --supports--> v1b\n- v13 --supports--> v15\n- v10 --refutes--> v01\n- v1e --refutes--> v0d\n- v13 --supports--> v1a\n- v10 --supports--> v14\n- v18 --supports--> v06\n- v12 --supports--> v0a\n- v03 --refutes--> v11\n- v06 --refutes--> v0c\n- v12 --refutes--> v02\n- v15 --supports--> v1e\n- v07 --refutes--> v03\n- v04 --supports--> v12\n- v05 --refutes--> v13\n- v0e --refutes--> v19\n- v16 --refutes--> v05\n- v0d --refutes--> v07\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"daeec35f889ddd0186eb60ad5e02ffe4b5c235e86534bf061e5364f25e4cc954","schema":"livegraph.boolean-propagation.v1","spec_hash":"fcfba904ee7ddd5c6657fa1e63ec25f6b34f09eda1c1bd107c3c0ec4cef34de3","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-520427d07e36f5f45e5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v1b --supports--> v09\n- v19 --supports--> v0c\n- v0d --supports--> v11\n- v12 --refutes--> v18\n- v17 --supports--> v04\n- v1e --refutes--> v00\n- v19 --supports--> v10\n- v09 --refutes--> v0d\n- v0c --supports--> v15\n- v08 --supports--> v19\n- v00 --refutes--> v1d\n- v0c --supports--> v16\n- v02 --refutes--> v0a\n- v09 --supports--> v0e\n- v1a --supports--> v0f\n- v0a --supports--> v05\n- v19 --supports--> v14\n- v1e --supports--> v13\n- v04 --refutes--> v1e\n- v0d --refutes--> v1c\n- v07 --supports--> v08\n- v1d --supports--> v03\n- v02 --refutes--> v01\n- v04 --refutes--> v06\n- v0f --supports--> v17\n- v03 --refutes--> v12\n- v0b --supports--> v1a\n- v0c --refutes--> v0b\n- v18 --supports--> v02\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":"d5b34efb9fc1c0d10224d4ab41f7ac075013fd5a62f4c7817ac767a09d67cf86","schema":"livegraph.boolean-propagation.v1","spec_hash":"520427d07e36f5f45e5b389427721466d9684731cdef9616579a7a13e3dd1dc8","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-392da7ee1c9dbb533b2e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v01\n- v0a --supports--> v19\n- v17 --refutes--> v14\n- v01 --supports--> v17\n- v16 --supports--> v0c\n- v19 --refutes--> v13\n- v05 --supports--> v09\n- v19 --refutes--> v08\n- v1e --refutes--> v16\n- v0f --refutes--> v12\n- v1b --supports--> v05\n- v11 --refutes--> v1c\n- v18 --supports--> v1e\n- v19 --supports--> v0b\n- v19 --supports--> v1a\n- v02 --supports--> v1b\n- v15 --refutes--> v0e\n- v09 --supports--> v10\n- v0e --supports--> v02\n- v11 --supports--> v0a\n- v17 --refutes--> v15\n- v04 --refutes--> v03\n- v11 --refutes--> v07\n- v0c --supports--> v11\n- v17 --supports--> v06\n- v10 --supports--> v04\n- v11 --refutes--> v1d\n- v12 --supports--> v0d\n- v05 --refutes--> v00\n- v04 --supports--> v0f\n\n[VERIFIED ROOT]\nv18 = 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":"9e408e06bd0debd22d3181beaeac39c95347744623f332a379886d35d39a6c81","schema":"livegraph.boolean-propagation.v1","spec_hash":"392da7ee1c9dbb533b2e39d297c67356156aff7fcdff9b57a783f830aa24bff6","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-0c2851fb85feed2e8e94","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --refutes--> v0b\n- v00 --refutes--> v17\n- v1c --refutes--> v09\n- v01 --supports--> v1c\n- v0b --refutes--> v01\n- v01 --supports--> v05\n- v1d --refutes--> v12\n- v02 --supports--> v03\n- v01 --supports--> v13\n- v1a --refutes--> v14\n- v1c --supports--> v0c\n- v03 --supports--> v1b\n- v14 --supports--> v0f\n- v0d --refutes--> v18\n- v0a --supports--> v0d\n- v19 --refutes--> v07\n- v09 --supports--> v19\n- v1b --refutes--> v1a\n- v12 --supports--> v06\n- v1e --supports--> v02\n- v00 --supports--> v16\n- v0d --supports--> v1e\n- v08 --refutes--> v0a\n- v08 --refutes--> v15\n- v0f --refutes--> v10\n- v0e --supports--> v08\n- v12 --refutes--> v00\n- v03 --refutes--> v11\n- v12 --refutes--> v04\n- v16 --supports--> v0e\n\n[VERIFIED ROOT]\nv1d = 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":"4f1f01dfbc3066c39061e650c0a267a67ee3070650c94a11867455cd4ea10ae9","schema":"livegraph.boolean-propagation.v1","spec_hash":"0c2851fb85feed2e8e94078cadb4a98c832fd6764054a208cb44880d78eae985","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-3092c393fbb835415f49","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v10\n- v0d --supports--> v17\n- v03 --refutes--> v05\n- v10 --refutes--> v14\n- v10 --supports--> v12\n- v08 --refutes--> v03\n- v18 --supports--> v1a\n- v03 --supports--> v04\n- v12 --refutes--> v1d\n- v0f --supports--> v0e\n- v16 --refutes--> v0b\n- v06 --supports--> v18\n- v1b --supports--> v19\n- v05 --refutes--> v11\n- v09 --supports--> v16\n- v1a --refutes--> v0d\n- v17 --refutes--> v0c\n- v0d --supports--> v15\n- v10 --refutes--> v1e\n- v0c --supports--> v1c\n- v0d --supports--> v01\n- v02 --refutes--> v09\n- v1c --supports--> v13\n- v1a --refutes--> v00\n- v05 --supports--> v1b\n- v12 --refutes--> v02\n- v0b --supports--> v06\n- v1a --supports--> v07\n- v19 --refutes--> v0f\n- v13 --refutes--> v0a\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90c7f9584126b58c97e9747ad46d531875a9e89db4d8fb38e93fdb3c9531145c","schema":"livegraph.boolean-propagation.v1","spec_hash":"3092c393fbb835415f49a11048d30ab2f48ed3b099cf565a475a483364747af0","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-22c24d189efb1e77bff8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v05\n- v0a --supports--> v00\n- v07 --supports--> v19\n- v08 --supports--> v11\n- v16 --refutes--> v0f\n- v1b --supports--> v1c\n- v0a --supports--> v03\n- v08 --refutes--> v15\n- v0d --supports--> v02\n- v0f --refutes--> v08\n- v04 --supports--> v06\n- v13 --supports--> v01\n- v1a --supports--> v0a\n- v06 --supports--> v1e\n- v07 --supports--> v09\n- v04 --supports--> v14\n- v0c --supports--> v18\n- v07 --refutes--> v1d\n- v1c --supports--> v0c\n- v0b --refutes--> v16\n- v03 --refutes--> v0d\n- v1c --refutes--> v10\n- v1e --supports--> v13\n- v19 --refutes--> v12\n- v13 --supports--> v07\n- v18 --refutes--> v0b\n- v12 --supports--> v1b\n- v02 --supports--> v04\n- v12 --refutes--> v17\n- v1a --supports--> v0e\n\n[VERIFIED ROOT]\nv1a = 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":"6ee191c437766b841191767b960dd5291012f107479796d431a584f011e4f8aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"22c24d189efb1e77bff8a769c1712803376f8b79af9bff485d1813b5942d4f30","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-37db7b513d03fdc84aed","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v0f\n- v16 --refutes--> v12\n- v1d --refutes--> v1c\n- v1e --refutes--> v06\n- v11 --refutes--> v0e\n- v1c --refutes--> v16\n- v0f --refutes--> v0b\n- v1b --supports--> v0a\n- v1e --supports--> v08\n- v06 --supports--> v01\n- v05 --supports--> v1b\n- v0e --refutes--> v09\n- v04 --refutes--> v00\n- v06 --refutes--> v0c\n- v1e --supports--> v17\n- v0d --supports--> v11\n- v13 --supports--> v19\n- v00 --supports--> v14\n- v0b --supports--> v15\n- v13 --refutes--> v02\n- v03 --refutes--> v1e\n- v0c --supports--> v04\n- v12 --supports--> v03\n- v16 --supports--> v10\n- v0e --supports--> v07\n- v0b --supports--> v0d\n- v13 --refutes--> v05\n- v00 --supports--> v1a\n- v0a --refutes--> v1d\n- v0e --supports--> v18\n\n[VERIFIED ROOT]\nv13 = 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":"b57f0f0fe0dd0d203b1bd95da560d342fc6b1560e5a8b5c5bf8c18d0f9f99786","schema":"livegraph.boolean-propagation.v1","spec_hash":"37db7b513d03fdc84aed752d2456cecfb260935637693c0e277ffddc1c947b64","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-e5a62d9ec03844fe566a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v00\n- v13 --refutes--> v03\n- v09 --refutes--> v12\n- v01 --refutes--> v0b\n- v0d --refutes--> v17\n- v1b --refutes--> v1a\n- v1c --supports--> v07\n- v04 --supports--> v05\n- v18 --supports--> v04\n- v06 --refutes--> v0a\n- v10 --refutes--> v0f\n- v19 --refutes--> v02\n- v19 --supports--> v06\n- v0d --supports--> v1e\n- v07 --refutes--> v0e\n- v0b --refutes--> v13\n- v07 --supports--> v10\n- v13 --supports--> v1b\n- v0d --supports--> v08\n- v1a --refutes--> v19\n- v10 --refutes--> v0d\n- v1e --supports--> v01\n- v05 --refutes--> v15\n- v1d --supports--> v09\n- v1b --refutes--> v11\n- v16 --refutes--> v18\n- v09 --supports--> v1c\n- v15 --refutes--> v1d\n- v10 --supports--> v14\n- v06 --supports--> v0c\n\n[VERIFIED ROOT]\nv16 = 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":"513dd26b2993eb8ea07a55cf6fc06dfffeadda84fa99362b7d7d4011a9a92582","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5a62d9ec03844fe566aebf757a8eb17393d9d35d357955b91b9ae3a8a3a644c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-bc5f519f44f8a62390c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v02 --supports--> v11\n- v19 --supports--> v02\n- v01 --supports--> v1d\n- v18 --refutes--> v0d\n- v19 --supports--> v16\n- v09 --supports--> v1a\n- v1d --refutes--> v04\n- v19 --supports--> v00\n- v13 --supports--> v07\n- v0a --refutes--> v0f\n- v0a --refutes--> v0c\n- v10 --refutes--> v19\n- v02 --supports--> v01\n- v1e --refutes--> v03\n- v06 --refutes--> v0b\n- v08 --supports--> v17\n- v0f --supports--> v1b\n- v06 --refutes--> v08\n- v18 --refutes--> v0e\n- v0e --refutes--> v15\n- v17 --supports--> v0a\n- v1b --refutes--> v18\n- v1a --supports--> v1e\n- v03 --supports--> v13\n- v07 --refutes--> v14\n- v15 --supports--> v09\n- v0f --refutes--> v05\n- v1d --supports--> v06\n- v17 --refutes--> v1c\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":"70fe7591e361040c5a9e335b0076514937a0911afb03483f707f257589723621","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc5f519f44f8a62390c164df5145e6610dbe83d114c490edde659f93a805ca81","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-89eae9e61eca659e0478","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v0b\n- v04 --refutes--> v01\n- v1d --refutes--> v00\n- v1a --supports--> v18\n- v0c --refutes--> v1d\n- v1c --supports--> v0c\n- v14 --supports--> v0f\n- v18 --refutes--> v08\n- v04 --refutes--> v16\n- v08 --supports--> v06\n- v0b --refutes--> v0d\n- v0a --supports--> v05\n- v05 --supports--> v13\n- v14 --supports--> v1a\n- v09 --refutes--> v1c\n- v1b --supports--> v15\n- v12 --supports--> v14\n- v13 --supports--> v1b\n- v08 --refutes--> v1e\n- v1c --supports--> v10\n- v03 --supports--> v11\n- v01 --refutes--> v0e\n- v1a --refutes--> v02\n- v1d --supports--> v07\n- v0b --supports--> v03\n- v01 --refutes--> v19\n- v03 --refutes--> v0a\n- v12 --supports--> v17\n- v07 --refutes--> v12\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ebb72bae1a49656ea04c01894e0f1ad6b6ea4d35885caadc1b833510b803d07a","schema":"livegraph.boolean-propagation.v1","spec_hash":"89eae9e61eca659e047832996c439769946533017d2329240c08082c7e32995f","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-d4d4d494a1a40fab5709","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v19 --refutes--> v18\n- v0c --refutes--> v11\n- v00 --refutes--> v08\n- v1c --supports--> v0d\n- v05 --refutes--> v1c\n- v16 --supports--> v17\n- v07 --refutes--> v1b\n- v1d --refutes--> v1e\n- v1d --supports--> v03\n- v02 --refutes--> v04\n- v1b --supports--> v0a\n- v13 --refutes--> v15\n- v0a --supports--> v09\n- v1d --refutes--> v10\n- v00 --supports--> v01\n- v12 --supports--> v14\n- v17 --refutes--> v0c\n- v0e --refutes--> v1a\n- v11 --supports--> v07\n- v00 --refutes--> v13\n- v0e --refutes--> v06\n- v02 --refutes--> v1d\n- v0e --refutes--> v16\n- v0a --supports--> v00\n- v15 --refutes--> v02\n- v1c --refutes--> v12\n- v11 --supports--> v0b\n- v18 --refutes--> v05\n- v14 --refutes--> v0e\n\n[VERIFIED ROOT]\nv19 = 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":"948bc440cc0a3269d6f4de4cc57efbbe5845e4750b269a25e2db2dc914aab576","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4d4d494a1a40fab57092b4b89cf32764ae21a3bf5e9124897c75a2d8e0c0de6","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v19\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-41a3293d3fef2eb3e9f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v16 --supports--> v18\n- v04 --refutes--> v15\n- v1c --refutes--> v01\n- v10 --supports--> v13\n- v1a --refutes--> v09\n- v1a --refutes--> v04\n- v0a --supports--> v02\n- v1d --refutes--> v03\n- v04 --supports--> v0c\n- v14 --refutes--> v1d\n- v04 --refutes--> v05\n- v0a --supports--> v12\n- v13 --refutes--> v1c\n- v19 --refutes--> v1a\n- v0d --supports--> v10\n- v16 --refutes--> v17\n- v0b --supports--> v0d\n- v02 --supports--> v0f\n- v0f --supports--> v0b\n- v18 --refutes--> v0e\n- v05 --refutes--> v0a\n- v1d --refutes--> v1b\n- v0b --refutes--> v1e\n- v03 --refutes--> v06\n- v00 --supports--> v11\n- v14 --supports--> v07\n- v18 --refutes--> v00\n- v1d --supports--> v08\n- v06 --refutes--> v19\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e038c8f97db92ebf903455bfd6589b66cf2722d972072e4920b0a12ad1bbfbdb","schema":"livegraph.boolean-propagation.v1","spec_hash":"41a3293d3fef2eb3e9f4cc3f763cad0b65174e920d94dd0b920cd3b54529e5e7","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"}],\"hop\":20,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-27e66f9021ef50117a6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v16 --supports--> v1e\n- v09 --supports--> v16\n- v04 --supports--> v03\n- v0e --supports--> v1d\n- v1e --refutes--> v13\n- v17 --refutes--> v07\n- v16 --refutes--> v1c\n- v08 --refutes--> v0b\n- v0c --supports--> v1a\n- v02 --refutes--> v04\n- v07 --refutes--> v0f\n- v1e --refutes--> v19\n- v14 --supports--> v0e\n- v11 --supports--> v1b\n- v02 --refutes--> v06\n- v09 --refutes--> v15\n- v07 --supports--> v14\n- v14 --supports--> v0a\n- v05 --refutes--> v09\n- v12 --supports--> v02\n- v09 --refutes--> v10\n- v0b --refutes--> v05\n- v1d --supports--> v0c\n- v08 --supports--> v18\n- v0c --refutes--> v11\n- v00 --supports--> v17\n- v19 --supports--> v00\n- v12 --supports--> v01\n- v1b --refutes--> v12\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"743acc5408fbb3324b2f78f6464417f94e7e1e2f0bb3f1cf93ec88912cf42f9f","schema":"livegraph.boolean-propagation.v1","spec_hash":"27e66f9021ef50117a6ce4f030419b7fb127b6a193d11848fa3def246f312166","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-8a4993dc7b14872ef111","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v1e --supports--> v17\n- v02 --refutes--> v04\n- v1a --refutes--> v1d\n- v03 --refutes--> v1a\n- v05 --refutes--> v12\n- v1d --refutes--> v09\n- v13 --supports--> v16\n- v08 --refutes--> v05\n- v05 --refutes--> v1c\n- v08 --supports--> v18\n- v04 --supports--> v11\n- v0b --supports--> v01\n- v1e --refutes--> v14\n- v12 --refutes--> v19\n- v03 --refutes--> v0f\n- v01 --supports--> v0d\n- v1d --supports--> v10\n- v1b --refutes--> v02\n- v15 --supports--> v06\n- v19 --refutes--> v0c\n- v16 --refutes--> v03\n- v01 --refutes--> v0a\n- v0d --supports--> v13\n- v14 --refutes--> v07\n- v09 --supports--> v1b\n- v16 --refutes--> v00\n- v15 --refutes--> v0b\n- v16 --refutes--> v0e\n- v14 --supports--> v08\n\n[VERIFIED ROOT]\nv1e = 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":"427d26e927f21b52abd5e63649245dad826ec20a286a1e086191b731aad495c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"8a4993dc7b14872ef1117a5a3104d6ef1804b71a59c1be99de9c84822d70e854","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v1e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-57d8b2cfd26620ced8db","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v10\n- v03 --supports--> v0a\n- v0f --refutes--> v1d\n- v01 --supports--> v0b\n- v04 --supports--> v09\n- v0b --refutes--> v18\n- v0c --supports--> v1b\n- v04 --refutes--> v14\n- v19 --refutes--> v13\n- v07 --supports--> v03\n- v11 --refutes--> v06\n- v0b --supports--> v12\n- v0f --supports--> v00\n- v15 --refutes--> v07\n- v03 --refutes--> v17\n- v04 --refutes--> v0d\n- v0c --supports--> v02\n- v0c --refutes--> v0f\n- v17 --supports--> v1a\n- v06 --refutes--> v1c\n- v17 --supports--> v0e\n- v13 --supports--> v08\n- v1c --refutes--> v01\n- v1e --supports--> v15\n- v11 --refutes--> v16\n- v08 --refutes--> v1e\n- v09 --refutes--> v11\n- v0c --refutes--> v05\n- v0e --refutes--> v0c\n- v00 --refutes--> v04\n\n[VERIFIED ROOT]\nv19 = 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":"feac9c762b22689ebce21008f14c4552530ec46b4f29526b89bdfff7cd7391da","schema":"livegraph.boolean-propagation.v1","spec_hash":"57d8b2cfd26620ced8db96933f37304a3a9b82f56c120a5ba279d9906748903c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-4575abc262f98ee7d0f5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v04 --supports--> v10\n- v05 --supports--> v16\n- v0d --refutes--> v0a\n- v16 --supports--> v1d\n- v17 --supports--> v02\n- v02 --supports--> v03\n- v16 --supports--> v09\n- v18 --supports--> v08\n- v1e --supports--> v1b\n- v0e --refutes--> v05\n- v18 --refutes--> v0e\n- v12 --supports--> v13\n- v01 --refutes--> v00\n- v14 --refutes--> v06\n- v13 --refutes--> v04\n- v04 --supports--> v1e\n- v01 --supports--> v14\n- v1b --refutes--> v11\n- v0f --refutes--> v12\n- v03 --refutes--> v01\n- v1d --supports--> v1a\n- v18 --supports--> v0b\n- v0c --refutes--> v1c\n- v0c --refutes--> v18\n- v12 --refutes--> v07\n- v1b --supports--> v19\n- v1b --supports--> v0c\n- v1a --refutes--> v17\n- v14 --supports--> v0d\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1e053495146755a1a20858329719af08a491a0c2a15d5bece3d4fa4f081de7d","schema":"livegraph.boolean-propagation.v1","spec_hash":"4575abc262f98ee7d0f528ccad1d31964cb18ad92d6c69e60c88ade5c3440a51","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-8e9cb7ffc0e2095b7794","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v07 --refutes--> v14\n- v1c --refutes--> v0f\n- v03 --supports--> v08\n- v00 --supports--> v03\n- v05 --supports--> v19\n- v0c --refutes--> v04\n- v0b --supports--> v01\n- v04 --refutes--> v1e\n- v13 --supports--> v1a\n- v03 --refutes--> v10\n- v1a --supports--> v1b\n- v17 --supports--> v15\n- v1e --refutes--> v12\n- v11 --refutes--> v0a\n- v14 --refutes--> v0b\n- v0d --supports--> v1c\n- v0d --refutes--> v02\n- v1e --refutes--> v07\n- v11 --refutes--> v16\n- v00 --refutes--> v1d\n- v1c --supports--> v06\n- v0a --refutes--> v0d\n- v08 --supports--> v11\n- v1a --supports--> v00\n- v15 --supports--> v13\n- v09 --refutes--> v0e\n- v0b --refutes--> v17\n- v05 --refutes--> v09\n- v04 --refutes--> v18\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":"7ec4f76d5acb24140a63b3d97997e26d2fedf5bab217b4b0b04ff6fbf9741534","schema":"livegraph.boolean-propagation.v1","spec_hash":"8e9cb7ffc0e2095b779495a046933877aba4e5ce96390d62310e06f1cc4d7732","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-910489ca3e4d3d6968d8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1a --supports--> v11\n- v1e --refutes--> v01\n- v11 --supports--> v12\n- v07 --refutes--> v19\n- v03 --refutes--> v16\n- v07 --supports--> v02\n- v11 --supports--> v04\n- v0e --refutes--> v09\n- v01 --supports--> v08\n- v03 --supports--> v0c\n- v12 --refutes--> v06\n- v05 --supports--> v00\n- v1a --supports--> v18\n- v11 --supports--> v0a\n- v01 --refutes--> v10\n- v0b --supports--> v1e\n- v17 --refutes--> v05\n- v13 --refutes--> v0e\n- v00 --refutes--> v07\n- v02 --supports--> v1b\n- v12 --supports--> v15\n- v0d --supports--> v13\n- v0f --supports--> v1a\n- v11 --refutes--> v14\n- v09 --refutes--> v0b\n- v08 --supports--> v0f\n- v15 --supports--> v1d\n- v0c --refutes--> v0d\n- v11 --refutes--> v1c\n\n[VERIFIED ROOT]\nv17 = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c415fb3bb7f8ec2e9206fe4dacf413b50f76734d9161161679d949fe8ec58cbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"910489ca3e4d3d6968d8af5ece8b885b23aaf38ea13f3ec030bd69d474499a35","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":20,"item_id":"lggrpo1-11ae82b83506a68352aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0f\n- v10 --supports--> v02\n- v11 --refutes--> v1d\n- v19 --supports--> v01\n- v18 --refutes--> v19\n- v07 --supports--> v0a\n- v08 --supports--> v1e\n- v00 --refutes--> v10\n- v1d --supports--> v07\n- v19 --refutes--> v05\n- v09 --supports--> v16\n- v0d --refutes--> v15\n- v17 --refutes--> v04\n- v1b --refutes--> v13\n- v16 --refutes--> v0d\n- v10 --supports--> v1b\n- v17 --supports--> v06\n- v04 --supports--> v1c\n- v0b --supports--> v09\n- v19 --refutes--> v0c\n- v13 --supports--> v12\n- v15 --supports--> v18\n- v08 --supports--> v03\n- v05 --supports--> v08\n- v19 --refutes--> v1a\n- v13 --supports--> v17\n- v1e --supports--> v00\n- v05 --refutes--> v0e\n- v0a --refutes--> v0b\n- v0a --supports--> v14\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bc815d286f6dd7f83372fc12236463b285f7498a9fcbe3175f8814ad2a103699","schema":"livegraph.boolean-propagation.v1","spec_hash":"11ae82b83506a68352aa05af4cb5dbe74acf6e2de34a7ec25dae53e7b4dc89cd","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":20,"item_id":"lggrpo1-ca25405273fe7b9ad0d9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v04\n- v04 --refutes--> v0b\n- v13 --refutes--> v07\n- v01 --supports--> v0c\n- v1a --supports--> v13\n- v0f --supports--> v1c\n- v16 --refutes--> v12\n- v01 --supports--> v1a\n- v15 --supports--> v08\n- v19 --supports--> v11\n- v11 --refutes--> v10\n- v14 --refutes--> v18\n- v08 --supports--> v1e\n- v0f --refutes--> v0a\n- v15 --supports--> v1b\n- v06 --refutes--> v00\n- v1e --refutes--> v17\n- v17 --supports--> v02\n- v0a --supports--> v01\n- v12 --supports--> v15\n- v10 --refutes--> v09\n- v09 --refutes--> v0f\n- v14 --supports--> v03\n- v0f --refutes--> v0e\n- v04 --refutes--> v05\n- v13 --supports--> v1d\n- v06 --refutes--> v19\n- v17 --supports--> v14\n- v05 --refutes--> v06\n- v13 --supports--> v0d\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8df35a2e13af5f316d7b7217a8f7465c799aa742564b887fcf50853e89f1c71f","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca25405273fe7b9ad0d9962910a10f9d70533871cce7b423f63aa79dbf83d068","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"}],\"hop\":20,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-956a08cde3391aac9735","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v22 --refutes--> v11\n- v17 --refutes--> v12\n- v1a --refutes--> v21\n- v24 --supports--> v07\n- v12 --refutes--> v03\n- v18 --refutes--> v0e\n- v19 --supports--> v1e\n- v1e --refutes--> v05\n- v01 --refutes--> v02\n- v00 --supports--> v1b\n- v1e --supports--> v24\n- v0d --supports--> v09\n- v1c --supports--> v10\n- v03 --supports--> v22\n- v21 --supports--> v17\n- v10 --supports--> v1a\n- v14 --supports--> v0f\n- v18 --refutes--> v0c\n- v11 --refutes--> v1d\n- v19 --supports--> v1f\n- v24 --refutes--> v0d\n- v20 --refutes--> v15\n- v0d --refutes--> v00\n- v16 --refutes--> v1c\n- v1d --refutes--> v0b\n- v18 --supports--> v19\n- v15 --supports--> v06\n- v02 --supports--> v16\n- v16 --refutes--> v0a\n- v14 --refutes--> v13\n- v00 --supports--> v01\n- v1d --refutes--> v08\n- v03 --supports--> v04\n- v0f --refutes--> v18\n- v23 --refutes--> v20\n- v1d --supports--> v23\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5d4fe115a6dc4c21c89c94f5f17c56fc26f8c17052bf56d0b3d05f9ebcc25142","schema":"livegraph.boolean-propagation.v1","spec_hash":"956a08cde3391aac97354940252b7191c7fc190fd3f1131939d817170187174f","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-61d0997d201d44f6b38b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v1b\n- v19 --refutes--> v0b\n- v0d --supports--> v18\n- v17 --refutes--> v0f\n- v0c --refutes--> v13\n- v18 --refutes--> v09\n- v20 --supports--> v12\n- v01 --refutes--> v20\n- v23 --supports--> v1e\n- v21 --supports--> v03\n- v12 --supports--> v0e\n- v15 --refutes--> v0c\n- v07 --refutes--> v14\n- v07 --refutes--> v01\n- v1f --supports--> v15\n- v21 --refutes--> v07\n- v0a --refutes--> v1f\n- v24 --refutes--> v11\n- v12 --supports--> v05\n- v13 --supports--> v1d\n- v0c --supports--> v06\n- v11 --supports--> v1c\n- v0b --refutes--> v10\n- v05 --supports--> v0d\n- v1a --refutes--> v17\n- v22 --supports--> v0a\n- v10 --refutes--> v21\n- v0d --supports--> v08\n- v05 --refutes--> v00\n- v1d --refutes--> v1a\n- v18 --refutes--> v23\n- v1f --refutes--> v02\n- v1e --refutes--> v24\n- v11 --refutes--> v22\n- v12 --refutes--> v16\n- v1a --supports--> v04\n\n[VERIFIED ROOT]\nv19 = 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":"a8eb48085b51ea4b82e5fd6b2daae5978e9093c8339e434e7d39c6f6bfc5267d","schema":"livegraph.boolean-propagation.v1","spec_hash":"61d0997d201d44f6b38b602448a8dfc271450afef7e3b24421ecf3c11ad27b7e","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-a0b3fe3043fe4a123910","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v11 --refutes--> v22\n- v06 --refutes--> v1e\n- v14 --supports--> v06\n- v19 --refutes--> v04\n- v17 --supports--> v24\n- v07 --refutes--> v12\n- v08 --refutes--> v02\n- v24 --refutes--> v21\n- v06 --refutes--> v0d\n- v06 --supports--> v0a\n- v10 --supports--> v14\n- v03 --refutes--> v1a\n- v06 --refutes--> v16\n- v1d --supports--> v1b\n- v1f --refutes--> v07\n- v21 --supports--> v05\n- v21 --supports--> v18\n- v19 --refutes--> v0c\n- v21 --refutes--> v1c\n- v01 --supports--> v23\n- v1b --supports--> v0b\n- v18 --refutes--> v09\n- v09 --supports--> v1f\n- v01 --refutes--> v00\n- v15 --refutes--> v03\n- v07 --supports--> v10\n- v0f --supports--> v19\n- v02 --supports--> v11\n- v03 --supports--> v08\n- v1b --refutes--> v17\n- v0f --refutes--> v0e\n- v13 --refutes--> v0f\n- v22 --supports--> v01\n- v21 --refutes--> v20\n- v04 --supports--> v1d\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":"faae7e05914f7459eeeafa903ddb6f9326486782a4d3a04c93ad6fff33bd2fc2","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0b3fe3043fe4a12391077a5fc14fe3dd53e5655d938120eea15b0a796d69b3b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-243ea539cccc5d72d6ae","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --supports--> v10\n- v1f --supports--> v11\n- v01 --refutes--> v22\n- v23 --supports--> v0d\n- v04 --supports--> v12\n- v14 --supports--> v15\n- v10 --refutes--> v18\n- v0b --refutes--> v1a\n- v0a --supports--> v0c\n- v1e --supports--> v20\n- v22 --supports--> v1d\n- v04 --refutes--> v09\n- v1b --refutes--> v07\n- v17 --refutes--> v13\n- v15 --refutes--> v0a\n- v21 --supports--> v1e\n- v0c --supports--> v05\n- v20 --supports--> v24\n- v20 --supports--> v02\n- v18 --supports--> v16\n- v0d --refutes--> v21\n- v14 --supports--> v0f\n- v20 --refutes--> v0b\n- v0d --supports--> v08\n- v21 --supports--> v00\n- v0d --supports--> v06\n- v1d --supports--> v17\n- v0b --supports--> v1b\n- v0c --supports--> v23\n- v1c --supports--> v01\n- v13 --supports--> v1f\n- v0e --supports--> v03\n- v16 --supports--> v1c\n- v1f --refutes--> v04\n- v20 --supports--> v19\n- v0e --refutes--> v14\n\n[VERIFIED ROOT]\nv0e = 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":"538e8fe24a15a25437b7012c988c407473ec60fbc1ff131330425abd98475092","schema":"livegraph.boolean-propagation.v1","spec_hash":"243ea539cccc5d72d6ae76653e9e4dcf0b117c1749c2cf4bfe5329ff8c30520e","spec_json":"{\"edges\":[{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-85f74b0ea59ca561671f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v01\n- v00 --refutes--> v17\n- v22 --refutes--> v14\n- v01 --supports--> v0a\n- v0e --supports--> v20\n- v03 --refutes--> v02\n- v20 --supports--> v08\n- v19 --refutes--> v1c\n- v1b --supports--> v13\n- v12 --supports--> v1a\n- v06 --refutes--> v1b\n- v23 --supports--> v0d\n- v03 --supports--> v0e\n- v22 --supports--> v18\n- v1c --supports--> v03\n- v13 --refutes--> v15\n- v17 --refutes--> v06\n- v1d --refutes--> v12\n- v03 --supports--> v24\n- v14 --supports--> v21\n- v0a --supports--> v0c\n- v0c --supports--> v16\n- v23 --refutes--> v0b\n- v12 --supports--> v1e\n- v0f --supports--> v22\n- v0f --supports--> v1f\n- v16 --refutes--> v00\n- v0f --supports--> v09\n- v08 --refutes--> v1d\n- v17 --refutes--> v07\n- v11 --supports--> v19\n- v06 --refutes--> v04\n- v13 --refutes--> v0f\n- v1a --supports--> v23\n- v23 --supports--> v10\n- v0a --supports--> v05\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34cb47fcd0da7cb61b136fdec28652e2a463e9b8ace8e26823ba54df9b23f817","schema":"livegraph.boolean-propagation.v1","spec_hash":"85f74b0ea59ca561671fb63d68e850835e3698426592168a351c293e08bcb14f","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-f002bd14fef81e583b5d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1a\n- v20 --supports--> v04\n- v01 --supports--> v16\n- v24 --refutes--> v17\n- v1e --refutes--> v11\n- v16 --refutes--> v0f\n- v24 --supports--> v02\n- v0a --supports--> v13\n- v1d --supports--> v05\n- v0c --supports--> v21\n- v08 --refutes--> v1b\n- v0d --refutes--> v19\n- v23 --refutes--> v1d\n- v1e --refutes--> v12\n- v1e --supports--> v22\n- v0a --supports--> v1e\n- v19 --supports--> v01\n- v1d --refutes--> v15\n- v1b --supports--> v06\n- v0e --refutes--> v0d\n- v1f --refutes--> v1c\n- v11 --supports--> v10\n- v05 --supports--> v1f\n- v0b --supports--> v23\n- v03 --supports--> v24\n- v09 --supports--> v08\n- v02 --supports--> v18\n- v04 --refutes--> v0a\n- v1f --supports--> v00\n- v1f --refutes--> v07\n- v21 --supports--> v09\n- v0f --refutes--> v20\n- v1f --refutes--> v0c\n- v0e --supports--> v14\n- v02 --refutes--> v0e\n- v1b --supports--> v03\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv10\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e26ff3e20b1cbc954cc7b99f01164f6fffec197cbe33546fa597319e38390438","schema":"livegraph.boolean-propagation.v1","spec_hash":"f002bd14fef81e583b5d9ba62cf522d3c8475f574e09de65766a6b4029270ad5","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-358e3aae35bad4999c11","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v10 --supports--> v06\n- v08 --refutes--> v13\n- v01 --refutes--> v07\n- v1f --supports--> v1a\n- v11 --refutes--> v18\n- v14 --refutes--> v05\n- v18 --refutes--> v14\n- v08 --refutes--> v0e\n- v10 --supports--> v01\n- v14 --refutes--> v0c\n- v13 --refutes--> v23\n- v19 --refutes--> v0a\n- v11 --supports--> v03\n- v0c --refutes--> v0f\n- v17 --refutes--> v1c\n- v1e --refutes--> v1f\n- v20 --refutes--> v1b\n- v13 --refutes--> v1e\n- v1b --refutes--> v10\n- v13 --supports--> v21\n- v13 --refutes--> v16\n- v09 --refutes--> v0b\n- v02 --refutes--> v24\n- v15 --supports--> v17\n- v0d --supports--> v00\n- v0d --supports--> v09\n- v1f --supports--> v04\n- v04 --refutes--> v11\n- v1c --refutes--> v22\n- v01 --refutes--> v08\n- v1c --supports--> v02\n- v14 --refutes--> v1d\n- v0b --refutes--> v19\n- v12 --supports--> v0d\n- v24 --supports--> v12\n\n[VERIFIED ROOT]\nv15 = 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":"2ca680f7b6a0973de505446959091447cc555983bce5e512e2ca5ca539209437","schema":"livegraph.boolean-propagation.v1","spec_hash":"358e3aae35bad4999c110e16cdc9eacbee4ffabc66d944c670b1c3064339a0cb","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-95f5dfb1c4b0dbc3ccab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1f\n- v13 --refutes--> v15\n- v13 --refutes--> v07\n- v08 --refutes--> v1a\n- v17 --supports--> v0c\n- v17 --supports--> v1b\n- v0e --supports--> v22\n- v16 --supports--> v09\n- v02 --supports--> v0d\n- v1a --supports--> v1e\n- v21 --supports--> v24\n- v04 --refutes--> v14\n- v0e --supports--> v18\n- v09 --supports--> v1d\n- v0b --supports--> v00\n- v06 --refutes--> v01\n- v1e --refutes--> v0b\n- v1c --refutes--> v02\n- v12 --supports--> v13\n- v19 --refutes--> v16\n- v0b --supports--> v06\n- v0c --refutes--> v04\n- v13 --supports--> v20\n- v13 --supports--> v10\n- v0f --refutes--> v17\n- v1c --refutes--> v11\n- v13 --refutes--> v0e\n- v23 --refutes--> v1c\n- v0d --supports--> v21\n- v13 --supports--> v05\n- v1f --supports--> v0f\n- v0f --refutes--> v0a\n- v1d --supports--> v08\n- v14 --supports--> v12\n- v1e --supports--> v03\n- v21 --supports--> v19\n\n[VERIFIED ROOT]\nv23 = TRUE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b03adfb02ae1e8faf963fcd7ca8e092836c6bc9604f601514820c9e09c052c22","schema":"livegraph.boolean-propagation.v1","spec_hash":"95f5dfb1c4b0dbc3ccab0688d3d66159df3bee5a97f6b9f6b8a80aa145fcd357","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-ecd50e236591ccdce950","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v05 --refutes--> v07\n- v08 --refutes--> v05\n- v05 --refutes--> v10\n- v00 --refutes--> v0f\n- v24 --supports--> v08\n- v08 --supports--> v02\n- v16 --supports--> v23\n- v0e --refutes--> v0c\n- v24 --supports--> v0b\n- v1b --supports--> v11\n- v00 --supports--> v09\n- v1e --supports--> v17\n- v1a --refutes--> v01\n- v12 --refutes--> v00\n- v23 --supports--> v12\n- v03 --supports--> v1a\n- v13 --refutes--> v1b\n- v11 --supports--> v0a\n- v1a --supports--> v1e\n- v15 --supports--> v04\n- v0c --supports--> v24\n- v17 --refutes--> v06\n- v1e --refutes--> v19\n- v14 --refutes--> v1c\n- v00 --refutes--> v1d\n- v18 --refutes--> v21\n- v20 --refutes--> v13\n- v06 --refutes--> v16\n- v1d --refutes--> v15\n- v10 --refutes--> v18\n- v15 --supports--> v0d\n- v1c --refutes--> v20\n- v15 --supports--> v0e\n- v1c --refutes--> v22\n- v11 --refutes--> v03\n- v13 --supports--> v1f\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"61d2ec85a82f1b41e3dab22cc82eaf50faf4d70b8e945e236fdcc611b53f6ca8","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecd50e236591ccdce9509a0bd96b4c9b16ab62eb4dfe56de82075475f14137f5","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-5e412f03143abac914ce","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v22\n- v07 --refutes--> v0a\n- v0d --supports--> v16\n- v10 --refutes--> v1f\n- v13 --supports--> v0c\n- v1d --refutes--> v0f\n- v03 --refutes--> v1c\n- v17 --refutes--> v19\n- v1b --refutes--> v13\n- v1f --supports--> v17\n- v0c --supports--> v03\n- v02 --refutes--> v0b\n- v18 --refutes--> v1b\n- v11 --supports--> v00\n- v04 --refutes--> v1e\n- v0b --refutes--> v04\n- v20 --refutes--> v11\n- v15 --refutes--> v1d\n- v1e --refutes--> v21\n- v11 --refutes--> v10\n- v0c --refutes--> v08\n- v0a --supports--> v1a\n- v09 --supports--> v23\n- v1b --refutes--> v01\n- v21 --supports--> v05\n- v03 --supports--> v24\n- v06 --refutes--> v15\n- v17 --refutes--> v07\n- v0a --refutes--> v09\n- v04 --supports--> v0e\n- v24 --supports--> v0d\n- v0f --supports--> v18\n- v15 --supports--> v14\n- v0d --supports--> v02\n- v17 --supports--> v12\n- v09 --supports--> v06\n\n[VERIFIED ROOT]\nv20 = 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":"38139f0795df1b144662ba952b5c9db3376fa5daf8089831718f2ce1c44300ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e412f03143abac914ce9801ff1a7806938ac2d7cf5bb65967c2a5fd56545f95","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v20\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-b45b888fb0d794ef0cf0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --supports--> v05\n- v1b --refutes--> v13\n- v0d --refutes--> v14\n- v0b --refutes--> v02\n- v17 --supports--> v08\n- v14 --refutes--> v0e\n- v12 --supports--> v19\n- v1c --refutes--> v21\n- v1b --refutes--> v0b\n- v19 --supports--> v01\n- v0e --supports--> v09\n- v20 --supports--> v1e\n- v08 --supports--> v0f\n- v0d --refutes--> v1f\n- v02 --supports--> v0a\n- v05 --refutes--> v11\n- v24 --supports--> v1a\n- v02 --refutes--> v15\n- v08 --supports--> v00\n- v22 --supports--> v1b\n- v15 --refutes--> v17\n- v0e --supports--> v22\n- v10 --refutes--> v0d\n- v16 --refutes--> v10\n- v01 --refutes--> v07\n- v1e --refutes--> v04\n- v11 --supports--> v24\n- v00 --supports--> v12\n- v04 --refutes--> v16\n- v05 --supports--> v1d\n- v19 --supports--> v18\n- v07 --refutes--> v1c\n- v17 --refutes--> v23\n- v14 --supports--> v06\n- v1c --refutes--> v03\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv20 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f24dea7c06ca58b5857ae65edc527958ff614926343803ba341da9c58bc59f14","schema":"livegraph.boolean-propagation.v1","spec_hash":"b45b888fb0d794ef0cf0cccaf94b7f277fbb19eb506031ef150dd40d44c441b8","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-944ec2b2e54398aed403","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v0e\n- v1e --refutes--> v10\n- v13 --refutes--> v1d\n- v15 --supports--> v22\n- v19 --refutes--> v21\n- v20 --refutes--> v18\n- v0a --supports--> v01\n- v0a --supports--> v04\n- v16 --supports--> v19\n- v14 --supports--> v09\n- v0b --refutes--> v1e\n- v19 --refutes--> v1a\n- v08 --refutes--> v23\n- v1c --supports--> v07\n- v13 --supports--> v11\n- v17 --refutes--> v1c\n- v1b --refutes--> v13\n- v18 --supports--> v0f\n- v20 --refutes--> v0c\n- v18 --refutes--> v24\n- v17 --supports--> v02\n- v09 --refutes--> v0a\n- v01 --refutes--> v16\n- v13 --supports--> v08\n- v23 --supports--> v12\n- v03 --refutes--> v20\n- v0d --supports--> v0b\n- v00 --refutes--> v14\n- v0f --supports--> v0d\n- v00 --supports--> v06\n- v22 --refutes--> v03\n- v1a --supports--> v15\n- v1c --refutes--> v00\n- v1b --supports--> v1f\n- v08 --refutes--> v05\n- v12 --refutes--> v17\n\n[VERIFIED ROOT]\nv1b = 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":"86bd1c41d30b62d4398f54dc4f073f0ca5ee6d942c5c0a4cc4e81ef46ed9ecce","schema":"livegraph.boolean-propagation.v1","spec_hash":"944ec2b2e54398aed4037cde1a96000fe44987b743b6aa357f52396a38c05e28","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v1b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-500dcbe5c05707b5d000","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v12\n- v21 --refutes--> v15\n- v18 --supports--> v0f\n- v1e --refutes--> v0b\n- v01 --supports--> v1b\n- v22 --refutes--> v0d\n- v21 --refutes--> v05\n- v13 --refutes--> v19\n- v1c --refutes--> v21\n- v21 --supports--> v04\n- v00 --supports--> v03\n- v14 --refutes--> v22\n- v16 --supports--> v0a\n- v10 --supports--> v1d\n- v23 --supports--> v13\n- v24 --refutes--> v01\n- v0b --supports--> v17\n- v10 --supports--> v1e\n- v03 --refutes--> v16\n- v1b --supports--> v10\n- v07 --refutes--> v02\n- v12 --supports--> v23\n- v0f --refutes--> v00\n- v1b --supports--> v1a\n- v17 --refutes--> v1f\n- v21 --supports--> v0e\n- v19 --supports--> v14\n- v08 --refutes--> v1c\n- v16 --refutes--> v11\n- v09 --refutes--> v08\n- v05 --supports--> v07\n- v1c --refutes--> v06\n- v08 --refutes--> v0c\n- v22 --refutes--> v18\n- v10 --refutes--> v20\n- v07 --refutes--> v24\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":"cee3c52b255d655b1c7c8160ab7557175d215b39e725abb6709a329efa2e4896","schema":"livegraph.boolean-propagation.v1","spec_hash":"500dcbe5c05707b5d00044e400c3aa82bcc7783e2dee6cfacb30c33493509b36","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-7cb2fbd59f4a618f1a72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v10\n- v04 --supports--> v1d\n- v12 --refutes--> v24\n- v16 --supports--> v22\n- v01 --supports--> v00\n- v13 --supports--> v12\n- v01 --refutes--> v0b\n- v09 --supports--> v1a\n- v12 --supports--> v20\n- v1e --supports--> v03\n- v02 --refutes--> v19\n- v20 --refutes--> v02\n- v14 --supports--> v0a\n- v0c --supports--> v05\n- v1e --supports--> v18\n- v00 --supports--> v1e\n- v20 --refutes--> v17\n- v1c --supports--> v0e\n- v1d --refutes--> v1c\n- v09 --refutes--> v15\n- v1f --supports--> v14\n- v0a --supports--> v01\n- v11 --refutes--> v1b\n- v0a --refutes--> v06\n- v0d --supports--> v13\n- v0c --supports--> v04\n- v1c --refutes--> v0d\n- v18 --supports--> v21\n- v1b --supports--> v1f\n- v21 --supports--> v23\n- v10 --supports--> v0c\n- v10 --supports--> v0f\n- v22 --refutes--> v11\n- v12 --refutes--> v07\n- v1c --refutes--> v08\n- v21 --supports--> v09\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"981f276006709559e2078ca791ce600715c1b3d36096d380cf545c767d93a95f","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cb2fbd59f4a618f1a721c7130da20905c05254db7bd70a399b441a70d92ffc8","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v22\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-77525b6b11b3ca467dc5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v1d --refutes--> v19\n- v22 --refutes--> v17\n- v1c --refutes--> v0e\n- v06 --supports--> v02\n- v18 --supports--> v1f\n- v1c --refutes--> v1a\n- v1d --supports--> v08\n- v0a --refutes--> v01\n- v13 --refutes--> v10\n- v0a --refutes--> v16\n- v0b --refutes--> v0c\n- v24 --supports--> v0b\n- v19 --refutes--> v14\n- v17 --supports--> v1e\n- v24 --refutes--> v07\n- v13 --supports--> v21\n- v02 --refutes--> v1c\n- v00 --refutes--> v03\n- v15 --refutes--> v12\n- v23 --refutes--> v22\n- v1a --supports--> v1d\n- v1b --supports--> v20\n- v0b --supports--> v09\n- v03 --supports--> v15\n- v20 --supports--> v24\n- v09 --supports--> v0f\n- v0a --refutes--> v05\n- v21 --supports--> v06\n- v13 --supports--> v04\n- v18 --refutes--> v13\n- v0a --refutes--> v18\n- v19 --refutes--> v0d\n- v0d --supports--> v23\n- v0f --refutes--> v0a\n- v17 --refutes--> v00\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e2d55426da9fad4c5c12b1778f02e50a58017eac23eb2520320f0da34e8fe58","schema":"livegraph.boolean-propagation.v1","spec_hash":"77525b6b11b3ca467dc5ee50a58ee0ba1180c84a78004d73c30d7a18fc7882a0","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-19a889a88071d5cf006d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v14 --refutes--> v24\n- v01 --supports--> v06\n- v1f --supports--> v00\n- v09 --refutes--> v18\n- v06 --refutes--> v0c\n- v11 --supports--> v21\n- v1a --refutes--> v19\n- v10 --refutes--> v02\n- v16 --supports--> v12\n- v1c --refutes--> v0f\n- v1e --refutes--> v1a\n- v1d --supports--> v09\n- v09 --refutes--> v22\n- v00 --refutes--> v23\n- v13 --supports--> v01\n- v0a --supports--> v1d\n- v10 --supports--> v11\n- v23 --refutes--> v17\n- v01 --supports--> v20\n- v00 --supports--> v08\n- v11 --refutes--> v0a\n- v15 --refutes--> v13\n- v07 --refutes--> v16\n- v14 --supports--> v0e\n- v15 --supports--> v0b\n- v01 --supports--> v03\n- v1b --supports--> v15\n- v0f --refutes--> v14\n- v06 --supports--> v1e\n- v22 --refutes--> v1c\n- v10 --supports--> v0d\n- v05 --refutes--> v07\n- v14 --refutes--> v05\n- v19 --supports--> v1f\n- v12 --supports--> v1b\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d18c89b05a8ed186c3730f1335dade74a2c6ef58aa3768449c0cab13c86b7a92","schema":"livegraph.boolean-propagation.v1","spec_hash":"19a889a88071d5cf006db19c9ba86b616bc404320125558d8ca35450d4e51e02","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-a18246bc0ce58d1d2cd9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v0f\n- v0f --supports--> v0d\n- v19 --refutes--> v11\n- v0a --refutes--> v0c\n- v21 --refutes--> v19\n- v22 --supports--> v03\n- v17 --supports--> v18\n- v1c --supports--> v05\n- v05 --supports--> v1d\n- v1b --refutes--> v13\n- v11 --supports--> v20\n- v1c --refutes--> v06\n- v03 --refutes--> v1c\n- v08 --supports--> v04\n- v02 --refutes--> v07\n- v04 --refutes--> v17\n- v1d --refutes--> v14\n- v09 --refutes--> v0b\n- v24 --refutes--> v23\n- v00 --refutes--> v22\n- v1b --refutes--> v0a\n- v17 --supports--> v0e\n- v23 --supports--> v00\n- v19 --refutes--> v12\n- v20 --refutes--> v1a\n- v0a --refutes--> v01\n- v14 --refutes--> v02\n- v15 --supports--> v08\n- v20 --supports--> v10\n- v1d --supports--> v16\n- v09 --refutes--> v1b\n- v20 --supports--> v1e\n- v0d --supports--> v21\n- v0a --supports--> v15\n- v05 --refutes--> v1f\n- v10 --supports--> v09\n\n[VERIFIED ROOT]\nv24 = 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":"3624468d1fecbe81eacaf4f479499b72a0e6cfe0e706af24e392a9653f51b50e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a18246bc0ce58d1d2cd9c6b0b3aa83c254a7f183b8e3ff96232b2f48accc6c34","spec_json":"{\"edges\":[{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v24\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-dc1251b0242ebd379206","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v18\n- v12 --supports--> v0d\n- v23 --supports--> v04\n- v20 --refutes--> v23\n- v0f --refutes--> v07\n- v12 --supports--> v11\n- v09 --refutes--> v17\n- v12 --supports--> v0b\n- v24 --refutes--> v0a\n- v20 --supports--> v1a\n- v1e --supports--> v00\n- v06 --supports--> v08\n- v0c --refutes--> v21\n- v1b --refutes--> v0c\n- v05 --supports--> v19\n- v22 --supports--> v15\n- v1e --refutes--> v0f\n- v18 --supports--> v1b\n- v0e --supports--> v13\n- v17 --supports--> v1e\n- v0e --supports--> v1f\n- v1f --refutes--> v1c\n- v0e --refutes--> v16\n- v1c --supports--> v06\n- v0a --supports--> v0e\n- v0d --refutes--> v02\n- v03 --supports--> v09\n- v07 --supports--> v12\n- v0d --supports--> v24\n- v1c --refutes--> v10\n- v04 --refutes--> v05\n- v06 --supports--> v22\n- v22 --refutes--> v01\n- v24 --supports--> v1d\n- v21 --refutes--> v14\n- v21 --refutes--> v03\n\n[VERIFIED ROOT]\nv20 = 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":"93cf4caf832954c848c390b7476b33871d08fcbb9ba3143b66467b66588e4e71","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc1251b0242ebd379206b82a5b2a60081ec2933802518e1ccd1d4cbed32a5a31","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-b1ffe0aaadb7fd7cde3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v0a\n- v1c --supports--> v15\n- v1c --supports--> v1d\n- v04 --refutes--> v09\n- v14 --supports--> v00\n- v0d --refutes--> v16\n- v1f --supports--> v05\n- v15 --supports--> v22\n- v22 --refutes--> v23\n- v1f --supports--> v14\n- v23 --supports--> v1a\n- v13 --supports--> v1f\n- v14 --supports--> v1c\n- v23 --refutes--> v0f\n- v10 --supports--> v17\n- v1e --supports--> v01\n- v21 --refutes--> v24\n- v1c --refutes--> v06\n- v16 --supports--> v12\n- v09 --supports--> v13\n- v02 --supports--> v08\n- v23 --refutes--> v19\n- v0c --refutes--> v04\n- v1e --refutes--> v0c\n- v08 --refutes--> v0d\n- v17 --supports--> v1e\n- v07 --refutes--> v18\n- v14 --supports--> v03\n- v16 --supports--> v10\n- v11 --refutes--> v20\n- v24 --supports--> v07\n- v11 --supports--> v1b\n- v1b --refutes--> v21\n- v0a --supports--> v11\n- v14 --refutes--> v0e\n- v21 --supports--> v0b\n\n[VERIFIED ROOT]\nv02 = 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":"2c5598abe7fb9c3c19d508e077f21e36018eea5367bb12b27cfc61b5eb7b61c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1ffe0aaadb7fd7cde3d8f01b15fecf67777667ce8460ce3bd93a665e748f5e0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-682e4818a645f01566dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v03 --supports--> v10\n- v05 --supports--> v0d\n- v1c --refutes--> v20\n- v0c --refutes--> v22\n- v24 --supports--> v05\n- v0d --refutes--> v19\n- v1e --refutes--> v13\n- v19 --supports--> v1e\n- v16 --refutes--> v1d\n- v1b --supports--> v1f\n- v1d --refutes--> v06\n- v1f --refutes--> v03\n- v22 --supports--> v09\n- v0f --refutes--> v12\n- v1c --refutes--> v15\n- v12 --supports--> v04\n- v13 --refutes--> v18\n- v21 --supports--> v0c\n- v0c --refutes--> v0e\n- v03 --supports--> v0b\n- v06 --refutes--> v0f\n- v1a --supports--> v01\n- v11 --supports--> v23\n- v13 --supports--> v07\n- v1c --supports--> v11\n- v17 --supports--> v1b\n- v14 --supports--> v24\n- v0b --refutes--> v1c\n- v03 --supports--> v08\n- v1a --refutes--> v02\n- v02 --supports--> v14\n- v13 --supports--> v17\n- v22 --refutes--> v1a\n- v12 --refutes--> v21\n- v05 --refutes--> v00\n- v06 --refutes--> v0a\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5f98c80d7d134a1cf32e75c044888e2a0d5b58f925d1d7565bfe7bc8154b52d9","schema":"livegraph.boolean-propagation.v1","spec_hash":"682e4818a645f01566dd046ed5153fb184c95aa645d1262b6ce8b7bb33e25d19","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-3eff4739b96c28ff0be0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v0e --refutes--> v1b\n- v1a --refutes--> v13\n- v0b --supports--> v11\n- v22 --refutes--> v23\n- v03 --supports--> v10\n- v16 --refutes--> v1e\n- v07 --supports--> v0c\n- v0d --refutes--> v06\n- v0e --supports--> v09\n- v01 --refutes--> v0e\n- v20 --supports--> v12\n- v18 --supports--> v19\n- v18 --supports--> v0b\n- v0e --refutes--> v21\n- v1e --supports--> v07\n- v0a --supports--> v1a\n- v1c --supports--> v03\n- v08 --supports--> v1f\n- v00 --supports--> v1d\n- v0f --refutes--> v1c\n- v12 --refutes--> v05\n- v24 --refutes--> v15\n- v1a --supports--> v20\n- v06 --supports--> v24\n- v1c --refutes--> v14\n- v0b --supports--> v01\n- v03 --refutes--> v18\n- v1f --refutes--> v17\n- v05 --supports--> v16\n- v20 --refutes--> v02\n- v22 --supports--> v08\n- v15 --refutes--> v00\n- v0c --supports--> v0f\n- v00 --supports--> v04\n- v1f --supports--> v0d\n\n[VERIFIED ROOT]\nv22 = 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":"79a3086eb6120b74f688120c5fcf770db24841b37c76c1dc7d22e9ac287df2a1","schema":"livegraph.boolean-propagation.v1","spec_hash":"3eff4739b96c28ff0be0db19cb4dd75322c4029d8eddb3b4a527c7d8c6f94a6a","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v22\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-238480b88d0640fa8dbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --refutes--> v0a\n- v11 --supports--> v23\n- v10 --refutes--> v03\n- v0e --refutes--> v09\n- v08 --supports--> v07\n- v0d --refutes--> v11\n- v0b --refutes--> v08\n- v10 --refutes--> v0d\n- v01 --supports--> v17\n- v07 --refutes--> v1d\n- v07 --refutes--> v19\n- v1c --supports--> v0c\n- v0f --supports--> v1a\n- v05 --supports--> v14\n- v1e --supports--> v1b\n- v00 --supports--> v12\n- v13 --refutes--> v0f\n- v15 --supports--> v13\n- v0e --supports--> v24\n- v00 --supports--> v04\n- v09 --refutes--> v21\n- v01 --refutes--> v1c\n- v00 --supports--> v18\n- v15 --supports--> v06\n- v14 --refutes--> v02\n- v1c --refutes--> v1f\n- v1d --supports--> v01\n- v02 --refutes--> v15\n- v1b --refutes--> v20\n- v15 --supports--> v16\n- v20 --supports--> v05\n- v04 --supports--> v0e\n- v11 --refutes--> v00\n- v21 --supports--> v1e\n- v1f --refutes--> v10\n- v1b --supports--> v22\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6243182d6f65c7a9a3e05a31497afe6efc4ca1d4ca233a75f8d3ab08e3555937","schema":"livegraph.boolean-propagation.v1","spec_hash":"238480b88d0640fa8dbe73895662a6118d590f401b70d6aa4c4ee1afe3134729","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v21\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-5acb96cd24c34e0281b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v21\n- v18 --refutes--> v03\n- v1b --supports--> v16\n- v24 --refutes--> v14\n- v21 --supports--> v1d\n- v02 --supports--> v22\n- v1b --refutes--> v13\n- v05 --supports--> v0f\n- v10 --supports--> v0c\n- v12 --supports--> v17\n- v11 --supports--> v09\n- v19 --refutes--> v24\n- v0e --refutes--> v0b\n- v23 --refutes--> v05\n- v18 --supports--> v1c\n- v00 --supports--> v19\n- v18 --refutes--> v20\n- v0d --refutes--> v08\n- v17 --supports--> v18\n- v04 --refutes--> v23\n- v21 --supports--> v02\n- v06 --supports--> v0d\n- v15 --refutes--> v00\n- v05 --supports--> v10\n- v0e --refutes--> v1f\n- v0a --supports--> v04\n- v1e --supports--> v0a\n- v1b --refutes--> v11\n- v04 --refutes--> v07\n- v10 --refutes--> v01\n- v1a --refutes--> v1b\n- v11 --supports--> v0e\n- v14 --refutes--> v1a\n- v08 --refutes--> v1e\n- v10 --refutes--> v12\n- v02 --refutes--> v15\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv1f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12139a4c2338123604930db500a007657bdc48b298a7cc59b136cecae87c4a41","schema":"livegraph.boolean-propagation.v1","spec_hash":"5acb96cd24c34e0281b0426d99924f49f6229f351923940e819ed920166843b4","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-d1b3f2800275f4da9de2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1d\n- v1c --refutes--> v15\n- v0b --supports--> v02\n- v17 --supports--> v1f\n- v10 --supports--> v1e\n- v18 --refutes--> v05\n- v0b --refutes--> v08\n- v16 --refutes--> v22\n- v02 --refutes--> v17\n- v06 --refutes--> v04\n- v09 --refutes--> v0a\n- v1b --refutes--> v09\n- v01 --supports--> v07\n- v1f --refutes--> v0c\n- v11 --refutes--> v06\n- v15 --refutes--> v11\n- v23 --refutes--> v10\n- v02 --supports--> v14\n- v19 --supports--> v1c\n- v0a --refutes--> v23\n- v1e --supports--> v01\n- v22 --supports--> v1b\n- v1e --refutes--> v0d\n- v17 --refutes--> v03\n- v20 --refutes--> v12\n- v11 --refutes--> v21\n- v1d --supports--> v13\n- v0f --supports--> v1a\n- v0f --refutes--> v0e\n- v07 --supports--> v19\n- v0c --refutes--> v18\n- v0f --supports--> v24\n- v0f --supports--> v20\n- v1d --supports--> v0f\n- v22 --supports--> v00\n- v06 --supports--> v0b\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":"d17ef54acf3957e99fd92fb8e5eb7c92afe5f013c89a7420700d8044b286f730","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1b3f2800275f4da9de2324a8922f97996bc2827b1f3fb61cde70796f9b21cbb","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-6968c67a1248b79f77b8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v12 --supports--> v08\n- v23 --refutes--> v1e\n- v07 --supports--> v13\n- v0c --refutes--> v1c\n- v23 --supports--> v18\n- v04 --supports--> v14\n- v01 --supports--> v1a\n- v1d --supports--> v11\n- v1c --supports--> v04\n- v1f --supports--> v07\n- v1f --supports--> v0f\n- v10 --refutes--> v0d\n- v14 --supports--> v21\n- v05 --refutes--> v0b\n- v0d --supports--> v01\n- v22 --refutes--> v05\n- v17 --supports--> v02\n- v07 --refutes--> v03\n- v11 --refutes--> v20\n- v15 --refutes--> v09\n- v0e --supports--> v1d\n- v11 --supports--> v0a\n- v1c --refutes--> v1b\n- v18 --supports--> v12\n- v14 --refutes--> v1f\n- v11 --refutes--> v22\n- v09 --supports--> v06\n- v12 --supports--> v0c\n- v1a --supports--> v15\n- v24 --refutes--> v10\n- v09 --supports--> v0e\n- v19 --refutes--> v16\n- v17 --refutes--> v24\n- v16 --supports--> v17\n- v11 --refutes--> v00\n- v07 --refutes--> v19\n\n[VERIFIED ROOT]\nv23 = 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":"23812d8a0ca57f12ad810de5761e6b97c0589c518248de2fdf015db63760e306","schema":"livegraph.boolean-propagation.v1","spec_hash":"6968c67a1248b79f77b87b338df2caaa5bfba7e1a8f8569f4e3381d730f39318","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-731b81ad54a4a837f8c4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v11\n- v17 --refutes--> v0c\n- v10 --refutes--> v22\n- v13 --supports--> v1d\n- v20 --supports--> v23\n- v04 --supports--> v05\n- v0c --refutes--> v09\n- v05 --refutes--> v08\n- v13 --refutes--> v16\n- v24 --refutes--> v01\n- v06 --supports--> v17\n- v14 --refutes--> v0d\n- v15 --supports--> v18\n- v15 --supports--> v12\n- v10 --supports--> v1a\n- v10 --supports--> v0e\n- v1a --supports--> v04\n- v15 --supports--> v0b\n- v00 --supports--> v07\n- v1f --supports--> v0a\n- v0c --supports--> v15\n- v1c --refutes--> v10\n- v05 --supports--> v19\n- v0d --refutes--> v1f\n- v24 --supports--> v03\n- v16 --supports--> v1c\n- v12 --refutes--> v14\n- v00 --supports--> v0f\n- v19 --supports--> v20\n- v1e --refutes--> v06\n- v1b --supports--> v00\n- v23 --supports--> v1e\n- v24 --supports--> v13\n- v0a --supports--> v1b\n- v07 --supports--> v21\n- v17 --refutes--> v02\n\n[VERIFIED ROOT]\nv24 = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3cb9e6e8e08be496000ebb802865896e3baf995ded9787f028ac18041fb6de17","schema":"livegraph.boolean-propagation.v1","spec_hash":"731b81ad54a4a837f8c4e7c1e6c4218e87bc4f9723bf7d84508bd6786f1c0fb0","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v24\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-6248fedf4a46a6337229","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v22\n- v0a --supports--> v0c\n- v11 --refutes--> v20\n- v04 --refutes--> v18\n- v20 --refutes--> v1a\n- v03 --refutes--> v19\n- v1e --supports--> v0f\n- v1d --refutes--> v24\n- v02 --refutes--> v16\n- v02 --supports--> v12\n- v03 --refutes--> v1e\n- v13 --refutes--> v21\n- v1b --supports--> v08\n- v1b --refutes--> v0a\n- v18 --supports--> v00\n- v22 --supports--> v10\n- v23 --supports--> v11\n- v1a --supports--> v17\n- v1d --refutes--> v14\n- v05 --refutes--> v0e\n- v10 --supports--> v05\n- v0a --supports--> v09\n- v07 --refutes--> v06\n- v13 --refutes--> v23\n- v1d --supports--> v13\n- v10 --refutes--> v1c\n- v1f --refutes--> v1d\n- v0b --refutes--> v1b\n- v0d --supports--> v03\n- v05 --refutes--> v02\n- v15 --supports--> v0b\n- v18 --supports--> v01\n- v12 --supports--> v0d\n- v06 --refutes--> v1f\n- v1a --refutes--> v04\n- v0a --supports--> v07\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40c3e2da82e1b5133fda8341b19d0a92f027fdad108307038237e7a6353aefcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"6248fedf4a46a6337229467e50fd8d609d0f90d3368c0fed6989450aebd0ee3e","spec_json":"{\"edges\":[{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-d44b1f08084d9333ca51","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0e\n- v16 --supports--> v1a\n- v04 --supports--> v10\n- v24 --supports--> v22\n- v1e --supports--> v0f\n- v10 --supports--> v01\n- v07 --supports--> v09\n- v1a --refutes--> v07\n- v1a --refutes--> v13\n- v21 --supports--> v17\n- v10 --supports--> v0b\n- v24 --refutes--> v21\n- v06 --supports--> v23\n- v23 --supports--> v02\n- v01 --supports--> v15\n- v21 --refutes--> v03\n- v18 --refutes--> v1f\n- v02 --refutes--> v05\n- v0e --supports--> v1c\n- v0a --supports--> v20\n- v1b --refutes--> v08\n- v0c --supports--> v11\n- v15 --refutes--> v1d\n- v0a --refutes--> v14\n- v15 --supports--> v24\n- v11 --supports--> v00\n- v1b --supports--> v04\n- v11 --supports--> v0d\n- v09 --refutes--> v06\n- v11 --supports--> v16\n- v14 --supports--> v1b\n- v05 --supports--> v12\n- v10 --refutes--> v19\n- v0e --supports--> v18\n- v0f --supports--> v0c\n- v18 --refutes--> v1e\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7fca60fbcbaef303fde1bfce7b1265e364ddc0e785f60274d2ea62a58bdf456a","schema":"livegraph.boolean-propagation.v1","spec_hash":"d44b1f08084d9333ca51c61eb7fa2b29ef00d3692143aa1aca440d3cc27d14ca","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-adbedf0674719edd2ae6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v15 --refutes--> v13\n- v1a --supports--> v20\n- v21 --refutes--> v12\n- v1e --refutes--> v0a\n- v03 --refutes--> v07\n- v06 --refutes--> v11\n- v14 --refutes--> v16\n- v20 --refutes--> v03\n- v00 --refutes--> v0d\n- v12 --supports--> v1f\n- v0e --supports--> v01\n- v12 --refutes--> v09\n- v19 --refutes--> v1b\n- v14 --supports--> v0c\n- v00 --refutes--> v18\n- v19 --refutes--> v23\n- v0a --refutes--> v19\n- v11 --refutes--> v24\n- v09 --supports--> v1a\n- v16 --supports--> v1c\n- v24 --refutes--> v1e\n- v08 --refutes--> v21\n- v1d --refutes--> v0e\n- v0f --refutes--> v1d\n- v1c --refutes--> v08\n- v0e --supports--> v0b\n- v14 --refutes--> v22\n- v08 --refutes--> v05\n- v15 --refutes--> v17\n- v03 --refutes--> v15\n- v19 --refutes--> v14\n- v18 --supports--> v0f\n- v02 --supports--> v10\n- v1a --supports--> v04\n- v0b --supports--> v06\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8080a9fc05ff17db26a41652cf06befa91e6ccbc4d5f6159a3976e9e02894f1a","schema":"livegraph.boolean-propagation.v1","spec_hash":"adbedf0674719edd2ae6fe92cb8bc2b5d3962a604816286c5f8809e48f85a777","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-647b2cfd0d8e96c5e975","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --refutes--> v0d\n- v10 --supports--> v0b\n- v18 --refutes--> v08\n- v05 --supports--> v15\n- v0a --refutes--> v0e\n- v12 --refutes--> v03\n- v0e --supports--> v06\n- v16 --supports--> v10\n- v1c --refutes--> v1b\n- v02 --supports--> v12\n- v0c --supports--> v1d\n- v17 --supports--> v18\n- v21 --refutes--> v17\n- v08 --refutes--> v0a\n- v01 --supports--> v05\n- v03 --supports--> v24\n- v14 --supports--> v20\n- v05 --refutes--> v1e\n- v14 --supports--> v16\n- v0a --supports--> v04\n- v10 --refutes--> v19\n- v16 --supports--> v11\n- v10 --supports--> v23\n- v0c --supports--> v14\n- v0a --supports--> v07\n- v1b --supports--> v02\n- v10 --refutes--> v13\n- v15 --supports--> v0c\n- v09 --supports--> v01\n- v1a --supports--> v00\n- v19 --supports--> v1c\n- v0e --supports--> v1f\n- v0e --supports--> v0f\n- v24 --refutes--> v22\n- v00 --supports--> v09\n- v0d --refutes--> v21\n\n[VERIFIED ROOT]\nv1a = 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":"36f8b49126c6b897ba47666e756089f9fec9ac95ef3b32b112fc8ce180f13f13","schema":"livegraph.boolean-propagation.v1","spec_hash":"647b2cfd0d8e96c5e975150b812c9467d9e7b49af1a1a90de7f6bb9d191f1a79","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-d3603797b74c0b312ef6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v06\n- v20 --refutes--> v04\n- v02 --refutes--> v0a\n- v09 --refutes--> v1e\n- v0c --refutes--> v1a\n- v20 --supports--> v21\n- v0e --supports--> v0b\n- v05 --refutes--> v09\n- v0c --refutes--> v15\n- v07 --refutes--> v11\n- v21 --refutes--> v1f\n- v0b --supports--> v1b\n- v1b --supports--> v01\n- v1b --refutes--> v23\n- v20 --supports--> v00\n- v0f --supports--> v03\n- v24 --supports--> v05\n- v14 --supports--> v07\n- v1a --refutes--> v10\n- v18 --supports--> v22\n- v1c --refutes--> v16\n- v03 --refutes--> v14\n- v13 --refutes--> v20\n- v19 --refutes--> v0f\n- v1a --refutes--> v18\n- v16 --refutes--> v08\n- v18 --supports--> v02\n- v13 --refutes--> v12\n- v1f --supports--> v0d\n- v11 --refutes--> v1c\n- v1d --refutes--> v24\n- v16 --refutes--> v1d\n- v23 --supports--> v0c\n- v07 --supports--> v17\n- v1f --supports--> v0e\n- v02 --supports--> v19\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"96a43400b478168211a4e662ccad4856141154bdd1d71cdcc6bc86a1fbb11247","schema":"livegraph.boolean-propagation.v1","spec_hash":"d3603797b74c0b312ef654b796a01ccb3dab681c45feb822dc42ebb047800118","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-f7aae12e3be86b6422f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v0e --refutes--> v18\n- v1d --refutes--> v21\n- v08 --supports--> v1b\n- v1d --refutes--> v24\n- v13 --supports--> v05\n- v10 --refutes--> v0a\n- v17 --refutes--> v0e\n- v0b --refutes--> v02\n- v00 --refutes--> v14\n- v1b --supports--> v10\n- v09 --supports--> v1d\n- v1a --refutes--> v03\n- v0c --supports--> v0d\n- v10 --supports--> v00\n- v14 --refutes--> v11\n- v1d --supports--> v1e\n- v02 --refutes--> v19\n- v0e --supports--> v13\n- v03 --refutes--> v20\n- v05 --refutes--> v12\n- v08 --supports--> v22\n- v0d --supports--> v1f\n- v08 --refutes--> v07\n- v01 --supports--> v04\n- v24 --refutes--> v17\n- v10 --supports--> v23\n- v0d --refutes--> v06\n- v20 --refutes--> v01\n- v1d --supports--> v0f\n- v11 --refutes--> v0b\n- v1c --supports--> v09\n- v19 --supports--> v16\n- v17 --supports--> v15\n- v19 --supports--> v1c\n- v04 --supports--> v08\n\n[VERIFIED ROOT]\nv0c = 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":"151c09567850bcf70c3bde43955069799da0eaa81abdeb0a576e858ab1c2a3a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7aae12e3be86b6422f4a666c9aff9e9ac447792185e503f3fb2a5a9a8f69364","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-aaf183957ab3c81a4413","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v16 --refutes--> v13\n- v17 --supports--> v22\n- v1c --supports--> v16\n- v1f --refutes--> v08\n- v03 --supports--> v18\n- v10 --supports--> v1c\n- v12 --supports--> v1a\n- v0e --refutes--> v17\n- v18 --supports--> v0e\n- v17 --refutes--> v1e\n- v15 --supports--> v19\n- v19 --refutes--> v10\n- v0d --supports--> v21\n- v02 --supports--> v1f\n- v1b --supports--> v07\n- v0e --supports--> v24\n- v11 --refutes--> v02\n- v02 --supports--> v23\n- v1a --supports--> v11\n- v02 --refutes--> v04\n- v0b --refutes--> v05\n- v1c --refutes--> v06\n- v03 --refutes--> v01\n- v0f --supports--> v0c\n- v1f --supports--> v09\n- v0d --supports--> v0b\n- v05 --supports--> v0f\n- v13 --refutes--> v00\n- v1d --supports--> v0d\n- v1e --refutes--> v1b\n- v07 --refutes--> v1d\n- v14 --refutes--> v12\n- v13 --refutes--> v20\n- v13 --refutes--> v03\n- v0f --refutes--> v14\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fa42278673e744320cbee069f0f172b1b358ab550f81e6d6f4a8666ef5d6ed1f","schema":"livegraph.boolean-propagation.v1","spec_hash":"aaf183957ab3c81a44135c627a45ed8a0ef06a2f4cc4b684c2fe7998bd02486c","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1f\"}],\"hop\":24,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-a2eb631784a5de4cb3da","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v05\n- v05 --supports--> v11\n- v1f --refutes--> v01\n- v23 --supports--> v21\n- v15 --refutes--> v20\n- v20 --supports--> v12\n- v14 --supports--> v07\n- v19 --supports--> v06\n- v15 --supports--> v0e\n- v0b --supports--> v1e\n- v1a --supports--> v0f\n- v1a --supports--> v0c\n- v24 --refutes--> v1c\n- v1f --supports--> v09\n- v1a --refutes--> v0a\n- v14 --refutes--> v19\n- v1f --supports--> v08\n- v11 --supports--> v10\n- v04 --refutes--> v23\n- v21 --supports--> v03\n- v0c --supports--> v0d\n- v0b --supports--> v16\n- v05 --refutes--> v13\n- v1c --supports--> v17\n- v06 --refutes--> v1d\n- v1e --supports--> v04\n- v09 --supports--> v18\n- v17 --refutes--> v14\n- v03 --refutes--> v15\n- v1d --refutes--> v02\n- v02 --supports--> v00\n- v17 --refutes--> v22\n- v1f --supports--> v1b\n- v20 --refutes--> v1f\n- v18 --refutes--> v1a\n- v11 --supports--> v24\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":"a8fb5cc113c0c961537e508a057978c653df9fbda9c1f6049b95a89ad9147d24","schema":"livegraph.boolean-propagation.v1","spec_hash":"a2eb631784a5de4cb3da2d7aca41c61632ecf1d8f71a0db7ffcbaedd62ff639b","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-e888203ded504887c003","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --supports--> v02\n- v1c --supports--> v0e\n- v17 --supports--> v1e\n- v19 --refutes--> v07\n- v14 --refutes--> v22\n- v22 --supports--> v0f\n- v0e --refutes--> v06\n- v21 --refutes--> v1b\n- v0d --refutes--> v17\n- v07 --supports--> v0c\n- v15 --supports--> v1f\n- v13 --supports--> v15\n- v1e --supports--> v0b\n- v22 --refutes--> v24\n- v1a --refutes--> v0d\n- v0b --refutes--> v04\n- v03 --supports--> v08\n- v23 --supports--> v16\n- v08 --refutes--> v0a\n- v08 --refutes--> v01\n- v0b --supports--> v1c\n- v1b --refutes--> v19\n- v1a --refutes--> v10\n- v15 --supports--> v11\n- v00 --supports--> v23\n- v13 --refutes--> v05\n- v15 --refutes--> v1d\n- v06 --supports--> v18\n- v09 --refutes--> v21\n- v07 --supports--> v1a\n- v23 --supports--> v13\n- v22 --supports--> v12\n- v01 --refutes--> v09\n- v20 --refutes--> v03\n- v1d --refutes--> v20\n- v06 --refutes--> v14\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2fa16684bae537693c378d61fae5eb8c5187ae9e8a27d06d1163d6e8af25a501","schema":"livegraph.boolean-propagation.v1","spec_hash":"e888203ded504887c0033d20218803d5931c9527cbfb7a3da0b4678e13f20bb3","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-fdb1a5b0d6907bb75b8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v12\n- v24 --refutes--> v16\n- v0f --refutes--> v0e\n- v20 --supports--> v0f\n- v02 --supports--> v09\n- v0d --refutes--> v0c\n- v24 --refutes--> v1d\n- v0a --refutes--> v14\n- v02 --refutes--> v01\n- v03 --refutes--> v00\n- v1a --refutes--> v0b\n- v19 --refutes--> v17\n- v0f --refutes--> v23\n- v24 --supports--> v04\n- v0c --refutes--> v24\n- v11 --refutes--> v1c\n- v1e --refutes--> v22\n- v18 --refutes--> v11\n- v1e --refutes--> v18\n- v05 --refutes--> v0d\n- v06 --refutes--> v05\n- v10 --supports--> v1a\n- v00 --supports--> v0a\n- v02 --supports--> v20\n- v1c --supports--> v03\n- v1b --supports--> v10\n- v17 --refutes--> v02\n- v02 --refutes--> v15\n- v0a --supports--> v07\n- v1f --supports--> v06\n- v0b --refutes--> v08\n- v0b --refutes--> v19\n- v23 --refutes--> v1f\n- v10 --refutes--> v13\n- v1e --supports--> v21\n- v14 --refutes--> v1b\n\n[VERIFIED ROOT]\nv1e = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b40747fe8e3fbafce00b775a61d80bafb3a6a42aeb811c449e1f2df99d4f6ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"fdb1a5b0d6907bb75b8ccfe2e05c346ea0b5e1fba91d1585b0dbf710eba75c81","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-5d2dce3c258dd48d1562","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v23\n- v1e --supports--> v1c\n- v0c --refutes--> v17\n- v18 --supports--> v01\n- v0b --supports--> v22\n- v24 --refutes--> v04\n- v0d --supports--> v14\n- v0f --refutes--> v18\n- v02 --supports--> v24\n- v07 --refutes--> v0b\n- v06 --refutes--> v00\n- v09 --supports--> v05\n- v0e --supports--> v10\n- v1e --supports--> v07\n- v18 --refutes--> v08\n- v12 --supports--> v09\n- v09 --supports--> v19\n- v11 --supports--> v02\n- v1f --supports--> v03\n- v18 --refutes--> v0d\n- v21 --refutes--> v12\n- v00 --refutes--> v21\n- v0b --refutes--> v0e\n- v16 --refutes--> v11\n- v00 --refutes--> v1a\n- v17 --refutes--> v0a\n- v14 --refutes--> v06\n- v0a --refutes--> v20\n- v0c --supports--> v13\n- v21 --supports--> v15\n- v05 --refutes--> v1e\n- v1d --refutes--> v1f\n- v1f --refutes--> v16\n- v05 --supports--> v1b\n- v20 --refutes--> v0f\n- v24 --supports--> v0c\n\n[VERIFIED ROOT]\nv1d = 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":"8e4b33706d61610bb191f68482235b78def595e3cc76753a71820afd79bb2024","schema":"livegraph.boolean-propagation.v1","spec_hash":"5d2dce3c258dd48d15624f53b080b93ad24947abe660d6987aa253c7c4d56d41","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-6a77d7b78c1b2baca082","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --supports--> v0d\n- v0b --supports--> v0a\n- v10 --supports--> v0f\n- v03 --supports--> v14\n- v14 --supports--> v12\n- v00 --supports--> v1c\n- v01 --supports--> v03\n- v1c --refutes--> v04\n- v12 --refutes--> v17\n- v0b --supports--> v24\n- v13 --supports--> v01\n- v0f --refutes--> v07\n- v0e --refutes--> v0b\n- v02 --refutes--> v13\n- v20 --supports--> v18\n- v17 --supports--> v00\n- v22 --supports--> v06\n- v1c --refutes--> v21\n- v18 --supports--> v0e\n- v01 --supports--> v1f\n- v04 --refutes--> v16\n- v11 --supports--> v20\n- v06 --refutes--> v23\n- v17 --supports--> v05\n- v0a --refutes--> v1b\n- v20 --supports--> v1d\n- v09 --refutes--> v02\n- v06 --supports--> v08\n- v14 --refutes--> v1e\n- v0f --supports--> v15\n- v16 --refutes--> v22\n- v1b --supports--> v10\n- v20 --supports--> v1a\n- v23 --refutes--> v11\n- v0b --supports--> v0c\n- v11 --refutes--> v19\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":"9a15aaf45420fff084ff86369f3ff68eaf7c65fde0b7b1a223299161b6736b37","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a77d7b78c1b2baca082e41458dabeb85dd28ecd2fa183355edccaf923059ca5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-fad7d6cad3e463c30310","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1f --supports--> v00\n- v07 --refutes--> v1e\n- v15 --refutes--> v09\n- v22 --refutes--> v06\n- v20 --supports--> v0b\n- v02 --supports--> v21\n- v24 --refutes--> v23\n- v05 --supports--> v20\n- v03 --supports--> v07\n- v19 --supports--> v0a\n- v0c --refutes--> v1d\n- v0f --refutes--> v04\n- v0c --refutes--> v0d\n- v02 --refutes--> v0c\n- v0e --refutes--> v08\n- v09 --refutes--> v1b\n- v09 --refutes--> v11\n- v1c --refutes--> v0f\n- v0e --refutes--> v12\n- v0f --supports--> v13\n- v12 --refutes--> v22\n- v0a --refutes--> v1a\n- v18 --refutes--> v0e\n- v00 --refutes--> v18\n- v1e --supports--> v01\n- v24 --refutes--> v15\n- v0c --supports--> v14\n- v0c --refutes--> v1f\n- v11 --supports--> v1c\n- v04 --refutes--> v19\n- v03 --supports--> v17\n- v05 --supports--> v10\n- v0a --supports--> v03\n- v22 --refutes--> v16\n- v06 --refutes--> v24\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54d10666f2b5e09f53ae3869ba4515ce83711f4d9fd2c732dd737dba431f134e","schema":"livegraph.boolean-propagation.v1","spec_hash":"fad7d6cad3e463c3031088ab7850314fce7444ab97878eb9765aac85ad0cb757","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-0f40d35c50b510c93ac6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1f\n- v20 --supports--> v06\n- v01 --refutes--> v14\n- v01 --refutes--> v00\n- v05 --supports--> v11\n- v17 --refutes--> v12\n- v06 --refutes--> v10\n- v09 --supports--> v03\n- v0c --refutes--> v1e\n- v1b --supports--> v0b\n- v24 --refutes--> v23\n- v07 --supports--> v17\n- v12 --supports--> v1b\n- v21 --supports--> v05\n- v16 --supports--> v21\n- v10 --supports--> v09\n- v02 --refutes--> v1c\n- v19 --supports--> v01\n- v15 --refutes--> v0e\n- v03 --refutes--> v16\n- v24 --refutes--> v20\n- v13 --supports--> v0a\n- v0e --refutes--> v02\n- v24 --supports--> v22\n- v15 --refutes--> v1d\n- v06 --supports--> v04\n- v15 --supports--> v0d\n- v02 --supports--> v07\n- v0c --refutes--> v1a\n- v1f --refutes--> v0f\n- v1f --supports--> v08\n- v1b --supports--> v0c\n- v00 --refutes--> v15\n- v03 --refutes--> v18\n- v0f --supports--> v13\n- v0a --supports--> v19\n\n[VERIFIED ROOT]\nv24 = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"99d9eb848c53df425f9c61d68570a52308248fdfbbaf9427deeb606624b30334","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f40d35c50b510c93ac63c93ac4bf509e2b5fceeaa552d05e87c702a58d1da24","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v24\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-70ebfd1b0732bd54605b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v24\n- v21 --supports--> v15\n- v10 --refutes--> v16\n- v05 --supports--> v0f\n- v14 --refutes--> v20\n- v02 --refutes--> v04\n- v17 --supports--> v1a\n- v19 --supports--> v0d\n- v0a --supports--> v17\n- v10 --supports--> v0a\n- v01 --supports--> v12\n- v0f --supports--> v23\n- v0f --refutes--> v09\n- v01 --supports--> v1d\n- v19 --supports--> v0b\n- v21 --supports--> v05\n- v02 --refutes--> v22\n- v06 --supports--> v21\n- v09 --supports--> v14\n- v1f --supports--> v11\n- v0d --supports--> v18\n- v1e --refutes--> v01\n- v1f --supports--> v07\n- v1b --refutes--> v02\n- v02 --supports--> v06\n- v20 --refutes--> v19\n- v1a --refutes--> v00\n- v05 --refutes--> v13\n- v12 --supports--> v1b\n- v1b --refutes--> v0c\n- v01 --supports--> v08\n- v0d --refutes--> v03\n- v18 --refutes--> v1f\n- v00 --refutes--> v1c\n- v07 --supports--> v10\n- v24 --supports--> v0e\n\n[VERIFIED ROOT]\nv1e = 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":"d6ab9eaf098045975947343b9b0e66c3340faf090fb6dfae99a1afde1984f117","schema":"livegraph.boolean-propagation.v1","spec_hash":"70ebfd1b0732bd54605b4de7977de4372cd5fe05e50ea640b035fb50aac6a44a","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-0aadb602d903e01d096d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v06 --refutes--> v05\n- v08 --supports--> v07\n- v21 --refutes--> v04\n- v09 --supports--> v1c\n- v1b --supports--> v1d\n- v23 --supports--> v19\n- v05 --supports--> v16\n- v07 --refutes--> v24\n- v22 --supports--> v0d\n- v0e --refutes--> v0b\n- v12 --refutes--> v15\n- v1c --supports--> v0c\n- v0b --supports--> v06\n- v05 --supports--> v14\n- v23 --supports--> v1e\n- v1e --refutes--> v1f\n- v05 --supports--> v03\n- v12 --refutes--> v08\n- v0a --refutes--> v10\n- v06 --supports--> v18\n- v24 --refutes--> v09\n- v14 --refutes--> v1b\n- v07 --refutes--> v17\n- v1d --refutes--> v23\n- v1a --supports--> v0e\n- v0d --supports--> v0a\n- v04 --refutes--> v1a\n- v24 --supports--> v00\n- v10 --supports--> v12\n- v0f --supports--> v21\n- v1c --supports--> v0f\n- v21 --refutes--> v01\n- v0a --refutes--> v20\n- v1f --refutes--> v13\n- v1e --refutes--> v11\n- v0b --supports--> v02\n\n[VERIFIED ROOT]\nv22 = 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":"0c724a1287c9fc43abda1946d116f189233f1dd7431cf84ff106dfbe5161f23d","schema":"livegraph.boolean-propagation.v1","spec_hash":"0aadb602d903e01d096d9b6ba546ca60f3c68178923f967fe23fdbc8ca6afe46","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v22\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-d09c3bf541ac4ba66ede","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v09\n- v12 --refutes--> v15\n- v01 --supports--> v1f\n- v1a --refutes--> v13\n- v16 --supports--> v02\n- v11 --refutes--> v0d\n- v20 --refutes--> v11\n- v17 --refutes--> v03\n- v21 --refutes--> v1b\n- v0c --supports--> v0e\n- v00 --refutes--> v1a\n- v11 --refutes--> v24\n- v24 --refutes--> v1e\n- v21 --refutes--> v0f\n- v08 --supports--> v00\n- v06 --supports--> v01\n- v05 --refutes--> v08\n- v09 --refutes--> v0c\n- v08 --supports--> v23\n- v12 --refutes--> v0a\n- v10 --supports--> v04\n- v07 --supports--> v12\n- v1f --supports--> v1c\n- v05 --refutes--> v1d\n- v24 --supports--> v06\n- v17 --supports--> v22\n- v02 --refutes--> v05\n- v17 --supports--> v16\n- v1c --refutes--> v0b\n- v0e --supports--> v21\n- v0f --refutes--> v14\n- v0f --refutes--> v07\n- v0a --refutes--> v17\n- v0c --supports--> v19\n- v06 --refutes--> v18\n- v0b --refutes--> v10\n\n[VERIFIED ROOT]\nv20 = 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":"90ceda0cb8bc15ccc957f0c1d0f8b594b9b9584061f25d961cb5edb7d2a22cd4","schema":"livegraph.boolean-propagation.v1","spec_hash":"d09c3bf541ac4ba66ede2c3277fdb287e6154b55c972266171245aef8372f0e2","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v20\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-8dbba61092c75f9f9f45","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v1e --refutes--> v12\n- v0a --refutes--> v14\n- v0a --supports--> v18\n- v19 --refutes--> v16\n- v14 --refutes--> v03\n- v01 --refutes--> v19\n- v23 --supports--> v00\n- v0c --refutes--> v0e\n- v16 --refutes--> v11\n- v1a --supports--> v10\n- v13 --supports--> v20\n- v06 --supports--> v1e\n- v0c --refutes--> v02\n- v20 --supports--> v04\n- v11 --refutes--> v0f\n- v12 --supports--> v01\n- v03 --supports--> v1f\n- v0a --refutes--> v17\n- v0d --refutes--> v24\n- v20 --refutes--> v09\n- v11 --refutes--> v07\n- v13 --supports--> v1d\n- v12 --refutes--> v1c\n- v16 --refutes--> v05\n- v21 --supports--> v06\n- v1f --supports--> v1b\n- v1b --supports--> v23\n- v02 --supports--> v13\n- v10 --refutes--> v0a\n- v0f --refutes--> v0d\n- v11 --refutes--> v22\n- v04 --supports--> v21\n- v02 --supports--> v15\n- v1a --refutes--> v08\n- v23 --refutes--> v0c\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c476509b9529796e5a2795edadb0c9a2cbb396f2f1f17153f734ca342f90cfcd","schema":"livegraph.boolean-propagation.v1","spec_hash":"8dbba61092c75f9f9f450b24bc25691c9348b18bec50e05a3efa3d0ee8a127c3","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-1012cf596ce0659dbc8c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v12 --supports--> v0f\n- v19 --supports--> v0d\n- v05 --refutes--> v1c\n- v1e --supports--> v06\n- v16 --refutes--> v24\n- v21 --supports--> v11\n- v1d --refutes--> v10\n- v0b --supports--> v1a\n- v18 --refutes--> v21\n- v01 --refutes--> v0a\n- v18 --supports--> v17\n- v15 --refutes--> v18\n- v08 --supports--> v03\n- v1d --refutes--> v12\n- v14 --refutes--> v20\n- v0a --supports--> v16\n- v19 --supports--> v1b\n- v0b --refutes--> v15\n- v00 --refutes--> v1d\n- v0a --refutes--> v13\n- v1c --supports--> v1e\n- v03 --refutes--> v0b\n- v1f --refutes--> v22\n- v19 --refutes--> v01\n- v20 --refutes--> v19\n- v21 --supports--> v05\n- v0f --refutes--> v09\n- v20 --supports--> v07\n- v06 --supports--> v02\n- v0e --supports--> v08\n- v06 --supports--> v23\n- v0f --supports--> v04\n- v23 --refutes--> v00\n- v09 --refutes--> v0c\n- v16 --refutes--> v1f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8c0cbb7351564180d2eebee004f600c605555c564d7659a9faba7458b860b8fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"1012cf596ce0659dbc8c50e89ae6a6c9ccedb374e11fbc87ef2f673d60031fbf","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-61515219ec6a4bb7a892","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v17\n- v06 --refutes--> v0b\n- v17 --supports--> v24\n- v06 --supports--> v16\n- v1c --refutes--> v0d\n- v24 --supports--> v08\n- v1d --supports--> v22\n- v04 --refutes--> v13\n- v01 --supports--> v1e\n- v09 --refutes--> v14\n- v21 --refutes--> v0e\n- v14 --refutes--> v0a\n- v08 --refutes--> v11\n- v18 --supports--> v1f\n- v20 --refutes--> v02\n- v0d --refutes--> v1a\n- v02 --refutes--> v1c\n- v07 --refutes--> v23\n- v01 --supports--> v1b\n- v0b --refutes--> v12\n- v1b --supports--> v07\n- v20 --refutes--> v19\n- v11 --supports--> v03\n- v17 --supports--> v0f\n- v23 --refutes--> v06\n- v14 --supports--> v21\n- v04 --supports--> v0c\n- v23 --refutes--> v05\n- v0d --refutes--> v15\n- v0e --refutes--> v20\n- v04 --refutes--> v01\n- v1b --refutes--> v10\n- v12 --supports--> v18\n- v1a --supports--> v1d\n- v03 --supports--> v04\n- v18 --supports--> v00\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":"55b320d895acc6fe6f37675c3d3054b5c7c929a6e88d7102b9fa14401f5192aa","schema":"livegraph.boolean-propagation.v1","spec_hash":"61515219ec6a4bb7a892f6f0cb8291beaf3c6ceabf31797ebd7d97fba1219c41","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-deab5454b5ca7c842500","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v17\n- v02 --refutes--> v21\n- v0a --supports--> v05\n- v1f --supports--> v1b\n- v0a --refutes--> v07\n- v03 --supports--> v0b\n- v20 --refutes--> v18\n- v04 --refutes--> v19\n- v0c --refutes--> v23\n- v20 --supports--> v0d\n- v1e --supports--> v22\n- v1a --refutes--> v12\n- v0b --supports--> v11\n- v18 --supports--> v24\n- v09 --supports--> v00\n- v18 --refutes--> v1c\n- v0c --refutes--> v1d\n- v09 --supports--> v1a\n- v20 --refutes--> v16\n- v0c --supports--> v02\n- v13 --refutes--> v20\n- v04 --supports--> v0f\n- v10 --refutes--> v03\n- v17 --supports--> v09\n- v1f --supports--> v06\n- v19 --supports--> v1f\n- v24 --supports--> v0a\n- v0e --supports--> v15\n- v15 --refutes--> v1e\n- v17 --supports--> v08\n- v07 --supports--> v04\n- v06 --refutes--> v10\n- v12 --supports--> v14\n- v21 --supports--> v0e\n- v14 --refutes--> v13\n- v0b --refutes--> v01\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":"7dff8b79b6af534d793c65f0fdfecc937d1b1af20ac56a04dbcb8138c244101d","schema":"livegraph.boolean-propagation.v1","spec_hash":"deab5454b5ca7c842500f7c3a786b56c28f1cc1f4e678e8e09b8cef14618d6a7","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-4a4f93fa26e442b465ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v24\n- v0c --refutes--> v21\n- v0e --refutes--> v1b\n- v13 --refutes--> v23\n- v0a --refutes--> v1a\n- v14 --supports--> v08\n- v0e --refutes--> v16\n- v19 --supports--> v03\n- v17 --supports--> v00\n- v03 --refutes--> v12\n- v00 --supports--> v20\n- v10 --refutes--> v18\n- v11 --supports--> v06\n- v1f --refutes--> v15\n- v24 --refutes--> v1f\n- v07 --refutes--> v0f\n- v1f --refutes--> v1c\n- v1c --refutes--> v11\n- v12 --refutes--> v0a\n- v1a --refutes--> v05\n- v02 --refutes--> v1e\n- v0f --supports--> v10\n- v16 --supports--> v0c\n- v02 --refutes--> v14\n- v08 --supports--> v07\n- v06 --refutes--> v19\n- v1a --supports--> v04\n- v13 --supports--> v01\n- v23 --refutes--> v0e\n- v0e --refutes--> v22\n- v00 --refutes--> v02\n- v13 --refutes--> v0b\n- v0a --supports--> v09\n- v04 --refutes--> v0d\n- v14 --refutes--> v1d\n- v0c --supports--> v17\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4f4cccf8e7312adc2add62856215cae39f9ed0fa19e4a80dbc24dd71c1f1f042","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a4f93fa26e442b465ac1093b8a8aa61e8728c12e5171045bd39a8e2ad5acfbd","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-33f5e5d9361616bbb685","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v18 --supports--> v20\n- v17 --refutes--> v14\n- v02 --refutes--> v07\n- v00 --refutes--> v02\n- v1e --refutes--> v01\n- v12 --supports--> v05\n- v1c --refutes--> v0b\n- v0c --supports--> v24\n- v01 --refutes--> v17\n- v07 --supports--> v0e\n- v18 --supports--> v00\n- v0d --refutes--> v11\n- v08 --refutes--> v03\n- v1d --supports--> v16\n- v06 --supports--> v1c\n- v0c --refutes--> v19\n- v0e --refutes--> v15\n- v11 --refutes--> v10\n- v15 --supports--> v1e\n- v04 --supports--> v18\n- v24 --refutes--> v21\n- v0a --supports--> v08\n- v18 --refutes--> v22\n- v07 --supports--> v1b\n- v13 --refutes--> v0c\n- v24 --refutes--> v0f\n- v12 --supports--> v1a\n- v17 --supports--> v0d\n- v0d --supports--> v23\n- v1c --refutes--> v0a\n- v1d --supports--> v13\n- v1a --refutes--> v06\n- v10 --refutes--> v12\n- v13 --supports--> v1f\n- v21 --supports--> v04\n\n[VERIFIED ROOT]\nv1d = 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":"808fc97d014693820bd42f04af6845c6253ec9980f568f7d59300eaa92bffcd4","schema":"livegraph.boolean-propagation.v1","spec_hash":"33f5e5d9361616bbb68585c888765abe4c6aaa8fed4a38fed0576b3267bd4f8e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-eb8885d5bb27f862e8f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v10\n- v16 --supports--> v15\n- v17 --refutes--> v0b\n- v17 --refutes--> v0f\n- v06 --supports--> v07\n- v01 --refutes--> v18\n- v1a --refutes--> v1c\n- v1b --refutes--> v04\n- v1a --supports--> v14\n- v10 --refutes--> v1f\n- v09 --supports--> v1e\n- v10 --supports--> v20\n- v1e --refutes--> v1b\n- v07 --supports--> v23\n- v13 --refutes--> v0c\n- v12 --supports--> v0a\n- v1d --supports--> v22\n- v18 --refutes--> v09\n- v13 --supports--> v01\n- v1b --refutes--> v02\n- v02 --supports--> v08\n- v06 --supports--> v0e\n- v0b --refutes--> v12\n- v24 --supports--> v1d\n- v0d --refutes--> v13\n- v0a --refutes--> v05\n- v11 --refutes--> v24\n- v02 --refutes--> v00\n- v02 --refutes--> v17\n- v0a --supports--> v21\n- v15 --refutes--> v06\n- v22 --supports--> v19\n- v20 --supports--> v1a\n- v1c --supports--> v11\n- v22 --refutes--> v0d\n- v20 --refutes--> v03\n\n[VERIFIED ROOT]\nv16 = 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":"af32848424a7593e79d098d6b3b76d5cbc419fa3a8cc8d246c951cd9099ee5b1","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb8885d5bb27f862e8f69915bfb5316e03c30916d336c5364752161e7278b3af","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-f7199be99faa64487440","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v03\n- v0c --supports--> v21\n- v1a --refutes--> v05\n- v14 --refutes--> v1d\n- v03 --refutes--> v1f\n- v0e --supports--> v23\n- v08 --supports--> v0f\n- v1a --supports--> v12\n- v11 --refutes--> v00\n- v13 --refutes--> v18\n- v15 --refutes--> v02\n- v17 --supports--> v22\n- v00 --supports--> v13\n- v0d --supports--> v0a\n- v1f --supports--> v15\n- v17 --supports--> v0c\n- v12 --refutes--> v16\n- v19 --refutes--> v08\n- v06 --refutes--> v14\n- v12 --supports--> v10\n- v12 --refutes--> v1e\n- v0d --supports--> v19\n- v08 --refutes--> v0e\n- v1b --refutes--> v1a\n- v07 --refutes--> v0d\n- v1c --refutes--> v09\n- v17 --supports--> v24\n- v13 --refutes--> v17\n- v11 --refutes--> v04\n- v23 --refutes--> v01\n- v06 --supports--> v0b\n- v1d --supports--> v11\n- v01 --supports--> v06\n- v1d --refutes--> v20\n- v02 --refutes--> v1c\n- v0c --supports--> v1b\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":"7a97e69142d0b12ba726813fcc223998284c8a3c5a3829caae7437e26bce99dc","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7199be99faa64487440cca3610500122108d2032183ad146547fa21d5e28ac0","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-2e6434b54cd983aa4539","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v01 --supports--> v13\n- v1f --supports--> v15\n- v0b --supports--> v0e\n- v0e --refutes--> v14\n- v0f --supports--> v22\n- v12 --refutes--> v0d\n- v1f --supports--> v0a\n- v05 --refutes--> v00\n- v12 --supports--> v1e\n- v17 --supports--> v05\n- v05 --supports--> v03\n- v05 --supports--> v24\n- v0f --refutes--> v21\n- v07 --refutes--> v1d\n- v20 --refutes--> v0b\n- v18 --refutes--> v09\n- v00 --supports--> v07\n- v19 --refutes--> v0c\n- v0c --refutes--> v18\n- v11 --supports--> v1f\n- v09 --refutes--> v06\n- v06 --supports--> v16\n- v06 --supports--> v01\n- v0c --refutes--> v23\n- v18 --refutes--> v1c\n- v10 --refutes--> v20\n- v12 --supports--> v1b\n- v15 --supports--> v17\n- v1d --refutes--> v0f\n- v12 --refutes--> v02\n- v14 --supports--> v1a\n- v21 --supports--> v08\n- v1e --supports--> v11\n- v14 --supports--> v04\n- v08 --supports--> v10\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"534d86089fbedfe1a189e212a545a03602a4cef9c13b6af3e104bd7e39b81265","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e6434b54cd983aa45397e64baba1c2b9ae3530ffebcec6dc99beb2a0a455657","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"}],\"hop\":24,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-3740469049d510974620","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v0f\n- v07 --refutes--> v05\n- v08 --supports--> v15\n- v19 --refutes--> v1d\n- v0e --refutes--> v09\n- v17 --refutes--> v0b\n- v1a --supports--> v20\n- v0b --supports--> v10\n- v17 --supports--> v1b\n- v0f --refutes--> v21\n- v09 --refutes--> v01\n- v20 --supports--> v02\n- v0b --refutes--> v1f\n- v17 --refutes--> v0d\n- v22 --refutes--> v17\n- v15 --refutes--> v0c\n- v0a --supports--> v04\n- v0b --refutes--> v16\n- v23 --refutes--> v06\n- v0b --refutes--> v11\n- v0c --refutes--> v1c\n- v13 --refutes--> v22\n- v00 --supports--> v1a\n- v09 --supports--> v03\n- v1d --refutes--> v0e\n- v21 --supports--> v08\n- v06 --supports--> v00\n- v20 --supports--> v12\n- v07 --refutes--> v23\n- v04 --refutes--> v24\n- v1c --supports--> v07\n- v04 --supports--> v13\n- v0b --refutes--> v18\n- v10 --refutes--> v19\n- v06 --supports--> v1e\n- v1d --supports--> v14\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv02\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c2a7f3a5d0cf177c0b9245d081523ab527535cd8e6ba9ba239bc9da96e1b956e","schema":"livegraph.boolean-propagation.v1","spec_hash":"3740469049d51097462020d8ff63cbaf4c234ff37595006be6762b9e37da187b","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-2ed903f0d6746717973b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v23\n- v07 --supports--> v19\n- v19 --supports--> v21\n- v0e --refutes--> v12\n- v1a --refutes--> v1c\n- v14 --supports--> v18\n- v1f --refutes--> v04\n- v0f --supports--> v0d\n- v0c --supports--> v0b\n- v10 --supports--> v06\n- v09 --supports--> v13\n- v14 --refutes--> v20\n- v14 --refutes--> v22\n- v14 --supports--> v11\n- v01 --supports--> v05\n- v0f --supports--> v03\n- v0d --supports--> v01\n- v1f --refutes--> v24\n- v1b --supports--> v1e\n- v1b --supports--> v14\n- v1c --supports--> v1b\n- v05 --supports--> v00\n- v00 --supports--> v1a\n- v0b --refutes--> v0e\n- v10 --supports--> v0f\n- v12 --supports--> v10\n- v23 --refutes--> v09\n- v11 --supports--> v16\n- v1d --refutes--> v07\n- v0b --supports--> v0a\n- v0e --refutes--> v02\n- v0e --refutes--> v15\n- v21 --refutes--> v1f\n- v1f --supports--> v08\n- v19 --refutes--> v17\n- v04 --refutes--> v0c\n\n[VERIFIED ROOT]\nv1d = 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":"d9914e9a80427fc654fecf1ef356641f1f1c299644636868658e91a0e1f42219","schema":"livegraph.boolean-propagation.v1","spec_hash":"2ed903f0d6746717973b948c0432ffeb9c0d8052ca799d9d017055f6af095890","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-4c8780e61d3836a68e73","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1f --supports--> v0c\n- v20 --refutes--> v1a\n- v06 --refutes--> v01\n- v0f --refutes--> v21\n- v02 --supports--> v1b\n- v09 --refutes--> v0b\n- v14 --refutes--> v06\n- v0c --supports--> v0f\n- v23 --supports--> v08\n- v20 --supports--> v10\n- v23 --refutes--> v0e\n- v16 --refutes--> v07\n- v23 --refutes--> v24\n- v12 --supports--> v09\n- v15 --refutes--> v0d\n- v22 --refutes--> v12\n- v0b --supports--> v04\n- v00 --supports--> v13\n- v23 --supports--> v1f\n- v0c --supports--> v1c\n- v23 --supports--> v1d\n- v20 --supports--> v19\n- v12 --supports--> v03\n- v10 --supports--> v1e\n- v00 --supports--> v11\n- v21 --refutes--> v14\n- v13 --refutes--> v18\n- v17 --refutes--> v16\n- v13 --refutes--> v05\n- v06 --supports--> v20\n- v1b --refutes--> v17\n- v04 --refutes--> v02\n- v1e --refutes--> v15\n- v07 --supports--> v0a\n- v13 --supports--> v23\n\n[VERIFIED ROOT]\nv22 = 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":"632def0acb1f523b77d31d225e95a2501076c90be7cfab7c0875bce560781721","schema":"livegraph.boolean-propagation.v1","spec_hash":"4c8780e61d3836a68e73f400e0f2ff6aecc81bb5be4c9db3349d835a4dec2815","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v22\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-7e4e942db2b37e741855","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v22 --supports--> v11\n- v16 --refutes--> v23\n- v01 --supports--> v0b\n- v24 --supports--> v0a\n- v04 --supports--> v13\n- v21 --supports--> v0f\n- v00 --refutes--> v17\n- v08 --refutes--> v14\n- v17 --refutes--> v03\n- v1c --refutes--> v1d\n- v14 --refutes--> v1f\n- v21 --refutes--> v0c\n- v0b --refutes--> v02\n- v08 --supports--> v20\n- v1c --refutes--> v09\n- v22 --refutes--> v1e\n- v12 --refutes--> v16\n- v14 --refutes--> v12\n- v04 --refutes--> v07\n- v08 --supports--> v1a\n- v06 --refutes--> v08\n- v1e --refutes--> v00\n- v10 --supports--> v06\n- v18 --refutes--> v1c\n- v17 --supports--> v01\n- v0f --refutes--> v24\n- v10 --refutes--> v05\n- v07 --supports--> v0d\n- v16 --supports--> v0e\n- v14 --supports--> v19\n- v23 --supports--> v04\n- v0a --refutes--> v15\n- v02 --supports--> v21\n- v15 --supports--> v10\n- v23 --supports--> v1b\n- v09 --supports--> v22\n\n[VERIFIED ROOT]\nv18 = 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":"a958da1800d91f407b565ca64cb5741e6beac65b03d268a5a85e681b713f8b6d","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e4e942db2b37e74185570620c330c896be8516a0d61af0dda4d5f395039bc09","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-fb7d73e2694bf4f99cb5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1d\n- v0e --supports--> v0a\n- v06 --refutes--> v1c\n- v20 --supports--> v05\n- v14 --refutes--> v01\n- v1d --supports--> v06\n- v12 --supports--> v18\n- v16 --refutes--> v08\n- v23 --refutes--> v04\n- v10 --supports--> v15\n- v0a --supports--> v0b\n- v1e --supports--> v0c\n- v11 --refutes--> v0d\n- v14 --supports--> v1b\n- v22 --supports--> v14\n- v16 --refutes--> v21\n- v10 --supports--> v22\n- v01 --refutes--> v12\n- v13 --supports--> v11\n- v1d --supports--> v17\n- v05 --supports--> v13\n- v0f --refutes--> v0e\n- v06 --refutes--> v20\n- v21 --supports--> v0f\n- v18 --supports--> v03\n- v14 --refutes--> v07\n- v23 --supports--> v1a\n- v0a --refutes--> v02\n- v00 --refutes--> v1e\n- v11 --supports--> v09\n- v19 --supports--> v00\n- v0d --supports--> v10\n- v24 --refutes--> v16\n- v18 --refutes--> v23\n- v1e --supports--> v24\n- v0b --refutes--> v1f\n\n[VERIFIED ROOT]\nv19 = 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":"5d8a3d088a4e44b40c20559355b29df53025a64d882ac9b47d9420048a808476","schema":"livegraph.boolean-propagation.v1","spec_hash":"fb7d73e2694bf4f99cb5086791486654fb225000ba01027c7c42985b26e1e959","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v19\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-1a62a4228856a4a19a34","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v0b\n- v09 --refutes--> v17\n- v18 --refutes--> v05\n- v15 --refutes--> v1f\n- v0b --refutes--> v01\n- v1a --refutes--> v20\n- v02 --refutes--> v24\n- v23 --refutes--> v12\n- v1f --refutes--> v04\n- v1a --refutes--> v11\n- v14 --refutes--> v16\n- v06 --refutes--> v0c\n- v24 --supports--> v1b\n- v21 --refutes--> v06\n- v24 --supports--> v1a\n- v09 --refutes--> v1c\n- v01 --supports--> v18\n- v18 --supports--> v1e\n- v24 --supports--> v0a\n- v1a --refutes--> v09\n- v1c --supports--> v21\n- v18 --refutes--> v0d\n- v04 --refutes--> v02\n- v16 --refutes--> v00\n- v22 --supports--> v14\n- v08 --supports--> v15\n- v12 --refutes--> v22\n- v1a --refutes--> v07\n- v21 --refutes--> v1d\n- v0c --supports--> v23\n- v15 --refutes--> v19\n- v01 --refutes--> v03\n- v0e --supports--> v08\n- v1f --refutes--> v10\n- v0d --refutes--> v0f\n- v00 --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":"8d77491b13a42cf4c34e2c3dd795d3a01768ef5146f6cffdcda85344b481b965","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a62a4228856a4a19a3414f151a462d92854d238d97c23eb3cacf55b10d123a7","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-0a00515ac552b0b9d326","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v20\n- v0a --refutes--> v08\n- v12 --refutes--> v19\n- v12 --refutes--> v03\n- v11 --refutes--> v21\n- v18 --supports--> v02\n- v0a --refutes--> v04\n- v14 --refutes--> v0d\n- v11 --refutes--> v24\n- v0e --supports--> v0c\n- v15 --refutes--> v0f\n- v15 --refutes--> v05\n- v18 --supports--> v09\n- v00 --refutes--> v07\n- v13 --refutes--> v1a\n- v14 --refutes--> v06\n- v02 --refutes--> v0b\n- v07 --refutes--> v22\n- v1d --refutes--> v11\n- v02 --refutes--> v14\n- v13 --refutes--> v1e\n- v1b --supports--> v1c\n- v05 --supports--> v1b\n- v08 --supports--> v12\n- v20 --supports--> v1f\n- v08 --supports--> v17\n- v1b --refutes--> v23\n- v01 --refutes--> v18\n- v14 --refutes--> v15\n- v21 --refutes--> v13\n- v16 --supports--> v1d\n- v19 --supports--> v0e\n- v0e --supports--> v16\n- v10 --supports--> v0a\n- v1f --supports--> v01\n- v22 --supports--> v10\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3aa06d0ed0370447a9ee4ef4aecdbbbec5750cd35cb5dfa09209fcb4d22704f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a00515ac552b0b9d326ee95808f2d1c367bd40e175be210d8b82ec782c2a85b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-45e60cbc66bf0f6db91f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v06\n- v1f --refutes--> v1b\n- v06 --supports--> v0c\n- v20 --supports--> v1e\n- v0c --supports--> v1f\n- v17 --supports--> v0d\n- v0b --supports--> v01\n- v13 --supports--> v05\n- v1c --supports--> v12\n- v08 --refutes--> v23\n- v1d --supports--> v0b\n- v14 --refutes--> v07\n- v15 --refutes--> v10\n- v09 --supports--> v13\n- v13 --refutes--> v04\n- v17 --refutes--> v1a\n- v02 --refutes--> v1d\n- v14 --supports--> v1c\n- v20 --refutes--> v09\n- v12 --refutes--> v00\n- v1b --supports--> v11\n- v0b --supports--> v03\n- v1d --supports--> v21\n- v0a --refutes--> v18\n- v18 --refutes--> v0f\n- v10 --refutes--> v20\n- v20 --supports--> v24\n- v1a --supports--> v08\n- v00 --supports--> v0a\n- v1b --refutes--> v15\n- v04 --refutes--> v16\n- v13 --refutes--> v22\n- v12 --refutes--> v0e\n- v03 --refutes--> v17\n- v16 --supports--> v14\n- v0b --supports--> v19\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":"cef8cc394726c7f04b0d22e3c49dff9fbd9f128d6f4d5fbf4317a8b6889aff65","schema":"livegraph.boolean-propagation.v1","spec_hash":"45e60cbc66bf0f6db91f3897c0bee76804368783be7babfe1d85a5bfd4abd58c","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"}],\"hop\":24,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-1cdd8471da6171ac3c4f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v05 --refutes--> v0c\n- v19 --refutes--> v0a\n- v1f --supports--> v08\n- v14 --supports--> v15\n- v14 --refutes--> v06\n- v08 --refutes--> v09\n- v20 --supports--> v14\n- v11 --supports--> v18\n- v19 --refutes--> v17\n- v02 --supports--> v1e\n- v11 --supports--> v07\n- v22 --refutes--> v02\n- v04 --refutes--> v1d\n- v14 --supports--> v01\n- v1e --supports--> v00\n- v21 --refutes--> v12\n- v24 --supports--> v0f\n- v09 --supports--> v24\n- v0f --refutes--> v1c\n- v09 --supports--> v03\n- v1c --supports--> v0e\n- v12 --supports--> v11\n- v19 --supports--> v23\n- v22 --supports--> v1b\n- v0e --supports--> v10\n- v23 --supports--> v05\n- v0e --supports--> v1a\n- v1e --refutes--> v19\n- v04 --supports--> v0d\n- v14 --refutes--> v04\n- v13 --refutes--> v1f\n- v18 --supports--> v20\n- v0b --supports--> v21\n- v04 --refutes--> v16\n- v16 --refutes--> v13\n\n[VERIFIED ROOT]\nv22 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"40b2a1cab75a2ee622dfd1e9c3e498afe98bb2a70144d58b8494215559155cee","schema":"livegraph.boolean-propagation.v1","spec_hash":"1cdd8471da6171ac3c4f92a29a98f827cb0fce263c0545cef4d8b98ad931a422","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v22\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-e4187ac093096c8a0165","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v07 --refutes--> v03\n- v00 --supports--> v0d\n- v04 --refutes--> v15\n- v18 --refutes--> v1e\n- v19 --supports--> v0b\n- v07 --supports--> v0e\n- v1b --supports--> v06\n- v0d --supports--> v0f\n- v1b --supports--> v1a\n- v0b --supports--> v09\n- v1a --refutes--> v10\n- v15 --refutes--> v1d\n- v15 --refutes--> v18\n- v04 --supports--> v13\n- v01 --supports--> v00\n- v0d --refutes--> v23\n- v09 --supports--> v1f\n- v04 --refutes--> v02\n- v05 --supports--> v07\n- v0f --supports--> v14\n- v08 --supports--> v20\n- v14 --supports--> v19\n- v12 --supports--> v1c\n- v10 --supports--> v11\n- v12 --refutes--> v16\n- v1c --refutes--> v01\n- v1e --refutes--> v12\n- v12 --supports--> v21\n- v11 --refutes--> v08\n- v10 --refutes--> v0a\n- v1f --refutes--> v1b\n- v0f --supports--> v22\n- v0c --supports--> v05\n- v07 --refutes--> v24\n- v00 --refutes--> v17\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e6fc4c02c0a9cc9a3d28503f594fa29be24d71c17448794626ad0f170dd3bc9b","schema":"livegraph.boolean-propagation.v1","spec_hash":"e4187ac093096c8a01656fe8c905a82e71b98ed1d37aba0b2cd8b8d5435caffa","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1f\"}],\"hop\":24,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-1d2c2f56c1dac8f67022","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v23 --refutes--> v06\n- v10 --refutes--> v03\n- v13 --refutes--> v1e\n- v1f --refutes--> v13\n- v0b --refutes--> v17\n- v0a --supports--> v05\n- v07 --refutes--> v09\n- v09 --refutes--> v23\n- v21 --supports--> v24\n- v16 --refutes--> v1a\n- v1a --refutes--> v0f\n- v17 --supports--> v00\n- v1c --refutes--> v11\n- v1a --supports--> v0b\n- v02 --supports--> v18\n- v14 --refutes--> v21\n- v1f --refutes--> v22\n- v14 --supports--> v08\n- v24 --supports--> v01\n- v1e --supports--> v04\n- v1d --refutes--> v10\n- v17 --refutes--> v0c\n- v1f --supports--> v19\n- v04 --supports--> v1d\n- v1d --supports--> v1b\n- v0c --refutes--> v14\n- v01 --supports--> v15\n- v20 --refutes--> v02\n- v23 --supports--> v0e\n- v1f --refutes--> v0d\n- v02 --refutes--> v1c\n- v06 --supports--> v16\n- v1c --refutes--> v0a\n- v05 --refutes--> v07\n- v10 --supports--> v20\n\n[VERIFIED ROOT]\nv1f = 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":"43f0fba2acf36daba3fb27516174a062afc90ba3d4969fe5a3f905a1ab8fb6a2","schema":"livegraph.boolean-propagation.v1","spec_hash":"1d2c2f56c1dac8f6702264f617f9740e79ea3996c840d09241dc486777ee3355","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-f2fa7295792a73e9b01e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --refutes--> v0a\n- v0b --refutes--> v0c\n- v00 --supports--> v16\n- v00 --refutes--> v14\n- v13 --refutes--> v00\n- v12 --refutes--> v03\n- v13 --supports--> v08\n- v0c --refutes--> v1c\n- v1b --supports--> v18\n- v18 --supports--> v23\n- v1c --supports--> v17\n- v10 --refutes--> v11\n- v07 --supports--> v01\n- v18 --supports--> v1d\n- v22 --supports--> v0e\n- v07 --refutes--> v10\n- v22 --refutes--> v0f\n- v04 --supports--> v12\n- v12 --refutes--> v05\n- v0a --refutes--> v1a\n- v21 --supports--> v1e\n- v1c --supports--> v20\n- v21 --refutes--> v13\n- v19 --supports--> v22\n- v02 --supports--> v0b\n- v20 --refutes--> v06\n- v10 --supports--> v0d\n- v06 --refutes--> v1b\n- v14 --refutes--> v15\n- v12 --supports--> v24\n- v23 --supports--> v07\n- v10 --refutes--> v09\n- v1a --refutes--> v02\n- v18 --supports--> v1f\n- v15 --supports--> v04\n- v0e --supports--> v21\n\n[VERIFIED ROOT]\nv19 = 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":"6215c33cf5bcb73af244a26ff2a4c30882e922254944b297ec51bef6516da6ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"f2fa7295792a73e9b01e7762d3225f1efaa2e5a8fc35ba1ea1c91092cb68f7a0","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-832a030ad1e3fedbfcc1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v16 --refutes--> v0c\n- v08 --refutes--> v21\n- v07 --supports--> v0d\n- v0c --refutes--> v01\n- v19 --refutes--> v02\n- v1b --supports--> v24\n- v1a --supports--> v0e\n- v11 --supports--> v07\n- v21 --supports--> v1a\n- v1c --refutes--> v15\n- v15 --supports--> v17\n- v24 --refutes--> v12\n- v0e --refutes--> v0a\n- v19 --refutes--> v04\n- v24 --supports--> v23\n- v19 --refutes--> v1b\n- v17 --refutes--> v13\n- v11 --supports--> v06\n- v00 --refutes--> v11\n- v24 --refutes--> v03\n- v0a --supports--> v1d\n- v0a --refutes--> v1c\n- v22 --refutes--> v09\n- v1e --supports--> v08\n- v24 --supports--> v16\n- v22 --refutes--> v0b\n- v00 --refutes--> v10\n- v18 --refutes--> v22\n- v0d --refutes--> v18\n- v1f --supports--> v19\n- v09 --refutes--> v05\n- v05 --refutes--> v1e\n- v09 --supports--> v14\n- v01 --refutes--> v00\n- v00 --refutes--> v0f\n\n[VERIFIED ROOT]\nv1f = 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":"1c678182b03d306149c4a02ad33de6a8bc59ba819390e36f3a875446a7ea9f38","schema":"livegraph.boolean-propagation.v1","spec_hash":"832a030ad1e3fedbfcc19c2c5254240811cf7bfbfb2c0a3cf4be0daf6971e571","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-03714c920033a9ec6f17","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v14 --refutes--> v03\n- v0a --refutes--> v1f\n- v0d --supports--> v11\n- v13 --refutes--> v22\n- v05 --supports--> v08\n- v1c --supports--> v16\n- v09 --refutes--> v15\n- v03 --supports--> v12\n- v21 --supports--> v13\n- v1b --refutes--> v1c\n- v23 --refutes--> v04\n- v23 --supports--> v19\n- v24 --supports--> v0d\n- v04 --supports--> v0f\n- v22 --supports--> v0a\n- v15 --refutes--> v1b\n- v1d --refutes--> v23\n- v21 --refutes--> v07\n- v03 --supports--> v05\n- v1a --refutes--> v18\n- v1a --supports--> v0b\n- v15 --refutes--> v06\n- v15 --supports--> v20\n- v0f --refutes--> v0c\n- v0f --supports--> v0e\n- v11 --supports--> v09\n- v08 --supports--> v21\n- v0f --supports--> v02\n- v1c --supports--> v01\n- v17 --supports--> v1a\n- v04 --supports--> v1e\n- v0e --supports--> v17\n- v16 --refutes--> v1d\n- v1f --supports--> v24\n- v16 --supports--> v00\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6c43e0e03ae28d32b1bde46872ced05466dff3be1248080c2d821db9b981ada2","schema":"livegraph.boolean-propagation.v1","spec_hash":"03714c920033a9ec6f17408838c9040cdc34fac557ca2f5e5fedf8344ab84d15","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-0792cceeefbbcda70aaa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v11\n- v1f --supports--> v03\n- v19 --supports--> v14\n- v01 --supports--> v07\n- v10 --refutes--> v06\n- v16 --refutes--> v1e\n- v01 --refutes--> v1b\n- v02 --refutes--> v0f\n- v0c --supports--> v0a\n- v0e --refutes--> v21\n- v22 --refutes--> v02\n- v1e --supports--> v0b\n- v0e --refutes--> v1d\n- v10 --supports--> v20\n- v11 --supports--> v13\n- v19 --supports--> v05\n- v0e --supports--> v0d\n- v0f --supports--> v08\n- v03 --supports--> v00\n- v05 --supports--> v09\n- v0d --refutes--> v0c\n- v0b --supports--> v1f\n- v12 --refutes--> v0e\n- v09 --refutes--> v01\n- v10 --supports--> v19\n- v1a --supports--> v15\n- v09 --supports--> v24\n- v15 --supports--> v04\n- v23 --supports--> v17\n- v11 --refutes--> v10\n- v08 --refutes--> v23\n- v22 --supports--> v1c\n- v01 --supports--> v16\n- v00 --refutes--> v18\n- v18 --supports--> v1a\n- v15 --supports--> v22\n\n[VERIFIED ROOT]\nv12 = 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":"bd851ac70821c74f7b054fa3d258f55c67033a9c36fe71cc9d5d44eb35af7754","schema":"livegraph.boolean-propagation.v1","spec_hash":"0792cceeefbbcda70aaa55faa9b6d01c31b142b38d0b7ff32ded4319f2e47490","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-70968f4970253c347b87","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --supports--> v04\n- v00 --refutes--> v08\n- v17 --supports--> v14\n- v23 --refutes--> v0f\n- v17 --refutes--> v21\n- v0c --supports--> v15\n- v13 --supports--> v23\n- v14 --refutes--> v0d\n- v22 --supports--> v1c\n- v24 --refutes--> v06\n- v24 --refutes--> v03\n- v05 --refutes--> v1d\n- v12 --supports--> v1f\n- v22 --supports--> v0b\n- v0d --refutes--> v1b\n- v18 --refutes--> v05\n- v09 --refutes--> v12\n- v22 --refutes--> v10\n- v05 --refutes--> v24\n- v16 --refutes--> v02\n- v1a --supports--> v00\n- v15 --refutes--> v13\n- v24 --refutes--> v11\n- v16 --supports--> v0e\n- v0d --supports--> v07\n- v0e --supports--> v1e\n- v0d --supports--> v19\n- v22 --supports--> v0a\n- v06 --refutes--> v1a\n- v22 --refutes--> v17\n- v1e --supports--> v22\n- v19 --refutes--> v09\n- v08 --supports--> v0c\n- v01 --refutes--> v20\n- v12 --refutes--> v18\n- v04 --refutes--> v16\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":"568ccc6265f72d3ea0a5c41d16a65f4d4020a90b2d038a327002e2efdcaac1d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"70968f4970253c347b87b90475664a74928d927191782d65bec8275fd7142bac","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v01\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-991b9c9f9f11c12e7a15","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1a --supports--> v0c\n- v00 --refutes--> v11\n- v03 --supports--> v17\n- v0c --refutes--> v00\n- v0e --supports--> v04\n- v16 --refutes--> v02\n- v12 --supports--> v1c\n- v24 --refutes--> v13\n- v13 --supports--> v20\n- v15 --supports--> v1b\n- v18 --refutes--> v10\n- v05 --supports--> v03\n- v02 --refutes--> v05\n- v0c --supports--> v1e\n- v04 --supports--> v09\n- v0d --supports--> v08\n- v07 --refutes--> v16\n- v16 --refutes--> v0b\n- v17 --refutes--> v1a\n- v18 --supports--> v24\n- v0a --supports--> v1f\n- v02 --refutes--> v22\n- v0a --refutes--> v0d\n- v1c --supports--> v21\n- v15 --refutes--> v0e\n- v1a --supports--> v23\n- v0a --refutes--> v1d\n- v21 --refutes--> v18\n- v13 --refutes--> v0a\n- v09 --supports--> v01\n- v11 --refutes--> v15\n- v07 --refutes--> v06\n- v0d --refutes--> v0f\n- v0e --supports--> v19\n- v08 --refutes--> v14\n\n[VERIFIED ROOT]\nv07 = 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":"06b345a9f876c66a3fc68bb762ce13d588b05bc227467bed5f2592a305200281","schema":"livegraph.boolean-propagation.v1","spec_hash":"991b9c9f9f11c12e7a15bf97c765edeb73ff6852b098212f201d246aa0c361bc","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-194797da99baa09282c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v1c --refutes--> v0a\n- v0d --refutes--> v14\n- v0f --supports--> v02\n- v0e --supports--> v08\n- v02 --supports--> v01\n- v0a --supports--> v17\n- v16 --refutes--> v13\n- v23 --refutes--> v1d\n- v0d --supports--> v24\n- v20 --supports--> v0d\n- v16 --supports--> v04\n- v18 --refutes--> v0f\n- v14 --refutes--> v23\n- v05 --supports--> v21\n- v12 --refutes--> v1a\n- v05 --refutes--> v1b\n- v1d --refutes--> v07\n- v0a --supports--> v05\n- v18 --refutes--> v1f\n- v00 --refutes--> v03\n- v1e --refutes--> v1c\n- v15 --supports--> v0c\n- v1a --supports--> v09\n- v01 --supports--> v15\n- v00 --refutes--> v16\n- v1d --supports--> v00\n- v09 --refutes--> v0b\n- v02 --supports--> v11\n- v10 --supports--> v19\n- v0c --supports--> v06\n- v19 --supports--> v18\n- v0e --supports--> v10\n- v21 --supports--> v22\n- v21 --supports--> v20\n- v0c --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a67c36a1d6e838ac9bbf17997f5f6282ff400d73b0ce5971f2980ab0034dac6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"194797da99baa09282c1660e488ad64199914c07b1439725a189cbee59f4ade1","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-4720a3a2973edeeb8e0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v0d --refutes--> v12\n- v17 --refutes--> v15\n- v22 --refutes--> v1a\n- v1d --supports--> v1e\n- v01 --supports--> v0d\n- v0d --supports--> v07\n- v0a --refutes--> v14\n- v1c --refutes--> v23\n- v17 --supports--> v02\n- v10 --supports--> v06\n- v19 --refutes--> v0e\n- v0c --refutes--> v19\n- v1c --supports--> v20\n- v0e --refutes--> v16\n- v23 --refutes--> v0a\n- v1f --refutes--> v03\n- v12 --supports--> v22\n- v18 --refutes--> v0c\n- v1f --supports--> v10\n- v0b --supports--> v09\n- v0a --supports--> v05\n- v15 --refutes--> v18\n- v10 --refutes--> v11\n- v1d --supports--> v08\n- v13 --supports--> v1f\n- v1a --supports--> v0b\n- v1c --supports--> v04\n- v06 --refutes--> v1c\n- v12 --supports--> v0f\n- v0b --refutes--> v24\n- v21 --refutes--> v13\n- v10 --refutes--> v00\n- v1e --supports--> v21\n- v16 --supports--> v01\n- v24 --refutes--> v1b\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04b15b265b5dc23351e8128dcbaf3e6a1b23cc1a40d991cde9ef6750be165af5","schema":"livegraph.boolean-propagation.v1","spec_hash":"4720a3a2973edeeb8e0cf138e84f0906d72dc4de2b31c2ea1a514f4e90960f0c","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-a90288672d8e9edb6ba9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0b\n- v04 --supports--> v14\n- v19 --refutes--> v20\n- v1d --supports--> v08\n- v23 --supports--> v00\n- v14 --refutes--> v24\n- v15 --refutes--> v0a\n- v10 --supports--> v1b\n- v0a --refutes--> v05\n- v0b --supports--> v19\n- v0e --supports--> v0f\n- v1d --refutes--> v21\n- v0f --supports--> v07\n- v18 --supports--> v16\n- v07 --refutes--> v12\n- v08 --refutes--> v0d\n- v09 --supports--> v1a\n- v00 --refutes--> v1d\n- v10 --supports--> v11\n- v11 --refutes--> v1e\n- v1a --refutes--> v1f\n- v0d --supports--> v1c\n- v11 --refutes--> v22\n- v18 --supports--> v15\n- v05 --refutes--> v0c\n- v1c --refutes--> v09\n- v20 --refutes--> v17\n- v1c --supports--> v01\n- v14 --refutes--> v0e\n- v04 --supports--> v02\n- v05 --refutes--> v03\n- v20 --supports--> v13\n- v1e --refutes--> v23\n- v20 --supports--> v10\n- v1f --refutes--> v18\n- v0f --refutes--> v06\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"570c643a5e001c7e49acac9a8be9cfd1e3fd0ccc21b019e1d0956de76db42bf3","schema":"livegraph.boolean-propagation.v1","spec_hash":"a90288672d8e9edb6ba9e1622e589de5f80e29e106477ace67cc8ddc8014a815","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-b203633f27b0ef8480e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v07\n- v0b --refutes--> v19\n- v12 --supports--> v1f\n- v1d --supports--> v03\n- v0a --supports--> v00\n- v0f --refutes--> v1e\n- v15 --refutes--> v05\n- v03 --supports--> v0e\n- v0a --refutes--> v02\n- v06 --supports--> v22\n- v01 --refutes--> v04\n- v01 --supports--> v15\n- v21 --supports--> v01\n- v1e --refutes--> v1c\n- v0c --refutes--> v21\n- v12 --refutes--> v0c\n- v1d --refutes--> v24\n- v09 --supports--> v1d\n- v13 --supports--> v0a\n- v19 --supports--> v09\n- v15 --refutes--> v1a\n- v1b --supports--> v13\n- v10 --supports--> v16\n- v1a --supports--> v14\n- v12 --supports--> v20\n- v0f --refutes--> v08\n- v10 --supports--> v18\n- v22 --supports--> v12\n- v23 --refutes--> v0b\n- v07 --refutes--> v10\n- v03 --refutes--> v06\n- v18 --supports--> v17\n- v0d --supports--> v23\n- v17 --refutes--> v1b\n- v0a --refutes--> v0f\n- v1a --supports--> v11\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9b2e63160fba447edb2420a84449d9f063edba6d636b1b9cfe5dad658e0845b","schema":"livegraph.boolean-propagation.v1","spec_hash":"b203633f27b0ef8480e340165cf8404a01eb27e8554db27a846ffc64ee1b9279","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-6ebb41d1d4eebe596872","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v23\n- v0b --supports--> v20\n- v0d --refutes--> v15\n- v04 --refutes--> v14\n- v14 --supports--> v0d\n- v06 --refutes--> v09\n- v01 --refutes--> v0e\n- v1f --supports--> v1b\n- v24 --refutes--> v22\n- v15 --supports--> v1f\n- v14 --supports--> v11\n- v12 --supports--> v03\n- v1b --supports--> v12\n- v03 --refutes--> v1c\n- v1c --supports--> v1a\n- v15 --refutes--> v21\n- v00 --refutes--> v07\n- v24 --supports--> v02\n- v23 --supports--> v01\n- v10 --refutes--> v1d\n- v16 --refutes--> v10\n- v0c --supports--> v0f\n- v0b --refutes--> v08\n- v1f --supports--> v05\n- v12 --refutes--> v13\n- v0e --supports--> v0c\n- v1b --supports--> v17\n- v0f --refutes--> v0a\n- v15 --refutes--> v18\n- v24 --refutes--> v1e\n- v0f --refutes--> v24\n- v10 --supports--> v04\n- v09 --refutes--> v0b\n- v0b --supports--> v00\n- v1a --refutes--> v19\n- v1e --supports--> v16\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"885cb3b644528c908808d3395563ad2f40a73b02556cf63fd01756788b21f357","schema":"livegraph.boolean-propagation.v1","spec_hash":"6ebb41d1d4eebe596872dc37e5727194733dca6afecf2dc9fe3f084469c8b56a","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-023f27500c189628769b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v24\n- v0e --supports--> v0d\n- v04 --refutes--> v0b\n- v07 --supports--> v0a\n- v15 --supports--> v0e\n- v1b --refutes--> v05\n- v09 --supports--> v06\n- v1a --supports--> v15\n- v24 --refutes--> v18\n- v13 --supports--> v1b\n- v18 --supports--> v0f\n- v1d --supports--> v16\n- v0e --supports--> v21\n- v0a --supports--> v1e\n- v1c --refutes--> v10\n- v17 --refutes--> v04\n- v22 --supports--> v1f\n- v05 --supports--> v22\n- v0f --refutes--> v19\n- v1c --supports--> v09\n- v16 --refutes--> v02\n- v0f --supports--> v03\n- v16 --supports--> v00\n- v0d --refutes--> v17\n- v16 --refutes--> v20\n- v1e --supports--> v1d\n- v22 --refutes--> v1a\n- v00 --supports--> v23\n- v1b --supports--> v11\n- v05 --refutes--> v14\n- v0c --supports--> v07\n- v23 --refutes--> v13\n- v06 --refutes--> v0c\n- v05 --supports--> v12\n- v23 --supports--> v01\n- v1e --supports--> v08\n\n[VERIFIED ROOT]\nv1c = 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":"a64e8bbed84aaff9eaf00a809ca3d1b25955d700009689c3e2e687b7d81c0aee","schema":"livegraph.boolean-propagation.v1","spec_hash":"023f27500c189628769b51e3daeb7ce12e5e11aaae1c9375289b432cc3c8d5e5","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-dd7b591568ed0b528fad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v12\n- v12 --supports--> v18\n- v20 --supports--> v15\n- v0f --supports--> v10\n- v18 --supports--> v0b\n- v0d --refutes--> v23\n- v06 --supports--> v0f\n- v0f --supports--> v0c\n- v03 --supports--> v09\n- v13 --supports--> v1a\n- v22 --supports--> v11\n- v1e --refutes--> v1f\n- v00 --supports--> v14\n- v0b --supports--> v1c\n- v01 --refutes--> v1d\n- v1f --supports--> v02\n- v21 --supports--> v17\n- v01 --refutes--> v05\n- v1f --refutes--> v00\n- v1d --supports--> v24\n- v15 --supports--> v22\n- v01 --supports--> v19\n- v17 --refutes--> v13\n- v0f --refutes--> v21\n- v18 --refutes--> v04\n- v24 --refutes--> v03\n- v09 --refutes--> v0d\n- v21 --supports--> v0a\n- v22 --supports--> v06\n- v1b --supports--> v16\n- v0d --supports--> v1e\n- v0f --supports--> v07\n- v16 --refutes--> v01\n- v14 --refutes--> v20\n- v0e --refutes--> v1b\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e84f362d1bd9fcab01e22be803ec9fe942dad1a19a6d3a2d51bd464155ba8c35","schema":"livegraph.boolean-propagation.v1","spec_hash":"dd7b591568ed0b528fadd46d736725d40acbc4b6feecd87db2f42adfff101b4f","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-dbdf30d9ada2218b53dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v20 --refutes--> v1f\n- v17 --refutes--> v1b\n- v0d --refutes--> v1a\n- v06 --supports--> v04\n- v05 --supports--> v10\n- v0b --supports--> v13\n- v0e --refutes--> v01\n- v0c --refutes--> v0e\n- v19 --supports--> v08\n- v1d --refutes--> v16\n- v00 --supports--> v1e\n- v0d --supports--> v03\n- v17 --supports--> v09\n- v11 --refutes--> v17\n- v24 --supports--> v14\n- v01 --refutes--> v19\n- v12 --refutes--> v05\n- v07 --refutes--> v00\n- v1c --supports--> v20\n- v12 --refutes--> v02\n- v20 --refutes--> v15\n- v14 --refutes--> v18\n- v17 --supports--> v23\n- v1e --supports--> v0d\n- v18 --refutes--> v0b\n- v05 --supports--> v11\n- v1f --refutes--> v22\n- v14 --supports--> v21\n- v16 --supports--> v06\n- v19 --supports--> v12\n- v1e --supports--> v0f\n- v22 --supports--> v07\n- v0b --refutes--> v1c\n- v22 --refutes--> v0a\n- v23 --refutes--> v24\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"719faeb2c8abc54e4f248eebb453fdeca0014687923bb7c75adced15535186a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbdf30d9ada2218b53dd4e1d74688764feab48db928d8115cc7bb2d88ef09f3f","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-48b4d9aa2827c5dbabe4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v17 --refutes--> v10\n- v16 --supports--> v12\n- v0f --refutes--> v07\n- v23 --refutes--> v1e\n- v20 --supports--> v02\n- v1d --refutes--> v20\n- v15 --supports--> v1c\n- v13 --supports--> v24\n- v0c --supports--> v17\n- v1c --refutes--> v1f\n- v0e --supports--> v06\n- v01 --refutes--> v08\n- v1f --supports--> v23\n- v12 --refutes--> v03\n- v21 --supports--> v13\n- v06 --supports--> v0a\n- v0f --refutes--> v14\n- v08 --refutes--> v1b\n- v1e --supports--> v0d\n- v06 --supports--> v0b\n- v1e --refutes--> v01\n- v13 --refutes--> v0e\n- v1d --refutes--> v04\n- v12 --supports--> v11\n- v14 --supports--> v21\n- v02 --refutes--> v0f\n- v17 --supports--> v18\n- v17 --refutes--> v1a\n- v03 --refutes--> v0c\n- v17 --supports--> v19\n- v1a --refutes--> v09\n- v0a --supports--> v16\n- v09 --refutes--> v15\n- v1c --supports--> v22\n- v06 --refutes--> v05\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c9ea8fd5cc20a9d3c77e5a2296faeff6023697594c91119bb9445e797b12c773","schema":"livegraph.boolean-propagation.v1","spec_hash":"48b4d9aa2827c5dbabe457cd6bf80c6cf2ee5c1a07b047d30b09e5d6674773d9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-a1a41db45cc6d4890b46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v17\n- v06 --supports--> v11\n- v11 --supports--> v05\n- v07 --refutes--> v19\n- v00 --refutes--> v02\n- v00 --refutes--> v07\n- v08 --refutes--> v0e\n- v19 --refutes--> v24\n- v0a --supports--> v0d\n- v0b --supports--> v1c\n- v23 --refutes--> v12\n- v20 --refutes--> v09\n- v1b --supports--> v0f\n- v23 --supports--> v04\n- v13 --supports--> v1d\n- v0c --supports--> v16\n- v14 --supports--> v1e\n- v0a --refutes--> v21\n- v1a --refutes--> v08\n- v09 --refutes--> v0b\n- v04 --refutes--> v15\n- v1d --refutes--> v00\n- v1b --refutes--> v0a\n- v16 --refutes--> v10\n- v1f --refutes--> v20\n- v14 --refutes--> v06\n- v22 --supports--> v14\n- v16 --refutes--> v1f\n- v08 --refutes--> v13\n- v20 --supports--> v18\n- v0d --refutes--> v23\n- v07 --refutes--> v03\n- v05 --supports--> v1a\n- v04 --supports--> v22\n- v0b --refutes--> v1b\n- v01 --refutes--> v0c\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"14f965ba7a2eaed695ac9ccdfe1d962d9c21549301f5a55b01ae279e9ed6c81b","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1a41db45cc6d4890b46d6c8bf919adec5a59834d6c328369e7a352c184e5d6e","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-1a40d295a6693bd66e42","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v0a --refutes--> v07\n- v22 --supports--> v05\n- v22 --refutes--> v0f\n- v0c --refutes--> v24\n- v1b --refutes--> v1f\n- v18 --refutes--> v21\n- v22 --refutes--> v18\n- v1c --refutes--> v0b\n- v21 --refutes--> v01\n- v1f --refutes--> v00\n- v04 --supports--> v15\n- v24 --refutes--> v12\n- v24 --supports--> v10\n- v01 --supports--> v13\n- v02 --refutes--> v04\n- v02 --supports--> v20\n- v19 --refutes--> v1c\n- v15 --supports--> v0d\n- v0c --refutes--> v08\n- v1a --supports--> v22\n- v13 --refutes--> v0c\n- v24 --supports--> v16\n- v07 --supports--> v02\n- v24 --refutes--> v23\n- v1f --supports--> v03\n- v1f --supports--> v14\n- v1c --supports--> v06\n- v06 --supports--> v0a\n- v11 --supports--> v0e\n- v17 --refutes--> v1d\n- v17 --refutes--> v11\n- v14 --refutes--> v09\n- v13 --supports--> v1e\n- v0d --supports--> v17\n- v10 --supports--> v19\n\n[VERIFIED ROOT]\nv1b = 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":"2cc0fed4009c71a7d28900bea9e72e4aee4a20c7ae87491585b447d68caafa18","schema":"livegraph.boolean-propagation.v1","spec_hash":"1a40d295a6693bd66e426961d0c0505aa686b2bad38657988de4175acb870455","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-c54ad970ab1426856aa8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --refutes--> v17\n- v0b --refutes--> v1f\n- v0d --refutes--> v04\n- v1c --supports--> v13\n- v0f --refutes--> v09\n- v16 --refutes--> v0a\n- v14 --supports--> v24\n- v22 --refutes--> v0d\n- v16 --supports--> v1a\n- v16 --supports--> v1e\n- v0b --supports--> v00\n- v19 --supports--> v06\n- v07 --refutes--> v01\n- v00 --refutes--> v1d\n- v08 --supports--> v18\n- v00 --refutes--> v05\n- v07 --refutes--> v10\n- v18 --supports--> v1b\n- v0d --refutes--> v03\n- v0a --supports--> v1c\n- v20 --refutes--> v22\n- v04 --refutes--> v0e\n- v22 --supports--> v0c\n- v13 --supports--> v08\n- v24 --supports--> v19\n- v21 --refutes--> v11\n- v0e --refutes--> v15\n- v04 --supports--> v23\n- v07 --supports--> v02\n- v0f --refutes--> v20\n- v06 --refutes--> v0b\n- v1d --refutes--> v16\n- v01 --supports--> v21\n- v0a --supports--> v12\n- v1b --refutes--> v07\n- v15 --supports--> v14\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":"e832f4e424100ae45b574d235990c315570d9ef93ccd9f95ac47784d7296882a","schema":"livegraph.boolean-propagation.v1","spec_hash":"c54ad970ab1426856aa8084fc795a00085274ce6797be69a9b7d81926e2bae26","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-e05b99d0d91168a329f9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v24\n- v1a --refutes--> v12\n- v22 --supports--> v21\n- v0f --supports--> v15\n- v00 --refutes--> v0d\n- v00 --refutes--> v10\n- v1b --refutes--> v0f\n- v03 --refutes--> v1d\n- v07 --refutes--> v0b\n- v0b --refutes--> v0e\n- v06 --supports--> v22\n- v14 --supports--> v09\n- v01 --refutes--> v20\n- v01 --supports--> v1a\n- v0c --refutes--> v1e\n- v00 --supports--> v01\n- v01 --supports--> v13\n- v1e --refutes--> v06\n- v16 --supports--> v17\n- v0a --supports--> v14\n- v07 --refutes--> v1f\n- v1c --supports--> v1b\n- v05 --supports--> v07\n- v0f --supports--> v03\n- v00 --supports--> v23\n- v11 --refutes--> v18\n- v17 --refutes--> v00\n- v19 --supports--> v1c\n- v0b --supports--> v08\n- v16 --refutes--> v04\n- v24 --supports--> v19\n- v24 --refutes--> v02\n- v09 --supports--> v11\n- v21 --refutes--> v05\n- v1d --supports--> v0a\n- v11 --refutes--> v0c\n\n[VERIFIED ROOT]\nv16 = 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":"2d6412e70b5565806f6491a50d2fb9fc6e014a56bebb8a2c5833f0e0ae3fcf71","schema":"livegraph.boolean-propagation.v1","spec_hash":"e05b99d0d91168a329f90753657c117dc0d749a4ce04584a4bfaf1ad92c8a125","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-f50720e71b1cf7c0158a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v1a\n- v23 --supports--> v04\n- v0a --refutes--> v07\n- v1f --supports--> v0b\n- v09 --supports--> v0a\n- v15 --supports--> v1b\n- v08 --refutes--> v13\n- v18 --supports--> v05\n- v0e --supports--> v09\n- v13 --refutes--> v1c\n- v20 --supports--> v14\n- v14 --supports--> v16\n- v07 --supports--> v23\n- v09 --refutes--> v19\n- v01 --refutes--> v1f\n- v1b --refutes--> v24\n- v16 --supports--> v08\n- v1d --refutes--> v15\n- v0b --refutes--> v0f\n- v0c --refutes--> v06\n- v22 --refutes--> v10\n- v06 --refutes--> v02\n- v1d --refutes--> v11\n- v24 --refutes--> v12\n- v08 --refutes--> v03\n- v22 --refutes--> v0d\n- v12 --supports--> v22\n- v02 --supports--> v00\n- v0b --supports--> v1d\n- v02 --supports--> v20\n- v15 --supports--> v1e\n- v13 --supports--> v0e\n- v0e --supports--> v17\n- v23 --supports--> v21\n- v0d --refutes--> v18\n- v18 --supports--> v0c\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4afec3d8c6ca3cedd1ac51976cc9d8720d8919a9fa6bbc4cd6245eb13c2b3cf9","schema":"livegraph.boolean-propagation.v1","spec_hash":"f50720e71b1cf7c0158a6173ad7f5cab9914bc542037c7f2d49c67a1c1f06f4e","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-2fc20be1d1715e2321e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --refutes--> v05\n- v1e --refutes--> v0f\n- v03 --refutes--> v06\n- v17 --refutes--> v12\n- v1e --refutes--> v02\n- v03 --refutes--> v1f\n- v16 --supports--> v13\n- v24 --supports--> v0b\n- v0e --refutes--> v0d\n- v01 --supports--> v09\n- v0c --refutes--> v10\n- v10 --refutes--> v15\n- v05 --refutes--> v16\n- v24 --supports--> v00\n- v08 --supports--> v1d\n- v10 --refutes--> v18\n- v03 --refutes--> v07\n- v14 --refutes--> v01\n- v01 --refutes--> v0a\n- v24 --supports--> v20\n- v17 --refutes--> v22\n- v12 --supports--> v14\n- v1c --supports--> v08\n- v21 --refutes--> v19\n- v16 --refutes--> v1c\n- v04 --refutes--> v11\n- v00 --refutes--> v04\n- v0d --refutes--> v0c\n- v1d --refutes--> v1e\n- v18 --refutes--> v1b\n- v1b --supports--> v03\n- v09 --refutes--> v21\n- v05 --refutes--> v23\n- v17 --supports--> v1a\n- v1f --supports--> v17\n- v21 --refutes--> v24\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":"1db6d276e1151ed190a1751c34f88f43d110024c4cdbc8223ccb4c2337a7938b","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fc20be1d1715e2321e6a49f51d8f02bbe40bcb9de19a917730cd35b94148e16","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-074bc90aed647fc1faf5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v20 --supports--> v00\n- v08 --supports--> v03\n- v1d --refutes--> v09\n- v1b --supports--> v1e\n- v0f --refutes--> v0a\n- v16 --supports--> v1a\n- v1b --refutes--> v23\n- v07 --refutes--> v11\n- v22 --refutes--> v1d\n- v17 --supports--> v0e\n- v13 --supports--> v0c\n- v04 --supports--> v21\n- v12 --supports--> v16\n- v06 --supports--> v07\n- v15 --supports--> v08\n- v0a --supports--> v1c\n- v02 --refutes--> v0d\n- v10 --refutes--> v1b\n- v1e --supports--> v22\n- v1d --refutes--> v01\n- v05 --refutes--> v13\n- v04 --supports--> v05\n- v1a --refutes--> v06\n- v00 --supports--> v18\n- v1d --supports--> v1f\n- v1a --refutes--> v19\n- v08 --supports--> v0b\n- v1c --supports--> v17\n- v03 --refutes--> v20\n- v1d --supports--> v0f\n- v16 --supports--> v14\n- v07 --supports--> v02\n- v17 --supports--> v24\n- v00 --supports--> v12\n- v0d --refutes--> v10\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"118c69a4038347b73b959abc49c1393db9e49b1ec1931a28c673805307ffcef7","schema":"livegraph.boolean-propagation.v1","spec_hash":"074bc90aed647fc1faf53c46bb8df1763a8283526e3dbe16b39176279c1989e1","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v22\"}],\"hop\":24,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-03f9a4516f15b516cd9e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v17\n- v11 --supports--> v06\n- v1e --refutes--> v13\n- v1b --refutes--> v00\n- v05 --refutes--> v20\n- v03 --supports--> v1b\n- v1f --refutes--> v11\n- v24 --supports--> v0c\n- v18 --supports--> v0e\n- v0c --refutes--> v04\n- v1d --refutes--> v03\n- v17 --supports--> v01\n- v0d --supports--> v18\n- v16 --supports--> v15\n- v06 --refutes--> v16\n- v06 --supports--> v10\n- v00 --refutes--> v1c\n- v1d --refutes--> v22\n- v0d --supports--> v09\n- v0b --refutes--> v0a\n- v14 --refutes--> v1e\n- v15 --refutes--> v21\n- v08 --supports--> v1a\n- v13 --supports--> v24\n- v17 --refutes--> v1d\n- v08 --supports--> v23\n- v0a --refutes--> v08\n- v08 --refutes--> v0d\n- v15 --supports--> v07\n- v07 --supports--> v0f\n- v00 --supports--> v1f\n- v16 --refutes--> v19\n- v0f --refutes--> v14\n- v0c --refutes--> v12\n- v0f --refutes--> v02\n- v05 --refutes--> v0b\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":"9fa6d19ea869fb38b030c8ef68e44daeb8496509a7abd1a6da4e8c75b6713e9e","schema":"livegraph.boolean-propagation.v1","spec_hash":"03f9a4516f15b516cd9ee4b22534ae08154bfce2f724a32f10f5f917937c08f5","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-922e2411cb561d23ab50","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v14 --refutes--> v1e\n- v0c --supports--> v08\n- v14 --refutes--> v23\n- v03 --supports--> v0a\n- v0f --supports--> v22\n- v0d --refutes--> v0b\n- v01 --refutes--> v24\n- v17 --supports--> v06\n- v0d --supports--> v1a\n- v16 --refutes--> v12\n- v11 --refutes--> v18\n- v05 --supports--> v03\n- v1d --refutes--> v20\n- v1a --refutes--> v1b\n- v18 --refutes--> v07\n- v09 --refutes--> v13\n- v14 --supports--> v1f\n- v06 --supports--> v00\n- v05 --supports--> v10\n- v18 --supports--> v02\n- v13 --supports--> v04\n- v24 --supports--> v0d\n- v09 --refutes--> v0e\n- v1d --supports--> v19\n- v0c --refutes--> v21\n- v13 --refutes--> v0f\n- v22 --supports--> v01\n- v12 --supports--> v17\n- v15 --supports--> v14\n- v1f --supports--> v1d\n- v18 --refutes--> v05\n- v00 --supports--> v0c\n- v22 --refutes--> v1c\n- v1d --supports--> v16\n- v08 --supports--> v11\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"841074c03d0d788f80854f4bb0ffbe86edbea0a4d536773004823a102e6351ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"922e2411cb561d23ab50b4dc45c46a64ce46976c42f391f844f48d2da82ef012","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-41c90900105a6e395029","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v13\n- v05 --supports--> v02\n- v18 --refutes--> v14\n- v09 --refutes--> v20\n- v01 --refutes--> v06\n- v12 --refutes--> v04\n- v18 --refutes--> v11\n- v0f --supports--> v22\n- v03 --supports--> v1f\n- v23 --supports--> v01\n- v01 --supports--> v10\n- v24 --refutes--> v1e\n- v1b --refutes--> v0e\n- v0e --refutes--> v08\n- v11 --supports--> v1c\n- v08 --supports--> v21\n- v1c --refutes--> v24\n- v07 --refutes--> v17\n- v0c --supports--> v23\n- v07 --refutes--> v0b\n- v1b --refutes--> v0d\n- v19 --refutes--> v00\n- v1e --supports--> v0a\n- v01 --refutes--> v1b\n- v19 --supports--> v15\n- v1e --refutes--> v12\n- v08 --supports--> v1a\n- v16 --refutes--> v07\n- v05 --supports--> v0f\n- v12 --refutes--> v1d\n- v04 --refutes--> v0c\n- v0b --supports--> v09\n- v09 --refutes--> v19\n- v00 --supports--> v18\n- v21 --refutes--> v03\n- v1f --supports--> v05\n\n[VERIFIED ROOT]\nv16 = TRUE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ad17168e1a7f0b5d93ed7fe1e3a69ed03d7a398d68297317c39df2d4ae18570c","schema":"livegraph.boolean-propagation.v1","spec_hash":"41c90900105a6e395029069c226abdc61ad81acd40cb10a5bf715957f1187206","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-c8cd105b94457ad9a71e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v03\n- v0c --refutes--> v04\n- v05 --refutes--> v16\n- v0e --supports--> v0d\n- v24 --refutes--> v12\n- v1d --supports--> v0c\n- v0a --supports--> v1a\n- v07 --supports--> v0f\n- v13 --supports--> v15\n- v1f --supports--> v05\n- v0d --supports--> v07\n- v03 --supports--> v24\n- v10 --refutes--> v0b\n- v04 --refutes--> v19\n- v08 --refutes--> v1f\n- v0f --supports--> v0a\n- v0a --refutes--> v09\n- v1e --supports--> v1b\n- v13 --supports--> v21\n- v11 --supports--> v14\n- v12 --refutes--> v01\n- v22 --supports--> v1d\n- v01 --refutes--> v13\n- v08 --supports--> v23\n- v06 --supports--> v22\n- v10 --supports--> v08\n- v18 --refutes--> v10\n- v0a --refutes--> v00\n- v0e --supports--> v02\n- v06 --refutes--> v1c\n- v19 --supports--> v11\n- v11 --refutes--> v0e\n- v1e --refutes--> v06\n- v18 --refutes--> v17\n- v12 --supports--> v20\n- v13 --refutes--> v1e\n\n[VERIFIED ROOT]\nv18 = 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":"b2a403b707772e8a097e5226174e991f293660b936a50a61833e57770900dad6","schema":"livegraph.boolean-propagation.v1","spec_hash":"c8cd105b94457ad9a71e84925896bbe02f5ca338e6bde673a3f17b4604a4beec","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-7882a09d032dd514784b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --supports--> v0d\n- v1a --refutes--> v13\n- v05 --refutes--> v16\n- v16 --refutes--> v12\n- v17 --refutes--> v08\n- v1b --refutes--> v23\n- v23 --refutes--> v04\n- v0a --refutes--> v0f\n- v03 --refutes--> v17\n- v21 --supports--> v00\n- v06 --refutes--> v15\n- v1e --supports--> v1f\n- v08 --supports--> v06\n- v21 --supports--> v0c\n- v0b --refutes--> v09\n- v1e --supports--> v10\n- v04 --refutes--> v01\n- v02 --supports--> v1c\n- v0a --refutes--> v14\n- v07 --refutes--> v0a\n- v0c --supports--> v0e\n- v0e --supports--> v24\n- v06 --supports--> v05\n- v24 --refutes--> v0b\n- v14 --refutes--> v21\n- v02 --refutes--> v20\n- v1d --supports--> v18\n- v04 --refutes--> v03\n- v12 --refutes--> v1a\n- v1a --supports--> v11\n- v18 --refutes--> v1e\n- v1f --refutes--> v02\n- v02 --supports--> v1b\n- v0b --supports--> v1d\n- v0b --refutes--> v22\n- v1f --supports--> v19\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f305f312a4a85a23e4c96a126c9c7cb95079b5141391f7d68e4e764b6c82135e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7882a09d032dd514784bc50cb3079db7bb97b24b5ab7b25a4a7a3c714f07fff0","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-5302bb46e6fba8f4e53d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v11 --supports--> v00\n- v1d --supports--> v21\n- v1a --supports--> v09\n- v1a --refutes--> v1d\n- v00 --refutes--> v16\n- v0d --refutes--> v0c\n- v10 --supports--> v19\n- v05 --supports--> v14\n- v1b --supports--> v01\n- v02 --supports--> v0d\n- v1b --refutes--> v1c\n- v06 --refutes--> v05\n- v01 --supports--> v03\n- v10 --supports--> v15\n- v04 --refutes--> v0a\n- v21 --supports--> v12\n- v05 --refutes--> v20\n- v1f --refutes--> v02\n- v04 --refutes--> v0f\n- v0e --refutes--> v23\n- v19 --supports--> v07\n- v0f --supports--> v1f\n- v1e --supports--> v06\n- v04 --refutes--> v18\n- v02 --supports--> v13\n- v17 --refutes--> v22\n- v03 --refutes--> v17\n- v17 --refutes--> v10\n- v23 --supports--> v1a\n- v1f --refutes--> v08\n- v02 --refutes--> v0b\n- v20 --refutes--> v11\n- v16 --supports--> v04\n- v19 --refutes--> v1e\n- v0d --refutes--> v24\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e05319eb66cf96ed3803bebe250ccd89d5ac2e6f55dd910aa3a5b52a1fb7812","schema":"livegraph.boolean-propagation.v1","spec_hash":"5302bb46e6fba8f4e53d9042a143d06909088e076b5a5996e29e8438c625525a","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-619780475b370e762e6c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --supports--> v0d\n- v12 --supports--> v04\n- v1a --refutes--> v21\n- v0c --supports--> v1c\n- v20 --refutes--> v08\n- v06 --supports--> v20\n- v1d --supports--> v01\n- v24 --supports--> v15\n- v23 --refutes--> v13\n- v19 --supports--> v0f\n- v24 --refutes--> v22\n- v1f --supports--> v11\n- v1d --refutes--> v09\n- v1e --refutes--> v18\n- v0d --supports--> v14\n- v04 --refutes--> v06\n- v03 --refutes--> v19\n- v20 --refutes--> v1b\n- v1d --supports--> v1e\n- v0a --supports--> v02\n- v07 --supports--> v1d\n- v22 --refutes--> v1a\n- v14 --refutes--> v12\n- v1e --supports--> v16\n- v0b --supports--> v07\n- v08 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v03\n- v1c --refutes--> v17\n- v1e --supports--> v1f\n- v1c --refutes--> v23\n- v23 --supports--> v24\n- v22 --supports--> v00\n- v0c --refutes--> v05\n- v11 --refutes--> v0c\n- v1a --refutes--> v10\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"928479128cddd3deb95f32ec33506787b6579bce29f0710fb189baa847c0df4f","schema":"livegraph.boolean-propagation.v1","spec_hash":"619780475b370e762e6c07663a6289b0a6f5496b442485c5491e4c7a51409b2e","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-2d2833f4cc25218effbe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v18 --supports--> v09\n- v05 --refutes--> v02\n- v01 --supports--> v00\n- v24 --supports--> v16\n- v03 --supports--> v0e\n- v18 --supports--> v17\n- v0c --refutes--> v24\n- v24 --refutes--> v13\n- v0b --refutes--> v10\n- v1f --refutes--> v0d\n- v1d --refutes--> v01\n- v01 --supports--> v03\n- v0d --supports--> v08\n- v1c --refutes--> v12\n- v0f --refutes--> v20\n- v05 --refutes--> v0f\n- v18 --supports--> v14\n- v0e --refutes--> v1f\n- v16 --refutes--> v1c\n- v21 --refutes--> v1a\n- v18 --refutes--> v0b\n- v1b --supports--> v05\n- v11 --supports--> v18\n- v0d --refutes--> v07\n- v19 --refutes--> v21\n- v12 --supports--> v11\n- v08 --supports--> v19\n- v1f --supports--> v04\n- v20 --refutes--> v0c\n- v22 --refutes--> v1b\n- v21 --supports--> v15\n- v1f --supports--> v06\n- v10 --supports--> v23\n- v23 --refutes--> v1d\n- v0d --refutes--> v0a\n\n[VERIFIED ROOT]\nv22 = 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":"d4ee2357ddb99919ae3359086e49323c2c3778dff5e803cfa5800c4bf8aae8a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d2833f4cc25218effbe4eb59fe0db5bedd6112f1770585081acc6fb78d79498","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v22\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-24545d2ff5adc00f1d69","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1d\n- v1b --refutes--> v20\n- v04 --refutes--> v0f\n- v1c --refutes--> v01\n- v18 --supports--> v15\n- v07 --refutes--> v1a\n- v09 --supports--> v10\n- v10 --supports--> v19\n- v17 --supports--> v0d\n- v17 --refutes--> v0a\n- v0b --supports--> v0c\n- v0a --supports--> v23\n- v18 --supports--> v00\n- v0c --refutes--> v1c\n- v0f --refutes--> v1e\n- v05 --refutes--> v24\n- v19 --refutes--> v18\n- v07 --supports--> v12\n- v20 --supports--> v13\n- v20 --refutes--> v22\n- v23 --refutes--> v08\n- v05 --refutes--> v1b\n- v12 --supports--> v02\n- v12 --refutes--> v0b\n- v23 --refutes--> v0e\n- v0b --supports--> v21\n- v05 --refutes--> v1f\n- v08 --refutes--> v14\n- v14 --supports--> v06\n- v00 --refutes--> v16\n- v1e --refutes--> v09\n- v01 --refutes--> v17\n- v22 --supports--> v07\n- v16 --refutes--> v05\n- v1b --refutes--> v03\n- v1c --supports--> v11\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":"37a5ccfc39a791db1645b249501c094e644468d5280c24f5f1e6217e7ca8ebac","schema":"livegraph.boolean-propagation.v1","spec_hash":"24545d2ff5adc00f1d6908a244afef9de117ad2ab5bd2ba983e8aef16dd2e539","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-db333e2aae7cf1a80abf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v07\n- v14 --supports--> v1b\n- v17 --supports--> v01\n- v07 --supports--> v16\n- v23 --supports--> v1f\n- v1e --supports--> v1c\n- v07 --refutes--> v09\n- v1e --refutes--> v22\n- v0a --supports--> v02\n- v1b --refutes--> v11\n- v1b --refutes--> v06\n- v22 --refutes--> v23\n- v0d --refutes--> v03\n- v0d --supports--> v10\n- v23 --refutes--> v15\n- v1b --refutes--> v0f\n- v1e --supports--> v24\n- v0e --refutes--> v20\n- v0f --refutes--> v0a\n- v21 --supports--> v17\n- v0a --supports--> v12\n- v0a --refutes--> v1e\n- v1f --supports--> v21\n- v00 --supports--> v04\n- v08 --refutes--> v0c\n- v1a --supports--> v14\n- v04 --supports--> v0b\n- v00 --supports--> v18\n- v22 --supports--> v19\n- v1d --supports--> v1a\n- v09 --refutes--> v1d\n- v0c --supports--> v0d\n- v1b --supports--> v05\n- v0b --refutes--> v13\n- v13 --refutes--> v08\n- v01 --refutes--> v0e\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bb4b2245ca7bbaac9ab4d64d9f41f6a12600bc7d38d87cbf57a98def63875572","schema":"livegraph.boolean-propagation.v1","spec_hash":"db333e2aae7cf1a80abfe08051c6f8bf9ff533cf342e4a4e5a922910b39be805","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-0a3d3ccf3f6da83280e0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1c --supports--> v05\n- v17 --supports--> v1f\n- v02 --refutes--> v22\n- v24 --supports--> v20\n- v18 --supports--> v02\n- v06 --refutes--> v1c\n- v17 --refutes--> v09\n- v00 --supports--> v24\n- v04 --supports--> v08\n- v1f --refutes--> v0c\n- v03 --supports--> v1a\n- v16 --refutes--> v0b\n- v22 --refutes--> v0f\n- v16 --supports--> v19\n- v07 --supports--> v06\n- v05 --supports--> v17\n- v12 --supports--> v23\n- v20 --supports--> v1b\n- v1f --refutes--> v1e\n- v18 --refutes--> v1d\n- v02 --refutes--> v11\n- v15 --supports--> v07\n- v21 --supports--> v16\n- v03 --refutes--> v18\n- v0b --refutes--> v04\n- v1b --refutes--> v01\n- v1c --supports--> v0d\n- v0b --refutes--> v14\n- v1f --refutes--> v21\n- v10 --supports--> v12\n- v03 --supports--> v0a\n- v04 --refutes--> v00\n- v13 --supports--> v10\n- v1b --refutes--> v03\n- v0f --supports--> v13\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"549370849693432421e65f385209810cd2db28772fad2c915123e8a2f4a96667","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a3d3ccf3f6da83280e0a2dee7d4925001c65615347e6ceb2b8da1084d249781","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-077a3e60fb32b98b8b84","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v04\n- v15 --supports--> v17\n- v1f --refutes--> v10\n- v1c --refutes--> v12\n- v12 --refutes--> v23\n- v11 --supports--> v1f\n- v01 --refutes--> v1d\n- v1b --refutes--> v11\n- v06 --supports--> v20\n- v0b --supports--> v07\n- v04 --refutes--> v0e\n- v1b --supports--> v09\n- v0a --supports--> v1e\n- v1f --refutes--> v05\n- v19 --refutes--> v01\n- v07 --supports--> v1a\n- v05 --supports--> v02\n- v12 --refutes--> v0b\n- v13 --refutes--> v16\n- v00 --supports--> v1b\n- v18 --supports--> v00\n- v21 --refutes--> v19\n- v20 --refutes--> v0c\n- v0c --refutes--> v0f\n- v1a --supports--> v0a\n- v15 --supports--> v06\n- v0c --supports--> v18\n- v0b --supports--> v08\n- v0e --refutes--> v1c\n- v05 --supports--> v13\n- v12 --refutes--> v24\n- v16 --supports--> v21\n- v05 --supports--> v14\n- v1a --supports--> v22\n- v0a --refutes--> v0d\n- v00 --supports--> v03\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"615a24e060af85678ff60cd1e0e69ddfd73dec9ec25b1dd83ca71642e69176bc","schema":"livegraph.boolean-propagation.v1","spec_hash":"077a3e60fb32b98b8b84c2cd9bd0125b37cb56c43820548753ee91a60ca0cd29","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"}],\"hop\":24,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-c167824b30fd4112fb38","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v23 --supports--> v0f\n- v09 --supports--> v13\n- v0a --supports--> v07\n- v06 --supports--> v1d\n- v11 --supports--> v1a\n- v08 --refutes--> v12\n- v22 --refutes--> v0e\n- v1d --supports--> v16\n- v24 --supports--> v22\n- v1d --supports--> v24\n- v17 --refutes--> v1b\n- v09 --refutes--> v15\n- v08 --refutes--> v19\n- v04 --supports--> v21\n- v04 --supports--> v0b\n- v1f --supports--> v01\n- v0e --refutes--> v23\n- v21 --refutes--> v18\n- v06 --supports--> v0c\n- v20 --supports--> v17\n- v1d --supports--> v1c\n- v00 --refutes--> v20\n- v23 --supports--> v14\n- v1f --refutes--> v03\n- v1f --supports--> v04\n- v0f --refutes--> v0a\n- v13 --refutes--> v00\n- v11 --supports--> v10\n- v1b --supports--> v02\n- v1e --refutes--> v05\n- v07 --refutes--> v1e\n- v1a --refutes--> v1f\n- v20 --refutes--> v0d\n- v1e --refutes--> v11\n- v12 --refutes--> v06\n\n[VERIFIED ROOT]\nv09 = 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":"274133a369057de22ee494a5e1c8cb5064556e385ac8171afaee822c1cb01684","schema":"livegraph.boolean-propagation.v1","spec_hash":"c167824b30fd4112fb38650645e606bc0d4649233d12b169cf0fcbd943e7c493","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-c97fa64c32b1f9aefc64","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --supports--> v07\n- v1c --supports--> v21\n- v17 --refutes--> v1b\n- v0a --supports--> v11\n- v1d --refutes--> v17\n- v0c --refutes--> v1e\n- v16 --supports--> v20\n- v0c --refutes--> v03\n- v11 --refutes--> v09\n- v1a --supports--> v0f\n- v06 --supports--> v04\n- v08 --supports--> v10\n- v13 --refutes--> v19\n- v00 --supports--> v1f\n- v1c --refutes--> v0c\n- v04 --refutes--> v0b\n- v0f --supports--> v15\n- v0f --refutes--> v01\n- v11 --refutes--> v0d\n- v07 --refutes--> v00\n- v05 --refutes--> v1d\n- v0c --supports--> v02\n- v15 --refutes--> v24\n- v08 --supports--> v14\n- v02 --supports--> v16\n- v17 --refutes--> v12\n- v15 --refutes--> v23\n- v20 --refutes--> v0e\n- v04 --supports--> v05\n- v11 --refutes--> v13\n- v22 --supports--> v0a\n- v10 --supports--> v22\n- v1c --refutes--> v18\n- v24 --refutes--> v06\n- v1b --supports--> v1c\n- v1f --supports--> v08\n\n[VERIFIED ROOT]\nv1a = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9571e46069288bedfb24dc4004ddf104f87d8e19d1d68f37b888cfcbec5e0997","schema":"livegraph.boolean-propagation.v1","spec_hash":"c97fa64c32b1f9aefc6463c4dd54b3dc5d8e2ed245a131a4ba259d0b81027fec","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-50627c3b445954d3e68d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1d --supports--> v08\n- v0b --refutes--> v17\n- v14 --supports--> v04\n- v12 --supports--> v06\n- v17 --refutes--> v11\n- v13 --supports--> v21\n- v14 --refutes--> v01\n- v18 --refutes--> v19\n- v06 --supports--> v20\n- v1b --supports--> v05\n- v18 --refutes--> v07\n- v20 --supports--> v0e\n- v15 --supports--> v0f\n- v0d --supports--> v0b\n- v07 --supports--> v0d\n- v1f --supports--> v14\n- v24 --refutes--> v1f\n- v1a --refutes--> v0c\n- v15 --refutes--> v10\n- v1d --refutes--> v12\n- v02 --supports--> v1e\n- v0d --supports--> v23\n- v07 --refutes--> v22\n- v1e --refutes--> v1d\n- v21 --supports--> v09\n- v24 --refutes--> v00\n- v1a --supports--> v02\n- v09 --refutes--> v15\n- v0e --supports--> v1b\n- v03 --refutes--> v1a\n- v0b --refutes--> v0a\n- v13 --supports--> v1c\n- v11 --refutes--> v13\n- v1b --supports--> v16\n- v1b --refutes--> v24\n\n[VERIFIED ROOT]\nv18 = 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":"aaa4252871b0e9df33da5e244a9d242ca28bd6666c369fcc1ca975ece07931c7","schema":"livegraph.boolean-propagation.v1","spec_hash":"50627c3b445954d3e68d3de2e39c54b2709bd8d453fa1f47ea307a43c386e1b9","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-288dc1aff6ed9ecb9c7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v15\n- v21 --supports--> v1f\n- v1d --supports--> v24\n- v0c --supports--> v20\n- v04 --refutes--> v0f\n- v18 --refutes--> v1d\n- v03 --refutes--> v02\n- v20 --supports--> v10\n- v19 --supports--> v05\n- v1a --supports--> v1c\n- v19 --refutes--> v1e\n- v1a --supports--> v07\n- v1f --refutes--> v08\n- v13 --supports--> v21\n- v23 --refutes--> v0b\n- v1f --supports--> v14\n- v06 --refutes--> v1a\n- v10 --supports--> v13\n- v0c --supports--> v09\n- v1b --refutes--> v22\n- v03 --refutes--> v12\n- v1c --supports--> v0e\n- v0f --refutes--> v03\n- v15 --supports--> v18\n- v08 --refutes--> v06\n- v22 --refutes--> v01\n- v03 --supports--> v0c\n- v0b --refutes--> v19\n- v1b --supports--> v0a\n- v18 --supports--> v17\n- v20 --refutes--> v16\n- v0e --supports--> v23\n- v24 --supports--> v04\n- v01 --supports--> v0d\n- v23 --refutes--> v11\n- v21 --refutes--> v00\n\n[VERIFIED ROOT]\nv1b = TRUE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc3c7929c5120e276fdda415dea6ea6db5cc6641b6295cdbe649d805263f60b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"288dc1aff6ed9ecb9c7a4086bce3e1bcbd627fab57502f32792e6d873d7b5ba3","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-1b8ff149d318f09f557b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v09\n- v1c --refutes--> v1d\n- v05 --refutes--> v22\n- v1a --supports--> v1e\n- v01 --refutes--> v19\n- v18 --supports--> v08\n- v06 --refutes--> v23\n- v0a --refutes--> v11\n- v02 --refutes--> v06\n- v0d --refutes--> v1b\n- v11 --refutes--> v1a\n- v0a --refutes--> v0c\n- v05 --supports--> v17\n- v07 --supports--> v01\n- v23 --supports--> v21\n- v0a --supports--> v04\n- v13 --refutes--> v14\n- v03 --supports--> v02\n- v1c --supports--> v24\n- v1b --refutes--> v0f\n- v0e --refutes--> v20\n- v12 --supports--> v18\n- v1b --supports--> v13\n- v16 --supports--> v07\n- v1d --supports--> v16\n- v0e --supports--> v0b\n- v01 --refutes--> v0a\n- v13 --refutes--> v10\n- v09 --refutes--> v0d\n- v20 --refutes--> v03\n- v22 --supports--> v15\n- v21 --supports--> v12\n- v23 --supports--> v00\n- v18 --refutes--> v1f\n- v08 --refutes--> v05\n- v22 --refutes--> v1c\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"009c88f56a22c572ffc3798b3ecabacea5d5caaa917e3403bf0fdf2283578592","schema":"livegraph.boolean-propagation.v1","spec_hash":"1b8ff149d318f09f557be316e217b5e2ef22dee579a24131d459957589c452b8","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-5e8ebe2f304f75b0bbaf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v05 --refutes--> v22\n- v11 --supports--> v08\n- v08 --supports--> v05\n- v18 --refutes--> v21\n- v1b --refutes--> v0b\n- v05 --refutes--> v01\n- v24 --refutes--> v1c\n- v0c --refutes--> v0a\n- v0b --refutes--> v10\n- v1c --supports--> v09\n- v06 --supports--> v0d\n- v18 --supports--> v1a\n- v12 --refutes--> v18\n- v0b --supports--> v20\n- v19 --refutes--> v06\n- v07 --supports--> v0f\n- v06 --refutes--> v1b\n- v1d --refutes--> v19\n- v15 --refutes--> v17\n- v0f --supports--> v03\n- v0a --refutes--> v07\n- v0b --refutes--> v16\n- v09 --supports--> v12\n- v14 --supports--> v15\n- v22 --refutes--> v24\n- v00 --refutes--> v1f\n- v03 --supports--> v00\n- v23 --refutes--> v1d\n- v05 --supports--> v13\n- v00 --refutes--> v14\n- v15 --refutes--> v23\n- v00 --refutes--> v04\n- v1b --supports--> v02\n- v0c --refutes--> v0e\n- v1a --supports--> v0c\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2df2945e530bc8fc2c6fd39513dff63115b1a30e8b9b6ffb3c2c6ca7d099dd1","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e8ebe2f304f75b0bbaf6018926b6d38a4429feb6ed46b0b8edee1f13bf251cc","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-db171b1410c9417cea0e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v04\n- v16 --supports--> v1b\n- v0b --refutes--> v12\n- v0c --supports--> v05\n- v20 --refutes--> v02\n- v10 --refutes--> v08\n- v20 --refutes--> v19\n- v14 --supports--> v0f\n- v20 --refutes--> v11\n- v03 --supports--> v18\n- v18 --refutes--> v14\n- v12 --supports--> v0d\n- v07 --supports--> v24\n- v14 --supports--> v00\n- v09 --supports--> v1f\n- v06 --refutes--> v23\n- v12 --supports--> v1c\n- v10 --supports--> v09\n- v04 --refutes--> v22\n- v1c --refutes--> v07\n- v1b --supports--> v0c\n- v21 --refutes--> v20\n- v1f --refutes--> v03\n- v0b --supports--> v15\n- v0a --refutes--> v06\n- v24 --supports--> v13\n- v00 --refutes--> v21\n- v24 --refutes--> v17\n- v24 --refutes--> v1d\n- v04 --refutes--> v10\n- v03 --supports--> v1a\n- v05 --supports--> v0b\n- v11 --refutes--> v0e\n- v0e --refutes--> v16\n- v0b --refutes--> v1e\n- v09 --supports--> v01\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4aa2a548b0b684cf50ab4e4ba727b0da3a62f1d7917bdbaeb574bf2b23f88aa9","schema":"livegraph.boolean-propagation.v1","spec_hash":"db171b1410c9417cea0e351997aa65bbd3cacf25a28b68eab7212901b84cba29","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-0dafbba0547da93203f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v0d --refutes--> v0c\n- v01 --refutes--> v14\n- v09 --refutes--> v05\n- v13 --supports--> v22\n- v1a --supports--> v0d\n- v1d --supports--> v1e\n- v13 --supports--> v06\n- v00 --supports--> v16\n- v22 --supports--> v00\n- v09 --refutes--> v17\n- v1d --supports--> v15\n- v24 --refutes--> v1d\n- v1e --supports--> v1c\n- v0c --refutes--> v09\n- v1d --supports--> v0f\n- v05 --refutes--> v1f\n- v16 --refutes--> v21\n- v24 --supports--> v1b\n- v0a --supports--> v13\n- v23 --refutes--> v07\n- v0b --supports--> v1a\n- v20 --supports--> v04\n- v1f --supports--> v23\n- v03 --supports--> v20\n- v10 --refutes--> v0e\n- v20 --supports--> v01\n- v07 --supports--> v0a\n- v13 --supports--> v11\n- v19 --refutes--> v02\n- v00 --refutes--> v18\n- v1e --refutes--> v12\n- v1e --supports--> v10\n- v14 --refutes--> v24\n- v19 --supports--> v0b\n- v0e --refutes--> v19\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8e772745e56e9a7a2a2a6b7537b618d72994c10fe697af2191bdfa69ad933323","schema":"livegraph.boolean-propagation.v1","spec_hash":"0dafbba0547da93203f1de1ab9b6f4c08d634744b898fd9cdf2b221edfff6f20","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-803f236dec60a71e3753","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v04 --supports--> v19\n- v02 --supports--> v11\n- v02 --refutes--> v05\n- v0a --refutes--> v09\n- v15 --supports--> v18\n- v1d --supports--> v08\n- v14 --supports--> v13\n- v05 --refutes--> v00\n- v1b --refutes--> v1f\n- v1f --refutes--> v1e\n- v0b --refutes--> v1d\n- v16 --refutes--> v21\n- v20 --refutes--> v1b\n- v24 --supports--> v1a\n- v20 --supports--> v0f\n- v08 --refutes--> v23\n- v01 --refutes--> v15\n- v17 --supports--> v1c\n- v14 --refutes--> v02\n- v22 --refutes--> v24\n- v18 --supports--> v0b\n- v06 --refutes--> v20\n- v1a --supports--> v04\n- v21 --refutes--> v03\n- v1a --supports--> v12\n- v00 --refutes--> v01\n- v08 --supports--> v06\n- v0a --supports--> v07\n- v03 --supports--> v22\n- v19 --refutes--> v0a\n- v19 --supports--> v0e\n- v00 --supports--> v0c\n- v0a --refutes--> v17\n- v1f --refutes--> v16\n- v03 --refutes--> v10\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b6d4283951479941fd742917eff8a55a9ab447e61c4221cfaacb7e0dc537bcbc","schema":"livegraph.boolean-propagation.v1","spec_hash":"803f236dec60a71e37539a18175f11bff404eb15f32c092932d209e972f15a95","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-ee8491cf5a1da83b33ec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v24\n- v1d --supports--> v03\n- v18 --refutes--> v05\n- v00 --refutes--> v15\n- v15 --refutes--> v1c\n- v10 --refutes--> v12\n- v10 --refutes--> v09\n- v20 --refutes--> v1d\n- v1c --refutes--> v04\n- v20 --supports--> v0e\n- v00 --supports--> v14\n- v05 --refutes--> v13\n- v0a --refutes--> v07\n- v0c --refutes--> v11\n- v05 --supports--> v23\n- v1c --refutes--> v1e\n- v16 --refutes--> v1f\n- v16 --supports--> v0f\n- v02 --supports--> v16\n- v01 --refutes--> v0c\n- v07 --supports--> v1b\n- v05 --refutes--> v10\n- v08 --refutes--> v19\n- v10 --supports--> v0d\n- v1e --refutes--> v17\n- v0d --supports--> v01\n- v17 --supports--> v21\n- v1d --supports--> v0a\n- v06 --supports--> v1a\n- v0b --refutes--> v20\n- v17 --supports--> v02\n- v1f --supports--> v0b\n- v1b --refutes--> v18\n- v19 --refutes--> v06\n- v19 --refutes--> v22\n- v11 --supports--> v08\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0396147bf5db43cddff9352a8feca1be536bdd6bfd62ac4f708a6c2688bdf2fa","schema":"livegraph.boolean-propagation.v1","spec_hash":"ee8491cf5a1da83b33ec978bd27780b860d71a34ba7d9e6238c2d1f6552787ba","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-e0de8f87669c1ba24ad8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v0d\n- v09 --supports--> v04\n- v01 --refutes--> v08\n- v02 --refutes--> v23\n- v19 --supports--> v0c\n- v0b --refutes--> v09\n- v20 --refutes--> v13\n- v24 --refutes--> v00\n- v18 --supports--> v14\n- v0d --refutes--> v01\n- v0f --supports--> v07\n- v12 --supports--> v1f\n- v1c --refutes--> v20\n- v1b --refutes--> v1a\n- v0e --refutes--> v18\n- v14 --refutes--> v05\n- v20 --supports--> v1d\n- v23 --refutes--> v0a\n- v1f --supports--> v1c\n- v0a --refutes--> v19\n- v01 --refutes--> v0b\n- v0f --refutes--> v12\n- v1f --refutes--> v15\n- v09 --supports--> v10\n- v10 --refutes--> v24\n- v0d --supports--> v06\n- v1a --refutes--> v11\n- v05 --supports--> v02\n- v24 --supports--> v1e\n- v1d --refutes--> v03\n- v1d --supports--> v1b\n- v12 --refutes--> v22\n- v16 --supports--> v0e\n- v02 --supports--> v17\n- v10 --supports--> v21\n- v00 --supports--> v0f\n\n[VERIFIED ROOT]\nv16 = 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":"6fba8e8ebdca670baf3bc68bd1c55f701b538382610031a5343658ad75eeb989","schema":"livegraph.boolean-propagation.v1","spec_hash":"e0de8f87669c1ba24ad828267ad9281eee49cad3f3edf68981e1be7588720b00","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-98b321f545558dd60f74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v09 --refutes--> v12\n- v11 --supports--> v1f\n- v02 --refutes--> v03\n- v20 --supports--> v18\n- v18 --supports--> v13\n- v21 --refutes--> v24\n- v09 --supports--> v11\n- v09 --refutes--> v1e\n- v01 --supports--> v08\n- v06 --refutes--> v01\n- v17 --refutes--> v1d\n- v02 --supports--> v00\n- v0f --refutes--> v05\n- v0b --supports--> v1a\n- v03 --refutes--> v0d\n- v13 --supports--> v09\n- v1c --refutes--> v0b\n- v01 --refutes--> v0c\n- v1b --supports--> v10\n- v07 --refutes--> v1b\n- v06 --refutes--> v15\n- v0c --supports--> v22\n- v17 --supports--> v20\n- v24 --supports--> v0f\n- v0a --supports--> v16\n- v16 --supports--> v14\n- v10 --supports--> v1c\n- v10 --supports--> v19\n- v14 --supports--> v07\n- v0e --supports--> v17\n- v0d --supports--> v21\n- v0c --supports--> v0e\n- v0b --refutes--> v02\n- v11 --supports--> v0a\n- v0f --refutes--> v04\n- v0f --supports--> v23\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":"aa5b12f63bcbb78230e1296481318e55b85f9965adb1af42088403d966af2de4","schema":"livegraph.boolean-propagation.v1","spec_hash":"98b321f545558dd60f74a227d805b26b77938b975008ffde71d16b3f63983ee3","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-3b883103a44d6e9ebcb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v16\n- v14 --supports--> v18\n- v22 --supports--> v0f\n- v13 --supports--> v14\n- v10 --refutes--> v09\n- v08 --supports--> v1d\n- v15 --refutes--> v20\n- v0a --supports--> v1a\n- v20 --supports--> v1b\n- v14 --supports--> v0e\n- v1b --supports--> v1c\n- v16 --refutes--> v24\n- v13 --refutes--> v0d\n- v20 --supports--> v21\n- v0f --supports--> v12\n- v24 --supports--> v08\n- v1e --refutes--> v11\n- v03 --refutes--> v22\n- v19 --supports--> v01\n- v0f --supports--> v06\n- v1b --refutes--> v17\n- v1e --supports--> v13\n- v04 --supports--> v15\n- v12 --supports--> v02\n- v00 --supports--> v05\n- v01 --refutes--> v23\n- v08 --supports--> v04\n- v1c --refutes--> v1e\n- v0b --supports--> v19\n- v19 --supports--> v0c\n- v18 --supports--> v00\n- v23 --supports--> v0a\n- v22 --refutes--> v07\n- v03 --supports--> v1f\n- v1a --refutes--> v03\n- v00 --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":"169ee910809b0581622dc9cabcd5585c82d3f2a91ce71b4123caaf6a69192ab0","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b883103a44d6e9ebcb3e5e45800d1884871923d04323b7265aeb46d16a91834","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-251d6b0f0a41d37140f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1b --refutes--> v16\n- v0b --supports--> v08\n- v11 --supports--> v0a\n- v17 --refutes--> v1f\n- v1a --refutes--> v20\n- v13 --supports--> v19\n- v14 --supports--> v01\n- v08 --supports--> v0e\n- v06 --refutes--> v24\n- v0d --supports--> v10\n- v1a --supports--> v1d\n- v1f --refutes--> v21\n- v11 --refutes--> v15\n- v0a --refutes--> v00\n- v09 --refutes--> v03\n- v06 --refutes--> v0c\n- v04 --refutes--> v13\n- v1b --supports--> v0d\n- v18 --refutes--> v0b\n- v07 --supports--> v12\n- v02 --refutes--> v11\n- v0e --supports--> v1a\n- v13 --refutes--> v1c\n- v03 --supports--> v04\n- v23 --supports--> v17\n- v06 --supports--> v02\n- v1f --refutes--> v1e\n- v14 --supports--> v1b\n- v0d --refutes--> v05\n- v04 --refutes--> v22\n- v0f --supports--> v23\n- v21 --supports--> v18\n- v13 --refutes--> v14\n- v12 --supports--> v06\n- v00 --supports--> v09\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a83d07d8ab2fdf2bf222b6eae8b722f52ebf484b8dcb574093546b426ae86f5d","schema":"livegraph.boolean-propagation.v1","spec_hash":"251d6b0f0a41d37140f07a70de20d7598e337129f27f68e7915f21535e85e960","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-0762ba6e7594d677ac6e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v0e\n- v20 --refutes--> v04\n- v0f --supports--> v06\n- v00 --refutes--> v08\n- v09 --supports--> v0d\n- v0a --refutes--> v1b\n- v0f --supports--> v02\n- v17 --supports--> v03\n- v22 --supports--> v1e\n- v09 --refutes--> v1c\n- v04 --supports--> v00\n- v05 --refutes--> v1d\n- v0d --supports--> v19\n- v21 --supports--> v18\n- v01 --refutes--> v0c\n- v03 --refutes--> v14\n- v05 --refutes--> v13\n- v1f --refutes--> v10\n- v08 --supports--> v24\n- v04 --supports--> v23\n- v10 --refutes--> v16\n- v1d --refutes--> v09\n- v0a --supports--> v21\n- v0b --supports--> v22\n- v19 --refutes--> v1a\n- v22 --refutes--> v20\n- v24 --supports--> v0a\n- v21 --refutes--> v11\n- v06 --refutes--> v1f\n- v11 --refutes--> v17\n- v0d --refutes--> v15\n- v03 --refutes--> v12\n- v0b --supports--> v07\n- v03 --supports--> v0f\n- v0c --refutes--> v0b\n- v0e --refutes--> v01\n\n[VERIFIED ROOT]\nv05 = 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":"68f0d42d3b786f983aa9c253cbceb11673e7d90e66043317ba382c953b7ed003","schema":"livegraph.boolean-propagation.v1","spec_hash":"0762ba6e7594d677ac6e13620d0e4b1018992f7dfcab5d465140efc2985bf0d9","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-cd9c20b7ddd9dcd96431","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v0f\n- v0b --refutes--> v24\n- v22 --supports--> v0d\n- v1d --refutes--> v05\n- v17 --refutes--> v23\n- v23 --supports--> v04\n- v0a --refutes--> v09\n- v16 --refutes--> v0a\n- v14 --refutes--> v01\n- v1c --refutes--> v19\n- v07 --supports--> v1b\n- v00 --refutes--> v15\n- v1f --refutes--> v22\n- v12 --refutes--> v1a\n- v00 --refutes--> v0e\n- v22 --refutes--> v00\n- v1b --refutes--> v1d\n- v17 --refutes--> v02\n- v10 --refutes--> v16\n- v0a --refutes--> v21\n- v21 --refutes--> v1c\n- v23 --refutes--> v1e\n- v15 --refutes--> v14\n- v01 --refutes--> v17\n- v06 --refutes--> v08\n- v05 --supports--> v03\n- v0f --refutes--> v07\n- v04 --refutes--> v12\n- v1c --refutes--> v13\n- v18 --refutes--> v10\n- v06 --supports--> v11\n- v06 --supports--> v1f\n- v15 --refutes--> v20\n- v05 --refutes--> v06\n- v16 --supports--> v0c\n- v24 --refutes--> v18\n\n[VERIFIED ROOT]\nv0b = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5921e398ccb1634981d118fc80d90d4680a89eb4708ee0b51c84ce4771dd6a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd9c20b7ddd9dcd96431c8204e97ed468e19ac6f17031ea7845d68d19fac7c49","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-84a2e9d3e7bfaf6c8377","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v04\n- v0f --refutes--> v14\n- v0e --supports--> v05\n- v21 --refutes--> v17\n- v1b --supports--> v21\n- v19 --supports--> v13\n- v0f --refutes--> v24\n- v07 --supports--> v1c\n- v13 --refutes--> v0f\n- v14 --supports--> v1b\n- v1a --supports--> v18\n- v1a --refutes--> v16\n- v10 --supports--> v19\n- v21 --supports--> v0c\n- v0f --refutes--> v1d\n- v03 --refutes--> v10\n- v07 --refutes--> v09\n- v10 --supports--> v22\n- v18 --supports--> v23\n- v19 --refutes--> v02\n- v10 --supports--> v1f\n- v11 --supports--> v08\n- v06 --refutes--> v0a\n- v04 --supports--> v15\n- v07 --refutes--> v06\n- v05 --refutes--> v12\n- v1e --supports--> v0d\n- v1e --refutes--> v03\n- v23 --refutes--> v0e\n- v17 --supports--> v01\n- v0f --supports--> v20\n- v08 --supports--> v07\n- v12 --supports--> v11\n- v00 --supports--> v0b\n- v0a --supports--> v1e\n\n[VERIFIED ROOT]\nv1a = 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":"d6015c2c4c8ebaec20f22ac74bf3c0bc3bc53a65c55d53006c4e62267b44f1b6","schema":"livegraph.boolean-propagation.v1","spec_hash":"84a2e9d3e7bfaf6c8377677bcc21f891dcbc7e0cb28603546d563ebdce7bf0ce","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-c93351554dec83ed59c3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v0d\n- v1e --refutes--> v05\n- v1e --refutes--> v1a\n- v1b --refutes--> v03\n- v21 --refutes--> v04\n- v0c --supports--> v1e\n- v14 --refutes--> v19\n- v11 --supports--> v22\n- v17 --refutes--> v16\n- v18 --supports--> v15\n- v01 --supports--> v1f\n- v1c --supports--> v09\n- v17 --refutes--> v1d\n- v12 --supports--> v0e\n- v07 --supports--> v1b\n- v1a --supports--> v14\n- v01 --refutes--> v1c\n- v0f --refutes--> v10\n- v10 --refutes--> v18\n- v23 --refutes--> v0c\n- v12 --supports--> v21\n- v04 --refutes--> v23\n- v22 --supports--> v02\n- v18 --supports--> v13\n- v24 --supports--> v17\n- v0a --refutes--> v01\n- v23 --supports--> v0b\n- v0d --supports--> v07\n- v16 --refutes--> v12\n- v1b --supports--> v08\n- v1b --refutes--> v0a\n- v18 --supports--> v11\n- v22 --refutes--> v00\n- v00 --supports--> v24\n- v11 --supports--> v20\n- v07 --supports--> v06\n\n[VERIFIED ROOT]\nv0f = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b48fd479d013f082ff62b3cdd61b46b24bb85b4496649329af376b7fa99b3f25","schema":"livegraph.boolean-propagation.v1","spec_hash":"c93351554dec83ed59c35448052cc1e8beae16a57ee1207d8ff027e087587f40","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-c0c5ebf3ce7f9fa199dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --supports--> v11\n- v24 --supports--> v1e\n- v21 --refutes--> v19\n- v06 --refutes--> v13\n- v23 --refutes--> v20\n- v1e --supports--> v0e\n- v08 --supports--> v17\n- v04 --refutes--> v15\n- v11 --supports--> v14\n- v04 --supports--> v0d\n- v24 --refutes--> v18\n- v0f --supports--> v08\n- v0a --supports--> v01\n- v0e --refutes--> v21\n- v00 --supports--> v0b\n- v0b --supports--> v0a\n- v1e --supports--> v22\n- v16 --refutes--> v23\n- v07 --supports--> v0f\n- v1d --supports--> v16\n- v01 --supports--> v1b\n- v21 --supports--> v09\n- v17 --refutes--> v00\n- v1d --supports--> v10\n- v0b --refutes--> v1a\n- v06 --refutes--> v03\n- v01 --refutes--> v1d\n- v0d --supports--> v0c\n- v0f --supports--> v02\n- v1e --supports--> v1c\n- v09 --supports--> v07\n- v0c --supports--> v1f\n- v13 --supports--> v24\n- v0b --supports--> v05\n- v12 --refutes--> v04\n- v14 --supports--> v06\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"34fdec5124f0266b0179147a98e21fb546b2b2c9ea9d4b854efdbce072bc38ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"c0c5ebf3ce7f9fa199dcd0826f01184bce7dc0656f2f5a6dadadf97aa3f8b245","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-d1353b49d6dd2bc4cadc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --refutes--> v12\n- v06 --supports--> v1d\n- v0f --supports--> v1b\n- v0a --refutes--> v15\n- v0e --supports--> v02\n- v15 --refutes--> v0d\n- v1a --refutes--> v14\n- v1b --refutes--> v0c\n- v0b --supports--> v09\n- v00 --supports--> v1c\n- v0d --refutes--> v06\n- v06 --refutes--> v1e\n- v23 --refutes--> v22\n- v1b --supports--> v1a\n- v24 --refutes--> v1f\n- v14 --refutes--> v23\n- v22 --supports--> v04\n- v03 --supports--> v0a\n- v1f --supports--> v00\n- v1b --refutes--> v07\n- v02 --refutes--> v21\n- v1b --supports--> v19\n- v1c --supports--> v18\n- v1b --supports--> v08\n- v18 --supports--> v11\n- v1e --refutes--> v0e\n- v24 --supports--> v17\n- v21 --refutes--> v20\n- v01 --refutes--> v0b\n- v05 --refutes--> v03\n- v02 --refutes--> v13\n- v18 --refutes--> v10\n- v0b --supports--> v24\n- v20 --refutes--> v0f\n- v21 --supports--> v16\n- v22 --refutes--> v01\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":"82571df5474b48df01c17aa7d061099dd9114ad07309e0e433d548804cd3444f","schema":"livegraph.boolean-propagation.v1","spec_hash":"d1353b49d6dd2bc4cadc0f2d4a026042194bbd6a936bbf9c39c86b01c7794794","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-584314b0c9e3533ae8b2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v01 --supports--> v03\n- v13 --refutes--> v10\n- v06 --supports--> v1e\n- v07 --refutes--> v04\n- v0e --refutes--> v1c\n- v00 --refutes--> v07\n- v02 --refutes--> v23\n- v20 --refutes--> v19\n- v05 --refutes--> v0e\n- v11 --supports--> v18\n- v13 --supports--> v08\n- v1d --refutes--> v0a\n- v0b --refutes--> v0d\n- v09 --supports--> v21\n- v1e --refutes--> v16\n- v0e --refutes--> v02\n- v14 --refutes--> v01\n- v0b --supports--> v15\n- v02 --refutes--> v00\n- v22 --supports--> v0f\n- v13 --supports--> v20\n- v24 --supports--> v1d\n- v19 --refutes--> v14\n- v1f --refutes--> v05\n- v0a --supports--> v12\n- v03 --refutes--> v0c\n- v11 --refutes--> v1b\n- v15 --supports--> v13\n- v0a --refutes--> v22\n- v1e --supports--> v24\n- v03 --refutes--> v11\n- v0f --refutes--> v09\n- v18 --refutes--> v06\n- v07 --supports--> v0b\n- v18 --refutes--> v1a\n- v0b --supports--> v17\n\n[VERIFIED ROOT]\nv1f = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae06306f80979e090d38d976d0a9475a1452ec2019b3af5e1d83a1d21d2c99d8","schema":"livegraph.boolean-propagation.v1","spec_hash":"584314b0c9e3533ae8b235bd7b58a86e3e25c46d4f2d162d87aed846d507acf3","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-2745ea88df4de330ffa1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v18 --refutes--> v0c\n- v1a --refutes--> v00\n- v0f --supports--> v11\n- v03 --supports--> v19\n- v11 --refutes--> v1a\n- v02 --refutes--> v1f\n- v16 --supports--> v06\n- v01 --refutes--> v04\n- v12 --supports--> v16\n- v21 --supports--> v13\n- v12 --supports--> v0a\n- v12 --refutes--> v09\n- v06 --supports--> v03\n- v19 --refutes--> v0f\n- v00 --refutes--> v1e\n- v18 --refutes--> v02\n- v1e --supports--> v07\n- v02 --refutes--> v0d\n- v17 --refutes--> v05\n- v01 --supports--> v18\n- v08 --supports--> v0b\n- v10 --refutes--> v20\n- v00 --refutes--> v23\n- v07 --supports--> v22\n- v0b --refutes--> v01\n- v05 --supports--> v1d\n- v24 --supports--> v17\n- v16 --refutes--> v1b\n- v0d --supports--> v10\n- v22 --refutes--> v24\n- v21 --refutes--> v1c\n- v11 --supports--> v0e\n- v05 --refutes--> v21\n- v20 --refutes--> v12\n- v05 --refutes--> v15\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"19c61174ca435e12a452d63ddc881d55f907dc2faaad1c52abaee10241949661","schema":"livegraph.boolean-propagation.v1","spec_hash":"2745ea88df4de330ffa1bce69d06814ceaf8e9cca27625fb2db05a31acb21e71","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-47d004dba36730b2b716","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v22\n- v1c --refutes--> v15\n- v03 --refutes--> v06\n- v0e --refutes--> v17\n- v05 --supports--> v1f\n- v1a --refutes--> v01\n- v1d --refutes--> v14\n- v0b --supports--> v1e\n- v1f --supports--> v19\n- v0e --supports--> v20\n- v16 --supports--> v18\n- v0b --refutes--> v21\n- v23 --supports--> v0f\n- v24 --refutes--> v07\n- v1d --supports--> v03\n- v1e --refutes--> v1c\n- v1d --supports--> v12\n- v13 --refutes--> v04\n- v0c --supports--> v13\n- v17 --refutes--> v1d\n- v10 --refutes--> v1b\n- v23 --refutes--> v0e\n- v00 --refutes--> v16\n- v24 --supports--> v23\n- v13 --refutes--> v11\n- v13 --supports--> v05\n- v19 --refutes--> v0a\n- v06 --refutes--> v0c\n- v15 --refutes--> v02\n- v1b --supports--> v08\n- v18 --supports--> v24\n- v1b --refutes--> v00\n- v1c --supports--> v0d\n- v1d --refutes--> v09\n- v01 --supports--> v10\n- v02 --supports--> v1a\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":"3a1cb9216fdbd0b3ca73d603002c3d9721a6a13bb2e5adc4a5fa66f6b5b98b39","schema":"livegraph.boolean-propagation.v1","spec_hash":"47d004dba36730b2b7160319ef0bccf56ceb20db4194d05cdf2736460bcd6451","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-2d098825b6d2f2607f9a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v18\n- v0a --supports--> v1f\n- v01 --supports--> v11\n- v08 --supports--> v19\n- v13 --refutes--> v01\n- v11 --refutes--> v0f\n- v15 --supports--> v07\n- v03 --refutes--> v0c\n- v0d --supports--> v20\n- v13 --supports--> v17\n- v0e --supports--> v02\n- v22 --refutes--> v12\n- v07 --supports--> v1c\n- v09 --refutes--> v08\n- v16 --refutes--> v09\n- v12 --supports--> v1e\n- v0b --refutes--> v06\n- v01 --refutes--> v05\n- v0e --supports--> v1d\n- v1c --supports--> v13\n- v08 --refutes--> v22\n- v24 --supports--> v03\n- v00 --supports--> v10\n- v0f --supports--> v0a\n- v02 --supports--> v1b\n- v13 --supports--> v21\n- v0c --supports--> v16\n- v18 --supports--> v14\n- v18 --supports--> v0e\n- v18 --refutes--> v1a\n- v1f --supports--> v0b\n- v10 --refutes--> v23\n- v0b --supports--> v24\n- v02 --refutes--> v15\n- v12 --refutes--> v04\n- v12 --supports--> v0d\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ec861ca5d0ac84d643d395710b8d99cef4a1ab9b1d87ac5652f75cd4c442598c","schema":"livegraph.boolean-propagation.v1","spec_hash":"2d098825b6d2f2607f9a90129abc056a5a9d81469df8c266991fdfe668f7da31","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-16715980e6069bbccecc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --supports--> v04\n- v1f --supports--> v19\n- v18 --refutes--> v0d\n- v1a --refutes--> v24\n- v1f --supports--> v14\n- v09 --refutes--> v00\n- v13 --refutes--> v16\n- v23 --refutes--> v1a\n- v1e --refutes--> v18\n- v13 --refutes--> v0f\n- v17 --refutes--> v06\n- v20 --supports--> v0e\n- v17 --refutes--> v0a\n- v22 --refutes--> v10\n- v00 --refutes--> v01\n- v20 --supports--> v1f\n- v10 --refutes--> v23\n- v02 --supports--> v11\n- v0f --refutes--> v20\n- v01 --refutes--> v17\n- v0a --supports--> v21\n- v04 --refutes--> v03\n- v12 --refutes--> v1d\n- v0c --supports--> v12\n- v22 --supports--> v1c\n- v02 --supports--> v1e\n- v1b --supports--> v22\n- v18 --refutes--> v13\n- v1b --refutes--> v08\n- v14 --refutes--> v1b\n- v1a --supports--> v0b\n- v07 --supports--> v15\n- v03 --refutes--> v02\n- v03 --refutes--> v05\n- v07 --supports--> v09\n- v12 --refutes--> v07\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"0a1fac30bc5dc319fafb1cde633be94b0558377b6684f0eb68046dbe7afae741","schema":"livegraph.boolean-propagation.v1","spec_hash":"16715980e6069bbcceccca42f432055fe815a9f2352aca1fed6e7da633c7b630","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v23\"}],\"hop\":24,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-b785d3f27ab0d42c701a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v01 --refutes--> v23\n- v10 --refutes--> v1b\n- v1f --supports--> v0d\n- v0f --supports--> v13\n- v0d --refutes--> v18\n- v02 --refutes--> v0b\n- v19 --supports--> v00\n- v0f --refutes--> v16\n- v00 --refutes--> v1f\n- v18 --supports--> v1a\n- v1a --refutes--> v0e\n- v23 --supports--> v02\n- v0c --refutes--> v03\n- v0c --supports--> v09\n- v1c --refutes--> v0f\n- v1a --supports--> v10\n- v04 --supports--> v0c\n- v22 --supports--> v1e\n- v1b --supports--> v15\n- v1b --supports--> v05\n- v19 --refutes--> v20\n- v0a --supports--> v19\n- v0f --refutes--> v1d\n- v02 --refutes--> v0a\n- v06 --supports--> v14\n- v09 --supports--> v21\n- v05 --refutes--> v06\n- v1f --refutes--> v12\n- v21 --supports--> v24\n- v06 --supports--> v17\n- v08 --refutes--> v22\n- v24 --supports--> v08\n- v13 --supports--> v11\n- v0d --refutes--> v07\n- v06 --refutes--> v04\n\n[VERIFIED ROOT]\nv1c = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9adbd4f21299576871a3b5995afb84b21ea96cd2a21a33675a3b7efd6228f79","schema":"livegraph.boolean-propagation.v1","spec_hash":"b785d3f27ab0d42c701a0191ef984c9c47c21b05ee4df011abe8fa30c5a49322","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-9ba50b5d52db088304d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v05 --supports--> v00\n- v1a --supports--> v10\n- v24 --supports--> v05\n- v00 --refutes--> v21\n- v06 --supports--> v09\n- v04 --supports--> v01\n- v0b --supports--> v12\n- v1b --refutes--> v04\n- v16 --refutes--> v22\n- v07 --refutes--> v14\n- v1e --refutes--> v17\n- v04 --refutes--> v06\n- v0f --refutes--> v16\n- v06 --supports--> v02\n- v10 --supports--> v0b\n- v12 --supports--> v0d\n- v0a --supports--> v18\n- v05 --refutes--> v23\n- v18 --supports--> v0f\n- v16 --supports--> v15\n- v0f --supports--> v03\n- v07 --refutes--> v19\n- v02 --supports--> v07\n- v0d --refutes--> v1b\n- v13 --supports--> v1f\n- v17 --supports--> v24\n- v19 --supports--> v1e\n- v0f --refutes--> v11\n- v0b --refutes--> v1d\n- v1f --refutes--> v1a\n- v18 --refutes--> v1c\n- v08 --supports--> v0e\n- v05 --supports--> v0c\n- v00 --refutes--> v08\n- v0e --refutes--> v0a\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"932decd60567a6f0aa46a8aa8f05f9198dda39503d76764b2e74a16a764a9482","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ba50b5d52db088304d6c318c991ad535503a8627846046e8eb6154fc2707068","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-7e8c6de2a20903baa3f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v0b\n- v15 --refutes--> v1c\n- v13 --supports--> v1e\n- v1f --supports--> v17\n- v1c --refutes--> v05\n- v06 --refutes--> v15\n- v20 --supports--> v03\n- v1b --supports--> v14\n- v04 --refutes--> v21\n- v0a --supports--> v13\n- v17 --supports--> v01\n- v03 --supports--> v07\n- v16 --supports--> v10\n- v19 --refutes--> v1a\n- v21 --refutes--> v16\n- v20 --refutes--> v02\n- v09 --refutes--> v04\n- v07 --supports--> v0d\n- v14 --supports--> v19\n- v11 --refutes--> v1f\n- v05 --supports--> v0a\n- v16 --supports--> v1d\n- v19 --refutes--> v24\n- v04 --refutes--> v22\n- v03 --supports--> v00\n- v1d --supports--> v06\n- v15 --supports--> v23\n- v06 --supports--> v0e\n- v07 --supports--> v12\n- v09 --supports--> v0f\n- v08 --refutes--> v18\n- v0d --refutes--> v11\n- v18 --supports--> v20\n- v11 --supports--> v0c\n- v1e --supports--> v08\n- v17 --refutes--> v1b\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e4c3bfd89846d0d4fcb770b501e21c502a2f4b5b1d18988395217bb79c1a30d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e8c6de2a20903baa3f61a4d42ae20ff7dca57d2fe271a611d8cd13ad6559ee3","spec_json":"{\"edges\":[{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v21\"}],\"hop\":24,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-e5f16d93c663417d72ab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --supports--> v10\n- v21 --supports--> v14\n- v1b --refutes--> v01\n- v09 --supports--> v13\n- v10 --supports--> v15\n- v1f --refutes--> v19\n- v07 --refutes--> v20\n- v0e --supports--> v0d\n- v15 --supports--> v1f\n- v04 --refutes--> v1d\n- v1f --supports--> v16\n- v03 --supports--> v02\n- v0e --supports--> v11\n- v00 --supports--> v0f\n- v23 --refutes--> v1b\n- v13 --refutes--> v00\n- v0c --supports--> v0e\n- v11 --refutes--> v24\n- v1b --refutes--> v0b\n- v06 --refutes--> v23\n- v0f --refutes--> v0c\n- v00 --supports--> v08\n- v15 --supports--> v1e\n- v23 --refutes--> v22\n- v09 --refutes--> v1c\n- v05 --supports--> v12\n- v18 --refutes--> v09\n- v01 --refutes--> v18\n- v16 --refutes--> v21\n- v21 --supports--> v07\n- v20 --refutes--> v03\n- v03 --supports--> v05\n- v0e --refutes--> v17\n- v1f --refutes--> v1a\n- v1f --supports--> v0a\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c04639310c50e5ecf45790ba1c2048756f7f2611b9a06afba91a7879592e0af","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5f16d93c663417d72ab9847b4bb621a8722cfdfb36d43ed2bb881f83a96909f","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":24,"item_id":"lggrpo1-7927349af8b2c556f6e2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --refutes--> v23\n- v0f --refutes--> v21\n- v0a --refutes--> v1a\n- v01 --supports--> v18\n- v23 --refutes--> v22\n- v16 --supports--> v09\n- v1b --refutes--> v05\n- v0e --supports--> v16\n- v08 --refutes--> v0f\n- v01 --refutes--> v19\n- v00 --supports--> v0d\n- v10 --refutes--> v00\n- v1a --supports--> v1e\n- v0f --refutes--> v1b\n- v13 --supports--> v1c\n- v19 --refutes--> v14\n- v00 --supports--> v12\n- v1a --supports--> v15\n- v1b --refutes--> v0c\n- v07 --supports--> v17\n- v04 --supports--> v1f\n- v09 --supports--> v0a\n- v1f --refutes--> v24\n- v15 --refutes--> v0b\n- v1b --refutes--> v06\n- v23 --refutes--> v11\n- v22 --supports--> v07\n- v0e --refutes--> v1d\n- v14 --supports--> v13\n- v06 --refutes--> v01\n- v15 --supports--> v20\n- v07 --supports--> v0e\n- v03 --refutes--> v10\n- v15 --supports--> v08\n- v23 --supports--> v02\n- v0d --refutes--> v04\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aabae0693614dd899f74a8e7c391f3da10f4a272d84800ab3bf42c69a031c8a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"7927349af8b2c556f6e2052019b14054198276df93f809bcfa0a646ba4198d5f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":24,"item_id":"lggrpo1-a010fac22765102f987f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v21 --refutes--> v1f\n- v19 --supports--> v10\n- v17 --refutes--> v12\n- v1d --refutes--> v15\n- v09 --refutes--> v1d\n- v23 --supports--> v13\n- v08 --refutes--> v14\n- v21 --refutes--> v0f\n- v0e --supports--> v0d\n- v18 --refutes--> v24\n- v22 --supports--> v1c\n- v06 --supports--> v1b\n- v1e --refutes--> v08\n- v0c --refutes--> v0b\n- v0a --refutes--> v22\n- v10 --supports--> v21\n- v06 --refutes--> v0c\n- v20 --supports--> v01\n- v00 --refutes--> v16\n- v0e --refutes--> v05\n- v24 --refutes--> v06\n- v00 --refutes--> v03\n- v1f --refutes--> v23\n- v23 --refutes--> v07\n- v05 --refutes--> v19\n- v17 --refutes--> v02\n- v20 --supports--> v0e\n- v12 --refutes--> v09\n- v1d --supports--> v04\n- v0e --refutes--> v11\n- v22 --supports--> v18\n- v1d --supports--> v0a\n- v14 --supports--> v20\n- v0b --refutes--> v1e\n- v16 --supports--> v17\n\n[VERIFIED ROOT]\nv00 = 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":"44526e23bc6cbf7f5350cfb745cf36f7328e3b9ccf704f28f686b0a4e41fbf69","schema":"livegraph.boolean-propagation.v1","spec_hash":"a010fac22765102f987f136545e3fc0a8b2aea5a16c100aff3a0594d051f842a","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v24\"}],\"hop\":24,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-9c9543969806828dbb4d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v25 --refutes--> v03\n- v1c --supports--> v1a\n- v28 --supports--> v10\n- v15 --supports--> v24\n- v0d --refutes--> v27\n- v13 --supports--> v0b\n- v1f --refutes--> v0c\n- v0c --supports--> v07\n- v0f --refutes--> v09\n- v1b --supports--> v08\n- v25 --refutes--> v26\n- v24 --supports--> v23\n- v26 --supports--> v16\n- v10 --supports--> v1b\n- v00 --supports--> v21\n- v07 --refutes--> v1c\n- v16 --refutes--> v14\n- v09 --refutes--> v1d\n- v23 --supports--> v0e\n- v1b --supports--> v19\n- v09 --supports--> v02\n- v11 --supports--> v00\n- v1e --supports--> v0f\n- v11 --refutes--> v22\n- v0b --supports--> v18\n- v14 --refutes--> v15\n- v05 --refutes--> v01\n- v27 --supports--> v28\n- v24 --supports--> v17\n- v26 --supports--> v20\n- v0e --refutes--> v29\n- v00 --supports--> v05\n- v15 --refutes--> v0a\n- v19 --refutes--> v1e\n- v09 --refutes--> v13\n- v26 --supports--> v2a\n- v25 --supports--> v06\n- v05 --refutes--> v1f\n- v12 --refutes--> v0d\n- v0b --supports--> v11\n- v29 --supports--> v12\n- v1f --refutes--> v04\n\n[VERIFIED ROOT]\nv25 = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ff549bb005412f19f7c04349de9a8708d658d394b280b3dd3f62765a39516fb","schema":"livegraph.boolean-propagation.v1","spec_hash":"9c9543969806828dbb4da42d0bdea29f7ee03cf5ab11d1ffc57f05ee996954fe","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v25\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-e425cd653f52f96f1c89","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --supports--> v24\n- v12 --refutes--> v0f\n- v08 --refutes--> v10\n- v27 --supports--> v00\n- v0c --refutes--> v15\n- v0c --refutes--> v1b\n- v01 --supports--> v29\n- v00 --supports--> v28\n- v24 --supports--> v04\n- v07 --refutes--> v18\n- v16 --refutes--> v1e\n- v1c --refutes--> v09\n- v1c --supports--> v17\n- v09 --refutes--> v16\n- v23 --refutes--> v02\n- v1a --supports--> v06\n- v04 --supports--> v14\n- v09 --refutes--> v20\n- v03 --supports--> v0d\n- v12 --supports--> v1f\n- v06 --refutes--> v23\n- v16 --refutes--> v05\n- v15 --refutes--> v19\n- v19 --refutes--> v0a\n- v1c --refutes--> v1d\n- v14 --supports--> v26\n- v02 --supports--> v13\n- v07 --refutes--> v2a\n- v22 --refutes--> v07\n- v07 --supports--> v12\n- v28 --refutes--> v22\n- v08 --refutes--> v1c\n- v27 --supports--> v0b\n- v11 --refutes--> v01\n- v13 --refutes--> v08\n- v23 --refutes--> v0e\n- v03 --supports--> v11\n- v00 --refutes--> v25\n- v14 --supports--> v0c\n- v0a --supports--> v1a\n- v29 --supports--> v21\n- v05 --supports--> v27\n\n[VERIFIED ROOT]\nv03 = TRUE\n\n[TARGET]\nv0f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"16e24d7a7611397b5dbcf548e1df61a26bebaa01b121de78092ef99a6206ff9c","schema":"livegraph.boolean-propagation.v1","spec_hash":"e425cd653f52f96f1c89851d78458df1fb8373c6e760b16291b29f6078b6a54d","spec_json":"{\"edges\":[{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v03\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-6c1ef0671a68f4d941c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v28\n- v07 --supports--> v21\n- v06 --supports--> v15\n- v12 --refutes--> v01\n- v10 --supports--> v04\n- v1e --refutes--> v1b\n- v24 --refutes--> v17\n- v1b --refutes--> v10\n- v06 --supports--> v0c\n- v01 --supports--> v16\n- v2a --supports--> v1e\n- v16 --refutes--> v0d\n- v20 --supports--> v23\n- v02 --supports--> v1f\n- v0c --refutes--> v27\n- v21 --supports--> v14\n- v20 --refutes--> v09\n- v1d --supports--> v0b\n- v20 --refutes--> v29\n- v0a --refutes--> v11\n- v11 --refutes--> v22\n- v26 --refutes--> v1c\n- v10 --supports--> v26\n- v00 --refutes--> v05\n- v16 --supports--> v00\n- v18 --refutes--> v24\n- v1f --refutes--> v12\n- v19 --refutes--> v02\n- v25 --refutes--> v1d\n- v11 --refutes--> v06\n- v14 --refutes--> v18\n- v24 --supports--> v2a\n- v1c --supports--> v20\n- v27 --refutes--> v25\n- v0a --refutes--> v1a\n- v08 --refutes--> v07\n- v20 --supports--> v0e\n- v00 --refutes--> v13\n- v00 --refutes--> v0a\n- v2a --supports--> v03\n- v0f --refutes--> v19\n- v1d --refutes--> v08\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1dffd4c9314dcfae9a70b6f697f76cd9f7832ff3fe193da2a3b50b4edfd48f28","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c1ef0671a68f4d941c5f896cf533bee9ffbeb2a42c41337f09bace242f1eb18","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-513f9d8f338f17a6a58c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v26 --supports--> v1d\n- v10 --refutes--> v02\n- v17 --supports--> v18\n- v04 --supports--> v06\n- v0a --supports--> v16\n- v26 --refutes--> v00\n- v21 --refutes--> v1e\n- v09 --supports--> v0c\n- v27 --refutes--> v24\n- v14 --refutes--> v11\n- v16 --supports--> v01\n- v02 --refutes--> v17\n- v17 --refutes--> v26\n- v0f --refutes--> v14\n- v0a --refutes--> v0e\n- v21 --supports--> v1c\n- v0c --refutes--> v22\n- v2a --supports--> v0a\n- v00 --supports--> v13\n- v13 --refutes--> v29\n- v1b --supports--> v05\n- v13 --refutes--> v25\n- v27 --supports--> v08\n- v05 --supports--> v0b\n- v19 --refutes--> v07\n- v0c --refutes--> v1a\n- v1c --refutes--> v10\n- v14 --refutes--> v09\n- v2a --supports--> v03\n- v1a --supports--> v19\n- v28 --supports--> v0f\n- v29 --refutes--> v2a\n- v01 --refutes--> v0d\n- v0c --supports--> v23\n- v2a --supports--> v12\n- v24 --refutes--> v1b\n- v07 --supports--> v04\n- v0d --refutes--> v28\n- v0f --refutes--> v1f\n- v05 --supports--> v20\n- v29 --supports--> v15\n- v0b --supports--> v21\n\n[VERIFIED ROOT]\nv27 = 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":"339a0e02e0010991cf09c7be2c148a7e4c16e927c835f7dbc164213d94c096bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"513f9d8f338f17a6a58c10128d3eaf640300beac5a30bb19af9cd0981bc572fb","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f139d746d3016967eb18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v27 --refutes--> v16\n- v1e --refutes--> v00\n- v11 --supports--> v06\n- v0d --supports--> v15\n- v15 --supports--> v08\n- v17 --refutes--> v0f\n- v0c --supports--> v05\n- v22 --refutes--> v1b\n- v0d --supports--> v01\n- v0a --supports--> v20\n- v08 --refutes--> v2a\n- v1a --supports--> v1f\n- v20 --supports--> v09\n- v22 --refutes--> v14\n- v08 --supports--> v25\n- v18 --supports--> v1e\n- v1b --refutes--> v1a\n- v10 --supports--> v27\n- v1c --refutes--> v10\n- v2a --refutes--> v0c\n- v10 --supports--> v03\n- v11 --supports--> v24\n- v28 --supports--> v11\n- v0f --supports--> v0d\n- v22 --supports--> v13\n- v09 --supports--> v22\n- v20 --supports--> v07\n- v27 --supports--> v0e\n- v09 --supports--> v21\n- v20 --refutes--> v04\n- v09 --refutes--> v19\n- v29 --refutes--> v26\n- v00 --refutes--> v1d\n- v1a --supports--> v28\n- v16 --refutes--> v29\n- v12 --refutes--> v0a\n- v05 --refutes--> v18\n- v02 --refutes--> v1c\n- v27 --supports--> v23\n- v1d --refutes--> v02\n- v2a --supports--> v0b\n- v24 --supports--> v17\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9d5f80a0965620b65171ef5edf47b901b1d2ddc5655a20217f503d2e63a1bc0","schema":"livegraph.boolean-propagation.v1","spec_hash":"f139d746d3016967eb18b78c174970f46a693512de1276554ab525140cfe6c86","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-7589c786f6905f22082b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v27 --supports--> v28\n- v0e --supports--> v02\n- v02 --supports--> v12\n- v00 --supports--> v1a\n- v21 --refutes--> v19\n- v1f --refutes--> v08\n- v03 --supports--> v25\n- v23 --supports--> v16\n- v28 --supports--> v11\n- v11 --supports--> v0f\n- v10 --refutes--> v00\n- v0d --supports--> v0e\n- v07 --refutes--> v04\n- v17 --refutes--> v1e\n- v06 --supports--> v26\n- v1d --refutes--> v1f\n- v14 --supports--> v01\n- v05 --refutes--> v27\n- v11 --supports--> v14\n- v22 --refutes--> v0a\n- v15 --supports--> v21\n- v25 --supports--> v07\n- v09 --refutes--> v17\n- v12 --refutes--> v0b\n- v16 --refutes--> v0c\n- v1a --refutes--> v15\n- v1f --supports--> v1c\n- v21 --supports--> v03\n- v22 --supports--> v10\n- v0e --supports--> v1b\n- v02 --refutes--> v18\n- v06 --refutes--> v09\n- v04 --refutes--> v0d\n- v10 --supports--> v24\n- v1f --supports--> v22\n- v02 --refutes--> v13\n- v14 --refutes--> v23\n- v09 --supports--> v20\n- v0c --supports--> v06\n- v1d --supports--> v2a\n- v1e --supports--> v1d\n- v0e --refutes--> v29\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":"d98182beba4315658a36e44f402c586b20598c2b1c328b0972c0f98c9f2d191b","schema":"livegraph.boolean-propagation.v1","spec_hash":"7589c786f6905f22082b0d3354edbb986416ad8dac23c75e9b35a7668263290c","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-90b66b9f333b28d2e94f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v0a\n- v0d --supports--> v05\n- v0a --refutes--> v22\n- v23 --supports--> v27\n- v03 --refutes--> v25\n- v1a --refutes--> v12\n- v1c --refutes--> v1e\n- v02 --supports--> v0f\n- v13 --supports--> v1c\n- v17 --refutes--> v28\n- v0e --refutes--> v1f\n- v12 --supports--> v15\n- v10 --supports--> v17\n- v28 --supports--> v07\n- v29 --refutes--> v1b\n- v20 --refutes--> v18\n- v20 --supports--> v04\n- v14 --refutes--> v13\n- v20 --supports--> v14\n- v03 --supports--> v0d\n- v10 --refutes--> v0b\n- v14 --supports--> v00\n- v15 --refutes--> v0c\n- v16 --supports--> v20\n- v0a --refutes--> v21\n- v14 --refutes--> v01\n- v24 --refutes--> v11\n- v08 --refutes--> v24\n- v08 --refutes--> v26\n- v0e --refutes--> v06\n- v1c --supports--> v19\n- v22 --supports--> v02\n- v23 --supports--> v1a\n- v07 --supports--> v08\n- v0c --supports--> v0e\n- v0f --refutes--> v03\n- v14 --refutes--> v09\n- v11 --refutes--> v2a\n- v11 --refutes--> v29\n- v1d --supports--> v10\n- v0e --supports--> v1d\n- v1b --refutes--> v16\n\n[VERIFIED ROOT]\nv23 = 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":"76b17379c6e12b143c0b6801ff118c8760ba9b73c8f6728e7be8e8418b113420","schema":"livegraph.boolean-propagation.v1","spec_hash":"90b66b9f333b28d2e94f8d489ffe4c269fa9536e5ec1f3c80c5ce4e0fcf1e96b","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-33035c53ac0e34eceb33","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v07 --refutes--> v0d\n- v08 --supports--> v05\n- v17 --supports--> v0c\n- v10 --supports--> v1d\n- v1d --refutes--> v02\n- v00 --refutes--> v14\n- v1a --supports--> v0b\n- v14 --supports--> v13\n- v06 --refutes--> v16\n- v03 --refutes--> v28\n- v1a --refutes--> v23\n- v1d --refutes--> v19\n- v0d --refutes--> v1c\n- v13 --supports--> v27\n- v0b --supports--> v09\n- v22 --refutes--> v17\n- v03 --supports--> v00\n- v05 --refutes--> v20\n- v10 --refutes--> v15\n- v24 --refutes--> v26\n- v1d --supports--> v06\n- v1e --supports--> v21\n- v0a --refutes--> v03\n- v1c --supports--> v0f\n- v17 --supports--> v01\n- v04 --refutes--> v29\n- v0b --supports--> v2a\n- v1f --refutes--> v1a\n- v26 --supports--> v07\n- v26 --supports--> v1b\n- v07 --refutes--> v18\n- v2a --supports--> v04\n- v04 --supports--> v08\n- v17 --refutes--> v10\n- v0f --refutes--> v0e\n- v13 --refutes--> v1e\n- v06 --supports--> v1f\n- v12 --supports--> v11\n- v0e --supports--> v22\n- v21 --refutes--> v24\n- v20 --refutes--> v12\n- v0f --supports--> v25\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":"9fed6042d8e6a280e331431c8cd6cbf895209691697d8b9a27c001fa3cc8f7fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"33035c53ac0e34eceb33bdaa1b66d38bc6ff945962b118cf6a364bfaa28f9b41","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-0a2d1171be04e0f01048","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0e\n- v1c --refutes--> v12\n- v1a --refutes--> v03\n- v06 --refutes--> v0a\n- v0d --supports--> v19\n- v1d --supports--> v16\n- v13 --supports--> v17\n- v0b --supports--> v01\n- v10 --refutes--> v21\n- v06 --supports--> v07\n- v02 --refutes--> v0c\n- v16 --refutes--> v14\n- v11 --refutes--> v27\n- v11 --supports--> v28\n- v24 --supports--> v29\n- v1c --supports--> v1d\n- v20 --supports--> v0f\n- v05 --supports--> v18\n- v0b --refutes--> v22\n- v1a --refutes--> v11\n- v10 --refutes--> v20\n- v26 --supports--> v13\n- v29 --supports--> v23\n- v0c --refutes--> v1e\n- v04 --refutes--> v15\n- v15 --refutes--> v05\n- v0e --refutes--> v08\n- v23 --refutes--> v04\n- v22 --refutes--> v0d\n- v10 --refutes--> v25\n- v16 --supports--> v10\n- v0f --supports--> v02\n- v06 --refutes--> v26\n- v0e --supports--> v24\n- v19 --refutes--> v1b\n- v0e --supports--> v09\n- v1e --refutes--> v00\n- v1b --supports--> v1a\n- v00 --refutes--> v0b\n- v11 --refutes--> v2a\n- v18 --refutes--> v06\n- v16 --supports--> v1f\n\n[VERIFIED ROOT]\nv1c = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"08e57b3c5450b7a7fb3057f2b55cf19997c293c3e8e0d1082e40c230c5ed3abc","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a2d1171be04e0f010485f5eb8eeb45ba817df7669a6a4972ba214f3be962bd4","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-e35646e7fc723cef002a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1f\n- v03 --supports--> v02\n- v17 --refutes--> v08\n- v20 --refutes--> v11\n- v0e --refutes--> v27\n- v23 --refutes--> v1d\n- v1a --supports--> v04\n- v21 --supports--> v19\n- v15 --refutes--> v1e\n- v05 --supports--> v20\n- v0b --refutes--> v17\n- v0e --refutes--> v24\n- v21 --supports--> v1a\n- v22 --refutes--> v07\n- v13 --supports--> v14\n- v27 --supports--> v22\n- v0b --refutes--> v18\n- v10 --supports--> v21\n- v01 --refutes--> v03\n- v28 --refutes--> v09\n- v1c --supports--> v28\n- v05 --refutes--> v12\n- v09 --refutes--> v23\n- v29 --refutes--> v0f\n- v0f --supports--> v25\n- v2a --refutes--> v01\n- v15 --refutes--> v10\n- v25 --supports--> v2a\n- v11 --refutes--> v0e\n- v25 --refutes--> v0c\n- v10 --refutes--> v26\n- v23 --supports--> v06\n- v20 --supports--> v1b\n- v13 --refutes--> v1c\n- v21 --refutes--> v0d\n- v22 --supports--> v0a\n- v04 --supports--> v05\n- v08 --supports--> v13\n- v23 --refutes--> v15\n- v00 --refutes--> v16\n- v16 --supports--> v0b\n- v22 --refutes--> v29\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv1f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4d10c2c945b63165e11000a88f8978edafeabb0248462523767fcd2e031ea143","schema":"livegraph.boolean-propagation.v1","spec_hash":"e35646e7fc723cef002afa2b0d6104bf808c120047928c96436f7a3a2d0fe1a8","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v00\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-2b454d48f2c4c1225e32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v24 --refutes--> v06\n- v08 --supports--> v10\n- v14 --supports--> v09\n- v27 --refutes--> v05\n- v1e --refutes--> v07\n- v06 --refutes--> v01\n- v15 --supports--> v16\n- v2a --supports--> v24\n- v23 --supports--> v2a\n- v0b --refutes--> v1e\n- v14 --supports--> v0e\n- v0f --supports--> v1c\n- v11 --supports--> v19\n- v06 --supports--> v21\n- v12 --supports--> v13\n- v26 --refutes--> v04\n- v26 --supports--> v15\n- v1b --refutes--> v1d\n- v28 --supports--> v0a\n- v13 --refutes--> v1f\n- v16 --supports--> v18\n- v18 --supports--> v28\n- v13 --supports--> v25\n- v1c --refutes--> v02\n- v11 --refutes--> v20\n- v19 --supports--> v29\n- v01 --refutes--> v27\n- v09 --refutes--> v12\n- v29 --supports--> v17\n- v00 --refutes--> v0f\n- v06 --supports--> v0d\n- v1a --refutes--> v08\n- v1c --refutes--> v1b\n- v1d --refutes--> v11\n- v29 --refutes--> v26\n- v27 --refutes--> v00\n- v01 --refutes--> v22\n- v1f --supports--> v0b\n- v28 --refutes--> v14\n- v23 --refutes--> v03\n- v1e --supports--> v1a\n\n[VERIFIED ROOT]\nv23 = 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":"5014c6625dbfb958771d4db698499f83aeca3fc29e2ea2282121ddf780a57585","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b454d48f2c4c1225e326cdabb8a1728eb0e84d5eb8096df202217bdfa38c51a","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-6da0ba397213e55a5019","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0f\n- v20 --supports--> v1a\n- v01 --supports--> v00\n- v20 --refutes--> v04\n- v16 --supports--> v0b\n- v00 --refutes--> v15\n- v22 --refutes--> v08\n- v11 --supports--> v1b\n- v22 --supports--> v18\n- v1d --supports--> v22\n- v14 --refutes--> v2a\n- v0f --refutes--> v26\n- v08 --refutes--> v0c\n- v0a --supports--> v07\n- v20 --refutes--> v17\n- v2a --supports--> v13\n- v0c --refutes--> v12\n- v27 --refutes--> v1f\n- v20 --supports--> v21\n- v03 --refutes--> v05\n- v27 --supports--> v06\n- v16 --refutes--> v19\n- v1c --supports--> v1e\n- v0f --supports--> v09\n- v15 --supports--> v1c\n- v12 --supports--> v14\n- v17 --supports--> v29\n- v1b --refutes--> v27\n- v10 --refutes--> v02\n- v24 --refutes--> v01\n- v1f --refutes--> v10\n- v0b --refutes--> v28\n- v07 --supports--> v20\n- v10 --supports--> v0a\n- v26 --supports--> v25\n- v26 --refutes--> v11\n- v17 --supports--> v0e\n- v28 --supports--> v1d\n- v12 --refutes--> v23\n- v1c --refutes--> v03\n- v03 --refutes--> v16\n- v17 --supports--> v0d\n\n[VERIFIED ROOT]\nv24 = 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":"58732ee308ebb13063ff177aecc7df15304f70c5a48b570138d6cc9dc754bc80","schema":"livegraph.boolean-propagation.v1","spec_hash":"6da0ba397213e55a501964ebaf5ead15db8984e5161aa0d4fa7707697fd798c8","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v24\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-e7fa0610e93ef1cbf4f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v21\n- v0c --supports--> v12\n- v26 --supports--> v00\n- v1d --refutes--> v15\n- v15 --refutes--> v19\n- v1b --refutes--> v17\n- v1f --refutes--> v10\n- v10 --refutes--> v16\n- v0b --supports--> v03\n- v10 --refutes--> v27\n- v12 --refutes--> v2a\n- v07 --supports--> v20\n- v22 --supports--> v01\n- v1a --supports--> v0e\n- v0a --refutes--> v1e\n- v0d --supports--> v22\n- v06 --supports--> v0c\n- v26 --refutes--> v04\n- v15 --supports--> v11\n- v15 --supports--> v02\n- v21 --supports--> v0d\n- v06 --refutes--> v13\n- v28 --refutes--> v0a\n- v0b --refutes--> v14\n- v27 --refutes--> v08\n- v11 --supports--> v09\n- v22 --supports--> v1a\n- v27 --refutes--> v28\n- v22 --refutes--> v25\n- v1a --supports--> v29\n- v00 --supports--> v24\n- v24 --supports--> v1f\n- v14 --supports--> v1d\n- v05 --refutes--> v1b\n- v1a --supports--> v0b\n- v2a --refutes--> v26\n- v17 --supports--> v18\n- v17 --supports--> v06\n- v12 --refutes--> v23\n- v2a --refutes--> v0f\n- v1e --refutes--> v1c\n- v20 --refutes--> v05\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"263c9dbb38345211d0d0e7cbed54200a329f09b92b42ebb2503ab82d11a2e118","schema":"livegraph.boolean-propagation.v1","spec_hash":"e7fa0610e93ef1cbf4f121509f93c33b1c284c17c7726805a5359f501cba6f44","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-4826e331d51a098021ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v22\n- v06 --refutes--> v0e\n- v15 --refutes--> v1e\n- v26 --refutes--> v00\n- v0b --refutes--> v29\n- v16 --refutes--> v1f\n- v11 --supports--> v12\n- v1f --refutes--> v02\n- v11 --supports--> v0f\n- v2a --refutes--> v07\n- v22 --refutes--> v09\n- v08 --supports--> v20\n- v1f --refutes--> v08\n- v22 --refutes--> v15\n- v1f --refutes--> v18\n- v0f --refutes--> v1b\n- v10 --refutes--> v1d\n- v0b --refutes--> v0c\n- v20 --refutes--> v26\n- v03 --refutes--> v05\n- v20 --supports--> v14\n- v1e --refutes--> v03\n- v0d --supports--> v1a\n- v07 --refutes--> v0a\n- v1f --supports--> v04\n- v05 --supports--> v06\n- v1b --refutes--> v17\n- v1a --supports--> v11\n- v1c --supports--> v13\n- v01 --supports--> v10\n- v1f --refutes--> v28\n- v0a --supports--> v0d\n- v0b --supports--> v01\n- v26 --refutes--> v21\n- v25 --supports--> v0b\n- v23 --supports--> v24\n- v0e --supports--> v2a\n- v1b --refutes--> v1c\n- v00 --supports--> v25\n- v11 --refutes--> v27\n- v23 --supports--> v16\n- v05 --refutes--> v19\n\n[VERIFIED ROOT]\nv23 = 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":"d72287e4708970d022aa415494b2138b598caa91c54eced4f63cc8ba3bf08ba5","schema":"livegraph.boolean-propagation.v1","spec_hash":"4826e331d51a098021efa4a5abc280b8451aabba9a997e912502c1ba4958fae4","spec_json":"{\"edges\":[{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-2075424fb498c335aadc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v26\n- v04 --refutes--> v12\n- v00 --supports--> v08\n- v05 --supports--> v0c\n- v11 --refutes--> v0b\n- v27 --supports--> v1a\n- v27 --supports--> v10\n- v06 --supports--> v0a\n- v01 --supports--> v1e\n- v19 --supports--> v05\n- v28 --supports--> v19\n- v14 --supports--> v23\n- v11 --supports--> v21\n- v23 --supports--> v17\n- v13 --supports--> v01\n- v11 --refutes--> v0e\n- v15 --refutes--> v02\n- v09 --refutes--> v22\n- v00 --supports--> v27\n- v06 --refutes--> v1f\n- v28 --supports--> v16\n- v1c --refutes--> v24\n- v01 --refutes--> v0d\n- v14 --supports--> v07\n- v10 --refutes--> v1c\n- v1d --supports--> v14\n- v25 --refutes--> v1d\n- v23 --supports--> v20\n- v0f --refutes--> v2a\n- v04 --refutes--> v29\n- v18 --supports--> v09\n- v2a --supports--> v13\n- v13 --refutes--> v03\n- v05 --supports--> v00\n- v17 --refutes--> v11\n- v0a --refutes--> v1b\n- v15 --refutes--> v25\n- v22 --refutes--> v06\n- v1e --refutes--> v15\n- v1c --refutes--> v0f\n- v1b --refutes--> v28\n- v12 --refutes--> v18\n\n[VERIFIED ROOT]\nv04 = FALSE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"352b49d8491517669e76431ffe7e4a00718ee64e8165673d4f673ed8c5359eba","schema":"livegraph.boolean-propagation.v1","spec_hash":"2075424fb498c335aadc9ef93418e5d091f09b5af71bda8398cf79fad25b58ce","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v04\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-a3c27491e999d3316cb8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v24\n- v20 --supports--> v13\n- v21 --refutes--> v23\n- v1c --refutes--> v1b\n- v17 --refutes--> v11\n- v0e --supports--> v20\n- v07 --supports--> v10\n- v27 --supports--> v14\n- v25 --supports--> v27\n- v02 --supports--> v26\n- v1a --supports--> v05\n- v05 --refutes--> v15\n- v02 --supports--> v28\n- v16 --refutes--> v07\n- v2a --supports--> v12\n- v1c --refutes--> v17\n- v24 --refutes--> v0d\n- v26 --refutes--> v0e\n- v07 --refutes--> v00\n- v0d --supports--> v18\n- v24 --supports--> v1d\n- v00 --refutes--> v08\n- v21 --supports--> v0f\n- v0f --refutes--> v16\n- v03 --refutes--> v22\n- v0d --refutes--> v06\n- v1e --supports--> v04\n- v14 --supports--> v09\n- v03 --refutes--> v2a\n- v20 --refutes--> v0c\n- v06 --refutes--> v1e\n- v15 --refutes--> v21\n- v1e --supports--> v1f\n- v0e --refutes--> v29\n- v11 --refutes--> v1a\n- v21 --supports--> v19\n- v09 --supports--> v03\n- v13 --supports--> v01\n- v1e --refutes--> v0a\n- v0e --refutes--> v0b\n- v00 --supports--> v25\n- v0a --refutes--> v1c\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5c1db10627a7ecf628841cf95221f36a1f619c2f6aac8ee902dd9f90d2703497","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3c27491e999d3316cb8ab11f8274aae3e61a1f46b03380119df55446568b640","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-2e5473c3d6885c251151","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v15\n- v14 --supports--> v29\n- v10 --refutes--> v0d\n- v13 --supports--> v09\n- v0c --supports--> v10\n- v05 --supports--> v0c\n- v10 --refutes--> v14\n- v01 --supports--> v12\n- v02 --refutes--> v24\n- v1b --supports--> v17\n- v29 --supports--> v03\n- v17 --supports--> v28\n- v1c --refutes--> v1b\n- v1e --supports--> v27\n- v2a --supports--> v01\n- v03 --supports--> v1f\n- v11 --refutes--> v02\n- v0a --supports--> v1a\n- v20 --supports--> v1d\n- v28 --refutes--> v0a\n- v1e --supports--> v2a\n- v09 --refutes--> v04\n- v14 --supports--> v18\n- v0e --refutes--> v00\n- v1a --refutes--> v05\n- v16 --supports--> v11\n- v12 --refutes--> v0f\n- v20 --supports--> v23\n- v05 --supports--> v06\n- v1f --refutes--> v20\n- v12 --supports--> v26\n- v02 --refutes--> v1c\n- v0c --supports--> v21\n- v00 --refutes--> v13\n- v2a --refutes--> v0b\n- v1d --refutes--> v08\n- v03 --supports--> v19\n- v26 --supports--> v16\n- v14 --refutes--> v07\n- v2a --refutes--> v25\n- v08 --refutes--> v0e\n- v05 --refutes--> v22\n\n[VERIFIED ROOT]\nv1e = 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":"f064a8a98f696c8079fb6aeced6400fcb8b897e2724879d68eb0036c81ecefca","schema":"livegraph.boolean-propagation.v1","spec_hash":"2e5473c3d6885c251151f53a6f745336e3dd93ff8d7ea017d5820dae3a688b2e","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-669149965a7c69f15d61","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v06\n- v19 --refutes--> v02\n- v22 --supports--> v04\n- v14 --refutes--> v1e\n- v26 --supports--> v27\n- v01 --supports--> v29\n- v27 --supports--> v0a\n- v1e --refutes--> v13\n- v01 --refutes--> v2a\n- v2a --refutes--> v20\n- v0e --refutes--> v01\n- v16 --supports--> v17\n- v00 --supports--> v1b\n- v1c --refutes--> v09\n- v1c --supports--> v05\n- v11 --supports--> v1c\n- v20 --supports--> v08\n- v27 --supports--> v0d\n- v06 --refutes--> v10\n- v14 --supports--> v21\n- v04 --supports--> v03\n- v0e --refutes--> v07\n- v08 --refutes--> v14\n- v09 --supports--> v26\n- v03 --refutes--> v00\n- v00 --refutes--> v19\n- v23 --supports--> v28\n- v16 --supports--> v1f\n- v0c --supports--> v22\n- v09 --refutes--> v1a\n- v0a --supports--> v15\n- v08 --refutes--> v0b\n- v15 --supports--> v12\n- v12 --refutes--> v23\n- v22 --refutes--> v1d\n- v13 --supports--> v11\n- v16 --supports--> v18\n- v02 --supports--> v0e\n- v24 --supports--> v16\n- v23 --refutes--> v24\n- v06 --supports--> v25\n- v1c --refutes--> v0f\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b9fbff5f3dfcb25594b6b6ed30d3549c3e63a2f9f0a9727a45afa44636469c8","schema":"livegraph.boolean-propagation.v1","spec_hash":"669149965a7c69f15d6175f995a649c8b62584e493df5db409146048eb42c228","spec_json":"{\"edges\":[{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-0f88bfbce4c3e19459bb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v14\n- v12 --refutes--> v22\n- v17 --supports--> v2a\n- v08 --supports--> v04\n- v05 --supports--> v16\n- v1f --supports--> v1c\n- v1a --supports--> v12\n- v19 --refutes--> v1b\n- v18 --supports--> v0f\n- v0d --refutes--> v08\n- v05 --refutes--> v02\n- v27 --supports--> v25\n- v25 --supports--> v0e\n- v1a --refutes--> v21\n- v20 --refutes--> v29\n- v1e --refutes--> v0a\n- v02 --supports--> v1f\n- v0d --refutes--> v06\n- v0e --refutes--> v18\n- v29 --supports--> v09\n- v26 --refutes--> v24\n- v29 --supports--> v23\n- v20 --supports--> v11\n- v0c --refutes--> v26\n- v0b --refutes--> v1d\n- v08 --supports--> v07\n- v26 --supports--> v20\n- v03 --supports--> v0d\n- v07 --refutes--> v05\n- v28 --supports--> v1a\n- v0f --refutes--> v0c\n- v19 --supports--> v15\n- v22 --refutes--> v17\n- v01 --refutes--> v0b\n- v0b --refutes--> v28\n- v2a --supports--> v27\n- v1c --refutes--> v19\n- v1e --refutes--> v03\n- v20 --refutes--> v10\n- v1b --supports--> v01\n- v20 --supports--> v13\n- v0f --supports--> v00\n\n[VERIFIED ROOT]\nv1e = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e4a500d652c19735a238889026cd1b9970a916913591043e0e6b80a70e2f5e4","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f88bfbce4c3e19459bb0cc355e188facbe5e51683b416a984e3f2dd9a7cd678","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-8d48a897873922ffcf86","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v27 --refutes--> v01\n- v28 --supports--> v18\n- v07 --supports--> v17\n- v14 --refutes--> v04\n- v1a --supports--> v06\n- v19 --refutes--> v23\n- v14 --refutes--> v2a\n- v0f --supports--> v08\n- v07 --refutes--> v09\n- v25 --supports--> v28\n- v1e --refutes--> v15\n- v06 --supports--> v0a\n- v16 --refutes--> v1b\n- v13 --supports--> v21\n- v1d --refutes--> v27\n- v13 --supports--> v0c\n- v24 --refutes--> v11\n- v25 --supports--> v10\n- v29 --refutes--> v0f\n- v05 --supports--> v02\n- v0a --supports--> v00\n- v08 --refutes--> v13\n- v1c --refutes--> v19\n- v15 --supports--> v07\n- v22 --supports--> v1e\n- v16 --refutes--> v20\n- v0d --refutes--> v29\n- v1b --supports--> v05\n- v26 --supports--> v25\n- v04 --supports--> v1a\n- v07 --supports--> v03\n- v19 --supports--> v16\n- v15 --refutes--> v1f\n- v1d --refutes--> v12\n- v28 --supports--> v14\n- v0f --refutes--> v0e\n- v02 --supports--> v0d\n- v0c --refutes--> v1d\n- v27 --refutes--> v26\n- v0a --refutes--> v22\n- v03 --refutes--> v24\n\n[VERIFIED ROOT]\nv1c = 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":"27718797f6e381137e36d07c70d7b8b808f749f7c0e46a2c63a3986b456f1509","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d48a897873922ffcf8641461a856047555b6c707aa4ec388fd5f36d9dada1ec","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-7e9210d6ea6969931067","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v08 --supports--> v24\n- v21 --supports--> v18\n- v25 --refutes--> v26\n- v0f --refutes--> v12\n- v21 --refutes--> v1c\n- v26 --refutes--> v09\n- v2a --supports--> v0e\n- v0a --refutes--> v21\n- v1b --supports--> v00\n- v13 --supports--> v1b\n- v11 --supports--> v2a\n- v24 --supports--> v05\n- v05 --supports--> v0a\n- v0e --supports--> v13\n- v00 --refutes--> v29\n- v1c --supports--> v23\n- v26 --refutes--> v04\n- v14 --refutes--> v0c\n- v12 --supports--> v0b\n- v1b --supports--> v15\n- v1b --supports--> v1e\n- v17 --refutes--> v06\n- v1a --supports--> v1f\n- v10 --refutes--> v01\n- v0b --refutes--> v14\n- v26 --supports--> v10\n- v1d --refutes--> v25\n- v29 --supports--> v16\n- v20 --refutes--> v17\n- v0f --supports--> v22\n- v1d --supports--> v02\n- v29 --refutes--> v20\n- v2a --refutes--> v07\n- v0a --refutes--> v19\n- v14 --supports--> v1a\n- v0a --supports--> v03\n- v0d --supports--> v11\n- v23 --supports--> v0d\n- v0a --supports--> v27\n- v13 --supports--> v28\n- v06 --refutes--> v1d\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv1f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"04e1e15e6a5b5ce0d682dd2eea2771962debca15791c1d970873fa7835d060df","schema":"livegraph.boolean-propagation.v1","spec_hash":"7e9210d6ea6969931067fdc42837000bdb1dbb077ac056a7b940390d0f0d6765","spec_json":"{\"edges\":[{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-057e79182acc12944909","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --supports--> v08\n- v02 --supports--> v1f\n- v21 --supports--> v16\n- v0b --refutes--> v24\n- v09 --refutes--> v11\n- v01 --supports--> v2a\n- v03 --refutes--> v07\n- v20 --supports--> v25\n- v05 --supports--> v20\n- v05 --refutes--> v06\n- v1d --refutes--> v19\n- v23 --supports--> v1c\n- v14 --supports--> v0c\n- v21 --supports--> v03\n- v0f --refutes--> v0b\n- v19 --refutes--> v17\n- v27 --refutes--> v0a\n- v14 --supports--> v28\n- v1a --supports--> v23\n- v02 --supports--> v12\n- v17 --supports--> v05\n- v1f --supports--> v10\n- v11 --refutes--> v27\n- v17 --refutes--> v1e\n- v04 --supports--> v01\n- v20 --supports--> v13\n- v03 --refutes--> v09\n- v2a --refutes--> v14\n- v29 --refutes--> v15\n- v0c --refutes--> v18\n- v27 --supports--> v29\n- v03 --supports--> v26\n- v18 --refutes--> v00\n- v29 --refutes--> v22\n- v1f --refutes--> v1b\n- v1f --supports--> v1d\n- v27 --refutes--> v0d\n- v24 --refutes--> v21\n- v1c --refutes--> v0e\n- v08 --supports--> v0f\n- v15 --refutes--> v1a\n- v0e --refutes--> 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":"c7fc291109e2aa09d78abfd2f60ccf6bf2d6c0399124cdc5583d34464741ce90","schema":"livegraph.boolean-propagation.v1","spec_hash":"057e79182acc129449092aa6e2c489310b4a57e4bd61eff655204918596e5d0d","spec_json":"{\"edges\":[{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-e5f27b6cc2fa27940a47","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v27 --refutes--> v1e\n- v1b --refutes--> v19\n- v15 --supports--> v09\n- v04 --supports--> v14\n- v19 --refutes--> v21\n- v29 --refutes--> v1f\n- v16 --refutes--> v18\n- v02 --supports--> v05\n- v09 --refutes--> v06\n- v29 --supports--> v1a\n- v0d --supports--> v08\n- v1d --supports--> v0c\n- v25 --refutes--> v0d\n- v02 --refutes--> v25\n- v0b --refutes--> v16\n- v1c --refutes--> v02\n- v09 --supports--> v0a\n- v22 --supports--> v29\n- v03 --supports--> v11\n- v21 --supports--> v07\n- v1a --refutes--> v03\n- v01 --refutes--> v23\n- v13 --refutes--> v17\n- v12 --refutes--> v1c\n- v1a --refutes--> v00\n- v07 --refutes--> v0b\n- v1d --refutes--> v22\n- v18 --supports--> v04\n- v17 --supports--> v10\n- v25 --supports--> v20\n- v27 --supports--> v1b\n- v1a --supports--> v26\n- v1d --supports--> v2a\n- v13 --supports--> v0e\n- v01 --refutes--> v1d\n- v16 --refutes--> v28\n- v1d --refutes--> v24\n- v14 --supports--> v12\n- v10 --refutes--> v15\n- v08 --refutes--> v01\n- v11 --supports--> v13\n- v17 --refutes--> v0f\n\n[VERIFIED ROOT]\nv27 = 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":"93afe436e6244ab38422e221edcf4e8794a5918e96661882f696b9851e6e4bda","schema":"livegraph.boolean-propagation.v1","spec_hash":"e5f27b6cc2fa27940a470bb79e8948ffa15c198f0c841467dec57cc163cc98cb","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-60cd62c74c6a43008774","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v1d --supports--> v26\n- v05 --supports--> v11\n- v09 --supports--> v1c\n- v29 --refutes--> v13\n- v10 --supports--> v24\n- v11 --supports--> v01\n- v0f --refutes--> v10\n- v10 --supports--> v1b\n- v0a --refutes--> v0c\n- v13 --supports--> v03\n- v28 --supports--> v29\n- v1d --supports--> v28\n- v1f --supports--> v1a\n- v0c --supports--> v25\n- v16 --supports--> v1f\n- v0b --refutes--> v1e\n- v15 --supports--> v21\n- v29 --supports--> v08\n- v29 --refutes--> v18\n- v20 --supports--> v22\n- v15 --supports--> v14\n- v0b --supports--> v0f\n- v0d --supports--> v12\n- v04 --supports--> v16\n- v20 --refutes--> v19\n- v12 --supports--> v0b\n- v0a --supports--> v07\n- v03 --supports--> v0a\n- v1b --refutes--> v2a\n- v06 --refutes--> v15\n- v19 --refutes--> v0d\n- v25 --supports--> v05\n- v29 --supports--> v0e\n- v1c --refutes--> v06\n- v11 --refutes--> v04\n- v13 --supports--> v17\n- v00 --supports--> v02\n- v1b --supports--> v00\n- v1f --refutes--> v23\n- v02 --supports--> v27\n- v1a --supports--> v09\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2bacc7326234ecc3318ec9a5f043f5c227335437fcdc74c6299df02b6f3575a9","schema":"livegraph.boolean-propagation.v1","spec_hash":"60cd62c74c6a43008774f3d120f5b92b1c735af6728fd09b38abbcb06eb6a2ef","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-5b814771b9a74dad0754","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v29 --supports--> v23\n- v03 --refutes--> v2a\n- v07 --supports--> v13\n- v1a --supports--> v0e\n- v15 --refutes--> v20\n- v12 --refutes--> v01\n- v1b --refutes--> v0d\n- v04 --refutes--> v03\n- v17 --supports--> v0c\n- v17 --refutes--> v08\n- v09 --supports--> v17\n- v1b --refutes--> v22\n- v15 --refutes--> v11\n- v11 --supports--> v24\n- v19 --supports--> v1a\n- v1f --supports--> v10\n- v10 --supports--> v15\n- v1b --refutes--> v14\n- v02 --supports--> v27\n- v07 --refutes--> v1c\n- v14 --supports--> v04\n- v0f --refutes--> v09\n- v25 --refutes--> v1f\n- v06 --supports--> v1b\n- v09 --supports--> v26\n- v01 --refutes--> v1d\n- v07 --refutes--> v05\n- v27 --refutes--> v25\n- v1a --refutes--> v00\n- v0e --supports--> v0a\n- v17 --refutes--> v1e\n- v11 --refutes--> v07\n- v21 --supports--> v19\n- v0b --refutes--> v06\n- v13 --supports--> v18\n- v18 --refutes--> v12\n- v06 --supports--> v28\n- v1d --refutes--> v0f\n- v27 --supports--> v16\n- v0c --supports--> v29\n- v03 --supports--> v02\n- v29 --supports--> v21\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":"0c690db58bbf3d74980b90b8c93d009e43321b8903fd2e06dd55c24a963f8a17","schema":"livegraph.boolean-propagation.v1","spec_hash":"5b814771b9a74dad075450cbd499381c64f1de8f78112be9433a3be70d7b2339","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-f929ef443dd03a59b48c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v14 --refutes--> v27\n- v25 --supports--> v19\n- v20 --refutes--> v13\n- v24 --refutes--> v03\n- v19 --supports--> v0a\n- v1c --refutes--> v25\n- v10 --refutes--> v00\n- v06 --refutes--> v22\n- v25 --refutes--> v29\n- v00 --refutes--> v11\n- v22 --refutes--> v08\n- v12 --refutes--> v1f\n- v03 --supports--> v17\n- v22 --supports--> v15\n- v00 --refutes--> v02\n- v26 --supports--> v01\n- v01 --supports--> v18\n- v11 --refutes--> v0d\n- v0d --refutes--> v1c\n- v1b --supports--> v0f\n- v20 --refutes--> v1b\n- v03 --supports--> v12\n- v22 --supports--> v1e\n- v04 --supports--> v2a\n- v06 --refutes--> v23\n- v1f --refutes--> v16\n- v27 --supports--> v0c\n- v0a --refutes--> v20\n- v0c --refutes--> v07\n- v0f --supports--> v26\n- v01 --refutes--> v0e\n- v16 --refutes--> v0b\n- v05 --supports--> v14\n- v27 --refutes--> v21\n- v01 --supports--> v24\n- v0b --supports--> v06\n- v08 --supports--> v04\n- v2a --supports--> v05\n- v1f --supports--> v09\n- v10 --refutes--> v1d\n- v05 --refutes--> v28\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90e8332ef4e795e405b017cd3ccab7040377360ba678f6ea8e4fe30a0299f159","schema":"livegraph.boolean-propagation.v1","spec_hash":"f929ef443dd03a59b48c8c7bab0def3e80b6cdb9416524604055fdf6a0d85de9","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v10\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-c90b78b25f99e4b9c0ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v25 --supports--> v09\n- v15 --supports--> v19\n- v17 --refutes--> v2a\n- v02 --refutes--> v23\n- v03 --supports--> v0e\n- v1c --supports--> v17\n- v15 --refutes--> v14\n- v10 --refutes--> v1e\n- v17 --refutes--> v00\n- v18 --refutes--> v0f\n- v03 --refutes--> v16\n- v23 --refutes--> v0b\n- v01 --refutes--> v28\n- v16 --refutes--> v12\n- v22 --supports--> v03\n- v15 --supports--> v21\n- v14 --refutes--> v06\n- v14 --supports--> v1d\n- v14 --refutes--> v18\n- v05 --refutes--> v20\n- v04 --refutes--> v25\n- v07 --supports--> v22\n- v1c --supports--> v1b\n- v0c --refutes--> v02\n- v2a --refutes--> v26\n- v29 --refutes--> v05\n- v01 --refutes--> v04\n- v08 --refutes--> v1c\n- v13 --supports--> v0d\n- v0a --refutes--> v07\n- v10 --supports--> v27\n- v2a --refutes--> v01\n- v27 --refutes--> v13\n- v1f --refutes--> v08\n- v09 --supports--> v0a\n- v1a --refutes--> v0c\n- v2a --supports--> v11\n- v12 --refutes--> v10\n- v29 --refutes--> v24\n- v20 --refutes--> v1f\n- v18 --refutes--> v29\n- v23 --supports--> v15\n\n[VERIFIED ROOT]\nv1a = 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":"07b8f74e2549ac8676a893addd297c81ec7dd43bb5db4c002b7c291f1ca3e2c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"c90b78b25f99e4b9c0ffa8c1d9a9d2c7005f7f915df26a72a3065ab70331dbe2","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-61cb89417a0a3193e81e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0b\n- v26 --supports--> v15\n- v0f --refutes--> v22\n- v04 --supports--> v21\n- v1e --refutes--> v0c\n- v15 --supports--> v1e\n- v13 --refutes--> v1b\n- v19 --refutes--> v07\n- v0a --supports--> v25\n- v0e --refutes--> v0f\n- v1d --refutes--> v09\n- v08 --refutes--> v17\n- v07 --supports--> v03\n- v1b --supports--> v1a\n- v0d --supports--> v28\n- v08 --supports--> v11\n- v01 --supports--> v16\n- v05 --supports--> v1c\n- v27 --supports--> v0a\n- v08 --supports--> v20\n- v16 --supports--> v04\n- v25 --supports--> v02\n- v04 --refutes--> v23\n- v18 --refutes--> v0d\n- v22 --refutes--> v12\n- v00 --supports--> v01\n- v1d --supports--> v0e\n- v22 --refutes--> v14\n- v19 --supports--> v1f\n- v21 --supports--> v18\n- v02 --supports--> v00\n- v0a --supports--> v29\n- v20 --supports--> v13\n- v1a --refutes--> v19\n- v14 --refutes--> v27\n- v0c --supports--> v08\n- v0a --supports--> v10\n- v00 --supports--> v24\n- v07 --supports--> v1d\n- v16 --refutes--> v2a\n- v08 --refutes--> v06\n- v05 --supports--> v26\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv28\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f66c82cd9fecb23abf3b7e27f556fe9b0ff77ec74e9b95771071d5bca1f48f32","schema":"livegraph.boolean-propagation.v1","spec_hash":"61cb89417a0a3193e81e0846924b4e717159ed2be2fbe74650d297c148319983","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v28\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-cd29eb1f3ab2053d93f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v10\n- v06 --refutes--> v25\n- v10 --supports--> v1d\n- v1e --refutes--> v0d\n- v18 --supports--> v0c\n- v1b --refutes--> v22\n- v00 --refutes--> v21\n- v1d --refutes--> v24\n- v1a --refutes--> v11\n- v05 --supports--> v03\n- v0b --supports--> v00\n- v29 --supports--> v02\n- v27 --supports--> v0f\n- v27 --refutes--> v28\n- v00 --refutes--> v09\n- v27 --refutes--> v19\n- v23 --refutes--> v1b\n- v13 --supports--> v18\n- v24 --refutes--> v0e\n- v1d --supports--> v0a\n- v29 --refutes--> v12\n- v2a --supports--> v04\n- v14 --supports--> v23\n- v1e --refutes--> v27\n- v03 --supports--> v26\n- v01 --refutes--> v1a\n- v03 --refutes--> v17\n- v0c --supports--> v05\n- v09 --supports--> v14\n- v25 --refutes--> v29\n- v28 --supports--> v2a\n- v16 --refutes--> v0b\n- v12 --supports--> v13\n- v1b --refutes--> v1e\n- v11 --supports--> v16\n- v14 --supports--> v1c\n- v0b --supports--> v20\n- v11 --supports--> v1f\n- v24 --supports--> v07\n- v1b --supports--> v15\n- v0e --supports--> v01\n- v23 --supports--> v08\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":"8f1c718eb20aaad1523ec4c70d30178380b8239e5bfb7191a706dc29efea8082","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd29eb1f3ab2053d93f1ad4a796c690dec443f1735a80ce1d9c6197d812bf884","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-0d3095869142fd192600","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v00 --refutes--> v17\n- v2a --supports--> v29\n- v27 --refutes--> v19\n- v11 --refutes--> v0c\n- v15 --refutes--> v08\n- v1c --refutes--> v12\n- v04 --supports--> v00\n- v29 --refutes--> v14\n- v02 --supports--> v28\n- v26 --supports--> v0e\n- v19 --supports--> v15\n- v15 --supports--> v0a\n- v20 --refutes--> v22\n- v0e --supports--> v1f\n- v02 --supports--> v05\n- v1f --refutes--> v16\n- v17 --supports--> v09\n- v1a --supports--> v1d\n- v1d --refutes--> v01\n- v1d --supports--> v06\n- v1c --supports--> v20\n- v01 --refutes--> v1b\n- v07 --refutes--> v27\n- v15 --supports--> v0d\n- v13 --supports--> v11\n- v15 --refutes--> v23\n- v25 --supports--> v21\n- v17 --refutes--> v03\n- v1f --refutes--> v02\n- v0c --refutes--> v2a\n- v23 --supports--> v24\n- v13 --supports--> v18\n- v22 --refutes--> v04\n- v10 --supports--> v1a\n- v21 --refutes--> v10\n- v01 --supports--> v07\n- v1c --supports--> v0b\n- v03 --refutes--> v13\n- v27 --supports--> v0f\n- v14 --supports--> v26\n- v11 --refutes--> v1e\n- v23 --refutes--> v1c\n\n[VERIFIED ROOT]\nv25 = 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":"b1132fe7897c131ed394258d0b12dff87102b926b9de42464ba7a4eab2103a84","schema":"livegraph.boolean-propagation.v1","spec_hash":"0d3095869142fd19260024de59c35b6d5017412f731cb785cdd87b3a0fbe78b5","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v25\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-fc87ddb22e90197c9d56","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v11\n- v2a --refutes--> v20\n- v0e --supports--> v05\n- v20 --refutes--> v16\n- v00 --supports--> v02\n- v25 --refutes--> v22\n- v19 --supports--> v28\n- v16 --supports--> v12\n- v01 --refutes--> v09\n- v07 --supports--> v24\n- v29 --refutes--> v03\n- v12 --supports--> v1f\n- v0e --refutes--> v17\n- v15 --refutes--> v1e\n- v0b --supports--> v13\n- v29 --refutes--> v27\n- v06 --supports--> v0f\n- v21 --supports--> v0a\n- v0b --refutes--> v0e\n- v22 --supports--> v01\n- v0e --refutes--> v2a\n- v06 --refutes--> v08\n- v0e --refutes--> v1b\n- v00 --supports--> v06\n- v12 --supports--> v0c\n- v1a --refutes--> v00\n- v23 --refutes--> v25\n- v15 --refutes--> v10\n- v04 --supports--> v15\n- v0e --refutes--> v1d\n- v03 --supports--> v18\n- v10 --supports--> v07\n- v24 --supports--> v14\n- v08 --supports--> v29\n- v18 --refutes--> v23\n- v14 --supports--> v1c\n- v19 --supports--> v0d\n- v21 --supports--> v04\n- v19 --supports--> v0b\n- v11 --supports--> v26\n- v1c --supports--> v1a\n- v12 --refutes--> v21\n\n[VERIFIED ROOT]\nv19 = TRUE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e46d301506dc61871a9615942cd418c6fa001439a8f4ea099a417715709ffeee","schema":"livegraph.boolean-propagation.v1","spec_hash":"fc87ddb22e90197c9d563d879c1fe3c57f4336212d6265f5a9325951c56968ee","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-1df39be3ac49b3669d74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --refutes--> v0b\n- v28 --supports--> v1c\n- v13 --refutes--> v1b\n- v19 --refutes--> v0d\n- v12 --supports--> v28\n- v15 --supports--> v00\n- v13 --supports--> v24\n- v1f --supports--> v15\n- v27 --refutes--> v03\n- v0d --supports--> v0f\n- v14 --supports--> v10\n- v28 --refutes--> v14\n- v09 --supports--> v0c\n- v0b --supports--> v1e\n- v05 --supports--> v1d\n- v1a --refutes--> v23\n- v21 --supports--> v26\n- v04 --refutes--> v21\n- v2a --refutes--> v17\n- v0d --supports--> v0a\n- v08 --refutes--> v13\n- v21 --supports--> v22\n- v0c --supports--> v01\n- v22 --refutes--> v1a\n- v1a --refutes--> v2a\n- v00 --supports--> v02\n- v10 --refutes--> v27\n- v0d --supports--> v07\n- v15 --supports--> v11\n- v16 --refutes--> v25\n- v0c --supports--> v0e\n- v17 --refutes--> v08\n- v06 --refutes--> v1f\n- v05 --refutes--> v04\n- v1f --refutes--> v18\n- v12 --refutes--> v29\n- v0d --refutes--> v05\n- v00 --supports--> v12\n- v25 --supports--> v06\n- v1b --supports--> v20\n- v1e --supports--> v16\n- v03 --refutes--> v09\n\n[VERIFIED ROOT]\nv19 = 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":"c43163e35115b97a4966133724f9369a5a22d273bae4370eb63420c02c3cb076","schema":"livegraph.boolean-propagation.v1","spec_hash":"1df39be3ac49b3669d74ceeada81ce9e8d35ab1d2e848f16d5c577d02365d248","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v19\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-17c494720e4cd9ec516d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v04\n- v0c --refutes--> v0d\n- v12 --refutes--> v0a\n- v15 --supports--> v14\n- v04 --refutes--> v1c\n- v04 --refutes--> v0c\n- v21 --supports--> v12\n- v0b --refutes--> v09\n- v12 --supports--> v25\n- v29 --refutes--> v08\n- v03 --refutes--> v19\n- v14 --refutes--> v1f\n- v02 --supports--> v21\n- v20 --supports--> v1e\n- v15 --refutes--> v23\n- v1e --supports--> v0e\n- v28 --refutes--> v1b\n- v14 --supports--> v06\n- v1d --refutes--> v07\n- v07 --refutes--> v10\n- v0e --supports--> v27\n- v2a --supports--> v28\n- v1b --supports--> v20\n- v0b --supports--> v18\n- v0f --refutes--> v02\n- v1f --refutes--> v17\n- v17 --supports--> v1a\n- v0d --supports--> v24\n- v29 --supports--> v15\n- v05 --supports--> v01\n- v07 --supports--> v16\n- v03 --refutes--> v05\n- v0f --refutes--> v13\n- v14 --supports--> v22\n- v12 --refutes--> v1d\n- v01 --refutes--> v26\n- v0d --refutes--> v29\n- v16 --supports--> v0b\n- v1a --supports--> v0f\n- v0e --supports--> v03\n- v1d --supports--> v00\n- v26 --refutes--> v11\n\n[VERIFIED ROOT]\nv2a = 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":"984a38b8b3a0e01b8ddf0d1451a4d43a891fb149bd62540f07b1926be753927a","schema":"livegraph.boolean-propagation.v1","spec_hash":"17c494720e4cd9ec516d3a0d715547b73d64e5f714bcba4b524cfb8b00946150","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v2a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-6c52cac245f946fa6a06","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v0a\n- v27 --supports--> v1f\n- v0e --refutes--> v22\n- v20 --supports--> v14\n- v1d --refutes--> v09\n- v1b --supports--> v0c\n- v14 --refutes--> v03\n- v18 --refutes--> v29\n- v1a --refutes--> v17\n- v18 --refutes--> v16\n- v13 --refutes--> v28\n- v15 --supports--> v05\n- v27 --refutes--> v08\n- v1c --supports--> v0d\n- v07 --refutes--> v23\n- v04 --supports--> v25\n- v03 --refutes--> v21\n- v2a --supports--> v24\n- v03 --refutes--> v00\n- v24 --supports--> v06\n- v23 --refutes--> v1e\n- v1b --refutes--> v11\n- v22 --refutes--> v15\n- v1a --refutes--> v18\n- v18 --supports--> v26\n- v04 --supports--> v12\n- v0b --refutes--> v07\n- v13 --refutes--> v1b\n- v06 --refutes--> v13\n- v11 --supports--> v0e\n- v05 --supports--> v27\n- v0a --refutes--> v1a\n- v06 --refutes--> v10\n- v09 --supports--> v1c\n- v09 --supports--> v19\n- v09 --refutes--> v0f\n- v00 --refutes--> v1d\n- v0d --supports--> v0b\n- v24 --refutes--> v02\n- v27 --supports--> v20\n- v16 --refutes--> v04\n- v04 --supports--> v01\n\n[VERIFIED ROOT]\nv2a = 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":"220e80c072edd062987caf11b2cead848465432952b1c21f1cd4f97e05f3138b","schema":"livegraph.boolean-propagation.v1","spec_hash":"6c52cac245f946fa6a06b09a25fcd77dd3c3648fc251726543fb69823dd8214f","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v2a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f0e1c39b5b9e41b6f195","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v01\n- v25 --refutes--> v03\n- v0f --supports--> v27\n- v0c --supports--> v1f\n- v22 --supports--> v04\n- v09 --refutes--> v00\n- v15 --refutes--> v21\n- v17 --refutes--> v11\n- v1a --supports--> v06\n- v0c --refutes--> v17\n- v14 --refutes--> v0d\n- v00 --supports--> v25\n- v18 --supports--> v1a\n- v11 --supports--> v1c\n- v27 --refutes--> v18\n- v11 --refutes--> v14\n- v23 --supports--> v16\n- v1d --supports--> v0e\n- v06 --refutes--> v1b\n- v0d --refutes--> v1d\n- v08 --refutes--> v0b\n- v0b --refutes--> v15\n- v16 --supports--> v24\n- v03 --refutes--> v2a\n- v1e --supports--> v0c\n- v04 --supports--> v13\n- v0e --supports--> v10\n- v0e --refutes--> v0f\n- v22 --supports--> v29\n- v27 --supports--> v02\n- v20 --supports--> v09\n- v24 --refutes--> v1e\n- v12 --refutes--> v08\n- v1a --supports--> v0a\n- v1d --refutes--> v26\n- v06 --refutes--> v07\n- v15 --refutes--> v22\n- v24 --refutes--> v19\n- v22 --supports--> v28\n- v1b --refutes--> v12\n- v03 --refutes--> v23\n- v00 --refutes--> v05\n\n[VERIFIED ROOT]\nv20 = 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":"af12ee920481f7990c725b2ed55539ff6234d02e29d8502de2203157c3b2def4","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0e1c39b5b9e41b6f19559e5ad9d2e2d949bfccb45c423a118dc6c1b4ea28542","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-033f48f7043f49cc03a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v04\n- v10 --supports--> v1a\n- v18 --supports--> v03\n- v08 --refutes--> v21\n- v17 --refutes--> v24\n- v1b --refutes--> v1f\n- v0c --supports--> v12\n- v0e --refutes--> v0f\n- v1b --refutes--> v01\n- v1c --supports--> v25\n- v0a --refutes--> v17\n- v0f --supports--> v11\n- v02 --refutes--> v29\n- v1a --supports--> v05\n- v1d --refutes--> v2a\n- v21 --refutes--> v18\n- v09 --refutes--> v1b\n- v2a --refutes--> v00\n- v11 --refutes--> v1d\n- v07 --supports--> v0e\n- v1b --supports--> v13\n- v00 --supports--> v10\n- v20 --supports--> v15\n- v02 --refutes--> v0d\n- v0a --refutes--> v16\n- v20 --refutes--> v0c\n- v00 --refutes--> v0b\n- v18 --refutes--> v22\n- v07 --supports--> v1e\n- v29 --refutes--> v07\n- v24 --supports--> v14\n- v10 --refutes--> v06\n- v27 --supports--> v0a\n- v05 --supports--> v09\n- v01 --supports--> v23\n- v12 --refutes--> v08\n- v04 --refutes--> v26\n- v26 --supports--> v27\n- v04 --refutes--> v28\n- v01 --refutes--> v20\n- v08 --refutes--> v19\n- v24 --refutes--> v02\n\n[VERIFIED ROOT]\nv1c = 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":"6d74e49971e7de093c230c28b69cee35294444164a3c70b7147112189113546c","schema":"livegraph.boolean-propagation.v1","spec_hash":"033f48f7043f49cc03a762f8d183c72e57d2a1daa718da3ba6285478291fb764","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-142712b3c1960e2fd9d7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v17 --refutes--> v21\n- v1a --supports--> v0b\n- v18 --supports--> v01\n- v0b --supports--> v05\n- v0b --refutes--> v11\n- v23 --refutes--> v08\n- v0e --refutes--> v1b\n- v1f --refutes--> v1e\n- v21 --refutes--> v27\n- v13 --supports--> v24\n- v24 --supports--> v0e\n- v15 --supports--> v28\n- v0b --refutes--> v06\n- v08 --refutes--> v2a\n- v04 --supports--> v1d\n- v1d --supports--> v10\n- v00 --supports--> v25\n- v19 --supports--> v20\n- v1e --refutes--> v29\n- v06 --supports--> v12\n- v0e --refutes--> v0a\n- v21 --refutes--> v0f\n- v18 --supports--> v16\n- v02 --refutes--> v1f\n- v2a --supports--> v04\n- v10 --refutes--> v1c\n- v0d --refutes--> v03\n- v12 --supports--> v0d\n- v26 --supports--> v09\n- v19 --refutes--> v26\n- v15 --supports--> v17\n- v0e --supports--> v22\n- v1c --refutes--> v0c\n- v0d --supports--> v19\n- v00 --refutes--> v02\n- v09 --refutes--> v13\n- v18 --supports--> v15\n- v1d --refutes--> v07\n- v0f --supports--> v23\n- v29 --refutes--> v18\n- v00 --refutes--> v14\n\n[VERIFIED ROOT]\nv1a = 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":"95f212de15a991a36e87e97e268ad0be18210fdd18c5ed6ed3a40b39a0c5b42d","schema":"livegraph.boolean-propagation.v1","spec_hash":"142712b3c1960e2fd9d736500fb3cf1035735d041da69db961e3761f43e2e35b","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-8f9d2d4bf055ba4a484e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v07\n- v05 --supports--> v10\n- v11 --supports--> v04\n- v19 --refutes--> v29\n- v1e --supports--> v0f\n- v00 --refutes--> v0a\n- v24 --refutes--> v2a\n- v0d --supports--> v09\n- v20 --supports--> v06\n- v16 --refutes--> v19\n- v04 --supports--> v25\n- v06 --refutes--> v28\n- v1c --refutes--> v05\n- v0e --refutes--> v14\n- v14 --supports--> v0c\n- v01 --refutes--> v1b\n- v0d --refutes--> v0b\n- v13 --refutes--> v1e\n- v1e --refutes--> v12\n- v29 --supports--> v0d\n- v05 --supports--> v03\n- v15 --supports--> v24\n- v02 --supports--> v08\n- v04 --supports--> v26\n- v16 --supports--> v1f\n- v08 --supports--> v13\n- v12 --supports--> v16\n- v0c --supports--> v01\n- v19 --refutes--> v17\n- v19 --supports--> v27\n- v24 --refutes--> v1a\n- v01 --refutes--> v00\n- v00 --supports--> v21\n- v25 --supports--> v0e\n- v14 --supports--> v23\n- v09 --supports--> v11\n- v1d --refutes--> v18\n- v1d --supports--> v1c\n- v22 --refutes--> v1d\n- v21 --refutes--> v20\n- v05 --refutes--> v15\n- v28 --refutes--> v22\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b21ef4d0230b4cc94d431d0b7174f2a0a56100ca35d9d74e8e9ef4d8a2c1a13b","schema":"livegraph.boolean-propagation.v1","spec_hash":"8f9d2d4bf055ba4a484e5b17845fcbfe6f82b55026c64864f216079c30d24dec","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f0083d865083a99460d5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v11\n- v22 --supports--> v1d\n- v19 --refutes--> v2a\n- v07 --refutes--> v01\n- v29 --supports--> v03\n- v28 --supports--> v0a\n- v19 --refutes--> v02\n- v03 --supports--> v15\n- v24 --supports--> v19\n- v04 --refutes--> v23\n- v20 --refutes--> v0d\n- v01 --refutes--> v27\n- v24 --supports--> v17\n- v0f --refutes--> v13\n- v11 --refutes--> v21\n- v18 --refutes--> v1e\n- v21 --refutes--> v24\n- v14 --supports--> v28\n- v23 --refutes--> v14\n- v0e --refutes--> v06\n- v1f --refutes--> v04\n- v00 --refutes--> v0e\n- v04 --supports--> v0c\n- v22 --supports--> v1b\n- v14 --supports--> v09\n- v01 --supports--> v1c\n- v22 --refutes--> v20\n- v0b --refutes--> v25\n- v20 --refutes--> v29\n- v16 --refutes--> v00\n- v27 --refutes--> v05\n- v28 --supports--> v22\n- v1a --refutes--> v18\n- v26 --refutes--> v0b\n- v0f --refutes--> v12\n- v1e --refutes--> v08\n- v0f --refutes--> v10\n- v03 --supports--> v16\n- v0e --supports--> v0f\n- v0b --refutes--> v07\n- v02 --supports--> v1a\n- v08 --refutes--> v1f\n\n[VERIFIED ROOT]\nv26 = 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":"589caa9753b44a126440b6965a6e61edd9fad8fb59afbd88fdc2f1dff135632f","schema":"livegraph.boolean-propagation.v1","spec_hash":"f0083d865083a99460d5c5a0d8869e7b4bb923a4ffc6ee99af24524d8cd30bd4","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v26\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-71776bd0628aa8b53e48","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2a --supports--> v1a\n- v05 --refutes--> v18\n- v16 --refutes--> v20\n- v06 --refutes--> v24\n- v05 --refutes--> v22\n- v12 --refutes--> v0c\n- v1e --refutes--> v17\n- v02 --supports--> v05\n- v12 --supports--> v11\n- v1d --refutes--> v28\n- v0b --refutes--> v0d\n- v17 --refutes--> v2a\n- v1c --refutes--> v25\n- v04 --supports--> v1f\n- v03 --refutes--> v12\n- v17 --supports--> v1b\n- v00 --refutes--> v0e\n- v00 --supports--> v09\n- v0c --refutes--> v21\n- v00 --supports--> v23\n- v21 --supports--> v04\n- v16 --refutes--> v01\n- v1c --supports--> v03\n- v29 --supports--> v02\n- v05 --supports--> v16\n- v00 --refutes--> v1d\n- v1e --supports--> v13\n- v0f --supports--> v1e\n- v10 --refutes--> v06\n- v27 --refutes--> v0a\n- v14 --refutes--> v1c\n- v16 --supports--> v15\n- v15 --refutes--> v0f\n- v12 --refutes--> v26\n- v1d --refutes--> v07\n- v0d --refutes--> v27\n- v1a --refutes--> v10\n- v1f --refutes--> v0b\n- v0a --refutes--> v19\n- v24 --refutes--> v14\n- v2a --refutes--> v08\n- v28 --supports--> v29\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"898be0dd6134c9b9df083781e35b0dbd5a5f0078d3773e421067893852e485c9","schema":"livegraph.boolean-propagation.v1","spec_hash":"71776bd0628aa8b53e486716268bab05a55a2c572dec4f3a745308bcb15b64de","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-8b6324dda7c978d5b766","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v22\n- v29 --supports--> v08\n- v27 --supports--> v28\n- v18 --refutes--> v02\n- v1b --supports--> v0a\n- v0c --refutes--> v18\n- v26 --supports--> v05\n- v16 --supports--> v19\n- v13 --supports--> v21\n- v01 --refutes--> v0b\n- v12 --supports--> v1d\n- v21 --refutes--> v23\n- v21 --supports--> v0f\n- v11 --refutes--> v12\n- v10 --supports--> v27\n- v07 --refutes--> v17\n- v18 --supports--> v11\n- v04 --refutes--> v20\n- v21 --supports--> v10\n- v04 --refutes--> v00\n- v03 --refutes--> v1f\n- v0a --supports--> v25\n- v1f --refutes--> v06\n- v00 --refutes--> v07\n- v16 --refutes--> v04\n- v06 --refutes--> v2a\n- v08 --refutes--> v14\n- v26 --refutes--> v1a\n- v25 --supports--> v26\n- v27 --supports--> v0e\n- v22 --refutes--> v16\n- v26 --supports--> v0c\n- v18 --supports--> v15\n- v1e --refutes--> v29\n- v17 --supports--> v1e\n- v01 --refutes--> v13\n- v0e --refutes--> v09\n- v08 --supports--> v01\n- v0a --refutes--> v24\n- v06 --refutes--> v0d\n- v1c --supports--> v03\n- v0e --refutes--> v1b\n\n[VERIFIED ROOT]\nv1c = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"335f7612a5015ff5e2061b2f16db97159b1313259615f72d467ec945c18355e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b6324dda7c978d5b766740f1811ce1230312f913d57108cc9caff87c0d04d7b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-b0fe722b4c84c821c727","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v25 --refutes--> v18\n- v11 --refutes--> v19\n- v06 --refutes--> v0d\n- v17 --refutes--> v1f\n- v28 --supports--> v02\n- v2a --supports--> v12\n- v0d --refutes--> v14\n- v10 --refutes--> v0e\n- v04 --supports--> v0f\n- v02 --supports--> v20\n- v20 --refutes--> v22\n- v08 --refutes--> v1c\n- v1c --supports--> v26\n- v27 --refutes--> v09\n- v1f --supports--> v08\n- v0f --refutes--> v0b\n- v0f --supports--> v13\n- v1a --refutes--> v25\n- v24 --refutes--> v0c\n- v23 --supports--> v1e\n- v1e --supports--> v1a\n- v02 --supports--> v16\n- v1b --supports--> v03\n- v0b --refutes--> v1b\n- v29 --refutes--> v21\n- v25 --refutes--> v24\n- v1c --refutes--> v2a\n- v10 --supports--> v1d\n- v14 --refutes--> v28\n- v25 --refutes--> v00\n- v1e --refutes--> v07\n- v0c --refutes--> v17\n- v22 --refutes--> v10\n- v1c --supports--> v01\n- v2a --supports--> v06\n- v27 --refutes--> v05\n- v21 --supports--> v23\n- v11 --supports--> v29\n- v0c --supports--> v0a\n- v1b --refutes--> v11\n- v05 --refutes--> v04\n- v24 --supports--> v15\n\n[VERIFIED ROOT]\nv27 = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b01239583dc1f2e64b1f81f0e5b695f140ca506f41e896a22532e6452d4ef57d","schema":"livegraph.boolean-propagation.v1","spec_hash":"b0fe722b4c84c821c7271059d81f84cc3eab491795151f7df0837ff166b4d81b","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-84d157e3ba984029a56d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v22 --refutes--> v26\n- v01 --refutes--> v15\n- v00 --refutes--> v0c\n- v0b --refutes--> v11\n- v21 --supports--> v1c\n- v1f --supports--> v05\n- v1f --supports--> v06\n- v1c --supports--> v07\n- v0b --supports--> v23\n- v01 --supports--> v13\n- v27 --refutes--> v00\n- v10 --supports--> v20\n- v14 --supports--> v10\n- v0e --supports--> v27\n- v17 --refutes--> v25\n- v0f --supports--> v1a\n- v06 --refutes--> v21\n- v1d --supports--> v19\n- v16 --supports--> v0f\n- v0c --refutes--> v22\n- v14 --refutes--> v08\n- v0e --refutes--> v28\n- v1c --supports--> v03\n- v23 --supports--> v16\n- v24 --refutes--> v01\n- v00 --supports--> v12\n- v26 --refutes--> v14\n- v17 --supports--> v04\n- v20 --refutes--> v0b\n- v0b --refutes--> v02\n- v07 --supports--> v0d\n- v1a --supports--> v1b\n- v1e --supports--> v1f\n- v1b --refutes--> v24\n- v10 --supports--> v09\n- v17 --supports--> v0a\n- v19 --supports--> v17\n- v07 --refutes--> v2a\n- v15 --supports--> v1d\n- v06 --refutes--> v29\n- v26 --supports--> v18\n- v2a --refutes--> v0e\n\n[VERIFIED ROOT]\nv1e = TRUE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"701b24853363616b4633dcd0f0a9de32145b8c8c8954c45c5bcd1cf36f5fad2f","schema":"livegraph.boolean-propagation.v1","spec_hash":"84d157e3ba984029a56dea0f8333918c5934b94cf14480c44e10b85d40b80134","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-5be0783d05b3d11cc494","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v11 --supports--> v14\n- v05 --supports--> v1f\n- v1e --supports--> v1d\n- v0d --supports--> v03\n- v0a --supports--> v1e\n- v29 --refutes--> v0a\n- v13 --supports--> v09\n- v19 --refutes--> v10\n- v1e --supports--> v08\n- v24 --supports--> v0b\n- v17 --supports--> v12\n- v17 --supports--> v0e\n- v1b --supports--> v02\n- v17 --refutes--> v20\n- v1a --refutes--> v0f\n- v18 --supports--> v23\n- v21 --supports--> v24\n- v1b --supports--> v06\n- v1f --refutes--> v16\n- v1d --supports--> v01\n- v09 --supports--> v00\n- v07 --refutes--> v27\n- v1e --refutes--> v04\n- v06 --refutes--> v29\n- v09 --refutes--> v25\n- v23 --refutes--> v28\n- v07 --supports--> v2a\n- v23 --refutes--> v1c\n- v00 --refutes--> v21\n- v07 --supports--> v1b\n- v11 --refutes--> v19\n- v17 --refutes--> v15\n- v10 --refutes--> v07\n- v0b --refutes--> v26\n- v0c --supports--> v05\n- v1a --refutes--> v11\n- v16 --supports--> v18\n- v26 --refutes--> v17\n- v03 --supports--> v0c\n- v15 --supports--> v0d\n- v00 --refutes--> v22\n- v1d --supports--> v13\n\n[VERIFIED ROOT]\nv1a = TRUE\n\n[TARGET]\nv28\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2b337cc7888ab9a65773bffbcf6277e0212b1e6cffc7955f9bcb15e3de942ffa","schema":"livegraph.boolean-propagation.v1","spec_hash":"5be0783d05b3d11cc494f616b15732e8554793e922ad6c867cdcf6b06a4ec4fc","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v28\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-33d6832017494065c93b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v0f --supports--> v19\n- v1d --supports--> v11\n- v1a --supports--> v18\n- v1e --refutes--> v03\n- v0a --supports--> v0c\n- v16 --refutes--> v14\n- v24 --refutes--> v0b\n- v05 --refutes--> v10\n- v28 --supports--> v0d\n- v21 --supports--> v06\n- v1b --refutes--> v16\n- v03 --refutes--> v1a\n- v0d --supports--> v27\n- v0a --supports--> v07\n- v1d --supports--> v20\n- v17 --supports--> v1e\n- v16 --refutes--> v23\n- v05 --supports--> v17\n- v09 --supports--> v25\n- v0a --refutes--> v1f\n- v20 --refutes--> v01\n- v18 --supports--> v0f\n- v06 --supports--> v24\n- v01 --supports--> v29\n- v0d --supports--> v21\n- v15 --supports--> v1c\n- v23 --supports--> v28\n- v0b --refutes--> v09\n- v28 --refutes--> v04\n- v0c --supports--> v2a\n- v09 --refutes--> v08\n- v0d --supports--> v13\n- v1e --refutes--> v12\n- v19 --refutes--> v00\n- v1c --refutes--> v1b\n- v21 --refutes--> v0e\n- v19 --supports--> v1d\n- v08 --supports--> v0a\n- v29 --supports--> v15\n- v2a --refutes--> v22\n- v1d --supports--> v26\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"924687249fcfd609fcda4ad5f415de703c8b0d344629d9acec6fab5b69950f6b","schema":"livegraph.boolean-propagation.v1","spec_hash":"33d6832017494065c93b67b41e2609fae6ba5df127ad9c439b038c0df6a89626","spec_json":"{\"edges\":[{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-43cf430be7168b8d988e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v27\n- v05 --refutes--> v1f\n- v07 --refutes--> v15\n- v22 --refutes--> v1a\n- v0a --refutes--> v19\n- v1a --refutes--> v1c\n- v0d --supports--> v17\n- v0b --refutes--> v26\n- v26 --refutes--> v23\n- v23 --refutes--> v09\n- v1a --refutes--> v1e\n- v1c --supports--> v0c\n- v15 --supports--> v00\n- v07 --supports--> v12\n- v01 --supports--> v0a\n- v04 --supports--> v0d\n- v18 --refutes--> v13\n- v0f --refutes--> v08\n- v04 --refutes--> v02\n- v00 --supports--> v2a\n- v08 --supports--> v0e\n- v00 --supports--> v25\n- v06 --supports--> v18\n- v0a --supports--> v0b\n- v17 --supports--> v0f\n- v28 --refutes--> v1b\n- v0d --refutes--> v10\n- v14 --refutes--> v29\n- v26 --supports--> v20\n- v1f --supports--> v22\n- v08 --supports--> v24\n- v1f --refutes--> v11\n- v08 --supports--> v07\n- v00 --supports--> v03\n- v28 --refutes--> v14\n- v0c --refutes--> v28\n- v29 --supports--> v04\n- v0b --refutes--> v1d\n- v27 --supports--> v05\n- v21 --supports--> v06\n- v29 --refutes--> v16\n- v2a --refutes--> v21\n\n[VERIFIED ROOT]\nv01 = 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":"19ee1ab01a0c68421861ea6166d566659011c1f7ff3de9a7aafb1ff263bfa04d","schema":"livegraph.boolean-propagation.v1","spec_hash":"43cf430be7168b8d988e6b0bda3c62f47ffdd2a45b432ba74757001028e6116b","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-04513d4473b205408cf8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v00\n- v21 --supports--> v15\n- v05 --supports--> v25\n- v1c --supports--> v1e\n- v09 --refutes--> v29\n- v05 --refutes--> v27\n- v27 --refutes--> v07\n- v19 --refutes--> v0d\n- v29 --supports--> v17\n- v16 --supports--> v04\n- v26 --supports--> v28\n- v0e --supports--> v09\n- v1c --refutes--> v13\n- v04 --refutes--> v03\n- v17 --supports--> v0f\n- v07 --supports--> v19\n- v00 --supports--> v0a\n- v0c --refutes--> v05\n- v1a --refutes--> v01\n- v1d --supports--> v16\n- v19 --supports--> v08\n- v15 --refutes--> v2a\n- v01 --refutes--> v26\n- v24 --supports--> v18\n- v28 --refutes--> v0c\n- v24 --supports--> v06\n- v03 --refutes--> v0e\n- v02 --supports--> v22\n- v02 --supports--> v24\n- v2a --refutes--> v1a\n- v19 --supports--> v23\n- v13 --refutes--> v12\n- v03 --refutes--> v0b\n- v01 --supports--> v10\n- v07 --refutes--> v1b\n- v29 --supports--> v11\n- v1f --refutes--> v02\n- v12 --refutes--> v1f\n- v20 --refutes--> v21\n- v20 --refutes--> v14\n- v00 --supports--> v20\n- v24 --refutes--> v1d\n\n[VERIFIED ROOT]\nv1c = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a1cd647ac9b664645d3a6a6a007db367e484a69201e09f35dd279202269e82e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"04513d4473b205408cf8e3536faff69956346d08d0b0394248d51cc7cfa32410","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-eacc5b87d7ad723e379e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --supports--> v1f\n- v13 --refutes--> v14\n- v1a --supports--> v29\n- v1d --refutes--> v0b\n- v23 --supports--> v0f\n- v0f --supports--> v12\n- v1f --supports--> v19\n- v06 --refutes--> v17\n- v0c --refutes--> v01\n- v0a --refutes--> v0e\n- v17 --supports--> v24\n- v1a --refutes--> v04\n- v0b --refutes--> v25\n- v28 --refutes--> v16\n- v1a --refutes--> v00\n- v1f --supports--> v21\n- v14 --refutes--> v2a\n- v15 --supports--> v0a\n- v15 --supports--> v1e\n- v25 --supports--> v09\n- v12 --refutes--> v06\n- v16 --supports--> v1c\n- v1f --refutes--> v11\n- v0a --refutes--> v27\n- v21 --refutes--> v15\n- v16 --supports--> v20\n- v13 --refutes--> v26\n- v05 --refutes--> v22\n- v0a --supports--> v18\n- v02 --supports--> v05\n- v03 --supports--> v13\n- v21 --refutes--> v08\n- v25 --supports--> v0d\n- v22 --supports--> v03\n- v06 --supports--> v07\n- v04 --supports--> v23\n- v24 --refutes--> v28\n- v01 --supports--> v02\n- v18 --refutes--> v0c\n- v10 --refutes--> v1d\n- v14 --refutes--> v10\n- v13 --refutes--> v1b\n\n[VERIFIED ROOT]\nv1a = 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":"14f8c80cd705805e58606e138a97c697d6ce66b45fa3823e1f019fe4f25945cd","schema":"livegraph.boolean-propagation.v1","spec_hash":"eacc5b87d7ad723e379e00e9426305fc7f9a0a6d931bf9284fab9405b6f70520","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-b9a30f8625c6e8e628fc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v0e\n- v09 --refutes--> v02\n- v03 --supports--> v05\n- v29 --refutes--> v0a\n- v23 --refutes--> v0c\n- v24 --refutes--> v1a\n- v17 --refutes--> v0b\n- v19 --supports--> v0d\n- v1a --refutes--> v07\n- v19 --supports--> v29\n- v11 --supports--> v18\n- v20 --refutes--> v1e\n- v22 --supports--> v06\n- v24 --refutes--> v13\n- v22 --refutes--> v19\n- v18 --refutes--> v28\n- v29 --refutes--> v14\n- v12 --refutes--> v25\n- v23 --supports--> v00\n- v1b --refutes--> v12\n- v1e --supports--> v04\n- v1c --supports--> v09\n- v1e --refutes--> v1f\n- v0b --refutes--> v03\n- v02 --supports--> v17\n- v07 --supports--> v22\n- v05 --supports--> v08\n- v12 --supports--> v1d\n- v05 --supports--> v24\n- v04 --refutes--> v1b\n- v0f --refutes--> v23\n- v26 --refutes--> v10\n- v14 --supports--> v26\n- v28 --supports--> v01\n- v1e --supports--> v21\n- v17 --refutes--> v2a\n- v25 --supports--> v0f\n- v0f --refutes--> v15\n- v10 --refutes--> v27\n- v05 --supports--> v16\n- v10 --supports--> v20\n- v01 --refutes--> v1c\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"038046f8b5e80513faad9647e99cad94ff20b0fe10db71cf0f7c42be516333d6","schema":"livegraph.boolean-propagation.v1","spec_hash":"b9a30f8625c6e8e628fcd6c9c0e33032ab360d6becb8d47d429503b86efd69d7","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-a5762c310d6a04d502a4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v11\n- v15 --supports--> v20\n- v1c --supports--> v25\n- v0d --supports--> v0b\n- v1c --supports--> v06\n- v1f --refutes--> v1d\n- v26 --refutes--> v22\n- v0a --refutes--> v12\n- v0c --supports--> v26\n- v0b --refutes--> v2a\n- v0b --supports--> v18\n- v04 --refutes--> v1c\n- v05 --refutes--> v0f\n- v22 --supports--> v14\n- v29 --supports--> v02\n- v00 --supports--> v04\n- v0b --supports--> v27\n- v1b --supports--> v03\n- v1a --refutes--> v19\n- v20 --refutes--> v1e\n- v25 --supports--> v21\n- v25 --refutes--> v28\n- v0b --supports--> v23\n- v1b --supports--> v1f\n- v08 --supports--> v29\n- v10 --supports--> v1b\n- v04 --supports--> v09\n- v0f --refutes--> v13\n- v24 --refutes--> v0d\n- v14 --refutes--> v01\n- v12 --supports--> v1a\n- v1f --supports--> v17\n- v23 --refutes--> v10\n- v0f --refutes--> v0e\n- v29 --supports--> v00\n- v01 --refutes--> v24\n- v17 --refutes--> v05\n- v28 --supports--> v16\n- v1e --supports--> v08\n- v28 --refutes--> v0c\n- v15 --supports--> v07\n- v0f --supports--> v0a\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"893ed732b6aa45bd572b103e3942dd8f5afff7c6573692a10e4ec5de683c3b98","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5762c310d6a04d502a435a818c7ca1a1fc15edc78799e1f1e4e6ea0d75368b5","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-16b5827aef557c5a90a8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v14 --supports--> v00\n- v04 --refutes--> v26\n- v26 --supports--> v23\n- v1a --supports--> v29\n- v10 --supports--> v0a\n- v21 --supports--> v13\n- v16 --refutes--> v10\n- v1d --supports--> v25\n- v2a --refutes--> v08\n- v12 --refutes--> v15\n- v28 --refutes--> v2a\n- v0a --supports--> v0b\n- v29 --supports--> v18\n- v1f --supports--> v1c\n- v18 --supports--> v1f\n- v0b --refutes--> v1a\n- v15 --supports--> v04\n- v0d --refutes--> v02\n- v1f --supports--> v27\n- v19 --supports--> v28\n- v1d --refutes--> v24\n- v14 --refutes--> v11\n- v2a --supports--> v01\n- v12 --supports--> v0c\n- v11 --refutes--> v12\n- v16 --refutes--> v06\n- v29 --refutes--> v22\n- v1d --refutes--> v1b\n- v0e --supports--> v21\n- v07 --refutes--> v16\n- v19 --refutes--> v05\n- v23 --supports--> v19\n- v1c --refutes--> v20\n- v0f --supports--> v09\n- v20 --refutes--> v17\n- v1d --refutes--> v0d\n- v12 --refutes--> v03\n- v0b --supports--> v1e\n- v2a --refutes--> v0e\n- v09 --supports--> v1d\n- v17 --supports--> v14\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":"892b0d349edb57a967cbad1b1030f8fc30bce7c80e3776583ef40e696ce42138","schema":"livegraph.boolean-propagation.v1","spec_hash":"16b5827aef557c5a90a8d1d20db34f1b9803fe3b83822076a1ac459c2d997f81","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-9642bce832ca465b5eee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --supports--> v07\n- v29 --refutes--> v22\n- v04 --supports--> v0c\n- v07 --refutes--> v0e\n- v17 --supports--> v0d\n- v23 --refutes--> v10\n- v24 --refutes--> v09\n- v02 --refutes--> v12\n- v0c --refutes--> v00\n- v23 --supports--> v13\n- v26 --refutes--> v21\n- v0e --refutes--> v28\n- v1a --refutes--> v1e\n- v09 --refutes--> v27\n- v20 --supports--> v26\n- v0c --refutes--> v11\n- v0f --supports--> v0a\n- v09 --refutes--> v20\n- v0d --supports--> v16\n- v23 --supports--> v1b\n- v2a --refutes--> v1a\n- v26 --refutes--> v14\n- v09 --refutes--> v01\n- v07 --supports--> v25\n- v28 --refutes--> v05\n- v1e --supports--> v18\n- v02 --supports--> v23\n- v00 --refutes--> v06\n- v26 --refutes--> v0b\n- v05 --supports--> v19\n- v00 --refutes--> v24\n- v18 --refutes--> v1f\n- v1c --supports--> v04\n- v05 --refutes--> v15\n- v16 --refutes--> v03\n- v1d --refutes--> v29\n- v03 --refutes--> v1d\n- v0f --refutes--> v02\n- v0b --refutes--> v2a\n- v1f --supports--> v08\n- v29 --refutes--> v1c\n- v10 --refutes--> v17\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv15\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"934c77afe4f29b28d28bbed455da8f9e24e5d00fabd03faab3e91ec18aaf4fdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"9642bce832ca465b5eeead90a4befb76c7c378da20d535f259e442d076a0ab98","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-19ec24cd178595cccd7a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0a\n- v0e --refutes--> v13\n- v23 --supports--> v08\n- v0c --supports--> v22\n- v03 --refutes--> v1a\n- v08 --supports--> v16\n- v26 --refutes--> v27\n- v27 --refutes--> v02\n- v12 --supports--> v15\n- v17 --refutes--> v0c\n- v16 --refutes--> v1c\n- v13 --supports--> v24\n- v1c --supports--> v0d\n- v14 --supports--> v01\n- v27 --refutes--> v11\n- v19 --refutes--> v14\n- v06 --refutes--> v0e\n- v1f --supports--> v23\n- v07 --refutes--> v0f\n- v28 --supports--> v21\n- v28 --refutes--> v19\n- v13 --refutes--> v03\n- v19 --supports--> v1b\n- v04 --supports--> v17\n- v1a --refutes--> v1f\n- v14 --refutes--> v18\n- v25 --supports--> v28\n- v1f --refutes--> v00\n- v18 --refutes--> v07\n- v0c --refutes--> v10\n- v0f --supports--> v12\n- v28 --refutes--> v1e\n- v16 --supports--> v20\n- v0d --refutes--> v29\n- v02 --refutes--> v06\n- v15 --supports--> v05\n- v08 --refutes--> v1d\n- v09 --supports--> v04\n- v0c --supports--> v26\n- v28 --supports--> v2a\n- v15 --supports--> v0b\n- v29 --supports--> v25\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b9ccc64839daa107c3fc895ffb845ef499b976fdacfdcbc443a5469f5b8c0323","schema":"livegraph.boolean-propagation.v1","spec_hash":"19ec24cd178595cccd7a0c2e6fa106ed52894a24db4209f87a26977321c58e25","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-324b82a51e991a993c3e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v19 --refutes--> v24\n- v09 --refutes--> v0f\n- v11 --supports--> v12\n- v29 --supports--> v13\n- v29 --supports--> v0d\n- v2a --refutes--> v0c\n- v00 --refutes--> v04\n- v24 --supports--> v18\n- v02 --supports--> v29\n- v11 --supports--> v10\n- v0a --refutes--> v02\n- v11 --supports--> v1f\n- v05 --refutes--> v07\n- v0c --supports--> v25\n- v14 --refutes--> v21\n- v13 --supports--> v03\n- v22 --supports--> v00\n- v21 --refutes--> v0e\n- v26 --refutes--> v08\n- v23 --refutes--> v19\n- v18 --refutes--> v16\n- v1c --supports--> v1b\n- v18 --refutes--> v01\n- v05 --refutes--> v23\n- v25 --refutes--> v11\n- v01 --refutes--> v0a\n- v02 --refutes--> v1d\n- v03 --supports--> v27\n- v24 --supports--> v0b\n- v24 --refutes--> v1a\n- v0f --supports--> v2a\n- v27 --supports--> v1c\n- v12 --supports--> v05\n- v19 --supports--> v17\n- v1e --supports--> v20\n- v26 --refutes--> v14\n- v00 --supports--> v09\n- v0e --refutes--> v06\n- v24 --refutes--> v28\n- v20 --supports--> v22\n- v09 --supports--> v15\n\n[VERIFIED ROOT]\nv26 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ae1466dd32b094b2f7f202905b8a70723fcd3005283caf98b7f8b56e3c9da62a","schema":"livegraph.boolean-propagation.v1","spec_hash":"324b82a51e991a993c3e5616e55e7cd4bd232500d5e60a99d904b2b430251492","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v26\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-06eb6bc7e713a6624bfe","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v01 --supports--> v11\n- v0c --refutes--> v25\n- v1e --supports--> v16\n- v1d --supports--> v1e\n- v11 --supports--> v04\n- v0e --refutes--> v13\n- v1b --refutes--> v22\n- v00 --supports--> v21\n- v12 --refutes--> v02\n- v02 --supports--> v19\n- v15 --supports--> v27\n- v1b --refutes--> v03\n- v13 --refutes--> v1f\n- v0f --supports--> v29\n- v04 --supports--> v1b\n- v1f --refutes--> v14\n- v04 --supports--> v23\n- v25 --supports--> v1d\n- v03 --refutes--> v1c\n- v1c --supports--> v12\n- v05 --supports--> v18\n- v20 --refutes--> v07\n- v1d --refutes--> v06\n- v26 --refutes--> v0f\n- v25 --supports--> v09\n- v18 --refutes--> v00\n- v27 --supports--> v0e\n- v10 --refutes--> v1a\n- v0f --refutes--> v10\n- v26 --refutes--> v0b\n- v03 --refutes--> v08\n- v19 --supports--> v0c\n- v05 --refutes--> v2a\n- v17 --supports--> v24\n- v1e --refutes--> v17\n- v21 --supports--> v26\n- v03 --supports--> v0a\n- v20 --supports--> v15\n- v05 --supports--> v0d\n- v19 --refutes--> v28\n- v24 --supports--> v05\n\n[VERIFIED ROOT]\nv01 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"290e9e14007a71d5bcbfe2049012e1cd8620db2554f5ebaf38af1812837f23ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"06eb6bc7e713a6624bfe9eaa6001835eba5d58cf6493d1d16ee67f1fb6394e34","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v01\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-e87bc9ff8bbfd5498ba2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v28\n- v08 --refutes--> v19\n- v25 --refutes--> v1d\n- v04 --refutes--> v10\n- v09 --supports--> v22\n- v14 --refutes--> v00\n- v19 --refutes--> v0e\n- v15 --refutes--> v02\n- v0c --supports--> v26\n- v07 --refutes--> v21\n- v01 --supports--> v1e\n- v2a --supports--> v0c\n- v14 --refutes--> v05\n- v29 --supports--> v12\n- v2a --supports--> v1b\n- v28 --refutes--> v25\n- v0f --supports--> v06\n- v22 --supports--> v1c\n- v28 --supports--> v17\n- v20 --refutes--> v11\n- v01 --refutes--> v18\n- v25 --supports--> v24\n- v02 --refutes--> v0f\n- v1d --supports--> v14\n- v21 --supports--> v29\n- v12 --supports--> v01\n- v11 --refutes--> v09\n- v05 --supports--> v1a\n- v10 --supports--> v2a\n- v10 --supports--> v0d\n- v1e --supports--> v15\n- v07 --supports--> v03\n- v16 --supports--> v20\n- v19 --refutes--> v16\n- v0f --supports--> v13\n- v29 --refutes--> v0b\n- v1c --refutes--> v0a\n- v0a --refutes--> v1f\n- v05 --supports--> v27\n- v0c --refutes--> v07\n- v13 --supports--> v08\n- v11 --supports--> v23\n\n[VERIFIED ROOT]\nv04 = TRUE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"039f314f8fb7d179538e863eec62e35cc24a6fd60bdaf7785e863b952bd76799","schema":"livegraph.boolean-propagation.v1","spec_hash":"e87bc9ff8bbfd5498ba229dc69c1e0e281ba7e721d91728f5412634c438fd7ac","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v04\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-90845ad5b66129cce0f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2a\n- v18 --refutes--> v03\n- v1d --supports--> v21\n- v25 --supports--> v0d\n- v1f --refutes--> v27\n- v02 --supports--> v0f\n- v16 --supports--> v09\n- v0f --supports--> v13\n- v25 --refutes--> v12\n- v0c --refutes--> v23\n- v25 --supports--> v00\n- v21 --supports--> v07\n- v0e --refutes--> v20\n- v26 --refutes--> v1d\n- v0b --refutes--> v22\n- v08 --supports--> v15\n- v21 --supports--> v10\n- v10 --supports--> v06\n- v01 --refutes--> v19\n- v09 --supports--> v0b\n- v1e --supports--> v1b\n- v22 --supports--> v11\n- v22 --refutes--> v24\n- v05 --supports--> v1a\n- v24 --refutes--> v0a\n- v02 --supports--> v17\n- v1c --refutes--> v01\n- v16 --refutes--> v04\n- v0e --refutes--> v26\n- v1e --refutes--> v18\n- v13 --refutes--> v29\n- v10 --supports--> v25\n- v00 --supports--> v02\n- v29 --supports--> v08\n- v27 --refutes--> v1e\n- v1c --refutes--> v14\n- v1e --supports--> v28\n- v15 --supports--> v1c\n- v1a --supports--> v0c\n- v19 --supports--> v05\n- v0c --supports--> v1f\n- v18 --refutes--> v16\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":"3e7b28db5031e7b5b46fceafb26aca92ee566bc2e654d8af249f46342b7b33cb","schema":"livegraph.boolean-propagation.v1","spec_hash":"90845ad5b66129cce0f840795778ea488a4fc1a86649214efa5ed67245021195","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-65dc39547593016f5295","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1f\n- v16 --refutes--> v0a\n- v11 --supports--> v1c\n- v04 --refutes--> v00\n- v04 --supports--> v08\n- v16 --supports--> v20\n- v12 --supports--> v03\n- v26 --supports--> v1e\n- v03 --refutes--> v18\n- v10 --refutes--> v05\n- v22 --supports--> v10\n- v0d --supports--> v16\n- v25 --supports--> v24\n- v0d --supports--> v02\n- v2a --refutes--> v14\n- v14 --refutes--> v0e\n- v14 --supports--> v0b\n- v1e --refutes--> v06\n- v06 --supports--> v1b\n- v16 --refutes--> v19\n- v1b --supports--> v09\n- v1e --refutes--> v01\n- v0e --refutes--> v15\n- v03 --refutes--> v0d\n- v0e --supports--> v13\n- v08 --supports--> v12\n- v19 --refutes--> v0f\n- v1d --refutes--> v29\n- v0f --supports--> v0c\n- v29 --supports--> v17\n- v19 --refutes--> v21\n- v07 --supports--> v26\n- v24 --refutes--> v22\n- v27 --refutes--> v07\n- v0f --supports--> v11\n- v05 --refutes--> v2a\n- v25 --refutes--> v28\n- v17 --supports--> v04\n- v13 --refutes--> v1d\n- v09 --supports--> v25\n- v06 --supports--> v23\n- v16 --refutes--> v1a\n\n[VERIFIED ROOT]\nv27 = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f028ca1f27ad228808888a60794625b434c77f62a9460e1df29a847a206e6670","schema":"livegraph.boolean-propagation.v1","spec_hash":"65dc39547593016f52952c7f65228efdf6864d22d8a84ffb68f4bb26defc80ee","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a472a9ad9457e94a69dd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v2a --supports--> v14\n- v17 --supports--> v26\n- v08 --refutes--> v0d\n- v27 --refutes--> v0c\n- v1d --supports--> v24\n- v1c --supports--> v20\n- v2a --supports--> v1b\n- v1c --supports--> v22\n- v17 --supports--> v1a\n- v0a --supports--> v11\n- v0d --supports--> v02\n- v2a --refutes--> v01\n- v27 --supports--> v0a\n- v0f --supports--> v2a\n- v23 --refutes--> v13\n- v13 --supports--> v18\n- v10 --supports--> v1f\n- v09 --refutes--> v10\n- v26 --refutes--> v00\n- v09 --refutes--> v15\n- v18 --refutes--> v0f\n- v12 --refutes--> v1d\n- v10 --supports--> v08\n- v1d --refutes--> v25\n- v20 --refutes--> v16\n- v0a --refutes--> v21\n- v05 --supports--> v23\n- v19 --supports--> v1c\n- v02 --supports--> v06\n- v29 --supports--> v17\n- v10 --supports--> v0e\n- v0d --refutes--> v1e\n- v06 --supports--> v19\n- v0f --supports--> v07\n- v04 --refutes--> v05\n- v24 --refutes--> v28\n- v1b --supports--> v27\n- v03 --refutes--> v04\n- v24 --supports--> v03\n- v11 --supports--> v29\n- v26 --refutes--> v09\n\n[VERIFIED ROOT]\nv0b = 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":"9741b1662e6ed85bc1f6f3139c6770d7dee9f423146fdcd45c19c8cc79a2d69c","schema":"livegraph.boolean-propagation.v1","spec_hash":"a472a9ad9457e94a69dd51ea1e9c9655e165beda4beb7bcbc85e8ed4aa2c749b","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-dadacbcf6efcf7184f0c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2a --refutes--> v24\n- v17 --supports--> v04\n- v1b --supports--> v2a\n- v0b --supports--> v1c\n- v13 --supports--> v0c\n- v21 --supports--> v19\n- v06 --refutes--> v14\n- v0b --supports--> v1d\n- v0d --supports--> v17\n- v20 --supports--> v01\n- v0d --refutes--> v07\n- v17 --refutes--> v0a\n- v12 --refutes--> v21\n- v05 --supports--> v26\n- v0b --refutes--> v1e\n- v1d --refutes--> v0f\n- v13 --refutes--> v1f\n- v0e --supports--> v11\n- v00 --refutes--> v29\n- v0e --supports--> v00\n- v0c --supports--> v02\n- v1d --refutes--> v20\n- v01 --refutes--> v05\n- v1a --supports--> v16\n- v12 --supports--> v25\n- v09 --refutes--> v0e\n- v14 --refutes--> v09\n- v00 --refutes--> v23\n- v17 --refutes--> v18\n- v02 --refutes--> v1a\n- v15 --supports--> v27\n- v27 --supports--> v06\n- v03 --refutes--> v13\n- v28 --refutes--> v15\n- v18 --supports--> v08\n- v01 --refutes--> v22\n- v24 --supports--> v12\n- v08 --refutes--> v03\n- v19 --supports--> v28\n- v02 --supports--> v10\n- v05 --supports--> v1b\n- v1a --supports--> v0b\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"db62b0fe5007571726c53b0aed52e65eea06fd41a7b2bd81031c7e8b3bcbc815","schema":"livegraph.boolean-propagation.v1","spec_hash":"dadacbcf6efcf7184f0cf631c2601f0a7265696e6f4e71b97177a98270890f5b","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-6391a78628989382ba72","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v18\n- v12 --supports--> v0d\n- v0e --supports--> v07\n- v09 --refutes--> v02\n- v14 --supports--> v1c\n- v0e --supports--> v09\n- v03 --supports--> v17\n- v26 --refutes--> v25\n- v0f --supports--> v13\n- v24 --supports--> v27\n- v13 --refutes--> v1f\n- v28 --refutes--> v1b\n- v04 --refutes--> v15\n- v1f --refutes--> v20\n- v28 --supports--> v08\n- v0d --refutes--> v0b\n- v22 --refutes--> v19\n- v08 --refutes--> v14\n- v06 --supports--> v01\n- v16 --refutes--> v28\n- v21 --supports--> v0a\n- v01 --refutes--> v0e\n- v29 --supports--> v11\n- v1c --supports--> v12\n- v17 --supports--> v04\n- v0a --refutes--> v03\n- v0d --supports--> v0f\n- v20 --refutes--> v05\n- v22 --supports--> v1a\n- v22 --refutes--> v06\n- v20 --refutes--> v26\n- v0e --supports--> v23\n- v27 --supports--> v29\n- v17 --supports--> v00\n- v14 --refutes--> v2a\n- v02 --refutes--> v1e\n- v01 --supports--> v0c\n- v11 --refutes--> v16\n- v15 --supports--> v22\n- v11 --refutes--> v1d\n- v08 --supports--> v10\n- v02 --refutes--> v24\n\n[VERIFIED ROOT]\nv21 = TRUE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f5cd081d1404efe8a2f9c2fa5ec10e62cba1fcd444df2fa59df6b49216b5f163","schema":"livegraph.boolean-propagation.v1","spec_hash":"6391a78628989382ba720ecc94f8e4c07d35df5661d6391226e0ec6af35b6a8c","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v21\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-6904f4c2428d612bdead","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v27 --supports--> v1d\n- v02 --supports--> v0c\n- v22 --supports--> v24\n- v0a --supports--> v2a\n- v14 --supports--> v0f\n- v16 --refutes--> v08\n- v1a --refutes--> v10\n- v10 --refutes--> v27\n- v16 --refutes--> v15\n- v09 --refutes--> v22\n- v0f --supports--> v0a\n- v0d --refutes--> v04\n- v06 --refutes--> v0b\n- v02 --refutes--> v26\n- v20 --refutes--> v18\n- v06 --supports--> v13\n- v06 --supports--> v25\n- v17 --refutes--> v19\n- v10 --refutes--> v05\n- v20 --refutes--> v14\n- v0a --refutes--> v11\n- v04 --supports--> v01\n- v21 --refutes--> v12\n- v0f --supports--> v1c\n- v21 --refutes--> v00\n- v1f --refutes--> v28\n- v22 --supports--> v03\n- v1d --refutes--> v20\n- v1b --supports--> v1f\n- v0e --supports--> v0d\n- v15 --supports--> v0e\n- v12 --refutes--> v02\n- v23 --refutes--> v09\n- v1f --refutes--> v07\n- v19 --refutes--> v1e\n- v03 --supports--> v17\n- v19 --supports--> v29\n- v0a --supports--> v1b\n- v28 --supports--> v23\n- v13 --refutes--> v16\n- v04 --refutes--> v21\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9dfb258782f0c546d49f59bc1028d58aaaacb0d29d9706710d4ea95a871a4b9","schema":"livegraph.boolean-propagation.v1","spec_hash":"6904f4c2428d612bdead99ff174ca57198ca896afe49fe63f6cd1fbc8167d765","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a5cfd3b7d3a27b3e4f9d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v12 --refutes--> v20\n- v28 --supports--> v0c\n- v24 --refutes--> v09\n- v19 --supports--> v0f\n- v06 --refutes--> v08\n- v0e --refutes--> v13\n- v22 --refutes--> v1b\n- v10 --supports--> v00\n- v12 --supports--> v26\n- v00 --supports--> v03\n- v03 --supports--> v1f\n- v26 --refutes--> v14\n- v02 --refutes--> v10\n- v14 --supports--> v0e\n- v0e --refutes--> v1e\n- v08 --supports--> v22\n- v0e --refutes--> v21\n- v18 --supports--> v28\n- v19 --refutes--> v11\n- v25 --refutes--> v1a\n- v17 --supports--> v06\n- v02 --refutes--> v1c\n- v01 --refutes--> v17\n- v09 --supports--> v04\n- v0d --refutes--> v05\n- v0a --supports--> v0d\n- v05 --refutes--> v25\n- v28 --supports--> v19\n- v15 --supports--> v27\n- v07 --supports--> v0a\n- v27 --refutes--> v18\n- v25 --refutes--> v29\n- v25 --refutes--> v12\n- v03 --supports--> v01\n- v28 --refutes--> v16\n- v06 --supports--> v2a\n- v1b --refutes--> v1d\n- v1d --supports--> v07\n- v06 --refutes--> v23\n- v11 --refutes--> v24\n- v0e --refutes--> v0b\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":"f8ee7545dc983cb903e45244c5b78447ae5a8115c00f1ac990b06f27bd18dd81","schema":"livegraph.boolean-propagation.v1","spec_hash":"a5cfd3b7d3a27b3e4f9d51141cf62f133866534bb27510405591c7b6a84fc745","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-9824db8045900bf0c92f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v21\n- v1b --supports--> v2a\n- v0e --refutes--> v0f\n- v20 --refutes--> v26\n- v22 --refutes--> v0d\n- v12 --refutes--> v1d\n- v11 --refutes--> v20\n- v0c --refutes--> v10\n- v21 --supports--> v08\n- v19 --refutes--> v05\n- v21 --refutes--> v11\n- v02 --supports--> v1f\n- v1e --refutes--> v17\n- v28 --supports--> v15\n- v28 --supports--> v1a\n- v0b --refutes--> v06\n- v17 --supports--> v25\n- v0b --refutes--> v03\n- v02 --supports--> v13\n- v20 --supports--> v04\n- v04 --refutes--> v1b\n- v29 --refutes--> v00\n- v03 --supports--> v29\n- v18 --supports--> v16\n- v01 --supports--> v27\n- v09 --supports--> v12\n- v2a --refutes--> v1c\n- v1a --supports--> v02\n- v27 --supports--> v28\n- v25 --supports--> v0e\n- v19 --refutes--> v18\n- v27 --supports--> v0a\n- v2a --refutes--> v1e\n- v2a --supports--> v24\n- v0e --refutes--> v22\n- v10 --refutes--> v01\n- v29 --supports--> v09\n- v2a --refutes--> v14\n- v13 --supports--> v0b\n- v23 --refutes--> v0c\n- v22 --refutes--> v19\n- v0b --supports--> v07\n\n[VERIFIED ROOT]\nv23 = 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":"8b9ac2bf21bd281ff61b51264941fbeed9f93321674bc22deb88d7fc9f48893c","schema":"livegraph.boolean-propagation.v1","spec_hash":"9824db8045900bf0c92f6889e23b23a2682401345655f9ff4ff879d2af89997d","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-7519365178f18b6b5fd5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v20\n- v01 --supports--> v28\n- v1a --refutes--> v29\n- v24 --supports--> v11\n- v0d --refutes--> v23\n- v15 --refutes--> v06\n- v02 --refutes--> v04\n- v15 --supports--> v1b\n- v03 --refutes--> v16\n- v17 --refutes--> v08\n- v13 --supports--> v18\n- v2a --refutes--> v0d\n- v13 --supports--> v01\n- v0e --supports--> v2a\n- v13 --refutes--> v14\n- v1f --refutes--> v1e\n- v02 --refutes--> v19\n- v0c --supports--> v25\n- v07 --refutes--> v05\n- v1f --refutes--> v07\n- v01 --supports--> v0f\n- v00 --supports--> v24\n- v23 --refutes--> v1d\n- v29 --supports--> v17\n- v03 --refutes--> v26\n- v0d --supports--> v27\n- v11 --refutes--> v12\n- v0a --refutes--> v1a\n- v07 --supports--> v09\n- v04 --refutes--> v0a\n- v08 --refutes--> v1f\n- v11 --supports--> v15\n- v05 --refutes--> v00\n- v20 --supports--> v1c\n- v01 --refutes--> v03\n- v24 --refutes--> v0b\n- v16 --supports--> v0e\n- v25 --supports--> v22\n- v16 --supports--> v10\n- v1c --refutes--> v02\n- v06 --supports--> v0c\n- v03 --supports--> v21\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"13f2c0b70f59fc985c9581c98cb4b759b6159afa046f64c0abd51ab612caa0d5","schema":"livegraph.boolean-propagation.v1","spec_hash":"7519365178f18b6b5fd5b09bdd0a298ea915a9e68cc28c0c833297cafbcd44e1","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-b177b8c61855ef9358f6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v12\n- v0e --refutes--> v28\n- v23 --supports--> v21\n- v00 --supports--> v0a\n- v20 --supports--> v10\n- v1a --refutes--> v0e\n- v01 --supports--> v29\n- v15 --refutes--> v03\n- v14 --refutes--> v08\n- v28 --refutes--> v0c\n- v26 --supports--> v2a\n- v20 --supports--> v18\n- v1f --refutes--> v20\n- v04 --refutes--> v15\n- v28 --refutes--> v06\n- v05 --refutes--> v26\n- v0f --supports--> v01\n- v22 --refutes--> v23\n- v15 --refutes--> v02\n- v14 --refutes--> v1b\n- v14 --refutes--> v1d\n- v24 --refutes--> v1f\n- v2a --refutes--> v1e\n- v1f --refutes--> v25\n- v12 --supports--> v05\n- v01 --supports--> v0d\n- v1e --refutes--> v14\n- v11 --refutes--> v27\n- v09 --supports--> v13\n- v16 --refutes--> v09\n- v0c --refutes--> v07\n- v1b --supports--> v24\n- v0b --refutes--> v17\n- v20 --supports--> v00\n- v02 --supports--> v19\n- v0c --refutes--> v22\n- v0b --refutes--> v1c\n- v09 --supports--> v0b\n- v23 --supports--> v0f\n- v1c --supports--> v11\n- v01 --refutes--> v16\n- v11 --supports--> v04\n\n[VERIFIED ROOT]\nv1a = 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":"29101ba784bca662cba87fb3e1367ed4f42e357c10c28e3fc2167d89f450d3b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"b177b8c61855ef9358f629610db8952ddba44e663340ae05e8e84f5fd64cf56e","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-e643df724b0bb1c35176","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v28 --supports--> v0a\n- v2a --supports--> v15\n- v12 --refutes--> v16\n- v1d --refutes--> v08\n- v07 --supports--> v01\n- v16 --supports--> v03\n- v1b --refutes--> v04\n- v19 --supports--> v02\n- v0f --refutes--> v21\n- v0b --supports--> v28\n- v0b --refutes--> v0d\n- v0b --refutes--> v13\n- v16 --refutes--> v27\n- v1c --supports--> v25\n- v02 --refutes--> v29\n- v03 --refutes--> v18\n- v15 --refutes--> v07\n- v0c --supports--> v22\n- v1a --refutes--> v1d\n- v15 --refutes--> v00\n- v01 --supports--> v06\n- v0b --supports--> v11\n- v25 --refutes--> v2a\n- v04 --supports--> v1a\n- v0c --refutes--> v20\n- v0c --refutes--> v1b\n- v21 --refutes--> v09\n- v1f --refutes--> v12\n- v10 --supports--> v23\n- v23 --refutes--> v1c\n- v12 --supports--> v17\n- v14 --supports--> v10\n- v06 --supports--> v1f\n- v2a --refutes--> v24\n- v12 --refutes--> v26\n- v08 --refutes--> v14\n- v0a --supports--> v19\n- v09 --supports--> v05\n- v02 --supports--> v0f\n- v08 --supports--> v0e\n- v23 --refutes--> v1e\n- v09 --supports--> v0c\n\n[VERIFIED ROOT]\nv0b = 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":"fbfc3850d57aa6e988a4c73b1e49944d67b5c64162a5f3070893b8fc2135c593","schema":"livegraph.boolean-propagation.v1","spec_hash":"e643df724b0bb1c35176a299c4ac68f854720eba6f96489c421f32db17cd381c","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-8c673cb91d42f42d1387","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v29\n- v22 --refutes--> v18\n- v01 --refutes--> v06\n- v0f --supports--> v17\n- v24 --refutes--> v0d\n- v14 --refutes--> v1b\n- v10 --supports--> v0c\n- v06 --supports--> v24\n- v24 --supports--> v03\n- v0e --refutes--> v20\n- v1f --refutes--> v04\n- v18 --refutes--> v0f\n- v1a --supports--> v12\n- v1c --refutes--> v10\n- v23 --refutes--> v0b\n- v0e --refutes--> v1f\n- v04 --refutes--> v16\n- v06 --refutes--> v08\n- v24 --supports--> v28\n- v23 --supports--> v2a\n- v23 --refutes--> v07\n- v0a --supports--> v19\n- v12 --supports--> v0a\n- v21 --supports--> v11\n- v16 --supports--> v1e\n- v1d --supports--> v01\n- v09 --supports--> v21\n- v11 --refutes--> v05\n- v21 --refutes--> v02\n- v07 --supports--> v1a\n- v24 --refutes--> v13\n- v16 --refutes--> v00\n- v25 --supports--> v14\n- v09 --supports--> v27\n- v0c --refutes--> v15\n- v10 --supports--> v26\n- v14 --supports--> v1d\n- v17 --supports--> v09\n- v0c --supports--> v0e\n- v00 --refutes--> v22\n- v05 --refutes--> v25\n- v19 --supports--> v1c\n\n[VERIFIED ROOT]\nv23 = 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":"24d064da9e847888c41f10b1fbdbd1827c14d14561816c0234024e863c6d5dc3","schema":"livegraph.boolean-propagation.v1","spec_hash":"8c673cb91d42f42d1387270016cb329ce6d53f11caec6fcc69b30a82629ea897","spec_json":"{\"edges\":[{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v25\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-7afd05fa1a7c1e1ae8b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v22 --supports--> v16\n- v1d --refutes--> v21\n- v1f --refutes--> v26\n- v22 --refutes--> v04\n- v14 --supports--> v13\n- v06 --supports--> v23\n- v27 --refutes--> v0b\n- v21 --refutes--> v08\n- v0f --supports--> v01\n- v17 --supports--> v12\n- v0a --supports--> v20\n- v0b --refutes--> v1a\n- v2a --supports--> v07\n- v26 --supports--> v10\n- v09 --refutes--> v1f\n- v0d --supports--> v06\n- v10 --refutes--> v17\n- v08 --refutes--> v14\n- v1d --refutes--> v1e\n- v11 --refutes--> v00\n- v27 --refutes--> v0e\n- v00 --refutes--> v0a\n- v09 --supports--> v0c\n- v27 --refutes--> v25\n- v16 --supports--> v0f\n- v20 --refutes--> v0d\n- v14 --supports--> v24\n- v2a --refutes--> v1c\n- v13 --supports--> v11\n- v19 --refutes--> v1b\n- v01 --supports--> v29\n- v29 --supports--> v27\n- v2a --supports--> v28\n- v17 --supports--> v2a\n- v1a --supports--> v1d\n- v1b --refutes--> v22\n- v0d --refutes--> v02\n- v1d --supports--> v05\n- v07 --supports--> v19\n- v0d --refutes--> v03\n- v08 --refutes--> v18\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e24d5c9c7c8cf036ef5613d5dfe9cbc8e3941b288782f249a9b3624e9ae18268","schema":"livegraph.boolean-propagation.v1","spec_hash":"7afd05fa1a7c1e1ae8b0f724a50f1a5253f517749c2758c849a77de28911bf92","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-43eb7330f2e6b85a46be","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v29\n- v19 --refutes--> v08\n- v15 --refutes--> v28\n- v29 --refutes--> v15\n- v0b --supports--> v12\n- v04 --supports--> v07\n- v03 --refutes--> v09\n- v03 --refutes--> v18\n- v26 --refutes--> v04\n- v0d --refutes--> v11\n- v13 --refutes--> v1d\n- v00 --supports--> v2a\n- v24 --refutes--> v1f\n- v1e --supports--> v23\n- v22 --refutes--> v26\n- v1d --refutes--> v20\n- v06 --supports--> v10\n- v0e --supports--> v0c\n- v1f --supports--> v0e\n- v06 --refutes--> v1b\n- v0f --refutes--> v01\n- v22 --supports--> v02\n- v11 --supports--> v03\n- v19 --refutes--> v06\n- v01 --refutes--> v27\n- v2a --supports--> v1e\n- v26 --supports--> v1c\n- v20 --refutes--> v05\n- v09 --refutes--> v17\n- v1d --supports--> v25\n- v11 --supports--> v21\n- v1e --supports--> v19\n- v0d --refutes--> v16\n- v28 --refutes--> v0f\n- v06 --refutes--> v1a\n- v0c --refutes--> v00\n- v27 --supports--> v0d\n- v0b --refutes--> v24\n- v15 --refutes--> v14\n- v10 --supports--> v0a\n- v09 --refutes--> v13\n- v07 --supports--> v0b\n\n[VERIFIED ROOT]\nv22 = 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":"91fb49f75cf071722c737f54c5fc9a834a32a5e5d67d303b0a053325955b75d1","schema":"livegraph.boolean-propagation.v1","spec_hash":"43eb7330f2e6b85a46be4697ec7358d37ef7c5a0c09baef451e24e14c035b5f7","spec_json":"{\"edges\":[{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v22\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-81d8d9ad1b68ed58a491","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v25 --refutes--> v08\n- v26 --refutes--> v05\n- v1a --refutes--> v24\n- v09 --refutes--> v00\n- v14 --supports--> v11\n- v13 --supports--> v04\n- v0d --supports--> v1f\n- v1c --supports--> v26\n- v05 --refutes--> v18\n- v03 --refutes--> v1e\n- v27 --supports--> v0f\n- v22 --refutes--> v03\n- v0d --refutes--> v13\n- v1a --refutes--> v1c\n- v10 --supports--> v14\n- v0b --supports--> v0a\n- v28 --refutes--> v17\n- v22 --refutes--> v16\n- v27 --refutes--> v09\n- v22 --supports--> v2a\n- v29 --refutes--> v0c\n- v23 --supports--> v0d\n- v0b --supports--> v15\n- v12 --supports--> v06\n- v08 --supports--> v12\n- v00 --supports--> v1a\n- v03 --refutes--> v0e\n- v06 --supports--> v01\n- v14 --refutes--> v07\n- v06 --supports--> v02\n- v0e --supports--> v25\n- v28 --refutes--> v21\n- v0c --supports--> v27\n- v27 --supports--> v1b\n- v11 --supports--> v22\n- v1d --supports--> v23\n- v01 --supports--> v20\n- v0b --supports--> v1d\n- v01 --supports--> v29\n- v18 --refutes--> v0b\n- v17 --refutes--> v10\n\n[VERIFIED ROOT]\nv28 = 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":"b64f2420b929fe6256a230cf84986c8dbc241f38c2cc88a8f52db1455f0e7972","schema":"livegraph.boolean-propagation.v1","spec_hash":"81d8d9ad1b68ed58a491edc1921fc1f578ca3bc728f5dca881dc9cbc1462e9b9","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v28\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-e427c8aab862237c47c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v15 --refutes--> v09\n- v26 --refutes--> v02\n- v10 --refutes--> v1f\n- v1c --refutes--> v12\n- v04 --refutes--> v0f\n- v22 --supports--> v25\n- v16 --supports--> v08\n- v19 --refutes--> v27\n- v00 --supports--> v2a\n- v0b --supports--> v20\n- v04 --refutes--> v14\n- v08 --supports--> v1c\n- v07 --supports--> v1b\n- v1a --refutes--> v22\n- v25 --refutes--> v03\n- v11 --supports--> v00\n- v23 --refutes--> v0e\n- v15 --refutes--> v07\n- v1e --refutes--> v05\n- v06 --refutes--> v1e\n- v0f --refutes--> v23\n- v17 --supports--> v24\n- v25 --refutes--> v1d\n- v05 --refutes--> v0d\n- v00 --refutes--> v0a\n- v26 --refutes--> v29\n- v22 --refutes--> v18\n- v25 --refutes--> v04\n- v19 --refutes--> v28\n- v05 --refutes--> v26\n- v01 --supports--> v06\n- v20 --supports--> v13\n- v13 --supports--> v15\n- v29 --refutes--> v10\n- v2a --supports--> v19\n- v0c --refutes--> v0b\n- v17 --refutes--> v16\n- v23 --supports--> v17\n- v19 --refutes--> v0c\n- v10 --supports--> v1a\n- v25 --supports--> v21\n- v1b --supports--> v01\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv12\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e70a1a0687d70e44373b239f8aa1653381e141819c1861a5185a2316e850b8b4","schema":"livegraph.boolean-propagation.v1","spec_hash":"e427c8aab862237c47c594690a9da2767c321799c303fbd21221da5727907ea2","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-ff5cdac93505dc7dceff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v29\n- v1a --refutes--> v28\n- v0e --refutes--> v1f\n- v0c --supports--> v22\n- v02 --supports--> v1c\n- v29 --supports--> v05\n- v04 --refutes--> v19\n- v1b --refutes--> v0b\n- v28 --supports--> v1e\n- v0b --refutes--> v07\n- v08 --supports--> v21\n- v17 --supports--> v08\n- v14 --supports--> v23\n- v23 --supports--> v01\n- v11 --supports--> v00\n- v0d --refutes--> v27\n- v05 --refutes--> v04\n- v18 --refutes--> v09\n- v05 --supports--> v20\n- v06 --supports--> v0c\n- v1e --supports--> v14\n- v1d --refutes--> v16\n- v1e --supports--> v03\n- v16 --refutes--> v0f\n- v1d --supports--> v15\n- v11 --supports--> v0d\n- v21 --refutes--> v18\n- v23 --refutes--> v02\n- v0f --refutes--> v17\n- v08 --supports--> v10\n- v0a --supports--> v2a\n- v22 --refutes--> v11\n- v1c --refutes--> v0a\n- v1e --refutes--> v13\n- v0e --refutes--> v25\n- v18 --supports--> v1a\n- v06 --supports--> v24\n- v27 --refutes--> v1b\n- v07 --refutes--> v0e\n- v04 --refutes--> v06\n- v08 --refutes--> v26\n- v16 --supports--> v12\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a2a247873c0c02e77ea768e7e65fdb2964183beea280080dea4585a3979d5de7","schema":"livegraph.boolean-propagation.v1","spec_hash":"ff5cdac93505dc7dceff06cc653975533d88c650be42c67686b029760e0cbd9f","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-2fad015a9d8660b4990a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v09 --refutes--> v03\n- v29 --refutes--> v08\n- v09 --supports--> v1b\n- v1c --supports--> v02\n- v1c --supports--> v24\n- v17 --supports--> v20\n- v0b --refutes--> v01\n- v18 --supports--> v19\n- v20 --refutes--> v26\n- v0d --refutes--> v13\n- v1a --supports--> v0b\n- v0c --supports--> v16\n- v10 --refutes--> v1a\n- v09 --refutes--> v12\n- v07 --supports--> v00\n- v14 --refutes--> v18\n- v1f --supports--> v0c\n- v06 --supports--> v17\n- v00 --refutes--> v15\n- v13 --supports--> v11\n- v20 --supports--> v0e\n- v05 --refutes--> v27\n- v09 --refutes--> v28\n- v05 --supports--> v23\n- v16 --refutes--> v29\n- v21 --refutes--> v1c\n- v04 --refutes--> v14\n- v09 --supports--> v21\n- v25 --refutes--> v05\n- v2a --refutes--> v1d\n- v2a --supports--> v0d\n- v02 --supports--> v06\n- v0b --supports--> v04\n- v13 --supports--> v0a\n- v0e --refutes--> v25\n- v22 --refutes--> v07\n- v0d --supports--> v0f\n- v13 --supports--> v22\n- v00 --supports--> v09\n- v19 --supports--> v1e\n- v19 --supports--> v2a\n\n[VERIFIED ROOT]\nv1f = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"904387ed5f5cf8a85d7ae32338161f3a9879b17bf78f2e2b09e3fba3099c2ae6","schema":"livegraph.boolean-propagation.v1","spec_hash":"2fad015a9d8660b4990a6e930600f262ee387c8c0682ddc6d9304a4e2a0a4899","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f3039b811342f5247da0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v28\n- v15 --refutes--> v07\n- v28 --supports--> v0d\n- v0b --refutes--> v25\n- v2a --refutes--> v18\n- v06 --refutes--> v00\n- v27 --refutes--> v12\n- v28 --supports--> v17\n- v17 --refutes--> v1f\n- v03 --refutes--> v2a\n- v16 --supports--> v11\n- v08 --refutes--> v04\n- v0b --refutes--> v27\n- v00 --refutes--> v08\n- v0a --refutes--> v0b\n- v11 --refutes--> v1b\n- v27 --refutes--> v06\n- v17 --refutes--> v29\n- v1a --supports--> v09\n- v18 --supports--> v02\n- v22 --supports--> v0c\n- v13 --supports--> v01\n- v0a --refutes--> v1e\n- v0f --supports--> v19\n- v28 --supports--> v20\n- v16 --refutes--> v1c\n- v08 --supports--> v14\n- v22 --supports--> v0a\n- v11 --supports--> v0f\n- v23 --supports--> v1d\n- v26 --refutes--> v16\n- v26 --refutes--> v10\n- v1a --refutes--> v15\n- v05 --refutes--> v23\n- v15 --supports--> v0e\n- v0e --supports--> v26\n- v24 --refutes--> v05\n- v02 --supports--> v1a\n- v21 --refutes--> v13\n- v14 --refutes--> v03\n- v1f --refutes--> v22\n- v23 --supports--> v21\n\n[VERIFIED ROOT]\nv24 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ee37cc8cf9ad1012c90826b2eabd90248ceb37488f8bbd22ffab71833f6cc73b","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3039b811342f5247da0a465d6d35fd7dd00e2e80f85287e1dbc45c3d708c11c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v24\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-91c7e5165f7e794a5e05","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v15\n- v11 --supports--> v27\n- v05 --refutes--> v13\n- v26 --refutes--> v18\n- v28 --supports--> v1b\n- v2a --refutes--> v25\n- v12 --supports--> v0a\n- v1d --refutes--> v02\n- v0c --refutes--> v23\n- v00 --supports--> v1c\n- v07 --refutes--> v06\n- v24 --supports--> v29\n- v23 --supports--> v00\n- v00 --supports--> v09\n- v22 --supports--> v07\n- v02 --supports--> v14\n- v21 --supports--> v03\n- v11 --refutes--> v21\n- v06 --supports--> v0b\n- v14 --refutes--> v12\n- v07 --refutes--> v01\n- v1f --supports--> v1d\n- v13 --supports--> v26\n- v04 --supports--> v05\n- v16 --supports--> v0d\n- v0e --refutes--> v17\n- v0b --refutes--> v28\n- v2a --supports--> v0c\n- v26 --supports--> v20\n- v29 --supports--> v0e\n- v24 --supports--> v1e\n- v12 --refutes--> v11\n- v17 --refutes--> v10\n- v04 --supports--> v08\n- v1c --supports--> v22\n- v16 --supports--> v04\n- v13 --supports--> v1a\n- v10 --supports--> v2a\n- v14 --supports--> v0f\n- v16 --refutes--> v19\n- v03 --refutes--> v24\n- v28 --refutes--> v16\n\n[VERIFIED ROOT]\nv1f = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2eb2331ca6d00926c31fe251d1b2580dc15659cd11cbebe3cbf9b925c3a445ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"91c7e5165f7e794a5e05220ee07bcf277f2dfbac9af575032560cc51d980cc73","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-6445a35580c0d0129c0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v28 --refutes--> v0c\n- v24 --refutes--> v08\n- v1f --refutes--> v04\n- v1c --refutes--> v05\n- v00 --refutes--> v17\n- v09 --refutes--> v12\n- v17 --supports--> v1c\n- v01 --supports--> v27\n- v00 --refutes--> v25\n- v0f --supports--> v1a\n- v0e --supports--> v09\n- v06 --refutes--> v0f\n- v20 --refutes--> v06\n- v1f --supports--> v21\n- v03 --refutes--> v2a\n- v05 --supports--> v22\n- v0a --refutes--> v00\n- v13 --refutes--> v1b\n- v28 --supports--> v15\n- v07 --supports--> v0b\n- v0c --refutes--> v23\n- v23 --refutes--> v0a\n- v02 --refutes--> v0d\n- v02 --supports--> v0e\n- v01 --supports--> v1e\n- v0f --refutes--> v26\n- v1b --refutes--> v28\n- v24 --refutes--> v07\n- v1c --supports--> v18\n- v00 --refutes--> v19\n- v04 --supports--> v02\n- v07 --supports--> v01\n- v22 --supports--> v20\n- v0f --supports--> v10\n- v14 --refutes--> v03\n- v04 --refutes--> v29\n- v17 --supports--> v1d\n- v1a --refutes--> v11\n- v2a --supports--> v13\n- v12 --supports--> v24\n- v1a --supports--> v1f\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7222d6258332fee7f7272818cc8817c3bc13917a51cc26cdc12ca0f4b083c415","schema":"livegraph.boolean-propagation.v1","spec_hash":"6445a35580c0d0129c0baa480030aece1ce09a7a1a56b72c96d0c21193ba186d","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-8dd439103ea3a573145d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v29 --supports--> v19\n- v17 --refutes--> v23\n- v21 --refutes--> v0d\n- v0e --supports--> v15\n- v0c --supports--> v03\n- v0e --refutes--> v26\n- v21 --supports--> v12\n- v1c --refutes--> v05\n- v15 --refutes--> v18\n- v09 --refutes--> v0f\n- v17 --refutes--> v01\n- v16 --supports--> v0a\n- v1c --supports--> v1d\n- v27 --refutes--> v0b\n- v27 --supports--> v08\n- v10 --supports--> v17\n- v00 --refutes--> v29\n- v05 --refutes--> v25\n- v17 --supports--> v02\n- v14 --refutes--> v1c\n- v22 --refutes--> v1e\n- v20 --refutes--> v24\n- v23 --supports--> v27\n- v1b --refutes--> v22\n- v1f --supports--> v21\n- v25 --refutes--> v09\n- v24 --supports--> v14\n- v0f --refutes--> v1a\n- v18 --refutes--> v1b\n- v13 --refutes--> v20\n- v23 --supports--> v07\n- v03 --refutes--> v10\n- v15 --supports--> v28\n- v29 --supports--> v13\n- v0f --supports--> v1f\n- v12 --refutes--> v2a\n- v00 --refutes--> v06\n- v1e --supports--> v0c\n- v10 --supports--> v11\n- v0a --refutes--> v00\n- v12 --refutes--> v0e\n\n[VERIFIED ROOT]\nv16 = 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":"da3feb80aabeb9ee7c239c7b487ce46f6572e4597e41a4ef4749b937eaea3ca9","schema":"livegraph.boolean-propagation.v1","spec_hash":"8dd439103ea3a573145da4225215c3c4a0c8c255f46ebf8fe1654dc92c35b8ea","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-9944ab268d74ce4e5e5f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v29\n- v24 --refutes--> v03\n- v25 --refutes--> v00\n- v15 --refutes--> v1d\n- v11 --supports--> v09\n- v05 --supports--> v01\n- v16 --supports--> v20\n- v0a --refutes--> v24\n- v1b --supports--> v2a\n- v21 --supports--> v25\n- v00 --refutes--> v07\n- v14 --supports--> v0d\n- v27 --refutes--> v28\n- v1d --supports--> v14\n- v14 --supports--> v1a\n- v16 --supports--> v13\n- v28 --supports--> v0e\n- v1e --supports--> v23\n- v03 --refutes--> v0b\n- v16 --supports--> v17\n- v22 --supports--> v1f\n- v08 --supports--> v16\n- v07 --supports--> v04\n- v19 --supports--> v21\n- v22 --refutes--> v06\n- v19 --supports--> v26\n- v04 --refutes--> v05\n- v09 --refutes--> v27\n- v1f --supports--> v19\n- v0d --supports--> v11\n- v0c --refutes--> v15\n- v0c --refutes--> v0f\n- v0a --supports--> v10\n- v23 --refutes--> v08\n- v20 --refutes--> v12\n- v2a --refutes--> v0c\n- v05 --supports--> v1b\n- v0d --supports--> v02\n- v20 --supports--> v22\n- v03 --supports--> v1e\n- v28 --refutes--> v18\n- v1b --supports--> v1c\n\n[VERIFIED ROOT]\nv0a = 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":"28e28923145b535a51c727053e30e5e40db886830d7839d41b1ddf2dc9cf147f","schema":"livegraph.boolean-propagation.v1","spec_hash":"9944ab268d74ce4e5e5fbeb893e4e7f11cb3491bfb4a8c5c398409fb21a99bed","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-674c82457897a1481410","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --supports--> v0b\n- v1f --refutes--> v0f\n- v0d --supports--> v1c\n- v17 --supports--> v18\n- v09 --refutes--> v11\n- v21 --refutes--> v1e\n- v26 --supports--> v01\n- v12 --refutes--> v19\n- v04 --supports--> v07\n- v21 --supports--> v29\n- v1e --refutes--> v20\n- v0a --refutes--> v1f\n- v0d --supports--> v15\n- v04 --supports--> v25\n- v04 --supports--> v0a\n- v0b --refutes--> v09\n- v22 --refutes--> v02\n- v1a --supports--> v05\n- v1c --refutes--> v04\n- v22 --supports--> v10\n- v26 --refutes--> v03\n- v2a --refutes--> v23\n- v17 --supports--> v1b\n- v1d --refutes--> v12\n- v2a --supports--> v0c\n- v23 --supports--> v1d\n- v19 --supports--> v22\n- v0f --refutes--> v13\n- v26 --refutes--> v0e\n- v0e --supports--> v14\n- v26 --refutes--> v08\n- v06 --refutes--> v24\n- v0d --refutes--> v16\n- v14 --supports--> v27\n- v06 --supports--> v17\n- v1e --refutes--> v28\n- v00 --supports--> v0d\n- v05 --refutes--> v21\n- v17 --supports--> v2a\n- v13 --supports--> v26\n- v02 --refutes--> v00\n- v27 --supports--> v1a\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12374555b809c66502e2c298c34f2bd7c21381089d5a6e8edf5a4d35016d3370","schema":"livegraph.boolean-propagation.v1","spec_hash":"674c82457897a148141047a686f5ea3378986522ed91c3e28b1d14f463ab714d","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f9a473188092d9038067","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1e\n- v16 --supports--> v20\n- v09 --refutes--> v19\n- v18 --supports--> v10\n- v27 --refutes--> v21\n- v05 --supports--> v27\n- v0c --supports--> v04\n- v16 --refutes--> v02\n- v0a --supports--> v15\n- v13 --refutes--> v1b\n- v0d --supports--> v22\n- v1b --supports--> v28\n- v20 --supports--> v0d\n- v01 --refutes--> v09\n- v1e --refutes--> v12\n- v1f --refutes--> v08\n- v00 --refutes--> v0b\n- v08 --supports--> v03\n- v03 --refutes--> v17\n- v16 --refutes--> v26\n- v07 --supports--> v25\n- v27 --refutes--> v16\n- v12 --supports--> v0c\n- v24 --supports--> v06\n- v23 --supports--> v05\n- v07 --supports--> v0f\n- v1c --refutes--> v24\n- v0f --supports--> v23\n- v22 --supports--> v11\n- v05 --refutes--> v0e\n- v15 --supports--> v01\n- v18 --supports--> v1f\n- v19 --supports--> v1c\n- v00 --refutes--> v14\n- v1b --refutes--> v07\n- v24 --supports--> v13\n- v07 --supports--> v29\n- v11 --refutes--> v00\n- v0c --supports--> v2a\n- v24 --supports--> v1a\n- v2a --refutes--> v1d\n- v2a --supports--> v18\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a347cf273aa60be23ff02d43a52f00791f2f6a272d38d3e0d803d96dd8610493","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9a473188092d90380673798ca36c93e911cd62eb9b09f93142203508face11c","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-3e903e31a8ca0c43181a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1d\n- v27 --refutes--> v1e\n- v0b --supports--> v15\n- v1d --supports--> v01\n- v25 --refutes--> v1b\n- v01 --supports--> v1f\n- v21 --supports--> v27\n- v00 --supports--> v12\n- v0d --supports--> v07\n- v03 --supports--> v13\n- v13 --refutes--> v24\n- v0b --refutes--> v1a\n- v23 --refutes--> v02\n- v1e --supports--> v25\n- v12 --refutes--> v05\n- v0c --refutes--> v09\n- v06 --refutes--> v22\n- v1f --refutes--> v23\n- v0f --supports--> v08\n- v0d --refutes--> v0b\n- v05 --supports--> v1c\n- v09 --supports--> v2a\n- v24 --supports--> v16\n- v26 --refutes--> v11\n- v11 --refutes--> v00\n- v19 --refutes--> v29\n- v05 --refutes--> v03\n- v25 --refutes--> v06\n- v2a --refutes--> v28\n- v22 --supports--> v0d\n- v1e --refutes--> v0a\n- v24 --refutes--> v17\n- v15 --refutes--> v26\n- v0f --refutes--> v04\n- v03 --supports--> v20\n- v23 --refutes--> v21\n- v19 --refutes--> v0f\n- v01 --supports--> v14\n- v04 --supports--> v18\n- v23 --refutes--> v10\n- v28 --refutes--> v19\n- v1d --refutes--> v0e\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"54b3a56887b0938d5fe5de92fb5187e7a3430e64b2bbcdfe8e1840f8e49fc195","schema":"livegraph.boolean-propagation.v1","spec_hash":"3e903e31a8ca0c43181a24e8145e2c1d5be25b5790b2a8e4554522780246af85","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a3b6d01376a402164654","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v28\n- v17 --supports--> v07\n- v23 --refutes--> v0b\n- v05 --supports--> v21\n- v0e --refutes--> v16\n- v1b --supports--> v01\n- v1a --supports--> v29\n- v06 --supports--> v05\n- v0b --refutes--> v1d\n- v24 --supports--> v06\n- v2a --supports--> v20\n- v16 --supports--> v09\n- v0a --supports--> v19\n- v27 --refutes--> v1f\n- v12 --supports--> v0e\n- v2a --supports--> v13\n- v13 --refutes--> v04\n- v13 --supports--> v1e\n- v0a --supports--> v18\n- v0d --refutes--> v0a\n- v27 --refutes--> v14\n- v18 --refutes--> v08\n- v22 --supports--> v00\n- v16 --supports--> v1a\n- v1f --refutes--> v26\n- v01 --supports--> v2a\n- v1a --refutes--> v0c\n- v00 --supports--> v27\n- v24 --refutes--> v10\n- v08 --refutes--> v15\n- v1d --refutes--> v0d\n- v00 --supports--> v03\n- v1c --refutes--> v22\n- v1f --refutes--> v17\n- v18 --refutes--> v25\n- v05 --refutes--> v1b\n- v17 --refutes--> v24\n- v1e --refutes--> v11\n- v0f --supports--> v12\n- v11 --supports--> v23\n- v27 --supports--> v02\n- v15 --supports--> v0f\n\n[VERIFIED ROOT]\nv1c = TRUE\n\n[TARGET]\nv29\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c5153cecc95d86cde6d2b0694955a650422885ea6c30ac18ab59d7838ebf0c90","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3b6d01376a402164654c1daaa60ebf759ce5a102e65bd16526a4efcb77e0f51","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v29\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-7a6ba06d707086d20a65","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v1b\n- v22 --refutes--> v14\n- v15 --supports--> v0c\n- v05 --refutes--> v2a\n- v2a --supports--> v1e\n- v02 --refutes--> v26\n- v19 --supports--> v1d\n- v04 --refutes--> v27\n- v0e --supports--> v0f\n- v0c --supports--> v1a\n- v1c --refutes--> v09\n- v1b --refutes--> v0b\n- v0f --refutes--> v28\n- v08 --refutes--> v05\n- v01 --supports--> v20\n- v26 --supports--> v03\n- v04 --supports--> v29\n- v07 --supports--> v0e\n- v23 --refutes--> v08\n- v1d --supports--> v25\n- v1e --refutes--> v24\n- v20 --supports--> v10\n- v22 --refutes--> v00\n- v23 --refutes--> v0a\n- v00 --refutes--> v07\n- v15 --refutes--> v1f\n- v1b --refutes--> v17\n- v1c --refutes--> v21\n- v25 --supports--> v04\n- v27 --supports--> v18\n- v05 --refutes--> v12\n- v11 --refutes--> v06\n- v28 --supports--> v19\n- v20 --refutes--> v15\n- v1c --supports--> v0d\n- v0c --supports--> v13\n- v26 --supports--> v11\n- v06 --refutes--> v16\n- v21 --refutes--> v01\n- v06 --supports--> v22\n- v18 --refutes--> v23\n- v1b --refutes--> v1c\n\n[VERIFIED ROOT]\nv02 = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7a23d73184c04d4ad0f83b9555ee8f629f4d9daef0f507cb310f418cc4c6591e","schema":"livegraph.boolean-propagation.v1","spec_hash":"7a6ba06d707086d20a65f7b2b94dc735e9e8582b4b561d4a3e626739443777ec","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v02\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f689009fbc9442d17d99","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v22\n- v1c --refutes--> v17\n- v19 --refutes--> v1f\n- v20 --refutes--> v07\n- v20 --refutes--> v0b\n- v10 --supports--> v1e\n- v0b --refutes--> v02\n- v28 --supports--> v11\n- v25 --refutes--> v0e\n- v08 --refutes--> v24\n- v0c --refutes--> v1a\n- v02 --supports--> v18\n- v08 --refutes--> v15\n- v1c --supports--> v23\n- v0f --supports--> v1c\n- v00 --supports--> v27\n- v1b --supports--> v29\n- v11 --supports--> v12\n- v27 --supports--> v25\n- v05 --refutes--> v26\n- v05 --refutes--> v1d\n- v05 --supports--> v19\n- v23 --refutes--> v04\n- v23 --refutes--> v00\n- v22 --refutes--> v20\n- v29 --supports--> v08\n- v02 --supports--> v0a\n- v13 --refutes--> v06\n- v24 --supports--> v10\n- v2a --refutes--> v03\n- v11 --supports--> v0f\n- v01 --refutes--> v09\n- v05 --supports--> v16\n- v0e --supports--> v2a\n- v1a --supports--> v13\n- v02 --refutes--> v05\n- v13 --supports--> v01\n- v19 --refutes--> v0d\n- v0d --supports--> v28\n- v03 --refutes--> v0c\n- v01 --refutes--> v14\n- v0c --refutes--> v21\n\n[VERIFIED ROOT]\nv1b = 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":"3c26bcfe1c9db5caaa18d4f2bbe8d72a17b352e0cb7300ee17777611bb28e08a","schema":"livegraph.boolean-propagation.v1","spec_hash":"f689009fbc9442d17d990f4f9d7302936051b2aead7d093cd32a947b4e847788","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-d769c69fabe7362adc18","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v18\n- v1f --refutes--> v03\n- v26 --supports--> v16\n- v08 --supports--> v09\n- v02 --supports--> v06\n- v06 --refutes--> v12\n- v15 --refutes--> v14\n- v1e --refutes--> v23\n- v0c --refutes--> v05\n- v25 --supports--> v01\n- v29 --refutes--> v26\n- v00 --supports--> v08\n- v26 --refutes--> v0e\n- v0a --supports--> v1e\n- v27 --supports--> v1b\n- v14 --refutes--> v0f\n- v27 --refutes--> v1a\n- v0a --refutes--> v24\n- v1d --supports--> v25\n- v10 --supports--> v11\n- v0c --refutes--> v0d\n- v0a --refutes--> v22\n- v1f --supports--> v02\n- v21 --supports--> v0c\n- v0f --refutes--> v13\n- v18 --refutes--> v1f\n- v1b --refutes--> v28\n- v16 --refutes--> v15\n- v13 --supports--> v19\n- v07 --supports--> v04\n- v0d --refutes--> v29\n- v11 --supports--> v00\n- v07 --refutes--> v17\n- v1b --refutes--> v1d\n- v23 --refutes--> v27\n- v13 --supports--> v07\n- v11 --supports--> v20\n- v0f --supports--> v2a\n- v11 --supports--> v0b\n- v12 --supports--> v10\n- v16 --refutes--> v1c\n- v08 --supports--> v21\n\n[VERIFIED ROOT]\nv0a = 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":"3284d04821ebd51226335175fe4a8b91074f93f0d0c8fc1f2c0dea6a55931f88","schema":"livegraph.boolean-propagation.v1","spec_hash":"d769c69fabe7362adc18a6bd6a47ed4c14b7c2d9cd28a01d99b5631c6ebbd62c","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-eb75b8960f9b9c6c0e7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1f\n- v05 --supports--> v0b\n- v1a --refutes--> v19\n- v22 --refutes--> v10\n- v0c --refutes--> v1d\n- v06 --refutes--> v21\n- v21 --refutes--> v26\n- v21 --refutes--> v20\n- v04 --refutes--> v0a\n- v18 --refutes--> v13\n- v03 --refutes--> v09\n- v26 --supports--> v1b\n- v03 --refutes--> v17\n- v1c --supports--> v15\n- v1b --refutes--> v23\n- v01 --refutes--> v18\n- v0c --refutes--> v07\n- v28 --supports--> v1a\n- v1e --supports--> v29\n- v27 --supports--> v01\n- v29 --supports--> v22\n- v21 --refutes--> v0e\n- v02 --supports--> v05\n- v0a --supports--> v0d\n- v00 --refutes--> v03\n- v1c --supports--> v04\n- v25 --refutes--> v02\n- v06 --refutes--> v11\n- v19 --supports--> v00\n- v19 --supports--> v16\n- v1e --refutes--> v14\n- v1d --refutes--> v28\n- v04 --refutes--> v0f\n- v09 --refutes--> v24\n- v0b --refutes--> v2a\n- v0b --supports--> v1e\n- v0a --refutes--> v27\n- v10 --supports--> v1c\n- v13 --supports--> v0c\n- v24 --supports--> v06\n- v00 --refutes--> v12\n- v09 --refutes--> v08\n\n[VERIFIED ROOT]\nv25 = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"47101377ab47d51e517cfda89226c1d6f3d9fab88b7ecfd4fde6939eb89692e2","schema":"livegraph.boolean-propagation.v1","spec_hash":"eb75b8960f9b9c6c0e7ed1fb4470241fcc7f2965b5a462414c56eede41068cf6","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v25\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-58926b73fc7176bc298e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v05\n- v20 --supports--> v0f\n- v13 --supports--> v17\n- v00 --supports--> v0c\n- v0a --supports--> v23\n- v05 --refutes--> v15\n- v2a --refutes--> v11\n- v02 --refutes--> v2a\n- v22 --refutes--> v1f\n- v1d --refutes--> v1a\n- v25 --refutes--> v01\n- v11 --refutes--> v1b\n- v12 --supports--> v27\n- v18 --supports--> v1e\n- v01 --refutes--> v13\n- v27 --refutes--> v00\n- v2a --refutes--> v0b\n- v14 --refutes--> v0d\n- v1a --refutes--> v29\n- v07 --refutes--> v10\n- v04 --refutes--> v26\n- v0c --refutes--> v08\n- v23 --refutes--> v20\n- v17 --supports--> v02\n- v11 --refutes--> v19\n- v00 --refutes--> v03\n- v16 --refutes--> v25\n- v10 --refutes--> v24\n- v18 --refutes--> v21\n- v05 --refutes--> v06\n- v0d --refutes--> v1d\n- v0c --refutes--> v1c\n- v0c --supports--> v28\n- v11 --refutes--> v07\n- v22 --refutes--> v18\n- v1e --supports--> v0e\n- v04 --supports--> v16\n- v20 --refutes--> v04\n- v1e --supports--> v09\n- v15 --refutes--> v14\n- v29 --supports--> v22\n- v1e --refutes--> v12\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"74cd33eb098618ccc95dba81083f7d40d0ff1d6bdd5db883d8f3c8281434bf57","schema":"livegraph.boolean-propagation.v1","spec_hash":"58926b73fc7176bc298e4f24f3580f9ba01e06717296f90a2ac9c3575b151a52","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-e088c8e3b19d47fdd9e3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2a --refutes--> v1c\n- v0a --supports--> v28\n- v13 --supports--> v16\n- v28 --supports--> v2a\n- v29 --refutes--> v20\n- v1b --supports--> v23\n- v0f --refutes--> v0d\n- v0a --supports--> v0b\n- v1b --supports--> v26\n- v09 --refutes--> v02\n- v12 --refutes--> v27\n- v04 --supports--> v0c\n- v00 --refutes--> v0e\n- v14 --refutes--> v29\n- v09 --refutes--> v05\n- v1d --refutes--> v14\n- v17 --refutes--> v12\n- v16 --refutes--> v09\n- v10 --refutes--> v03\n- v22 --refutes--> v1e\n- v27 --refutes--> v25\n- v08 --refutes--> v06\n- v25 --refutes--> v08\n- v00 --supports--> v1d\n- v09 --supports--> v22\n- v25 --refutes--> v21\n- v24 --refutes--> v17\n- v1e --refutes--> v1b\n- v2a --supports--> v04\n- v0d --supports--> v00\n- v17 --supports--> v1f\n- v20 --refutes--> v0a\n- v18 --refutes--> v13\n- v17 --supports--> v07\n- v10 --supports--> v11\n- v06 --supports--> v0f\n- v1b --refutes--> v10\n- v03 --refutes--> v01\n- v1a --refutes--> v19\n- v03 --supports--> v24\n- v1a --refutes--> v18\n- v03 --supports--> v15\n\n[VERIFIED ROOT]\nv1a = 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":"b1f143d28b1a468a51d4ccc4caece453b93dea1795cfbfc357cb645823e7eefb","schema":"livegraph.boolean-propagation.v1","spec_hash":"e088c8e3b19d47fdd9e37e4651a0216af5f91d61d9f471b06fdfadf2f0606f35","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-d4d2f2e66fb32587625e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v04 --refutes--> v0b\n- v01 --refutes--> v05\n- v25 --refutes--> v21\n- v17 --refutes--> v28\n- v09 --refutes--> v1a\n- v29 --refutes--> v0a\n- v28 --refutes--> v13\n- v13 --refutes--> v08\n- v27 --refutes--> v20\n- v0b --supports--> v26\n- v11 --supports--> v23\n- v1f --refutes--> v19\n- v0f --supports--> v1c\n- v1e --refutes--> v03\n- v01 --refutes--> v29\n- v26 --refutes--> v07\n- v0d --refutes--> v17\n- v0e --supports--> v1b\n- v1f --refutes--> v0f\n- v1d --refutes--> v0c\n- v22 --supports--> v01\n- v02 --supports--> v1d\n- v1f --supports--> v18\n- v29 --supports--> v06\n- v1d --supports--> v2a\n- v15 --supports--> v14\n- v08 --refutes--> v15\n- v0f --supports--> v10\n- v0f --refutes--> v00\n- v10 --supports--> v1e\n- v11 --refutes--> v25\n- v2a --supports--> v11\n- v25 --refutes--> v09\n- v27 --refutes--> v24\n- v0a --supports--> v04\n- v26 --refutes--> v0e\n- v0b --supports--> v12\n- v1a --refutes--> v22\n- v1b --supports--> v0d\n- v22 --supports--> v16\n- v03 --refutes--> v27\n- v27 --supports--> v02\n\n[VERIFIED ROOT]\nv1f = 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":"ea2fa1a4eeecd79892877593fa4b0bc8a5d3d6f110df0434ad840b4c7acb2f63","schema":"livegraph.boolean-propagation.v1","spec_hash":"d4d2f2e66fb32587625e271a8b53423cf58e74301444893c5d2a1c744815d141","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a1f55c59e103c066d03a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v26\n- v00 --supports--> v1b\n- v05 --refutes--> v1e\n- v2a --supports--> v27\n- v0f --refutes--> v1c\n- v0e --supports--> v20\n- v1a --supports--> v21\n- v1f --supports--> v18\n- v1c --supports--> v24\n- v19 --refutes--> v22\n- v1e --supports--> v2a\n- v1b --supports--> v0f\n- v27 --supports--> v1d\n- v1d --refutes--> v0b\n- v16 --refutes--> v08\n- v1d --supports--> v10\n- v28 --supports--> v16\n- v0a --refutes--> v01\n- v25 --refutes--> v00\n- v1b --refutes--> v07\n- v29 --refutes--> v0d\n- v0a --supports--> v09\n- v03 --supports--> v1a\n- v10 --supports--> v12\n- v19 --supports--> v17\n- v0e --supports--> v28\n- v0a --refutes--> v23\n- v08 --supports--> v03\n- v0d --supports--> v13\n- v06 --supports--> v02\n- v10 --supports--> v06\n- v29 --supports--> v0c\n- v18 --supports--> v04\n- v13 --supports--> v19\n- v24 --supports--> v05\n- v02 --supports--> v0a\n- v18 --refutes--> v15\n- v1a --refutes--> v14\n- v15 --refutes--> v0e\n- v19 --supports--> v11\n- v1a --refutes--> v25\n- v19 --refutes--> v1f\n\n[VERIFIED ROOT]\nv29 = FALSE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"63860d137ff9f6baa553435a54a59c854760b5f07509b75b9f98643fef44a6ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"a1f55c59e103c066d03a25e28db6812f96610c089d31bb5f60e4b9399bad5b81","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v29\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-9615f314b73f8c1e8eaf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --supports--> v0d\n- v00 --refutes--> v16\n- v12 --supports--> v0b\n- v19 --refutes--> v28\n- v10 --supports--> v1b\n- v0e --supports--> v17\n- v1d --supports--> v19\n- v1b --refutes--> v18\n- v1f --supports--> v06\n- v1e --refutes--> v29\n- v0f --refutes--> v02\n- v26 --refutes--> v01\n- v0b --refutes--> v1a\n- v10 --supports--> v11\n- v15 --refutes--> v08\n- v22 --supports--> v15\n- v1e --supports--> v04\n- v14 --refutes--> v05\n- v04 --supports--> v10\n- v21 --supports--> v0c\n- v00 --supports--> v27\n- v1f --refutes--> v03\n- v25 --supports--> v2a\n- v1b --supports--> v0e\n- v07 --supports--> v1c\n- v19 --supports--> v1e\n- v14 --refutes--> v12\n- v25 --supports--> v26\n- v0f --refutes--> v1f\n- v1a --refutes--> v07\n- v0a --refutes--> v22\n- v0d --supports--> v09\n- v22 --supports--> v20\n- v09 --supports--> v25\n- v13 --supports--> v1d\n- v17 --supports--> v14\n- v07 --refutes--> v00\n- v03 --refutes--> v0a\n- v15 --refutes--> v24\n- v08 --refutes--> v21\n- v27 --refutes--> v0f\n- v1f --refutes--> v23\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv01\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"60757ee2440f0dc999ef2108f8d409cd53e2031ff5b62f761e03682e586e6b24","schema":"livegraph.boolean-propagation.v1","spec_hash":"9615f314b73f8c1e8eaf315b2a70d7d07a63bb1eff1bcfcf80170a2faa95147c","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a906191000652612ad71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v28\n- v29 --refutes--> v20\n- v22 --supports--> v25\n- v13 --refutes--> v01\n- v0f --refutes--> v2a\n- v17 --supports--> v0d\n- v22 --refutes--> v04\n- v1b --supports--> v08\n- v1c --supports--> v1a\n- v29 --supports--> v15\n- v12 --refutes--> v21\n- v28 --refutes--> v22\n- v09 --supports--> v06\n- v25 --refutes--> v18\n- v06 --refutes--> v1c\n- v01 --refutes--> v27\n- v2a --supports--> v16\n- v03 --refutes--> v17\n- v10 --refutes--> v24\n- v13 --refutes--> v14\n- v27 --supports--> v05\n- v19 --refutes--> v07\n- v26 --refutes--> v13\n- v25 --supports--> v02\n- v19 --refutes--> v26\n- v0b --supports--> v1e\n- v05 --refutes--> v10\n- v1e --refutes--> v11\n- v23 --supports--> v0b\n- v1e --supports--> v0a\n- v29 --refutes--> v12\n- v10 --refutes--> v03\n- v21 --refutes--> v0c\n- v1c --supports--> v29\n- v26 --supports--> v00\n- v03 --refutes--> v0e\n- v0c --supports--> v19\n- v0f --refutes--> v1d\n- v16 --supports--> v09\n- v18 --supports--> v1f\n- v11 --refutes--> v1b\n- v18 --supports--> v0f\n\n[VERIFIED ROOT]\nv23 = 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":"286fd4389d94a22539f88731364515c7e41687c99262a38383a65eb66ad045f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"a906191000652612ad717f1c321771d5b724aa219e0ba82be65d2728571d0c45","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-6b8817b25a840cd5e0d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v15 --supports--> v06\n- v12 --supports--> v05\n- v22 --refutes--> v09\n- v04 --refutes--> v07\n- v14 --refutes--> v1c\n- v22 --refutes--> v0d\n- v1d --refutes--> v27\n- v14 --supports--> v03\n- v0c --supports--> v02\n- v1d --refutes--> v08\n- v0d --supports--> v0c\n- v10 --refutes--> v18\n- v11 --supports--> v20\n- v07 --refutes--> v00\n- v26 --supports--> v0f\n- v12 --refutes--> v22\n- v17 --refutes--> v23\n- v1c --refutes--> v25\n- v21 --refutes--> v04\n- v06 --refutes--> v14\n- v1a --refutes--> v29\n- v1f --supports--> v24\n- v01 --refutes--> v16\n- v2a --refutes--> v15\n- v01 --supports--> v11\n- v24 --refutes--> v1a\n- v17 --refutes--> v12\n- v0a --supports--> v17\n- v29 --refutes--> v26\n- v27 --supports--> v10\n- v0a --supports--> v13\n- v22 --supports--> v28\n- v18 --refutes--> v2a\n- v20 --supports--> v0e\n- v07 --refutes--> v01\n- v0c --refutes--> v0b\n- v0d --refutes--> v1e\n- v0f --supports--> v1d\n- v0e --refutes--> v1f\n- v1c --refutes--> v1b\n- v0c --refutes--> v21\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"725878f8b44923b2aa7c2693729ffafaae01a34474f6817aee935fca155bf049","schema":"livegraph.boolean-propagation.v1","spec_hash":"6b8817b25a840cd5e0d40945e2a046bd7dea8adb69ded0d0c770149ebc90e200","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-5ead1d1d42266982ff32","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v11 --refutes--> v13\n- v0f --supports--> v0d\n- v1e --refutes--> v14\n- v05 --supports--> v12\n- v18 --refutes--> v20\n- v05 --refutes--> v2a\n- v0b --refutes--> v09\n- v07 --supports--> v24\n- v2a --supports--> v1e\n- v28 --supports--> v04\n- v04 --supports--> v1a\n- v21 --supports--> v1f\n- v13 --supports--> v0b\n- v27 --supports--> v16\n- v18 --supports--> v08\n- v0d --refutes--> v0a\n- v04 --supports--> v05\n- v0b --supports--> v26\n- v10 --supports--> v25\n- v1f --refutes--> v28\n- v1f --refutes--> v1b\n- v18 --supports--> v0c\n- v23 --supports--> v0e\n- v24 --refutes--> v0f\n- v14 --supports--> v1d\n- v0a --supports--> v23\n- v24 --refutes--> v29\n- v1d --supports--> v03\n- v0e --supports--> v17\n- v20 --refutes--> v21\n- v09 --supports--> v18\n- v03 --supports--> v10\n- v16 --refutes--> v07\n- v0e --refutes--> v01\n- v1c --supports--> v06\n- v23 --supports--> v19\n- v1c --supports--> v11\n- v1d --supports--> v22\n- v00 --refutes--> v1c\n- v1c --supports--> v15\n- v00 --supports--> v02\n\n[VERIFIED ROOT]\nv27 = FALSE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7bed141533de5f2a9084d520128cdf435d61bfbc470a142f027fcc4b011a51ff","schema":"livegraph.boolean-propagation.v1","spec_hash":"5ead1d1d42266982ff32f3a6cf585d68382a8d14c267ed954508d19305c375ea","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-301e59487b9f9b06a1b9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0e --refutes--> v13\n- v1b --supports--> v25\n- v13 --supports--> v09\n- v15 --refutes--> v19\n- v23 --refutes--> v01\n- v20 --refutes--> v0d\n- v1b --refutes--> v12\n- v23 --supports--> v0a\n- v15 --refutes--> v1a\n- v20 --supports--> v03\n- v16 --supports--> v0b\n- v0c --refutes--> v2a\n- v18 --supports--> v1e\n- v14 --refutes--> v29\n- v1d --refutes--> v07\n- v20 --refutes--> v06\n- v08 --supports--> v22\n- v19 --refutes--> v0f\n- v0b --supports--> v08\n- v0d --refutes--> v04\n- v1c --refutes--> v14\n- v1b --refutes--> v26\n- v2a --supports--> v15\n- v13 --refutes--> v23\n- v23 --refutes--> v18\n- v17 --refutes--> v1f\n- v08 --supports--> v1c\n- v24 --refutes--> v21\n- v0d --refutes--> v27\n- v1f --supports--> v20\n- v05 --refutes--> v24\n- v1e --supports--> v1b\n- v1d --refutes--> v11\n- v29 --supports--> v0c\n- v24 --refutes--> v00\n- v02 --supports--> v16\n- v11 --supports--> v0e\n- v21 --refutes--> v28\n- v1b --refutes--> v17\n- v0f --refutes--> v05\n- v04 --refutes--> v02\n- v16 --supports--> v10\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv28\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fb2654cfca06b14beb5b7ef2b91a007c015a668c7e3e291387a8bb8c25275a42","schema":"livegraph.boolean-propagation.v1","spec_hash":"301e59487b9f9b06a1b956840a7adbf0969ff84e3150ba8125d8eff225d46fca","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v28\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-382dea2631e8efe2a305","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v25\n- v1b --refutes--> v02\n- v1a --supports--> v20\n- v2a --refutes--> v24\n- v00 --supports--> v13\n- v08 --supports--> v29\n- v28 --refutes--> v1d\n- v17 --supports--> v08\n- v0b --refutes--> v2a\n- v18 --refutes--> v1e\n- v12 --refutes--> v05\n- v28 --supports--> v21\n- v00 --refutes--> v04\n- v00 --refutes--> v1c\n- v22 --supports--> v19\n- v11 --supports--> v0e\n- v08 --supports--> v1a\n- v25 --supports--> v18\n- v19 --refutes--> v17\n- v19 --supports--> v07\n- v20 --supports--> v1b\n- v0c --supports--> v03\n- v03 --supports--> v12\n- v18 --supports--> v09\n- v24 --supports--> v1f\n- v11 --supports--> v22\n- v1f --supports--> v28\n- v23 --refutes--> v11\n- v12 --supports--> v0d\n- v15 --refutes--> v23\n- v1e --supports--> v0a\n- v1f --refutes--> v10\n- v00 --refutes--> v27\n- v22 --refutes--> v14\n- v1b --refutes--> v06\n- v06 --refutes--> v0c\n- v01 --refutes--> v15\n- v20 --supports--> v26\n- v0a --refutes--> v01\n- v27 --refutes--> v0b\n- v23 --refutes--> v16\n- v0f --refutes--> v00\n\n[VERIFIED ROOT]\nv0f = 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":"9f2c80bf66ce07b49b3949a27e5e25a40a0fe6a7d97bbe86204545fa8479e2e8","schema":"livegraph.boolean-propagation.v1","spec_hash":"382dea2631e8efe2a305b930ccfa69f6fdf987a82926a3d081e5f28938f2a830","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0f\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-f564f4e4efc2bd2515eb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v0a --supports--> v1c\n- v16 --supports--> v24\n- v00 --supports--> v08\n- v08 --refutes--> v07\n- v19 --refutes--> v23\n- v10 --supports--> v05\n- v1c --refutes--> v20\n- v0a --supports--> v06\n- v19 --refutes--> v1f\n- v0e --supports--> v0a\n- v14 --supports--> v29\n- v0e --supports--> v11\n- v29 --supports--> v12\n- v0f --supports--> v15\n- v12 --refutes--> v02\n- v0c --supports--> v09\n- v18 --supports--> v04\n- v26 --supports--> v1e\n- v02 --refutes--> v00\n- v18 --supports--> v22\n- v1b --refutes--> v0c\n- v13 --refutes--> v1d\n- v05 --supports--> v21\n- v00 --supports--> v27\n- v20 --refutes--> v13\n- v0c --refutes--> v18\n- v0f --refutes--> v16\n- v03 --refutes--> v19\n- v0c --refutes--> v25\n- v21 --supports--> v1b\n- v22 --refutes--> v17\n- v0b --supports--> v26\n- v0c --supports--> v2a\n- v0a --supports--> v01\n- v24 --supports--> v1a\n- v07 --refutes--> v0b\n- v1a --refutes--> v28\n- v00 --supports--> v0d\n- v26 --refutes--> v0e\n- v19 --supports--> v10\n- v13 --supports--> v0f\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv28\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"347a67d680cac121558b9f51b7c265f98936c0731228ffbc6acbeae0a593322e","schema":"livegraph.boolean-propagation.v1","spec_hash":"f564f4e4efc2bd2515eb6c11d7f589475ad9a8ee98eac34b08cce74e5d0c1e4e","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v28\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-48a66a62e3e07fea8e2b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v11\n- v28 --refutes--> v14\n- v12 --refutes--> v22\n- v20 --refutes--> v29\n- v22 --supports--> v09\n- v17 --refutes--> v07\n- v2a --supports--> v1b\n- v28 --supports--> v0b\n- v28 --supports--> v18\n- v20 --refutes--> v27\n- v24 --supports--> v03\n- v17 --refutes--> v1f\n- v25 --refutes--> v12\n- v1f --supports--> v0e\n- v23 --refutes--> v1e\n- v21 --supports--> v15\n- v09 --supports--> v2a\n- v0d --refutes--> v1d\n- v08 --refutes--> v1a\n- v0c --refutes--> v24\n- v2a --supports--> v0d\n- v03 --supports--> v08\n- v21 --supports--> v01\n- v12 --supports--> v00\n- v02 --supports--> v0f\n- v15 --supports--> v05\n- v29 --supports--> v28\n- v0d --refutes--> v21\n- v02 --supports--> v20\n- v18 --refutes--> v10\n- v1a --supports--> v06\n- v22 --refutes--> v13\n- v10 --refutes--> v26\n- v17 --refutes--> v19\n- v04 --refutes--> v0c\n- v1c --supports--> v16\n- v16 --supports--> v25\n- v1f --refutes--> v1c\n- v11 --supports--> v17\n- v1e --supports--> v04\n- v0c --supports--> v0a\n- v05 --supports--> v02\n\n[VERIFIED ROOT]\nv23 = TRUE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8407b5ba5eb3af4046a9092908da8ab38f94dc3555b5943c7edb47a82e0fe1ed","schema":"livegraph.boolean-propagation.v1","spec_hash":"48a66a62e3e07fea8e2b422a175b46f50a45b7332c1419797bfcbb574355fdb5","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-e085ce57b4e6aedb16f3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2a --supports--> v19\n- v28 --refutes--> v1b\n- v07 --refutes--> v00\n- v24 --refutes--> v1d\n- v10 --supports--> v13\n- v1a --supports--> v07\n- v1d --refutes--> v10\n- v23 --refutes--> v20\n- v0a --supports--> v21\n- v21 --supports--> v29\n- v01 --refutes--> v03\n- v1c --supports--> v2a\n- v1c --supports--> v11\n- v28 --refutes--> v0e\n- v19 --supports--> v0d\n- v16 --supports--> v28\n- v02 --refutes--> v08\n- v1e --refutes--> v16\n- v1e --refutes--> v06\n- v21 --refutes--> v22\n- v0c --supports--> v25\n- v00 --refutes--> v12\n- v1c --refutes--> v04\n- v27 --refutes--> v1e\n- v1a --refutes--> v18\n- v0e --supports--> v1f\n- v08 --refutes--> v0c\n- v26 --refutes--> v15\n- v09 --supports--> v23\n- v10 --refutes--> v0a\n- v15 --refutes--> v02\n- v0d --refutes--> v27\n- v22 --supports--> v26\n- v08 --supports--> v17\n- v25 --refutes--> v09\n- v0c --supports--> v0b\n- v0e --supports--> v24\n- v01 --supports--> v05\n- v12 --supports--> v01\n- v01 --supports--> v14\n- v00 --refutes--> v0f\n- v23 --refutes--> v1a\n\n[VERIFIED ROOT]\nv1c = 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":"bd652f429be5938c8eaebca1fa24d84bffe60d04f2393c2d61f353b51cfe5679","schema":"livegraph.boolean-propagation.v1","spec_hash":"e085ce57b4e6aedb16f3c7fac3b1a57b197212a03658dda758f4ae813b78338f","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-21aea025b9314fe901f8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0c --refutes--> v0b\n- v03 --supports--> v25\n- v0e --refutes--> v0c\n- v04 --supports--> v18\n- v1f --supports--> v11\n- v22 --supports--> v1d\n- v09 --refutes--> v12\n- v27 --refutes--> v01\n- v13 --supports--> v02\n- v06 --supports--> v22\n- v1e --refutes--> v2a\n- v03 --refutes--> v1a\n- v04 --supports--> v14\n- v09 --refutes--> v1e\n- v15 --supports--> v20\n- v1e --supports--> v0d\n- v28 --supports--> v17\n- v27 --refutes--> v26\n- v29 --supports--> v08\n- v20 --supports--> v06\n- v1d --supports--> v03\n- v13 --supports--> v1b\n- v28 --refutes--> v0e\n- v19 --refutes--> v04\n- v08 --supports--> v28\n- v0b --refutes--> v16\n- v24 --refutes--> v15\n- v1e --refutes--> v0a\n- v1f --supports--> v19\n- v1a --refutes--> v13\n- v2a --refutes--> v0f\n- v07 --supports--> v1f\n- v23 --refutes--> v09\n- v1d --refutes--> v00\n- v03 --refutes--> v10\n- v14 --refutes--> v29\n- v0f --refutes--> v07\n- v0b --refutes--> v21\n- v26 --supports--> v23\n- v27 --supports--> v05\n- v16 --refutes--> v24\n- v24 --supports--> v1c\n\n[VERIFIED ROOT]\nv27 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"fd02598eb445e892865b10e8be5b791ac4a1f1f104208c3b5200f67f7b7d8446","schema":"livegraph.boolean-propagation.v1","spec_hash":"21aea025b9314fe901f8c5048e562d98295ee7aecf674eee431a6c03f77c5234","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-3b14b8a0661b7e8df3d4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v21\n- v14 --refutes--> v0a\n- v15 --supports--> v00\n- v13 --supports--> v03\n- v0e --supports--> v13\n- v27 --refutes--> v1c\n- v04 --refutes--> v07\n- v25 --refutes--> v09\n- v0d --refutes--> v1f\n- v1d --supports--> v14\n- v25 --supports--> v1d\n- v07 --supports--> v29\n- v1e --supports--> v18\n- v05 --refutes--> v10\n- v21 --refutes--> v04\n- v20 --refutes--> v02\n- v11 --refutes--> v27\n- v1b --refutes--> v20\n- v18 --supports--> v05\n- v0e --refutes--> v23\n- v0a --supports--> v17\n- v24 --refutes--> v0f\n- v02 --supports--> v01\n- v18 --refutes--> v2a\n- v1c --refutes--> v1b\n- v1b --supports--> v22\n- v03 --refutes--> v16\n- v15 --refutes--> v0c\n- v27 --refutes--> v06\n- v01 --supports--> v1a\n- v00 --refutes--> v11\n- v11 --refutes--> v0b\n- v10 --supports--> v25\n- v0d --refutes--> v28\n- v16 --supports--> v12\n- v07 --supports--> v08\n- v01 --supports--> v19\n- v12 --refutes--> v1e\n- v05 --refutes--> v26\n- v01 --supports--> v0d\n- v07 --supports--> v0e\n- v0f --supports--> v15\n\n[VERIFIED ROOT]\nv24 = 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":"fbf3d80304d43aeee50194ada4f80a66c8c61638013b2560f33256fce58f0fd3","schema":"livegraph.boolean-propagation.v1","spec_hash":"3b14b8a0661b7e8df3d4311d1b4b9e93fd7101d0020c56ef7b8079dad371a4ae","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v24\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-9f827478b703a1975cc2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v29\n- v24 --supports--> v0e\n- v09 --refutes--> v06\n- v08 --supports--> v10\n- v2a --refutes--> v0a\n- v00 --refutes--> v22\n- v0c --supports--> v1f\n- v16 --refutes--> v01\n- v05 --supports--> v25\n- v1b --supports--> v2a\n- v00 --refutes--> v18\n- v28 --refutes--> v05\n- v26 --refutes--> v11\n- v1e --supports--> v1b\n- v14 --supports--> v0c\n- v18 --supports--> v24\n- v0a --supports--> v15\n- v20 --refutes--> v1a\n- v04 --refutes--> v27\n- v19 --refutes--> v09\n- v26 --refutes--> v14\n- v1f --refutes--> v1d\n- v13 --supports--> v1c\n- v25 --refutes--> v13\n- v18 --supports--> v17\n- v16 --supports--> v0b\n- v05 --supports--> v0d\n- v12 --supports--> v19\n- v13 --refutes--> v21\n- v24 --supports--> v16\n- v0b --refutes--> v1e\n- v24 --refutes--> v03\n- v15 --supports--> v04\n- v1f --refutes--> v07\n- v04 --supports--> v26\n- v2a --refutes--> v0f\n- v1c --refutes--> v08\n- v10 --refutes--> v20\n- v1b --refutes--> v02\n- v00 --refutes--> v23\n- v29 --refutes--> v28\n- v20 --refutes--> v12\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":"a9e8d8176fca962f8877be0ce05ae6d5aca2774a9a15d0b491c05ded2fcf121b","schema":"livegraph.boolean-propagation.v1","spec_hash":"9f827478b703a1975cc2ce970bbdd81f3917c59245d1b445ef8703e3171a63cc","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-284a3266f0e79070854a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v12 --supports--> v0a\n- v27 --refutes--> v17\n- v20 --supports--> v2a\n- v10 --supports--> v19\n- v11 --supports--> v14\n- v20 --supports--> v00\n- v1a --supports--> v22\n- v06 --refutes--> v02\n- v25 --refutes--> v20\n- v00 --refutes--> v13\n- v23 --refutes--> v27\n- v08 --supports--> v1f\n- v0a --supports--> v18\n- v0e --supports--> v1d\n- v11 --refutes--> v0e\n- v25 --supports--> v0b\n- v0d --refutes--> v21\n- v0e --supports--> v1c\n- v0c --supports--> v07\n- v1f --supports--> v15\n- v19 --refutes--> v26\n- v27 --supports--> v24\n- v0e --supports--> v1e\n- v12 --refutes--> v04\n- v0d --supports--> v0c\n- v29 --supports--> v0d\n- v06 --supports--> v05\n- v00 --refutes--> v0f\n- v05 --supports--> v10\n- v1d --refutes--> v29\n- v13 --supports--> v23\n- v20 --refutes--> v28\n- v07 --supports--> v01\n- v18 --refutes--> v09\n- v27 --supports--> v03\n- v15 --refutes--> v06\n- v22 --supports--> v11\n- v12 --refutes--> v16\n- v29 --supports--> v1b\n- v03 --refutes--> v08\n- v09 --supports--> v25\n\n[VERIFIED ROOT]\nv1a = FALSE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2989afbc204472997d413635eb555a3df4fa2a7b55d1955c2d5dda31146f9c6f","schema":"livegraph.boolean-propagation.v1","spec_hash":"284a3266f0e79070854a18296e9817977d60a87cdfd4aff359f5282f32b16a49","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-bf13e78e0bf6526cad5c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v29 --refutes--> v19\n- v00 --supports--> v23\n- v0a --supports--> v1b\n- v10 --supports--> v28\n- v0c --supports--> v26\n- v09 --refutes--> v24\n- v11 --supports--> v15\n- v0a --refutes--> v07\n- v2a --refutes--> v27\n- v05 --supports--> v0b\n- v16 --supports--> v0c\n- v29 --supports--> v08\n- v20 --supports--> v02\n- v17 --refutes--> v20\n- v07 --refutes--> v05\n- v04 --refutes--> v1d\n- v08 --supports--> v11\n- v0c --refutes--> v01\n- v04 --refutes--> v17\n- v02 --refutes--> v2a\n- v11 --refutes--> v22\n- v23 --refutes--> v29\n- v1f --refutes--> v13\n- v1a --refutes--> v21\n- v0b --supports--> v06\n- v21 --supports--> v18\n- v0c --supports--> v0d\n- v06 --refutes--> v0f\n- v18 --refutes--> v04\n- v0a --refutes--> v14\n- v1c --supports--> v1a\n- v2a --supports--> v25\n- v0d --supports--> v03\n- v09 --supports--> v0a\n- v0f --supports--> v16\n- v0e --refutes--> v09\n- v02 --refutes--> v1e\n- v17 --refutes--> v12\n- v03 --supports--> v10\n- v1f --refutes--> v1c\n- v2a --supports--> v00\n- v10 --supports--> v1f\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb469269640c67b567067ec9f99ff03d9aa7c4cfd76eaa00381bdefa19e991d7","schema":"livegraph.boolean-propagation.v1","spec_hash":"bf13e78e0bf6526cad5c03beed10372efcf968d8825620eb206fbc2a6758e2ff","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-b1f1a6f9cf3e390ca7c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v01 --supports--> v0a\n- v24 --supports--> v2a\n- v03 --supports--> v1b\n- v15 --refutes--> v10\n- v0d --supports--> v00\n- v1f --supports--> v0c\n- v20 --refutes--> v17\n- v14 --supports--> v20\n- v1a --supports--> v12\n- v20 --supports--> v03\n- v03 --supports--> v06\n- v1e --supports--> v07\n- v16 --supports--> v09\n- v2a --supports--> v21\n- v24 --refutes--> v0e\n- v26 --supports--> v0b\n- v00 --refutes--> v16\n- v1f --refutes--> v28\n- v23 --supports--> v22\n- v0c --refutes--> v24\n- v29 --refutes--> v01\n- v15 --refutes--> v04\n- v09 --supports--> v13\n- v01 --refutes--> v1c\n- v13 --refutes--> v1f\n- v02 --refutes--> v05\n- v18 --refutes--> v0d\n- v21 --supports--> v29\n- v03 --refutes--> v08\n- v22 --supports--> v25\n- v27 --supports--> v15\n- v01 --supports--> v27\n- v12 --supports--> v23\n- v11 --refutes--> v02\n- v15 --refutes--> v11\n- v02 --supports--> v14\n- v22 --refutes--> v1e\n- v0b --supports--> v18\n- v07 --refutes--> v26\n- v03 --refutes--> v1d\n- v01 --supports--> v0f\n\n[VERIFIED ROOT]\nv1a = 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":"6c2326c82f3567c1c500d8157bff54e263db29543b9f4fd0d7ec3e80cc69e9e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"b1f1a6f9cf3e390ca7c837f9b8180d96c7c24cc424e24ea32346ccba7cd70757","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-f1ec4b018d78d6c535a7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v22 --refutes--> v1b\n- v2a --supports--> v27\n- v18 --refutes--> v15\n- v19 --supports--> v0c\n- v2a --refutes--> v16\n- v1a --refutes--> v02\n- v1b --supports--> v09\n- v12 --refutes--> v23\n- v0c --supports--> v22\n- v1c --refutes--> v17\n- v03 --supports--> v07\n- v06 --refutes--> v0f\n- v08 --refutes--> v03\n- v0d --refutes--> v21\n- v14 --refutes--> v25\n- v22 --refutes--> v01\n- v02 --refutes--> v10\n- v28 --refutes--> v20\n- v28 --supports--> v13\n- v00 --refutes--> v1a\n- v0e --supports--> v12\n- v05 --supports--> v0a\n- v07 --refutes--> v05\n- v07 --supports--> v1e\n- v06 --supports--> v0d\n- v20 --refutes--> v06\n- v27 --supports--> v24\n- v27 --supports--> v00\n- v15 --supports--> v1f\n- v21 --refutes--> v08\n- v19 --supports--> v11\n- v02 --supports--> v26\n- v23 --refutes--> v1c\n- v1b --refutes--> v2a\n- v0a --refutes--> v18\n- v10 --supports--> v1d\n- v1c --supports--> v14\n- v25 --refutes--> v19\n- v15 --supports--> v29\n- v0c --refutes--> v04\n- v15 --refutes--> v0b\n\n[VERIFIED ROOT]\nv28 = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"993efe1d1b8297b369556f8502bffbaa09a39343d129f5f37d59c2d93c0c4332","schema":"livegraph.boolean-propagation.v1","spec_hash":"f1ec4b018d78d6c535a70ce4c426ea57729811e3515c9a0d36f224334e53ae85","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v28\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-209c66add9f71be937f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v27\n- v21 --refutes--> v0f\n- v1e --supports--> v11\n- v0f --supports--> v20\n- v18 --supports--> v10\n- v04 --supports--> v12\n- v1a --refutes--> v24\n- v1d --supports--> v03\n- v20 --supports--> v0b\n- v04 --supports--> v01\n- v29 --supports--> v14\n- v2a --refutes--> v08\n- v17 --supports--> v1f\n- v22 --supports--> v04\n- v19 --refutes--> v17\n- v12 --supports--> v0d\n- v12 --supports--> v1d\n- v09 --refutes--> v26\n- v05 --supports--> v06\n- v16 --supports--> v19\n- v1e --refutes--> v23\n- v14 --supports--> v1a\n- v00 --refutes--> v22\n- v29 --supports--> v25\n- v27 --supports--> v0e\n- v15 --supports--> v28\n- v1d --supports--> v15\n- v07 --refutes--> v2a\n- v11 --refutes--> v21\n- v10 --supports--> v16\n- v08 --refutes--> v02\n- v1e --supports--> v13\n- v24 --supports--> v1e\n- v28 --refutes--> v05\n- v1e --refutes--> v0c\n- v0f --refutes--> v0a\n- v26 --supports--> v00\n- v20 --supports--> v07\n- v07 --refutes--> v1c\n- v05 --refutes--> v29\n- v24 --supports--> v1b\n- v17 --supports--> v09\n\n[VERIFIED ROOT]\nv18 = 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":"a06396f8bac6f50f902b81325022fb452ba4247f7cb0f525edfc4f6ce41a3fb0","schema":"livegraph.boolean-propagation.v1","spec_hash":"209c66add9f71be937f44886827166a6e9114725a9431361e21e143d4090454c","spec_json":"{\"edges\":[{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-1019f9be2dae2fcf49f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v16\n- v16 --supports--> v21\n- v1c --refutes--> v05\n- v05 --refutes--> v0c\n- v21 --refutes--> v1a\n- v0c --supports--> v09\n- v0b --supports--> v15\n- v02 --supports--> v01\n- v08 --supports--> v22\n- v06 --supports--> v1b\n- v29 --refutes--> v28\n- v28 --refutes--> v26\n- v27 --refutes--> v10\n- v17 --refutes--> v25\n- v15 --supports--> v0a\n- v12 --refutes--> v08\n- v1b --refutes--> v11\n- v1b --supports--> v0e\n- v1a --refutes--> v1c\n- v0f --refutes--> v29\n- v1b --supports--> v18\n- v26 --refutes--> v0b\n- v00 --supports--> v12\n- v23 --refutes--> v17\n- v11 --supports--> v19\n- v22 --refutes--> v20\n- v14 --supports--> v27\n- v27 --supports--> v06\n- v19 --supports--> v2a\n- v1b --supports--> v1f\n- v02 --refutes--> v23\n- v0a --supports--> v02\n- v17 --refutes--> v1e\n- v09 --supports--> v0f\n- v0b --supports--> v03\n- v0b --supports--> v0d\n- v12 --refutes--> v1d\n- v1e --supports--> v00\n- v12 --supports--> v07\n- v29 --refutes--> v24\n- v0b --refutes--> v13\n- v00 --supports--> v04\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9615b4dca101b36c676c0db9f426df951344ee68af9fa0f6ff028e283624f086","schema":"livegraph.boolean-propagation.v1","spec_hash":"1019f9be2dae2fcf49f10cc3a9d853ca21bdb1647ddea5e5d3fb06e744702dc6","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-6cd8d7c2f4c7f6defa7e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v14\n- v00 --supports--> v0e\n- v08 --supports--> v21\n- v17 --refutes--> v0c\n- v0d --refutes--> v06\n- v24 --supports--> v27\n- v07 --supports--> v05\n- v10 --refutes--> v17\n- v00 --refutes--> v23\n- v20 --refutes--> v25\n- v19 --refutes--> v0b\n- v0c --supports--> v08\n- v07 --refutes--> v01\n- v0e --refutes--> v20\n- v2a --refutes--> v1e\n- v0b --refutes--> v13\n- v06 --supports--> v2a\n- v0a --supports--> v04\n- v2a --refutes--> v16\n- v21 --refutes--> v11\n- v27 --refutes--> v19\n- v29 --supports--> v0f\n- v1d --refutes--> v07\n- v25 --supports--> v1d\n- v1e --refutes--> v00\n- v04 --refutes--> v28\n- v25 --supports--> v26\n- v02 --refutes--> v1f\n- v21 --refutes--> v03\n- v01 --supports--> v29\n- v0b --refutes--> v09\n- v22 --supports--> v10\n- v03 --supports--> v1a\n- v02 --supports--> v1b\n- v28 --refutes--> v12\n- v29 --supports--> v24\n- v02 --refutes--> v22\n- v1a --refutes--> v0a\n- v2a --refutes--> v18\n- v0b --refutes--> v02\n- v28 --supports--> v15\n- v01 --refutes--> v1c\n\n[VERIFIED ROOT]\nv0d = 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":"ba26c073d76b0e438699029a5004b72f30b6675c2c9d1697aae5f8b9964352e0","schema":"livegraph.boolean-propagation.v1","spec_hash":"6cd8d7c2f4c7f6defa7eed6dc0c5495c9e10a5889003621ec0e3a686da7d1004","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-15566c7412d6e41aff62","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --supports--> v16\n- v20 --refutes--> v29\n- v12 --refutes--> v21\n- v19 --refutes--> v06\n- v22 --refutes--> v0a\n- v0a --supports--> v08\n- v0c --refutes--> v03\n- v1c --supports--> v15\n- v24 --supports--> v1b\n- v1c --refutes--> v1f\n- v09 --supports--> v26\n- v19 --supports--> v25\n- v25 --supports--> v0e\n- v26 --refutes--> v07\n- v1c --refutes--> v28\n- v05 --refutes--> v01\n- v23 --supports--> v17\n- v0a --refutes--> v1c\n- v27 --refutes--> v05\n- v13 --supports--> v20\n- v0b --refutes--> v22\n- v12 --supports--> v1d\n- v21 --supports--> v27\n- v23 --supports--> v02\n- v01 --supports--> v0c\n- v20 --supports--> v04\n- v11 --supports--> v23\n- v04 --supports--> v14\n- v1b --supports--> v0f\n- v15 --supports--> v13\n- v2a --supports--> v0b\n- v21 --supports--> v10\n- v07 --supports--> v1a\n- v16 --refutes--> v19\n- v04 --supports--> v1e\n- v03 --refutes--> v09\n- v12 --refutes--> v18\n- v0e --supports--> v12\n- v03 --refutes--> v00\n- v23 --supports--> v24\n- v07 --supports--> v2a\n- v1b --refutes--> v0d\n\n[VERIFIED ROOT]\nv11 = TRUE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"530672247db377c5635aba83a69787503d939eaedd68bee58a56aeb0acb3026e","schema":"livegraph.boolean-propagation.v1","spec_hash":"15566c7412d6e41aff627f564c5d1e581fdfa2b1271b649ea1bdcde1fc50f1e6","spec_json":"{\"edges\":[{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v11\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-cd4026f4bf51d30905fa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1a\n- v25 --refutes--> v06\n- v11 --supports--> v03\n- v14 --supports--> v0c\n- v25 --refutes--> v0e\n- v23 --refutes--> v09\n- v03 --supports--> v2a\n- v0f --supports--> v02\n- v1e --refutes--> v23\n- v09 --refutes--> v22\n- v10 --supports--> v1f\n- v01 --refutes--> v28\n- v29 --refutes--> v15\n- v26 --refutes--> v05\n- v15 --supports--> v00\n- v21 --refutes--> v01\n- v0b --refutes--> v1d\n- v0b --refutes--> v07\n- v18 --refutes--> v11\n- v22 --supports--> v0a\n- v1a --refutes--> v12\n- v1a --supports--> v27\n- v1b --supports--> v17\n- v18 --supports--> v08\n- v0d --refutes--> v26\n- v01 --refutes--> v04\n- v15 --refutes--> v18\n- v24 --refutes--> v21\n- v21 --supports--> v19\n- v1c --refutes--> v1b\n- v03 --supports--> v16\n- v0a --supports--> v1c\n- v01 --refutes--> v14\n- v1d --refutes--> v0f\n- v05 --supports--> v25\n- v1b --supports--> v29\n- v25 --refutes--> v13\n- v06 --refutes--> v0b\n- v10 --supports--> v1e\n- v27 --supports--> v24\n- v14 --supports--> v10\n- v1c --supports--> v20\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e1a57afb46c627ad785eb14ed794db88603b9e24ee0668ab6f0db28cafc2e9cf","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd4026f4bf51d30905fa036b3f457cb14e86e0822dc301f7dc43048d2b9ca580","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-5a39ef6e03c15dcb3c54","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v22\n- v1c --refutes--> v19\n- v04 --supports--> v0a\n- v02 --supports--> v0d\n- v12 --refutes--> v20\n- v16 --refutes--> v25\n- v27 --refutes--> v10\n- v12 --supports--> v06\n- v1d --supports--> v1e\n- v04 --supports--> v23\n- v0c --refutes--> v11\n- v08 --refutes--> v02\n- v25 --supports--> v09\n- v00 --refutes--> v14\n- v0e --refutes--> v15\n- v05 --supports--> v08\n- v2a --supports--> v28\n- v18 --refutes--> v03\n- v07 --supports--> v1b\n- v18 --refutes--> v0c\n- v0f --refutes--> v05\n- v2a --refutes--> v26\n- v04 --supports--> v1a\n- v0e --refutes--> v0f\n- v27 --refutes--> v24\n- v12 --supports--> v18\n- v00 --refutes--> v21\n- v09 --refutes--> v04\n- v19 --refutes--> v0e\n- v28 --refutes--> v01\n- v0a --refutes--> v07\n- v24 --refutes--> v13\n- v0b --supports--> v12\n- v0e --supports--> v17\n- v11 --refutes--> v29\n- v00 --refutes--> v27\n- v09 --refutes--> v1f\n- v22 --supports--> v1d\n- v02 --refutes--> v16\n- v13 --refutes--> v0b\n- v1d --refutes--> v1c\n- v1b --refutes--> v00\n\n[VERIFIED ROOT]\nv2a = TRUE\n\n[TARGET]\nv29\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e5e84f08d413d74f1127441a0ff572ddbe0c2c286688c0fe44000b084a8e4629","schema":"livegraph.boolean-propagation.v1","spec_hash":"5a39ef6e03c15dcb3c5468e70a157bf6e7a229570ee61c4ee46a80e130d51d14","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v2a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v29\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-5803ab23d0561944cac2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v17\n- v27 --refutes--> v25\n- v01 --supports--> v29\n- v22 --supports--> v00\n- v00 --refutes--> v1f\n- v01 --supports--> v10\n- v24 --supports--> v0a\n- v02 --supports--> v28\n- v16 --supports--> v13\n- v10 --refutes--> v1e\n- v26 --refutes--> v19\n- v09 --supports--> v15\n- v17 --supports--> v1c\n- v1e --refutes--> v0c\n- v19 --supports--> v04\n- v0a --supports--> v12\n- v22 --supports--> v1a\n- v1c --supports--> v06\n- v0c --supports--> v08\n- v14 --supports--> v11\n- v16 --supports--> v20\n- v09 --refutes--> v05\n- v09 --refutes--> v18\n- v09 --supports--> v0f\n- v1b --supports--> v22\n- v02 --supports--> v24\n- v1f --refutes--> v16\n- v06 --supports--> v14\n- v04 --supports--> v09\n- v1f --supports--> v23\n- v12 --refutes--> v1d\n- v16 --refutes--> v02\n- v03 --refutes--> v27\n- v11 --refutes--> v0e\n- v03 --refutes--> v2a\n- v18 --refutes--> v03\n- v00 --supports--> v0d\n- v25 --supports--> v01\n- v0e --supports--> v26\n- v10 --supports--> v07\n- v21 --supports--> v1b\n- v00 --refutes--> v0b\n\n[VERIFIED ROOT]\nv21 = 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":"29666b6d4e7f26c638451126ff39b121a75566a32042dc8a45f58dc99d6c6b6e","schema":"livegraph.boolean-propagation.v1","spec_hash":"5803ab23d0561944cac22e70011555ed22d17944e0df8abfbd6b18bd2df3be84","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"}],\"hop\":28,\"root\":\"v21\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-a4f437d0deed2e1f8625","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2a\n- v00 --refutes--> v18\n- v27 --supports--> v01\n- v23 --refutes--> v0b\n- v25 --refutes--> v14\n- v04 --supports--> v1b\n- v25 --refutes--> v1d\n- v02 --refutes--> v20\n- v1b --supports--> v10\n- v28 --supports--> v22\n- v2a --supports--> v1f\n- v0e --supports--> v17\n- v22 --refutes--> v23\n- v27 --supports--> v07\n- v08 --supports--> v19\n- v06 --refutes--> v11\n- v25 --supports--> v0d\n- v21 --supports--> v16\n- v1a --refutes--> v24\n- v0c --supports--> v26\n- v28 --supports--> v03\n- v28 --refutes--> v29\n- v15 --refutes--> v25\n- v19 --refutes--> v15\n- v1c --supports--> v1a\n- v17 --refutes--> v21\n- v0d --supports--> v00\n- v19 --refutes--> v13\n- v18 --supports--> v27\n- v10 --refutes--> v05\n- v19 --refutes--> v12\n- v18 --supports--> v09\n- v06 --refutes--> v02\n- v27 --refutes--> v0e\n- v10 --supports--> v0f\n- v22 --supports--> v1e\n- v24 --supports--> v08\n- v16 --refutes--> v0c\n- v26 --refutes--> v04\n- v0e --refutes--> v0a\n- v10 --supports--> v28\n- v1f --refutes--> v06\n\n[VERIFIED ROOT]\nv1c = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33c0ee7b7c1d5e5ced1aca19e04bfb491001b78583988c78c4766708003e377e","schema":"livegraph.boolean-propagation.v1","spec_hash":"a4f437d0deed2e1f86254a49a4435741aebb2d9b3de8804b33a975bcaeada1bc","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-cf0f49d7db01a5b15205","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v22\n- v26 --supports--> v05\n- v15 --supports--> v27\n- v0e --refutes--> v12\n- v0a --refutes--> v20\n- v07 --refutes--> v06\n- v0e --supports--> v14\n- v07 --refutes--> v0c\n- v1a --supports--> v21\n- v16 --refutes--> v28\n- v0c --refutes--> v1a\n- v27 --refutes--> v11\n- v09 --supports--> v1d\n- v13 --supports--> v15\n- v02 --supports--> v17\n- v12 --refutes--> v02\n- v08 --supports--> v19\n- v29 --supports--> v09\n- v05 --refutes--> v2a\n- v17 --refutes--> v10\n- v16 --supports--> v07\n- v05 --supports--> v04\n- v18 --supports--> v24\n- v0b --supports--> v0d\n- v22 --refutes--> v26\n- v24 --supports--> v1c\n- v03 --supports--> v00\n- v0f --supports--> v18\n- v29 --supports--> v23\n- v20 --supports--> v03\n- v08 --refutes--> v29\n- v03 --refutes--> v0e\n- v12 --supports--> v01\n- v22 --supports--> v1f\n- v0e --supports--> v1b\n- v1c --refutes--> v16\n- v1d --supports--> v0a\n- v0b --refutes--> v25\n- v1a --refutes--> v0b\n- v04 --supports--> v13\n- v27 --refutes--> v0f\n- v0b --refutes--> v1e\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4a6ec55048a3ca2400f308f932b57caf57f51cdd86d2e6987ba5cdb831e81591","schema":"livegraph.boolean-propagation.v1","spec_hash":"cf0f49d7db01a5b15205735a6a106615702bd5660d37efbe69cb1c2312a832da","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-fe7adf177232b57ba09a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v20\n- v05 --supports--> v18\n- v1f --refutes--> v25\n- v1f --supports--> v0d\n- v06 --refutes--> v03\n- v02 --supports--> v13\n- v03 --refutes--> v11\n- v14 --refutes--> v15\n- v0c --supports--> v1f\n- v26 --supports--> v05\n- v07 --supports--> v1a\n- v06 --supports--> v1c\n- v01 --supports--> v22\n- v13 --refutes--> v07\n- v17 --refutes--> v27\n- v15 --supports--> v17\n- v14 --supports--> v0e\n- v04 --supports--> v24\n- v05 --refutes--> v04\n- v01 --supports--> v02\n- v0b --supports--> v10\n- v1e --supports--> v26\n- v27 --supports--> v08\n- v07 --supports--> v0c\n- v0f --supports--> v01\n- v29 --supports--> v21\n- v24 --supports--> v09\n- v1d --refutes--> v14\n- v1b --supports--> v06\n- v23 --supports--> v0f\n- v21 --refutes--> v0a\n- v12 --supports--> v2a\n- v00 --refutes--> v23\n- v03 --supports--> v16\n- v0a --refutes--> v1d\n- v27 --supports--> v00\n- v03 --supports--> v0b\n- v27 --refutes--> v19\n- v2a --supports--> v29\n- v1d --supports--> v28\n- v0b --refutes--> v12\n- v24 --refutes--> v1b\n\n[VERIFIED ROOT]\nv1e = FALSE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1e787a6281c4904939858af0fcb18ca88064265a7b76233d0cdbbba0d67997e5","schema":"livegraph.boolean-propagation.v1","spec_hash":"fe7adf177232b57ba09aa4091816a87b087e46457f6c269bcb1e592b6ad6687b","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v1e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-846783202210fae8fb8f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v29\n- v1b --refutes--> v1c\n- v08 --refutes--> v15\n- v15 --supports--> v02\n- v21 --supports--> v08\n- v1c --supports--> v10\n- v1f --supports--> v24\n- v27 --supports--> v1b\n- v1d --supports--> v13\n- v1f --supports--> v28\n- v0b --refutes--> v06\n- v00 --supports--> v1e\n- v23 --refutes--> v26\n- v17 --refutes--> v23\n- v18 --refutes--> v0b\n- v25 --refutes--> v12\n- v22 --refutes--> v05\n- v03 --refutes--> v22\n- v1d --supports--> v0f\n- v28 --refutes--> v0d\n- v11 --refutes--> v19\n- v04 --refutes--> v11\n- v20 --supports--> v0a\n- v26 --refutes--> v04\n- v20 --refutes--> v09\n- v07 --refutes--> v25\n- v1a --supports--> v27\n- v1c --refutes--> v0c\n- v09 --refutes--> v00\n- v15 --refutes--> v1f\n- v21 --refutes--> v14\n- v00 --refutes--> v18\n- v19 --supports--> v0e\n- v1c --refutes--> v21\n- v08 --supports--> v16\n- v06 --supports--> v07\n- v0d --refutes--> v17\n- v1d --supports--> v03\n- v05 --refutes--> v20\n- v11 --refutes--> v2a\n- v05 --refutes--> v01\n- v19 --refutes--> v1d\n\n[VERIFIED ROOT]\nv1a = 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":"d6a84927301231b5957f067899b51b914187420c27be8c05c52be9f74be89eed","schema":"livegraph.boolean-propagation.v1","spec_hash":"846783202210fae8fb8f0aa3cebab479ed6a8bd4a83869a24c8c1ba1120ec5a0","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v1a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-3f0858564afbe2cb679f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v26 --refutes--> v09\n- v24 --supports--> v03\n- v22 --supports--> v13\n- v2a --refutes--> v08\n- v27 --refutes--> v00\n- v1f --supports--> v1d\n- v22 --supports--> v0d\n- v1c --refutes--> v21\n- v28 --refutes--> v11\n- v07 --refutes--> v1a\n- v17 --supports--> v29\n- v1e --supports--> v19\n- v03 --supports--> v0c\n- v19 --supports--> v0e\n- v02 --refutes--> v22\n- v0d --supports--> v1e\n- v23 --supports--> v16\n- v01 --supports--> v14\n- v06 --refutes--> v0a\n- v1d --supports--> v01\n- v2a --refutes--> v20\n- v0e --refutes--> v28\n- v00 --supports--> v17\n- v1f --supports--> v04\n- v22 --refutes--> v05\n- v03 --refutes--> v02\n- v20 --supports--> v15\n- v0a --refutes--> v23\n- v27 --refutes--> v25\n- v12 --refutes--> v0b\n- v26 --supports--> v1c\n- v14 --refutes--> v2a\n- v0b --supports--> v18\n- v11 --supports--> v07\n- v07 --refutes--> v06\n- v23 --refutes--> v27\n- v28 --supports--> v1b\n- v1c --refutes--> v1f\n- v18 --supports--> v26\n- v17 --supports--> v12\n- v01 --supports--> v0f\n- v19 --refutes--> v10\n\n[VERIFIED ROOT]\nv24 = 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":"74a9309ee08cd2f7ce409e376e03171b30e112b08fd3bd6880315690027e84a0","schema":"livegraph.boolean-propagation.v1","spec_hash":"3f0858564afbe2cb679f7da2ba83c6e130d918624109c72c35cdadab17bf7dae","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v24\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-0a4cdf5845648c5583d2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1f\n- v1c --supports--> v0a\n- v0d --supports--> v10\n- v14 --supports--> v1d\n- v21 --refutes--> v0c\n- v1d --refutes--> v0b\n- v11 --refutes--> v27\n- v0a --supports--> v23\n- v27 --refutes--> v08\n- v12 --refutes--> v29\n- v09 --refutes--> v28\n- v16 --refutes--> v22\n- v0a --supports--> v04\n- v17 --supports--> v0d\n- v13 --refutes--> v20\n- v25 --refutes--> v0f\n- v18 --refutes--> v19\n- v21 --supports--> v15\n- v0e --supports--> v12\n- v23 --supports--> v26\n- v12 --supports--> v03\n- v23 --refutes--> v1e\n- v22 --refutes--> v1c\n- v15 --supports--> v00\n- v19 --refutes--> v14\n- v28 --refutes--> v16\n- v22 --refutes--> v2a\n- v10 --supports--> v21\n- v0d --supports--> v01\n- v0b --refutes--> v09\n- v23 --refutes--> v05\n- v06 --refutes--> v25\n- v0e --refutes--> v07\n- v23 --supports--> v06\n- v02 --supports--> v0e\n- v17 --supports--> v24\n- v03 --supports--> v13\n- v25 --refutes--> v02\n- v1a --supports--> v17\n- v20 --refutes--> v1a\n- v27 --refutes--> v1b\n- v15 --supports--> v11\n\n[VERIFIED ROOT]\nv18 = 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":"740bd02363d940f836cda31b91103d46d5b1939a4a1ed8b76cfaea5f2be247a8","schema":"livegraph.boolean-propagation.v1","spec_hash":"0a4cdf5845648c5583d2395a76246352285b6ac60d5ef1984ff6cde06b09c874","spec_json":"{\"edges\":[{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v18\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-536d482ff91700ccc273","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v28 --supports--> v16\n- v0a --refutes--> v19\n- v02 --supports--> v0b\n- v13 --refutes--> v11\n- v1d --refutes--> v0a\n- v10 --refutes--> v29\n- v23 --supports--> v13\n- v00 --supports--> v2a\n- v04 --refutes--> v24\n- v0a --refutes--> v20\n- v0d --refutes--> v07\n- v15 --refutes--> v1f\n- v19 --supports--> v0c\n- v03 --refutes--> v0d\n- v24 --refutes--> v28\n- v13 --supports--> v03\n- v22 --supports--> v09\n- v27 --refutes--> v25\n- v01 --supports--> v1b\n- v07 --refutes--> v1a\n- v22 --supports--> v00\n- v08 --supports--> v27\n- v21 --supports--> v01\n- v0e --supports--> v10\n- v00 --refutes--> v1c\n- v10 --refutes--> v1d\n- v0b --supports--> v0e\n- v1f --refutes--> v14\n- v24 --supports--> v0f\n- v17 --supports--> v06\n- v0b --refutes--> v05\n- v0b --supports--> v1e\n- v25 --supports--> v02\n- v14 --supports--> v17\n- v28 --supports--> v23\n- v0c --refutes--> v22\n- v21 --refutes--> v18\n- v25 --supports--> v26\n- v07 --supports--> v15\n- v01 --supports--> v12\n- v12 --refutes--> v04\n- v17 --supports--> v08\n\n[VERIFIED ROOT]\nv21 = FALSE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f4be7bfd0b89f481f0bae4b81635fcb8888da9791f8b29e6e660818f9d85999","schema":"livegraph.boolean-propagation.v1","spec_hash":"536d482ff91700ccc273386ad630e8d2991396826a46c5b1eb8ec22a18bc421f","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v28\"}],\"hop\":28,\"root\":\"v21\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-62236586fa8f6c7617b0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --supports--> v19\n- v00 --supports--> v16\n- v03 --refutes--> v14\n- v01 --supports--> v1b\n- v22 --refutes--> v28\n- v12 --refutes--> v20\n- v0d --refutes--> v0c\n- v04 --refutes--> v10\n- v26 --supports--> v03\n- v09 --supports--> v27\n- v07 --refutes--> v0a\n- v24 --refutes--> v1a\n- v0e --refutes--> v1d\n- v07 --supports--> v26\n- v0b --supports--> v0d\n- v25 --refutes--> v04\n- v18 --supports--> v0f\n- v03 --supports--> v1c\n- v10 --refutes--> v18\n- v15 --refutes--> v1e\n- v27 --supports--> v12\n- v02 --refutes--> v0b\n- v0d --supports--> v15\n- v0d --refutes--> v29\n- v1c --supports--> v09\n- v10 --refutes--> v17\n- v23 --refutes--> v00\n- v18 --refutes--> v2a\n- v06 --refutes--> v23\n- v19 --supports--> v22\n- v1a --refutes--> v08\n- v23 --refutes--> v05\n- v2a --supports--> v06\n- v20 --supports--> v02\n- v27 --refutes--> v11\n- v01 --refutes--> v1f\n- v04 --supports--> v13\n- v28 --supports--> v25\n- v00 --refutes--> v01\n- v01 --refutes--> v24\n- v1a --supports--> v0e\n- v27 --supports--> v21\n\n[VERIFIED ROOT]\nv07 = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"774219ffba547075c506590e690cd75622d9cefef54edf4dd12cb121905c2943","schema":"livegraph.boolean-propagation.v1","spec_hash":"62236586fa8f6c7617b0a6ba30c96a829e00ae52005ef3fe20f0d07e9278ad71","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-2b2e229db60abe3a41c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v20\n- v1d --supports--> v0e\n- v18 --refutes--> v26\n- v16 --supports--> v07\n- v0b --refutes--> v05\n- v15 --refutes--> v23\n- v1d --supports--> v27\n- v14 --supports--> v11\n- v09 --supports--> v1c\n- v09 --refutes--> v1e\n- v24 --refutes--> v22\n- v16 --refutes--> v06\n- v0d --supports--> v1a\n- v0d --supports--> v13\n- v21 --supports--> v28\n- v19 --supports--> v0b\n- v22 --refutes--> v14\n- v19 --supports--> v0f\n- v07 --supports--> v10\n- v28 --refutes--> v1d\n- v1e --supports--> v24\n- v00 --supports--> v16\n- v18 --refutes--> v00\n- v29 --supports--> v02\n- v0a --supports--> v2a\n- v19 --supports--> v0c\n- v27 --refutes--> v18\n- v1b --refutes--> v0a\n- v14 --supports--> v19\n- v01 --supports--> v04\n- v15 --refutes--> v0d\n- v0d --refutes--> v03\n- v03 --supports--> v1f\n- v1f --supports--> v29\n- v20 --refutes--> v09\n- v07 --supports--> v12\n- v12 --supports--> v17\n- v0b --refutes--> v01\n- v04 --refutes--> v1b\n- v29 --refutes--> v25\n- v12 --supports--> v08\n- v25 --supports--> v21\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f0e0eae03031889c4aa9184424a427f21a41fe0c97dd76ccd3e0e9b840bf0aff","schema":"livegraph.boolean-propagation.v1","spec_hash":"2b2e229db60abe3a41c160533b756cb35090861e5391933e1dd0315f3948c097","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v29\"}],\"hop\":28,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-bed782717009aae342c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1d\n- v17 --refutes--> v15\n- v01 --refutes--> v23\n- v21 --supports--> v10\n- v1e --supports--> v0a\n- v10 --supports--> v26\n- v0a --refutes--> v28\n- v29 --supports--> v1e\n- v11 --refutes--> v1b\n- v12 --supports--> v19\n- v25 --supports--> v06\n- v13 --refutes--> v22\n- v04 --supports--> v21\n- v03 --supports--> v0f\n- v1f --supports--> v00\n- v23 --supports--> v0c\n- v18 --refutes--> v0e\n- v14 --refutes--> v04\n- v29 --supports--> v0d\n- v1a --supports--> v17\n- v23 --refutes--> v24\n- v14 --supports--> v16\n- v0e --supports--> v07\n- v25 --refutes--> v08\n- v0c --supports--> v18\n- v1c --supports--> v1a\n- v15 --refutes--> v01\n- v21 --supports--> v20\n- v1f --refutes--> v11\n- v04 --refutes--> v09\n- v2a --refutes--> v03\n- v10 --refutes--> v2a\n- v13 --supports--> v14\n- v11 --supports--> v12\n- v0a --supports--> v1f\n- v08 --refutes--> v29\n- v1d --refutes--> v25\n- v2a --refutes--> v05\n- v27 --supports--> v13\n- v0a --refutes--> v0b\n- v0f --supports--> v1c\n- v18 --supports--> v02\n\n[VERIFIED ROOT]\nv27 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c6a65af41abe9406882d9b346e588dae214c362d80e0dc9b7a83c722af253cc4","schema":"livegraph.boolean-propagation.v1","spec_hash":"bed782717009aae342c895272aa9de33b436e96afb9efb8cc8a9bca3ff7055c7","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v27\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-112163b9cca4be28d155","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v0b\n- v25 --supports--> v01\n- v02 --supports--> v16\n- v0b --refutes--> v02\n- v0b --supports--> v27\n- v0a --supports--> v06\n- v13 --supports--> v08\n- v23 --refutes--> v25\n- v1e --supports--> v2a\n- v28 --refutes--> v0e\n- v1d --refutes--> v09\n- v04 --refutes--> v21\n- v09 --supports--> v1c\n- v0c --supports--> v1e\n- v2a --supports--> v26\n- v0e --supports--> v12\n- v09 --supports--> v20\n- v25 --refutes--> v14\n- v01 --supports--> v17\n- v22 --supports--> v23\n- v03 --supports--> v05\n- v0e --refutes--> v24\n- v12 --refutes--> v13\n- v29 --supports--> v00\n- v1f --refutes--> v07\n- v0a --refutes--> v15\n- v17 --supports--> v1d\n- v20 --supports--> v0c\n- v21 --supports--> v22\n- v16 --refutes--> v03\n- v00 --refutes--> v1f\n- v01 --supports--> v0f\n- v08 --supports--> v19\n- v23 --supports--> v10\n- v0e --refutes--> v11\n- v03 --supports--> v1b\n- v0a --supports--> v29\n- v1f --supports--> v28\n- v17 --refutes--> v18\n- v00 --refutes--> v0d\n- v1b --supports--> v04\n- v1b --supports--> v1a\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"482c3132c09d9eeda3e876cd19b27fac5e8fd002e646e69d6e388a9add0431b3","schema":"livegraph.boolean-propagation.v1","spec_hash":"112163b9cca4be28d1554d25705dd4ee2b3e778d881797be79d29f2913301871","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-2c8a3371ee0da33e45f0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v08 --refutes--> v11\n- v02 --supports--> v1b\n- v21 --refutes--> v08\n- v24 --refutes--> v07\n- v13 --supports--> v1f\n- v1a --supports--> v0a\n- v28 --refutes--> v1e\n- v05 --refutes--> v21\n- v1f --supports--> v0e\n- v25 --supports--> v00\n- v10 --refutes--> v09\n- v0b --supports--> v10\n- v1b --supports--> v25\n- v1c --supports--> v15\n- v24 --supports--> v18\n- v20 --refutes--> v26\n- v16 --supports--> v0c\n- v27 --refutes--> v2a\n- v16 --supports--> v0b\n- v1a --refutes--> v06\n- v0e --refutes--> v28\n- v0e --supports--> v0f\n- v27 --refutes--> v14\n- v15 --refutes--> v03\n- v13 --supports--> v22\n- v10 --refutes--> v01\n- v1e --refutes--> v04\n- v17 --supports--> v13\n- v12 --supports--> v19\n- v24 --refutes--> v27\n- v2a --supports--> v0d\n- v06 --supports--> v17\n- v12 --supports--> v1d\n- v00 --supports--> v1a\n- v1e --supports--> v20\n- v12 --supports--> v1c\n- v0d --supports--> v16\n- v09 --refutes--> v12\n- v13 --supports--> v29\n- v1e --supports--> v23\n- v26 --refutes--> v24\n- v08 --supports--> v02\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4fcfc21c4a102c7b0441598f6d67104422ab0a38973a2109744be40c2f190270","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c8a3371ee0da33e45f091d356f992dc548af80fc2e1b25d784ec5f8dcc53968","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":28,"item_id":"lggrpo1-ca4c950f5105c7560e25","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --refutes--> v0c\n- v03 --refutes--> v02\n- v15 --supports--> v0e\n- v24 --supports--> v0a\n- v0c --refutes--> v26\n- v1f --refutes--> v1d\n- v0f --refutes--> v11\n- v27 --refutes--> v1f\n- v07 --supports--> v0f\n- v04 --refutes--> v01\n- v09 --refutes--> v17\n- v16 --refutes--> v27\n- v00 --supports--> v1c\n- v1e --refutes--> v16\n- v00 --supports--> v1b\n- v24 --refutes--> v22\n- v26 --refutes--> v15\n- v13 --refutes--> v14\n- v1e --refutes--> v20\n- v0e --refutes--> v13\n- v18 --refutes--> v21\n- v23 --refutes--> v2a\n- v03 --refutes--> v23\n- v0d --supports--> v1e\n- v04 --refutes--> v0b\n- v01 --supports--> v08\n- v1a --supports--> v18\n- v00 --refutes--> v25\n- v0f --refutes--> v03\n- v04 --refutes--> v29\n- v2a --supports--> v06\n- v25 --supports--> v04\n- v08 --supports--> v0d\n- v18 --refutes--> v05\n- v1f --refutes--> v10\n- v14 --refutes--> v1a\n- v06 --supports--> v24\n- v1a --supports--> v19\n- v18 --supports--> v09\n- v14 --refutes--> v28\n- v1a --supports--> v12\n- v22 --refutes--> v00\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3082b8ad7d97a1bb15663008633458b8a252e66e0dfb32512df61a22cefbc00a","schema":"livegraph.boolean-propagation.v1","spec_hash":"ca4c950f5105c7560e25a3702973dead32d34ccdfad25d75d29d112a0e4240a8","spec_json":"{\"edges\":[{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":28,"item_id":"lggrpo1-f7148475f49a28fb3fb8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v0a --supports--> v14\n- v07 --refutes--> v11\n- v01 --refutes--> v21\n- v2a --supports--> v1c\n- v21 --refutes--> v17\n- v0f --supports--> v07\n- v28 --supports--> v15\n- v0f --refutes--> v03\n- v0e --supports--> v18\n- v14 --supports--> v19\n- v1d --refutes--> v08\n- v1b --refutes--> v16\n- v0c --supports--> v1b\n- v08 --refutes--> v0c\n- v1f --supports--> v1e\n- v1c --refutes--> v23\n- v21 --supports--> v05\n- v09 --supports--> v2a\n- v15 --refutes--> v0a\n- v07 --supports--> v13\n- v1e --refutes--> v20\n- v24 --supports--> v22\n- v1c --refutes--> v04\n- v05 --refutes--> v29\n- v24 --refutes--> v25\n- v29 --supports--> v1f\n- v04 --refutes--> v0d\n- v0d --supports--> v06\n- v00 --supports--> v09\n- v0e --supports--> v12\n- v1a --refutes--> v01\n- v14 --supports--> v1a\n- v18 --refutes--> v0b\n- v1d --supports--> v26\n- v1e --supports--> v24\n- v1b --refutes--> v28\n- v11 --refutes--> v1d\n- v01 --supports--> v02\n- v06 --refutes--> v0f\n- v18 --supports--> v00\n- v1b --refutes--> v27\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f5e38de9d33d990375c226c646814bf7387a131b3a974df79a778d9d1c9092f3","schema":"livegraph.boolean-propagation.v1","spec_hash":"f7148475f49a28fb3fb8516490b513522d5781c5c55e12875b44591b2e15133d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2a\"}],\"hop\":28,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-cdad99ad9a7a17408229","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --refutes--> v12\n- v1f --refutes--> v18\n- v1c --supports--> v2a\n- v28 --refutes--> v14\n- v1c --refutes--> v07\n- v0a --supports--> v29\n- v04 --refutes--> v1e\n- v15 --supports--> v01\n- v2c --refutes--> v21\n- v1a --refutes--> v09\n- v05 --supports--> v24\n- v13 --supports--> v11\n- v1e --supports--> v1f\n- v02 --refutes--> v26\n- v08 --refutes--> v0c\n- v00 --refutes--> v2b\n- v01 --refutes--> v1a\n- v20 --refutes--> v13\n- v2b --supports--> v27\n- v1d --supports--> v2f\n- v28 --refutes--> v02\n- v17 --refutes--> v04\n- v26 --refutes--> v0b\n- v0c --refutes--> v25\n- v2b --refutes--> v23\n- v0a --refutes--> v06\n- v2c --supports--> v20\n- v10 --supports--> v15\n- v26 --refutes--> v30\n- v27 --supports--> v08\n- v13 --refutes--> v10\n- v12 --refutes--> v05\n- v26 --refutes--> v1c\n- v17 --supports--> v0f\n- v0e --refutes--> v2c\n- v01 --supports--> v0d\n- v20 --supports--> v2d\n- v05 --supports--> v17\n- v2e --refutes--> v00\n- v1f --refutes--> v16\n- v03 --supports--> v2e\n- v22 --supports--> v0a\n- v06 --refutes--> v1b\n- v2b --refutes--> v19\n- v2f --refutes--> v22\n- v1a --supports--> v28\n- v1f --supports--> v0e\n- v2a --refutes--> v03\n\n[VERIFIED ROOT]\nv1d = TRUE\n\n[TARGET]\nv25\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"dc259e89f5ef3e020f948312b494eaf798e77afe81f1aaa4d08e76ed6457ab24","schema":"livegraph.boolean-propagation.v1","spec_hash":"cdad99ad9a7a174082297497dbcfb84842f9a9a1b9ae414023e5b785beb9fcc9","spec_json":"{\"edges\":[{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v25\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-17b8ccffb4d89b707ab4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v25\n- v0c --supports--> v01\n- v2b --supports--> v0e\n- v10 --supports--> v03\n- v06 --refutes--> v27\n- v0d --refutes--> v1e\n- v0d --refutes--> v2b\n- v2a --supports--> v09\n- v24 --supports--> v0c\n- v09 --refutes--> v24\n- v2f --supports--> v02\n- v07 --supports--> v04\n- v2c --refutes--> v13\n- v13 --refutes--> v14\n- v12 --supports--> v00\n- v05 --supports--> v1d\n- v26 --supports--> v10\n- v0e --refutes--> v2e\n- v30 --refutes--> v1c\n- v0c --refutes--> v0b\n- v06 --refutes--> v1b\n- v29 --supports--> v08\n- v12 --refutes--> v0f\n- v1b --refutes--> v23\n- v02 --supports--> v21\n- v12 --refutes--> v0d\n- v01 --refutes--> v17\n- v25 --supports--> v06\n- v08 --refutes--> v11\n- v24 --supports--> v15\n- v03 --refutes--> v22\n- v2d --supports--> v16\n- v08 --refutes--> v19\n- v2c --supports--> v18\n- v03 --supports--> v12\n- v16 --refutes--> v2a\n- v06 --supports--> v1a\n- v0c --supports--> v28\n- v25 --supports--> v20\n- v01 --supports--> v30\n- v07 --refutes--> v1f\n- v19 --supports--> v2c\n- v2e --refutes--> v29\n- v1c --refutes--> v05\n- v02 --supports--> v0a\n- v14 --refutes--> v2f\n- v21 --refutes--> v2d\n- v1b --refutes--> v26\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eebb40577142f0165e6b0e369c7cd791509b74eba2db1436da77daf38818e41d","schema":"livegraph.boolean-propagation.v1","spec_hash":"17b8ccffb4d89b707ab4bbbb87b9585c00b1f500a516b84a62aa2d7dfb2c25bf","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-fefcc8dd8e90413477ac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v15\n- v28 --refutes--> v17\n- v11 --refutes--> v23\n- v05 --refutes--> v1a\n- v06 --refutes--> v0b\n- v1a --refutes--> v2e\n- v2b --supports--> v03\n- v1e --refutes--> v30\n- v15 --refutes--> v11\n- v2e --supports--> v1e\n- v1f --refutes--> v02\n- v14 --refutes--> v22\n- v02 --refutes--> v00\n- v2b --refutes--> v04\n- v0e --refutes--> v16\n- v0f --refutes--> v28\n- v20 --refutes--> v08\n- v25 --supports--> v01\n- v1b --refutes--> v0f\n- v22 --refutes--> v05\n- v07 --refutes--> v0d\n- v1e --supports--> v0a\n- v02 --refutes--> v10\n- v13 --supports--> v2b\n- v1b --refutes--> v27\n- v07 --supports--> v1b\n- v2f --supports--> v0c\n- v0e --supports--> v18\n- v20 --supports--> v1d\n- v03 --supports--> v06\n- v22 --supports--> v24\n- v2c --supports--> v25\n- v1e --supports--> v12\n- v11 --refutes--> v0e\n- v1d --supports--> v1c\n- v2a --supports--> v2c\n- v1c --supports--> v29\n- v30 --refutes--> v09\n- v1c --refutes--> v19\n- v29 --supports--> v26\n- v01 --supports--> v2f\n- v0c --supports--> v13\n- v06 --refutes--> v20\n- v13 --supports--> v21\n- v2c --supports--> v2d\n- v26 --refutes--> v1f\n- v30 --refutes--> v07\n- v28 --supports--> v2a\n\n[VERIFIED ROOT]\nv14 = 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":"975deb3c81a0edc5da06e33de00af49f30d2059de87fc69356c338887dbd6178","schema":"livegraph.boolean-propagation.v1","spec_hash":"fefcc8dd8e90413477ac2d274ff1166e8d703e9f2831f4f744dff6fdb883a147","spec_json":"{\"edges\":[{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-4200a27a5760e77c166f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v29\n- v0b --supports--> v1f\n- v07 --supports--> v14\n- v0c --supports--> v19\n- v1e --supports--> v13\n- v06 --refutes--> v04\n- v02 --refutes--> v1b\n- v03 --supports--> v12\n- v1d --refutes--> v17\n- v22 --refutes--> v09\n- v19 --refutes--> v30\n- v22 --supports--> v0f\n- v1b --refutes--> v1a\n- v02 --refutes--> v00\n- v25 --refutes--> v07\n- v12 --refutes--> v27\n- v15 --refutes--> v25\n- v23 --refutes--> v2e\n- v12 --refutes--> v2a\n- v1a --supports--> v01\n- v0d --supports--> v10\n- v26 --supports--> v03\n- v2f --refutes--> v1e\n- v29 --supports--> v2d\n- v0f --supports--> v24\n- v10 --refutes--> v0b\n- v12 --refutes--> v18\n- v08 --refutes--> v2b\n- v04 --supports--> v15\n- v19 --refutes--> v1c\n- v13 --supports--> v20\n- v2d --refutes--> v0e\n- v18 --refutes--> v22\n- v01 --supports--> v08\n- v1d --refutes--> v26\n- v26 --refutes--> v28\n- v14 --supports--> v0d\n- v0b --supports--> v1d\n- v08 --supports--> v0c\n- v0c --refutes--> v0a\n- v19 --refutes--> v05\n- v20 --supports--> v02\n- v2e --refutes--> v06\n- v22 --supports--> v11\n- v1b --refutes--> v2c\n- v07 --refutes--> v16\n- v0f --supports--> v2f\n- v2d --refutes--> v21\n\n[VERIFIED ROOT]\nv23 = 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":"6efecbc36be3ad7ffb9c7c9b55b545c3d1edb724ff7404eb02bbd68064748d39","schema":"livegraph.boolean-propagation.v1","spec_hash":"4200a27a5760e77c166f8956067ae2c4562a7aa9b1d6da0a0f8fa76dcac5d7d0","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-e8ede22645328ef33f43","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v07 --refutes--> v03\n- v16 --refutes--> v06\n- v2d --refutes--> v19\n- v2a --supports--> v1a\n- v17 --supports--> v1f\n- v07 --supports--> v30\n- v1e --refutes--> v18\n- v1c --refutes--> v2e\n- v2c --refutes--> v27\n- v24 --refutes--> v09\n- v04 --refutes--> v20\n- v20 --refutes--> v1b\n- v2b --refutes--> v01\n- v00 --supports--> v2b\n- v29 --supports--> v25\n- v1a --supports--> v13\n- v1c --supports--> v17\n- v11 --supports--> v0f\n- v12 --supports--> v0e\n- v0a --supports--> v16\n- v2c --refutes--> v0d\n- v2f --supports--> v2d\n- v24 --supports--> v10\n- v16 --refutes--> v2c\n- v18 --supports--> v04\n- v17 --supports--> v24\n- v16 --supports--> v05\n- v30 --refutes--> v15\n- v0e --refutes--> v2a\n- v0c --supports--> v2f\n- v1b --supports--> v11\n- v2c --supports--> v12\n- v2f --supports--> v22\n- v23 --supports--> v02\n- v28 --refutes--> v00\n- v2d --refutes--> v21\n- v23 --supports--> v26\n- v1d --supports--> v29\n- v11 --supports--> v1d\n- v30 --supports--> v0b\n- v07 --supports--> v08\n- v26 --refutes--> v07\n- v01 --supports--> v23\n- v25 --refutes--> v0a\n- v21 --refutes--> v1c\n- v29 --refutes--> v14\n- v24 --supports--> v28\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"51136e8979da871e1d04b992c31415b348be8032abda06df398d9c3bff992900","schema":"livegraph.boolean-propagation.v1","spec_hash":"e8ede22645328ef33f4342cb6a275f9ca08d2ff669c46aa9c28aa7938a2248b1","spec_json":"{\"edges\":[{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-97370d31f27aed6649ef","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v02 --supports--> v09\n- v20 --refutes--> v2c\n- v26 --supports--> v04\n- v27 --refutes--> v0c\n- v2e --supports--> v29\n- v09 --supports--> v12\n- v0d --supports--> v16\n- v20 --refutes--> v11\n- v0f --supports--> v0a\n- v0c --supports--> v18\n- v28 --supports--> v27\n- v26 --refutes--> v00\n- v2f --refutes--> v1a\n- v0f --refutes--> v1e\n- v2d --supports--> v15\n- v1a --supports--> v08\n- v07 --refutes--> v28\n- v18 --refutes--> v2f\n- v29 --refutes--> v14\n- v14 --refutes--> v02\n- v22 --supports--> v2b\n- v15 --supports--> v24\n- v08 --supports--> v13\n- v27 --refutes--> v01\n- v26 --supports--> v30\n- v15 --supports--> v0e\n- v22 --supports--> v26\n- v16 --refutes--> v03\n- v00 --supports--> v25\n- v24 --supports--> v07\n- v12 --refutes--> v21\n- v00 --refutes--> v20\n- v20 --refutes--> v10\n- v16 --supports--> v1c\n- v13 --refutes--> v2e\n- v12 --supports--> v1b\n- v10 --supports--> v05\n- v08 --supports--> v1d\n- v19 --refutes--> v0f\n- v00 --supports--> v06\n- v0d --refutes--> v0b\n- v05 --supports--> v19\n- v0d --refutes--> v17\n- v1f --refutes--> v2d\n- v21 --refutes--> v22\n- v2a --refutes--> v1f\n- v23 --refutes--> v2a\n- v0f --supports--> v0d\n\n[VERIFIED ROOT]\nv23 = 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":"b5c3a87253a2d3941f2781087ad19c8fe892ebc5925fccd5370469524271c092","schema":"livegraph.boolean-propagation.v1","spec_hash":"97370d31f27aed6649efe4a752c9c4afeecec318ec82eaedbe09bcd37639583f","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-dc8d048ea37883da2b0b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v27\n- v2e --refutes--> v26\n- v08 --refutes--> v2f\n- v06 --refutes--> v08\n- v19 --refutes--> v05\n- v03 --refutes--> v1e\n- v26 --supports--> v1b\n- v09 --refutes--> v2d\n- v0b --refutes--> v0a\n- v03 --refutes--> v17\n- v14 --refutes--> v0c\n- v22 --refutes--> v2c\n- v20 --supports--> v1a\n- v24 --supports--> v29\n- v0d --supports--> v0e\n- v10 --refutes--> v22\n- v1d --refutes--> v02\n- v16 --supports--> v14\n- v1c --refutes--> v15\n- v2f --supports--> v12\n- v28 --supports--> v03\n- v0f --supports--> v09\n- v1e --refutes--> v1c\n- v1e --refutes--> v25\n- v02 --supports--> v20\n- v13 --refutes--> v0f\n- v29 --refutes--> v19\n- v1c --supports--> v01\n- v22 --refutes--> v11\n- v0f --supports--> v07\n- v1a --refutes--> v06\n- v30 --supports--> v00\n- v0e --refutes--> v24\n- v19 --refutes--> v16\n- v30 --refutes--> v0d\n- v18 --refutes--> v10\n- v15 --refutes--> v13\n- v11 --supports--> v1d\n- v2f --refutes--> v2a\n- v10 --refutes--> v1f\n- v11 --refutes--> v2b\n- v12 --supports--> v30\n- v26 --refutes--> v0b\n- v0c --refutes--> v2e\n- v0b --supports--> v23\n- v1a --refutes--> v04\n- v22 --refutes--> v21\n- v09 --refutes--> v18\n\n[VERIFIED ROOT]\nv28 = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3945d672fafeaf3854aa3b12c372d40813fefd46e284d9113fb34d18463d0bc7","schema":"livegraph.boolean-propagation.v1","spec_hash":"dc8d048ea37883da2b0b4cfd092bc296c251bce28217a253a206dc8d23a28f61","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v28\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-22303640b5bf9b7659bd","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v21\n- v2d --supports--> v0a\n- v00 --supports--> v1a\n- v18 --supports--> v11\n- v07 --refutes--> v23\n- v22 --refutes--> v2f\n- v20 --refutes--> v17\n- v02 --refutes--> v2e\n- v2b --refutes--> v15\n- v1c --supports--> v29\n- v0c --supports--> v07\n- v2e --supports--> v24\n- v0b --supports--> v1b\n- v2c --supports--> v00\n- v12 --refutes--> v30\n- v21 --refutes--> v20\n- v00 --refutes--> v2d\n- v09 --supports--> v12\n- v10 --refutes--> v25\n- v19 --supports--> v09\n- v04 --refutes--> v0b\n- v2d --supports--> v28\n- v10 --refutes--> v0f\n- v1b --refutes--> v19\n- v21 --supports--> v1d\n- v2b --refutes--> v01\n- v24 --refutes--> v1c\n- v27 --refutes--> v2c\n- v18 --refutes--> v14\n- v05 --refutes--> v27\n- v30 --supports--> v1e\n- v28 --refutes--> v2a\n- v10 --refutes--> v18\n- v05 --supports--> v03\n- v0c --refutes--> v08\n- v1e --refutes--> v05\n- v28 --refutes--> v0e\n- v23 --supports--> v22\n- v30 --refutes--> v16\n- v1e --refutes--> v06\n- v02 --supports--> v13\n- v14 --supports--> v1f\n- v29 --supports--> v2b\n- v1f --refutes--> v02\n- v02 --refutes--> v26\n- v2f --refutes--> v04\n- v07 --supports--> v0d\n- v20 --supports--> v0c\n\n[VERIFIED ROOT]\nv10 = TRUE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d80a8f944e06d81974376595b9b1dcc387fd1aaa7a0df702c85af8209d63e8ae","schema":"livegraph.boolean-propagation.v1","spec_hash":"22303640b5bf9b7659bd2992fb26dccc13b640d66a6df558b27477561f9a60c5","spec_json":"{\"edges\":[{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v10\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-6664c73e7c7b82be8ade","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2c --refutes--> v0e\n- v1f --refutes--> v29\n- v03 --refutes--> v25\n- v06 --supports--> v2e\n- v13 --refutes--> v24\n- v19 --supports--> v1f\n- v2c --supports--> v1b\n- v01 --refutes--> v03\n- v2c --supports--> v1e\n- v17 --supports--> v20\n- v0b --refutes--> v1a\n- v0c --supports--> v27\n- v07 --refutes--> v02\n- v23 --supports--> v08\n- v21 --refutes--> v11\n- v08 --supports--> v06\n- v22 --supports--> v07\n- v2e --supports--> v2c\n- v17 --refutes--> v26\n- v09 --refutes--> v01\n- v09 --supports--> v2f\n- v21 --supports--> v09\n- v20 --refutes--> v22\n- v1c --supports--> v19\n- v2e --supports--> v0d\n- v0c --refutes--> v0f\n- v2c --refutes--> v0a\n- v13 --refutes--> v2d\n- v25 --refutes--> v0c\n- v1c --supports--> v15\n- v0f --supports--> v05\n- v25 --refutes--> v10\n- v05 --supports--> v18\n- v29 --refutes--> v21\n- v16 --refutes--> v2b\n- v1e --refutes--> v14\n- v24 --supports--> v16\n- v1d --refutes--> v0b\n- v1a --supports--> v17\n- v20 --supports--> v2a\n- v00 --supports--> v30\n- v12 --refutes--> v1c\n- v05 --supports--> v13\n- v02 --refutes--> v23\n- v24 --refutes--> v28\n- v1e --supports--> v00\n- v30 --refutes--> v12\n- v09 --refutes--> v04\n\n[VERIFIED ROOT]\nv1d = FALSE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7821a9b3d6b3de0df6e1b1da06ce1bfab20a7d6c95b1188c11dd62a54893bc5e","schema":"livegraph.boolean-propagation.v1","spec_hash":"6664c73e7c7b82be8adee2eb3ac4cb92601f82d89dd735bcfe9e49de335086f8","spec_json":"{\"edges\":[{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-924ba93e48796363b1f4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --refutes--> v09\n- v03 --supports--> v22\n- v1d --supports--> v1b\n- v25 --refutes--> v04\n- v16 --refutes--> v0e\n- v21 --supports--> v27\n- v06 --supports--> v21\n- v16 --refutes--> v00\n- v0b --refutes--> v2d\n- v11 --refutes--> v1a\n- v23 --supports--> v0a\n- v03 --refutes--> v13\n- v2d --supports--> v03\n- v25 --refutes--> v06\n- v30 --refutes--> v16\n- v16 --supports--> v1f\n- v12 --refutes--> v14\n- v16 --refutes--> v2a\n- v01 --supports--> v08\n- v26 --refutes--> v19\n- v26 --refutes--> v28\n- v02 --supports--> v20\n- v1c --refutes--> v01\n- v09 --refutes--> v25\n- v2b --refutes--> v30\n- v0d --supports--> v2c\n- v0f --supports--> v2f\n- v1a --refutes--> v26\n- v2c --refutes--> v29\n- v01 --supports--> v15\n- v0b --supports--> v2e\n- v17 --supports--> v23\n- v1e --refutes--> v10\n- v28 --supports--> v1d\n- v27 --refutes--> v0b\n- v2f --supports--> v02\n- v17 --refutes--> v05\n- v11 --refutes--> v07\n- v13 --refutes--> v11\n- v12 --refutes--> v0c\n- v16 --supports--> v12\n- v1e --refutes--> v0f\n- v08 --supports--> v1e\n- v29 --refutes--> v1c\n- v1b --refutes--> v18\n- v1b --supports--> v2b\n- v0c --supports--> v0d\n- v29 --refutes--> v24\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3277bae0d2d1a1f35b9e4adfa04c5333890fa5a5f7b6c5652e014b72fefa6693","schema":"livegraph.boolean-propagation.v1","spec_hash":"924ba93e48796363b1f4ca07079a3690176ebde6ca71e54bf5295338e2379266","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-a3bd26bedef95f6c049a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v30 --refutes--> v1b\n- v27 --refutes--> v24\n- v1b --refutes--> v0f\n- v1a --supports--> v25\n- v07 --supports--> v01\n- v1e --supports--> v28\n- v19 --refutes--> v2b\n- v00 --supports--> v1e\n- v2d --supports--> v30\n- v14 --refutes--> v00\n- v03 --refutes--> v13\n- v12 --refutes--> v2c\n- v2e --supports--> v1f\n- v27 --supports--> v19\n- v23 --refutes--> v07\n- v1b --refutes--> v1a\n- v22 --refutes--> v26\n- v0a --supports--> v09\n- v1d --refutes--> v10\n- v0b --supports--> v03\n- v15 --supports--> v18\n- v28 --supports--> v2e\n- v1d --supports--> v04\n- v17 --supports--> v23\n- v01 --supports--> v1d\n- v29 --supports--> v06\n- v1e --refutes--> v0e\n- v22 --supports--> v2a\n- v26 --supports--> v27\n- v14 --refutes--> v1c\n- v2e --supports--> v21\n- v0c --refutes--> v0a\n- v2c --refutes--> v0d\n- v14 --supports--> v16\n- v07 --supports--> v05\n- v03 --supports--> v12\n- v20 --refutes--> v08\n- v15 --supports--> v2d\n- v10 --supports--> v2f\n- v0d --supports--> v14\n- v10 --supports--> v0b\n- v19 --supports--> v20\n- v08 --refutes--> v0c\n- v0c --supports--> v11\n- v1a --refutes--> v02\n- v09 --refutes--> v15\n- v29 --supports--> v22\n- v21 --supports--> v29\n\n[VERIFIED ROOT]\nv17 = 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":"cf001b6e0d78c2631fcba9f28369ead0c16be1121fa953c5428947e69ca3af67","schema":"livegraph.boolean-propagation.v1","spec_hash":"a3bd26bedef95f6c049a774d5c7dabf7b39bef5643f5dd3153588e6bdd7ecdc9","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-8b994f7b0d5fdf768297","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v0f --refutes--> v2f\n- v1d --supports--> v21\n- v28 --refutes--> v14\n- v09 --refutes--> v10\n- v19 --refutes--> v30\n- v2e --supports--> v16\n- v2b --refutes--> v27\n- v09 --supports--> v17\n- v05 --supports--> v01\n- v26 --refutes--> v25\n- v26 --refutes--> v0b\n- v05 --supports--> v03\n- v29 --supports--> v2c\n- v21 --refutes--> v09\n- v00 --supports--> v2a\n- v0a --supports--> v20\n- v17 --refutes--> v28\n- v06 --refutes--> v08\n- v12 --refutes--> v1a\n- v04 --refutes--> v22\n- v0a --refutes--> v19\n- v04 --refutes--> v1b\n- v0c --supports--> v1c\n- v0d --supports--> v13\n- v0b --refutes--> v0c\n- v1c --supports--> v0e\n- v28 --supports--> v07\n- v08 --refutes--> v04\n- v07 --supports--> v0d\n- v04 --refutes--> v0f\n- v0b --supports--> v18\n- v1c --refutes--> v2b\n- v27 --refutes--> v23\n- v1d --supports--> v1f\n- v1a --supports--> v00\n- v2c --supports--> v2d\n- v2a --supports--> v29\n- v23 --refutes--> v24\n- v06 --refutes--> v15\n- v16 --refutes--> v26\n- v24 --refutes--> v12\n- v2c --supports--> v06\n- v09 --refutes--> v11\n- v03 --supports--> v1d\n- v30 --supports--> v2e\n- v06 --supports--> v02\n- v0f --refutes--> v05\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d14031388fee9a33b7775a3c0e443355490c26e9fa24152756f88ee9a2f6bd11","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b994f7b0d5fdf768297f871ea6cf2306cff79d1cda3af15b6c805c0782bea77","spec_json":"{\"edges\":[{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-7c920ac7bc7dfe418525","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --refutes--> v0d\n- v1c --refutes--> v11\n- v0e --refutes--> v2f\n- v18 --refutes--> v25\n- v14 --refutes--> v30\n- v20 --supports--> v21\n- v1c --supports--> v22\n- v1b --supports--> v24\n- v03 --supports--> v09\n- v14 --supports--> v2e\n- v17 --refutes--> v05\n- v28 --refutes--> v06\n- v01 --refutes--> v0c\n- v18 --refutes--> v0b\n- v20 --supports--> v1d\n- v19 --refutes--> v29\n- v26 --supports--> v08\n- v11 --refutes--> v2d\n- v08 --supports--> v1b\n- v28 --supports--> v1e\n- v14 --refutes--> v13\n- v1f --refutes--> v1a\n- v0d --supports--> v02\n- v30 --supports--> v15\n- v00 --supports--> v0e\n- v23 --refutes--> v1c\n- v19 --supports--> v2c\n- v0d --refutes--> v28\n- v1d --refutes--> v00\n- v07 --refutes--> v2b\n- v11 --refutes--> v26\n- v01 --refutes--> v27\n- v25 --supports--> v17\n- v12 --refutes--> v1f\n- v30 --refutes--> v03\n- v0f --refutes--> v04\n- v16 --supports--> v23\n- v1d --supports--> v2a\n- v17 --supports--> v14\n- v00 --refutes--> v0a\n- v09 --supports--> v0f\n- v1a --supports--> v16\n- v2b --supports--> v19\n- v19 --refutes--> v12\n- v1e --supports--> v01\n- v2f --supports--> v18\n- v0c --refutes--> v10\n- v10 --supports--> v20\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":"76f12888085873bed4de5a7611a327fffe014e42a292fa6e6ce5bdd7b4a31562","schema":"livegraph.boolean-propagation.v1","spec_hash":"7c920ac7bc7dfe418525604efd6f9925301c44634538d7555cb0b2551ae1b46f","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v07\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-2c347cc008ad4d1e1c74","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v11 --supports--> v1c\n- v2e --supports--> v27\n- v1c --refutes--> v1f\n- v02 --refutes--> v2b\n- v14 --supports--> v09\n- v1b --refutes--> v0e\n- v1c --supports--> v24\n- v2c --supports--> v11\n- v2b --refutes--> v06\n- v07 --refutes--> v26\n- v1b --refutes--> v0f\n- v21 --supports--> v00\n- v17 --supports--> v22\n- v1e --refutes--> v10\n- v05 --supports--> v12\n- v0d --refutes--> v29\n- v09 --refutes--> v0a\n- v12 --refutes--> v30\n- v17 --supports--> v19\n- v00 --refutes--> v07\n- v0f --refutes--> v2d\n- v03 --refutes--> v21\n- v2a --refutes--> v2c\n- v1c --refutes--> v0b\n- v03 --refutes--> v20\n- v01 --refutes--> v02\n- v2b --refutes--> v2e\n- v0c --refutes--> v0d\n- v1e --supports--> v01\n- v30 --supports--> v1d\n- v28 --refutes--> v15\n- v03 --supports--> v13\n- v1a --refutes--> v1b\n- v27 --refutes--> v17\n- v2e --refutes--> v18\n- v29 --supports--> v23\n- v25 --refutes--> v1e\n- v23 --refutes--> v1a\n- v0f --supports--> v2a\n- v15 --refutes--> v03\n- v17 --supports--> v14\n- v1d --supports--> v16\n- v05 --supports--> v08\n- v29 --refutes--> v2f\n- v1b --refutes--> v04\n- v16 --supports--> v28\n- v09 --refutes--> v05\n\n[VERIFIED ROOT]\nv25 = TRUE\n\n[TARGET]\nv1f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4afdf3b55c2492eba671a54987b5c74d75700304740254239bc47725608e7507","schema":"livegraph.boolean-propagation.v1","spec_hash":"2c347cc008ad4d1e1c74885faf22f376453647cf5f42a53f9e42c4ae61ce4e5b","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-f67f433d2abfb9416aec","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v2d\n- v04 --supports--> v03\n- v0f --refutes--> v15\n- v00 --refutes--> v13\n- v1c --supports--> v02\n- v16 --refutes--> v1f\n- v10 --refutes--> v18\n- v1d --supports--> v2c\n- v02 --refutes--> v2a\n- v1d --supports--> v17\n- v2e --supports--> v05\n- v0a --refutes--> v0d\n- v07 --supports--> v1c\n- v13 --supports--> v23\n- v0e --refutes--> v0a\n- v16 --refutes--> v0e\n- v22 --refutes--> v06\n- v29 --refutes--> v07\n- v19 --supports--> v10\n- v0f --refutes--> v14\n- v25 --supports--> v29\n- v19 --supports--> v30\n- v09 --refutes--> v04\n- v23 --supports--> v2f\n- v17 --refutes--> v21\n- v23 --refutes--> v08\n- v25 --refutes--> v1e\n- v29 --refutes--> v20\n- v26 --supports--> v2e\n- v21 --supports--> v0f\n- v0d --supports--> v26\n- v2d --refutes--> v28\n- v11 --supports--> v09\n- v0f --supports--> v24\n- v12 --refutes--> v16\n- v2f --supports--> v12\n- v18 --supports--> v00\n- v04 --refutes--> v19\n- v0e --supports--> v1b\n- v06 --refutes--> v1d\n- v0c --supports--> v11\n- v2f --refutes--> v0b\n- v1c --refutes--> v1a\n- v06 --supports--> v01\n- v2f --supports--> v2b\n- v07 --refutes--> v27\n- v05 --refutes--> v22\n- v28 --supports--> v0c\n\n[VERIFIED ROOT]\nv25 = 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":"be63e37a5de8d4e6a4cc7a8cad4dd021d8ba19724bd2aad6ec2be6a22d0e0c75","schema":"livegraph.boolean-propagation.v1","spec_hash":"f67f433d2abfb9416aec221f8f767a6af3b1259402f6add9557fdb6976796e95","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-e977e9e9bf06a646d471","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v2a\n- v11 --refutes--> v04\n- v30 --supports--> v1c\n- v1e --refutes--> v03\n- v2f --refutes--> v1b\n- v1e --supports--> v29\n- v21 --refutes--> v2f\n- v18 --refutes--> v0a\n- v2e --refutes--> v0f\n- v01 --supports--> v0c\n- v2c --refutes--> v2e\n- v19 --supports--> v0e\n- v1a --refutes--> v23\n- v1e --supports--> v05\n- v1a --supports--> v16\n- v27 --supports--> v10\n- v04 --supports--> v30\n- v10 --refutes--> v13\n- v26 --refutes--> v2c\n- v19 --refutes--> v26\n- v15 --supports--> v18\n- v06 --supports--> v1a\n- v20 --refutes--> v17\n- v1f --supports--> v2b\n- v12 --refutes--> v14\n- v1f --supports--> v15\n- v12 --supports--> v1f\n- v03 --refutes--> v09\n- v2f --supports--> v0d\n- v23 --supports--> v28\n- v1b --supports--> v27\n- v23 --supports--> v1e\n- v01 --refutes--> v24\n- v0a --refutes--> v2d\n- v17 --refutes--> v02\n- v2d --refutes--> v1d\n- v1d --refutes--> v21\n- v00 --refutes--> v06\n- v30 --supports--> v07\n- v25 --refutes--> v12\n- v1c --refutes--> v00\n- v10 --supports--> v01\n- v01 --supports--> v0b\n- v03 --refutes--> v25\n- v02 --refutes--> v11\n- v18 --supports--> v08\n- v0f --supports--> v20\n- v1f --refutes--> v22\n\n[VERIFIED ROOT]\nv19 = FALSE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3b66d7114002e25d611288ae894be571e781bae55b2fe731afbfadcb497b8fbe","schema":"livegraph.boolean-propagation.v1","spec_hash":"e977e9e9bf06a646d4713d0df53ec57af850ee77143adaf339f70190c38799a9","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v19\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-f35be5fc6d83fe555170","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v27 --supports--> v0d\n- v28 --supports--> v25\n- v19 --refutes--> v1e\n- v14 --refutes--> v2e\n- v03 --refutes--> v30\n- v2d --supports--> v23\n- v26 --refutes--> v15\n- v17 --refutes--> v1f\n- v30 --supports--> v04\n- v20 --supports--> v13\n- v05 --supports--> v11\n- v21 --supports--> v0c\n- v15 --supports--> v22\n- v30 --refutes--> v0a\n- v16 --refutes--> v0e\n- v14 --supports--> v17\n- v0d --refutes--> v01\n- v0e --refutes--> v06\n- v17 --refutes--> v1a\n- v21 --refutes--> v12\n- v22 --refutes--> v21\n- v21 --refutes--> v1b\n- v1f --supports--> v19\n- v2f --supports--> v0f\n- v09 --refutes--> v2d\n- v05 --refutes--> v24\n- v23 --supports--> v18\n- v00 --supports--> v29\n- v2b --refutes--> v16\n- v13 --supports--> v28\n- v06 --supports--> v07\n- v30 --supports--> v2f\n- v19 --refutes--> v26\n- v06 --supports--> v09\n- v27 --refutes--> v2c\n- v20 --supports--> v08\n- v2a --refutes--> v00\n- v02 --supports--> v14\n- v23 --supports--> v1c\n- v28 --supports--> v02\n- v1d --refutes--> v10\n- v21 --supports--> v0b\n- v00 --supports--> v20\n- v18 --supports--> v2a\n- v0c --refutes--> v05\n- v11 --refutes--> v27\n- v2f --refutes--> v1d\n\n[VERIFIED ROOT]\nv2b = 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":"773f63f9c027aecd56d1b7b44a85d09843864b77d0ead8f32b6a8d06b98b3bc0","schema":"livegraph.boolean-propagation.v1","spec_hash":"f35be5fc6d83fe555170df992bb8c272f52c1399c71738843d412a223690e776","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2b\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-b3ae5fb0ff9f8faae816","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --refutes--> v05\n- v13 --refutes--> v16\n- v03 --refutes--> v1e\n- v0b --refutes--> v27\n- v04 --supports--> v0d\n- v1e --refutes--> v12\n- v09 --supports--> v04\n- v21 --supports--> v26\n- v12 --supports--> v07\n- v1c --refutes--> v22\n- v02 --refutes--> v20\n- v0f --supports--> v23\n- v20 --supports--> v21\n- v22 --supports--> v0a\n- v04 --supports--> v10\n- v2b --supports--> v02\n- v30 --supports--> v2d\n- v07 --refutes--> v30\n- v22 --refutes--> v01\n- v26 --refutes--> v0f\n- v01 --supports--> v1b\n- v06 --supports--> v14\n- v25 --refutes--> v18\n- v10 --refutes--> v17\n- v14 --refutes--> v00\n- v1d --refutes--> v2a\n- v2c --refutes--> v08\n- v02 --supports--> v29\n- v21 --refutes--> v11\n- v17 --refutes--> v13\n- v1f --supports--> v0c\n- v15 --supports--> v1d\n- v2a --refutes--> v03\n- v25 --refutes--> v2f\n- v14 --refutes--> v2b\n- v0c --refutes--> v15\n- v0c --refutes--> v1a\n- v24 --supports--> v09\n- v13 --supports--> v0b\n- v1b --supports--> v06\n- v1d --supports--> v28\n- v09 --refutes--> v0e\n- v25 --refutes--> v1c\n- v30 --refutes--> v2c\n- v1c --refutes--> v2e\n- v0f --supports--> v1f\n- v08 --supports--> v24\n- v12 --supports--> v19\n\n[VERIFIED ROOT]\nv25 = FALSE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6b65c2865642e3c0f5c530deeae6d6334c29df9bab77f1ac0da77179eef29efa","schema":"livegraph.boolean-propagation.v1","spec_hash":"b3ae5fb0ff9f8faae8165964055a962b01d6d488199090cb52289227b192a2f8","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-22597117cbe67e327966","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --supports--> v0c\n- v22 --refutes--> v02\n- v25 --supports--> v1a\n- v2a --supports--> v07\n- v2e --supports--> v10\n- v07 --supports--> v18\n- v24 --supports--> v2d\n- v29 --refutes--> v24\n- v14 --supports--> v03\n- v24 --supports--> v28\n- v1d --supports--> v20\n- v05 --refutes--> v1f\n- v12 --refutes--> v26\n- v04 --supports--> v0f\n- v1e --supports--> v0e\n- v0d --refutes--> v1e\n- v22 --supports--> v00\n- v0c --supports--> v21\n- v2c --refutes--> v29\n- v17 --refutes--> v23\n- v22 --supports--> v09\n- v1d --refutes--> v05\n- v14 --supports--> v1c\n- v14 --supports--> v0b\n- v16 --refutes--> v04\n- v2f --supports--> v2a\n- v26 --supports--> v27\n- v05 --supports--> v2f\n- v0e --refutes--> v2e\n- v04 --refutes--> v08\n- v00 --supports--> v13\n- v13 --refutes--> v16\n- v12 --supports--> v2b\n- v30 --supports--> v06\n- v1a --supports--> v0d\n- v24 --refutes--> v19\n- v21 --supports--> v30\n- v0f --refutes--> v15\n- v26 --supports--> v14\n- v0f --supports--> v1d\n- v01 --refutes--> v25\n- v18 --supports--> v01\n- v06 --refutes--> v12\n- v1a --refutes--> v11\n- v2e --refutes--> v0a\n- v29 --supports--> v1b\n- v0a --refutes--> v17\n- v23 --supports--> v2c\n\n[VERIFIED ROOT]\nv22 = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"44874dffd075ffc289fbdb2ff8d6f3d28c3e4fb797049895bd887d202cbb3ec2","schema":"livegraph.boolean-propagation.v1","spec_hash":"22597117cbe67e327966d254ca4a730c5070222bc42b81006b71ffcfb6ad8cd9","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v22\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-7cb09909ded17fbc1bff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v1f\n- v22 --supports--> v19\n- v20 --refutes--> v12\n- v06 --supports--> v07\n- v29 --refutes--> v05\n- v19 --supports--> v03\n- v1f --supports--> v29\n- v0f --refutes--> v15\n- v0f --refutes--> v04\n- v08 --refutes--> v18\n- v26 --supports--> v16\n- v0c --refutes--> v0e\n- v08 --supports--> v10\n- v2e --refutes--> v30\n- v24 --refutes--> v0d\n- v2e --supports--> v23\n- v20 --refutes--> v25\n- v2f --supports--> v2d\n- v2a --refutes--> v0a\n- v1f --supports--> v01\n- v1e --supports--> v2e\n- v02 --refutes--> v1b\n- v22 --refutes--> v1c\n- v26 --refutes--> v17\n- v07 --refutes--> v26\n- v1a --refutes--> v14\n- v11 --supports--> v08\n- v16 --supports--> v0b\n- v09 --refutes--> v20\n- v21 --supports--> v13\n- v2a --refutes--> v2b\n- v24 --refutes--> v1d\n- v13 --refutes--> v0f\n- v02 --supports--> v09\n- v22 --refutes--> v00\n- v30 --supports--> v2a\n- v15 --supports--> v0c\n- v2d --supports--> v24\n- v1d --supports--> v28\n- v03 --refutes--> v1a\n- v03 --supports--> v27\n- v29 --supports--> v06\n- v28 --refutes--> v21\n- v0b --supports--> v11\n- v2a --refutes--> v2c\n- v25 --supports--> v1e\n- v0c --refutes--> v02\n- v18 --supports--> v2f\n\n[VERIFIED ROOT]\nv22 = 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":"0f360c7843db4883cf1072b64120eea8b6b89ca9b196db61d6d886205d2648da","schema":"livegraph.boolean-propagation.v1","spec_hash":"7cb09909ded17fbc1bff545c2690450b1122db0c9d8b5ba97f906e6f7d50e9ec","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v22\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-4e668d49ae4d845cd264","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v29 --refutes--> v10\n- v2e --refutes--> v00\n- v27 --refutes--> v07\n- v26 --supports--> v17\n- v08 --supports--> v1b\n- v1b --supports--> v11\n- v1f --refutes--> v23\n- v1b --refutes--> v26\n- v00 --refutes--> v30\n- v2f --supports--> v0a\n- v1b --supports--> v1a\n- v04 --supports--> v0e\n- v30 --refutes--> v0f\n- v09 --supports--> v1f\n- v17 --supports--> v09\n- v0c --supports--> v28\n- v05 --refutes--> v0d\n- v21 --supports--> v12\n- v07 --supports--> v2e\n- v2b --refutes--> v18\n- v2f --refutes--> v05\n- v2d --refutes--> v1d\n- v01 --refutes--> v1e\n- v1d --refutes--> v29\n- v30 --supports--> v0c\n- v19 --refutes--> v14\n- v24 --supports--> v22\n- v05 --refutes--> v19\n- v22 --refutes--> v2b\n- v1e --supports--> v02\n- v16 --supports--> v24\n- v0c --refutes--> v16\n- v04 --refutes--> v27\n- v2a --supports--> v2f\n- v18 --supports--> v03\n- v1b --refutes--> v20\n- v16 --refutes--> v2c\n- v00 --supports--> v15\n- v03 --refutes--> v0b\n- v14 --supports--> v04\n- v1e --supports--> v2d\n- v1b --supports--> v25\n- v30 --supports--> v13\n- v21 --refutes--> v1c\n- v1c --supports--> v06\n- v10 --refutes--> v21\n- v1f --refutes--> v01\n\n[VERIFIED ROOT]\nv2a = 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":"d6de0b354ef2e6b8dcfdb7fbe9d97b638927a6a12da809d5615b0213dc9cd72a","schema":"livegraph.boolean-propagation.v1","spec_hash":"4e668d49ae4d845cd2649ef1004dfa8188a9075c25fc997aed6bed315b5e5f7b","spec_json":"{\"edges\":[{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-f4bd29f3d37873caf299","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1f\n- v01 --supports--> v11\n- v13 --refutes--> v0c\n- v14 --refutes--> v2c\n- v28 --supports--> v17\n- v15 --supports--> v1d\n- v11 --supports--> v0e\n- v16 --supports--> v1b\n- v21 --supports--> v1a\n- v07 --refutes--> v10\n- v23 --supports--> v04\n- v1e --refutes--> v16\n- v10 --supports--> v03\n- v03 --refutes--> v20\n- v30 --refutes--> v01\n- v1e --supports--> v0f\n- v2b --refutes--> v1c\n- v0e --supports--> v29\n- v17 --refutes--> v00\n- v1d --supports--> v2b\n- v24 --supports--> v13\n- v18 --supports--> v05\n- v02 --refutes--> v0b\n- v17 --supports--> v18\n- v2c --refutes--> v08\n- v12 --supports--> v2f\n- v22 --refutes--> v30\n- v17 --supports--> v0a\n- v20 --supports--> v14\n- v1b --refutes--> v12\n- v08 --refutes--> v25\n- v0b --refutes--> v0d\n- v24 --supports--> v19\n- v12 --supports--> v22\n- v10 --supports--> v06\n- v1d --refutes--> v2e\n- v0d --refutes--> v2d\n- v0d --refutes--> v28\n- v18 --supports--> v24\n- v22 --refutes--> v2a\n- v23 --supports--> v07\n- v30 --supports--> v26\n- v2b --supports--> v1e\n- v09 --supports--> v21\n- v12 --refutes--> v27\n- v29 --supports--> v02\n- v13 --supports--> v23\n- v25 --supports--> v09\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d398621edc734547ec2dd3da2b5d455fd51a7333cbc9e371299fd853e322b38d","schema":"livegraph.boolean-propagation.v1","spec_hash":"f4bd29f3d37873caf299aa2f4046db4736915d84c8fdb4e968e07857d20829af","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-343d42af9de2a21dd0e4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v29\n- v2d --supports--> v12\n- v10 --refutes--> v15\n- v2e --supports--> v04\n- v00 --refutes--> v1b\n- v2f --refutes--> v13\n- v02 --supports--> v27\n- v0b --refutes--> v03\n- v0e --supports--> v08\n- v30 --supports--> v09\n- v26 --refutes--> v16\n- v0b --supports--> v19\n- v0c --refutes--> v00\n- v13 --supports--> v23\n- v12 --supports--> v0a\n- v16 --refutes--> v2a\n- v14 --supports--> v28\n- v15 --refutes--> v05\n- v18 --refutes--> v11\n- v00 --supports--> v2e\n- v27 --refutes--> v2d\n- v07 --refutes--> v06\n- v0a --refutes--> v18\n- v16 --supports--> v22\n- v13 --supports--> v0d\n- v13 --supports--> v2b\n- v06 --refutes--> v14\n- v1c --supports--> v07\n- v24 --supports--> v0b\n- v07 --supports--> v17\n- v14 --supports--> v02\n- v1c --refutes--> v1a\n- v06 --supports--> v25\n- v1c --supports--> v0f\n- v04 --refutes--> v1c\n- v08 --refutes--> v24\n- v1c --supports--> v1e\n- v11 --refutes--> v10\n- v1d --refutes--> v1f\n- v05 --refutes--> v2f\n- v1d --supports--> v0e\n- v2a --supports--> v30\n- v18 --refutes--> v2c\n- v09 --refutes--> v20\n- v1d --refutes--> v01\n- v14 --supports--> v21\n- v20 --refutes--> v0c\n- v23 --supports--> v1d\n\n[VERIFIED ROOT]\nv26 = 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":"a78fe83f5875ab02ea47a70e05edee73ad4aec15de039006f9a1189b5956f70e","schema":"livegraph.boolean-propagation.v1","spec_hash":"343d42af9de2a21dd0e4ea4b0b6a45cc9adbbdadcfb813fd15d748868e20e91d","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v26\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-d118fcee269bfdade4ad","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --supports--> v05\n- v2d --refutes--> v0a\n- v06 --refutes--> v10\n- v1c --refutes--> v18\n- v21 --supports--> v19\n- v10 --supports--> v17\n- v24 --refutes--> v1d\n- v01 --supports--> v04\n- v07 --refutes--> v2c\n- v2a --supports--> v01\n- v1a --supports--> v1f\n- v13 --supports--> v26\n- v0e --refutes--> v2b\n- v26 --supports--> v1b\n- v0e --refutes--> v30\n- v30 --refutes--> v24\n- v27 --supports--> v2e\n- v25 --supports--> v12\n- v0b --refutes--> v13\n- v1d --supports--> v1a\n- v17 --supports--> v22\n- v28 --supports--> v1c\n- v30 --refutes--> v23\n- v25 --refutes--> v0c\n- v18 --refutes--> v25\n- v1e --supports--> v00\n- v1a --supports--> v20\n- v26 --refutes--> v06\n- v0a --supports--> v0f\n- v06 --refutes--> v16\n- v0f --supports--> v2f\n- v28 --refutes--> v02\n- v24 --supports--> v03\n- v0c --refutes--> v2a\n- v04 --supports--> v2d\n- v07 --supports--> v21\n- v00 --supports--> v14\n- v22 --supports--> v07\n- v00 --supports--> v27\n- v21 --refutes--> v0e\n- v06 --supports--> v11\n- v15 --refutes--> v1e\n- v08 --supports--> v0b\n- v30 --refutes--> v29\n- v2f --refutes--> v15\n- v15 --refutes--> v09\n- v17 --refutes--> v0d\n- v1f --supports--> v28\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e35ee741a66ab2dc385de179babe807707265e8112291de5109b63f37fe60428","schema":"livegraph.boolean-propagation.v1","spec_hash":"d118fcee269bfdade4ad100f3a0909ea9d1c30ebe982e0b68b93f85b5370df29","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v08\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-3d4534c8e5976daee2aa","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --refutes--> v02\n- v0c --supports--> v29\n- v14 --refutes--> v26\n- v16 --supports--> v0d\n- v2d --refutes--> v30\n- v22 --supports--> v2e\n- v04 --refutes--> v1e\n- v25 --supports--> v11\n- v0a --supports--> v00\n- v1b --refutes--> v22\n- v2b --refutes--> v09\n- v06 --supports--> v2a\n- v26 --refutes--> v06\n- v28 --supports--> v04\n- v13 --supports--> v08\n- v19 --supports--> v0e\n- v1e --supports--> v13\n- v06 --supports--> v0b\n- v21 --supports--> v20\n- v10 --refutes--> v27\n- v0b --refutes--> v2b\n- v11 --refutes--> v16\n- v03 --refutes--> v21\n- v04 --refutes--> v1f\n- v28 --supports--> v07\n- v18 --supports--> v28\n- v1e --refutes--> v1d\n- v13 --supports--> v17\n- v12 --supports--> v01\n- v2f --supports--> v19\n- v11 --refutes--> v2c\n- v15 --supports--> v05\n- v30 --refutes--> v0f\n- v15 --refutes--> v24\n- v21 --refutes--> v14\n- v0c --refutes--> v12\n- v1c --refutes--> v10\n- v09 --supports--> v2f\n- v12 --refutes--> v1a\n- v12 --refutes--> v23\n- v00 --supports--> v2d\n- v12 --refutes--> v0a\n- v15 --supports--> v03\n- v0d --supports--> v18\n- v0e --refutes--> v25\n- v22 --refutes--> v15\n- v30 --refutes--> v1c\n- v17 --refutes--> v0c\n\n[VERIFIED ROOT]\nv1b = FALSE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"2f271cc5db327d60b4ce7760e1daebb895964f916989a054a02c0d7be5f5a001","schema":"livegraph.boolean-propagation.v1","spec_hash":"3d4534c8e5976daee2aa45683ab7105ab8a6a890e3e31a83a5810b4d0b15ba8c","spec_json":"{\"edges\":[{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-60859ed669aaf6731e71","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2c --supports--> v0c\n- v24 --supports--> v25\n- v1f --supports--> v00\n- v19 --refutes--> v01\n- v02 --supports--> v17\n- v20 --refutes--> v0a\n- v2d --refutes--> v04\n- v28 --supports--> v2a\n- v18 --refutes--> v10\n- v0e --supports--> v26\n- v2c --refutes--> v03\n- v19 --supports--> v05\n- v02 --supports--> v22\n- v06 --refutes--> v1e\n- v12 --refutes--> v1f\n- v23 --refutes--> v06\n- v1b --refutes--> v0e\n- v16 --supports--> v20\n- v13 --refutes--> v28\n- v18 --supports--> v19\n- v0a --refutes--> v2f\n- v0c --supports--> v2d\n- v15 --supports--> v18\n- v28 --supports--> v15\n- v2f --refutes--> v23\n- v25 --refutes--> v2c\n- v0b --refutes--> v30\n- v2b --refutes--> v0d\n- v0a --supports--> v1a\n- v09 --supports--> v1b\n- v02 --refutes--> v14\n- v2b --refutes--> v16\n- v0f --supports--> v29\n- v07 --refutes--> v0b\n- v0e --supports--> v12\n- v2b --supports--> v08\n- v2f --supports--> v21\n- v04 --refutes--> v0f\n- v01 --supports--> v02\n- v1e --supports--> v2e\n- v30 --supports--> v09\n- v11 --refutes--> v24\n- v0b --refutes--> v1c\n- v09 --refutes--> v1d\n- v1f --refutes--> v27\n- v0f --supports--> v13\n- v02 --refutes--> v07\n- v00 --supports--> v2b\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3df73486c34840437619d6d4cee7334d0acaba58503bc9cc4952c2ca256b8c37","schema":"livegraph.boolean-propagation.v1","spec_hash":"60859ed669aaf6731e71e121c1b3f6b074a2af03acac3e974d72cecba3539169","spec_json":"{\"edges\":[{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-af5315be99dd45df76e6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2b --refutes--> v1f\n- v2a --refutes--> v12\n- v0a --refutes--> v11\n- v0b --supports--> v0f\n- v0c --supports--> v2f\n- v25 --refutes--> v06\n- v0e --refutes--> v0b\n- v1f --refutes--> v14\n- v21 --supports--> v29\n- v29 --refutes--> v0e\n- v1d --supports--> v01\n- v13 --supports--> v2d\n- v1e --refutes--> v03\n- v0e --supports--> v15\n- v11 --refutes--> v20\n- v03 --refutes--> v0c\n- v2a --supports--> v00\n- v0d --supports--> v23\n- v0c --refutes--> v22\n- v00 --refutes--> v1c\n- v11 --supports--> v2b\n- v10 --refutes--> v02\n- v1a --supports--> v27\n- v0b --refutes--> v1d\n- v1f --refutes--> v17\n- v0b --supports--> v18\n- v06 --refutes--> v2a\n- v17 --refutes--> v19\n- v16 --refutes--> v1a\n- v10 --supports--> v2c\n- v05 --refutes--> v04\n- v26 --supports--> v10\n- v27 --supports--> v2e\n- v13 --supports--> v1b\n- v27 --refutes--> v30\n- v23 --supports--> v16\n- v27 --supports--> v05\n- v17 --refutes--> v1e\n- v13 --refutes--> v07\n- v26 --refutes--> v24\n- v01 --refutes--> v08\n- v08 --supports--> v25\n- v0c --supports--> v09\n- v1c --supports--> v13\n- v2c --supports--> v0d\n- v07 --refutes--> v0a\n- v17 --refutes--> v28\n- v04 --refutes--> v21\n\n[VERIFIED ROOT]\nv26 = FALSE\n\n[TARGET]\nv2f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"18723987691c8f26381e4ea5371f257f64f97342312664d8b6266515de749e28","schema":"livegraph.boolean-propagation.v1","spec_hash":"af5315be99dd45df76e6a493eeb0da17c455a396a9caf37f4e34e4999e29ac69","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2c\"}],\"hop\":32,\"root\":\"v26\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-99ab7461976e8be8f347","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v13 --refutes--> v04\n- v0a --refutes--> v19\n- v1a --supports--> v02\n- v2d --refutes--> v22\n- v05 --supports--> v30\n- v17 --refutes--> v0b\n- v26 --refutes--> v16\n- v08 --supports--> v1d\n- v30 --refutes--> v2f\n- v13 --supports--> v20\n- v18 --refutes--> v07\n- v21 --supports--> v1a\n- v2b --supports--> v29\n- v2e --supports--> v2c\n- v0d --refutes--> v09\n- v2a --refutes--> v0c\n- v1a --refutes--> v1e\n- v11 --refutes--> v26\n- v1e --refutes--> v2b\n- v22 --supports--> v0f\n- v1b --refutes--> v2e\n- v1f --refutes--> v0a\n- v1b --supports--> v28\n- v16 --supports--> v1b\n- v2c --refutes--> v1c\n- v2b --supports--> v08\n- v2f --refutes--> v2a\n- v09 --supports--> v06\n- v0f --supports--> v01\n- v2a --refutes--> v03\n- v17 --supports--> v1f\n- v0a --supports--> v05\n- v1c --refutes--> v10\n- v00 --supports--> v0d\n- v0f --supports--> v11\n- v17 --refutes--> v23\n- v26 --supports--> v27\n- v21 --supports--> v14\n- v07 --supports--> v17\n- v2b --refutes--> v24\n- v0c --refutes--> v0e\n- v2f --supports--> v15\n- v08 --supports--> v00\n- v10 --refutes--> v13\n- v12 --refutes--> v21\n- v2f --refutes--> v25\n- v09 --refutes--> v18\n- v04 --supports--> v12\n\n[VERIFIED ROOT]\nv2d = 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":"025039e1934eec6cecf22826900042e7968cac8980c7ebccbfda7df127d3a653","schema":"livegraph.boolean-propagation.v1","spec_hash":"99ab7461976e8be8f347200292b90e9ed33c7a2b822c7c0720e908f5af50d6b5","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-187aff298e70e02db16b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v22\n- v09 --refutes--> v2f\n- v16 --supports--> v21\n- v17 --refutes--> v1a\n- v21 --supports--> v04\n- v1a --supports--> v0a\n- v06 --refutes--> v05\n- v0a --refutes--> v11\n- v24 --refutes--> v30\n- v28 --refutes--> v13\n- v1b --supports--> v10\n- v0b --supports--> v14\n- v00 --supports--> v0b\n- v0b --supports--> v0f\n- v2a --supports--> v02\n- v13 --supports--> v0d\n- v28 --supports--> v2c\n- v10 --supports--> v2d\n- v06 --refutes--> v1f\n- v0a --supports--> v03\n- v08 --refutes--> v27\n- v1f --refutes--> v1b\n- v1a --refutes--> v07\n- v21 --refutes--> v2b\n- v13 --supports--> v1c\n- v20 --refutes--> v17\n- v09 --refutes--> v24\n- v1c --refutes--> v0c\n- v10 --supports--> v29\n- v29 --supports--> v2a\n- v1e --supports--> v28\n- v02 --refutes--> v0e\n- v2b --refutes--> v00\n- v19 --refutes--> v20\n- v1d --refutes--> v09\n- v10 --refutes--> v01\n- v21 --supports--> v23\n- v10 --supports--> v2e\n- v25 --refutes--> v18\n- v0e --refutes--> v16\n- v24 --refutes--> v08\n- v14 --supports--> v26\n- v03 --supports--> v1d\n- v29 --refutes--> v15\n- v0c --refutes--> v12\n- v26 --supports--> v19\n- v0c --refutes--> v06\n- v22 --refutes--> v25\n\n[VERIFIED ROOT]\nv1e = 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":"c5b884e2f12d1a3720abaa506599d1029cf23cff51f6b3e75a61f6de2995a0ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"187aff298e70e02db16b7d340f8d852fb115830fe5df0a4a86ff87362f467654","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2b\"}],\"hop\":32,\"root\":\"v1e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-6d31d03d968dc29bceb3","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v23\n- v05 --supports--> v16\n- v2f --supports--> v29\n- v15 --refutes--> v25\n- v1a --supports--> v01\n- v02 --refutes--> v24\n- v1f --refutes--> v0c\n- v01 --supports--> v04\n- v2a --refutes--> v2c\n- v20 --refutes--> v13\n- v03 --supports--> v21\n- v24 --supports--> v2e\n- v21 --refutes--> v20\n- v08 --refutes--> v1e\n- v0e --refutes--> v27\n- v20 --refutes--> v1c\n- v05 --supports--> v2d\n- v0c --supports--> v2f\n- v15 --refutes--> v1b\n- v26 --refutes--> v05\n- v08 --refutes--> v12\n- v19 --supports--> v0d\n- v2a --supports--> v30\n- v07 --refutes--> v18\n- v25 --refutes--> v07\n- v27 --refutes--> v0b\n- v26 --supports--> v28\n- v19 --refutes--> v00\n- v2e --refutes--> v03\n- v17 --supports--> v08\n- v27 --refutes--> v1f\n- v03 --refutes--> v06\n- v01 --refutes--> v10\n- v30 --refutes--> v0a\n- v2b --supports--> v0f\n- v23 --supports--> v0e\n- v0e --refutes--> v09\n- v2a --refutes--> v1d\n- v10 --supports--> v26\n- v05 --supports--> v22\n- v10 --refutes--> v14\n- v08 --supports--> v11\n- v29 --supports--> v1a\n- v12 --supports--> v2a\n- v0a --refutes--> v19\n- v2b --supports--> v02\n- v00 --refutes--> v2b\n- v2d --refutes--> v17\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv13\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e069cb292ac0017b0b4d4bb7af05cf016d78d85b6186314f801d0f90c73f0b64","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d31d03d968dc29bceb30be0a16bcde439c14ff437cd3730afb7b49b255c4d0a","spec_json":"{\"edges\":[{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v13\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-6314a62fc7958a98695c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v22\n- v22 --supports--> v0e\n- v2a --supports--> v28\n- v14 --refutes--> v13\n- v2d --supports--> v23\n- v08 --refutes--> v1d\n- v16 --supports--> v00\n- v24 --refutes--> v1c\n- v2f --supports--> v1e\n- v14 --supports--> v17\n- v10 --supports--> v24\n- v2f --supports--> v19\n- v00 --supports--> v05\n- v2f --supports--> v18\n- v01 --refutes--> v07\n- v26 --supports--> v03\n- v1a --supports--> v2b\n- v29 --supports--> v10\n- v0c --refutes--> v14\n- v18 --supports--> v12\n- v13 --supports--> v06\n- v2c --refutes--> v09\n- v1a --supports--> v0a\n- v13 --refutes--> v1b\n- v12 --supports--> v11\n- v07 --refutes--> v26\n- v01 --refutes--> v27\n- v2c --refutes--> v2f\n- v04 --supports--> v21\n- v05 --supports--> v0c\n- v2a --refutes--> v1f\n- v2b --refutes--> v2c\n- v2d --supports--> v04\n- v15 --supports--> v25\n- v06 --supports--> v1a\n- v25 --refutes--> v2a\n- v03 --supports--> v0f\n- v21 --refutes--> v16\n- v01 --refutes--> v20\n- v0e --refutes--> v15\n- v12 --supports--> v01\n- v28 --refutes--> v2d\n- v0f --supports--> v29\n- v00 --supports--> v2e\n- v2c --supports--> v0d\n- v1c --supports--> v0b\n- v2d --supports--> v02\n- v0e --refutes--> v30\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b13b30b5e2a047998551b90885b2f1aaf35dc21854c920de8e73207dde14c20f","schema":"livegraph.boolean-propagation.v1","spec_hash":"6314a62fc7958a98695c843dc68098d5c298a5bb4a1df3b8491b8ed6ec1b80ee","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-454957418dd4394530a2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v11\n- v10 --refutes--> v1b\n- v1b --supports--> v0d\n- v1b --supports--> v30\n- v2f --refutes--> v28\n- v0e --supports--> v12\n- v28 --supports--> v0f\n- v15 --supports--> v16\n- v0f --supports--> v1d\n- v04 --supports--> v07\n- v0f --refutes--> v23\n- v00 --supports--> v03\n- v2f --refutes--> v1e\n- v08 --supports--> v22\n- v0f --supports--> v2a\n- v2f --refutes--> v2b\n- v2e --refutes--> v0b\n- v2d --refutes--> v20\n- v24 --supports--> v26\n- v12 --supports--> v1a\n- v21 --refutes--> v10\n- v07 --refutes--> v0e\n- v1c --supports--> v06\n- v07 --supports--> v17\n- v22 --supports--> v2e\n- v11 --refutes--> v02\n- v12 --supports--> v09\n- v0e --supports--> v19\n- v09 --supports--> v0a\n- v29 --supports--> v13\n- v21 --refutes--> v27\n- v0c --supports--> v04\n- v06 --refutes--> v29\n- v30 --refutes--> v05\n- v1f --supports--> v2d\n- v0c --supports--> v18\n- v1d --refutes--> v21\n- v09 --refutes--> v1f\n- v01 --refutes--> v1c\n- v04 --refutes--> v25\n- v0b --refutes--> v24\n- v26 --refutes--> v01\n- v16 --refutes--> v2f\n- v30 --supports--> v0c\n- v00 --supports--> v08\n- v0f --refutes--> v14\n- v28 --supports--> v2c\n- v02 --refutes--> v15\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"106acae717f1f65c26bc620a87d37992e7008754a9ee9710dd5079476beda7bb","schema":"livegraph.boolean-propagation.v1","spec_hash":"454957418dd4394530a2e30456f0accda7394cf5e3b478cff2f95d1e667243b3","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-d0057c83b78709852787","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v12\n- v20 --supports--> v0a\n- v28 --supports--> v0e\n- v24 --refutes--> v1e\n- v30 --refutes--> v05\n- v04 --refutes--> v02\n- v0c --refutes--> v2a\n- v28 --refutes--> v19\n- v2c --refutes--> v07\n- v2a --supports--> v23\n- v26 --refutes--> v0c\n- v0d --refutes--> v27\n- v22 --refutes--> v18\n- v1f --supports--> v20\n- v0a --refutes--> v26\n- v1b --refutes--> v14\n- v2a --refutes--> v1a\n- v30 --supports--> v1d\n- v1c --supports--> v22\n- v01 --supports--> v04\n- v27 --supports--> v28\n- v00 --refutes--> v1b\n- v0a --refutes--> v03\n- v28 --supports--> v0b\n- v2a --supports--> v00\n- v08 --supports--> v0d\n- v1e --refutes--> v25\n- v1f --supports--> v11\n- v12 --refutes--> v09\n- v04 --refutes--> v15\n- v13 --supports--> v01\n- v25 --refutes--> v30\n- v19 --supports--> v2e\n- v10 --supports--> v2c\n- v2d --supports--> v24\n- v19 --refutes--> v29\n- v06 --supports--> v10\n- v12 --supports--> v2f\n- v12 --supports--> v0f\n- v06 --supports--> v21\n- v2c --supports--> v17\n- v02 --supports--> v1c\n- v25 --supports--> v2b\n- v30 --supports--> v16\n- v05 --refutes--> v08\n- v2e --refutes--> v1f\n- v09 --refutes--> v13\n- v22 --refutes--> v06\n\n[VERIFIED ROOT]\nv2d = 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":"49dc9c315e57f017331f288b45470ae151963580e81ca0506f301ca622c9adb3","schema":"livegraph.boolean-propagation.v1","spec_hash":"d0057c83b787098527876785125d42480213f063f57df0afebdb2a4b378443ac","spec_json":"{\"edges\":[{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-cfe3cf9b315b137fdbac","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v29\n- v30 --refutes--> v27\n- v02 --supports--> v21\n- v13 --supports--> v03\n- v26 --refutes--> v24\n- v1d --supports--> v0f\n- v19 --refutes--> v11\n- v24 --refutes--> v09\n- v01 --refutes--> v1e\n- v24 --supports--> v17\n- v2c --refutes--> v2d\n- v06 --refutes--> v30\n- v2e --supports--> v1c\n- v1b --refutes--> v2b\n- v04 --supports--> v0c\n- v20 --refutes--> v07\n- v1f --supports--> v18\n- v2d --supports--> v2f\n- v11 --supports--> v2e\n- v01 --supports--> v14\n- v05 --refutes--> v26\n- v0b --refutes--> v20\n- v2b --refutes--> v01\n- v1c --supports--> v0d\n- v1f --supports--> v04\n- v1c --supports--> v12\n- v21 --supports--> v05\n- v0c --supports--> v2a\n- v1d --supports--> v16\n- v1a --refutes--> v06\n- v13 --supports--> v1d\n- v09 --refutes--> v28\n- v0e --refutes--> v19\n- v0c --refutes--> v25\n- v2d --refutes--> v1a\n- v0c --supports--> v0b\n- v0f --supports--> v22\n- v1c --refutes--> v08\n- v08 --supports--> v23\n- v1f --refutes--> v15\n- v24 --refutes--> v00\n- v28 --supports--> v2c\n- v30 --refutes--> v1b\n- v0f --refutes--> v0e\n- v1c --refutes--> v0a\n- v23 --refutes--> v1f\n- v10 --refutes--> v02\n- v07 --supports--> v10\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":"1d1821b42622f38c4ee09444368ffc567ca505ca20eba750882d4418adbcc135","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfe3cf9b315b137fdbac92160be6d85afb1b7e09c34e210538d01a8f7a8823ae","spec_json":"{\"edges\":[{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-9ce5bfc45b3a6362702b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v16 --supports--> v11\n- v25 --supports--> v07\n- v12 --supports--> v1d\n- v2a --refutes--> v23\n- v18 --refutes--> v0a\n- v24 --refutes--> v2d\n- v1c --supports--> v16\n- v19 --refutes--> v1f\n- v28 --supports--> v19\n- v02 --supports--> v2f\n- v03 --refutes--> v15\n- v1e --supports--> v24\n- v07 --refutes--> v08\n- v07 --supports--> v2e\n- v18 --supports--> v26\n- v28 --supports--> v09\n- v07 --refutes--> v02\n- v20 --supports--> v01\n- v02 --refutes--> v17\n- v2c --refutes--> v1b\n- v22 --refutes--> v10\n- v27 --refutes--> v2b\n- v0b --refutes--> v00\n- v2d --refutes--> v18\n- v1d --supports--> v25\n- v0e --supports--> v12\n- v2b --supports--> v0e\n- v2c --refutes--> v2a\n- v26 --supports--> v21\n- v0b --supports--> v22\n- v28 --refutes--> v1a\n- v04 --refutes--> v1c\n- v28 --supports--> v30\n- v21 --refutes--> v20\n- v26 --refutes--> v0c\n- v22 --supports--> v06\n- v23 --supports--> v28\n- v15 --refutes--> v1e\n- v02 --refutes--> v29\n- v11 --refutes--> v0b\n- v01 --refutes--> v0d\n- v10 --refutes--> v03\n- v05 --supports--> v04\n- v01 --supports--> v27\n- v0b --refutes--> v14\n- v22 --refutes--> v13\n- v2f --supports--> v2c\n- v19 --refutes--> v0f\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv1f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90d924b2e847e13cf6bbc19f14ac73688f6b6e9f6d313b3dae4e37b3685a2d44","schema":"livegraph.boolean-propagation.v1","spec_hash":"9ce5bfc45b3a6362702b4784e857dfc12dce81d2a4a3612e25500745a841b27f","spec_json":"{\"edges\":[{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-b541758a4774628c63ff","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v28 --refutes--> v2b\n- v22 --supports--> v17\n- v1c --refutes--> v2f\n- v1b --supports--> v27\n- v30 --refutes--> v0d\n- v28 --refutes--> v21\n- v0e --supports--> v05\n- v2e --supports--> v0c\n- v0f --supports--> v1e\n- v26 --refutes--> v20\n- v21 --refutes--> v19\n- v24 --supports--> v22\n- v01 --supports--> v0b\n- v0e --refutes--> v00\n- v16 --refutes--> v09\n- v16 --supports--> v1f\n- v12 --refutes--> v16\n- v21 --supports--> v13\n- v23 --supports--> v30\n- v22 --refutes--> v1d\n- v05 --supports--> v2e\n- v1e --refutes--> v0a\n- v0c --supports--> v2a\n- v09 --refutes--> v1c\n- v2d --refutes--> v08\n- v06 --supports--> v26\n- v20 --supports--> v2c\n- v20 --supports--> v01\n- v08 --refutes--> v06\n- v0a --supports--> v2d\n- v17 --refutes--> v25\n- v25 --supports--> v23\n- v0b --refutes--> v24\n- v28 --supports--> v18\n- v24 --supports--> v02\n- v2a --refutes--> v11\n- v0d --supports--> v28\n- v11 --supports--> v0f\n- v1b --refutes--> v07\n- v2a --refutes--> v15\n- v17 --supports--> v29\n- v20 --refutes--> v03\n- v2e --refutes--> v04\n- v1a --refutes--> v12\n- v10 --refutes--> v1b\n- v1b --supports--> v1a\n- v21 --supports--> v10\n- v2e --supports--> v14\n\n[VERIFIED ROOT]\nv0e = FALSE\n\n[TARGET]\nv2f\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"94cbafaf421b7f6c374c1531a00ec7123d032b093906a749ea91438ac9166e42","schema":"livegraph.boolean-propagation.v1","spec_hash":"b541758a4774628c63ffd65544ee3038a1165df8ec27d98ac441c12b850faf33","spec_json":"{\"edges\":[{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-055a6d3b3e8745ee52cb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2e --supports--> v1e\n- v23 --refutes--> v02\n- v22 --supports--> v2e\n- v03 --refutes--> v2d\n- v1a --refutes--> v1f\n- v0d --supports--> v01\n- v03 --refutes--> v22\n- v0e --supports--> v12\n- v2b --supports--> v11\n- v0b --refutes--> v0c\n- v03 --supports--> v26\n- v04 --refutes--> v1a\n- v15 --refutes--> v09\n- v15 --supports--> v2c\n- v2f --supports--> v1b\n- v21 --supports--> v20\n- v19 --supports--> v0b\n- v09 --supports--> v16\n- v0b --supports--> v15\n- v10 --refutes--> v05\n- v01 --supports--> v0f\n- v18 --supports--> v06\n- v2b --refutes--> v0d\n- v02 --supports--> v21\n- v02 --refutes--> v0a\n- v24 --refutes--> v04\n- v12 --supports--> v08\n- v15 --supports--> v29\n- v12 --refutes--> v1d\n- v30 --supports--> v2b\n- v1a --supports--> v17\n- v22 --refutes--> v13\n- v2f --refutes--> v10\n- v1d --supports--> v14\n- v05 --supports--> v19\n- v2a --supports--> v23\n- v0f --refutes--> v25\n- v06 --supports--> v03\n- v2b --supports--> v27\n- v19 --supports--> v28\n- v2e --refutes--> v2f\n- v1d --supports--> v30\n- v1c --refutes--> v0e\n- v25 --refutes--> v2a\n- v09 --refutes--> v00\n- v17 --supports--> v1c\n- v20 --supports--> v18\n- v07 --supports--> v24\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":"6a50bc3b99c12c8e2044d42a65e3710ba782987c3e0358a9ea8b044c52f69381","schema":"livegraph.boolean-propagation.v1","spec_hash":"055a6d3b3e8745ee52cb939d2db95093ab618eec5a9bf3137f647778022d4692","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-e79ed9b2308f0d9debf5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v11 --supports--> v30\n- v0e --supports--> v29\n- v25 --supports--> v11\n- v05 --supports--> v17\n- v2b --refutes--> v07\n- v12 --refutes--> v04\n- v00 --refutes--> v0f\n- v0a --refutes--> v1a\n- v12 --supports--> v14\n- v11 --refutes--> v09\n- v20 --refutes--> v0c\n- v30 --supports--> v27\n- v0f --refutes--> v18\n- v06 --refutes--> v24\n- v16 --supports--> v2f\n- v0e --refutes--> v2d\n- v0e --supports--> v28\n- v30 --refutes--> v21\n- v1b --supports--> v19\n- v04 --supports--> v0a\n- v1c --refutes--> v25\n- v1a --supports--> v0e\n- v2b --supports--> v1d\n- v1a --supports--> v22\n- v18 --supports--> v16\n- v24 --supports--> v12\n- v25 --refutes--> v0b\n- v20 --refutes--> v05\n- v29 --supports--> v1b\n- v1b --refutes--> v26\n- v15 --supports--> v00\n- v02 --refutes--> v1e\n- v08 --refutes--> v02\n- v2c --refutes--> v20\n- v23 --refutes--> v2a\n- v23 --refutes--> v10\n- v05 --supports--> v08\n- v06 --supports--> v0d\n- v1e --refutes--> v03\n- v2f --refutes--> v06\n- v15 --refutes--> v2e\n- v1b --refutes--> v23\n- v10 --refutes--> v1c\n- v2b --supports--> v1f\n- v27 --refutes--> v01\n- v03 --supports--> v2b\n- v13 --refutes--> v2c\n\n[VERIFIED ROOT]\nv13 = 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":"cbec94c2ce91bcc262aaa738c8b9b1694a1722f1c6bfff992d08ec3a4375b3cc","schema":"livegraph.boolean-propagation.v1","spec_hash":"e79ed9b2308f0d9debf52f17d87d67d74f7fa60d3b07c4ea40016688971e4c3e","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-7d683605b06a36e348b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --refutes--> v23\n- v29 --refutes--> v13\n- v25 --refutes--> v05\n- v0c --supports--> v0e\n- v24 --supports--> v11\n- v23 --refutes--> v01\n- v29 --refutes--> v03\n- v26 --refutes--> v00\n- v0c --refutes--> v10\n- v0d --refutes--> v04\n- v13 --supports--> v1f\n- v0a --refutes--> v1d\n- v2e --supports--> v09\n- v12 --supports--> v16\n- v19 --supports--> v25\n- v10 --refutes--> v2a\n- v17 --supports--> v24\n- v25 --supports--> v14\n- v15 --supports--> v19\n- v12 --refutes--> v1a\n- v01 --refutes--> v0d\n- v00 --refutes--> v1c\n- v1a --supports--> v0b\n- v11 --supports--> v26\n- v0d --refutes--> v02\n- v2f --supports--> v2b\n- v07 --refutes--> v1e\n- v20 --refutes--> v17\n- v24 --refutes--> v21\n- v2e --refutes--> v2d\n- v0a --supports--> v2c\n- v30 --supports--> v08\n- v25 --supports--> v20\n- v2a --refutes--> v0f\n- v2c --refutes--> v07\n- v0b --refutes--> v0c\n- v2b --refutes--> v15\n- v1f --refutes--> v18\n- v23 --refutes--> v06\n- v01 --supports--> v1b\n- v0c --supports--> v22\n- v0f --supports--> v2e\n- v1c --refutes--> v28\n- v15 --refutes--> v27\n- v30 --refutes--> v2f\n- v07 --supports--> v12\n- v02 --refutes--> v30\n- v28 --refutes--> v0a\n\n[VERIFIED ROOT]\nv29 = 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":"791933fa208f010082f33f2d6dae812ddbabd4eb6f21cb49de1d26c297d0530a","schema":"livegraph.boolean-propagation.v1","spec_hash":"7d683605b06a36e348b5df5095336b5897e4317e56b5c301f7ba96d13cc2ed7b","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v29\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-dbc8bdaeb47addfde1e5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v26 --refutes--> v19\n- v1a --supports--> v16\n- v0c --supports--> v2b\n- v2a --supports--> v07\n- v04 --refutes--> v20\n- v0a --refutes--> v10\n- v20 --supports--> v09\n- v2d --supports--> v24\n- v02 --supports--> v26\n- v20 --supports--> v2e\n- v29 --supports--> v12\n- v17 --refutes--> v28\n- v29 --supports--> v1d\n- v16 --refutes--> v2f\n- v28 --supports--> v06\n- v2f --supports--> v27\n- v03 --supports--> v0f\n- v1b --refutes--> v2d\n- v25 --supports--> v01\n- v22 --refutes--> v0d\n- v13 --supports--> v0a\n- v0c --refutes--> v14\n- v2c --refutes--> v1a\n- v15 --refutes--> v02\n- v10 --refutes--> v0e\n- v1c --refutes--> v0c\n- v28 --supports--> v2c\n- v0c --refutes--> v18\n- v22 --refutes--> v17\n- v03 --refutes--> v22\n- v19 --refutes--> v1f\n- v1b --supports--> v1e\n- v25 --supports--> v00\n- v27 --supports--> v13\n- v20 --refutes--> v30\n- v10 --refutes--> v04\n- v0c --refutes--> v0b\n- v2e --refutes--> v1b\n- v1d --refutes--> v21\n- v00 --refutes--> v23\n- v02 --refutes--> v05\n- v0b --supports--> v03\n- v20 --refutes--> v11\n- v23 --refutes--> v1c\n- v24 --refutes--> v29\n- v2a --refutes--> v25\n- v1c --refutes--> v08\n- v1f --refutes--> v2a\n\n[VERIFIED ROOT]\nv15 = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d157e83424bf9a95cd9a21668e44070ae8c4d558a320ee8e0825fe9522b76215","schema":"livegraph.boolean-propagation.v1","spec_hash":"dbc8bdaeb47addfde1e5806e472241ccd8a01ea13c96e3ea1a6cb30e7ba4d282","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v15\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-e031fd36b4493d06015d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1a\n- v1e --supports--> v19\n- v1d --supports--> v11\n- v28 --supports--> v21\n- v17 --refutes--> v03\n- v1e --supports--> v2a\n- v26 --refutes--> v0c\n- v0e --supports--> v1d\n- v17 --refutes--> v0d\n- v0c --supports--> v29\n- v2e --refutes--> v2f\n- v08 --refutes--> v18\n- v25 --refutes--> v01\n- v01 --refutes--> v17\n- v30 --refutes--> v08\n- v1b --supports--> v0e\n- v0f --supports--> v25\n- v18 --supports--> v00\n- v30 --supports--> v04\n- v07 --supports--> v24\n- v16 --refutes--> v1e\n- v12 --refutes--> v30\n- v25 --supports--> v1c\n- v10 --supports--> v06\n- v22 --refutes--> v13\n- v10 --supports--> v22\n- v2c --refutes--> v0b\n- v0e --supports--> v15\n- v2f --refutes--> v23\n- v27 --refutes--> v2d\n- v22 --refutes--> v28\n- v2a --supports--> v27\n- v02 --refutes--> v0a\n- v07 --refutes--> v10\n- v11 --supports--> v02\n- v2b --refutes--> v07\n- v00 --supports--> v0f\n- v2f --supports--> v12\n- v28 --supports--> v14\n- v0a --supports--> v16\n- v2c --refutes--> v2e\n- v02 --supports--> v20\n- v07 --supports--> v1f\n- v29 --supports--> v05\n- v2b --refutes--> v09\n- v27 --supports--> v26\n- v21 --refutes--> v2c\n- v0d --refutes--> v1b\n\n[VERIFIED ROOT]\nv2b = 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":"99f788ef68d7c3d20c798d2193230b4325d58ea4a3949096e335a165f5b881f7","schema":"livegraph.boolean-propagation.v1","spec_hash":"e031fd36b4493d06015dc0f05a74fcbeda7184ef86d17b289c05bb5794c0a3fd","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-b28b3bcaa842c3c9af46","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v16\n- v2b --supports--> v09\n- v17 --supports--> v0b\n- v05 --supports--> v15\n- v03 --refutes--> v02\n- v0b --refutes--> v00\n- v16 --refutes--> v25\n- v02 --refutes--> v1a\n- v0c --refutes--> v2e\n- v25 --supports--> v2b\n- v10 --refutes--> v2c\n- v0e --refutes--> v0c\n- v2c --supports--> v1f\n- v1d --supports--> v0a\n- v2f --supports--> v1c\n- v0f --supports--> v22\n- v15 --refutes--> v2d\n- v13 --refutes--> v08\n- v2d --refutes--> v17\n- v05 --refutes--> v14\n- v2b --refutes--> v2f\n- v20 --refutes--> v0e\n- v1d --supports--> v06\n- v2a --supports--> v03\n- v07 --supports--> v21\n- v0c --supports--> v0f\n- v0e --supports--> v1e\n- v0b --refutes--> v23\n- v30 --refutes--> v12\n- v0a --supports--> v19\n- v2c --supports--> v04\n- v0a --supports--> v01\n- v29 --refutes--> v0d\n- v29 --supports--> v1b\n- v00 --refutes--> v1d\n- v02 --supports--> v20\n- v19 --supports--> v07\n- v2f --supports--> v18\n- v27 --supports--> v29\n- v13 --supports--> v27\n- v28 --refutes--> v2a\n- v1f --refutes--> v28\n- v0c --supports--> v24\n- v0d --supports--> v10\n- v12 --refutes--> v26\n- v07 --supports--> v13\n- v1c --supports--> v05\n- v0e --refutes--> v11\n\n[VERIFIED ROOT]\nv30 = TRUE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"636b61c3f0dade666ce82a96ce6e47598f22389677865f8b7ed24c4cb350395c","schema":"livegraph.boolean-propagation.v1","spec_hash":"b28b3bcaa842c3c9af46b07afd985cc4beca1520888d27152f334e5d5b9ca527","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v30\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-1e4bc4329ab389db513e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2b --supports--> v01\n- v0d --supports--> v16\n- v22 --supports--> v08\n- v28 --supports--> v0e\n- v00 --refutes--> v10\n- v20 --refutes--> v22\n- v03 --refutes--> v12\n- v0a --refutes--> v2d\n- v08 --supports--> v07\n- v11 --supports--> v24\n- v25 --supports--> v1e\n- v1a --supports--> v28\n- v04 --refutes--> v0f\n- v21 --supports--> v04\n- v29 --refutes--> v09\n- v1c --refutes--> v11\n- v00 --refutes--> v1b\n- v27 --supports--> v13\n- v2a --refutes--> v15\n- v27 --supports--> v2f\n- v0e --supports--> v17\n- v07 --supports--> v1f\n- v1f --supports--> v0c\n- v18 --supports--> v03\n- v12 --supports--> v2c\n- v15 --refutes--> v25\n- v2f --refutes--> v2b\n- v1f --refutes--> v02\n- v1e --refutes--> v05\n- v01 --refutes--> v0b\n- v25 --refutes--> v19\n- v0e --supports--> v30\n- v0c --supports--> v00\n- v2b --refutes--> v23\n- v1e --supports--> v1d\n- v1d --supports--> v18\n- v00 --refutes--> v1c\n- v12 --supports--> v2e\n- v2d --refutes--> v29\n- v29 --supports--> v2a\n- v0d --supports--> v27\n- v21 --refutes--> v14\n- v0e --supports--> v0a\n- v24 --refutes--> v21\n- v04 --refutes--> v1a\n- v2f --supports--> v06\n- v0b --refutes--> v20\n- v01 --supports--> v26\n\n[VERIFIED ROOT]\nv0d = TRUE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"78ff8dbc9148ecb49ef56e335555aeb4baf02d3cba8a8ef7eb3a5613f163e5af","schema":"livegraph.boolean-propagation.v1","spec_hash":"1e4bc4329ab389db513e69667d872a2ca972aaa236c480d1b4b52772c01e0284","spec_json":"{\"edges\":[{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-e2ea67c35032c97f628a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v2e --supports--> v1d\n- v07 --refutes--> v25\n- v1e --supports--> v2c\n- v29 --refutes--> v0c\n- v0e --supports--> v10\n- v25 --refutes--> v1a\n- v13 --refutes--> v07\n- v1e --refutes--> v0d\n- v2a --refutes--> v17\n- v23 --supports--> v1b\n- v18 --supports--> v19\n- v00 --refutes--> v14\n- v1c --refutes--> v0b\n- v1e --refutes--> v03\n- v15 --refutes--> v0e\n- v1b --refutes--> v20\n- v2c --supports--> v2f\n- v1e --supports--> v05\n- v15 --refutes--> v27\n- v10 --supports--> v00\n- v01 --refutes--> v18\n- v04 --supports--> v2b\n- v02 --refutes--> v30\n- v24 --supports--> v01\n- v1b --supports--> v2d\n- v2a --refutes--> v22\n- v25 --supports--> v1f\n- v17 --refutes--> v1c\n- v13 --refutes--> v28\n- v00 --refutes--> v2a\n- v16 --supports--> v04\n- v0b --supports--> v2e\n- v12 --supports--> v29\n- v30 --supports--> v11\n- v0c --supports--> v06\n- v2e --refutes--> v0f\n- v08 --refutes--> v24\n- v19 --refutes--> v1e\n- v29 --refutes--> v21\n- v1b --refutes--> v13\n- v25 --refutes--> v26\n- v2b --supports--> v23\n- v2e --refutes--> v08\n- v11 --refutes--> v0a\n- v11 --supports--> v15\n- v1a --refutes--> v02\n- v2f --refutes--> v12\n\n[VERIFIED ROOT]\nv16 = 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":"4f80bd37c6b2beedcdcb59d5a3bac2a07e6689354b3ed22f67679b22f22a22b2","schema":"livegraph.boolean-propagation.v1","spec_hash":"e2ea67c35032c97f628a9afacf690428f1d886bac99847e055662815f1cfbff9","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v06\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-af44d4af47b9d84c49dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v28\n- v1e --refutes--> v2e\n- v2b --refutes--> v15\n- v2f --refutes--> v18\n- v24 --refutes--> v29\n- v0f --refutes--> v30\n- v04 --supports--> v06\n- v0f --refutes--> v16\n- v1b --refutes--> v21\n- v19 --supports--> v0e\n- v26 --refutes--> v04\n- v01 --supports--> v0c\n- v18 --supports--> v10\n- v09 --supports--> v0a\n- v23 --supports--> v2a\n- v2a --supports--> v1a\n- v27 --supports--> v02\n- v00 --supports--> v09\n- v14 --refutes--> v13\n- v0c --refutes--> v14\n- v21 --refutes--> v08\n- v1c --supports--> v11\n- v20 --supports--> v2b\n- v09 --supports--> v05\n- v1f --refutes--> v00\n- v2d --refutes--> v26\n- v2a --refutes--> v25\n- v01 --supports--> v07\n- v24 --refutes--> v1e\n- v0d --supports--> v24\n- v04 --refutes--> v27\n- v11 --supports--> v1f\n- v27 --supports--> v2f\n- v03 --supports--> v01\n- v0d --refutes--> v1d\n- v0c --supports--> v17\n- v0d --supports--> v2c\n- v12 --supports--> v23\n- v15 --supports--> v1b\n- v2a --refutes--> v2d\n- v19 --supports--> v0f\n- v1e --supports--> v22\n- v05 --supports--> v0d\n- v13 --supports--> v1c\n- v16 --refutes--> v20\n- v18 --refutes--> v19\n- v08 --supports--> v03\n- v23 --refutes--> v0b\n\n[VERIFIED ROOT]\nv12 = FALSE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"6f8f9ca8fd48148c4ca7b2bd13880ce91b9a18649e14a249f9363222d2a1fa80","schema":"livegraph.boolean-propagation.v1","spec_hash":"af44d4af47b9d84c49dc1469a6e5e92abe99d80813ba4d19963ad9b17b3c34d8","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v12\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-7df7798049033ae7addc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v21 --supports--> v16\n- v02 --refutes--> v0a\n- v1a --refutes--> v1d\n- v05 --supports--> v07\n- v18 --supports--> v25\n- v14 --refutes--> v10\n- v1d --supports--> v2f\n- v30 --supports--> v1a\n- v1d --supports--> v12\n- v2c --supports--> v30\n- v2f --supports--> v21\n- v26 --supports--> v06\n- v09 --supports--> v24\n- v17 --refutes--> v13\n- v1d --refutes--> v15\n- v22 --refutes--> v1e\n- v10 --refutes--> v1f\n- v06 --refutes--> v1b\n- v03 --refutes--> v2a\n- v09 --refutes--> v22\n- v1e --refutes--> v08\n- v0b --refutes--> v1c\n- v13 --supports--> v0e\n- v10 --supports--> v0b\n- v03 --refutes--> v01\n- v03 --refutes--> v11\n- v0f --refutes--> v05\n- v16 --refutes--> v17\n- v17 --refutes--> v04\n- v05 --refutes--> v23\n- v1c --supports--> v03\n- v28 --supports--> v0f\n- v1b --refutes--> v20\n- v0c --supports--> v28\n- v02 --refutes--> v2c\n- v07 --refutes--> v14\n- v13 --refutes--> v00\n- v2e --refutes--> v02\n- v00 --refutes--> v26\n- v07 --refutes--> v2d\n- v1e --supports--> v2e\n- v20 --refutes--> v18\n- v2e --supports--> v19\n- v2e --supports--> v0d\n- v25 --refutes--> v27\n- v11 --supports--> v09\n- v2e --refutes--> v2b\n- v1d --supports--> v29\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"02d823957e09204d6d175ee18cd9e47042e7ec2cd29d8990deeda568f6edfd36","schema":"livegraph.boolean-propagation.v1","spec_hash":"7df7798049033ae7addc296ac22a6a256fc206837052795e9e26dcdc3ac8355e","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-5e2a16a470f0ff211fbb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1c\n- v05 --refutes--> v2f\n- v13 --refutes--> v01\n- v27 --refutes--> v20\n- v12 --refutes--> v27\n- v20 --supports--> v03\n- v03 --supports--> v04\n- v13 --refutes--> v0e\n- v30 --supports--> v09\n- v07 --supports--> v0c\n- v15 --refutes--> v23\n- v0b --supports--> v17\n- v15 --refutes--> v2d\n- v01 --supports--> v16\n- v2d --refutes--> v26\n- v18 --supports--> v19\n- v2e --supports--> v0d\n- v12 --refutes--> v1b\n- v04 --supports--> v21\n- v27 --refutes--> v1f\n- v0a --refutes--> v12\n- v2e --supports--> v18\n- v2e --refutes--> v00\n- v0c --refutes--> v10\n- v09 --refutes--> v2c\n- v2b --supports--> v13\n- v1a --refutes--> v06\n- v13 --supports--> v29\n- v1c --supports--> v24\n- v28 --supports--> v02\n- v18 --supports--> v2b\n- v17 --refutes--> v2e\n- v02 --supports--> v11\n- v09 --supports--> v05\n- v14 --supports--> v2a\n- v05 --refutes--> v14\n- v05 --refutes--> v08\n- v0c --supports--> v15\n- v1e --refutes--> v30\n- v24 --supports--> v1e\n- v0c --refutes--> v0f\n- v21 --refutes--> v0b\n- v2e --supports--> v1d\n- v26 --refutes--> v1a\n- v2e --refutes--> v22\n- v25 --refutes--> v07\n- v02 --supports--> v0a\n- v06 --refutes--> v28\n\n[VERIFIED ROOT]\nv25 = TRUE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4e8b7e17457548a07165e4890a149fe2db90151f7e980720f892572135732fdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"5e2a16a470f0ff211fbbd373a54023557858210351173dc30e67976680ce6bf1","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-6671ca9dac7b735af022","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v2b\n- v2e --refutes--> v2a\n- v24 --supports--> v29\n- v1c --refutes--> v11\n- v16 --refutes--> v22\n- v0e --supports--> v1d\n- v2c --refutes--> v0c\n- v22 --refutes--> v14\n- v28 --refutes--> v13\n- v21 --refutes--> v1a\n- v2f --refutes--> v24\n- v0c --supports--> v27\n- v03 --supports--> v2c\n- v09 --supports--> v06\n- v24 --supports--> v1f\n- v02 --refutes--> v25\n- v00 --supports--> v16\n- v1f --refutes--> v23\n- v12 --supports--> v0b\n- v06 --refutes--> v03\n- v01 --supports--> v18\n- v01 --supports--> v15\n- v13 --refutes--> v2d\n- v17 --refutes--> v0f\n- v22 --refutes--> v26\n- v27 --refutes--> v07\n- v12 --supports--> v01\n- v22 --refutes--> v0e\n- v2d --supports--> v04\n- v25 --supports--> v2f\n- v27 --refutes--> v28\n- v09 --supports--> v20\n- v2e --refutes--> v1e\n- v02 --refutes--> v30\n- v06 --supports--> v10\n- v1d --supports--> v08\n- v2d --refutes--> v02\n- v0a --supports--> v2e\n- v1b --supports--> v09\n- v2b --supports--> v17\n- v18 --supports--> v05\n- v11 --refutes--> v1b\n- v2e --supports--> v21\n- v21 --supports--> v1c\n- v23 --refutes--> v12\n- v17 --refutes--> v19\n- v02 --refutes--> v0d\n- v1d --supports--> v0a\n\n[VERIFIED ROOT]\nv00 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2baf7b398f7225f2c16a5857514c60c4f487b41ffa7aa32c9aa3b04a97cdc29","schema":"livegraph.boolean-propagation.v1","spec_hash":"6671ca9dac7b735af022ecb2b3df60268b349bbdd230dbe1d4c8943ae04730b0","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v00\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-98a9c20038a92976697b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v0c\n- v25 --supports--> v19\n- v13 --refutes--> v2f\n- v1f --refutes--> v20\n- v07 --refutes--> v23\n- v26 --refutes--> v22\n- v1c --refutes--> v15\n- v2a --supports--> v24\n- v23 --supports--> v2e\n- v1c --supports--> v28\n- v03 --supports--> v1b\n- v05 --refutes--> v29\n- v02 --refutes--> v1c\n- v2d --refutes--> v0b\n- v13 --supports--> v2b\n- v04 --supports--> v1f\n- v06 --supports--> v00\n- v0d --supports--> v01\n- v1c --refutes--> v26\n- v00 --refutes--> v30\n- v0e --supports--> v2a\n- v0c --supports--> v16\n- v17 --supports--> v2d\n- v01 --refutes--> v11\n- v0c --refutes--> v18\n- v19 --refutes--> v17\n- v0a --refutes--> v0e\n- v06 --supports--> v09\n- v14 --refutes--> v1d\n- v26 --refutes--> v27\n- v03 --supports--> v02\n- v30 --supports--> v1e\n- v10 --supports--> v06\n- v2d --supports--> v2c\n- v13 --supports--> v05\n- v2a --refutes--> v25\n- v30 --supports--> v0d\n- v1d --supports--> v08\n- v2c --refutes--> v04\n- v16 --supports--> v13\n- v01 --refutes--> v12\n- v27 --refutes--> v0a\n- v2e --supports--> v10\n- v1d --refutes--> v03\n- v20 --supports--> v0f\n- v02 --supports--> v21\n- v05 --supports--> v14\n- v01 --supports--> v1a\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":"eef42dc2e09b39cb68c45e037bf4ca9ca30e47571748684a5e8666b86ff0716d","schema":"livegraph.boolean-propagation.v1","spec_hash":"98a9c20038a92976697b68d5c4fb7c83d0bf6ecdbf4d368a1260fb6db98840c8","spec_json":"{\"edges\":[{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-6d3575831f316ab6025d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2f --supports--> v12\n- v06 --refutes--> v2e\n- v19 --refutes--> v05\n- v0f --supports--> v2c\n- v0e --refutes--> v26\n- v30 --refutes--> v07\n- v06 --refutes--> v11\n- v29 --supports--> v22\n- v0c --refutes--> v25\n- v27 --supports--> v18\n- v25 --refutes--> v29\n- v0e --supports--> v24\n- v2c --supports--> v1e\n- v2a --refutes--> v0b\n- v29 --refutes--> v30\n- v07 --refutes--> v1c\n- v08 --refutes--> v1f\n- v08 --supports--> v15\n- v21 --refutes--> v04\n- v2a --refutes--> v13\n- v18 --refutes--> v28\n- v14 --supports--> v2a\n- v2d --supports--> v09\n- v13 --refutes--> v10\n- v23 --supports--> v19\n- v2d --supports--> v06\n- v1f --refutes--> v2f\n- v0e --refutes--> v2b\n- v06 --refutes--> v0d\n- v25 --refutes--> v03\n- v02 --supports--> v0f\n- v1b --supports--> v08\n- v24 --refutes--> v02\n- v17 --refutes--> v20\n- v24 --supports--> v00\n- v1d --refutes--> v01\n- v16 --supports--> v14\n- v1c --supports--> v1b\n- v20 --refutes--> v21\n- v2f --supports--> v2d\n- v27 --refutes--> v0a\n- v28 --supports--> v16\n- v0d --supports--> v1d\n- v1d --supports--> v17\n- v21 --supports--> v1a\n- v04 --refutes--> v23\n- v13 --refutes--> v0e\n- v19 --refutes--> v27\n\n[VERIFIED ROOT]\nv0c = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"49dde8ac5b662455b0970ab5d63003e5ccb22ebf77258ac3c426cd9b614f1518","schema":"livegraph.boolean-propagation.v1","spec_hash":"6d3575831f316ab6025dcc6a1e7e67e24e75d9b44a95c14d60528e71699de473","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-3fcb6ce5dac990ee4192","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1a --supports--> v04\n- v17 --refutes--> v11\n- v20 --supports--> v16\n- v1b --refutes--> v12\n- v17 --refutes--> v0d\n- v07 --supports--> v23\n- v14 --refutes--> v08\n- v2e --supports--> v1e\n- v09 --supports--> v2c\n- v28 --refutes--> v2f\n- v2c --supports--> v28\n- v12 --supports--> v2d\n- v29 --supports--> v13\n- v26 --refutes--> v19\n- v16 --refutes--> v0c\n- v28 --supports--> v14\n- v17 --refutes--> v21\n- v06 --supports--> v17\n- v12 --supports--> v1f\n- v13 --refutes--> v01\n- v04 --refutes--> v00\n- v08 --supports--> v0a\n- v16 --refutes--> v02\n- v2d --refutes--> v03\n- v08 --refutes--> v29\n- v0f --supports--> v2b\n- v16 --refutes--> v15\n- v20 --refutes--> v18\n- v27 --refutes--> v1a\n- v2e --supports--> v0f\n- v26 --refutes--> v05\n- v19 --refutes--> v1b\n- v2b --supports--> v0b\n- v2d --supports--> v20\n- v0b --supports--> v26\n- v29 --refutes--> v1d\n- v27 --refutes--> v22\n- v10 --supports--> v24\n- v09 --refutes--> v0e\n- v2a --supports--> v09\n- v07 --refutes--> v1c\n- v0c --supports--> v10\n- v24 --supports--> v30\n- v26 --supports--> v25\n- v30 --refutes--> v2a\n- v23 --supports--> v2e\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv27 = 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":"1cc02f8d7ba59e16e8ac8fd4c0e9facd71a631bc5f4f872be748c71af291a845","schema":"livegraph.boolean-propagation.v1","spec_hash":"3fcb6ce5dac990ee41924df1015acd81a43ec596226be4ff071bae2686921f83","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v27\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-a7616454c9035ae9e83c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --refutes--> v2e\n- v2e --refutes--> v1c\n- v1d --refutes--> v18\n- v28 --refutes--> v2b\n- v0f --supports--> v10\n- v2e --supports--> v0b\n- v05 --supports--> v04\n- v29 --refutes--> v07\n- v15 --supports--> v14\n- v02 --refutes--> v11\n- v24 --supports--> v1e\n- v14 --supports--> v01\n- v0d --refutes--> v27\n- v1c --supports--> v19\n- v02 --refutes--> v2c\n- v1f --supports--> v17\n- v0f --supports--> v20\n- v0a --supports--> v12\n- v27 --supports--> v0a\n- v29 --supports--> v03\n- v26 --refutes--> v0f\n- v04 --refutes--> v1b\n- v28 --supports--> v2d\n- v02 --supports--> v2f\n- v0c --supports--> v09\n- v12 --supports--> v24\n- v17 --supports--> v02\n- v01 --refutes--> v16\n- v2a --refutes--> v0c\n- v22 --supports--> v1d\n- v1c --refutes--> v08\n- v1d --refutes--> v2a\n- v10 --refutes--> v1a\n- v25 --refutes--> v22\n- v12 --supports--> v0e\n- v1a --refutes--> v05\n- v1b --refutes--> v1f\n- v03 --supports--> v13\n- v16 --supports--> v28\n- v08 --supports--> v29\n- v1c --supports--> v00\n- v2a --supports--> v23\n- v2a --refutes--> v30\n- v29 --refutes--> v06\n- v05 --supports--> v21\n- v13 --refutes--> v26\n- v0c --supports--> v15\n- v2f --supports--> v25\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"aab2247248270d585c07f9e9268f8576823d7e0f6964ada298d333787db38ad8","schema":"livegraph.boolean-propagation.v1","spec_hash":"a7616454c9035ae9e83c725917a71872d94a274668002ba14835ba7d6b1ab2cc","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0d\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-de14e8817b6eb49d4cfb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1e\n- v28 --supports--> v23\n- v26 --supports--> v0d\n- v13 --refutes--> v01\n- v14 --supports--> v12\n- v10 --refutes--> v1b\n- v00 --refutes--> v09\n- v2d --refutes--> v20\n- v2b --supports--> v2a\n- v1d --refutes--> v07\n- v1e --supports--> v2b\n- v23 --refutes--> v24\n- v0a --refutes--> v0c\n- v0b --refutes--> v0a\n- v1a --refutes--> v13\n- v1b --supports--> v06\n- v18 --supports--> v21\n- v29 --refutes--> v1f\n- v29 --supports--> v2f\n- v2b --refutes--> v05\n- v0d --supports--> v1a\n- v01 --refutes--> v00\n- v26 --refutes--> v15\n- v09 --supports--> v19\n- v2a --refutes--> v18\n- v19 --refutes--> v16\n- v1a --refutes--> v04\n- v25 --refutes--> v29\n- v0a --refutes--> v26\n- v29 --refutes--> v30\n- v2b --refutes--> v2c\n- v2d --refutes--> v0e\n- v30 --refutes--> v0b\n- v16 --refutes--> v10\n- v18 --refutes--> v25\n- v02 --supports--> v1d\n- v1c --supports--> v02\n- v0e --refutes--> v1c\n- v1b --refutes--> v28\n- v2a --refutes--> v2e\n- v24 --supports--> v2d\n- v14 --refutes--> v11\n- v0e --supports--> v17\n- v07 --supports--> v27\n- v02 --refutes--> v0f\n- v26 --refutes--> v08\n- v18 --supports--> v22\n- v27 --supports--> v03\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"893a679f54906f5d815c7eeb760e2c2511bbd41f7513812edb0122bfd35fb25b","schema":"livegraph.boolean-propagation.v1","spec_hash":"de14e8817b6eb49d4cfbe754c1b9ca9d9f81491cbcf4da1c711408f0557596b3","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v14\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-12561bb28e409a92bac4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v10 --supports--> v08\n- v1e --supports--> v1b\n- v04 --refutes--> v2c\n- v1a --refutes--> v0d\n- v01 --refutes--> v14\n- v25 --supports--> v12\n- v05 --supports--> v2f\n- v1c --refutes--> v05\n- v2c --supports--> v2a\n- v2d --refutes--> v09\n- v16 --supports--> v06\n- v09 --supports--> v2b\n- v1e --refutes--> v30\n- v15 --refutes--> v1a\n- v01 --supports--> v1d\n- v0b --supports--> v0f\n- v08 --refutes--> v11\n- v06 --refutes--> v23\n- v23 --supports--> v17\n- v14 --refutes--> v24\n- v29 --supports--> v0b\n- v0a --supports--> v16\n- v0e --refutes--> v01\n- v29 --supports--> v00\n- v21 --supports--> v0e\n- v0e --supports--> v03\n- v06 --supports--> v0c\n- v1b --refutes--> v1f\n- v19 --refutes--> v2e\n- v11 --supports--> v2d\n- v23 --supports--> v26\n- v04 --refutes--> v22\n- v0f --supports--> v10\n- v23 --supports--> v13\n- v20 --supports--> v1c\n- v0d --supports--> v0a\n- v2d --refutes--> v07\n- v1c --refutes--> v18\n- v17 --refutes--> v19\n- v2a --supports--> v21\n- v05 --refutes--> v1e\n- v2d --refutes--> v02\n- v2b --supports--> v15\n- v14 --supports--> v29\n- v1b --refutes--> v25\n- v16 --refutes--> v27\n- v20 --supports--> v28\n- v12 --refutes--> v04\n\n[VERIFIED ROOT]\nv20 = TRUE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e2c1944c15dfea2b6384610371f7bad4e6c285e8d68902089bbce54b3779d803","schema":"livegraph.boolean-propagation.v1","spec_hash":"12561bb28e409a92bac45ee1c49b6b16d7d181027c109d684dbe95ffe3d5eb0d","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2d\"}],\"hop\":32,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-d03100591c5f26241182","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2b --refutes--> v00\n- v0a --refutes--> v13\n- v27 --refutes--> v08\n- v1a --supports--> v16\n- v08 --refutes--> v0e\n- v19 --refutes--> v06\n- v1a --supports--> v07\n- v1c --supports--> v0d\n- v21 --refutes--> v1b\n- v00 --supports--> v05\n- v26 --refutes--> v1c\n- v2d --refutes--> v19\n- v17 --refutes--> v25\n- v18 --refutes--> v21\n- v20 --supports--> v24\n- v05 --refutes--> v2e\n- v0b --supports--> v14\n- v08 --supports--> v1e\n- v2f --refutes--> v02\n- v24 --refutes--> v15\n- v18 --supports--> v12\n- v2a --refutes--> v04\n- v08 --refutes--> v23\n- v18 --supports--> v1d\n- v07 --supports--> v17\n- v17 --supports--> v18\n- v2f --refutes--> v26\n- v09 --supports--> v30\n- v04 --refutes--> v28\n- v13 --supports--> v11\n- v04 --supports--> v0a\n- v2b --supports--> v01\n- v18 --supports--> v2c\n- v0e --supports--> v2a\n- v2f --refutes--> v1f\n- v30 --refutes--> v20\n- v11 --refutes--> v0b\n- v2e --refutes--> v2d\n- v14 --supports--> v03\n- v0d --supports--> v22\n- v22 --refutes--> v27\n- v13 --refutes--> v29\n- v03 --refutes--> v2b\n- v2a --refutes--> v0f\n- v04 --supports--> v0c\n- v24 --supports--> v2f\n- v06 --refutes--> v1a\n- v0b --refutes--> v10\n\n[VERIFIED ROOT]\nv09 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a8b679589ab2717369fe3dbbc147c02c5363fe1bbee70c1a7327b3baa6e0e4f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"d03100591c5f262411825d3bfe0d7680b9bb61c76d1eb1aa9fa28fe647608a97","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v09\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-418ff6087a7eb9a820b5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2e --supports--> v0e\n- v03 --refutes--> v2c\n- v18 --supports--> v2e\n- v30 --supports--> v1c\n- v1d --supports--> v23\n- v11 --refutes--> v18\n- v10 --refutes--> v09\n- v14 --supports--> v12\n- v2e --refutes--> v19\n- v2c --supports--> v21\n- v1a --supports--> v2a\n- v21 --supports--> v00\n- v00 --supports--> v2f\n- v25 --refutes--> v27\n- v11 --refutes--> v26\n- v02 --supports--> v24\n- v21 --supports--> v28\n- v03 --supports--> v05\n- v0b --supports--> v1f\n- v1e --supports--> v04\n- v2d --refutes--> v1d\n- v25 --supports--> v02\n- v20 --refutes--> v01\n- v23 --supports--> v16\n- v24 --supports--> v20\n- v20 --supports--> v2b\n- v2e --supports--> v29\n- v1c --refutes--> v1a\n- v0d --refutes--> v07\n- v20 --refutes--> v0d\n- v2a --supports--> v11\n- v1b --refutes--> v17\n- v0b --supports--> v03\n- v2d --supports--> v13\n- v2f --refutes--> v2d\n- v0d --refutes--> v1b\n- v1e --refutes--> v30\n- v00 --refutes--> v15\n- v09 --supports--> v0c\n- v29 --refutes--> v0a\n- v17 --refutes--> v10\n- v02 --refutes--> v0f\n- v08 --refutes--> v14\n- v0c --supports--> v1e\n- v2f --refutes--> v22\n- v12 --supports--> v25\n- v0a --supports--> v0b\n- v30 --refutes--> v06\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"9b1fa821a1d5d20a4a711caa7cd88907b8c3ea5a66a0276efb50f53ca8f508e9","schema":"livegraph.boolean-propagation.v1","spec_hash":"418ff6087a7eb9a820b54144bf4249154fba99885171d5fb884f487124a29754","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v08\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-bc8d3b3b6d20b21d2273","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2a --supports--> v1d\n- v2d --refutes--> v0b\n- v16 --supports--> v1e\n- v24 --refutes--> v01\n- v15 --refutes--> v0e\n- v04 --supports--> v02\n- v19 --supports--> v10\n- v1b --supports--> v28\n- v0e --supports--> v1c\n- v0e --refutes--> v21\n- v1a --supports--> v25\n- v1d --supports--> v0c\n- v02 --refutes--> v1a\n- v21 --refutes--> v17\n- v21 --supports--> v1f\n- v11 --supports--> v07\n- v1d --supports--> v13\n- v14 --supports--> v04\n- v09 --supports--> v1b\n- v28 --refutes--> v19\n- v24 --refutes--> v0d\n- v13 --supports--> v16\n- v0b --refutes--> v2f\n- v0f --refutes--> v05\n- v24 --supports--> v0f\n- v0b --refutes--> v15\n- v08 --refutes--> v18\n- v0a --refutes--> v11\n- v1e --refutes--> v20\n- v05 --supports--> v2a\n- v13 --supports--> v22\n- v27 --supports--> v09\n- v16 --refutes--> v2e\n- v27 --refutes--> v29\n- v16 --refutes--> v23\n- v1e --supports--> v00\n- v2d --supports--> v2b\n- v26 --refutes--> v06\n- v1e --supports--> v27\n- v25 --refutes--> v12\n- v06 --supports--> v08\n- v03 --supports--> v26\n- v07 --supports--> v03\n- v02 --refutes--> v2c\n- v1f --refutes--> v14\n- v19 --refutes--> v2d\n- v02 --supports--> v30\n- v12 --refutes--> v0a\n\n[VERIFIED ROOT]\nv24 = 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":"da0b43ba8b09eb3e391c97cdd7b91d4c681c82a7a65021be6cbbf90fb15b702a","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc8d3b3b6d20b21d227357b12dfe0cf51cb55dbdb05ae1888e90a97ae5fe4916","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2d\"}],\"hop\":32,\"root\":\"v24\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-f9dda69325d4c0e95dee","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v17\n- v18 --supports--> v10\n- v22 --supports--> v14\n- v1a --refutes--> v29\n- v22 --supports--> v0d\n- v1e --supports--> v27\n- v26 --refutes--> v18\n- v0e --supports--> v15\n- v2e --supports--> v25\n- v1c --refutes--> v1f\n- v2e --supports--> v1e\n- v1f --refutes--> v0a\n- v20 --refutes--> v2a\n- v07 --refutes--> v16\n- v04 --refutes--> v30\n- v15 --supports--> v2c\n- v30 --supports--> v1a\n- v29 --refutes--> v22\n- v23 --supports--> v19\n- v03 --refutes--> v21\n- v06 --refutes--> v20\n- v2e --refutes--> v0b\n- v1d --refutes--> v12\n- v17 --refutes--> v04\n- v2e --refutes--> v24\n- v1b --supports--> v09\n- v0a --refutes--> v28\n- v27 --refutes--> v0e\n- v2a --refutes--> v2e\n- v1c --refutes--> v13\n- v0f --refutes--> v0c\n- v1d --supports--> v01\n- v23 --supports--> v1d\n- v11 --supports--> v00\n- v09 --refutes--> v03\n- v11 --refutes--> v2f\n- v28 --supports--> v11\n- v09 --supports--> v05\n- v21 --refutes--> v07\n- v2f --supports--> v2b\n- v1a --supports--> v02\n- v14 --refutes--> v1c\n- v03 --refutes--> v2d\n- v10 --refutes--> v0f\n- v1d --refutes--> v06\n- v0f --refutes--> v1b\n- v11 --supports--> v08\n- v2f --supports--> v26\n\n[VERIFIED ROOT]\nv23 = 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":"de14418e68f271d3718f3cfabfe0277041315acfc52865f962b76decc461873e","schema":"livegraph.boolean-propagation.v1","spec_hash":"f9dda69325d4c0e95dee261537c88ca19333265925d00488f74cd81f115a06d1","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-85f72b3a1531ae449049","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --supports--> v03\n- v11 --refutes--> v23\n- v2c --refutes--> v22\n- v26 --refutes--> v04\n- v23 --refutes--> v21\n- v0d --refutes--> v26\n- v26 --refutes--> v27\n- v13 --refutes--> v07\n- v02 --supports--> v05\n- v14 --refutes--> v2c\n- v1e --supports--> v1b\n- v2f --supports--> v28\n- v21 --refutes--> v17\n- v19 --supports--> v0c\n- v0e --refutes--> v18\n- v2e --supports--> v02\n- v2d --supports--> v12\n- v1f --refutes--> v08\n- v2c --refutes--> v29\n- v1e --supports--> v11\n- v17 --refutes--> v2d\n- v1e --refutes--> v0f\n- v1d --refutes--> v0e\n- v2a --refutes--> v13\n- v21 --refutes--> v01\n- v0c --supports--> v2e\n- v20 --supports--> v00\n- v26 --refutes--> v1e\n- v1a --refutes--> v06\n- v12 --refutes--> v09\n- v28 --supports--> v30\n- v16 --supports--> v0d\n- v0c --supports--> v1c\n- v18 --refutes--> v15\n- v17 --supports--> v2b\n- v1e --refutes--> v24\n- v30 --supports--> v10\n- v1d --supports--> v25\n- v30 --supports--> v1d\n- v13 --refutes--> v14\n- v0b --supports--> v19\n- v20 --refutes--> v2a\n- v09 --refutes--> v0b\n- v06 --refutes--> v16\n- v02 --refutes--> v20\n- v08 --supports--> v2f\n- v0a --refutes--> v1a\n- v18 --supports--> v0a\n\n[VERIFIED ROOT]\nv1f = FALSE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"583d1df7379448e07ad540b7e8531df7b0406c3c8d4a814f6b06e16f0d38b5a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"85f72b3a1531ae44904995850a4a00e65f38ccc44e3a48022969eba48c85bf6a","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-4a899a63cf02b0b182fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v25 --supports--> v12\n- v18 --refutes--> v03\n- v2f --supports--> v07\n- v04 --refutes--> v0b\n- v25 --refutes--> v2b\n- v0d --refutes--> v20\n- v29 --refutes--> v24\n- v30 --refutes--> v1c\n- v02 --supports--> v2d\n- v10 --supports--> v2e\n- v16 --supports--> v10\n- v1f --supports--> v06\n- v28 --supports--> v1b\n- v22 --supports--> v1a\n- v30 --refutes--> v2c\n- v14 --refutes--> v1e\n- v07 --refutes--> v0c\n- v28 --supports--> v00\n- v1a --refutes--> v0e\n- v29 --refutes--> v30\n- v30 --refutes--> v26\n- v03 --refutes--> v1f\n- v06 --refutes--> v02\n- v17 --refutes--> v2a\n- v09 --refutes--> v11\n- v01 --supports--> v27\n- v11 --refutes--> v04\n- v23 --supports--> v21\n- v08 --supports--> v28\n- v10 --refutes--> v13\n- v23 --refutes--> v1d\n- v17 --refutes--> v09\n- v2d --supports--> v29\n- v14 --supports--> v22\n- v20 --supports--> v05\n- v06 --refutes--> v0a\n- v16 --supports--> v0f\n- v0b --supports--> v23\n- v1b --refutes--> v18\n- v13 --supports--> v14\n- v0e --refutes--> v01\n- v01 --supports--> v08\n- v07 --refutes--> v0d\n- v05 --refutes--> v16\n- v26 --supports--> v25\n- v13 --supports--> v15\n- v23 --supports--> v2f\n- v1a --refutes--> v19\n\n[VERIFIED ROOT]\nv17 = 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":"48fd3f6a359be83504c5982fc464a7ab8cd40945bd65636122d3b4ddd5b4d116","schema":"livegraph.boolean-propagation.v1","spec_hash":"4a899a63cf02b0b182fb06484d58716a312f5d8d8fbd583174baa859beb1c94d","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-b548743035a8a9b6c470","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2c --refutes--> v00\n- v13 --refutes--> v0b\n- v2f --refutes--> v25\n- v21 --refutes--> v1c\n- v14 --refutes--> v0e\n- v1b --refutes--> v2c\n- v09 --refutes--> v1b\n- v0a --refutes--> v21\n- v08 --supports--> v16\n- v2d --refutes--> v09\n- v1d --refutes--> v30\n- v0e --supports--> v06\n- v2e --refutes--> v04\n- v01 --supports--> v0f\n- v1e --refutes--> v1f\n- v05 --refutes--> v0d\n- v1d --supports--> v02\n- v0e --supports--> v22\n- v06 --refutes--> v2e\n- v24 --supports--> v17\n- v26 --supports--> v1a\n- v02 --supports--> v23\n- v0c --refutes--> v19\n- v0b --refutes--> v29\n- v29 --supports--> v27\n- v1c --refutes--> v2f\n- v24 --supports--> v1d\n- v07 --refutes--> v15\n- v18 --refutes--> v01\n- v05 --supports--> v2d\n- v1a --refutes--> v08\n- v04 --refutes--> v13\n- v06 --supports--> v2a\n- v1f --refutes--> v0a\n- v2f --supports--> v07\n- v07 --supports--> v10\n- v01 --supports--> v0c\n- v02 --supports--> v2b\n- v0e --supports--> v28\n- v08 --supports--> v24\n- v19 --supports--> v1e\n- v02 --refutes--> v14\n- v29 --supports--> v20\n- v07 --refutes--> v26\n- v1d --supports--> v03\n- v00 --supports--> v12\n- v12 --supports--> v18\n- v24 --refutes--> v11\n\n[VERIFIED ROOT]\nv05 = FALSE\n\n[TARGET]\nv20\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7036c0bc096ebd6cbb730c3bf3436bba0ace63108737aa713973ae905f3967ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"b548743035a8a9b6c47051409bc810bede919f4c7042d798671da016d07d84cd","spec_json":"{\"edges\":[{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v05\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v20\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-ecee229167569ad28876","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2b\n- v08 --refutes--> v02\n- v30 --supports--> v1e\n- v0b --refutes--> v01\n- v17 --supports--> v1a\n- v21 --supports--> v1b\n- v2b --refutes--> v08\n- v2c --refutes--> v07\n- v23 --refutes--> v16\n- v20 --supports--> v00\n- v0a --refutes--> v11\n- v2c --supports--> v0f\n- v05 --refutes--> v29\n- v17 --supports--> v22\n- v25 --supports--> v03\n- v0a --refutes--> v28\n- v15 --supports--> v26\n- v28 --refutes--> v1f\n- v27 --refutes--> v30\n- v0e --supports--> v1c\n- v1f --refutes--> v0d\n- v23 --supports--> v10\n- v1e --supports--> v17\n- v2e --refutes--> v13\n- v1d --supports--> v2c\n- v1b --refutes--> v27\n- v0c --supports--> v12\n- v19 --supports--> v05\n- v03 --supports--> v20\n- v23 --supports--> v0a\n- v21 --refutes--> v2a\n- v0e --supports--> v2d\n- v05 --supports--> v18\n- v0e --supports--> v14\n- v2c --refutes--> v04\n- v13 --refutes--> v25\n- v17 --refutes--> v0b\n- v06 --refutes--> v15\n- v06 --supports--> v24\n- v18 --supports--> v21\n- v20 --refutes--> v19\n- v26 --refutes--> v2e\n- v0d --refutes--> v0e\n- v25 --supports--> v2f\n- v02 --refutes--> v09\n- v07 --supports--> v0c\n- v0c --refutes--> v06\n- v14 --supports--> v1d\n\n[VERIFIED ROOT]\nv23 = 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":"30f175d71763fa67411c71be636efac8be9c1c97819e068add660be7937e2eff","schema":"livegraph.boolean-propagation.v1","spec_hash":"ecee229167569ad2887687dfb1e9e921bd461dcfefe769109323230b256f91f6","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-79a2771336157b86a0dc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v13\n- v30 --supports--> v2e\n- v0e --supports--> v30\n- v2d --supports--> v10\n- v15 --refutes--> v28\n- v22 --refutes--> v29\n- v0d --refutes--> v1c\n- v1a --supports--> v1f\n- v1f --refutes--> v2d\n- v18 --supports--> v04\n- v2f --supports--> v1b\n- v04 --refutes--> v1a\n- v04 --supports--> v27\n- v02 --refutes--> v05\n- v22 --refutes--> v12\n- v23 --refutes--> v09\n- v19 --refutes--> v2a\n- v2b --refutes--> v0c\n- v1c --supports--> v18\n- v17 --refutes--> v26\n- v2b --supports--> v03\n- v25 --supports--> v0e\n- v02 --supports--> v21\n- v16 --supports--> v2b\n- v1e --refutes--> v0a\n- v03 --supports--> v15\n- v01 --refutes--> v20\n- v02 --supports--> v24\n- v06 --refutes--> v1e\n- v1d --supports--> v14\n- v07 --refutes--> v02\n- v12 --supports--> v17\n- v06 --refutes--> v0b\n- v1b --supports--> v23\n- v17 --refutes--> v19\n- v15 --refutes--> v0f\n- v2d --refutes--> v22\n- v00 --refutes--> v06\n- v25 --refutes--> v11\n- v28 --refutes--> v1d\n- v15 --supports--> v2c\n- v01 --supports--> v0d\n- v18 --refutes--> v08\n- v2a --refutes--> v25\n- v1e --supports--> v16\n- v09 --refutes--> v01\n- v2e --supports--> v00\n- v1d --refutes--> v07\n\n[VERIFIED ROOT]\nv2f = FALSE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b43600420128a738cf3865a8ebd6c67747c2f4a1ce01ca28325395f398dcc4ca","schema":"livegraph.boolean-propagation.v1","spec_hash":"79a2771336157b86a0dcbd696c4061c088a3511b79c5a9cf47bc997d2debaae1","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-6a86256216103bdffb35","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v2a --refutes--> v1d\n- v0d --refutes--> v25\n- v2f --refutes--> v03\n- v16 --supports--> v07\n- v2a --supports--> v0d\n- v25 --supports--> v08\n- v16 --refutes--> v15\n- v11 --supports--> v1b\n- v10 --supports--> v28\n- v10 --supports--> v27\n- v20 --refutes--> v11\n- v1f --refutes--> v14\n- v07 --supports--> v1c\n- v28 --supports--> v30\n- v1b --supports--> v2a\n- v05 --supports--> v26\n- v11 --supports--> v22\n- v18 --refutes--> v20\n- v1e --refutes--> v23\n- v2c --refutes--> v12\n- v21 --refutes--> v1a\n- v06 --supports--> v2f\n- v21 --supports--> v0e\n- v2a --refutes--> v02\n- v25 --refutes--> v00\n- v13 --refutes--> v2b\n- v1c --supports--> v17\n- v26 --supports--> v0b\n- v0a --refutes--> v24\n- v10 --supports--> v04\n- v01 --refutes--> v21\n- v2f --supports--> v2d\n- v1e --refutes--> v16\n- v08 --supports--> v06\n- v03 --supports--> v1f\n- v2e --refutes--> v1e\n- v16 --supports--> v19\n- v12 --refutes--> v10\n- v0b --supports--> v13\n- v0c --refutes--> v01\n- v1a --refutes--> v09\n- v29 --supports--> v2e\n- v1c --supports--> v0a\n- v1a --supports--> v2c\n- v28 --refutes--> v05\n- v1e --refutes--> v0f\n- v14 --supports--> v29\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4870bbf3f8551143802d7695ffa8590d0b77f6cc8b080ce87c66b68f58c61903","schema":"livegraph.boolean-propagation.v1","spec_hash":"6a86256216103bdffb354abedf2bbeaf3da78e4e5fbd1dc38e82ac52b125505d","spec_json":"{\"edges\":[{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-c20a34eb5e6099642dab","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v1a\n- v20 --supports--> v22\n- v02 --supports--> v2d\n- v30 --supports--> v21\n- v28 --refutes--> v0d\n- v30 --supports--> v16\n- v15 --refutes--> v1d\n- v16 --supports--> v0c\n- v14 --supports--> v10\n- v19 --refutes--> v02\n- v05 --supports--> v11\n- v12 --supports--> v2c\n- v11 --refutes--> v1e\n- v19 --refutes--> v08\n- v23 --supports--> v0b\n- v0d --supports--> v2b\n- v2f --supports--> v13\n- v07 --refutes--> v2a\n- v20 --supports--> v27\n- v23 --refutes--> v17\n- v10 --refutes--> v18\n- v18 --refutes--> v04\n- v0f --supports--> v2f\n- v00 --refutes--> v07\n- v29 --supports--> v1b\n- v04 --refutes--> v00\n- v1d --refutes--> v30\n- v13 --refutes--> v05\n- v2c --supports--> v09\n- v02 --supports--> v28\n- v19 --refutes--> v1c\n- v27 --supports--> v19\n- v14 --supports--> v0e\n- v16 --supports--> v29\n- v25 --refutes--> v14\n- v03 --supports--> v15\n- v2c --refutes--> v23\n- v04 --refutes--> v0a\n- v2b --supports--> v24\n- v11 --supports--> v25\n- v1f --supports--> v20\n- v18 --refutes--> v2e\n- v10 --supports--> v06\n- v1b --refutes--> v0f\n- v30 --refutes--> v26\n- v17 --refutes--> v1f\n- v1a --supports--> v01\n- v2b --refutes--> v03\n\n[VERIFIED ROOT]\nv12 = 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":"f1f38960608776f73c124151283be50b05eb7f9339283cf935c8a89e676f91ee","schema":"livegraph.boolean-propagation.v1","spec_hash":"c20a34eb5e6099642dab244f18277b596b70200968a5c0e64005343249a11741","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-f3f45e4aaac10d5c63c1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v00\n- v1c --refutes--> v21\n- v0d --refutes--> v30\n- v1d --supports--> v20\n- v28 --supports--> v13\n- v0b --refutes--> v07\n- v02 --refutes--> v14\n- v27 --refutes--> v1b\n- v0b --refutes--> v25\n- v0f --supports--> v1e\n- v20 --supports--> v0f\n- v1a --refutes--> v12\n- v01 --refutes--> v27\n- v00 --refutes--> v15\n- v1b --supports--> v11\n- v05 --supports--> v10\n- v24 --refutes--> v23\n- v30 --supports--> v1d\n- v15 --supports--> v01\n- v0b --refutes--> v0d\n- v30 --refutes--> v19\n- v1b --refutes--> v0a\n- v01 --refutes--> v16\n- v17 --supports--> v06\n- v29 --refutes--> v02\n- v0d --refutes--> v09\n- v1b --supports--> v2a\n- v01 --refutes--> v18\n- v0c --supports--> v17\n- v0f --refutes--> v0c\n- v21 --refutes--> v28\n- v2f --supports--> v04\n- v13 --supports--> v03\n- v17 --refutes--> v24\n- v22 --refutes--> v08\n- v05 --refutes--> v1f\n- v13 --supports--> v29\n- v14 --supports--> v22\n- v00 --refutes--> v2d\n- v12 --supports--> v1c\n- v2c --refutes--> v2b\n- v30 --supports--> v2e\n- v1c --refutes--> v26\n- v11 --supports--> v1a\n- v15 --refutes--> v0e\n- v2b --refutes--> v05\n- v04 --supports--> v0b\n- v10 --refutes--> v2f\n\n[VERIFIED ROOT]\nv2c = 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":"dee9b63115896b56b9bac59c582cc76c817f4df713ad5df067a0d0db787077da","schema":"livegraph.boolean-propagation.v1","spec_hash":"f3f45e4aaac10d5c63c13ff0ac071c722a25750de64434313968d4eb0d3aa320","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-c18874d205f0c02c815b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2e\n- v28 --refutes--> v11\n- v24 --refutes--> v1b\n- v17 --supports--> v03\n- v0b --refutes--> v29\n- v0a --refutes--> v19\n- v21 --refutes--> v24\n- v03 --supports--> v21\n- v0a --refutes--> v08\n- v26 --supports--> v1e\n- v14 --refutes--> v04\n- v2b --refutes--> v18\n- v17 --refutes--> v01\n- v24 --refutes--> v2a\n- v09 --refutes--> v10\n- v00 --supports--> v1d\n- v1f --supports--> v12\n- v27 --supports--> v30\n- v22 --refutes--> v28\n- v06 --supports--> v17\n- v03 --refutes--> v0d\n- v0e --supports--> v15\n- v18 --supports--> v0b\n- v0b --supports--> v23\n- v21 --refutes--> v07\n- v19 --refutes--> v20\n- v2d --supports--> v2f\n- v04 --supports--> v25\n- v29 --supports--> v1f\n- v1c --supports--> v05\n- v11 --refutes--> v2c\n- v0f --supports--> v06\n- v22 --refutes--> v1a\n- v1c --supports--> v26\n- v24 --refutes--> v22\n- v30 --refutes--> v2d\n- v0e --refutes--> v0f\n- v02 --refutes--> v27\n- v19 --supports--> v00\n- v13 --supports--> v16\n- v2b --refutes--> v0c\n- v04 --supports--> v0a\n- v12 --refutes--> v14\n- v26 --refutes--> v0e\n- v2e --refutes--> v02\n- v2d --refutes--> v1c\n- v2c --supports--> v2b\n- v1d --supports--> v13\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv16\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8f64e314fbd3b82f9a9e897fa05dc958c8905fc68219857f18ee1d63419ef9a4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c18874d205f0c02c815be33138e1720675cec3edcad1ca76a31bb10730c371aa","spec_json":"{\"edges\":[{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-77bc6e11f359f8ec5906","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2d --refutes--> v14\n- v27 --refutes--> v2a\n- v00 --supports--> v03\n- v2e --refutes--> v17\n- v29 --refutes--> v20\n- v12 --supports--> v1d\n- v16 --refutes--> v21\n- v0b --supports--> v0a\n- v1e --supports--> v2f\n- v2c --supports--> v06\n- v28 --refutes--> v30\n- v2a --supports--> v22\n- v20 --supports--> v12\n- v0f --refutes--> v23\n- v15 --refutes--> v28\n- v21 --supports--> v08\n- v24 --refutes--> v15\n- v16 --refutes--> v18\n- v08 --supports--> v0b\n- v12 --refutes--> v19\n- v26 --supports--> v0c\n- v11 --refutes--> v1b\n- v17 --refutes--> v27\n- v20 --supports--> v1a\n- v09 --supports--> v25\n- v16 --supports--> v01\n- v11 --supports--> v2e\n- v0f --supports--> v0d\n- v2c --refutes--> v1c\n- v04 --refutes--> v1f\n- v06 --supports--> v2d\n- v26 --refutes--> v04\n- v26 --refutes--> v2b\n- v2f --supports--> v11\n- v16 --supports--> v07\n- v12 --supports--> v2c\n- v22 --refutes--> v02\n- v05 --refutes--> v29\n- v1f --supports--> v24\n- v0b --refutes--> v05\n- v28 --refutes--> v16\n- v10 --supports--> v0f\n- v03 --supports--> v10\n- v14 --supports--> v09\n- v20 --refutes--> v13\n- v02 --supports--> v26\n- v09 --supports--> v00\n- v06 --refutes--> v0e\n\n[VERIFIED ROOT]\nv1e = 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":"391b8938e090926f60cea28558f17a53144db7c7b191dc8983ceb8c35667441a","schema":"livegraph.boolean-propagation.v1","spec_hash":"77bc6e11f359f8ec590685d2e1047526e5cff3f8312e1db4c7329cd7d93a1438","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v1e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-0f6cb994d79d598b0e95","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --refutes--> v07\n- v30 --refutes--> v14\n- v20 --supports--> v18\n- v01 --supports--> v28\n- v18 --refutes--> v03\n- v26 --supports--> v0e\n- v24 --refutes--> v05\n- v10 --supports--> v20\n- v2a --supports--> v01\n- v00 --refutes--> v30\n- v18 --refutes--> v06\n- v00 --refutes--> v04\n- v09 --refutes--> v0a\n- v2b --supports--> v1d\n- v29 --refutes--> v1c\n- v05 --supports--> v10\n- v19 --refutes--> v2e\n- v2d --refutes--> v23\n- v22 --supports--> v15\n- v01 --supports--> v21\n- v18 --supports--> v2a\n- v1c --supports--> v0d\n- v09 --supports--> v2f\n- v10 --refutes--> v13\n- v28 --supports--> v0f\n- v0f --refutes--> v2d\n- v2b --refutes--> v08\n- v0c --refutes--> v12\n- v14 --refutes--> v1a\n- v22 --supports--> v17\n- v2e --supports--> v27\n- v08 --supports--> v16\n- v18 --refutes--> v1f\n- v00 --refutes--> v25\n- v23 --supports--> v29\n- v0d --supports--> v22\n- v27 --supports--> v00\n- v28 --refutes--> v11\n- v17 --supports--> v0c\n- v26 --supports--> v1e\n- v07 --supports--> v02\n- v0a --supports--> v24\n- v0f --supports--> v2c\n- v07 --refutes--> v2b\n- v18 --supports--> v1b\n- v17 --supports--> v0b\n- v12 --supports--> v19\n- v16 --supports--> v26\n\n[VERIFIED ROOT]\nv09 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d71b46e1da933a5eaf42c725b199bade5acd1c48a4d2eb9cc2927e34c2626060","schema":"livegraph.boolean-propagation.v1","spec_hash":"0f6cb994d79d598b0e95287f34a2d1959613e68308fb3ef97373f1c3599e09ec","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v09\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-8fd2a70b01639a01f325","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v1b\n- v0d --supports--> v20\n- v28 --supports--> v0b\n- v17 --refutes--> v0c\n- v1f --supports--> v2c\n- v2b --supports--> v0a\n- v1b --refutes--> v14\n- v26 --supports--> v04\n- v13 --refutes--> v01\n- v08 --refutes--> v23\n- v29 --refutes--> v22\n- v0f --supports--> v21\n- v0f --supports--> v2d\n- v1b --refutes--> v1a\n- v15 --supports--> v28\n- v22 --refutes--> v2b\n- v23 --refutes--> v03\n- v16 --supports--> v1c\n- v1a --supports--> v2a\n- v1d --refutes--> v0d\n- v26 --supports--> v25\n- v12 --refutes--> v24\n- v05 --refutes--> v11\n- v2f --supports--> v17\n- v30 --refutes--> v27\n- v28 --refutes--> v2f\n- v27 --refutes--> v1f\n- v25 --supports--> v09\n- v13 --supports--> v15\n- v0c --supports--> v08\n- v2f --refutes--> v18\n- v03 --supports--> v05\n- v06 --refutes--> v12\n- v0d --supports--> v19\n- v1f --refutes--> v29\n- v05 --supports--> v0e\n- v1a --supports--> v07\n- v03 --supports--> v10\n- v0f --refutes--> v02\n- v11 --refutes--> v1d\n- v24 --refutes--> v26\n- v19 --supports--> v30\n- v29 --refutes--> v2e\n- v02 --refutes--> v16\n- v16 --refutes--> v13\n- v19 --refutes--> v1e\n- v2f --refutes--> v00\n- v09 --supports--> v0f\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d0fb9c17f0ca349cf7ccdce85c931d89e8e5e211f0eb4c231aea67742dd8e27b","schema":"livegraph.boolean-propagation.v1","spec_hash":"8fd2a70b01639a01f325a7f8f7cdf128ce8363d502fae1c9c5d2516483a2fecf","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v06\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-04dacf8cdad68c627335","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2d\n- v0a --supports--> v30\n- v13 --supports--> v2e\n- v18 --refutes--> v19\n- v17 --supports--> v03\n- v2f --refutes--> v2c\n- v2c --refutes--> v29\n- v20 --refutes--> v28\n- v16 --supports--> v27\n- v18 --refutes--> v1a\n- v0c --refutes--> v05\n- v23 --supports--> v0e\n- v15 --refutes--> v1d\n- v0e --supports--> v15\n- v04 --refutes--> v2b\n- v05 --supports--> v01\n- v07 --supports--> v18\n- v2a --supports--> v0f\n- v10 --supports--> v1c\n- v18 --supports--> v02\n- v2b --refutes--> v2f\n- v20 --supports--> v14\n- v15 --supports--> v09\n- v21 --refutes--> v0d\n- v27 --supports--> v0b\n- v1e --supports--> v21\n- v12 --supports--> v0a\n- v01 --refutes--> v16\n- v12 --refutes--> v22\n- v20 --supports--> v26\n- v10 --refutes--> v04\n- v03 --refutes--> v13\n- v27 --supports--> v11\n- v17 --supports--> v1f\n- v2d --supports--> v17\n- v19 --refutes--> v20\n- v2c --refutes--> v24\n- v06 --refutes--> v2a\n- v2f --refutes--> v08\n- v21 --supports--> v00\n- v11 --refutes--> v1e\n- v15 --refutes--> v10\n- v21 --refutes--> v1b\n- v2e --supports--> v23\n- v24 --refutes--> v0c\n- v0f --supports--> v12\n- v2a --supports--> v25\n- v00 --supports--> v07\n\n[VERIFIED ROOT]\nv06 = TRUE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33ec602bd6638afca3b938064cf0605cbc7157c0d4440e9517c61686ac896b9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"04dacf8cdad68c627335807556bf1e32e9658aa5372c38f31b9b7d42e0f72e60","spec_json":"{\"edges\":[{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v06\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-4035d78f0879d461daf4","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v03 --refutes--> v04\n- v05 --supports--> v06\n- v0b --supports--> v1d\n- v00 --refutes--> v11\n- v26 --supports--> v21\n- v20 --supports--> v1c\n- v2c --refutes--> v18\n- v08 --refutes--> v0e\n- v19 --refutes--> v07\n- v20 --refutes--> v24\n- v0d --refutes--> v1b\n- v17 --refutes--> v10\n- v09 --refutes--> v08\n- v2d --refutes--> v03\n- v1a --supports--> v13\n- v2f --supports--> v02\n- v01 --supports--> v0f\n- v0f --supports--> v2b\n- v23 --supports--> v1a\n- v1b --refutes--> v00\n- v1f --supports--> v20\n- v2f --refutes--> v0c\n- v1a --supports--> v09\n- v08 --supports--> v1e\n- v23 --supports--> v15\n- v25 --supports--> v30\n- v02 --supports--> v2a\n- v26 --supports--> v25\n- v18 --supports--> v2f\n- v16 --supports--> v27\n- v00 --refutes--> v2c\n- v25 --supports--> v14\n- v2a --supports--> v1f\n- v0f --refutes--> v23\n- v2e --refutes--> v2d\n- v04 --refutes--> v26\n- v27 --supports--> v17\n- v05 --supports--> v01\n- v03 --refutes--> v12\n- v0b --refutes--> v0d\n- v02 --refutes--> v29\n- v0a --supports--> v28\n- v14 --supports--> v05\n- v14 --refutes--> v22\n- v1c --refutes--> v2e\n- v17 --refutes--> v0a\n- v07 --refutes--> v16\n\n[VERIFIED ROOT]\nv19 = TRUE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cf693c135bcb53ba577122881ffd93f6508ec8e11926eb659f437df109059227","schema":"livegraph.boolean-propagation.v1","spec_hash":"4035d78f0879d461daf4e67f0fb0a97ee99bede0c4836849d5b8caa8ed6540f8","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-a6bc367bae1ea91251c5","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v29\n- v06 --refutes--> v08\n- v07 --supports--> v30\n- v1e --refutes--> v28\n- v0c --refutes--> v2e\n- v02 --supports--> v0d\n- v30 --refutes--> v04\n- v0b --refutes--> v17\n- v09 --refutes--> v0e\n- v0f --refutes--> v2b\n- v1e --supports--> v22\n- v2a --supports--> v13\n- v2d --refutes--> v26\n- v2b --refutes--> v1f\n- v2b --supports--> v2c\n- v21 --supports--> v2a\n- v28 --refutes--> v14\n- v17 --supports--> v0f\n- v1c --supports--> v11\n- v13 --supports--> v2d\n- v2b --refutes--> v24\n- v20 --refutes--> v01\n- v2e --supports--> v19\n- v30 --refutes--> v0b\n- v26 --supports--> v16\n- v2d --refutes--> v12\n- v05 --refutes--> v03\n- v20 --refutes--> v1b\n- v1c --refutes--> v25\n- v28 --refutes--> v1a\n- v18 --supports--> v0a\n- v10 --refutes--> v1e\n- v1d --refutes--> v1c\n- v27 --refutes--> v02\n- v0a --supports--> v15\n- v05 --supports--> v23\n- v00 --supports--> v27\n- v06 --refutes--> v0c\n- v01 --refutes--> v06\n- v28 --supports--> v2f\n- v05 --supports--> v10\n- v02 --supports--> v07\n- v2f --refutes--> v18\n- v2e --supports--> v21\n- v15 --supports--> v1d\n- v11 --supports--> v09\n- v26 --refutes--> v05\n- v0e --supports--> v00\n\n[VERIFIED ROOT]\nv20 = TRUE\n\n[TARGET]\nv2c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f83e910f6c82c1ffe88b661919cdfd1b8637ec2805adb0e514055e8e99d2d9c1","schema":"livegraph.boolean-propagation.v1","spec_hash":"a6bc367bae1ea91251c5c3b30a8bd15df151603f17f7e51352fa7666ee0bfe0d","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-41aa34900f32211649d0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v14\n- v23 --refutes--> v0f\n- v12 --supports--> v2f\n- v10 --refutes--> v1e\n- v1d --refutes--> v2d\n- v28 --refutes--> v2a\n- v13 --refutes--> v2b\n- v02 --supports--> v18\n- v24 --refutes--> v1b\n- v1b --supports--> v17\n- v27 --refutes--> v0d\n- v11 --supports--> v16\n- v22 --refutes--> v05\n- v2c --refutes--> v15\n- v0e --refutes--> v1a\n- v03 --supports--> v06\n- v01 --refutes--> v20\n- v28 --supports--> v19\n- v02 --refutes--> v27\n- v2b --refutes--> v1d\n- v14 --supports--> v02\n- v25 --supports--> v11\n- v27 --supports--> v22\n- v25 --refutes--> v30\n- v15 --supports--> v0b\n- v20 --refutes--> v2e\n- v27 --supports--> v1c\n- v05 --supports--> v03\n- v2c --supports--> v0c\n- v1e --supports--> v0a\n- v1b --refutes--> v01\n- v1e --supports--> v08\n- v1a --supports--> v13\n- v06 --supports--> v09\n- v2d --refutes--> v07\n- v15 --refutes--> v10\n- v03 --refutes--> v04\n- v03 --supports--> v21\n- v00 --refutes--> v24\n- v2a --refutes--> v2c\n- v11 --refutes--> v28\n- v2f --supports--> v25\n- v03 --refutes--> v26\n- v07 --supports--> v23\n- v10 --refutes--> v29\n- v23 --supports--> v00\n- v2e --supports--> v12\n- v07 --supports--> v1f\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"30d7d7e8cd0eed49b709e5ce7f9ce398f6358e409c15a68414ba7ada54cf7ea9","schema":"livegraph.boolean-propagation.v1","spec_hash":"41aa34900f32211649d008a9b2c1ce80751ba5ff88ec055b4445a57f93fba751","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-866b92952b5afc2a336f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --refutes--> v01\n- v26 --refutes--> v03\n- v2b --supports--> v12\n- v11 --supports--> v06\n- v17 --supports--> v26\n- v1b --supports--> v2f\n- v10 --refutes--> v0f\n- v2a --refutes--> v24\n- v07 --refutes--> v11\n- v0c --refutes--> v00\n- v21 --supports--> v25\n- v03 --supports--> v07\n- v05 --supports--> v1b\n- v15 --refutes--> v13\n- v16 --supports--> v2b\n- v2a --refutes--> v30\n- v10 --supports--> v22\n- v0e --supports--> v0d\n- v2b --refutes--> v19\n- v18 --supports--> v1f\n- v01 --supports--> v2c\n- v01 --refutes--> v0c\n- v25 --supports--> v16\n- v18 --refutes--> v08\n- v14 --refutes--> v17\n- v2f --supports--> v0b\n- v0f --supports--> v1d\n- v0d --supports--> v15\n- v2e --supports--> v0e\n- v1a --refutes--> v2a\n- v00 --refutes--> v05\n- v11 --supports--> v04\n- v0d --supports--> v09\n- v26 --refutes--> v20\n- v06 --supports--> v21\n- v18 --supports--> v1e\n- v19 --supports--> v18\n- v19 --supports--> v29\n- v16 --supports--> v23\n- v0f --supports--> v1c\n- v1c --supports--> v27\n- v0e --supports--> v2d\n- v30 --refutes--> v2e\n- v0a --supports--> v14\n- v25 --refutes--> v28\n- v0b --refutes--> v1a\n- v15 --refutes--> v10\n- v1f --refutes--> v02\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"951157f07d70be23354232779c43b66cc3db2f1f0f6e99068bd7150506e0a705","schema":"livegraph.boolean-propagation.v1","spec_hash":"866b92952b5afc2a336f4af394aed491644c7b15e0e7fb6d4cdbff768d29715d","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-5c7db264a4b3a7327708","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v06 --refutes--> v04\n- v1a --refutes--> v25\n- v09 --supports--> v1b\n- v21 --supports--> v23\n- v30 --supports--> v0d\n- v14 --refutes--> v19\n- v13 --refutes--> v18\n- v2b --supports--> v0e\n- v0d --refutes--> v05\n- v2a --refutes--> v0b\n- v15 --supports--> v2a\n- v2c --refutes--> v30\n- v05 --refutes--> v11\n- v28 --refutes--> v15\n- v1f --supports--> v20\n- v20 --refutes--> v28\n- v11 --supports--> v2e\n- v25 --supports--> v16\n- v11 --refutes--> v27\n- v0e --supports--> v01\n- v00 --refutes--> v0f\n- v21 --refutes--> v24\n- v2b --refutes--> v0c\n- v30 --supports--> v22\n- v2b --refutes--> v02\n- v0b --refutes--> v1a\n- v03 --refutes--> v00\n- v0a --refutes--> v09\n- v04 --supports--> v2c\n- v15 --supports--> v1d\n- v2b --refutes--> v10\n- v17 --supports--> v1f\n- v1a --supports--> v1e\n- v08 --refutes--> v1c\n- v27 --supports--> v07\n- v01 --supports--> v03\n- v25 --supports--> v08\n- v0d --supports--> v12\n- v21 --supports--> v2b\n- v09 --refutes--> v21\n- v07 --refutes--> v17\n- v1c --refutes--> v13\n- v29 --supports--> v14\n- v03 --refutes--> v26\n- v17 --supports--> v2d\n- v0f --supports--> v06\n- v14 --refutes--> v2f\n\n[VERIFIED ROOT]\nv29 = 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":"2435b0b93cf3dfc0e60a5b4329fad1cb0f36b2e1a626e400586036e189d3fc4a","schema":"livegraph.boolean-propagation.v1","spec_hash":"5c7db264a4b3a73277082fd02dad25216fcb40f64c188d69bc51db21f947c1f6","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v29\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v18\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-c2521171ad1f48b97552","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v13\n- v18 --supports--> v25\n- v11 --refutes--> v19\n- v1f --supports--> v21\n- v12 --refutes--> v08\n- v0c --supports--> v2a\n- v0c --refutes--> v1c\n- v05 --supports--> v15\n- v05 --refutes--> v12\n- v09 --supports--> v20\n- v20 --refutes--> v1b\n- v06 --refutes--> v2c\n- v28 --supports--> v14\n- v13 --supports--> v05\n- v28 --supports--> v09\n- v1f --refutes--> v27\n- v1e --refutes--> v29\n- v1d --refutes--> v07\n- v1f --refutes--> v06\n- v28 --refutes--> v04\n- v1d --supports--> v0d\n- v0e --refutes--> v17\n- v00 --refutes--> v0a\n- v06 --refutes--> v30\n- v25 --supports--> v02\n- v1c --refutes--> v2f\n- v2e --supports--> v0b\n- v29 --supports--> v11\n- v16 --supports--> v28\n- v07 --supports--> v16\n- v1b --refutes--> v10\n- v0e --supports--> v2d\n- v00 --refutes--> v0c\n- v0b --refutes--> v00\n- v09 --refutes--> v2b\n- v19 --refutes--> v18\n- v25 --refutes--> v1d\n- v2f --refutes--> v0f\n- v26 --supports--> v24\n- v2d --refutes--> v1f\n- v0c --supports--> v22\n- v1b --refutes--> v03\n- v08 --refutes--> v0e\n- v24 --supports--> v2e\n- v26 --refutes--> v23\n- v0e --refutes--> v01\n- v30 --refutes--> v1a\n- v10 --supports--> v26\n\n[VERIFIED ROOT]\nv1e = FALSE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e4d2bdc98501271b34f250db67b0435677e93b6689cfba66cf9a9509cafd66f4","schema":"livegraph.boolean-propagation.v1","spec_hash":"c2521171ad1f48b97552e6d11d4042837cfd5caed49bd49b76bb8ee3856d6a67","spec_json":"{\"edges\":[{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-20bd9450f98e6b3a5b1f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v19 --refutes--> v08\n- v1f --supports--> v06\n- v14 --refutes--> v1d\n- v0f --supports--> v26\n- v15 --supports--> v0b\n- v0c --refutes--> v24\n- v10 --supports--> v19\n- v06 --refutes--> v14\n- v20 --refutes--> v00\n- v17 --supports--> v23\n- v1d --supports--> v30\n- v15 --supports--> v10\n- v1c --refutes--> v1e\n- v15 --refutes--> v22\n- v13 --supports--> v09\n- v29 --refutes--> v0a\n- v06 --supports--> v2f\n- v30 --refutes--> v07\n- v12 --refutes--> v28\n- v16 --supports--> v17\n- v17 --supports--> v27\n- v1b --supports--> v02\n- v01 --refutes--> v2c\n- v1b --refutes--> v18\n- v13 --supports--> v12\n- v0c --supports--> v25\n- v0d --supports--> v20\n- v05 --refutes--> v1b\n- v30 --refutes--> v2d\n- v28 --supports--> v0e\n- v0e --refutes--> v0d\n- v01 --supports--> v1f\n- v10 --refutes--> v2b\n- v1c --refutes--> v05\n- v26 --supports--> v15\n- v27 --refutes--> v01\n- v00 --supports--> v11\n- v18 --refutes--> v1a\n- v2d --refutes--> v21\n- v08 --supports--> v1c\n- v2d --refutes--> v0c\n- v26 --refutes--> v03\n- v0c --supports--> v2a\n- v29 --supports--> v04\n- v2e --supports--> v29\n- v11 --refutes--> v16\n- v29 --supports--> v0f\n- v25 --supports--> v2e\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv1a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"3aa13cc728d99cec8c313debcf02244d4dcc133aaf4d46debffb6cc6c3d6af77","schema":"livegraph.boolean-propagation.v1","spec_hash":"20bd9450f98e6b3a5b1f1f99b9c0afbfc3ce8c93f4f93d0cb0e33b9e9878d6cf","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-8d5b8e36a96cc4b4f89c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2a\n- v0c --refutes--> v12\n- v23 --refutes--> v17\n- v16 --refutes--> v09\n- v09 --refutes--> v27\n- v04 --refutes--> v2b\n- v19 --refutes--> v13\n- v23 --supports--> v06\n- v30 --refutes--> v2e\n- v1c --refutes--> v0e\n- v0f --supports--> v11\n- v07 --supports--> v25\n- v0f --refutes--> v18\n- v02 --supports--> v2d\n- v24 --refutes--> v2f\n- v27 --supports--> v1d\n- v03 --supports--> v02\n- v0a --refutes--> v30\n- v16 --supports--> v0d\n- v13 --supports--> v1b\n- v1a --refutes--> v28\n- v00 --refutes--> v2c\n- v29 --supports--> v0f\n- v24 --supports--> v0a\n- v20 --supports--> v00\n- v30 --supports--> v1e\n- v1d --supports--> v10\n- v1a --refutes--> v05\n- v0c --supports--> v03\n- v00 --refutes--> v15\n- v1f --refutes--> v26\n- v0a --refutes--> v0b\n- v26 --refutes--> v19\n- v1a --supports--> v21\n- v15 --supports--> v14\n- v25 --refutes--> v29\n- v00 --supports--> v01\n- v14 --refutes--> v1c\n- v11 --supports--> v0c\n- v13 --supports--> v04\n- v1e --supports--> v1a\n- v10 --refutes--> v24\n- v0e --refutes--> v22\n- v23 --refutes--> v1f\n- v2d --supports--> v23\n- v23 --refutes--> v08\n- v22 --refutes--> v16\n- v1a --supports--> v07\n\n[VERIFIED ROOT]\nv20 = TRUE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"67db661bb240d1f97d54daa03743e578c0eacef29a37a1e6ed395d3f25eead9a","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d5b8e36a96cc4b4f89c1029bca0d0cdf4ceee52383331af89c15e093e337f45","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v20\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-067840caebe1d5a4eac7","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --refutes--> v21\n- v04 --supports--> v25\n- v2c --supports--> v12\n- v21 --refutes--> v0d\n- v1e --supports--> v03\n- v17 --supports--> v2c\n- v30 --supports--> v23\n- v2f --supports--> v20\n- v02 --refutes--> v06\n- v10 --supports--> v2b\n- v1d --refutes--> v0c\n- v12 --supports--> v08\n- v2f --refutes--> v01\n- v25 --supports--> v16\n- v1b --supports--> v0a\n- v0a --supports--> v00\n- v23 --supports--> v0b\n- v27 --refutes--> v26\n- v05 --supports--> v1b\n- v11 --refutes--> v28\n- v07 --supports--> v04\n- v0e --refutes--> v15\n- v23 --supports--> v14\n- v02 --supports--> v2e\n- v25 --supports--> v11\n- v00 --supports--> v30\n- v0d --supports--> v1f\n- v01 --supports--> v27\n- v06 --supports--> v09\n- v0f --supports--> v2f\n- v12 --supports--> v18\n- v0b --supports--> v1c\n- v13 --refutes--> v07\n- v10 --supports--> v29\n- v0e --supports--> v05\n- v28 --refutes--> v0e\n- v0d --refutes--> v24\n- v1b --supports--> v2d\n- v13 --refutes--> v2a\n- v09 --supports--> v1e\n- v2b --supports--> v0f\n- v27 --refutes--> v17\n- v0d --supports--> v1d\n- v1d --refutes--> v10\n- v23 --supports--> v1a\n- v08 --supports--> v02\n- v30 --refutes--> v22\n- v07 --refutes--> v19\n\n[VERIFIED ROOT]\nv13 = FALSE\n\n[TARGET]\nv03\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b5a9b58337796673986e3f31d3d3e873517f63d82e178248f60ac7306bac7570","schema":"livegraph.boolean-propagation.v1","spec_hash":"067840caebe1d5a4eac7d376774d95bec9b44f76d1469d6ff6d2d8b00f0442c2","spec_json":"{\"edges\":[{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v13\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-8d68d98fc242d87a1588","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v29 --refutes--> v23\n- v05 --refutes--> v13\n- v00 --supports--> v03\n- v24 --refutes--> v07\n- v0d --refutes--> v18\n- v25 --supports--> v28\n- v21 --refutes--> v2a\n- v05 --supports--> v02\n- v2e --refutes--> v17\n- v1b --supports--> v29\n- v00 --refutes--> v27\n- v17 --refutes--> v04\n- v0e --refutes--> v1f\n- v16 --supports--> v22\n- v03 --refutes--> v15\n- v2b --refutes--> v0b\n- v23 --supports--> v2d\n- v14 --supports--> v12\n- v13 --refutes--> v24\n- v0e --supports--> v0a\n- v18 --refutes--> v00\n- v22 --refutes--> v25\n- v10 --refutes--> v0d\n- v18 --supports--> v26\n- v25 --refutes--> v0e\n- v21 --supports--> v16\n- v1f --refutes--> v10\n- v23 --refutes--> v1a\n- v1e --supports--> v05\n- v15 --supports--> v20\n- v1b --refutes--> v0f\n- v25 --supports--> v30\n- v0c --supports--> v2e\n- v12 --refutes--> v2f\n- v12 --supports--> v11\n- v17 --refutes--> v1e\n- v06 --refutes--> v1b\n- v20 --supports--> v19\n- v06 --supports--> v09\n- v1e --refutes--> v01\n- v2c --refutes--> v0c\n- v24 --supports--> v21\n- v2f --refutes--> v2b\n- v23 --refutes--> v1c\n- v1a --refutes--> v08\n- v2f --refutes--> v1d\n- v0b --supports--> v06\n- v08 --refutes--> v2c\n\n[VERIFIED ROOT]\nv14 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"8066b0d70b64526616e37fb159b8e53643fcc88e26946f44a40591fca9ecdb4b","schema":"livegraph.boolean-propagation.v1","spec_hash":"8d68d98fc242d87a1588527e18b36ff811767b5a47584337d70c86f22bd44ba5","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-c3629a841b20be55e835","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2f --refutes--> v0e\n- v22 --supports--> v25\n- v18 --refutes--> v2a\n- v08 --supports--> v06\n- v0d --supports--> v01\n- v0f --refutes--> v18\n- v04 --refutes--> v00\n- v09 --refutes--> v1a\n- v24 --refutes--> v02\n- v16 --refutes--> v11\n- v23 --refutes--> v2f\n- v1d --supports--> v2d\n- v23 --refutes--> v2b\n- v2c --supports--> v0a\n- v29 --refutes--> v2e\n- v2a --refutes--> v12\n- v10 --refutes--> v16\n- v01 --supports--> v1e\n- v20 --refutes--> v0f\n- v27 --supports--> v0d\n- v24 --refutes--> v22\n- v15 --supports--> v1b\n- v2d --supports--> v1c\n- v0b --supports--> v29\n- v24 --refutes--> v14\n- v2a --supports--> v23\n- v19 --refutes--> v21\n- v08 --refutes--> v17\n- v09 --supports--> v28\n- v09 --refutes--> v10\n- v1b --refutes--> v03\n- v15 --refutes--> v30\n- v01 --supports--> v13\n- v26 --refutes--> v09\n- v0c --refutes--> v07\n- v18 --supports--> v1f\n- v04 --supports--> v1d\n- v05 --refutes--> v0c\n- v2f --supports--> v19\n- v1b --supports--> v05\n- v21 --refutes--> v04\n- v2d --supports--> v24\n- v01 --supports--> v26\n- v17 --supports--> v20\n- v2e --refutes--> v08\n- v22 --refutes--> v15\n- v11 --supports--> v0b\n- v07 --supports--> v2c\n\n[VERIFIED ROOT]\nv27 = 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":"dc8cbd59531227eaaad2c6771ea5db6e9fc83e76544caf8c5756c6c33b5cd1c0","schema":"livegraph.boolean-propagation.v1","spec_hash":"c3629a841b20be55e8359c312c2f63fb126ee85dd3cca9b8c07615fb9a62f468","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v27\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-82b0a552414ecedcecb9","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v29\n- v06 --refutes--> v19\n- v05 --refutes--> v24\n- v09 --supports--> v2a\n- v1f --refutes--> v02\n- v20 --supports--> v22\n- v2e --refutes--> v1e\n- v0a --supports--> v09\n- v20 --refutes--> v01\n- v30 --supports--> v26\n- v1a --refutes--> v11\n- v01 --refutes--> v28\n- v24 --supports--> v04\n- v0c --supports--> v23\n- v04 --refutes--> v0c\n- v2e --supports--> v0d\n- v0c --supports--> v1a\n- v0e --refutes--> v00\n- v17 --supports--> v15\n- v25 --supports--> v03\n- v1e --supports--> v2b\n- v05 --refutes--> v13\n- v26 --supports--> v2e\n- v19 --refutes--> v12\n- v19 --refutes--> v27\n- v24 --refutes--> v0b\n- v18 --refutes--> v21\n- v21 --supports--> v0f\n- v01 --supports--> v05\n- v1e --supports--> v1c\n- v11 --refutes--> v10\n- v2a --supports--> v20\n- v16 --supports--> v06\n- v10 --refutes--> v2f\n- v29 --refutes--> v08\n- v08 --refutes--> v17\n- v30 --supports--> v14\n- v00 --supports--> v16\n- v02 --supports--> v30\n- v30 --refutes--> v07\n- v0c --refutes--> v1b\n- v19 --supports--> v1f\n- v21 --refutes--> v0a\n- v17 --refutes--> v2c\n- v10 --supports--> v25\n- v1d --supports--> v0e\n- v30 --supports--> v2d\n- v2c --refutes--> v1d\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"38b408b52bd46850767ecc92e6f77da3335fed65adf8a064c9e4274f9295375b","schema":"livegraph.boolean-propagation.v1","spec_hash":"82b0a552414ecedcecb9e714d09ea0ca53a692572381a1b4d716f8504713adac","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-31a60f79dae3ca96016a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v05\n- v15 --supports--> v2c\n- v14 --supports--> v27\n- v0b --refutes--> v2d\n- v1e --refutes--> v28\n- v30 --refutes--> v2a\n- v24 --supports--> v16\n- v29 --supports--> v1d\n- v05 --supports--> v14\n- v02 --supports--> v12\n- v2f --supports--> v00\n- v2f --refutes--> v10\n- v0a --refutes--> v29\n- v22 --refutes--> v19\n- v02 --supports--> v11\n- v0c --refutes--> v24\n- v03 --refutes--> v06\n- v06 --supports--> v04\n- v27 --refutes--> v0c\n- v2d --supports--> v2f\n- v25 --refutes--> v0f\n- v09 --refutes--> v08\n- v23 --refutes--> v0e\n- v15 --supports--> v0a\n- v0f --refutes--> v1b\n- v23 --refutes--> v09\n- v1e --refutes--> v26\n- v10 --refutes--> v15\n- v0d --supports--> v18\n- v09 --refutes--> v30\n- v21 --supports--> v1c\n- v11 --refutes--> v23\n- v2b --supports--> v22\n- v0f --supports--> v1a\n- v1b --refutes--> v2e\n- v26 --supports--> v0d\n- v09 --refutes--> v20\n- v1c --refutes--> v03\n- v0d --refutes--> v07\n- v0d --supports--> v21\n- v22 --supports--> v1e\n- v0a --supports--> v1f\n- v06 --supports--> v25\n- v11 --refutes--> v13\n- v29 --supports--> v2b\n- v2a --refutes--> v0b\n- v2e --refutes--> v17\n- v05 --refutes--> v01\n\n[VERIFIED ROOT]\nv02 = 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":"cddf6b20e6d4ab66b75f5211834b84b87c30aea9de724a5f2268f21b09fa1bff","schema":"livegraph.boolean-propagation.v1","spec_hash":"31a60f79dae3ca96016afa397074e9c8f99a7eb984ea98a251acad788821c832","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v16\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-f703f11fd555ccbfe3f1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0a --refutes--> v11\n- v0e --refutes--> v18\n- v30 --refutes--> v1b\n- v2c --refutes--> v05\n- v0b --supports--> v24\n- v12 --supports--> v2f\n- v11 --refutes--> v1c\n- v08 --supports--> v09\n- v27 --refutes--> v2a\n- v0f --supports--> v26\n- v10 --refutes--> v2e\n- v2f --refutes--> v19\n- v2a --refutes--> v1a\n- v30 --supports--> v10\n- v23 --supports--> v2b\n- v15 --refutes--> v29\n- v13 --refutes--> v07\n- v20 --refutes--> v30\n- v0b --supports--> v02\n- v16 --supports--> v22\n- v0d --supports--> v1d\n- v12 --supports--> v28\n- v00 --supports--> v25\n- v11 --supports--> v27\n- v0a --refutes--> v04\n- v1e --refutes--> v21\n- v23 --refutes--> v0b\n- v1d --refutes--> v1e\n- v03 --refutes--> v13\n- v15 --supports--> v0c\n- v2a --refutes--> v0e\n- v2a --supports--> v1f\n- v25 --supports--> v14\n- v2f --refutes--> v2d\n- v06 --supports--> v20\n- v2d --refutes--> v2c\n- v2e --refutes--> v0f\n- v10 --refutes--> v01\n- v17 --refutes--> v00\n- v05 --supports--> v08\n- v0f --supports--> v15\n- v21 --supports--> v03\n- v15 --supports--> v0a\n- v25 --supports--> v0d\n- v08 --refutes--> v06\n- v07 --supports--> v16\n- v22 --refutes--> v12\n- v0e --refutes--> v23\n\n[VERIFIED ROOT]\nv17 = FALSE\n\n[TARGET]\nv24\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f4c88d22e8ad9e081c0d838b1d402d655943bdaecf82d8435e40fa74bc3975a5","schema":"livegraph.boolean-propagation.v1","spec_hash":"f703f11fd555ccbfe3f1dc9be0c504854822c3cd19202300baa2ff2c0509c0db","spec_json":"{\"edges\":[{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v17\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v24\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-00566281efb34b35096c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v00\n- v10 --refutes--> v12\n- v14 --supports--> v19\n- v0a --refutes--> v18\n- v19 --refutes--> v05\n- v1f --supports--> v16\n- v1c --refutes--> v0c\n- v2a --supports--> v08\n- v06 --supports--> v25\n- v1e --supports--> v2a\n- v0c --refutes--> v26\n- v28 --supports--> v0d\n- v12 --supports--> v0f\n- v0c --supports--> v2e\n- v07 --refutes--> v1b\n- v1a --supports--> v2d\n- v19 --supports--> v27\n- v1a --supports--> v1d\n- v0e --supports--> v1e\n- v2d --refutes--> v13\n- v2c --supports--> v10\n- v23 --refutes--> v1f\n- v15 --supports--> v09\n- v0b --refutes--> v06\n- v10 --refutes--> v17\n- v14 --refutes--> v22\n- v24 --supports--> v14\n- v1e --refutes--> v01\n- v0a --supports--> v02\n- v21 --refutes--> v28\n- v07 --refutes--> v0a\n- v05 --supports--> v03\n- v03 --refutes--> v0e\n- v08 --refutes--> v15\n- v28 --supports--> v29\n- v13 --refutes--> v2c\n- v20 --refutes--> v21\n- v24 --refutes--> v04\n- v25 --refutes--> v20\n- v0f --refutes--> v24\n- v14 --refutes--> v11\n- v24 --supports--> v30\n- v02 --supports--> v23\n- v16 --supports--> v0b\n- v28 --refutes--> v2b\n- v28 --refutes--> v1c\n- v2e --supports--> v1a\n- v1a --refutes--> v2f\n\n[VERIFIED ROOT]\nv07 = FALSE\n\n[TARGET]\nv09\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"faaa7dead6b44ce9a538707a5ca983512d49fb4a6d579758a1f7327f43806555","schema":"livegraph.boolean-propagation.v1","spec_hash":"00566281efb34b35096c7f430f7ed71d86eda62535cbc08f6dc7a01c0af54f48","spec_json":"{\"edges\":[{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v07\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v09\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-33b66c1b69679a05a9d6","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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- v1f --supports--> v2b\n- v0e --refutes--> v19\n- v12 --refutes--> v08\n- v0d --supports--> v29\n- v19 --supports--> v16\n- v13 --supports--> v1c\n- v22 --refutes--> v2a\n- v16 --refutes--> v13\n- v09 --supports--> v06\n- v20 --refutes--> v07\n- v05 --refutes--> v15\n- v1f --supports--> v2d\n- v27 --refutes--> v26\n- v08 --supports--> v04\n- v00 --refutes--> v17\n- v2e --refutes--> v23\n- v21 --refutes--> v0b\n- v15 --supports--> v12\n- v21 --supports--> v27\n- v27 --refutes--> v1d\n- v23 --supports--> v25\n- v1b --supports--> v0a\n- v23 --supports--> v05\n- v29 --refutes--> v21\n- v04 --supports--> v00\n- v12 --supports--> v1e\n- v26 --supports--> v18\n- v1a --refutes--> v2e\n- v0c --refutes--> v0f\n- v1c --supports--> v0d\n- v2d --refutes--> v0c\n- v08 --refutes--> v28\n- v21 --supports--> v02\n- v00 --refutes--> v30\n- v24 --refutes--> v11\n- v21 --supports--> v01\n- v24 --refutes--> v1a\n- v0f --supports--> v09\n- v1f --supports--> v10\n- v17 --refutes--> v1b\n- v2a --refutes--> v1f\n- v09 --supports--> v2c\n- v0c --supports--> v14\n- v18 --supports--> v22\n- v20 --supports--> v24\n- v17 --supports--> v03\n- v19 --refutes--> v2f\n\n[VERIFIED ROOT]\nv20 = FALSE\n\n[TARGET]\nv2c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"01c9332429b60542ccc8521d05111317d9e008d431acd2e147a91f3dd0e5201e","schema":"livegraph.boolean-propagation.v1","spec_hash":"33b66c1b69679a05a9d6b6625385a40c4f38aa21eb2082e3ab07644d1784aa5e","spec_json":"{\"edges\":[{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v20\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-ed4d43de85016daf4961","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2f --refutes--> v10\n- v26 --refutes--> v1e\n- v15 --refutes--> v04\n- v1b --refutes--> v27\n- v2a --refutes--> v2d\n- v21 --refutes--> v2b\n- v29 --supports--> v18\n- v1c --refutes--> v16\n- v05 --supports--> v0d\n- v21 --supports--> v02\n- v04 --refutes--> v30\n- v0d --supports--> v21\n- v22 --refutes--> v05\n- v00 --refutes--> v1f\n- v22 --supports--> v14\n- v06 --refutes--> v1b\n- v20 --supports--> v28\n- v30 --refutes--> v09\n- v24 --refutes--> v1d\n- v11 --refutes--> v24\n- v2a --refutes--> v0b\n- v04 --supports--> v0e\n- v09 --supports--> v0f\n- v29 --supports--> v12\n- v06 --refutes--> v0c\n- v26 --refutes--> v08\n- v16 --supports--> v2e\n- v27 --refutes--> v00\n- v21 --refutes--> v06\n- v28 --refutes--> v2f\n- v2e --supports--> v2a\n- v1a --refutes--> v26\n- v11 --refutes--> v03\n- v22 --refutes--> v19\n- v01 --refutes--> v1c\n- v2c --refutes--> v25\n- v0a --supports--> v07\n- v15 --supports--> v23\n- v2c --refutes--> v01\n- v1d --refutes--> v22\n- v18 --supports--> v1a\n- v00 --supports--> v15\n- v1e --refutes--> v2c\n- v07 --refutes--> v20\n- v09 --supports--> v13\n- v2f --refutes--> v17\n- v0f --supports--> v29\n- v2f --refutes--> v11\n\n[VERIFIED ROOT]\nv0a = FALSE\n\n[TARGET]\nv2d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"98038eef05db63a3973a7c715188d1da24eae72c434bb96cbe43577ac2c3d7a6","schema":"livegraph.boolean-propagation.v1","spec_hash":"ed4d43de85016daf4961dfaf2b7bb2e1b555fae0efdc220bb78302bc4ff580ae","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0a\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-55223c111e04f83c7788","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v20 --refutes--> v2d\n- v1d --supports--> v1a\n- v1c --refutes--> v1f\n- v1d --refutes--> v0a\n- v0b --refutes--> v14\n- v1f --supports--> v23\n- v30 --refutes--> v21\n- v09 --supports--> v2f\n- v28 --refutes--> v24\n- v21 --refutes--> v2b\n- v05 --refutes--> v10\n- v00 --refutes--> v04\n- v10 --supports--> v1d\n- v2f --refutes--> v0f\n- v10 --refutes--> v07\n- v2f --supports--> v03\n- v11 --supports--> v18\n- v2f --supports--> v22\n- v10 --supports--> v13\n- v2e --supports--> v00\n- v00 --supports--> v28\n- v1a --supports--> v08\n- v14 --supports--> v11\n- v28 --refutes--> v1c\n- v25 --refutes--> v09\n- v20 --refutes--> v06\n- v18 --refutes--> v2e\n- v27 --refutes--> v25\n- v01 --refutes--> v20\n- v25 --supports--> v1e\n- v1f --supports--> v17\n- v25 --refutes--> v29\n- v1f --refutes--> v26\n- v02 --refutes--> v16\n- v11 --supports--> v1b\n- v06 --supports--> v05\n- v16 --supports--> v27\n- v0c --supports--> v02\n- v12 --refutes--> v0d\n- v22 --supports--> v19\n- v19 --supports--> v0b\n- v1d --supports--> v2c\n- v06 --supports--> v15\n- v08 --supports--> v30\n- v0d --refutes--> v0c\n- v26 --supports--> v01\n- v2b --supports--> v0e\n- v01 --refutes--> v2a\n\n[VERIFIED ROOT]\nv12 = TRUE\n\n[TARGET]\nv0e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1a907471d51c7a548ffc6932b3fb3ab1d5693d90116a2a5b256fb5ed26d4f38f","schema":"livegraph.boolean-propagation.v1","spec_hash":"55223c111e04f83c7788ac07990e9a6e9615d36c6d2316c1aa7bcc5190af8443","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v12\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-a9b2c9a61376b0409d6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v22 --supports--> v24\n- v16 --supports--> v2c\n- v25 --refutes--> v17\n- v2c --refutes--> v19\n- v1a --refutes--> v2e\n- v14 --supports--> v07\n- v2b --refutes--> v26\n- v10 --supports--> v28\n- v09 --supports--> v25\n- v05 --supports--> v04\n- v1b --supports--> v02\n- v03 --supports--> v1f\n- v18 --refutes--> v29\n- v01 --supports--> v2b\n- v1d --refutes--> v1a\n- v05 --supports--> v03\n- v18 --refutes--> v0c\n- v0d --refutes--> v0f\n- v15 --refutes--> v1e\n- v2a --supports--> v21\n- v03 --refutes--> v13\n- v1e --refutes--> v09\n- v1a --supports--> v10\n- v16 --refutes--> v0a\n- v23 --supports--> v18\n- v1a --refutes--> v11\n- v21 --refutes--> v00\n- v0c --supports--> v30\n- v1a --supports--> v1c\n- v07 --refutes--> v0b\n- v25 --supports--> v01\n- v02 --supports--> v2f\n- v30 --refutes--> v14\n- v02 --refutes--> v0e\n- v22 --supports--> v15\n- v21 --refutes--> v05\n- v0b --supports--> v2a\n- v0f --refutes--> v16\n- v19 --supports--> v2d\n- v28 --refutes--> v12\n- v03 --refutes--> v06\n- v2b --refutes--> v0d\n- v2f --supports--> v20\n- v1f --refutes--> v1d\n- v10 --supports--> v08\n- v27 --refutes--> v22\n- v2f --refutes--> v23\n- v19 --refutes--> v1b\n\n[VERIFIED ROOT]\nv27 = 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":"425668f6d9b05f260f5a016e4affffbd8f241339180ba8fb684389b6ac93d046","schema":"livegraph.boolean-propagation.v1","spec_hash":"a9b2c9a61376b0409d6ac3ddb3f71ada09bae45c66606f77a5e703c62731fac3","spec_json":"{\"edges\":[{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v27\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v12\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-b4c7bc69a4e5cbb3b12e","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v24\n- v15 --supports--> v2c\n- v15 --supports--> v20\n- v10 --refutes--> v07\n- v04 --refutes--> v05\n- v02 --refutes--> v03\n- v2a --refutes--> v08\n- v17 --refutes--> v1a\n- v00 --supports--> v06\n- v1a --supports--> v19\n- v29 --refutes--> v1d\n- v2b --refutes--> v21\n- v17 --supports--> v1e\n- v01 --supports--> v26\n- v01 --refutes--> v0f\n- v1f --refutes--> v29\n- v27 --supports--> v1f\n- v0e --supports--> v10\n- v03 --supports--> v30\n- v05 --supports--> v11\n- v1a --refutes--> v1c\n- v05 --refutes--> v2e\n- v00 --refutes--> v14\n- v2c --supports--> v1b\n- v12 --supports--> v0d\n- v11 --supports--> v28\n- v0e --supports--> v0c\n- v1f --refutes--> v0a\n- v14 --supports--> v18\n- v2c --refutes--> v16\n- v2d --refutes--> v02\n- v08 --refutes--> v01\n- v1c --refutes--> v13\n- v1d --refutes--> v15\n- v12 --supports--> v2b\n- v1b --supports--> v0b\n- v0f --refutes--> v12\n- v0b --refutes--> v25\n- v11 --supports--> v2f\n- v30 --refutes--> v2a\n- v22 --refutes--> v2d\n- v12 --refutes--> v09\n- v21 --supports--> v04\n- v11 --supports--> v23\n- v14 --supports--> v0e\n- v23 --supports--> v27\n- v0b --refutes--> v00\n- v13 --supports--> v22\n\n[VERIFIED ROOT]\nv17 = 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":"0a058348e589270f5a7c1a23a602ec28f1cb30e37660cca0c5b85b7a54ab8d41","schema":"livegraph.boolean-propagation.v1","spec_hash":"b4c7bc69a4e5cbb3b12e762346a1b1061688e7321aede1a9661c15c076acedbc","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v17\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-be1216a50c7a26db2f09","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v1c\n- v09 --refutes--> v20\n- v08 --refutes--> v26\n- v1e --supports--> v21\n- v00 --supports--> v09\n- v2e --refutes--> v1e\n- v22 --refutes--> v0c\n- v09 --refutes--> v25\n- v25 --refutes--> v2a\n- v08 --refutes--> v27\n- v0f --supports--> v2c\n- v2d --refutes--> v08\n- v08 --refutes--> v13\n- v0f --refutes--> v07\n- v16 --supports--> v2f\n- v18 --refutes--> v19\n- v1e --refutes--> v1f\n- v04 --refutes--> v1d\n- v02 --supports--> v16\n- v05 --supports--> v0d\n- v1c --refutes--> v0b\n- v16 --refutes--> v1a\n- v07 --supports--> v12\n- v15 --supports--> v18\n- v01 --refutes--> v15\n- v1a --supports--> v22\n- v0b --supports--> v01\n- v13 --refutes--> v03\n- v13 --refutes--> v00\n- v05 --supports--> v24\n- v23 --refutes--> v29\n- v1d --supports--> v05\n- v12 --refutes--> v2e\n- v07 --supports--> v1b\n- v13 --supports--> v11\n- v25 --supports--> v0e\n- v28 --refutes--> v2d\n- v14 --supports--> v2b\n- v0d --refutes--> v14\n- v02 --refutes--> v0a\n- v1d --supports--> v30\n- v21 --refutes--> v23\n- v22 --refutes--> v06\n- v2b --supports--> v17\n- v0c --refutes--> v10\n- v17 --supports--> v28\n- v0c --refutes--> v04\n- v2a --supports--> v0f\n\n[VERIFIED ROOT]\nv02 = TRUE\n\n[TARGET]\nv19\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"12c6731b8bcb740b6d50455dbd80ba31fcdf29f210aa936904128d5aac013bdd","schema":"livegraph.boolean-propagation.v1","spec_hash":"be1216a50c7a26db2f093165d1114dbd37e06390f501e07a0017b3afe17ab13d","spec_json":"{\"edges\":[{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v02\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v19\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-8b4e8ba90125cb4a5696","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --refutes--> v18\n- v2d --supports--> v0d\n- v13 --refutes--> v01\n- v22 --supports--> v11\n- v09 --supports--> v2b\n- v03 --supports--> v0e\n- v07 --refutes--> v30\n- v09 --supports--> v17\n- v28 --supports--> v07\n- v13 --refutes--> v14\n- v30 --supports--> v1e\n- v0a --refutes--> v02\n- v25 --supports--> v1a\n- v15 --refutes--> v2c\n- v16 --supports--> v25\n- v27 --supports--> v2a\n- v2d --supports--> v04\n- v11 --refutes--> v21\n- v08 --refutes--> v16\n- v1b --supports--> v05\n- v15 --supports--> v0b\n- v1a --supports--> v19\n- v1d --refutes--> v00\n- v0f --refutes--> v2d\n- v0c --supports--> v03\n- v28 --supports--> v23\n- v0f --supports--> v06\n- v12 --refutes--> v27\n- v1e --refutes--> v1b\n- v16 --refutes--> v20\n- v04 --supports--> v12\n- v1d --supports--> v26\n- v09 --supports--> v13\n- v26 --refutes--> v22\n- v02 --supports--> v15\n- v19 --refutes--> v0c\n- v19 --supports--> v10\n- v04 --supports--> v1c\n- v30 --supports--> v2f\n- v03 --supports--> v24\n- v04 --supports--> v1f\n- v2a --refutes--> v08\n- v1a --supports--> v29\n- v21 --refutes--> v0a\n- v2c --refutes--> v28\n- v05 --supports--> v0f\n- v18 --supports--> v09\n- v26 --supports--> v2e\n\n[VERIFIED ROOT]\nv1d = 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":"ce6371830a5b5cfa18b0fac1a209bc07b08c17f750ef70187499912acc3a3b8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"8b4e8ba90125cb4a5696d044c6995519736f26b3d866ef3363ff186976794f03","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-29afd7510bfac6d10a3d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v26\n- v0c --supports--> v02\n- v1e --supports--> v1d\n- v1a --refutes--> v28\n- v07 --supports--> v0c\n- v00 --refutes--> v18\n- v2b --supports--> v24\n- v00 --refutes--> v1b\n- v15 --supports--> v21\n- v20 --refutes--> v16\n- v02 --supports--> v2a\n- v2f --supports--> v11\n- v25 --refutes--> v14\n- v29 --supports--> v0b\n- v1d --refutes--> v0e\n- v1c --refutes--> v12\n- v06 --supports--> v01\n- v05 --refutes--> v1c\n- v11 --supports--> v2d\n- v1f --supports--> v2b\n- v06 --refutes--> v2e\n- v17 --refutes--> v1f\n- v0c --supports--> v30\n- v06 --refutes--> v15\n- v2a --refutes--> v29\n- v18 --refutes--> v25\n- v15 --supports--> v27\n- v1c --supports--> v03\n- v15 --refutes--> v0f\n- v1a --refutes--> v20\n- v29 --refutes--> v05\n- v0d --refutes--> v19\n- v14 --refutes--> v09\n- v2b --refutes--> v06\n- v22 --supports--> v13\n- v1d --supports--> v2f\n- v2b --refutes--> v04\n- v19 --supports--> v1e\n- v20 --refutes--> v17\n- v2c --refutes--> v0d\n- v23 --refutes--> v22\n- v13 --refutes--> v2c\n- v15 --supports--> v0a\n- v12 --supports--> v00\n- v0f --supports--> v08\n- v2d --refutes--> v07\n- v0c --refutes--> v10\n- v09 --supports--> v1a\n\n[VERIFIED ROOT]\nv23 = 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":"1da7f32c9e3e73cc0c5022aef27124654a5d08960c832bc777ee638e19f1951d","schema":"livegraph.boolean-propagation.v1","spec_hash":"29afd7510bfac6d10a3d7ec92eefce968c5f3b2fc774d3ce4fbff8eb48e14ecd","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v08\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-dec109a2b62d42b3dc5a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1f --refutes--> v1d\n- v2e --supports--> v0d\n- v09 --refutes--> v14\n- v2b --refutes--> v30\n- v11 --supports--> v12\n- v09 --supports--> v0c\n- v08 --refutes--> v02\n- v0a --refutes--> v04\n- v23 --refutes--> v24\n- v1a --refutes--> v23\n- v0e --refutes--> v05\n- v26 --refutes--> v22\n- v23 --refutes--> v2f\n- v20 --refutes--> v15\n- v09 --supports--> v28\n- v19 --refutes--> v16\n- v1c --refutes--> v2d\n- v1a --supports--> v00\n- v08 --supports--> v06\n- v24 --refutes--> v19\n- v0a --supports--> v29\n- v0e --supports--> v11\n- v18 --supports--> v07\n- v0e --supports--> v13\n- v12 --supports--> v08\n- v22 --supports--> v25\n- v14 --refutes--> v21\n- v1d --supports--> v2b\n- v0e --refutes--> v0f\n- v07 --supports--> v2a\n- v15 --supports--> v0a\n- v03 --refutes--> v1a\n- v07 --refutes--> v10\n- v25 --supports--> v01\n- v07 --supports--> v1f\n- v0d --supports--> v0b\n- v1c --refutes--> v27\n- v25 --supports--> v0e\n- v06 --supports--> v20\n- v0e --refutes--> v2c\n- v1b --refutes--> v09\n- v16 --supports--> v2e\n- v21 --refutes--> v18\n- v23 --supports--> v1e\n- v30 --supports--> v26\n- v1a --refutes--> v17\n- v2d --refutes--> v1b\n- v29 --supports--> v03\n\n[VERIFIED ROOT]\nv1c = 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":"0f57540db441ca48f64ed4be394f950beeda68f27767b4f3ee74a08a1400c561","schema":"livegraph.boolean-propagation.v1","spec_hash":"dec109a2b62d42b3dc5a4e6f8999f2b5142a9be04099c6ea0f83b5b0ee9cbdc7","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-e44f0556d255d4bac065","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2f --supports--> v07\n- v06 --refutes--> v29\n- v0b --supports--> v12\n- v08 --supports--> v16\n- v19 --supports--> v1f\n- v10 --supports--> v19\n- v22 --refutes--> v1b\n- v0b --refutes--> v28\n- v30 --refutes--> v00\n- v10 --supports--> v13\n- v29 --refutes--> v26\n- v1a --refutes--> v15\n- v21 --supports--> v10\n- v16 --refutes--> v03\n- v1b --refutes--> v2d\n- v2b --supports--> v30\n- v27 --refutes--> v01\n- v15 --refutes--> v1e\n- v15 --refutes--> v27\n- v0f --supports--> v0e\n- v09 --refutes--> v06\n- v25 --supports--> v14\n- v11 --refutes--> v21\n- v07 --supports--> v05\n- v19 --refutes--> v0c\n- v2b --refutes--> v2a\n- v0f --supports--> v08\n- v05 --refutes--> v22\n- v2b --refutes--> v1c\n- v2d --supports--> v20\n- v18 --supports--> v1d\n- v0c --refutes--> v2f\n- v16 --refutes--> v24\n- v29 --supports--> v2b\n- v11 --supports--> v23\n- v17 --refutes--> v25\n- v16 --supports--> v0a\n- v2e --supports--> v0f\n- v2d --refutes--> v0b\n- v22 --supports--> v02\n- v04 --supports--> v09\n- v22 --refutes--> v2c\n- v1d --refutes--> v2e\n- v30 --refutes--> v17\n- v12 --refutes--> v04\n- v01 --supports--> v11\n- v1b --supports--> v0d\n- v0a --refutes--> v1a\n\n[VERIFIED ROOT]\nv18 = 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":"3c7753cab5bf04c1a6e6e65eedb89da775093fb7f1bf479734912a70aea5f355","schema":"livegraph.boolean-propagation.v1","spec_hash":"e44f0556d255d4bac06539050682acbf4708bdfa19ad4b12d88449f7c3df98d9","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-bc7f2d52ac902c2ec897","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v27 --refutes--> v21\n- v0d --refutes--> v0c\n- v22 --refutes--> v0a\n- v1f --supports--> v1b\n- v27 --refutes--> v07\n- v2d --supports--> v2a\n- v2d --supports--> v0f\n- v16 --refutes--> v1a\n- v2c --refutes--> v06\n- v1a --supports--> v17\n- v2f --supports--> v13\n- v26 --refutes--> v03\n- v16 --supports--> v24\n- v19 --supports--> v2e\n- v0b --supports--> v29\n- v01 --supports--> v09\n- v2e --supports--> v28\n- v25 --refutes--> v20\n- v14 --refutes--> v1e\n- v1b --supports--> v0d\n- v1c --refutes--> v2c\n- v22 --supports--> v05\n- v30 --supports--> v15\n- v27 --refutes--> v08\n- v00 --refutes--> v26\n- v20 --refutes--> v2d\n- v0e --refutes--> v1d\n- v09 --supports--> v11\n- v07 --refutes--> v02\n- v30 --refutes--> v00\n- v1d --supports--> v01\n- v14 --refutes--> v18\n- v09 --refutes--> v30\n- v2d --refutes--> v10\n- v06 --supports--> v19\n- v14 --supports--> v27\n- v04 --refutes--> v2f\n- v17 --supports--> v22\n- v2e --refutes--> v25\n- v0b --supports--> v2b\n- v12 --supports--> v14\n- v2c --refutes--> v23\n- v03 --refutes--> v1f\n- v22 --refutes--> v04\n- v2f --refutes--> v0e\n- v0f --refutes--> v0b\n- v0c --supports--> v12\n- v07 --supports--> v1c\n\n[VERIFIED ROOT]\nv16 = FALSE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d30449ec6bca2cc7f642d586ac698de28f84ac8673671e2412c83881be10d542","schema":"livegraph.boolean-propagation.v1","spec_hash":"bc7f2d52ac902c2ec8972ea642027dc13c82beaa9e5622e6713d4a67505cff8b","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v16\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-9cbe1713d1537e022956","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v25 --supports--> v16\n- v11 --refutes--> v14\n- v23 --supports--> v05\n- v21 --refutes--> v1d\n- v20 --supports--> v06\n- v17 --refutes--> v0d\n- v1b --refutes--> v22\n- v14 --supports--> v02\n- v19 --supports--> v12\n- v08 --supports--> v2f\n- v10 --supports--> v17\n- v18 --supports--> v0b\n- v0b --refutes--> v28\n- v1c --refutes--> v07\n- v30 --refutes--> v1e\n- v28 --supports--> v04\n- v05 --supports--> v11\n- v22 --supports--> v0e\n- v2b --refutes--> v26\n- v00 --supports--> v2c\n- v06 --refutes--> v00\n- v1c --supports--> v30\n- v10 --refutes--> v13\n- v2c --supports--> v18\n- v1f --supports--> v2d\n- v1f --refutes--> v15\n- v22 --refutes--> v29\n- v1e --refutes--> v24\n- v1f --refutes--> v27\n- v29 --refutes--> v1f\n- v0d --refutes--> v21\n- v08 --supports--> v2a\n- v1f --supports--> v0f\n- v21 --refutes--> v19\n- v24 --refutes--> v0c\n- v20 --supports--> v2e\n- v1f --refutes--> v09\n- v22 --supports--> v03\n- v28 --refutes--> v01\n- v2b --refutes--> v1b\n- v19 --supports--> v1c\n- v25 --supports--> v1a\n- v2f --supports--> v2b\n- v0c --refutes--> v23\n- v04 --supports--> v10\n- v02 --supports--> v08\n- v20 --refutes--> v0a\n- v1a --refutes--> v20\n\n[VERIFIED ROOT]\nv25 = 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":"b22f261cb2cb5b0460ff149381fa0c4ce86ed56c2f93338525b39cbc87a640fe","schema":"livegraph.boolean-propagation.v1","spec_hash":"9cbe1713d1537e022956ad8740653728c73fbfdd79586f7079a39d22c87925f6","spec_json":"{\"edges\":[{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v0f\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-68ca96445ef4a74c3ce2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0d --refutes--> v05\n- v11 --refutes--> v02\n- v2a --supports--> v07\n- v24 --refutes--> v19\n- v2f --supports--> v30\n- v17 --refutes--> v13\n- v1c --supports--> v29\n- v06 --supports--> v22\n- v06 --supports--> v15\n- v16 --refutes--> v1c\n- v18 --refutes--> v03\n- v11 --supports--> v2b\n- v29 --refutes--> v21\n- v11 --supports--> v27\n- v03 --supports--> v12\n- v1b --supports--> v0a\n- v25 --supports--> v0c\n- v1d --refutes--> v01\n- v12 --supports--> v2e\n- v0e --supports--> v06\n- v09 --refutes--> v1a\n- v22 --refutes--> v04\n- v07 --supports--> v14\n- v2e --refutes--> v0b\n- v25 --supports--> v26\n- v1b --supports--> v28\n- v18 --supports--> v08\n- v1f --refutes--> v17\n- v1f --refutes--> v10\n- v24 --supports--> v23\n- v19 --refutes--> v2d\n- v01 --supports--> v18\n- v01 --supports--> v1e\n- v1a --refutes--> v16\n- v20 --supports--> v24\n- v0f --supports--> v11\n- v29 --refutes--> v1d\n- v0b --refutes--> v2a\n- v2d --refutes--> v0d\n- v2b --supports--> v1b\n- v16 --refutes--> v2c\n- v13 --refutes--> v20\n- v22 --refutes--> v25\n- v16 --refutes--> v00\n- v2f --refutes--> v09\n- v25 --supports--> v0f\n- v1b --refutes--> v1f\n- v14 --supports--> v0e\n\n[VERIFIED ROOT]\nv2f = 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":"0d9d22936c584b9019f9c785fd5e8fd80b22a1282ae804ca5fdab40cbcda02ac","schema":"livegraph.boolean-propagation.v1","spec_hash":"68ca96445ef4a74c3ce2465ea3dcb97eab098b633eeb4902a441fc55f6eabb55","spec_json":"{\"edges\":[{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v2f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-7f3dbba39d05c758a089","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2f\n- v19 --refutes--> v2a\n- v2e --refutes--> v2b\n- v12 --supports--> v24\n- v2a --refutes--> v04\n- v27 --refutes--> v10\n- v0d --supports--> v00\n- v1c --supports--> v05\n- v16 --supports--> v02\n- v01 --refutes--> v19\n- v22 --supports--> v29\n- v0e --refutes--> v2e\n- v18 --supports--> v20\n- v10 --supports--> v0b\n- v2f --supports--> v25\n- v2e --supports--> v17\n- v19 --supports--> v06\n- v1a --supports--> v01\n- v0f --refutes--> v18\n- v08 --supports--> v0f\n- v22 --refutes--> v30\n- v16 --refutes--> v1b\n- v04 --supports--> v22\n- v21 --refutes--> v09\n- v22 --supports--> v1e\n- v14 --refutes--> v21\n- v20 --supports--> v0d\n- v13 --supports--> v0e\n- v2c --supports--> v1c\n- v1c --refutes--> v1d\n- v16 --supports--> v11\n- v22 --refutes--> v03\n- v04 --supports--> v26\n- v09 --refutes--> v1f\n- v0c --refutes--> v0a\n- v2a --refutes--> v2d\n- v0f --refutes--> v15\n- v09 --supports--> v13\n- v00 --supports--> v14\n- v12 --supports--> v07\n- v00 --supports--> v28\n- v2f --supports--> v23\n- v23 --supports--> v27\n- v03 --supports--> v12\n- v05 --supports--> v1a\n- v0a --refutes--> v2c\n- v10 --refutes--> v16\n- v07 --refutes--> v08\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":"59868e15095fa95eff9a7b3ff82633725804a015db120d32dbd7781fd5d50905","schema":"livegraph.boolean-propagation.v1","spec_hash":"7f3dbba39d05c758a08952ce6b17f836f58c2ed876af869a3a6a51f5608af23b","spec_json":"{\"edges\":[{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0c\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v02\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-3c0de8ef831885cff43b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v29 --refutes--> v11\n- v25 --supports--> v26\n- v00 --refutes--> v13\n- v2d --supports--> v1c\n- v25 --refutes--> v10\n- v13 --supports--> v01\n- v11 --refutes--> v0d\n- v0b --refutes--> v27\n- v27 --refutes--> v06\n- v18 --supports--> v1f\n- v06 --supports--> v1e\n- v1f --refutes--> v20\n- v2f --supports--> v08\n- v09 --supports--> v2a\n- v14 --supports--> v19\n- v12 --refutes--> v23\n- v1a --supports--> v05\n- v0a --supports--> v21\n- v27 --supports--> v16\n- v15 --supports--> v2b\n- v01 --refutes--> v1a\n- v2f --refutes--> v03\n- v19 --supports--> v04\n- v26 --refutes--> v2c\n- v2b --supports--> v29\n- v1c --refutes--> v25\n- v23 --supports--> v0c\n- v2a --supports--> v2f\n- v20 --supports--> v00\n- v20 --supports--> v24\n- v17 --refutes--> v30\n- v19 --supports--> v22\n- v23 --refutes--> v18\n- v0a --refutes--> v02\n- v2f --refutes--> v07\n- v1e --refutes--> v14\n- v19 --supports--> v1b\n- v15 --supports--> v1d\n- v00 --supports--> v0f\n- v2c --supports--> v17\n- v21 --refutes--> v0b\n- v17 --refutes--> v2e\n- v11 --refutes--> v09\n- v2d --supports--> v0e\n- v2a --refutes--> v28\n- v04 --supports--> v2d\n- v17 --supports--> v12\n- v07 --refutes--> v0a\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"33fa6a770b0b7a8e662dad87c6746f23585afe7c2c61561d26ae4eeb4ea9f464","schema":"livegraph.boolean-propagation.v1","spec_hash":"3c0de8ef831885cff43b5da9e8a674027ce8c5f35e6db2568c377cf5121bf041","spec_json":"{\"edges\":[{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-9fcaff9ac09ab86aa192","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2f --refutes--> v1f\n- v2e --supports--> v22\n- v2d --supports--> v18\n- v23 --refutes--> v2b\n- v1c --supports--> v01\n- v29 --refutes--> v07\n- v0f --refutes--> v15\n- v08 --refutes--> v05\n- v2c --supports--> v16\n- v14 --refutes--> v13\n- v07 --supports--> v12\n- v06 --refutes--> v29\n- v10 --refutes--> v2a\n- v01 --supports--> v06\n- v23 --supports--> v00\n- v26 --refutes--> v0e\n- v08 --supports--> v23\n- v20 --refutes--> v1e\n- v24 --refutes--> v25\n- v23 --refutes--> v27\n- v07 --supports--> v1a\n- v11 --refutes--> v2c\n- v13 --supports--> v0a\n- v04 --refutes--> v26\n- v27 --supports--> v19\n- v1e --supports--> v08\n- v26 --refutes--> v2f\n- v19 --supports--> v0c\n- v23 --supports--> v28\n- v12 --supports--> v11\n- v1f --refutes--> v1c\n- v13 --refutes--> v03\n- v16 --supports--> v20\n- v22 --refutes--> v1b\n- v1b --supports--> v1d\n- v03 --refutes--> v10\n- v15 --refutes--> v14\n- v2f --refutes--> v09\n- v0b --refutes--> v2e\n- v29 --supports--> v17\n- v11 --refutes--> v30\n- v1b --supports--> v2d\n- v2c --refutes--> v0d\n- v19 --supports--> v24\n- v2d --supports--> v04\n- v01 --supports--> v02\n- v25 --refutes--> v0f\n- v14 --supports--> v21\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv2a\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"bdafb044220c8d48c6b0c19b8152c0c1555618b1f88762139a19613622057502","schema":"livegraph.boolean-propagation.v1","spec_hash":"9fcaff9ac09ab86aa1921e76ecdec530b1daaba979f00b7777d68bd82b4da3d7","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2a\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-1be588c76b9f889ec18d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v2b\n- v14 --supports--> v1b\n- v17 --refutes--> v13\n- v03 --supports--> v0d\n- v01 --supports--> v27\n- v1c --supports--> v20\n- v2b --refutes--> v1e\n- v0e --refutes--> v00\n- v2f --refutes--> v12\n- v1a --supports--> v15\n- v2d --refutes--> v03\n- v24 --refutes--> v16\n- v06 --supports--> v07\n- v17 --supports--> v19\n- v2c --refutes--> v30\n- v0a --refutes--> v10\n- v1b --refutes--> v05\n- v26 --supports--> v09\n- v28 --supports--> v04\n- v22 --supports--> v2e\n- v2c --supports--> v1a\n- v1e --refutes--> v1f\n- v10 --refutes--> v14\n- v17 --supports--> v0c\n- v22 --refutes--> v1d\n- v27 --refutes--> v2f\n- v08 --supports--> v26\n- v1d --refutes--> v2d\n- v17 --supports--> v0e\n- v07 --supports--> v25\n- v27 --refutes--> v29\n- v2a --supports--> v0a\n- v25 --supports--> v22\n- v03 --supports--> v21\n- v21 --refutes--> v2c\n- v1d --refutes--> v11\n- v2f --supports--> v18\n- v00 --supports--> v28\n- v1a --supports--> v02\n- v0f --refutes--> v17\n- v1b --refutes--> v0b\n- v1c --refutes--> v2a\n- v24 --supports--> v08\n- v05 --refutes--> v06\n- v2f --refutes--> v1c\n- v24 --supports--> v23\n- v1e --refutes--> v01\n- v09 --supports--> v0f\n\n[VERIFIED ROOT]\nv24 = 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":"37a5ac1c5e5617b0cf90045756c683c46f70af52c486ba69d533a0b26dd0ced7","schema":"livegraph.boolean-propagation.v1","spec_hash":"1be588c76b9f889ec18dc9a7a62e4655ca3d3e90cb6737efd8101ae24cf10b4f","spec_json":"{\"edges\":[{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v24\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v15\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-64a854db13d0fbfe7f82","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2d --refutes--> v03\n- v2a --refutes--> v17\n- v16 --refutes--> v09\n- v2b --supports--> v0f\n- v04 --supports--> v1b\n- v13 --refutes--> v06\n- v01 --refutes--> v30\n- v09 --refutes--> v28\n- v2d --refutes--> v29\n- v09 --refutes--> v1f\n- v29 --refutes--> v07\n- v0e --supports--> v08\n- v22 --refutes--> v0e\n- v0d --refutes--> v2a\n- v10 --supports--> v26\n- v2a --refutes--> v05\n- v29 --refutes--> v23\n- v18 --refutes--> v24\n- v0a --supports--> v2c\n- v26 --supports--> v25\n- v1e --refutes--> v2e\n- v1f --supports--> v0c\n- v19 --supports--> v2f\n- v13 --supports--> v2b\n- v27 --refutes--> v11\n- v0f --supports--> v10\n- v0e --refutes--> v20\n- v13 --refutes--> v21\n- v00 --refutes--> v1a\n- v11 --refutes--> v00\n- v25 --supports--> v0a\n- v26 --refutes--> v15\n- v0a --supports--> v01\n- v05 --supports--> v27\n- v19 --refutes--> v14\n- v27 --supports--> v1c\n- v10 --supports--> v0b\n- v18 --refutes--> v13\n- v08 --supports--> v19\n- v0e --refutes--> v12\n- v1a --refutes--> v16\n- v07 --supports--> v04\n- v14 --supports--> v0d\n- v0c --supports--> v1e\n- v02 --refutes--> v2d\n- v2e --supports--> v02\n- v01 --supports--> v22\n- v2a --supports--> v1d\n\n[VERIFIED ROOT]\nv18 = FALSE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"303843c52db48832f6efbf0fa644665fba4413aae34c7f72f9586031bfb9ff2d","schema":"livegraph.boolean-propagation.v1","spec_hash":"64a854db13d0fbfe7f82dbc227540698825ea42f5b92e8a992a5188fbc5e9bcb","spec_json":"{\"edges\":[{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-8cf270ac9e4e2158a7c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v23 --supports--> v01\n- v05 --supports--> v0f\n- v03 --refutes--> v16\n- v27 --supports--> v03\n- v14 --refutes--> v2b\n- v0e --refutes--> v1a\n- v17 --supports--> v29\n- v1a --refutes--> v17\n- v09 --refutes--> v1f\n- v1b --supports--> v12\n- v14 --supports--> v00\n- v11 --refutes--> v28\n- v2e --refutes--> v2a\n- v30 --supports--> v11\n- v17 --supports--> v2d\n- v23 --supports--> v15\n- v22 --supports--> v1e\n- v08 --refutes--> v0c\n- v2c --refutes--> v08\n- v17 --supports--> v2f\n- v0c --refutes--> v14\n- v09 --supports--> v04\n- v1c --refutes--> v0a\n- v21 --supports--> v24\n- v06 --supports--> v27\n- v16 --refutes--> v22\n- v06 --supports--> v0d\n- v0f --refutes--> v2e\n- v23 --supports--> v26\n- v0e --supports--> v07\n- v13 --refutes--> v02\n- v30 --supports--> v19\n- v0f --refutes--> v20\n- v1c --supports--> v13\n- v00 --refutes--> v05\n- v21 --refutes--> v1c\n- v2e --refutes--> v1d\n- v2d --refutes--> v09\n- v12 --supports--> v30\n- v2a --refutes--> v18\n- v1a --supports--> v0b\n- v10 --supports--> v21\n- v18 --supports--> v1b\n- v25 --supports--> v0e\n- v13 --supports--> v25\n- v11 --refutes--> v06\n- v26 --supports--> v10\n- v1f --refutes--> v2c\n\n[VERIFIED ROOT]\nv23 = FALSE\n\n[TARGET]\nv1e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"90fc178dcef021a4a327a24aa5b21f60ee1337916662087e9690ec9bb189017d","schema":"livegraph.boolean-propagation.v1","spec_hash":"8cf270ac9e4e2158a7c0ac1c50ced88b46d8320abf863642f5f8d0840c1090d6","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-14994c523a00b78aa7d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2f\n- v1a --refutes--> v00\n- v30 --supports--> v2a\n- v2a --refutes--> v2d\n- v0d --supports--> v23\n- v1f --supports--> v07\n- v05 --refutes--> v20\n- v02 --refutes--> v18\n- v08 --refutes--> v01\n- v2e --refutes--> v06\n- v06 --supports--> v0c\n- v26 --supports--> v13\n- v2f --refutes--> v0e\n- v15 --supports--> v09\n- v2d --supports--> v2b\n- v00 --supports--> v08\n- v22 --supports--> v1f\n- v0c --refutes--> v1a\n- v1a --supports--> v0f\n- v05 --refutes--> v0b\n- v1b --refutes--> v29\n- v03 --refutes--> v0d\n- v1e --refutes--> v25\n- v20 --supports--> v22\n- v23 --refutes--> v02\n- v1c --supports--> v28\n- v28 --refutes--> v17\n- v1c --supports--> v19\n- v08 --refutes--> v21\n- v27 --refutes--> v1e\n- v17 --supports--> v26\n- v02 --supports--> v30\n- v05 --refutes--> v12\n- v1f --supports--> v11\n- v24 --supports--> v03\n- v1e --refutes--> v2e\n- v2d --refutes--> v1d\n- v15 --refutes--> v10\n- v1d --refutes--> v1b\n- v15 --supports--> v27\n- v21 --refutes--> v05\n- v00 --supports--> v04\n- v29 --supports--> v1c\n- v0d --supports--> v2c\n- v0e --refutes--> v24\n- v20 --refutes--> v16\n- v26 --supports--> v15\n- v2e --refutes--> v14\n\n[VERIFIED ROOT]\nv0a = TRUE\n\n[TARGET]\nv07\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"85d7e654398d7fceae457d7d459353ec333b3f25666a8b0263de619e173aeb8f","schema":"livegraph.boolean-propagation.v1","spec_hash":"14994c523a00b78aa7d1c6c3b9653415a58af38aeeb36f07a9903c779b9aa613","spec_json":"{\"edges\":[{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0a\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v07\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-133c176f9bfc137f2b31","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v24 --supports--> v16\n- v23 --refutes--> v1d\n- v16 --refutes--> v1e\n- v15 --refutes--> v21\n- v19 --supports--> v13\n- v02 --supports--> v2c\n- v2c --refutes--> v27\n- v13 --supports--> v18\n- v29 --supports--> v23\n- v14 --supports--> v05\n- v0d --supports--> v0f\n- v2f --refutes--> v03\n- v07 --refutes--> v0e\n- v2a --refutes--> v1c\n- v05 --refutes--> v2a\n- v1e --refutes--> v0a\n- v1a --refutes--> v06\n- v2c --refutes--> v30\n- v23 --refutes--> v0b\n- v1b --refutes--> v0d\n- v0d --supports--> v26\n- v25 --refutes--> v12\n- v16 --refutes--> v2e\n- v05 --supports--> v1f\n- v15 --refutes--> v19\n- v1c --supports--> v11\n- v07 --supports--> v29\n- v19 --refutes--> v09\n- v01 --supports--> v08\n- v23 --supports--> v2b\n- v11 --supports--> v1a\n- v13 --refutes--> v02\n- v30 --refutes--> v07\n- v29 --refutes--> v22\n- v0f --refutes--> v01\n- v0b --supports--> v0c\n- v06 --refutes--> v1b\n- v2d --supports--> v25\n- v0b --supports--> v20\n- v24 --supports--> v17\n- v0a --supports--> v15\n- v24 --supports--> v00\n- v12 --refutes--> v10\n- v01 --refutes--> v24\n- v20 --refutes--> v2d\n- v10 --refutes--> v28\n- v25 --refutes--> v04\n- v28 --supports--> v2f\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":"dbcc34968da951ef455d51de3d0ae0ddf99b23b36c65a46a9414389a516d790e","schema":"livegraph.boolean-propagation.v1","spec_hash":"133c176f9bfc137f2b31e98a42baccc82ba2ef76a56322d10d6c562e86315c91","spec_json":"{\"edges\":[{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v14\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-46217bb0694da0146cbf","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1a --supports--> v2c\n- v04 --supports--> v2a\n- v0d --refutes--> v0c\n- v15 --refutes--> v0d\n- v24 --refutes--> v00\n- v00 --supports--> v06\n- v25 --supports--> v0a\n- v18 --supports--> v20\n- v28 --supports--> v1b\n- v03 --supports--> v25\n- v23 --refutes--> v2d\n- v08 --supports--> v1a\n- v0f --refutes--> v19\n- v0f --supports--> v10\n- v0c --refutes--> v05\n- v2f --supports--> v22\n- v10 --refutes--> v13\n- v1b --refutes--> v08\n- v20 --supports--> v1c\n- v17 --refutes--> v28\n- v11 --refutes--> v01\n- v25 --refutes--> v04\n- v09 --supports--> v03\n- v11 --supports--> v24\n- v2f --refutes--> v2e\n- v03 --refutes--> v1d\n- v16 --supports--> v23\n- v26 --supports--> v14\n- v23 --refutes--> v15\n- v27 --refutes--> v09\n- v2a --supports--> v2b\n- v28 --refutes--> v0e\n- v0c --supports--> v1f\n- v16 --supports--> v1e\n- v13 --refutes--> v16\n- v0f --refutes--> v12\n- v0d --supports--> v30\n- v1a --supports--> v07\n- v29 --refutes--> v27\n- v23 --refutes--> v21\n- v2c --refutes--> v2f\n- v27 --supports--> v0b\n- v26 --refutes--> v02\n- v00 --supports--> v26\n- v14 --refutes--> v0f\n- v2b --supports--> v18\n- v1c --supports--> v11\n- v1f --supports--> v17\n\n[VERIFIED ROOT]\nv29 = TRUE\n\n[TARGET]\nv2e\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c4d9b7146285ba73295c40f28b96212cb31f6a2f456414e224f082e76b5767b8","schema":"livegraph.boolean-propagation.v1","spec_hash":"46217bb0694da0146cbf8b88e9836a15d7ac5a6f50b7c0025676a4c1efe220d5","spec_json":"{\"edges\":[{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v29\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2e\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-abf57fbcb3e4ef67558f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v28 --supports--> v15\n- v1e --refutes--> v2f\n- v0b --supports--> v0f\n- v19 --refutes--> v06\n- v0b --supports--> v18\n- v25 --supports--> v03\n- v00 --refutes--> v1f\n- v1b --supports--> v09\n- v11 --supports--> v24\n- v1f --refutes--> v10\n- v18 --refutes--> v17\n- v1c --refutes--> v25\n- v01 --supports--> v0a\n- v12 --refutes--> v19\n- v29 --supports--> v14\n- v25 --refutes--> v13\n- v1d --refutes--> v11\n- v25 --supports--> v2b\n- v17 --supports--> v30\n- v0b --supports--> v20\n- v19 --refutes--> v00\n- v10 --supports--> v23\n- v0a --refutes--> v1c\n- v23 --supports--> v22\n- v2a --refutes--> v1b\n- v15 --refutes--> v01\n- v09 --refutes--> v26\n- v22 --supports--> v21\n- v25 --supports--> v2d\n- v22 --supports--> v29\n- v24 --refutes--> v07\n- v14 --supports--> v05\n- v16 --supports--> v0e\n- v2f --refutes--> v1d\n- v02 --supports--> v04\n- v01 --supports--> v0d\n- v09 --refutes--> v16\n- v12 --refutes--> v2c\n- v08 --refutes--> v12\n- v24 --refutes--> v2a\n- v1b --refutes--> v27\n- v30 --supports--> v1e\n- v2a --supports--> v1a\n- v28 --refutes--> v2e\n- v16 --supports--> v08\n- v18 --supports--> v0c\n- v05 --supports--> v28\n- v13 --supports--> v02\n\n[VERIFIED ROOT]\nv0b = FALSE\n\n[TARGET]\nv04\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"de4f6032329645cf5d7edf69b865e85c28d001b802fe36740364e6194c03846c","schema":"livegraph.boolean-propagation.v1","spec_hash":"abf57fbcb3e4ef67558f9dcf57654d18eee11fcdd2dbf69840218fbeb48f6a46","spec_json":"{\"edges\":[{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v04\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-837adab0a6915e18e447","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v30\n- v2f --supports--> v00\n- v18 --refutes--> v04\n- v11 --supports--> v20\n- v20 --refutes--> v16\n- v10 --refutes--> v1c\n- v09 --supports--> v0b\n- v21 --supports--> v2d\n- v27 --refutes--> v23\n- v2c --supports--> v0c\n- v03 --supports--> v2e\n- v1f --refutes--> v1e\n- v25 --supports--> v0a\n- v1f --refutes--> v28\n- v1b --refutes--> v09\n- v26 --refutes--> v1a\n- v2e --refutes--> v08\n- v28 --supports--> v05\n- v1a --supports--> v18\n- v2e --refutes--> v17\n- v1c --supports--> v06\n- v20 --refutes--> v03\n- v24 --supports--> v21\n- v0d --refutes--> v29\n- v2b --refutes--> v1b\n- v0b --refutes--> v0f\n- v11 --supports--> v01\n- v11 --refutes--> v15\n- v05 --refutes--> v27\n- v21 --supports--> v2f\n- v17 --supports--> v2c\n- v0c --refutes--> v19\n- v1f --refutes--> v02\n- v12 --refutes--> v14\n- v1c --refutes--> v0d\n- v1d --supports--> v24\n- v1c --supports--> v13\n- v14 --supports--> v1f\n- v17 --refutes--> v07\n- v0e --supports--> v22\n- v25 --supports--> v10\n- v0c --refutes--> v1d\n- v0d --supports--> v11\n- v22 --refutes--> v2a\n- v00 --refutes--> v26\n- v22 --refutes--> v25\n- v18 --supports--> v2b\n- v0f --refutes--> v12\n\n[VERIFIED ROOT]\nv0e = TRUE\n\n[TARGET]\nv23\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"a3ed90cd16bb2c727f9d762d589fcc3814d110633fa7e4d66a1c46399c4617f5","schema":"livegraph.boolean-propagation.v1","spec_hash":"837adab0a6915e18e4470f47a45503a6f2f1fb7497da59f3575820d27a61280c","spec_json":"{\"edges\":[{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v0e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v23\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-a0881bcca52d86922167","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v25\n- v10 --refutes--> v20\n- v04 --supports--> v10\n- v29 --supports--> v0b\n- v19 --refutes--> v1e\n- v17 --supports--> v13\n- v14 --refutes--> v07\n- v20 --supports--> v0d\n- v23 --supports--> v11\n- v01 --refutes--> v2c\n- v03 --supports--> v18\n- v22 --supports--> v29\n- v1d --refutes--> v02\n- v25 --refutes--> v24\n- v0a --refutes--> v15\n- v00 --refutes--> v28\n- v0c --supports--> v0a\n- v13 --supports--> v2f\n- v22 --supports--> v16\n- v2d --supports--> v0e\n- v1a --supports--> v03\n- v0f --supports--> v1f\n- v1f --refutes--> v0c\n- v22 --supports--> v12\n- v19 --supports--> v00\n- v28 --refutes--> v30\n- v17 --refutes--> v08\n- v0d --supports--> v0f\n- v01 --refutes--> v23\n- v06 --supports--> v21\n- v2e --refutes--> v1b\n- v0a --supports--> v01\n- v0d --supports--> v2b\n- v03 --supports--> v04\n- v24 --refutes--> v17\n- v21 --refutes--> v19\n- v06 --supports--> v2a\n- v27 --supports--> v09\n- v1d --supports--> v26\n- v21 --supports--> v1c\n- v09 --refutes--> v06\n- v28 --supports--> v2e\n- v13 --refutes--> v22\n- v11 --supports--> v14\n- v2d --supports--> v1a\n- v29 --supports--> v1d\n- v1d --supports--> v2d\n- v07 --refutes--> v27\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv1b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c174017962fb416c9877130e8104645e0d718649e53015c160e64c0f1080ed6a","schema":"livegraph.boolean-propagation.v1","spec_hash":"a0881bcca52d8692216784498469e2d5bef560fb877d309d9f9064cc0be9694c","spec_json":"{\"edges\":[{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-493c5796c2bf719c1538","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v11 --refutes--> v1f\n- v06 --supports--> v30\n- v2e --supports--> v29\n- v0c --refutes--> v0a\n- v26 --supports--> v2e\n- v07 --refutes--> v2b\n- v18 --supports--> v24\n- v09 --refutes--> v0d\n- v29 --supports--> v22\n- v13 --supports--> v1e\n- v18 --supports--> v00\n- v13 --refutes--> v14\n- v09 --supports--> v0e\n- v1d --supports--> v02\n- v28 --refutes--> v1a\n- v0c --supports--> v03\n- v22 --supports--> v2a\n- v0a --supports--> v1d\n- v26 --refutes--> v25\n- v0f --supports--> v28\n- v28 --refutes--> v0c\n- v2e --supports--> v20\n- v02 --refutes--> v18\n- v05 --supports--> v0f\n- v22 --refutes--> v1b\n- v0e --supports--> v23\n- v23 --supports--> v10\n- v1f --supports--> v07\n- v0c --refutes--> v2c\n- v1c --supports--> v04\n- v1b --refutes--> v09\n- v27 --supports--> v2d\n- v0a --refutes--> v17\n- v23 --refutes--> v2f\n- v1e --refutes--> v08\n- v28 --supports--> v16\n- v2b --refutes--> v27\n- v12 --supports--> v11\n- v2d --refutes--> v1c\n- v2f --supports--> v21\n- v1e --supports--> v26\n- v04 --supports--> v06\n- v12 --refutes--> v19\n- v27 --refutes--> v0b\n- v04 --refutes--> v15\n- v30 --refutes--> v01\n- v21 --refutes--> v12\n\n[VERIFIED ROOT]\nv13 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"cb4f771c5660ff23924132c946fc544efd98bd96a9fe563fbf5e0bdad996cd2c","schema":"livegraph.boolean-propagation.v1","spec_hash":"493c5796c2bf719c1538333fb40a5d24ae7677187b5d3bdef06b2487c3588dc1","spec_json":"{\"edges\":[{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v13\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v00\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-d2c29456cb0a274c446b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2b --refutes--> v11\n- v25 --supports--> v02\n- v2f --supports--> v0f\n- v27 --refutes--> v1c\n- v18 --supports--> v01\n- v0d --refutes--> v21\n- v05 --supports--> v0e\n- v04 --supports--> v1d\n- v1e --supports--> v07\n- v23 --refutes--> v10\n- v23 --supports--> v1a\n- v30 --refutes--> v2a\n- v04 --supports--> v22\n- v0a --refutes--> v0b\n- v24 --supports--> v13\n- v12 --refutes--> v17\n- v2f --supports--> v00\n- v08 --refutes--> v27\n- v26 --refutes--> v05\n- v2a --refutes--> v18\n- v09 --supports--> v1e\n- v21 --supports--> v26\n- v1c --supports--> v16\n- v23 --refutes--> v0c\n- v1e --refutes--> v2d\n- v16 --refutes--> v14\n- v1d --refutes--> v06\n- v1e --supports--> v23\n- v0b --refutes--> v08\n- v10 --supports--> v28\n- v17 --refutes--> v25\n- v12 --supports--> v15\n- v03 --supports--> v29\n- v05 --refutes--> v1b\n- v04 --supports--> v20\n- v2e --supports--> v0a\n- v29 --supports--> v04\n- v05 --refutes--> v2e\n- v2e --supports--> v19\n- v1f --refutes--> v2f\n- v02 --supports--> v03\n- v2b --supports--> v12\n- v10 --refutes--> v0d\n- v16 --supports--> v1f\n- v24 --refutes--> v30\n- v06 --supports--> v09\n- v0f --supports--> v24\n- v1d --supports--> v2c\n\n[VERIFIED ROOT]\nv2b = 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":"5f89db97af2d33687f50eb93fbe6eafad30cc4178ff6266d4bd23fd0879ea211","schema":"livegraph.boolean-propagation.v1","spec_hash":"d2c29456cb0a274c446beb67125b0369ccdc9e803bb845c63d7387ba967eb38e","spec_json":"{\"edges\":[{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2b\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v01\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-d6ac75aaa71d8a22660f","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v0f --supports--> v00\n- v19 --supports--> v10\n- v29 --supports--> v20\n- v1a --supports--> v05\n- v2a --refutes--> v0a\n- v1f --refutes--> v25\n- v20 --refutes--> v2e\n- v30 --supports--> v04\n- v2b --refutes--> v1a\n- v1b --refutes--> v22\n- v1b --supports--> v11\n- v08 --refutes--> v02\n- v07 --refutes--> v0f\n- v22 --supports--> v2a\n- v2e --refutes--> v08\n- v30 --refutes--> v26\n- v01 --supports--> v19\n- v1b --refutes--> v16\n- v25 --supports--> v12\n- v2d --supports--> v06\n- v2e --supports--> v1d\n- v08 --refutes--> v27\n- v30 --supports--> v1e\n- v29 --refutes--> v14\n- v30 --supports--> v0d\n- v05 --refutes--> v17\n- v1a --refutes--> v2c\n- v0a --supports--> v0c\n- v17 --refutes--> v1b\n- v23 --supports--> v24\n- v20 --refutes--> v13\n- v15 --refutes--> v01\n- v12 --refutes--> v2b\n- v1e --supports--> v29\n- v02 --refutes--> v18\n- v09 --refutes--> v30\n- v03 --supports--> v23\n- v05 --refutes--> v2f\n- v1f --supports--> v28\n- v0b --supports--> v07\n- v0c --supports--> v21\n- v03 --refutes--> v1c\n- v23 --refutes--> v0b\n- v21 --supports--> v09\n- v02 --supports--> v2d\n- v06 --supports--> v03\n- v15 --supports--> v0e\n- v00 --refutes--> v15\n\n[VERIFIED ROOT]\nv1f = 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":"22c62ba832ece588277ff9f75686d7765682bb1804126863a3981d27eed17007","schema":"livegraph.boolean-propagation.v1","spec_hash":"d6ac75aaa71d8a22660fbdd061cbe946cbe06e8f0efb92487db1edc3fe53beb6","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v30\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v1f\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v10\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-a31826943f8b223f9dd2","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2e --refutes--> v11\n- v1d --refutes--> v00\n- v26 --refutes--> v1a\n- v2a --refutes--> v0f\n- v14 --supports--> v0e\n- v03 --refutes--> v20\n- v01 --supports--> v2a\n- v14 --refutes--> v1b\n- v21 --refutes--> v1d\n- v1c --refutes--> v2e\n- v09 --supports--> v17\n- v0e --refutes--> v18\n- v0d --refutes--> v14\n- v24 --supports--> v1f\n- v08 --refutes--> v0d\n- v0a --supports--> v12\n- v25 --refutes--> v01\n- v2a --refutes--> v2b\n- v06 --supports--> v2d\n- v17 --supports--> v04\n- v1f --refutes--> v07\n- v21 --refutes--> v05\n- v2b --refutes--> v22\n- v20 --refutes--> v09\n- v00 --refutes--> v2c\n- v23 --supports--> v26\n- v2c --supports--> v27\n- v26 --supports--> v06\n- v2b --refutes--> v30\n- v18 --refutes--> v21\n- v1f --supports--> v13\n- v04 --supports--> v24\n- v12 --supports--> v0c\n- v09 --refutes--> v02\n- v2d --supports--> v1c\n- v19 --supports--> v25\n- v09 --refutes--> v16\n- v18 --refutes--> v2f\n- v11 --supports--> v19\n- v1d --supports--> v1e\n- v12 --refutes--> v10\n- v1d --refutes--> v28\n- v10 --supports--> v08\n- v03 --supports--> v15\n- v1f --refutes--> v29\n- v14 --refutes--> v0b\n- v2b --refutes--> v0a\n- v07 --supports--> v23\n\n[VERIFIED ROOT]\nv03 = FALSE\n\n[TARGET]\nv27\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1ccc73bac5a565c3a3f3e4e5e1944701ca5a6a74f652788287784a2a27504662","schema":"livegraph.boolean-propagation.v1","spec_hash":"a31826943f8b223f9dd29970c7e7e6545d6d4896e41e730035514e23e1ee56ed","spec_json":"{\"edges\":[{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v03\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v27\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-a02ecf46dddaae05f2d1","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2c --supports--> v1e\n- v13 --supports--> v2f\n- v2b --supports--> v06\n- v0e --supports--> v30\n- v07 --supports--> v25\n- v1b --refutes--> v2a\n- v23 --refutes--> v01\n- v0d --refutes--> v24\n- v10 --refutes--> v0b\n- v2c --supports--> v1c\n- v12 --supports--> v00\n- v23 --refutes--> v1b\n- v2b --supports--> v0a\n- v2d --refutes--> v02\n- v0f --refutes--> v13\n- v04 --refutes--> v15\n- v08 --refutes--> v2e\n- v0d --supports--> v29\n- v28 --supports--> v1a\n- v11 --supports--> v1f\n- v00 --supports--> v27\n- v04 --refutes--> v19\n- v08 --supports--> v2b\n- v2a --refutes--> v12\n- v30 --refutes--> v2c\n- v17 --refutes--> v28\n- v25 --refutes--> v23\n- v0c --refutes--> v22\n- v29 --refutes--> v03\n- v1f --refutes--> v07\n- v2a --refutes--> v14\n- v07 --refutes--> v1d\n- v08 --supports--> v21\n- v1a --refutes--> v20\n- v2f --refutes--> v05\n- v05 --supports--> v0e\n- v04 --supports--> v26\n- v03 --refutes--> v17\n- v02 --refutes--> v10\n- v27 --supports--> v0c\n- v1e --supports--> v04\n- v1a --refutes--> v0f\n- v19 --supports--> v08\n- v1b --supports--> v09\n- v0e --supports--> v18\n- v2b --refutes--> v11\n- v1b --supports--> v16\n- v10 --refutes--> v0d\n\n[VERIFIED ROOT]\nv2d = TRUE\n\n[TARGET]\nv22\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"4311fe7180ceef9a6c7493cc84a4ccb615f87735ca6f767650d3b835fd27a723","schema":"livegraph.boolean-propagation.v1","spec_hash":"a02ecf46dddaae05f2d1553101fd81d03fc1b1bbae1a903090b6a5c928db0431","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1f\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v22\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-bfcbce0b6b2a3cc8cc5b","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2b --supports--> v11\n- v20 --refutes--> v22\n- v28 --supports--> v02\n- v02 --supports--> v19\n- v16 --refutes--> v1e\n- v22 --refutes--> v21\n- v10 --supports--> v1c\n- v15 --refutes--> v0d\n- v2a --supports--> v08\n- v0e --refutes--> v07\n- v21 --supports--> v14\n- v23 --refutes--> v10\n- v1d --supports--> v2f\n- v23 --supports--> v2e\n- v14 --supports--> v28\n- v09 --supports--> v01\n- v07 --supports--> v25\n- v1d --supports--> v13\n- v30 --refutes--> v09\n- v1a --refutes--> v30\n- v01 --supports--> v20\n- v2d --supports--> v0f\n- v16 --refutes--> v0e\n- v19 --supports--> v0a\n- v08 --supports--> v2b\n- v00 --supports--> v15\n- v1a --supports--> v04\n- v2a --refutes--> v05\n- v15 --refutes--> v06\n- v25 --supports--> v23\n- v1b --supports--> v12\n- v13 --refutes--> v1f\n- v19 --supports--> v00\n- v16 --supports--> v0b\n- v27 --supports--> v03\n- v27 --refutes--> v16\n- v11 --refutes--> v2c\n- v11 --supports--> v27\n- v15 --supports--> v26\n- v10 --supports--> v17\n- v15 --supports--> v18\n- v29 --refutes--> v2a\n- v18 --supports--> v29\n- v12 --refutes--> v1d\n- v08 --supports--> v24\n- v0f --refutes--> v1b\n- v13 --supports--> v1a\n- v2d --refutes--> v0c\n\n[VERIFIED ROOT]\nv2d = TRUE\n\n[TARGET]\nv1c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73e89a28f5317836e747981b754da933521fabc1e0c0d13a12025ebb5dbe9a97","schema":"livegraph.boolean-propagation.v1","spec_hash":"bfcbce0b6b2a3cc8cc5b09ff6ba9a4fdead8ae88d3f94aeaba8d531e9bbe7d3c","spec_json":"{\"edges\":[{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v01\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v1f\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2d\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1c\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-670bbe8dc5905ad961c8","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v2c --supports--> v06\n- v08 --supports--> v2e\n- v2e --refutes--> v0c\n- v26 --refutes--> v28\n- v2b --refutes--> v00\n- v27 --supports--> v1e\n- v05 --refutes--> v10\n- v27 --supports--> v29\n- v1d --refutes--> v01\n- v0e --refutes--> v02\n- v2f --supports--> v17\n- v15 --supports--> v0b\n- v1a --supports--> v15\n- v0c --supports--> v24\n- v27 --refutes--> v18\n- v2b --refutes--> v0e\n- v22 --supports--> v03\n- v07 --supports--> v0f\n- v1b --refutes--> v04\n- v12 --supports--> v1f\n- v2a --refutes--> v2c\n- v02 --supports--> v09\n- v18 --refutes--> v16\n- v1b --refutes--> v1c\n- v0c --refutes--> v11\n- v14 --refutes--> v23\n- v0b --supports--> v19\n- v0b --refutes--> v07\n- v01 --refutes--> v27\n- v28 --refutes--> v30\n- v2f --refutes--> v14\n- v06 --supports--> v1d\n- v07 --supports--> v0a\n- v18 --refutes--> v20\n- v2a --refutes--> v0d\n- v2d --refutes--> v1a\n- v0b --refutes--> v21\n- v24 --refutes--> v12\n- v30 --refutes--> v2f\n- v07 --refutes--> v26\n- v18 --supports--> v08\n- v23 --refutes--> v2b\n- v1f --supports--> v2d\n- v1c --refutes--> v2a\n- v2b --refutes--> v13\n- v05 --refutes--> v25\n- v02 --supports--> v22\n- v25 --supports--> v1b\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":"134f21b8ad2da4b49956a9e63e4f7b35207aaf5a70bc8f0af7e45c9f1a533a01","schema":"livegraph.boolean-propagation.v1","spec_hash":"670bbe8dc5905ad961c82a49efa23b966d09641a94cdedd3d87418dec31a4c72","spec_json":"{\"edges\":[{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v05\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v03\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-5af7430211038e4f3c58","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v17 --refutes--> v1b\n- v2e --refutes--> v0e\n- v0c --supports--> v03\n- v04 --supports--> v2c\n- v18 --supports--> v1f\n- v0b --refutes--> v28\n- v03 --refutes--> v1d\n- v1f --supports--> v1e\n- v04 --refutes--> v30\n- v25 --refutes--> v2a\n- v0e --refutes--> v1c\n- v2d --supports--> v21\n- v0f --supports--> v05\n- v30 --supports--> v20\n- v19 --refutes--> v00\n- v17 --supports--> v22\n- v15 --refutes--> v18\n- v12 --refutes--> v04\n- v0e --refutes--> v01\n- v2a --refutes--> v12\n- v2b --refutes--> v27\n- v1f --supports--> v11\n- v0b --supports--> v0f\n- v17 --refutes--> v10\n- v2f --refutes--> v16\n- v30 --refutes--> v23\n- v0c --refutes--> v07\n- v03 --refutes--> v15\n- v26 --refutes--> v2e\n- v13 --refutes--> v2b\n- v16 --refutes--> v13\n- v00 --supports--> v17\n- v14 --supports--> v02\n- v2b --supports--> v06\n- v1c --refutes--> v0d\n- v1e --supports--> v09\n- v02 --refutes--> v2d\n- v00 --refutes--> v08\n- v2a --refutes--> v0a\n- v0d --refutes--> v2f\n- v10 --refutes--> v24\n- v2a --supports--> v1a\n- v09 --supports--> v0b\n- v10 --supports--> v26\n- v23 --refutes--> v0c\n- v12 --supports--> v29\n- v2b --refutes--> v25\n- v05 --refutes--> v14\n\n[VERIFIED ROOT]\nv19 = TRUE\n\n[TARGET]\nv21\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"1b05d7228376fcc717e200982d2e9c093a0e68b7a8b2e1d7f32224c812d2d1f0","schema":"livegraph.boolean-propagation.v1","spec_hash":"5af7430211038e4f3c58e424cc0dc112890d7142944e7a4dde9a225cedbafa13","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v19\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v21\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-d5f253e038949e65ea6a","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1c --supports--> v20\n- v10 --refutes--> v24\n- v1b --refutes--> v28\n- v2e --supports--> v12\n- v0c --refutes--> v2a\n- v0e --refutes--> v0f\n- v28 --refutes--> v0d\n- v07 --refutes--> v29\n- v26 --refutes--> v27\n- v28 --refutes--> v13\n- v2c --supports--> v09\n- v10 --refutes--> v14\n- v24 --supports--> v30\n- v27 --refutes--> v15\n- v07 --supports--> v25\n- v20 --refutes--> v0e\n- v02 --supports--> v1c\n- v07 --supports--> v22\n- v1e --supports--> v1d\n- v27 --refutes--> v1a\n- v12 --refutes--> v0c\n- v08 --refutes--> v1e\n- v16 --refutes--> v18\n- v30 --refutes--> v01\n- v1d --supports--> v19\n- v13 --supports--> v02\n- v04 --refutes--> v0a\n- v06 --supports--> v2d\n- v0a --refutes--> v11\n- v0f --supports--> v08\n- v15 --supports--> v23\n- v16 --refutes--> v10\n- v1d --refutes--> v07\n- v25 --supports--> v2c\n- v2e --supports--> v05\n- v0b --supports--> v26\n- v2c --refutes--> v21\n- v30 --supports--> v2b\n- v28 --supports--> v1f\n- v2b --refutes--> v0b\n- v15 --refutes--> v04\n- v2d --refutes--> v1b\n- v1c --refutes--> v00\n- v0b --supports--> v03\n- v02 --supports--> v2f\n- v09 --refutes--> v16\n- v2a --refutes--> v06\n- v2d --supports--> v17\n\n[VERIFIED ROOT]\nv2e = 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":"f91a8ff9d0f02b6851b0309a3a70b5770a75c663390ee677e96857e8b1013e18","schema":"livegraph.boolean-propagation.v1","spec_hash":"d5f253e038949e65ea6a9144ee8526b4c9b7ce782c970b56a2e50ead1085571e","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2e\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-84ca0f3d6814db46b07c","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1b --supports--> v15\n- v1d --refutes--> v22\n- v05 --refutes--> v02\n- v16 --refutes--> v29\n- v30 --supports--> v24\n- v0d --refutes--> v19\n- v0c --refutes--> v14\n- v25 --refutes--> v23\n- v1d --refutes--> v04\n- v01 --refutes--> v0f\n- v1c --refutes--> v25\n- v0a --refutes--> v1a\n- v15 --supports--> v30\n- v2e --refutes--> v0e\n- v0f --supports--> v06\n- v1d --refutes--> v2a\n- v1a --supports--> v0b\n- v2e --refutes--> v27\n- v1a --refutes--> v0d\n- v04 --refutes--> v12\n- v23 --refutes--> v10\n- v2c --refutes--> v2e\n- v12 --supports--> v17\n- v04 --supports--> v28\n- v0c --refutes--> v26\n- v2e --refutes--> v21\n- v10 --refutes--> v07\n- v14 --supports--> v1b\n- v13 --refutes--> v0a\n- v11 --supports--> v1d\n- v2e --refutes--> v09\n- v21 --supports--> v08\n- v20 --refutes--> v13\n- v16 --supports--> v2c\n- v13 --supports--> v2b\n- v02 --supports--> v20\n- v06 --refutes--> v11\n- v06 --refutes--> v18\n- v1b --refutes--> v1e\n- v23 --supports--> v1f\n- v02 --refutes--> v2f\n- v10 --refutes--> v03\n- v10 --supports--> v2d\n- v19 --refutes--> v00\n- v08 --refutes--> v01\n- v03 --supports--> v16\n- v00 --supports--> v1c\n- v24 --supports--> v05\n\n[VERIFIED ROOT]\nv0c = TRUE\n\n[TARGET]\nv17\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"b2782cf402b3698288c254c1198296d14f5454cbef55e47dcecdbbc318de8d19","schema":"livegraph.boolean-propagation.v1","spec_hash":"84ca0f3d6814db46b07c4fde360ca91eefa9c2381c50939df9c05b0927bc38fa","spec_json":"{\"edges\":[{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v29\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v2e\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v0c\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v17\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-665cd46af25c16e3e42d","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v21\n- v0f --refutes--> v1c\n- v2c --supports--> v1f\n- v2d --refutes--> v0c\n- v1c --supports--> v05\n- v23 --supports--> v28\n- v13 --supports--> v2d\n- v16 --refutes--> v03\n- v0c --supports--> v11\n- v18 --supports--> v27\n- v17 --refutes--> v10\n- v10 --supports--> v09\n- v1a --refutes--> v13\n- v1b --supports--> v29\n- v26 --supports--> v1d\n- v1d --refutes--> v06\n- v02 --refutes--> v30\n- v26 --refutes--> v22\n- v24 --refutes--> v19\n- v0c --supports--> v25\n- v30 --refutes--> v01\n- v0d --refutes--> v12\n- v1c --supports--> v26\n- v1b --supports--> v0f\n- v19 --refutes--> v18\n- v01 --supports--> v1a\n- v06 --supports--> v0a\n- v1f --supports--> v2b\n- v0d --supports--> v20\n- v06 --refutes--> v2a\n- v16 --refutes--> v1b\n- v2d --supports--> v1e\n- v26 --supports--> v15\n- v0e --refutes--> v07\n- v09 --supports--> v0b\n- v18 --refutes--> v14\n- v01 --supports--> v04\n- v28 --supports--> v24\n- v0c --refutes--> v2e\n- v07 --refutes--> v2c\n- v0b --supports--> v0e\n- v0c --refutes--> v00\n- v10 --supports--> v2f\n- v12 --supports--> v17\n- v00 --supports--> v16\n- v14 --supports--> v02\n- v0a --refutes--> v0d\n- v24 --refutes--> v08\n\n[VERIFIED ROOT]\nv23 = TRUE\n\n[TARGET]\nv2b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"73555619cb9e9f416cafebedb13687a3e244a02b9c85afe04a5b4f30d7f537ba","schema":"livegraph.boolean-propagation.v1","spec_hash":"665cd46af25c16e3e42d538e72e394baea3e0b0a92d5a5a64f86eea2e2e5e750","spec_json":"{\"edges\":[{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v23\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v2b\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-1665adde21a53f2034af","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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- v1b --refutes--> v13\n- v03 --refutes--> v08\n- v2f --refutes--> v0a\n- v10 --supports--> v27\n- v25 --refutes--> v0c\n- v06 --refutes--> v2e\n- v24 --supports--> v1a\n- v14 --refutes--> v0f\n- v0f --refutes--> v06\n- v2d --supports--> v03\n- v04 --supports--> v14\n- v30 --refutes--> v21\n- v15 --refutes--> v2d\n- v1a --supports--> v00\n- v07 --refutes--> v30\n- v0c --refutes--> v01\n- v00 --refutes--> v02\n- v1d --refutes--> v07\n- v20 --refutes--> v1b\n- v06 --supports--> v29\n- v18 --refutes--> v1d\n- v18 --supports--> v1f\n- v02 --supports--> v26\n- v03 --supports--> v19\n- v1c --supports--> v10\n- v0f --refutes--> v16\n- v02 --refutes--> v22\n- v2b --refutes--> v2f\n- v0d --supports--> v0b\n- v15 --supports--> v1e\n- v0c --refutes--> v04\n- v21 --supports--> v28\n- v21 --refutes--> v2b\n- v09 --supports--> v25\n- v05 --refutes--> v1c\n- v0b --supports--> v23\n- v2b --supports--> v2a\n- v20 --supports--> v2c\n- v2f --supports--> v05\n- v13 --refutes--> v18\n- v03 --refutes--> v09\n- v30 --refutes--> v17\n- v2e --supports--> v0d\n- v23 --supports--> v20\n- v12 --supports--> v24\n- v10 --refutes--> v12\n- v24 --supports--> v0e\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv26\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e42ff9c53ea7e7e0e7fad21e31d5431f2471eab473c74ae804fd91a24276334a","schema":"livegraph.boolean-propagation.v1","spec_hash":"1665adde21a53f2034afce7276352c108d48397d7e66b0de552621edf37300e5","spec_json":"{\"edges\":[{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v04\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v0b\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v1d\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v24\"},{\"dst\":\"v0c\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v15\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v26\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-41c86fab82d6f21dc4fb","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean 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--> v2c\n- v16 --supports--> v0a\n- v05 --refutes--> v09\n- v0d --refutes--> v04\n- v26 --refutes--> v06\n- v20 --refutes--> v01\n- v03 --refutes--> v27\n- v29 --refutes--> v07\n- v27 --supports--> v17\n- v29 --refutes--> v02\n- v17 --supports--> v00\n- v03 --refutes--> v0e\n- v07 --refutes--> v21\n- v13 --refutes--> v18\n- v06 --refutes--> v15\n- v1c --refutes--> v14\n- v07 --refutes--> v13\n- v2b --supports--> v0b\n- v20 --supports--> v0c\n- v2e --supports--> v30\n- v2c --refutes--> v16\n- v12 --refutes--> v24\n- v0f --supports--> v2b\n- v0b --refutes--> v10\n- v1b --supports--> v05\n- v18 --supports--> v2d\n- v30 --refutes--> v2a\n- v25 --refutes--> v11\n- v00 --refutes--> v0f\n- v17 --refutes--> v19\n- v01 --supports--> v08\n- v0a --supports--> v25\n- v09 --refutes--> v2f\n- v04 --refutes--> v03\n- v11 --refutes--> v1d\n- v14 --refutes--> v12\n- v0b --refutes--> v20\n- v06 --supports--> v0d\n- v29 --supports--> v1a\n- v24 --refutes--> v1b\n- v23 --supports--> v1f\n- v00 --refutes--> v1e\n- v2e --supports--> v28\n- v26 --refutes--> v22\n- v30 --refutes--> v23\n- v23 --supports--> v1c\n- v09 --supports--> v29\n- v18 --refutes--> v26\n\n[VERIFIED ROOT]\nv2e = TRUE\n\n[TARGET]\nv1d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d9469ca26e92108d8e3e0c5caf48b66ec4cb30741eb74c4399a34cdfcfc27fd0","schema":"livegraph.boolean-propagation.v1","spec_hash":"41c86fab82d6f21dc4fb92a16387e7602af8160195d00003dfb7fc83aabdb039","spec_json":"{\"edges\":[{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v01\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v05\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v0a\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v0f\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v17\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v17\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v20\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v29\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0b\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v30\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v2e\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v1d\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-cd1f00b6ace3d67c75c0","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1d --supports--> v05\n- v28 --supports--> v02\n- v04 --refutes--> v0a\n- v27 --refutes--> v17\n- v15 --refutes--> v03\n- v02 --refutes--> v01\n- v04 --refutes--> v19\n- v2f --refutes--> v0b\n- v19 --refutes--> v21\n- v18 --supports--> v15\n- v30 --refutes--> v2f\n- v2f --refutes--> v2a\n- v1b --supports--> v00\n- v11 --refutes--> v18\n- v00 --supports--> v0f\n- v07 --supports--> v1b\n- v22 --refutes--> v16\n- v25 --supports--> v2d\n- v27 --supports--> v06\n- v03 --refutes--> v12\n- v16 --refutes--> v10\n- v2d --supports--> v1f\n- v2c --refutes--> v0e\n- v1d --supports--> v11\n- v19 --refutes--> v09\n- v1e --supports--> v07\n- v0d --supports--> v2e\n- v06 --supports--> v22\n- v29 --supports--> v2c\n- v2e --supports--> v04\n- v2c --refutes--> v23\n- v04 --refutes--> v13\n- v22 --refutes--> v26\n- v1a --supports--> v29\n- v12 --refutes--> v1e\n- v2d --refutes--> v28\n- v22 --supports--> v0c\n- v03 --refutes--> v2b\n- v0b --refutes--> v14\n- v19 --supports--> v1a\n- v27 --refutes--> v20\n- v08 --supports--> v27\n- v28 --refutes--> v1c\n- v01 --refutes--> v1d\n- v00 --refutes--> v08\n- v10 --supports--> v0d\n- v25 --refutes--> v24\n- v23 --supports--> v30\n\n[VERIFIED ROOT]\nv25 = 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":"0b02b58733f802b65a46e912b05b9671c3eb04a79ebf2fe112563a94a5967435","schema":"livegraph.boolean-propagation.v1","spec_hash":"cd1f00b6ace3d67c75c0421bd5649707b24a838d3646a6365f65df40fd01e1e4","spec_json":"{\"edges\":[{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v00\"},{\"dst\":\"v1d\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v0a\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v27\",\"relation\":\"supports\",\"src\":\"v08\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v0b\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v09\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v19\"},{\"dst\":\"v1a\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v11\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v25\",\"root_value\":true,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v14\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-d07c49da63d79dd0fbfc","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v2a\n- v23 --refutes--> v13\n- v0a --refutes--> v23\n- v27 --supports--> v29\n- v00 --refutes--> v01\n- v1f --supports--> v19\n- v2d --refutes--> v1a\n- v0d --refutes--> v18\n- v01 --refutes--> v2c\n- v1b --refutes--> v27\n- v01 --refutes--> v07\n- v14 --refutes--> v03\n- v1f --supports--> v1d\n- v09 --refutes--> v1e\n- v02 --refutes--> v2d\n- v15 --refutes--> v0b\n- v1f --supports--> v0e\n- v23 --supports--> v02\n- v0a --refutes--> v16\n- v09 --refutes--> v17\n- v26 --supports--> v0f\n- v21 --supports--> v14\n- v25 --refutes--> v00\n- v12 --supports--> v04\n- v2d --refutes--> v2e\n- v03 --supports--> v1f\n- v0c --refutes--> v26\n- v20 --supports--> v05\n- v04 --refutes--> v15\n- v2f --refutes--> v11\n- v0e --refutes--> v24\n- v0e --supports--> v10\n- v15 --refutes--> v2b\n- v30 --supports--> v09\n- v2c --refutes--> v1c\n- v1c --supports--> v12\n- v09 --supports--> v1b\n- v22 --supports--> v21\n- v10 --supports--> v25\n- v11 --supports--> v0c\n- v27 --supports--> v08\n- v03 --refutes--> v06\n- v18 --refutes--> v2f\n- v27 --refutes--> v20\n- v27 --supports--> v28\n- v2e --supports--> v30\n- v2b --refutes--> v0d\n- v26 --supports--> v0a\n\n[VERIFIED ROOT]\nv22 = 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":"736808dc3134cde80eaf34f9d9f49b89523c29397b5432ea11545d6b09d3d126","schema":"livegraph.boolean-propagation.v1","spec_hash":"d07c49da63d79dd0fbfc7c6ba845b4f65ae0a7186c3ffc2f4e21f2c3836c4fa1","spec_json":"{\"edges\":[{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v01\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v06\",\"relation\":\"refutes\",\"src\":\"v03\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v09\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v0a\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v18\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v24\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v10\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v2f\",\"relation\":\"refutes\",\"src\":\"v18\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v1c\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v20\"},{\"dst\":\"v14\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v13\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v0f\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v08\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v28\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v27\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v2e\"},{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v30\"}],\"hop\":32,\"root\":\"v22\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v05\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"FALSE","hop":32,"item_id":"lggrpo1-910bbb43ebd007feeb21","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a 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--> v19\n- v21 --supports--> v12\n- v0e --supports--> v21\n- v13 --refutes--> v14\n- v1e --supports--> v16\n- v25 --refutes--> v17\n- v0e --supports--> v05\n- v03 --supports--> v24\n- v2b --supports--> v1b\n- v15 --refutes--> v1a\n- v16 --refutes--> v03\n- v28 --supports--> v1d\n- v0d --supports--> v1f\n- v12 --refutes--> v26\n- v26 --refutes--> v04\n- v22 --refutes--> v10\n- v1a --refutes--> v00\n- v14 --supports--> v2f\n- v26 --refutes--> v28\n- v1e --refutes--> v0f\n- v2c --supports--> v2b\n- v10 --supports--> v06\n- v18 --supports--> v2a\n- v16 --supports--> v1c\n- v2e --supports--> v0c\n- v2a --refutes--> v2e\n- v21 --refutes--> v2d\n- v14 --supports--> v0d\n- v21 --refutes--> v20\n- v24 --refutes--> v23\n- v22 --supports--> v0a\n- v02 --refutes--> v07\n- v1d --supports--> v2c\n- v25 --refutes--> v1e\n- v19 --supports--> v30\n- v0d --refutes--> v0b\n- v13 --supports--> v29\n- v00 --refutes--> v11\n- v24 --refutes--> v01\n- v06 --supports--> v13\n- v28 --refutes--> v08\n- v23 --refutes--> v22\n- v19 --supports--> v25\n- v07 --refutes--> v15\n- v1f --supports--> v0e\n- v02 --supports--> v09\n- v1b --refutes--> v02\n- v07 --refutes--> v27\n\n[VERIFIED ROOT]\nv18 = 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":"07d997cf73101178f41e9119a9026bae0bd52d761dd35efa4a6344c90a641082","schema":"livegraph.boolean-propagation.v1","spec_hash":"910bbb43ebd007feeb2138762a117a2214b41219f3fa982df4f4be96f0fc53eb","spec_json":"{\"edges\":[{\"dst\":\"v11\",\"relation\":\"refutes\",\"src\":\"v00\"},{\"dst\":\"v07\",\"relation\":\"refutes\",\"src\":\"v02\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v02\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v15\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v07\"},{\"dst\":\"v19\",\"relation\":\"refutes\",\"src\":\"v0c\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v0d\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v05\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v21\",\"relation\":\"supports\",\"src\":\"v0e\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v10\"},{\"dst\":\"v26\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v13\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0d\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v14\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v15\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v16\"},{\"dst\":\"v1c\",\"relation\":\"supports\",\"src\":\"v16\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v18\"},{\"dst\":\"v25\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v19\"},{\"dst\":\"v00\",\"relation\":\"refutes\",\"src\":\"v1a\"},{\"dst\":\"v02\",\"relation\":\"refutes\",\"src\":\"v1b\"},{\"dst\":\"v2c\",\"relation\":\"supports\",\"src\":\"v1d\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v16\",\"relation\":\"supports\",\"src\":\"v1e\"},{\"dst\":\"v0e\",\"relation\":\"supports\",\"src\":\"v1f\"},{\"dst\":\"v20\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v2d\",\"relation\":\"refutes\",\"src\":\"v21\"},{\"dst\":\"v12\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v22\"},{\"dst\":\"v22\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v23\",\"relation\":\"refutes\",\"src\":\"v24\"},{\"dst\":\"v17\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v1e\",\"relation\":\"refutes\",\"src\":\"v25\"},{\"dst\":\"v04\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v26\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v28\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v28\"},{\"dst\":\"v2e\",\"relation\":\"refutes\",\"src\":\"v2a\"},{\"dst\":\"v1b\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v2b\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v2e\"}],\"hop\":32,\"root\":\"v18\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v11\"}","split":"eval_ood","view":"graph"}
{"generator":"livegraph-grpo-v1","gold":"TRUE","hop":32,"item_id":"lggrpo1-cfcc65e658e9fd676127","prompt":[{"content":"You are LiveGraph. Follow directed graph edges. The relation 'supports' preserves a Boolean truth value and 'refutes' negates it. Output exactly one token: TRUE or FALSE.","role":"system"},{"content":"[DIRECTED GRAPH; EDGE ORDER IS ARBITRARY]\n- v1e --refutes--> v0d\n- v2f --refutes--> v0e\n- v12 --supports--> v22\n- v2d --refutes--> v10\n- v2f --supports--> v18\n- v1b --supports--> v2a\n- v12 --refutes--> v0b\n- v2f --supports--> v19\n- v11 --refutes--> v01\n- v1a --supports--> v0c\n- v15 --supports--> v00\n- v07 --supports--> v24\n- v26 --supports--> v15\n- v2b --refutes--> v21\n- v06 --refutes--> v1a\n- v29 --supports--> v09\n- v23 --supports--> v1f\n- v11 --supports--> v2d\n- v27 --refutes--> v1c\n- v2c --supports--> v2e\n- v21 --supports--> v26\n- v1c --refutes--> v2c\n- v2b --supports--> v17\n- v2c --supports--> v0a\n- v12 --refutes--> v16\n- v23 --refutes--> v28\n- v06 --supports--> v30\n- v13 --supports--> v04\n- v25 --supports--> v07\n- v0d --supports--> v2f\n- v0f --refutes--> v2b\n- v2d --refutes--> v05\n- v08 --refutes--> v14\n- v0e --refutes--> v25\n- v2d --supports--> v02\n- v07 --supports--> v23\n- v0c --supports--> v1e\n- v03 --supports--> v20\n- v22 --refutes--> v27\n- v09 --refutes--> v08\n- v04 --refutes--> v1b\n- v2a --supports--> v06\n- v25 --supports--> v1d\n- v10 --refutes--> v12\n- v15 --supports--> v13\n- v2c --refutes--> v03\n- v03 --supports--> v29\n- v14 --refutes--> v0f\n\n[VERIFIED ROOT]\nv11 = FALSE\n\n[TARGET]\nv28\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d28b1a0139df984ee709a5965ec586c91f5ac61651002fd28ab27b040e8a023d","schema":"livegraph.boolean-propagation.v1","spec_hash":"cfcc65e658e9fd676127fe7b879812923b44ed99ac4b37bb872419dce75923a8","spec_json":"{\"edges\":[{\"dst\":\"v20\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v29\",\"relation\":\"supports\",\"src\":\"v03\"},{\"dst\":\"v1b\",\"relation\":\"refutes\",\"src\":\"v04\"},{\"dst\":\"v1a\",\"relation\":\"refutes\",\"src\":\"v06\"},{\"dst\":\"v30\",\"relation\":\"supports\",\"src\":\"v06\"},{\"dst\":\"v23\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v24\",\"relation\":\"supports\",\"src\":\"v07\"},{\"dst\":\"v14\",\"relation\":\"refutes\",\"src\":\"v08\"},{\"dst\":\"v08\",\"relation\":\"refutes\",\"src\":\"v09\"},{\"dst\":\"v1e\",\"relation\":\"supports\",\"src\":\"v0c\"},{\"dst\":\"v2f\",\"relation\":\"supports\",\"src\":\"v0d\"},{\"dst\":\"v25\",\"relation\":\"refutes\",\"src\":\"v0e\"},{\"dst\":\"v2b\",\"relation\":\"refutes\",\"src\":\"v0f\"},{\"dst\":\"v12\",\"relation\":\"refutes\",\"src\":\"v10\"},{\"dst\":\"v01\",\"relation\":\"refutes\",\"src\":\"v11\"},{\"dst\":\"v2d\",\"relation\":\"supports\",\"src\":\"v11\"},{\"dst\":\"v0b\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v16\",\"relation\":\"refutes\",\"src\":\"v12\"},{\"dst\":\"v22\",\"relation\":\"supports\",\"src\":\"v12\"},{\"dst\":\"v04\",\"relation\":\"supports\",\"src\":\"v13\"},{\"dst\":\"v0f\",\"relation\":\"refutes\",\"src\":\"v14\"},{\"dst\":\"v00\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v13\",\"relation\":\"supports\",\"src\":\"v15\"},{\"dst\":\"v0c\",\"relation\":\"supports\",\"src\":\"v1a\"},{\"dst\":\"v2a\",\"relation\":\"supports\",\"src\":\"v1b\"},{\"dst\":\"v2c\",\"relation\":\"refutes\",\"src\":\"v1c\"},{\"dst\":\"v0d\",\"relation\":\"refutes\",\"src\":\"v1e\"},{\"dst\":\"v26\",\"relation\":\"supports\",\"src\":\"v21\"},{\"dst\":\"v27\",\"relation\":\"refutes\",\"src\":\"v22\"},{\"dst\":\"v28\",\"relation\":\"refutes\",\"src\":\"v23\"},{\"dst\":\"v1f\",\"relation\":\"supports\",\"src\":\"v23\"},{\"dst\":\"v07\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v1d\",\"relation\":\"supports\",\"src\":\"v25\"},{\"dst\":\"v15\",\"relation\":\"supports\",\"src\":\"v26\"},{\"dst\":\"v1c\",\"relation\":\"refutes\",\"src\":\"v27\"},{\"dst\":\"v09\",\"relation\":\"supports\",\"src\":\"v29\"},{\"dst\":\"v06\",\"relation\":\"supports\",\"src\":\"v2a\"},{\"dst\":\"v21\",\"relation\":\"refutes\",\"src\":\"v2b\"},{\"dst\":\"v17\",\"relation\":\"supports\",\"src\":\"v2b\"},{\"dst\":\"v03\",\"relation\":\"refutes\",\"src\":\"v2c\"},{\"dst\":\"v0a\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v2e\",\"relation\":\"supports\",\"src\":\"v2c\"},{\"dst\":\"v05\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v10\",\"relation\":\"refutes\",\"src\":\"v2d\"},{\"dst\":\"v02\",\"relation\":\"supports\",\"src\":\"v2d\"},{\"dst\":\"v0e\",\"relation\":\"refutes\",\"src\":\"v2f\"},{\"dst\":\"v18\",\"relation\":\"supports\",\"src\":\"v2f\"},{\"dst\":\"v19\",\"relation\":\"supports\",\"src\":\"v2f\"}],\"hop\":32,\"root\":\"v11\",\"root_value\":false,\"schema\":\"livegraph.boolean-propagation.v1\",\"target\":\"v28\"}","split":"eval_ood","view":"graph"}