{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v11\n- v11 --refutes--> v0a\n- v0a --refutes--> v12\n- v12 --refutes--> v17\n- v17 --supports--> v0d\n- v0d --refutes--> v16\n- v16 --refutes--> v01\n- v01 --refutes--> v1e\n- v1e --supports--> v1b\n- v1b --supports--> v00\n- v00 --refutes--> v0e\n- v0e --supports--> v19\n- v19 --refutes--> v1d\n- v1d --supports--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v08 --supports--> v10\n- v0c --supports--> v03\n- v11 --refutes--> v18\n- v16 --supports--> v02\n- v16 --supports--> v0b\n- v16 --supports--> v1c\n- v1b --refutes--> v0f\n- v1b --supports--> v13\n- v1d --supports--> v14\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":"ba0c51cddfb1ccf4c903d335d26e954c037a7aa5ebf362250496f55f461af289","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v00\n- v00 --refutes--> v14\n- v14 --refutes--> v19\n- v19 --supports--> v18\n- v18 --supports--> v16\n- v16 --supports--> v11\n- v11 --supports--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v1d\n- v1d --supports--> v1b\n- v1b --supports--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v1c\n- v1c --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v02 --refutes--> v0a\n- v0b --refutes--> v05\n- v0e --supports--> v13\n- v11 --supports--> v17\n- v14 --refutes--> v04\n- v15 --supports--> v1e\n- v16 --refutes--> v10\n- v19 --refutes--> v0f\n- v1d --supports--> v03\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":"49dfa10688ef39ee22a9261e9be9a9c16cd20d305afffffa9a9c48deef6b50c0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v19\n- v19 --supports--> v13\n- v13 --supports--> v02\n- v02 --refutes--> v15\n- v15 --refutes--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v1b\n- v1b --supports--> v1c\n- v1c --supports--> v11\n- v11 --refutes--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v09\n- v09 --refutes--> v1e\n- v1e --supports--> v0e\n- v0e --refutes--> v1a\n- v1a --supports--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v02 --supports--> v0c\n- v04 --supports--> v17\n- v0b --supports--> v0a\n- v0e --supports--> v18\n- v10 --refutes--> v14\n- v15 --refutes--> v0f\n- v15 --supports--> v0d\n- v1a --refutes--> v16\n- v1e --supports--> v12\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":"786a7937ff9d4c49e731f6984b403a92dfb45ce8162759e18baf8aa37c6d6d5c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --supports--> v18\n- v18 --supports--> v02\n- v02 --refutes--> v15\n- v15 --supports--> v1c\n- v1c --supports--> v1e\n- v1e --supports--> v19\n- v19 --refutes--> v07\n- v07 --refutes--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v04\n- v04 --supports--> v10\n- v10 --refutes--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0e\n- v05 --supports--> v08\n- v06 --supports--> v1b\n- v07 --supports--> v00\n- v09 --supports--> v11\n- v0c --supports--> v03\n- v0d --supports--> v16\n- v19 --refutes--> v0a\n- v19 --refutes--> v13\n- v1e --supports--> v1a\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":"05f2015b8368b8cff3298cbecb9d1c3bd722a9f358474253440b121a768ff72a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v1b\n- v1b --supports--> v01\n- v01 --supports--> v09\n- v09 --supports--> v04\n- v04 --refutes--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v1d\n- v1d --refutes--> v14\n- v14 --refutes--> v06\n- v06 --supports--> v1a\n- v1a --supports--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v19\n- v19 --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v1c\n- v06 --supports--> v02\n- v0b --refutes--> v15\n- v0b --supports--> v08\n- v0d --refutes--> v1e\n- v17 --refutes--> v11\n- v1a --refutes--> v0e\n- v1a --supports--> v07\n- v1a --supports--> v16\n- v1d --supports--> v13\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":"8b23e15d2f9e430c5e8a73973d525c9d3718a15033a276d0fca57e7412022c7e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v1c\n- v1c --supports--> v04\n- v04 --supports--> v18\n- v18 --supports--> v1d\n- v1d --refutes--> v10\n- v10 --refutes--> v1b\n- v1b --refutes--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v0a\n- v0a --supports--> v1a\n- v1a --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v13\n- v03 --refutes--> v12\n- v07 --refutes--> v05\n- v0e --refutes--> v0f\n- v10 --refutes--> v15\n- v11 --supports--> v16\n- v18 --refutes--> v17\n- v19 --refutes--> v1e\n- v19 --supports--> v06\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":"7d72f7565b4cf26346b42df9ba77122eb35d00efb57f625b806da5f718735097","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v07\n- v07 --supports--> v1c\n- v1c --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v0e\n- v0e --refutes--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v1d\n- v1d --refutes--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v1b\n- v1b --supports--> v0f\n- v0f --refutes--> v1a\n- v1a --supports--> v04\n- v04 --supports--> v1e\n- v1e --refutes--> v17\n- v17 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v14\n- v14 --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v10\n- v04 --refutes--> v0b\n- v06 --supports--> v16\n- v07 --refutes--> v12\n- v0c --supports--> v09\n- v11 --supports--> v15\n- v13 --refutes--> v03\n- v17 --supports--> v08\n- v1b --supports--> v00\n- v1c --supports--> v05\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":"aafab22c30ad5573d141fff9f9335a7b9ac862882ccb44f785670cd1ddb029ca","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v11\n- v11 --supports--> v1b\n- v1b --supports--> v00\n- v00 --refutes--> v10\n- v10 --refutes--> v06\n- v06 --refutes--> v18\n- v18 --supports--> v12\n- v12 --supports--> v07\n- v07 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v15\n- v15 --supports--> v0d\n- v0d --refutes--> v13\n- v13 --supports--> v02\n- v02 --supports--> v16\n- v16 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v02 --supports--> v17\n- v0c --refutes--> v0b\n- v0d --refutes--> v0a\n- v0e --refutes--> v19\n- v0f --refutes--> v03\n- v12 --refutes--> v05\n- v16 --refutes--> v1e\n- v18 --supports--> v1c\n- v1d --supports--> v1a\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":"e261e9a2be9070428c4fc1cd17f74f59445e269333bd2bc55d34a271037790a2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v05\n- v05 --refutes--> v1d\n- v1d --supports--> v15\n- v15 --supports--> v1b\n- v1b --refutes--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v12\n- v12 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v07\n- v07 --supports--> v0e\n- v0e --refutes--> v19\n- v19 --supports--> v0a\n- v0a --supports--> v17\n- v17 --supports--> v1c\n- v1c --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v16\n- v0a --refutes--> v0c\n- v0d --supports--> v18\n- v0e --supports--> v13\n- v10 --supports--> v00\n- v10 --supports--> v02\n- v14 --refutes--> v06\n- v15 --supports--> v1a\n- v17 --supports--> v01\n- v1c --refutes--> v1e\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":"2d090f98ad978a29f382f9779504fab346ebd5ef4cc318e056feb725bee99419","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v08\n- v08 --supports--> v10\n- v10 --supports--> v13\n- v13 --refutes--> v19\n- v19 --refutes--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v11\n- v11 --supports--> v1e\n- v1e --refutes--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --supports--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v1a\n- v1a --refutes--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v1b\n- v1b --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v02\n- v12 --refutes--> v01\n- v12 --supports--> v05\n- v17 --refutes--> v1c\n- v17 --supports--> v0a\n- v19 --refutes--> v07\n- v19 --supports--> v0d\n- v1a --refutes--> v0b\n- v1e --refutes--> v0f\n- v1e --supports--> v1d\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":"5537b4e45d8105142ce9ec1ebc11ab8577ad6dd0d0786b2db343fe461816c84b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v18\n- v18 --refutes--> v13\n- v13 --refutes--> v12\n- v12 --refutes--> v15\n- v15 --supports--> v0f\n- v0f --supports--> v01\n- v01 --refutes--> v07\n- v07 --supports--> v14\n- v14 --refutes--> v19\n- v19 --supports--> v1b\n- v1b --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v04\n- v04 --supports--> v03\n- v03 --supports--> v17\n- v17 --supports--> v1a\n- v1a --supports--> v16\n- v16 --refutes--> v10\n- v10 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1d\n- v09 --refutes--> v00\n- v0b --refutes--> v0e\n- v0f --supports--> v0c\n- v12 --refutes--> v02\n- v13 --refutes--> v06\n- v16 --supports--> v1c\n- v1a --refutes--> v1e\n- v1a --supports--> v11\n- v1b --supports--> v0d\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":"5e031d7c1738e6f2068554cf6937e12aca948f088e67c40b7c666e5fd4e97d50","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v16\n- v16 --supports--> v1a\n- v1a --supports--> v12\n- v12 --supports--> v14\n- v14 --supports--> v05\n- v05 --supports--> v18\n- v18 --refutes--> v0a\n- v0a --supports--> v17\n- v17 --refutes--> v02\n- v02 --supports--> v10\n- v10 --refutes--> v11\n- v11 --supports--> v1d\n- v1d --supports--> v1c\n- v1c --supports--> v0b\n- v0b --supports--> v0e\n- v0e --refutes--> v13\n- v13 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v0c\n- v05 --supports--> v01\n- v07 --supports--> v03\n- v0a --refutes--> v19\n- v0b --supports--> v1e\n- v0d --refutes--> v1b\n- v0d --supports--> v00\n- v0d --supports--> v06\n- v17 --supports--> v09\n- v1c --refutes--> v04\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":"c2b0f988fc9a01669a3a264e66ffea983d99601b50ba7a807a7b47e7a683b5ed","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v17\n- v17 --supports--> v1b\n- v1b --refutes--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v1a\n- v1a --supports--> v03\n- v03 --refutes--> v1d\n- v1d --supports--> v06\n- v06 --supports--> v1c\n- v1c --supports--> v01\n- v01 --refutes--> v1e\n- v1e --supports--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v08\n- v08 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --refutes--> v11\n- v11 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0a\n- v03 --supports--> v0d\n- v06 --supports--> v04\n- v07 --supports--> v09\n- v11 --refutes--> v0f\n- v12 --refutes--> v02\n- v17 --supports--> v18\n- v1c --supports--> v0e\n- v1d --supports--> v13\n- v1e --supports--> v14\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":"b2d4129bfa630a0ed1b30d56ef24218782fa65be5dc60698fa9c094366a91ae1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v0c\n- v0c --refutes--> v13\n- v13 --refutes--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v17\n- v17 --supports--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v18\n- v18 --supports--> v1d\n- v1d --refutes--> v00\n- v00 --supports--> v1b\n- v1b --supports--> v1e\n- v1e --supports--> v16\n- v16 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v06\n- v08 --refutes--> v14\n- v0b --supports--> v15\n- v12 --supports--> v03\n- v16 --supports--> v09\n- v18 --refutes--> v01\n- v18 --refutes--> v0e\n- v18 --refutes--> v19\n- v1d --supports--> v0f\n- v1e --refutes--> v0a\n\n[VERIFIED ROOT]\nv0d = FALSE\n\n[TARGET]\nv05\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"83ee11913e1c9dd8f07891cae65899ad47b0b0362a032540bf5d5280c703508f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v1c\n- v1c --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v12\n- v12 --refutes--> v1b\n- v1b --supports--> v11\n- v11 --supports--> v05\n- v05 --supports--> v07\n- v07 --supports--> v17\n- v17 --supports--> v1a\n- v1a --supports--> v14\n- v14 --supports--> v19\n- v19 --refutes--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v0e\n- v0e --supports--> v0f\n- v0f --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v0e --refutes--> v01\n- v0e --refutes--> v13\n- v0e --supports--> v0b\n- v0f --refutes--> v04\n- v14 --supports--> v03\n- v15 --supports--> v08\n- v19 --supports--> v18\n- v1b --refutes--> v10\n- v1b --supports--> v16\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":"f9e559e37c7e1af7ca31075eb485e749db98f5f1ea32cc52e102a82dd37786f7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v17\n- v17 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v1b\n- v1b --supports--> v14\n- v14 --supports--> v18\n- v18 --supports--> v0a\n- v0a --refutes--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v1c\n- v1c --refutes--> v15\n- v15 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v05\n- v06 --refutes--> v11\n- v08 --refutes--> v0f\n- v08 --refutes--> v1d\n- v0c --supports--> v0b\n- v12 --refutes--> v1a\n- v12 --supports--> v01\n- v14 --refutes--> v02\n- v1b --supports--> v16\n- v1c --refutes--> v19\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":"3d8120ad52dc0265e1b6925ee122db5bd3b1e3bf011002721f7e43011f46ca74","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v1e\n- v1e --supports--> v16\n- v16 --supports--> v15\n- v15 --refutes--> v06\n- v06 --supports--> v05\n- v05 --supports--> v03\n- v03 --supports--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v08\n- v08 --supports--> v02\n- v02 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --supports--> v13\n- v13 --supports--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v1b\n- v1b --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0d\n- v03 --supports--> v1d\n- v04 --refutes--> v01\n- v0a --supports--> v0e\n- v0c --refutes--> v18\n- v0c --supports--> v10\n- v0c --supports--> v1a\n- v14 --refutes--> v09\n- v15 --refutes--> v17\n- v15 --supports--> v12\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":"fef2bd23834feb1e7f36786b9c66755ac26ef731012198badc683493a4ce3d5a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v1c\n- v1c --refutes--> v18\n- v18 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v11\n- v11 --supports--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v0b\n- v0b --refutes--> v15\n- v15 --refutes--> v16\n- v16 --refutes--> v1b\n- v1b --supports--> v17\n- v17 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v14\n- v07 --supports--> v02\n- v0b --refutes--> v13\n- v0e --refutes--> v19\n- v0f --refutes--> v1d\n- v0f --supports--> v1e\n- v11 --refutes--> v0c\n- v11 --supports--> v09\n- v17 --supports--> v12\n- v18 --refutes--> v0a\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":"e2a664e906fb6ff10c6782ed73eb4f61c22e3ae9656d08a303c4df117f63cc0f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0a\n- v0a --supports--> v02\n- v02 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v1e\n- v1e --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v17\n- v17 --supports--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v12\n- v12 --supports--> v14\n- v14 --supports--> v13\n- v13 --supports--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v07 --refutes--> v1b\n- v08 --refutes--> v19\n- v08 --supports--> v06\n- v15 --refutes--> v0e\n- v15 --refutes--> v11\n- v15 --refutes--> v1d\n- v17 --refutes--> v05\n- v17 --refutes--> v0f\n- v1e --supports--> v03\n\n[VERIFIED ROOT]\nv08 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"ea1c7a093ee300768b80093befc64fd177d9816e81ae816d7636d7e540d45798","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v1e\n- v1e --refutes--> v0c\n- v0c --supports--> v1d\n- v1d --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v11\n- v11 --supports--> v1a\n- v1a --refutes--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v19\n- v19 --supports--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v13\n- v13 --refutes--> v03\n- v03 --refutes--> v18\n- v18 --refutes--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v1c\n- v02 --supports--> v14\n- v03 --supports--> v15\n- v0b --refutes--> v00\n- v0c --refutes--> v16\n- v0e --supports--> v10\n- v13 --supports--> v09\n- v17 --supports--> v04\n- v1e --supports--> v0d\n- v1e --supports--> v12\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":"6cd384d556956914864e97bec18cee2bcaf018d7aae925f98b95a2047b7ec881","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v11\n- v11 --refutes--> v1a\n- v1a --supports--> v05\n- v05 --refutes--> v1d\n- v1d --supports--> v06\n- v06 --supports--> v18\n- v18 --supports--> v08\n- v08 --refutes--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v19\n- v19 --refutes--> v17\n- v17 --refutes--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v1e\n- v1e --supports--> v13\n- v13 --refutes--> v15\n- v15 --supports--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v0c\n- v0c --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v01\n- v04 --supports--> v10\n- v0c --refutes--> v0d\n- v11 --supports--> v07\n- v17 --supports--> v0b\n- v1a --supports--> v0f\n- v1d --refutes--> v0e\n- v1d --refutes--> v1b\n- v1e --supports--> v00\n- v1e --supports--> v0a\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":"60340c145c6d024770c277665f67d1407906a2c3390eb09dc711382ade435006","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v01\n- v01 --refutes--> v1a\n- v1a --refutes--> v19\n- v19 --refutes--> v1e\n- v1e --supports--> v1b\n- v1b --refutes--> v05\n- v05 --supports--> v1c\n- v1c --refutes--> v07\n- v07 --supports--> v12\n- v12 --supports--> v06\n- v06 --supports--> v18\n- v18 --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --supports--> v15\n- v15 --refutes--> v11\n- v11 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0b\n- v07 --supports--> v08\n- v07 --supports--> v1d\n- v11 --refutes--> v00\n- v11 --supports--> v09\n- v16 --supports--> v0f\n- v19 --supports--> v17\n- v1a --refutes--> v0a\n- v1e --refutes--> v03\n- v1e --supports--> v13\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":"50add9d9b0d17dc5437b6f8d9e05902a552597440fd7ec327a7a213e6de9b3c0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v03\n- v03 --supports--> v19\n- v19 --supports--> v14\n- v14 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v1a\n- v1a --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v1e\n- v1e --refutes--> v01\n- v01 --supports--> v18\n- v18 --supports--> v0c\n- v0c --supports--> v1d\n- v1d --supports--> v10\n- v10 --supports--> v05\n- v05 --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v17\n- v01 --supports--> v1c\n- v03 --refutes--> v15\n- v07 --supports--> v00\n- v07 --supports--> v0e\n- v08 --supports--> v11\n- v0a --refutes--> v16\n- v14 --supports--> v04\n- v1e --refutes--> v13\n- v1e --supports--> v09\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":"5b4332f32c145f3fcafdf080347b88d776f2daf04de71fd66d2f6c20283f859c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v11\n- v11 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --supports--> v15\n- v15 --refutes--> v0e\n- v0e --refutes--> v05\n- v05 --supports--> v06\n- v06 --supports--> v12\n- v12 --refutes--> v16\n- v16 --refutes--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v0c\n- v0c --supports--> v13\n- v13 --refutes--> v17\n- v17 --supports--> v0f\n- v0f --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v14\n- v07 --refutes--> v1a\n- v0a --refutes--> v00\n- v0b --refutes--> v19\n- v0b --refutes--> v1c\n- v11 --supports--> v03\n- v12 --refutes--> v1e\n- v15 --supports--> v1b\n- v15 --supports--> v1d\n- v16 --refutes--> v01\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":"d7ce1ec32c11ba2f943fe67a39e42cd1c0a317db7288f56d65f0794b973ffcbc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v14\n- v14 --refutes--> v06\n- v06 --supports--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v08\n- v08 --supports--> v1b\n- v1b --supports--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v1d\n- v1d --supports--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v13\n- v13 --supports--> v07\n- v07 --supports--> v03\n- v03 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v04\n- v05 --refutes--> v01\n- v06 --supports--> v0c\n- v07 --refutes--> v02\n- v07 --supports--> v16\n- v08 --supports--> v1a\n- v12 --refutes--> v0b\n- v13 --supports--> v10\n- v15 --supports--> v0a\n- v1b --refutes--> v1e\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":"98f1e27f6b6a0c673d86d7197a1d4cfb324857eec8658db18b18193a4e34e21f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0b\n- v0b --supports--> v16\n- v16 --supports--> v1b\n- v1b --refutes--> v1e\n- v1e --refutes--> v17\n- v17 --refutes--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v11\n- v11 --supports--> v15\n- v15 --supports--> v03\n- v03 --supports--> v1c\n- v1c --refutes--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v08\n- v08 --supports--> v0e\n- v0e --supports--> v14\n- v14 --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v06\n- v07 --supports--> v1a\n- v0d --supports--> v1d\n- v11 --refutes--> v0f\n- v15 --refutes--> v13\n- v16 --refutes--> v18\n- v19 --supports--> v05\n- v1b --refutes--> v12\n- v1b --supports--> v10\n- v1c --supports--> v09\n\n[VERIFIED ROOT]\nv01 = TRUE\n\n[TARGET]\nv00\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"5dd1fd8f0cdfdcb25e13e75e40b8dc57dea3ef39be17c65d098691abc9786a4f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --supports--> v1c\n- v1c --refutes--> v10\n- v10 --refutes--> v17\n- v17 --refutes--> v08\n- v08 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v07\n- v07 --supports--> v11\n- v11 --refutes--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v19\n- v19 --supports--> v02\n- v02 --supports--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v01\n- v07 --refutes--> v1d\n- v0b --supports--> v1a\n- v0b --supports--> v1e\n- v0c --refutes--> v16\n- v0c --refutes--> v18\n- v0e --supports--> v05\n- v12 --supports--> v0d\n- v14 --refutes--> v06\n- v17 --supports--> v0f\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":"ee229d9818e25bc59f0dff0146a22fc6b8a509306771f14eb8318e31c46640f4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v00\n- v00 --refutes--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --supports--> v1b\n- v1b --refutes--> v16\n- v16 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v1e\n- v1e --supports--> v1d\n- v1d --supports--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v12\n- v12 --refutes--> v18\n- v18 --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0b\n- v11 --supports--> v0e\n- v16 --refutes--> v01\n- v18 --supports--> v02\n- v1b --refutes--> v17\n- v1b --supports--> v04\n- v1b --supports--> v0a\n- v1e --refutes--> v0d\n- v1e --supports--> v14\n- v1e --supports--> v19\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":"1544ff6432aa39059e4347bb302d065debc3515db3c12736d964824589295100","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v1e\n- v1e --refutes--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v1d\n- v1d --supports--> v18\n- v18 --supports--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v14\n- v14 --refutes--> v10\n- v10 --supports--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v1c\n- v1c --supports--> v1b\n- v1b --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --refutes--> v11\n- v05 --supports--> v19\n- v06 --supports--> v09\n- v0c --supports--> v0f\n- v10 --supports--> v13\n- v15 --supports--> v04\n- v1d --refutes--> v16\n- v1e --refutes--> v0e\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":"834bbd265ed03631d9d478d362556c94edcf35ea48b91492c4f5347b206fbd29","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v11\n- v11 --refutes--> v1e\n- v1e --refutes--> v1d\n- v1d --refutes--> v18\n- v18 --supports--> v19\n- v19 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v08\n- v08 --supports--> v07\n- v07 --supports--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v13\n- v13 --supports--> v1a\n- v1a --refutes--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v05 --supports--> v0e\n- v07 --refutes--> v1b\n- v0d --supports--> v02\n- v12 --refutes--> v0a\n- v12 --supports--> v04\n- v13 --refutes--> v1c\n- v17 --supports--> v16\n- v19 --supports--> v0f\n- v1e --supports--> v15\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":"9a5fc03c063bf06c23651c1e400cf54b960c7f9a1280c7b13a1506c79835a9af","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v1b\n- v1b --refutes--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v0d\n- v0d --refutes--> v19\n- v19 --refutes--> v10\n- v10 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --supports--> v14\n- v14 --refutes--> v17\n- v17 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v1a\n- v1a --refutes--> v11\n- v11 --refutes--> v03\n- v03 --supports--> v18\n- v18 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v09\n- v00 --supports--> v1d\n- v03 --refutes--> v0a\n- v04 --supports--> v01\n- v06 --refutes--> v1c\n- v0b --supports--> v16\n- v10 --refutes--> v12\n- v10 --supports--> v0e\n- v11 --supports--> v13\n- v17 --supports--> v02\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":"96c58901ec512af12de83191d4e1471a513536beffb76f64ac262a3de821ff33","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v16\n- v16 --refutes--> v19\n- v19 --refutes--> v18\n- v18 --refutes--> v1c\n- v1c --supports--> v07\n- v07 --refutes--> v1a\n- v1a --supports--> v13\n- v13 --supports--> v06\n- v06 --supports--> v10\n- v10 --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --supports--> v04\n- v04 --supports--> v1e\n- v1e --supports--> v12\n- v12 --refutes--> v0e\n- v0e --refutes--> v11\n- v11 --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v04 --refutes--> v05\n- v04 --supports--> v01\n- v07 --refutes--> v1d\n- v0e --refutes--> v09\n- v0f --supports--> v0c\n- v15 --supports--> v03\n- v16 --refutes--> v08\n- v1b --refutes--> v02\n- v1b --refutes--> v14\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":"a0ccaa881c6439ae047fdf4d0d03f75d16c6194e9c1504d776940d3e7e6af624","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v1a\n- v1a --supports--> v08\n- v08 --supports--> v09\n- v09 --supports--> v07\n- v07 --supports--> v16\n- v16 --refutes--> v1e\n- v1e --refutes--> v18\n- v18 --supports--> v0e\n- v0e --supports--> v1c\n- v1c --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v1d\n- v1d --supports--> v0f\n- v0f --refutes--> v0a\n- v0a --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v03\n- v04 --supports--> v0d\n- v04 --supports--> v19\n- v09 --refutes--> v12\n- v0e --supports--> v15\n- v10 --supports--> v13\n- v14 --refutes--> v1b\n- v1c --refutes--> v01\n- v1c --refutes--> v11\n- v1e --refutes--> v17\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":"bc66a87a6865440c88f64584614df8ea79d79b1357722966cca6b01e7034cfc0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v03\n- v03 --supports--> v19\n- v19 --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --supports--> v00\n- v00 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v1c\n- v1c --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v14\n- v14 --supports--> v18\n- v18 --supports--> v0d\n- v0d --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v11\n- v08 --supports--> v1a\n- v0a --refutes--> v04\n- v0c --refutes--> v13\n- v0d --supports--> v17\n- v10 --refutes--> v05\n- v12 --supports--> v16\n- v19 --refutes--> v01\n- v1b --refutes--> v15\n- v1b --refutes--> v1e\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":"172d4730db528ff5986d411dc53677ed6fea3c20229b7fd642556a1627d511c9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v05\n- v05 --refutes--> v17\n- v17 --supports--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v02\n- v02 --supports--> v0c\n- v0c --supports--> v16\n- v16 --supports--> v09\n- v09 --refutes--> v13\n- v13 --supports--> v0b\n- v0b --supports--> v19\n- v19 --refutes--> v1e\n- v1e --refutes--> v03\n- v03 --supports--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1b\n- v00 --supports--> v07\n- v03 --supports--> v1a\n- v08 --refutes--> v1d\n- v0a --supports--> v18\n- v0b --supports--> v01\n- v0c --refutes--> v12\n- v0e --refutes--> v15\n- v0f --refutes--> v14\n- v17 --refutes--> v1c\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":"cc78a4ef3f430152d3c5fb475248ceedd4d8ea742b45b2035bfa4e145c20128c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v04\n- v04 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v1c\n- v1c --supports--> v05\n- v05 --refutes--> v15\n- v15 --refutes--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v12\n- v12 --refutes--> v1e\n- v1e --supports--> v1d\n- v1d --refutes--> v1a\n- v1a --supports--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v10\n- v10 --supports--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v16\n- v0e --refutes--> v1b\n- v0f --refutes--> v00\n- v0f --supports--> v06\n- v12 --refutes--> v02\n- v13 --refutes--> v14\n- v18 --refutes--> v0a\n- v18 --refutes--> v0b\n- v1a --refutes--> v09\n- v1a --supports--> v17\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":"2b2409b05fe2a7a8ff58195ac49f45253c2accb5baf9e124aa4ecfc5eac39987","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v18\n- v18 --refutes--> v14\n- v14 --supports--> v15\n- v15 --supports--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v1c\n- v1c --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --refutes--> v17\n- v17 --supports--> v07\n- v07 --supports--> v1a\n- v1a --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v1e\n- v06 --refutes--> v0f\n- v07 --supports--> v12\n- v0e --refutes--> v13\n- v11 --supports--> v0b\n- v15 --supports--> v03\n- v18 --refutes--> v10\n- v18 --supports--> v16\n- v1c --refutes--> v0a\n- v1d --supports--> v19\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":"c5dcc6b69efd32ed77a80e4ac476b7ebee06a819cb0807f2769c63e7cdae150a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v00\n- v00 --supports--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v01\n- v01 --refutes--> v12\n- v12 --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v14\n- v14 --supports--> v1d\n- v1d --supports--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v1b\n- v1b --supports--> v16\n- v16 --supports--> v19\n- v19 --supports--> v15\n- v15 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v17\n- v02 --supports--> v0a\n- v04 --supports--> v0b\n- v09 --supports--> v13\n- v0f --supports--> v1e\n- v12 --refutes--> v1a\n- v14 --supports--> v0c\n- v16 --supports--> v07\n- v19 --supports--> v0d\n- v1c --refutes--> v18\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":"b9c15651c4619c197925f0ab56d92da3ff691123f7f9e478608deb7b5e0a6757","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v19\n- v19 --refutes--> v15\n- v15 --refutes--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v1d\n- v1d --refutes--> v1a\n- v1a --supports--> v07\n- v07 --supports--> v06\n- v06 --supports--> v17\n- v17 --supports--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v11\n- v11 --supports--> v00\n- v00 --supports--> v1e\n- v1e --supports--> v03\n- v03 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v16\n- v07 --refutes--> v05\n- v0d --supports--> v1b\n- v0e --refutes--> v14\n- v11 --supports--> v08\n- v15 --supports--> v12\n- v15 --supports--> v13\n- v18 --supports--> v04\n- v1a --refutes--> v01\n- v1e --supports--> v1c\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":"6734773782a201100fba2e2d22cf6b9cb56de41c02e0c1b668208761e2ea1a1c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v1e\n- v1e --supports--> v03\n- v03 --supports--> v0f\n- v0f --refutes--> v1c\n- v1c --supports--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v16\n- v16 --supports--> v17\n- v17 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --supports--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v14\n- v03 --refutes--> v07\n- v06 --refutes--> v13\n- v09 --supports--> v0b\n- v10 --supports--> v0e\n- v11 --supports--> v18\n- v16 --supports--> v00\n- v17 --supports--> v1a\n- v1c --refutes--> v08\n- v1e --supports--> v1b\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":"898acab850b74d6b76bee08dc6e212c691fa4ddbb94992e472f9fb6a0c11587f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v19\n- v19 --supports--> v02\n- v02 --refutes--> v1a\n- v1a --supports--> v12\n- v12 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v16\n- v16 --supports--> v06\n- v06 --supports--> v18\n- v18 --refutes--> v1b\n- v1b --refutes--> v0e\n- v0e --supports--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v1d\n- v1d --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v06 --supports--> v1c\n- v07 --refutes--> v0a\n- v0c --refutes--> v10\n- v0d --supports--> v17\n- v0f --supports--> v0b\n- v12 --refutes--> v05\n- v13 --supports--> v14\n- v16 --supports--> v15\n- v1a --refutes--> v11\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":"df173c5b1e25954ce4984cdb13e358cf064ec982087be95ba7bf946910fc5003","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v1d\n- v1d --supports--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v1b\n- v1b --supports--> v13\n- v13 --refutes--> v18\n- v18 --supports--> v16\n- v16 --refutes--> v17\n- v17 --refutes--> v0c\n- v0c --refutes--> v19\n- v19 --supports--> v15\n- v15 --refutes--> v09\n- v09 --supports--> v07\n- v07 --supports--> v1e\n- v1e --supports--> v1a\n- v1a --refutes--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v1c\n- v09 --refutes--> v0f\n- v12 --refutes--> v00\n- v12 --supports--> v0a\n- v15 --supports--> v0d\n- v18 --refutes--> v02\n- v19 --supports--> v04\n- v1a --supports--> v10\n- v1e --refutes--> v05\n- v1e --supports--> v14\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":"832af5bc9112d8bf094ea4790bf51aed45cba5f7668beb68c5df401d7a89912d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v10\n- v10 --refutes--> v1c\n- v1c --refutes--> v11\n- v11 --refutes--> v1e\n- v1e --refutes--> v1b\n- v1b --supports--> v17\n- v17 --supports--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --supports--> v1a\n- v1a --supports--> v1d\n- v1d --refutes--> v19\n- v19 --supports--> v16\n- v16 --refutes--> v14\n- v14 --refutes--> v0c\n- v0c --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v13\n- v05 --supports--> v0a\n- v07 --supports--> v06\n- v11 --refutes--> v04\n- v12 --refutes--> v08\n- v14 --refutes--> v0f\n- v16 --refutes--> v18\n- v16 --supports--> v0d\n- v1b --supports--> v09\n- v1d --refutes--> v01\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":"ef61d411b9b80edb0b46af38b6975707fb07ab20dd660efc7579efbb60f75498","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v13\n- v13 --supports--> v1d\n- v1d --refutes--> v18\n- v18 --supports--> v10\n- v10 --refutes--> v08\n- v08 --supports--> v14\n- v14 --refutes--> v15\n- v15 --supports--> v17\n- v17 --supports--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v1a\n- v1a --refutes--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v01\n- v01 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v0e\n- v0e --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1e\n- v00 --supports--> v19\n- v01 --supports--> v09\n- v02 --supports--> v05\n- v04 --supports--> v0c\n- v0d --refutes--> v06\n- v0e --supports--> v0a\n- v10 --refutes--> v16\n- v13 --refutes--> v03\n- v1a --refutes--> v1b\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":"7f7e20c2b9bfbf17bc9bae6bbaa31a1ea75e3a22ea1cdcd27efccf294a1c5022","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --supports--> v07\n- v07 --supports--> v1d\n- v1d --supports--> v04\n- v04 --supports--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v1e\n- v1e --refutes--> v12\n- v12 --refutes--> v1b\n- v1b --refutes--> v16\n- v16 --refutes--> v0d\n- v0d --refutes--> v18\n- v18 --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v14\n- v14 --supports--> v19\n- v19 --supports--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v04 --supports--> v0a\n- v07 --refutes--> v02\n- v09 --refutes--> v01\n- v18 --refutes--> v15\n- v19 --supports--> v10\n- v1d --refutes--> v06\n- v1d --refutes--> v0b\n- v1d --refutes--> v1c\n- v1d --supports--> v11\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":"d7000180506e9bda216f05880cfffc70edfbbbe94a092dcbddfb525fbb6f0bc7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v06\n- v06 --supports--> v1a\n- v1a --refutes--> v14\n- v14 --supports--> v03\n- v03 --supports--> v1b\n- v1b --refutes--> v1c\n- v1c --refutes--> v18\n- v18 --supports--> v19\n- v19 --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v12\n- v12 --supports--> v1e\n- v1e --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v1d\n- v1d --supports--> v16\n- v16 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v02 --refutes--> v05\n- v06 --supports--> v09\n- v07 --supports--> v0d\n- v0b --refutes--> v0e\n- v0b --supports--> v13\n- v19 --supports--> v10\n- v1a --supports--> v11\n- v1c --refutes--> v04\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":"82a9ae6d8b27317d9f16cba85e36b3b2402660d2768df6d50f29f3aed1a9beb0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v1c\n- v1c --refutes--> v0a\n- v0a --supports--> v1d\n- v1d --supports--> v10\n- v10 --supports--> v0c\n- v0c --refutes--> v1a\n- v1a --supports--> v1e\n- v1e --supports--> v19\n- v19 --refutes--> v0b\n- v0b --supports--> v12\n- v12 --refutes--> v1b\n- v1b --refutes--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v11\n- v11 --supports--> v05\n- v05 --refutes--> v17\n- v17 --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v15\n- v0c --refutes--> v18\n- v0c --supports--> v14\n- v0f --refutes--> v0e\n- v10 --refutes--> v01\n- v10 --refutes--> v06\n- v10 --refutes--> v08\n- v11 --refutes--> v0d\n- v13 --refutes--> v03\n- v17 --refutes--> v09\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":"e2027d8b3e1b270e1b84c96ec74c103ad12190430c1eac59c4f96ba3aff1c7dc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v14\n- v14 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v19\n- v19 --refutes--> v00\n- v00 --supports--> v18\n- v18 --refutes--> v11\n- v11 --supports--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v15\n- v15 --refutes--> v08\n- v08 --supports--> v17\n- v17 --refutes--> v1e\n- v1e --supports--> v1a\n- v1a --supports--> v0f\n- v0f --refutes--> v13\n- v13 --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --supports--> v03\n- v03 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v06 --supports--> v0b\n- v0a --refutes--> v1d\n- v0f --refutes--> v10\n- v12 --supports--> v16\n- v15 --refutes--> v07\n- v18 --refutes--> v01\n- v19 --refutes--> v0e\n- v1c --supports--> v09\n- v1c --supports--> v1b\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":"19f87a7d2169b7c7d33f985c5972ed98a63cd110e630a000c6313430a9e11332","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v1a\n- v1a --supports--> v06\n- v06 --supports--> v10\n- v10 --supports--> v17\n- v17 --supports--> v1e\n- v1e --refutes--> v1c\n- v1c --refutes--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v00\n- v00 --refutes--> v1d\n- v1d --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v19\n- v19 --supports--> v14\n- v14 --refutes--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v1b\n- v1b --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v01 --refutes--> v11\n- v05 --refutes--> v15\n- v06 --refutes--> v07\n- v0a --refutes--> v08\n- v0a --supports--> v12\n- v0c --refutes--> v04\n- v0e --refutes--> v16\n- v10 --refutes--> v18\n- v1b --refutes--> v0f\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":"89de84ec18873f47fdc4e77e52c5bbd75ae4882d8e891ec8acfa72a5a0117aee","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v1d\n- v1d --supports--> v05\n- v05 --supports--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v1a\n- v1a --refutes--> v14\n- v14 --refutes--> v1c\n- v1c --supports--> v0e\n- v0e --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v1e\n- v1e --supports--> v0c\n- v0c --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v13\n- v13 --refutes--> v18\n- v18 --supports--> v12\n- v12 --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v07\n- v04 --supports--> v19\n- v06 --supports--> v0b\n- v0e --supports--> v0f\n- v10 --refutes--> v16\n- v12 --supports--> v08\n- v13 --supports--> v1b\n- v14 --refutes--> v11\n- v18 --refutes--> v03\n- v1e --supports--> v15\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":"1b578ce6b5aa655b03682af41e785ee7081c0cef59477938794109ac81fb04b6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0d\n- v0d --supports--> v1c\n- v1c --supports--> v02\n- v02 --supports--> v14\n- v14 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v1d\n- v1d --refutes--> v13\n- v13 --supports--> v1a\n- v1a --supports--> v12\n- v12 --refutes--> v11\n- v11 --supports--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v19\n- v19 --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --refutes--> v0f\n- v0f --refutes--> v18\n- v18 --refutes--> v03\n- v03 --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v10\n- v03 --supports--> v08\n- v0b --refutes--> v17\n- v0d --supports--> v00\n- v0f --supports--> v1e\n- v11 --supports--> v05\n- v11 --supports--> v16\n- v13 --supports--> v06\n- v14 --supports--> v0c\n- v15 --supports--> v1b\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":"cf52db908f4223e1baa9dcaec66e92a21b49ddffacacd280425a34dd0b9d3693","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v1c\n- v1c --refutes--> v1b\n- v1b --refutes--> v19\n- v19 --refutes--> v18\n- v18 --supports--> v08\n- v08 --supports--> v14\n- v14 --supports--> v0b\n- v0b --supports--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v03\n- v03 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v13\n- v13 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0a\n- v02 --refutes--> v16\n- v02 --supports--> v01\n- v05 --supports--> v11\n- v05 --supports--> v12\n- v06 --supports--> v17\n- v07 --refutes--> v0d\n- v08 --supports--> v15\n- v09 --refutes--> v0f\n- v1a --refutes--> v1d\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":"5640f1a0e8a7a91cb8c2a7b4328b6c4de6289d3a1d2e900ab947aba3ac25abcd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v0b\n- v0b --supports--> v17\n- v17 --refutes--> v16\n- v16 --refutes--> v1e\n- v1e --refutes--> v1a\n- v1a --supports--> v02\n- v02 --supports--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v18\n- v18 --supports--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v19\n- v19 --supports--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v1c\n- v1c --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v05\n- v09 --refutes--> v15\n- v0b --supports--> v1b\n- v0d --supports--> v06\n- v0f --refutes--> v14\n- v16 --refutes--> v0e\n- v18 --refutes--> v0c\n- v18 --supports--> v13\n- v1c --refutes--> v03\n- v1d --refutes--> v10\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":"42d048b15b5019185e5bc601dd872d28560e816b0171fb6493887f3512c06cee","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v00\n- v00 --supports--> v0b\n- v0b --refutes--> v1a\n- v1a --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --supports--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v19\n- v19 --supports--> v1c\n- v1c --supports--> v03\n- v03 --supports--> v16\n- v16 --refutes--> v1e\n- v1e --refutes--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v10\n- v10 --supports--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v00 --supports--> v0d\n- v05 --supports--> v08\n- v0a --refutes--> v09\n- v11 --refutes--> v07\n- v11 --supports--> v15\n- v14 --supports--> v1b\n- v19 --refutes--> v1d\n- v1a --refutes--> v17\n- v1a --supports--> v0e\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":"5adf390a8941cef3f456aff407bd247f19c8988186b5eb3d731f4af22fd51a52","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v0e\n- v0e --supports--> v18\n- v18 --refutes--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v1c\n- v1c --refutes--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v15\n- v15 --refutes--> v1d\n- v1d --refutes--> v13\n- v13 --refutes--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v01\n- v01 --supports--> v1e\n- v1e --supports--> v11\n- v11 --refutes--> v0c\n- v0c --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v03 --supports--> v0f\n- v06 --refutes--> v0d\n- v07 --refutes--> v0b\n- v07 --supports--> v1b\n- v09 --refutes--> v12\n- v0c --supports--> v05\n- v14 --refutes--> v16\n- v15 --refutes--> v17\n- v15 --supports--> v19\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":"60f37fae1d12fd73d25bec890ad62abbc0bb3390ff9051eff0dae66484537236","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v02\n- v02 --refutes--> v17\n- v17 --supports--> v19\n- v19 --refutes--> v01\n- v01 --refutes--> v04\n- v04 --supports--> v10\n- v10 --refutes--> v15\n- v15 --supports--> v06\n- v06 --supports--> v0e\n- v0e --supports--> v0f\n- v0f --supports--> v1b\n- v1b --refutes--> v1c\n- v1c --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v12\n- v12 --refutes--> v1d\n- v1d --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v01 --refutes--> v07\n- v01 --supports--> v08\n- v04 --refutes--> v09\n- v17 --refutes--> v13\n- v17 --supports--> v1a\n- v19 --supports--> v16\n- v19 --supports--> v1e\n- v1c --refutes--> v0b\n- v1c --refutes--> v0d\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":"16562da1b047b682bd1356a5ae664c1d6158a8441ed9c36c4e8b5a9d8e5d02ec","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v02\n- v02 --refutes--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v0d\n- v0d --supports--> v1a\n- v1a --supports--> v16\n- v16 --supports--> v1e\n- v1e --refutes--> v12\n- v12 --refutes--> v1c\n- v1c --refutes--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v11\n- v11 --refutes--> v03\n- v03 --refutes--> v17\n- v17 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v1b\n- v05 --refutes--> v1d\n- v06 --refutes--> v10\n- v06 --refutes--> v13\n- v0e --refutes--> v01\n- v0f --supports--> v0b\n- v16 --supports--> v14\n- v1a --supports--> v15\n- v1c --refutes--> v19\n- v1e --refutes--> v0c\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":"52bff6976b0b4f11a13fa743b428f4b7d43038789c4161ec767af8d1ef85aabd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v16\n- v16 --supports--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v1c\n- v1c --refutes--> v07\n- v07 --supports--> v09\n- v09 --supports--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v1a\n- v1a --refutes--> v05\n- v05 --refutes--> v19\n- v19 --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --supports--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v11\n- v11 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v02 --refutes--> v17\n- v02 --supports--> v1b\n- v04 --supports--> v03\n- v07 --refutes--> v15\n- v08 --refutes--> v14\n- v0c --supports--> v12\n- v19 --refutes--> v0b\n- v19 --supports--> v18\n- v1e --refutes--> v0e\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":"bca73bb9752041809674b9d24d74e351fe5662b75d1dbafd0ba8c2035d46b995","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v17\n- v17 --supports--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v12\n- v12 --refutes--> v1b\n- v1b --supports--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v1c\n- v1c --supports--> v0c\n- v0c --refutes--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v19\n- v19 --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v01 --refutes--> v07\n- v09 --supports--> v1d\n- v0c --supports--> v05\n- v12 --refutes--> v0d\n- v13 --supports--> v14\n- v17 --supports--> v0b\n- v19 --supports--> v04\n- v1a --supports--> v10\n- v1b --supports--> v1e\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":"0b174a50562a5e204b656e834c79330cbddc636b0d47541cf504f2c1e2a3c822","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v1c\n- v1c --refutes--> v0b\n- v0b --supports--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v14\n- v14 --refutes--> v1e\n- v1e --refutes--> v15\n- v15 --refutes--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v01\n- v01 --supports--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v19\n- v05 --supports--> v00\n- v05 --supports--> v13\n- v08 --supports--> v0e\n- v0a --supports--> v11\n- v0c --supports--> v10\n- v0c --supports--> v12\n- v0d --supports--> v1a\n- v0f --supports--> v1d\n- v14 --refutes--> v18\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":"80f0c432ba16c1b95bbf529cbe9178059201ebbf31293f63d1a1d6fd3934b724","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v17\n- v17 --supports--> v07\n- v07 --refutes--> v1d\n- v1d --supports--> v15\n- v15 --refutes--> v1c\n- v1c --supports--> v0f\n- v0f --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v13\n- v13 --supports--> v18\n- v18 --refutes--> v1e\n- v1e --refutes--> v1a\n- v1a --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v10\n- v04 --supports--> v12\n- v07 --supports--> v0d\n- v08 --supports--> v09\n- v0b --refutes--> v01\n- v0e --supports--> v03\n- v0f --supports--> v1b\n- v17 --supports--> v02\n- v18 --refutes--> v05\n- v18 --refutes--> v19\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":"0c21662f76fb7272665f5439082ceb2fa26b7dfaf31393f10a654b4266576d55","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v18\n- v18 --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v04\n- v04 --refutes--> v14\n- v14 --refutes--> v09\n- v09 --refutes--> v13\n- v13 --supports--> v1e\n- v1e --refutes--> v0f\n- v0f --supports--> v0c\n- v0c --refutes--> v15\n- v15 --supports--> v0d\n- v0d --refutes--> v1b\n- v1b --refutes--> v1a\n- v1a --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0e\n- v01 --refutes--> v11\n- v04 --refutes--> v07\n- v04 --supports--> v10\n- v0d --refutes--> v08\n- v13 --refutes--> v0a\n- v14 --supports--> v16\n- v1a --refutes--> v19\n- v1a --supports--> v0b\n- v1e --refutes--> v02\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":"f59ae2cfdbefc234d801733f612d4beb4d81fcb2bedb4240ebf1777a76552ecf","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v15\n- v15 --supports--> v19\n- v19 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v18\n- v18 --supports--> v02\n- v02 --refutes--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v16\n- v16 --supports--> v08\n- v08 --supports--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v17\n- v17 --refutes--> v12\n- v12 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v06\n- v02 --refutes--> v1d\n- v08 --refutes--> v13\n- v09 --supports--> v0c\n- v0d --supports--> v11\n- v0f --refutes--> v00\n- v12 --refutes--> v1b\n- v18 --supports--> v0b\n- v19 --supports--> v0a\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":"1f44022b7c7a59a82834147c2e036d307a295477076abccc0db6104bb954c3c4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v14\n- v14 --supports--> v00\n- v00 --supports--> v1b\n- v1b --refutes--> v01\n- v01 --refutes--> v19\n- v19 --supports--> v11\n- v11 --supports--> v17\n- v17 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v1c\n- v1c --supports--> v15\n- v15 --supports--> v16\n- v16 --supports--> v1a\n- v1a --refutes--> v1d\n- v1d --supports--> v13\n- v13 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v05\n- v04 --supports--> v0a\n- v0f --refutes--> v07\n- v0f --supports--> v02\n- v0f --supports--> v0d\n- v0f --supports--> v10\n- v0f --supports--> v1e\n- v12 --supports--> v09\n- v13 --supports--> v0b\n- v1c --refutes--> v0c\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":"02ea2eff2109f59c80f998e3c4cdec35616435af130a8c8dbb89a4cf56421b4d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v0b\n- v0b --supports--> v1a\n- v1a --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v1e\n- v1e --refutes--> v16\n- v16 --supports--> v19\n- v19 --supports--> v1d\n- v1d --refutes--> v12\n- v12 --refutes--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v08\n- v0a --refutes--> v1b\n- v0a --supports--> v0c\n- v0e --supports--> v05\n- v0f --supports--> v18\n- v13 --supports--> v04\n- v19 --refutes--> v15\n- v1a --supports--> v01\n- v1d --refutes--> v03\n- v1e --supports--> v17\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":"5a7ee15c524838886e00a57278100c801f585aeea6306d26f16dfc29b6b76b1e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v18\n- v18 --refutes--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v1d\n- v1d --refutes--> v0e\n- v0e --refutes--> v1c\n- v1c --refutes--> v16\n- v16 --refutes--> v12\n- v12 --refutes--> v17\n- v17 --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v11\n- v11 --supports--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v19\n- v19 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v05\n- v06 --supports--> v15\n- v0e --supports--> v09\n- v0e --supports--> v14\n- v16 --supports--> v04\n- v17 --supports--> v0b\n- v19 --refutes--> v0a\n- v19 --supports--> v1b\n- v1c --refutes--> v13\n- v1c --refutes--> v1e\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":"f76dbd4abdf0e05bb2614f24a71423c3143b829c1463d7183e8c6e4c485dca8c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v1d\n- v1d --supports--> v1a\n- v1a --refutes--> v10\n- v10 --supports--> v17\n- v17 --refutes--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --refutes--> v1e\n- v1e --supports--> v06\n- v06 --refutes--> v19\n- v19 --refutes--> v0b\n- v0b --supports--> v11\n- v11 --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v1c\n- v1c --supports--> v12\n- v12 --supports--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v06 --supports--> v09\n- v06 --supports--> v13\n- v0c --supports--> v14\n- v0d --refutes--> v16\n- v11 --supports--> v00\n- v12 --refutes--> v04\n- v1a --refutes--> v03\n- v1a --refutes--> v18\n- v1a --refutes--> v1b\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":"8470bd8f4a2d3a61fe712ac6d5e6f5e34d309f532de3c546977d4705709b5b36","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0a\n- v0a --refutes--> v18\n- v18 --refutes--> v16\n- v16 --supports--> v03\n- v03 --supports--> v01\n- v01 --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v12\n- v12 --supports--> v07\n- v07 --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v1e\n- v1e --supports--> v05\n- v05 --supports--> v19\n- v19 --supports--> v17\n- v17 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v1a\n- v0a --refutes--> v04\n- v0c --supports--> v0e\n- v12 --refutes--> v1d\n- v15 --supports--> v06\n- v16 --refutes--> v0f\n- v16 --supports--> v1c\n- v17 --supports--> v0b\n- v17 --supports--> v11\n- v1b --refutes--> v0d\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":"15227c7bcf23345e5e74ef7d268e04a3f610bd34fa0cf2b809e145cbff1f64a0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v02\n- v02 --refutes--> v08\n- v08 --supports--> v1b\n- v1b --supports--> v0b\n- v0b --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v1d\n- v1d --supports--> v04\n- v04 --refutes--> v14\n- v14 --supports--> v17\n- v17 --supports--> v10\n- v10 --refutes--> v1c\n- v1c --supports--> v16\n- v16 --refutes--> v0f\n- v0f --supports--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v09\n- v0b --refutes--> v0c\n- v0b --supports--> v1e\n- v0e --supports--> v11\n- v0e --supports--> v13\n- v12 --supports--> v0a\n- v16 --refutes--> v1a\n- v17 --refutes--> v03\n- v1c --refutes--> v01\n- v1d --supports--> v19\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":"45881d42de2eb7dd59a1cbec40e09d0d27ed6c7f32d28b793ae8b44c2198b337","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v09\n- v09 --supports--> v02\n- v02 --refutes--> v14\n- v14 --supports--> v17\n- v17 --supports--> v1a\n- v1a --supports--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v11\n- v11 --refutes--> v0d\n- v0d --supports--> v1d\n- v1d --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v1e\n- v1e --refutes--> v1b\n- v1b --supports--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v16\n- v16 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v1c\n- v04 --supports--> v13\n- v06 --supports--> v10\n- v07 --supports--> v15\n- v09 --supports--> v0c\n- v14 --refutes--> v0f\n- v18 --supports--> v01\n- v18 --supports--> v0a\n- v18 --supports--> v19\n- v1d --supports--> v05\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":"84217f539ae40c3ce065bb936f891be8620c9fb6591752b7f36d54dd0ce022d3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v13\n- v13 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v0e\n- v0e --supports--> v14\n- v14 --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v17\n- v17 --supports--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v1e\n- v1e --refutes--> v1a\n- v1a --refutes--> v04\n- v04 --supports--> v07\n- v07 --supports--> v09\n- v09 --supports--> v1d\n- v1d --supports--> v1c\n- v1c --supports--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v10\n- v08 --supports--> v15\n- v09 --refutes--> v02\n- v09 --refutes--> v0b\n- v0a --supports--> v16\n- v0c --refutes--> v18\n- v0c --supports--> v01\n- v0c --supports--> v19\n- v17 --supports--> v06\n- v1d --refutes--> v1b\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":"e6b526d5354865b8bf4c55e09a3ff6c0c2d803cf570b78a1bb06844db21e30a8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v0d\n- v0d --supports--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v16\n- v16 --supports--> v1e\n- v1e --refutes--> v1a\n- v1a --refutes--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v19\n- v19 --supports--> v1c\n- v1c --refutes--> v1d\n- v1d --supports--> v1b\n- v1b --supports--> v13\n- v13 --supports--> v0e\n- v0e --refutes--> v14\n- v14 --refutes--> v15\n- v15 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0b\n- v01 --supports--> v04\n- v07 --supports--> v05\n- v0a --refutes--> v03\n- v0d --supports--> v0c\n- v0f --refutes--> v02\n- v14 --refutes--> v10\n- v1a --refutes--> v06\n- v1a --refutes--> v17\n- v1d --supports--> v09\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":"c1ca06ca26edf24d26213d87adab8b3388db4d7fcc07daa486b1f5517abb528a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v0a\n- v0a --supports--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v1d\n- v1d --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --supports--> v1a\n- v1a --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v0d\n- v0d --supports--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v1c\n- v1c --refutes--> v19\n- v19 --supports--> v01\n- v01 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v17\n- v05 --refutes--> v08\n- v06 --refutes--> v16\n- v06 --supports--> v13\n- v0a --refutes--> v11\n- v0e --refutes--> v04\n- v14 --supports--> v02\n- v15 --refutes--> v18\n- v19 --supports--> v09\n- v1c --refutes--> v10\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":"845d1778af8bcb4b9b04419255df70ec7cab9da74e4657d1546d22033c59188f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v16\n- v16 --refutes--> v0d\n- v0d --refutes--> v0e\n- v0e --refutes--> v1e\n- v1e --supports--> v1c\n- v1c --supports--> v05\n- v05 --refutes--> v03\n- v03 --refutes--> v11\n- v11 --refutes--> v12\n- v12 --refutes--> v00\n- v00 --supports--> v06\n- v06 --supports--> v1d\n- v1d --refutes--> v1b\n- v1b --supports--> v14\n- v14 --refutes--> v0b\n- v0b --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v07\n- v03 --refutes--> v01\n- v03 --supports--> v17\n- v08 --supports--> v09\n- v08 --supports--> v1a\n- v0e --supports--> v0f\n- v11 --supports--> v10\n- v15 --refutes--> v0a\n- v15 --supports--> v18\n- v1d --refutes--> v19\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":"34a341fbbec7a2d297295ae555dd6760438d9776889bf19d6fd1ad0e52b5cb49","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v0c\n- v0c --refutes--> v19\n- v19 --supports--> v11\n- v11 --refutes--> v1d\n- v1d --supports--> v15\n- v15 --supports--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v17\n- v17 --supports--> v1e\n- v1e --supports--> v18\n- v18 --supports--> v07\n- v07 --refutes--> v1c\n- v1c --refutes--> v0f\n- v0f --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v13\n- v09 --refutes--> v06\n- v09 --supports--> v1a\n- v0b --refutes--> v16\n- v0c --supports--> v05\n- v12 --refutes--> v1b\n- v15 --refutes--> v0d\n- v15 --supports--> v08\n- v17 --supports--> v00\n- v19 --supports--> v0a\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":"651db2878fecec46b43aa132135bb9526cd9051dbb26a31b5ecc83590c5a81db","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v06\n- v06 --refutes--> v0a\n- v0a --refutes--> v1d\n- v1d --refutes--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v1a\n- v1a --refutes--> v00\n- v00 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v16\n- v16 --refutes--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v0b\n- v0b --refutes--> v17\n- v17 --supports--> v0d\n- v0d --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v15\n- v09 --refutes--> v01\n- v0a --refutes--> v19\n- v0a --supports--> v02\n- v0a --supports--> v05\n- v0e --refutes--> v1e\n- v14 --refutes--> v03\n- v17 --supports--> v10\n- v17 --supports--> v1b\n- v1a --refutes--> v18\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":"bf15a7030478bee708ffdd572415f6654bc1d5485c277563602073d7130ff225","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v1d\n- v1d --refutes--> v01\n- v01 --supports--> v1c\n- v1c --refutes--> v17\n- v17 --supports--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v16\n- v16 --supports--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v1b\n- v1b --supports--> v0f\n- v0f --supports--> v19\n- v19 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1e\n- v01 --supports--> v13\n- v0b --supports--> v02\n- v0b --supports--> v1a\n- v0e --supports--> v07\n- v0f --refutes--> v09\n- v15 --refutes--> v10\n- v16 --refutes--> v18\n- v16 --supports--> v14\n- v19 --refutes--> v12\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":"6244bc2bdbfb2ba77b14f306b1bbbfbb14f060db73d31d18e36bda34e6f9253a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v06\n- v06 --supports--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v18\n- v18 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --refutes--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v16\n- v16 --supports--> v1e\n- v1e --refutes--> v01\n- v01 --refutes--> v19\n- v19 --supports--> v02\n- v02 --refutes--> v1d\n- v1d --refutes--> v1c\n- v1c --supports--> v14\n- v14 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v0e\n- v0a --supports--> v00\n- v0a --supports--> v10\n- v0a --supports--> v17\n- v0a --supports--> v1b\n- v0d --supports--> v03\n- v13 --refutes--> v1a\n- v14 --refutes--> v09\n- v15 --supports--> v0b\n- v19 --refutes--> v04\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":"00cb853d3507f49798b608f234a4ce7a8bad37b1be5369060038f2433fcad928","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v1d\n- v1d --refutes--> v16\n- v16 --supports--> v0a\n- v0a --supports--> v17\n- v17 --supports--> v05\n- v05 --supports--> v07\n- v07 --supports--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v19\n- v19 --refutes--> v06\n- v06 --supports--> v1a\n- v1a --refutes--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v00\n- v00 --supports--> v1c\n- v1c --supports--> v0e\n- v0e --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v01 --supports--> v04\n- v0a --supports--> v08\n- v10 --refutes--> v03\n- v10 --refutes--> v12\n- v15 --supports--> v14\n- v16 --supports--> v11\n- v19 --refutes--> v13\n- v19 --refutes--> v1e\n- v1a --supports--> v09\n\n[VERIFIED ROOT]\nv0f = TRUE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"7c84427993aab9d8d7087e9b06151eb28b301ec934a8cc045bfbab26d2f536a9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v0f\n- v0f --refutes--> v11\n- v11 --refutes--> v1c\n- v1c --supports--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v0b\n- v0b --supports--> v08\n- v08 --refutes--> v1e\n- v1e --supports--> v18\n- v18 --refutes--> v05\n- v05 --refutes--> v19\n- v19 --refutes--> v1d\n- v1d --supports--> v16\n- v16 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v08 --refutes--> v01\n- v08 --refutes--> v13\n- v0c --supports--> v0a\n- v0c --supports--> v14\n- v0e --refutes--> v04\n- v0f --refutes--> v10\n- v17 --supports--> v1a\n- v18 --refutes--> v15\n- v1c --supports--> v12\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":"9455ba274cd408aa5975ee62b110708aa55725878d41329e10a4a975da7a451a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v15\n- v15 --supports--> v19\n- v19 --supports--> v10\n- v10 --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --supports--> v17\n- v17 --refutes--> v0a\n- v0a --supports--> v1e\n- v1e --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1a\n- v08 --refutes--> v1c\n- v0b --supports--> v07\n- v0c --refutes--> v12\n- v0c --refutes--> v1b\n- v0c --supports--> v18\n- v0e --refutes--> v02\n- v0e --refutes--> v03\n- v10 --refutes--> v04\n- v17 --supports--> v1d\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":"1a8c6299dfda2c09923ad76de96540600eb7aa4a00ee63dbcfab29503ccb7494","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v18\n- v18 --refutes--> v1b\n- v1b --supports--> v12\n- v12 --supports--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --refutes--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v08\n- v08 --supports--> v03\n- v03 --supports--> v0a\n- v0a --supports--> v15\n- v15 --supports--> v1e\n- v1e --supports--> v19\n- v19 --refutes--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v0c\n- v0c --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v13\n- v05 --supports--> v0d\n- v07 --supports--> v00\n- v08 --refutes--> v1a\n- v08 --supports--> v11\n- v0c --refutes--> v04\n- v12 --refutes--> v1d\n- v19 --supports--> v16\n- v1b --supports--> v1c\n- v1e --refutes--> v14\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":"914bd74b058b9f512ad49e9cbad88ca840e9c037c1a6f1671f221d5805de0f1d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v1d\n- v1d --refutes--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v17\n- v17 --supports--> v1b\n- v1b --refutes--> v13\n- v13 --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v1e\n- v1e --refutes--> v12\n- v12 --refutes--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v18\n- v18 --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v08\n- v02 --refutes--> v11\n- v06 --supports--> v16\n- v0b --refutes--> v19\n- v0c --supports--> v07\n- v0f --supports--> v14\n- v13 --refutes--> v03\n- v18 --refutes--> v05\n- v1d --refutes--> v1c\n- v1e --refutes--> v10\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":"d9cc001c8547ac4fc04b8dfc313d6277690c56287e624eed81cab73f2dcec402","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v10\n- v10 --refutes--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v15\n- v15 --supports--> v04\n- v04 --supports--> v17\n- v17 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v1e\n- v1e --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v1d\n- v1d --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v01 --supports--> v0b\n- v0d --refutes--> v16\n- v0e --refutes--> v13\n- v11 --refutes--> v19\n- v11 --supports--> v03\n- v15 --refutes--> v1a\n- v17 --refutes--> v14\n- v1d --refutes--> v05\n- v1e --refutes--> v18\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":"656dcbb89b6cad3934c642ab8983d53d2702d2171b4ac59c5e29496256ae819f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v0e\n- v0e --refutes--> v10\n- v10 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v08\n- v08 --refutes--> v1d\n- v1d --refutes--> v11\n- v11 --refutes--> v15\n- v15 --refutes--> v1b\n- v1b --refutes--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v06\n- v06 --refutes--> v18\n- v18 --refutes--> v01\n- v01 --refutes--> v12\n- v12 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v1c\n- v0a --refutes--> v04\n- v0f --supports--> v0b\n- v10 --refutes--> v0d\n- v11 --refutes--> v13\n- v11 --refutes--> v16\n- v11 --supports--> v0c\n- v11 --supports--> v17\n- v1b --supports--> v19\n- v1d --supports--> v05\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":"e98802f639494dc328905d0003b9bac15cdb9fea4b8903eb03d81510765c7faf","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --supports--> v17\n- v17 --refutes--> v16\n- v16 --supports--> v10\n- v10 --supports--> v05\n- v05 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v1e\n- v1e --refutes--> v19\n- v19 --refutes--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v14\n- v14 --refutes--> v04\n- v04 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v1b\n- v05 --supports--> v0c\n- v08 --supports--> v03\n- v09 --refutes--> v13\n- v11 --refutes--> v01\n- v12 --supports--> v06\n- v14 --supports--> v0e\n- v17 --refutes--> v0d\n- v18 --refutes--> v1d\n- v19 --refutes--> v15\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":"f2bd89dfcf898f680ae89ebdd5c8a164dd2dba612ce8f168377a865ecd889beb","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v1c\n- v1c --supports--> v13\n- v13 --supports--> v11\n- v11 --supports--> v0b\n- v0b --supports--> v0a\n- v0a --supports--> v1e\n- v1e --refutes--> v19\n- v19 --refutes--> v04\n- v04 --supports--> v1a\n- v1a --supports--> v06\n- v06 --supports--> v07\n- v07 --supports--> v1d\n- v1d --supports--> v1b\n- v1b --supports--> v09\n- v09 --supports--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v02\n- v06 --supports--> v03\n- v0a --refutes--> v0e\n- v0c --supports--> v14\n- v0f --refutes--> v0d\n- v18 --supports--> v16\n- v18 --supports--> v17\n- v1b --refutes--> v10\n- v1b --supports--> v01\n- v1e --supports--> v15\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":"e0c2a546704af1faa92ef924ab43e72eaa0b1b2a6960b8237316c8d7fa14b45e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v1a\n- v1a --refutes--> v08\n- v08 --supports--> v1c\n- v1c --supports--> v18\n- v18 --supports--> v0a\n- v0a --refutes--> v17\n- v17 --supports--> v0c\n- v0c --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --supports--> v1e\n- v1e --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v1b\n- v1b --supports--> v0f\n- v0f --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v01\n- v03 --supports--> v1d\n- v04 --refutes--> v00\n- v06 --refutes--> v13\n- v07 --refutes--> v0e\n- v0d --refutes--> v16\n- v0d --supports--> v15\n- v12 --refutes--> v14\n- v18 --supports--> v09\n- v1e --refutes--> v10\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":"a0e309f4733f68580737dcc670ed7de0f1ca26ab0663079b19ade247ec269ba7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v02\n- v02 --refutes--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v18\n- v18 --supports--> v17\n- v17 --supports--> v10\n- v10 --refutes--> v1b\n- v1b --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --refutes--> v1e\n- v1e --supports--> v0c\n- v0c --refutes--> v03\n- v03 --refutes--> v1a\n- v1a --supports--> v0e\n- v0e --refutes--> v1c\n- v1c --supports--> v09\n- v09 --refutes--> v15\n- v15 --refutes--> v05\n- v05 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v07\n- v05 --supports--> v16\n- v0e --refutes--> v1d\n- v11 --refutes--> v06\n- v12 --refutes--> v00\n- v12 --supports--> v14\n- v17 --refutes--> v0f\n- v1a --refutes--> v19\n- v1c --refutes--> v04\n- v1c --refutes--> v13\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":"ba1a37e5dfd34122fd17d629b9f3a0bd37abf81f1469ad6dd5aede9d594ef6dc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v1b\n- v1b --refutes--> v02\n- v02 --supports--> v04\n- v04 --supports--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --refutes--> v18\n- v18 --refutes--> v07\n- v07 --refutes--> v1e\n- v1e --supports--> v0d\n- v0d --refutes--> v1d\n- v1d --supports--> v19\n- v19 --supports--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v15\n- v15 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v08\n- v06 --refutes--> v1c\n- v06 --supports--> v17\n- v0c --supports--> v13\n- v0f --supports--> v12\n- v19 --refutes--> v10\n- v19 --supports--> v14\n- v1b --refutes--> v16\n- v1e --refutes--> v03\n- v1e --supports--> v1a\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":"d16ec7c233b00c22215a29a9653545affaf9b1825f5f67fd89e585748c60dd1b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v15\n- v15 --refutes--> v1a\n- v1a --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v18\n- v18 --refutes--> v04\n- v04 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v1d\n- v1d --supports--> v0b\n- v0b --supports--> v11\n- v11 --refutes--> v02\n- v02 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v14\n- v05 --refutes--> v1b\n- v06 --refutes--> v0c\n- v06 --supports--> v01\n- v06 --supports--> v16\n- v08 --supports--> v1c\n- v0e --supports--> v0d\n- v0f --supports--> v19\n- v15 --refutes--> v09\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":"2b241fb2c821bfe50fcda02430789389ebc65517c81e2f34205e6f54a3da43fd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v18\n- v18 --supports--> v17\n- v17 --refutes--> v09\n- v09 --supports--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v15\n- v15 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v12\n- v12 --supports--> v16\n- v16 --refutes--> v19\n- v19 --supports--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v05\n- v09 --supports--> v14\n- v0e --supports--> v08\n- v0f --supports--> v07\n- v12 --refutes--> v06\n- v12 --supports--> v01\n- v15 --refutes--> v1b\n- v18 --supports--> v1e\n- v19 --refutes--> v1d\n- v1a --refutes--> v13\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":"4c09dc411de8071153818c1382f48e184d91f8b4d52b6b5101dc628224bd2036","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v07\n- v07 --refutes--> v1d\n- v1d --refutes--> v15\n- v15 --supports--> v14\n- v14 --supports--> v16\n- v16 --refutes--> v1c\n- v1c --refutes--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v1e\n- v1e --refutes--> v01\n- v01 --supports--> v03\n- v03 --supports--> v19\n- v19 --supports--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v17\n- v17 --supports--> v13\n- v13 --supports--> v12\n- v12 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v0b\n- v09 --refutes--> v0c\n- v10 --refutes--> v08\n- v10 --refutes--> v11\n- v14 --supports--> v06\n- v15 --refutes--> v0d\n- v1c --refutes--> v05\n- v1d --refutes--> v1b\n- v1e --refutes--> v18\n- v1e --supports--> v1a\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":"ae8c085f3fd4002bba42f062541f2655376e39c268d6a23a656340b039e8f7f2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v03\n- v03 --refutes--> v19\n- v19 --supports--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v0c\n- v0c --refutes--> v06\n- v06 --supports--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v18\n- v18 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v16\n- v06 --supports--> v1b\n- v07 --refutes--> v05\n- v08 --refutes--> v04\n- v14 --refutes--> v12\n- v15 --supports--> v1d\n- v15 --supports--> v1e\n- v19 --refutes--> v0a\n- v19 --supports--> v17\n- v1c --refutes--> v0f\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":"821740a5a1d62e626b9c061bfda5cbf899053a0647d7423009df53205e4cd071","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v03\n- v03 --supports--> v0e\n- v0e --refutes--> v16\n- v16 --refutes--> v07\n- v07 --refutes--> v19\n- v19 --supports--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v10\n- v10 --refutes--> v13\n- v13 --refutes--> v17\n- v17 --refutes--> v1a\n- v1a --supports--> v18\n- v18 --refutes--> v1c\n- v1c --refutes--> v0b\n- v0b --supports--> v1d\n- v1d --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1b\n- v01 --supports--> v0d\n- v04 --supports--> v0a\n- v07 --refutes--> v0f\n- v07 --supports--> v08\n- v10 --supports--> v15\n- v14 --refutes--> v1e\n- v16 --supports--> v02\n- v17 --supports--> v12\n- v1c --supports--> v11\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":"138a732396134ee4b76e4e4d1d3e41ec8ffa9b95c4396398920161dad6fb75ea","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v13\n- v13 --refutes--> v02\n- v02 --refutes--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v07\n- v07 --supports--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v1c\n- v1c --refutes--> v01\n- v01 --supports--> v17\n- v17 --supports--> v10\n- v10 --supports--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --refutes--> v1d\n- v1d --refutes--> v19\n- v19 --refutes--> v08\n- v08 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0b\n- v01 --refutes--> v14\n- v01 --supports--> v18\n- v0e --refutes--> v05\n- v0e --refutes--> v15\n- v12 --refutes--> v16\n- v12 --supports--> v06\n- v1b --supports--> v1a\n- v1d --refutes--> v11\n- v1d --supports--> v04\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":"fde3f3555251eee45c8b39ead2fbe5af5a515c59fd059ccd130367ee146f3685","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v03\n- v03 --refutes--> v1d\n- v1d --supports--> v08\n- v08 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v15\n- v15 --refutes--> v1e\n- v1e --supports--> v06\n- v06 --refutes--> v19\n- v19 --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v14\n- v14 --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --refutes--> v10\n- v10 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0d\n- v05 --refutes--> v01\n- v07 --refutes--> v1c\n- v08 --refutes--> v0f\n- v08 --refutes--> v16\n- v0c --supports--> v09\n- v12 --refutes--> v13\n- v14 --supports--> v17\n- v14 --supports--> v18\n- v1e --supports--> v0b\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":"a10bb104370ef6b09bebefc05b1ed1638f3d8b8346f31beb2ef1f1a285190f95","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v0d\n- v0d --refutes--> v17\n- v17 --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v1b\n- v1b --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v05\n- v05 --supports--> v1a\n- v1a --refutes--> v16\n- v16 --supports--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v1c\n- v1c --supports--> v0a\n- v0a --supports--> v10\n- v10 --refutes--> v15\n- v15 --supports--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v1d\n- v07 --supports--> v18\n- v0a --supports--> v06\n- v0b --supports--> v13\n- v0e --refutes--> v1e\n- v10 --refutes--> v0f\n- v12 --refutes--> v14\n- v12 --supports--> v11\n- v17 --supports--> v04\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":"14b6fba4da9aa6fe1df386c00c1a03c8379ee5385da4f1fa83b96d9e0d3ee153","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v1a\n- v1a --refutes--> v10\n- v10 --refutes--> v04\n- v04 --supports--> v1e\n- v1e --refutes--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v0f\n- v0f --refutes--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v19\n- v19 --supports--> v11\n- v11 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v1d\n- v0c --refutes--> v13\n- v0f --refutes--> v0e\n- v12 --supports--> v02\n- v15 --supports--> v14\n- v15 --supports--> v1c\n- v19 --refutes--> v09\n- v1a --refutes--> v05\n- v1b --refutes--> v03\n- v1b --refutes--> v16\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":"cdfc52908c094be49b7321771fd079f37ac83efb00b71ec368332c62b5f2f74b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v19\n- v19 --refutes--> v0e\n- v0e --supports--> v1c\n- v1c --supports--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v1d\n- v1d --refutes--> v01\n- v01 --refutes--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v02\n- v02 --supports--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v15\n- v15 --refutes--> v04\n- v04 --supports--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v1b\n- v1b --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v17\n- v03 --refutes--> v06\n- v04 --supports--> v0b\n- v08 --refutes--> v1e\n- v08 --supports--> v13\n- v0d --refutes--> v18\n- v0f --supports--> v1a\n- v15 --refutes--> v00\n- v1c --refutes--> v16\n- v1c --supports--> v0a\n\n[VERIFIED ROOT]\nv08 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f17d1df5316f5f87ac997883bb9bbdac04927ffc42cec8c072d2cbac2dcc1d1a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v12\n- v12 --refutes--> v04\n- v04 --supports--> v18\n- v18 --supports--> v1a\n- v1a --refutes--> v03\n- v03 --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v19\n- v19 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v17\n- v17 --refutes--> v0d\n- v0d --refutes--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v05\n- v05 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1e\n- v01 --supports--> v1c\n- v06 --supports--> v13\n- v07 --supports--> v0c\n- v0f --supports--> v08\n- v0f --supports--> v0a\n- v11 --supports--> v10\n- v17 --refutes--> v02\n- v17 --refutes--> v15\n- v1a --refutes--> v1b\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":"4676987ed7d37d864d3443ac204fa0a22e9913d6a532f134e792e2ee05a8d82c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v11\n- v11 --supports--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v05\n- v05 --refutes--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --supports--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --supports--> v1c\n- v1c --refutes--> v1d\n- v1d --refutes--> v0e\n- v0e --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v0b --supports--> v16\n- v0d --refutes--> v00\n- v0e --supports--> v1a\n- v11 --refutes--> v04\n- v11 --refutes--> v19\n- v1b --refutes--> v06\n- v1b --refutes--> v1e\n- v1d --refutes--> v14\n- v1d --supports--> v18\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":"9bf374d1506ce52307ce7ff800126285897f3e6ee9ca9a5526c1197a6cba07f7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v0b\n- v0b --supports--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v19\n- v19 --refutes--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v07\n- v07 --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v1a\n- v1a --refutes--> v18\n- v18 --supports--> v11\n- v11 --supports--> v1b\n- v1b --supports--> v14\n- v14 --supports--> v10\n- v10 --refutes--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v1e\n- v1e --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v17\n- v05 --refutes--> v13\n- v07 --refutes--> v03\n- v07 --refutes--> v04\n- v0d --supports--> v09\n- v10 --supports--> v06\n- v15 --refutes--> v0e\n- v16 --supports--> v12\n- v18 --supports--> v08\n- v1e --refutes--> v01\n\n[VERIFIED ROOT]\nv00 = FALSE\n\n[TARGET]\nv0c\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"adbec7b1d2d53db729a92d1867c7210fe9770acbc2148fcd664d3e0f57ddbb80","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v1d\n- v1d --supports--> v0c\n- v0c --supports--> v1b\n- v1b --supports--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v05\n- v05 --refutes--> v1a\n- v1a --refutes--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v14\n- v14 --supports--> v16\n- v16 --supports--> v18\n- v18 --refutes--> v08\n- v08 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0a\n- v0c --supports--> v12\n- v0d --supports--> v00\n- v11 --supports--> v10\n- v13 --supports--> v0b\n- v13 --supports--> v17\n- v18 --supports--> v1e\n- v19 --refutes--> v03\n- v1a --refutes--> v07\n- v1a --supports--> v09\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":"3f6b87de39b1cf516c7379ca8394e23d128083c9dbfb7a7612912eed402bc2ff","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v15\n- v15 --refutes--> v17\n- v17 --supports--> v13\n- v13 --supports--> v0b\n- v0b --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --supports--> v0f\n- v0f --supports--> v16\n- v16 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --refutes--> v0d\n- v0d --supports--> v1b\n- v1b --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v1c\n- v1c --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v18\n- v18 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v1a\n- v08 --supports--> v02\n- v0d --refutes--> v01\n- v12 --supports--> v19\n- v15 --refutes--> v07\n- v15 --supports--> v09\n- v17 --refutes--> v1e\n- v18 --refutes--> v10\n- v18 --supports--> v03\n- v1c --refutes--> v04\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":"e9dec5bfee248e88cef6617b584bb527745a6e1a4b2eb6c1c4e6ef5d2cfb08e3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v02\n- v02 --refutes--> v1d\n- v1d --supports--> v05\n- v05 --supports--> v1e\n- v1e --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v1c\n- v1c --refutes--> v13\n- v13 --refutes--> v18\n- v18 --refutes--> v15\n- v15 --supports--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v0c\n- v0c --refutes--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v16\n- v16 --refutes--> v03\n- v03 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v04\n- v05 --refutes--> v01\n- v09 --supports--> v07\n- v0a --refutes--> v17\n- v12 --refutes--> v06\n- v13 --refutes--> v10\n- v14 --refutes--> v0e\n- v18 --supports--> v19\n- v1c --supports--> v1b\n- v1d --supports--> v11\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":"c682b60afbac7ed8eb987b28662e61ec2ba0f47ac66311925e271c8cc0f2dffd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v1e\n- v1e --refutes--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v0d\n- v0d --refutes--> v1c\n- v1c --supports--> v14\n- v14 --refutes--> v1b\n- v1b --supports--> v08\n- v08 --supports--> v05\n- v05 --supports--> v07\n- v07 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --supports--> v19\n- v09 --supports--> v1d\n- v18 --refutes--> v03\n- v18 --refutes--> v17\n- v18 --supports--> v13\n- v1b --refutes--> v1a\n- v1e --refutes--> v06\n- v1e --refutes--> v0c\n- v1e --refutes--> v10\n- v1e --supports--> v12\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":"4d8e59a538bf9509aacedab0477ab16f7a0848d35c6a5f78754592e809dec65b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v1b\n- v1b --supports--> v17\n- v17 --refutes--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --supports--> v15\n- v15 --refutes--> v04\n- v04 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v1c\n- v1c --refutes--> v02\n- v02 --refutes--> v12\n- v12 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v1d\n- v1d --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0c\n- v00 --refutes--> v0d\n- v02 --supports--> v16\n- v04 --supports--> v01\n- v06 --refutes--> v1e\n- v09 --supports--> v05\n- v10 --refutes--> v1a\n- v12 --refutes--> v19\n- v14 --refutes--> v18\n- v1d --refutes--> v13\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":"4480fed28775c60ad6c88d145df961c7f6477fb0ff57a3157c77db64372da768","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v0e\n- v0e --refutes--> v19\n- v19 --refutes--> v09\n- v09 --refutes--> v18\n- v18 --supports--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v14\n- v14 --supports--> v1d\n- v1d --supports--> v16\n- v16 --refutes--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v15\n- v15 --supports--> v1e\n- v1e --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v03\n- v03 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v11\n- v09 --supports--> v08\n- v09 --supports--> v1c\n- v10 --refutes--> v01\n- v12 --refutes--> v02\n- v12 --supports--> v1b\n- v13 --supports--> v1a\n- v15 --refutes--> v00\n- v16 --refutes--> v17\n- v18 --refutes--> v0b\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":"eb414f4bd0c3eb83dc9d23887ad055835457a57eee9827f99722e5bd3bfc9231","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v08\n- v08 --supports--> v19\n- v19 --supports--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v1a\n- v1a --supports--> v0f\n- v0f --supports--> v17\n- v17 --supports--> v04\n- v04 --refutes--> v1e\n- v1e --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --supports--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v18\n- v18 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --refutes--> v1b\n- v1b --supports--> v09\n- v09 --refutes--> v0d\n- v0d --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v04 --refutes--> v06\n- v09 --supports--> v0e\n- v0a --supports--> v05\n- v0c --supports--> v15\n- v0c --supports--> v16\n- v0d --refutes--> v1c\n- v19 --supports--> v10\n- v19 --supports--> v14\n- v1e --supports--> v13\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":"de96c741e798a7696eacfb384b3f30ad9162582a9ed50f833457fbb3e48f54d8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v1e\n- v1e --refutes--> v16\n- v16 --supports--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v1a\n- v1a --refutes--> v01\n- v01 --supports--> v17\n- v17 --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v1b\n- v1b --supports--> v05\n- v05 --supports--> v09\n- v09 --supports--> v10\n- v10 --supports--> v04\n- v04 --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v03\n- v05 --refutes--> v00\n- v11 --refutes--> v07\n- v11 --refutes--> v1c\n- v11 --refutes--> v1d\n- v17 --refutes--> v14\n- v17 --supports--> v06\n- v19 --refutes--> v08\n- v19 --refutes--> v13\n- v19 --supports--> v0b\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":"178d89cdb671b5d55806ddf3ca48f205f9269509697714773349bd28ba2965f4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v12\n- v12 --refutes--> v00\n- v00 --supports--> v16\n- v16 --supports--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v0a\n- v0a --supports--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v1b\n- v1b --refutes--> v1a\n- v1a --refutes--> v14\n- v14 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v1c\n- v1c --refutes--> v09\n- v09 --supports--> v19\n- v19 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v17\n- v01 --supports--> v05\n- v01 --supports--> v13\n- v03 --refutes--> v11\n- v08 --refutes--> v15\n- v0d --refutes--> v18\n- v0f --refutes--> v10\n- v12 --refutes--> v04\n- v12 --supports--> v06\n- v1c --supports--> v0c\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":"565215f3a5fc8f575def1bef756eb4e77fd9edcffcddafb4dc99e8fce0f774f6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v12\n- v12 --refutes--> v02\n- v02 --refutes--> v09\n- v09 --supports--> v16\n- v16 --refutes--> v0b\n- v0b --supports--> v06\n- v06 --supports--> v18\n- v18 --supports--> v1a\n- v1a --refutes--> v0d\n- v0d --supports--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v1c\n- v1c --supports--> v13\n- v13 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v04\n- v05 --refutes--> v11\n- v0d --supports--> v01\n- v0d --supports--> v15\n- v0f --supports--> v0e\n- v10 --refutes--> v14\n- v10 --refutes--> v1e\n- v12 --refutes--> v1d\n- v1a --refutes--> v00\n- v1a --supports--> v07\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":"7f762cc4ffddb6c04ae9f1bbcc1bb961110d867ebd2b55f655764f6aa842ec86","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v0a\n- v0a --supports--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v02\n- v02 --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v1e\n- v1e --supports--> v13\n- v13 --supports--> v07\n- v07 --supports--> v19\n- v19 --refutes--> v12\n- v12 --supports--> v1b\n- v1b --supports--> v1c\n- v1c --supports--> v0c\n- v0c --supports--> v18\n- v18 --refutes--> v0b\n- v0b --refutes--> v16\n- v16 --refutes--> v0f\n- v0f --refutes--> v08\n- v08 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v04 --supports--> v14\n- v07 --refutes--> v1d\n- v07 --supports--> v09\n- v08 --supports--> v11\n- v0a --supports--> v00\n- v12 --refutes--> v17\n- v13 --supports--> v01\n- v1a --supports--> v0e\n- v1c --refutes--> v10\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":"a48d78b8c21aaf2813095efd1accfc1c834c097f84f6da42372ca3c8c2d53905","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v0a\n- v0a --refutes--> v1d\n- v1d --refutes--> v1c\n- v1c --refutes--> v16\n- v16 --refutes--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v1e\n- v1e --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v06 --supports--> v01\n- v0b --supports--> v15\n- v0e --refutes--> v09\n- v0e --supports--> v07\n- v13 --refutes--> v02\n- v13 --supports--> v19\n- v16 --supports--> v10\n- v1e --supports--> v08\n- v1e --supports--> v17\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":"c1d678dadd26ec10e3894698db3d367ab893885884d6433e25d171ceb91b43ba","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v18\n- v18 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v15\n- v15 --refutes--> v1d\n- v1d --supports--> v09\n- v09 --supports--> v1c\n- v1c --supports--> v07\n- v07 --supports--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v13\n- v13 --supports--> v1b\n- v1b --refutes--> v1a\n- v1a --refutes--> v19\n- v19 --supports--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0a\n- v07 --refutes--> v0e\n- v09 --refutes--> v12\n- v0d --refutes--> v17\n- v0d --supports--> v08\n- v10 --refutes--> v0f\n- v10 --supports--> v14\n- v13 --refutes--> v03\n- v19 --refutes--> v02\n- v1b --refutes--> v11\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":"ef77a3f64c24e799908ee806a9f358dab9e6397966de922ba02ea6c4b07962fd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v19\n- v19 --supports--> v02\n- v02 --supports--> v01\n- v01 --supports--> v1d\n- v1d --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v17\n- v17 --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --supports--> v1b\n- v1b --refutes--> v18\n- v18 --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v1e\n- v1e --refutes--> v03\n- v03 --supports--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v11\n- v06 --refutes--> v0b\n- v06 --supports--> v12\n- v0a --refutes--> v0c\n- v0f --refutes--> v05\n- v17 --refutes--> v1c\n- v18 --refutes--> v0d\n- v19 --supports--> v00\n- v19 --supports--> v16\n- v1d --refutes--> v04\n\n[VERIFIED ROOT]\nv10 = FALSE\n\n[TARGET]\nv14\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"eeb50515806650ba9051960c4baf43078c4e0eae5a4eebf1cbccf8656c951aa3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v1c\n- v1c --supports--> v0c\n- v0c --refutes--> v1d\n- v1d --supports--> v07\n- v07 --refutes--> v12\n- v12 --supports--> v14\n- v14 --supports--> v1a\n- v1a --supports--> v18\n- v18 --refutes--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v04\n- v04 --refutes--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v13\n- v13 --supports--> v1b\n- v1b --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v19\n- v03 --supports--> v11\n- v04 --refutes--> v16\n- v08 --refutes--> v1e\n- v0b --refutes--> v0d\n- v12 --supports--> v17\n- v14 --supports--> v0f\n- v1a --refutes--> v02\n- v1c --supports--> v10\n- v1d --refutes--> v00\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":"7eb91d3b9068fdc0a77b4619f22b4a3ff81f343a505964d8d5f41f14d5926d4d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v18\n- v18 --refutes--> v05\n- v05 --refutes--> v1c\n- v1c --refutes--> v12\n- v12 --supports--> v14\n- v14 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v17\n- v17 --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --supports--> v07\n- v07 --refutes--> v1b\n- v1b --supports--> v0a\n- v0a --supports--> v00\n- v00 --refutes--> v13\n- v13 --refutes--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v1d\n- v1d --refutes--> v10\n- v10 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v00 --supports--> v01\n- v02 --refutes--> v04\n- v0a --supports--> v09\n- v0e --refutes--> v06\n- v0e --refutes--> v1a\n- v11 --supports--> v0b\n- v1c --supports--> v0d\n- v1d --refutes--> v1e\n- v1d --supports--> v03\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":"1a8bad9be667f077ecec38ea80e75a8bf2a7303ae4a7bce6c92b8f5ebb358f78","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v18\n- v18 --refutes--> v00\n- v00 --supports--> v11\n- v11 --refutes--> v14\n- v14 --refutes--> v1d\n- v1d --refutes--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v19\n- v19 --refutes--> v1a\n- v1a --refutes--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v10\n- v10 --supports--> v13\n- v13 --refutes--> v1c\n- v1c --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v15\n- v04 --supports--> v0c\n- v0a --supports--> v12\n- v0b --refutes--> v1e\n- v14 --supports--> v07\n- v16 --refutes--> v17\n- v18 --refutes--> v0e\n- v1a --refutes--> v09\n- v1d --refutes--> v1b\n- v1d --supports--> v08\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":"bc2ca8f2ca072069f839923bff4c03b531e2977ada46aa4aa3a528c15636677d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v16\n- v16 --supports--> v1e\n- v1e --refutes--> v19\n- v19 --supports--> v00\n- v00 --supports--> v17\n- v17 --refutes--> v07\n- v07 --supports--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v1d\n- v1d --supports--> v0c\n- v0c --refutes--> v11\n- v11 --supports--> v1b\n- v1b --refutes--> v12\n- v12 --supports--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v06\n- v07 --refutes--> v0f\n- v08 --supports--> v18\n- v09 --refutes--> v10\n- v09 --refutes--> v15\n- v0c --supports--> v1a\n- v12 --supports--> v01\n- v14 --supports--> v0a\n- v16 --refutes--> v1c\n- v1e --refutes--> v13\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":"40d92ba610faecdcd6d33ba47063ea04691ecde7f3ed6a118c3d9efbf4e3968b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --refutes--> v14\n- v14 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v19\n- v19 --refutes--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v0d\n- v0d --supports--> v13\n- v13 --supports--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v1a\n- v1a --refutes--> v1d\n- v1d --refutes--> v09\n- v09 --supports--> v1b\n- v1b --refutes--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v03 --refutes--> v0f\n- v05 --refutes--> v1c\n- v08 --supports--> v18\n- v14 --refutes--> v07\n- v15 --supports--> v06\n- v16 --refutes--> v00\n- v16 --refutes--> v0e\n- v1d --supports--> v10\n- v1e --supports--> v17\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":"8d87e225510192571552da1a07d15c9a3b8f5a151bf044be93b03bcc73b3ad6f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v13\n- v13 --supports--> v08\n- v08 --refutes--> v1e\n- v1e --supports--> v15\n- v15 --refutes--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v17\n- v17 --supports--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v06\n- v06 --refutes--> v1c\n- v1c --refutes--> v01\n- v01 --supports--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0a\n- v04 --refutes--> v0d\n- v04 --refutes--> v14\n- v0b --refutes--> v18\n- v0c --refutes--> v05\n- v0c --supports--> v02\n- v0c --supports--> v1b\n- v0f --refutes--> v1d\n- v11 --refutes--> v16\n- v17 --supports--> v1a\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":"ef5fe057f9912dd68af5d688fdf15de9ccb6150125d3c5b7bcfa2c357ae1873f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v12\n- v12 --supports--> v13\n- v13 --refutes--> v04\n- v04 --supports--> v1e\n- v1e --supports--> v1b\n- v1b --supports--> v0c\n- v0c --refutes--> v18\n- v18 --refutes--> v0e\n- v0e --refutes--> v05\n- v05 --supports--> v16\n- v16 --supports--> v1d\n- v1d --supports--> v1a\n- v1a --refutes--> v17\n- v17 --supports--> v02\n- v02 --supports--> v03\n- v03 --refutes--> v01\n- v01 --supports--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v00\n- v04 --supports--> v10\n- v0c --refutes--> v1c\n- v12 --refutes--> v07\n- v14 --refutes--> v06\n- v16 --supports--> v09\n- v18 --supports--> v08\n- v18 --supports--> v0b\n- v1b --refutes--> v11\n- v1b --supports--> v19\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":"ee268cb22b71ca9e28c9e688f67fb7d5bb86c94aaa17fc9a01e275f445cb403d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v1e\n- v1e --refutes--> v07\n- v07 --refutes--> v14\n- v14 --refutes--> v0b\n- v0b --refutes--> v17\n- v17 --supports--> v15\n- v15 --supports--> v13\n- v13 --supports--> v1a\n- v1a --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v08\n- v08 --supports--> v11\n- v11 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --supports--> v1c\n- v1c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1d\n- v03 --refutes--> v10\n- v04 --refutes--> v18\n- v05 --supports--> v19\n- v0b --supports--> v01\n- v0d --refutes--> v02\n- v11 --refutes--> v16\n- v1a --supports--> v1b\n- v1c --refutes--> v0f\n- v1e --refutes--> v12\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":"19ae301e08c6962e98146bff3fe10a04e6344db1f869fe1cd40787c56ee501d5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --supports--> v13\n- v13 --refutes--> v0e\n- v0e --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v1e\n- v1e --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v0f\n- v0f --supports--> v1a\n- v1a --supports--> v11\n- v11 --supports--> v12\n- v12 --supports--> v15\n- v15 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v10\n- v02 --supports--> v1b\n- v03 --refutes--> v16\n- v07 --refutes--> v19\n- v11 --refutes--> v14\n- v11 --refutes--> v1c\n- v11 --supports--> v04\n- v11 --supports--> v0a\n- v12 --refutes--> v06\n- v1a --supports--> v18\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":"e6acff94814a5cd6332926229454f08ef7762c1ea9ee72c0104e808090558f33","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v1d\n- v1d --supports--> v07\n- v07 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v16\n- v16 --refutes--> v0d\n- v0d --refutes--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v19\n- v19 --refutes--> v05\n- v05 --supports--> v08\n- v08 --supports--> v1e\n- v1e --supports--> v00\n- v00 --refutes--> v10\n- v10 --supports--> v1b\n- v1b --refutes--> v13\n- v13 --supports--> v17\n- v17 --refutes--> v04\n- v04 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v0e\n- v08 --supports--> v03\n- v0a --supports--> v14\n- v10 --supports--> v02\n- v13 --supports--> v12\n- v15 --refutes--> v0f\n- v17 --supports--> v06\n- v19 --refutes--> v0c\n- v19 --refutes--> v1a\n- v19 --supports--> v01\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":"9d8b8cc43221b07cc9efdfcb87601233075378413d8e8a5efade2aa41ba569ec","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v12\n- v12 --supports--> v15\n- v15 --supports--> v08\n- v08 --supports--> v1e\n- v1e --refutes--> v17\n- v17 --supports--> v14\n- v14 --supports--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v19\n- v19 --supports--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --refutes--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v1a\n- v1a --supports--> v13\n- v13 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0c\n- v04 --refutes--> v0b\n- v06 --refutes--> v00\n- v0f --refutes--> v0e\n- v0f --supports--> v1c\n- v13 --supports--> v0d\n- v13 --supports--> v1d\n- v14 --refutes--> v18\n- v15 --supports--> v1b\n- v17 --supports--> v02\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":"cfe62f0876381b5b6dde38cad767981518ce30afeee5b3eb4d3b98429a5bc493","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v0f\n- v0f --refutes--> v18\n- v18 --supports--> v19\n- v19 --supports--> v1e\n- v1e --supports--> v24\n- v24 --refutes--> v0d\n- v0d --refutes--> v00\n- v00 --supports--> v01\n- v01 --refutes--> v02\n- v02 --supports--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v10\n- v10 --supports--> v1a\n- v1a --refutes--> v21\n- v21 --supports--> v17\n- v17 --refutes--> v12\n- v12 --refutes--> v03\n- v03 --supports--> v22\n- v22 --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --supports--> v23\n- v23 --refutes--> v20\n- v20 --refutes--> v15\n- v15 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1b\n- v03 --supports--> v04\n- v0d --supports--> v09\n- v14 --refutes--> v13\n- v16 --refutes--> v0a\n- v18 --refutes--> v0c\n- v18 --refutes--> v0e\n- v19 --supports--> v1f\n- v1d --refutes--> v08\n- v1d --refutes--> v0b\n- v1e --refutes--> v05\n- v24 --supports--> v07\n\n[VERIFIED ROOT]\nv14 = FALSE\n\n[TARGET]\nv06\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"e152ca4691e174343df7ab347987ba3ce2940923a33f5905f6d7810e6632293f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --refutes--> v21\n- v21 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v20\n- v20 --supports--> v12\n- v12 --supports--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v18\n- v18 --refutes--> v23\n- v23 --supports--> v1e\n- v1e --refutes--> v24\n- v24 --refutes--> v11\n- v11 --refutes--> v22\n- v22 --supports--> v0a\n- v0a --refutes--> v1f\n- v1f --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v13\n- v13 --supports--> v1d\n- v1d --refutes--> v1a\n- v1a --refutes--> v17\n- v17 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v00\n- v07 --refutes--> v14\n- v0c --supports--> v06\n- v0d --supports--> v08\n- v11 --supports--> v1c\n- v12 --refutes--> v16\n- v12 --supports--> v0e\n- v17 --refutes--> v1b\n- v18 --refutes--> v09\n- v1a --supports--> v04\n- v1f --refutes--> v02\n- v21 --supports--> v03\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":"79ccf84a1721ecb5c5f777b0ea15ebf8c2d939992f25fe7f9dce8d9d09e2e317","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v0f\n- v0f --supports--> v19\n- v19 --refutes--> v04\n- v04 --supports--> v1d\n- v1d --supports--> v1b\n- v1b --refutes--> v17\n- v17 --supports--> v24\n- v24 --refutes--> v21\n- v21 --supports--> v18\n- v18 --refutes--> v09\n- v09 --supports--> v1f\n- v1f --refutes--> v07\n- v07 --supports--> v10\n- v10 --supports--> v14\n- v14 --supports--> v06\n- v06 --supports--> v0a\n- v0a --supports--> v15\n- v15 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v11\n- v11 --refutes--> v22\n- v22 --supports--> v01\n- v01 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v23\n- v03 --refutes--> v1a\n- v06 --refutes--> v0d\n- v06 --refutes--> v16\n- v06 --refutes--> v1e\n- v07 --refutes--> v12\n- v0f --refutes--> v0e\n- v19 --refutes--> v0c\n- v1b --supports--> v0b\n- v21 --refutes--> v1c\n- v21 --refutes--> v20\n- v21 --supports--> v05\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":"d2a5bd76bec01730f3d58e0370a314729ea84611ce3b56bc65be68d48b5bcca1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v14\n- v14 --supports--> v15\n- v15 --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v23\n- v23 --supports--> v0d\n- v0d --refutes--> v21\n- v21 --supports--> v1e\n- v1e --supports--> v20\n- v20 --refutes--> v0b\n- v0b --supports--> v1b\n- v1b --refutes--> v07\n- v07 --supports--> v10\n- v10 --refutes--> v18\n- v18 --supports--> v16\n- v16 --supports--> v1c\n- v1c --supports--> v01\n- v01 --refutes--> v22\n- v22 --supports--> v1d\n- v1d --supports--> v17\n- v17 --refutes--> v13\n- v13 --supports--> v1f\n- v1f --refutes--> v04\n- v04 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v09\n- v0b --refutes--> v1a\n- v0c --supports--> v05\n- v0d --supports--> v06\n- v0d --supports--> v08\n- v0e --supports--> v03\n- v14 --supports--> v0f\n- v1f --supports--> v11\n- v20 --supports--> v02\n- v20 --supports--> v19\n- v20 --supports--> v24\n- v21 --supports--> v00\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":"9a83506b93406c9b87fa93d286274fff563c9e8b7a9749e8f63d2e804941f972","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v19\n- v19 --refutes--> v1c\n- v1c --supports--> v03\n- v03 --supports--> v0e\n- v0e --supports--> v20\n- v20 --supports--> v08\n- v08 --refutes--> v1d\n- v1d --refutes--> v12\n- v12 --supports--> v1a\n- v1a --supports--> v23\n- v23 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v16\n- v16 --refutes--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --supports--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v22\n- v22 --refutes--> v14\n- v14 --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v02\n- v03 --supports--> v24\n- v06 --refutes--> v04\n- v0a --supports--> v05\n- v0f --supports--> v09\n- v0f --supports--> v1f\n- v12 --supports--> v1e\n- v13 --refutes--> v15\n- v17 --refutes--> v07\n- v22 --supports--> v18\n- v23 --refutes--> v0b\n- v23 --supports--> v10\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":"c01da5b8d65ffbc1450d9995f50885f9482a74fa14ad12a0b959b5d1f8bc223f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v23\n- v23 --refutes--> v1d\n- v1d --supports--> v05\n- v05 --supports--> v1f\n- v1f --refutes--> v0c\n- v0c --supports--> v21\n- v21 --supports--> v09\n- v09 --supports--> v08\n- v08 --refutes--> v1b\n- v1b --supports--> v03\n- v03 --supports--> v24\n- v24 --supports--> v02\n- v02 --refutes--> v0e\n- v0e --refutes--> v0d\n- v0d --refutes--> v19\n- v19 --supports--> v01\n- v01 --supports--> v16\n- v16 --refutes--> v0f\n- v0f --refutes--> v20\n- v20 --supports--> v04\n- v04 --refutes--> v0a\n- v0a --supports--> v1e\n- v1e --refutes--> v11\n- v11 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v18\n- v02 --supports--> v1a\n- v0a --supports--> v13\n- v0e --supports--> v14\n- v1b --supports--> v06\n- v1d --refutes--> v15\n- v1e --refutes--> v12\n- v1e --supports--> v22\n- v1f --refutes--> v07\n- v1f --refutes--> v1c\n- v1f --supports--> v00\n- v24 --refutes--> v17\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":"fab971ff52ee576cc71512ea80be9d5daf7aec176e91fcfbbef63742b3b61a3a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v17\n- v17 --refutes--> v1c\n- v1c --supports--> v02\n- v02 --refutes--> v24\n- v24 --supports--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v19\n- v19 --refutes--> v0a\n- v0a --refutes--> v20\n- v20 --refutes--> v1b\n- v1b --refutes--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v13\n- v13 --refutes--> v1e\n- v1e --refutes--> v1f\n- v1f --supports--> v04\n- v04 --refutes--> v11\n- v11 --refutes--> v18\n- v18 --refutes--> v14\n- v14 --refutes--> v0c\n- v0c --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v08 --refutes--> v0e\n- v0d --supports--> v00\n- v10 --supports--> v06\n- v11 --supports--> v03\n- v13 --refutes--> v16\n- v13 --refutes--> v23\n- v13 --supports--> v21\n- v14 --refutes--> v05\n- v14 --refutes--> v1d\n- v1c --refutes--> v22\n- v1f --supports--> v1a\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":"ca63aeddedbaeebf01129d60021d32815a93806b53e1f086e2c25e1507d8600e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v1c\n- v1c --refutes--> v02\n- v02 --supports--> v0d\n- v0d --supports--> v21\n- v21 --supports--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v09\n- v09 --supports--> v1d\n- v1d --supports--> v08\n- v08 --refutes--> v1a\n- v1a --supports--> v1e\n- v1e --refutes--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v01\n- v01 --supports--> v1f\n- v1f --supports--> v0f\n- v0f --refutes--> v17\n- v17 --supports--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v14\n- v14 --supports--> v12\n- v12 --supports--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v00\n- v0e --supports--> v18\n- v0f --refutes--> v0a\n- v13 --refutes--> v07\n- v13 --refutes--> v15\n- v13 --supports--> v05\n- v13 --supports--> v10\n- v13 --supports--> v20\n- v17 --supports--> v1b\n- v1c --refutes--> v11\n- v1e --supports--> v03\n- v21 --supports--> v24\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":"ae9272c2def07ce951f13ddeecbe612928817490f9c3dcd635f91a70bc57462e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v1c\n- v1c --refutes--> v20\n- v20 --refutes--> v13\n- v13 --refutes--> v1b\n- v1b --supports--> v11\n- v11 --refutes--> v03\n- v03 --supports--> v1a\n- v1a --supports--> v1e\n- v1e --supports--> v17\n- v17 --refutes--> v06\n- v06 --refutes--> v16\n- v16 --supports--> v23\n- v23 --supports--> v12\n- v12 --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --refutes--> v15\n- v15 --supports--> v0e\n- v0e --refutes--> v0c\n- v0c --supports--> v24\n- v24 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v00 --supports--> v09\n- v05 --refutes--> v07\n- v08 --supports--> v02\n- v11 --supports--> v0a\n- v13 --supports--> v1f\n- v15 --supports--> v04\n- v15 --supports--> v0d\n- v1a --refutes--> v01\n- v1c --refutes--> v22\n- v1e --refutes--> v19\n- v24 --supports--> v0b\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":"a65e4899515980df23891a1b18b69e4907fc27cc2ef2858b34bd017094d92642","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v1f\n- v1f --supports--> v17\n- v17 --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v15\n- v15 --refutes--> v1d\n- v1d --refutes--> v0f\n- v0f --supports--> v18\n- v18 --refutes--> v1b\n- v1b --refutes--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v24\n- v24 --supports--> v0d\n- v0d --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v1e\n- v1e --refutes--> v21\n- v21 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v1c\n- v04 --supports--> v0e\n- v09 --supports--> v23\n- v0a --supports--> v1a\n- v0c --refutes--> v08\n- v0d --supports--> v16\n- v11 --supports--> v00\n- v15 --supports--> v14\n- v17 --refutes--> v19\n- v17 --supports--> v12\n- v1b --refutes--> v01\n- v1d --supports--> v22\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":"474e8e0766d224056e3529bd307e2776a315b9c4961b6ea54db5d2dd1b6be7ae","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --supports--> v1e\n- v1e --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v22\n- v22 --supports--> v1b\n- v1b --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v15\n- v15 --refutes--> v17\n- v17 --supports--> v08\n- v08 --supports--> v00\n- v00 --supports--> v12\n- v12 --supports--> v19\n- v19 --supports--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v1c\n- v1c --supports--> v05\n- v05 --refutes--> v11\n- v11 --supports--> v24\n- v24 --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0c\n- v02 --supports--> v0a\n- v05 --supports--> v1d\n- v08 --supports--> v0f\n- v0d --refutes--> v1f\n- v0e --supports--> v09\n- v14 --supports--> v06\n- v17 --refutes--> v23\n- v19 --supports--> v18\n- v1b --refutes--> v13\n- v1c --refutes--> v03\n- v1c --refutes--> v21\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":"f72ffb896c29978092eaed772f54c89a014cd69491c919ea2730467ccc645a39","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --refutes--> v13\n- v13 --supports--> v08\n- v08 --refutes--> v23\n- v23 --supports--> v12\n- v12 --refutes--> v17\n- v17 --refutes--> v1c\n- v1c --refutes--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v09\n- v09 --refutes--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v16\n- v16 --supports--> v19\n- v19 --refutes--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v22\n- v22 --refutes--> v03\n- v03 --refutes--> v20\n- v20 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v1e\n- v1e --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v08 --refutes--> v05\n- v0a --supports--> v04\n- v13 --refutes--> v1d\n- v13 --supports--> v11\n- v17 --supports--> v02\n- v18 --refutes--> v24\n- v19 --refutes--> v21\n- v1b --supports--> v1f\n- v1c --supports--> v07\n- v1e --supports--> v0e\n- v20 --refutes--> v0c\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":"3e8ea5338757a5205fe8bb78437fdcf86e34c7acb575349fb0715b1901535198","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v08\n- v08 --refutes--> v1c\n- v1c --refutes--> v21\n- v21 --refutes--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v24\n- v24 --refutes--> v01\n- v01 --supports--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v1e\n- v1e --refutes--> v0b\n- v0b --supports--> v17\n- v17 --supports--> v12\n- v12 --supports--> v23\n- v23 --supports--> v13\n- v13 --refutes--> v19\n- v19 --supports--> v14\n- v14 --refutes--> v22\n- v22 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --refutes--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v02\n- v08 --refutes--> v0c\n- v10 --refutes--> v20\n- v10 --supports--> v1d\n- v16 --refutes--> v11\n- v17 --refutes--> v1f\n- v1b --supports--> v1a\n- v1c --refutes--> v06\n- v21 --refutes--> v15\n- v21 --supports--> v04\n- v21 --supports--> v0e\n- v22 --refutes--> v0d\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":"cf5b97cb30eb257d68d24ad370c89595834558b8f006911106943275979b2868","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v22\n- v22 --refutes--> v11\n- v11 --refutes--> v1b\n- v1b --supports--> v1f\n- v1f --supports--> v14\n- v14 --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v00\n- v00 --supports--> v1e\n- v1e --supports--> v18\n- v18 --supports--> v21\n- v21 --supports--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v10\n- v10 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v1d\n- v1d --refutes--> v1c\n- v1c --refutes--> v0d\n- v0d --supports--> v13\n- v13 --supports--> v12\n- v12 --supports--> v20\n- v20 --refutes--> v02\n- v02 --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0b\n- v09 --refutes--> v15\n- v0a --refutes--> v06\n- v0c --supports--> v05\n- v10 --supports--> v0f\n- v12 --refutes--> v07\n- v12 --refutes--> v24\n- v1c --refutes--> v08\n- v1c --supports--> v0e\n- v1e --supports--> v03\n- v20 --refutes--> v17\n- v21 --supports--> v23\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":"5c7a9f55156eaded1ce4346d0a779df564967b9537b1b532a8c5980fe3156b62","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v1b\n- v1b --supports--> v20\n- v20 --supports--> v24\n- v24 --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v0a\n- v0a --refutes--> v18\n- v18 --refutes--> v13\n- v13 --supports--> v21\n- v21 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v1a\n- v1a --supports--> v1d\n- v1d --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --supports--> v23\n- v23 --refutes--> v22\n- v22 --refutes--> v17\n- v17 --refutes--> v00\n- v00 --refutes--> v03\n- v03 --supports--> v15\n- v15 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v01\n- v0a --refutes--> v05\n- v0a --refutes--> v16\n- v0b --refutes--> v0c\n- v13 --refutes--> v10\n- v13 --supports--> v04\n- v17 --supports--> v1e\n- v18 --supports--> v1f\n- v19 --refutes--> v14\n- v1c --refutes--> v0e\n- v1d --supports--> v08\n- v24 --refutes--> v07\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":"7b45d3203470947fbe08f047f6a0e316319b4bf684b8b3a25a66c79454d6ec3e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --supports--> v11\n- v11 --refutes--> v0a\n- v0a --supports--> v1d\n- v1d --supports--> v09\n- v09 --refutes--> v22\n- v22 --refutes--> v1c\n- v1c --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --refutes--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v16\n- v16 --supports--> v12\n- v12 --supports--> v1b\n- v1b --supports--> v15\n- v15 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v1e\n- v1e --refutes--> v1a\n- v1a --refutes--> v19\n- v19 --supports--> v1f\n- v1f --supports--> v00\n- v00 --refutes--> v23\n- v23 --refutes--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v01 --supports--> v03\n- v01 --supports--> v20\n- v06 --refutes--> v04\n- v06 --refutes--> v0c\n- v09 --refutes--> v18\n- v10 --refutes--> v02\n- v10 --supports--> v0d\n- v11 --supports--> v21\n- v14 --refutes--> v24\n- v14 --supports--> v0e\n- v15 --supports--> v0b\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":"f7a74befa2ab4398fbfa9857bd98aa5a6203f03b4d365aebaae9eb182bbae9f6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --refutes--> v23\n- v23 --supports--> v00\n- v00 --refutes--> v22\n- v22 --supports--> v03\n- v03 --refutes--> v1c\n- v1c --supports--> v05\n- v05 --supports--> v1d\n- v1d --refutes--> v14\n- v14 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v21\n- v21 --refutes--> v19\n- v19 --refutes--> v11\n- v11 --supports--> v20\n- v20 --supports--> v10\n- v10 --supports--> v09\n- v09 --refutes--> v1b\n- v1b --refutes--> v0a\n- v0a --supports--> v15\n- v15 --supports--> v08\n- v08 --supports--> v04\n- v04 --refutes--> v17\n- v17 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v07\n- v05 --refutes--> v1f\n- v09 --refutes--> v0b\n- v0a --refutes--> v01\n- v0a --refutes--> v0c\n- v17 --supports--> v18\n- v19 --refutes--> v12\n- v1b --refutes--> v13\n- v1c --refutes--> v06\n- v1d --supports--> v16\n- v20 --refutes--> v1a\n- v20 --supports--> v1e\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":"cafd84259e0bb8b7f70658da134827f7925814097235905a13e5acea4ef88c70","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --refutes--> v23\n- v23 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v19\n- v19 --supports--> v18\n- v18 --supports--> v1b\n- v1b --refutes--> v0c\n- v0c --refutes--> v21\n- v21 --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v17\n- v17 --supports--> v1e\n- v1e --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v24\n- v24 --refutes--> v0a\n- v0a --supports--> v0e\n- v0e --supports--> v1f\n- v1f --refutes--> v1c\n- v1c --supports--> v06\n- v06 --supports--> v22\n- v22 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v08\n- v0d --refutes--> v02\n- v0e --refutes--> v16\n- v0e --supports--> v13\n- v12 --supports--> v0b\n- v12 --supports--> v11\n- v1c --refutes--> v10\n- v1e --supports--> v00\n- v20 --supports--> v1a\n- v21 --refutes--> v14\n- v22 --refutes--> v01\n- v24 --supports--> v1d\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":"c916b73087de007fb22be0aa5ee63abd6b8af99c5caf08acc5d4fe4b7335f828","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v16\n- v16 --supports--> v10\n- v10 --supports--> v17\n- v17 --supports--> v1e\n- v1e --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v09\n- v09 --supports--> v13\n- v13 --supports--> v1f\n- v1f --supports--> v14\n- v14 --supports--> v1c\n- v1c --supports--> v15\n- v15 --supports--> v22\n- v22 --refutes--> v23\n- v23 --supports--> v1a\n- v1a --supports--> v0a\n- v0a --supports--> v11\n- v11 --supports--> v1b\n- v1b --refutes--> v21\n- v21 --refutes--> v24\n- v24 --supports--> v07\n- v07 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v11 --refutes--> v20\n- v14 --refutes--> v0e\n- v14 --supports--> v00\n- v14 --supports--> v03\n- v16 --supports--> v12\n- v1c --refutes--> v06\n- v1c --supports--> v1d\n- v1e --supports--> v01\n- v1f --supports--> v05\n- v21 --supports--> v0b\n- v23 --refutes--> v0f\n- v23 --refutes--> v19\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":"086d270ee4a0229ebba13246c6b8bb49b20b23c4fbd235ce3e0bb4387120d3a9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v1d\n- v1d --refutes--> v06\n- v06 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v21\n- v21 --supports--> v0c\n- v0c --refutes--> v22\n- v22 --refutes--> v1a\n- v1a --refutes--> v02\n- v02 --supports--> v14\n- v14 --supports--> v24\n- v24 --supports--> v05\n- v05 --supports--> v0d\n- v0d --refutes--> v19\n- v19 --supports--> v1e\n- v1e --refutes--> v13\n- v13 --supports--> v17\n- v17 --supports--> v1b\n- v1b --supports--> v1f\n- v1f --refutes--> v03\n- v03 --supports--> v0b\n- v0b --refutes--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v08\n- v03 --supports--> v10\n- v05 --refutes--> v00\n- v06 --refutes--> v0a\n- v0c --refutes--> v0e\n- v12 --supports--> v04\n- v13 --refutes--> v18\n- v13 --supports--> v07\n- v1a --supports--> v01\n- v1c --refutes--> v15\n- v1c --refutes--> v20\n- v22 --supports--> v09\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":"55b5f081ed9ff07d086862dc40f3f565770800d206ac430b50ae53e5cb554de5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --supports--> v08\n- v08 --supports--> v1f\n- v1f --supports--> v0d\n- v0d --refutes--> v06\n- v06 --supports--> v24\n- v24 --refutes--> v15\n- v15 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --supports--> v1a\n- v1a --supports--> v20\n- v20 --supports--> v12\n- v12 --refutes--> v05\n- v05 --supports--> v16\n- v16 --refutes--> v1e\n- v1e --supports--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v0f\n- v0f --refutes--> v1c\n- v1c --supports--> v03\n- v03 --refutes--> v18\n- v18 --supports--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v00 --supports--> v1d\n- v03 --supports--> v10\n- v0b --supports--> v11\n- v0e --refutes--> v1b\n- v0e --refutes--> v21\n- v18 --supports--> v19\n- v1a --refutes--> v13\n- v1c --refutes--> v14\n- v1f --refutes--> v17\n- v20 --refutes--> v02\n- v22 --refutes--> v23\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":"85906ee403a054a63976e639d84b5704877d797e7ba670adc9d54a1f5c5a41e5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v1d\n- v1d --supports--> v01\n- v01 --refutes--> v1c\n- v1c --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v11\n- v11 --refutes--> v00\n- v00 --supports--> v04\n- v04 --supports--> v0e\n- v0e --refutes--> v09\n- v09 --refutes--> v21\n- v21 --supports--> v1e\n- v1e --supports--> v1b\n- v1b --refutes--> v20\n- v20 --supports--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v02\n- v02 --refutes--> v15\n- v15 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v00 --supports--> v18\n- v01 --supports--> v17\n- v07 --refutes--> v19\n- v0e --supports--> v24\n- v10 --refutes--> v03\n- v11 --supports--> v23\n- v15 --supports--> v06\n- v15 --supports--> v16\n- v1b --supports--> v22\n- v1c --supports--> v0c\n- v1f --refutes--> v0a\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":"4ec4da4e35ad1e26cf1a07434af3dd9157f5e742ca5239e5893678fc4cfe63bb","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v0d\n- v0d --refutes--> v08\n- v08 --refutes--> v1e\n- v1e --supports--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v23\n- v23 --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v17\n- v17 --supports--> v18\n- v18 --refutes--> v03\n- v03 --refutes--> v21\n- v21 --supports--> v02\n- v02 --refutes--> v15\n- v15 --refutes--> v00\n- v00 --supports--> v19\n- v19 --refutes--> v24\n- v24 --refutes--> v14\n- v14 --refutes--> v1a\n- v1a --refutes--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v0e\n- v0e --refutes--> v1f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v22\n- v04 --refutes--> v07\n- v05 --supports--> v0f\n- v0e --refutes--> v0b\n- v10 --refutes--> v01\n- v10 --supports--> v0c\n- v11 --supports--> v09\n- v18 --refutes--> v20\n- v18 --supports--> v1c\n- v1b --refutes--> v13\n- v1b --supports--> v16\n- v21 --supports--> v1d\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":"93560488f22bac3b6115ccfcb0b29bc837f105276ed5d79838b229339d077fa6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v22\n- v22 --supports--> v1b\n- v1b --refutes--> v09\n- v09 --refutes--> v0a\n- v0a --refutes--> v23\n- v23 --refutes--> v10\n- v10 --supports--> v1e\n- v1e --supports--> v01\n- v01 --supports--> v07\n- v07 --supports--> v19\n- v19 --supports--> v1c\n- v1c --refutes--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v0b\n- v0b --supports--> v02\n- v02 --refutes--> v17\n- v17 --supports--> v1f\n- v1f --refutes--> v0c\n- v0c --refutes--> v18\n- v18 --refutes--> v1d\n- v1d --supports--> v0f\n- v0f --supports--> v20\n- v20 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v14\n- v06 --refutes--> v04\n- v0b --refutes--> v08\n- v0f --refutes--> v0e\n- v0f --supports--> v1a\n- v0f --supports--> v24\n- v11 --refutes--> v21\n- v17 --refutes--> v03\n- v18 --refutes--> v05\n- v1d --supports--> v13\n- v1e --refutes--> v0d\n- v22 --supports--> v00\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":"3887384b47c47681301fba6cd8d8efd0f9851293e24f004c7f3b1df875d11011","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v18\n- v18 --supports--> v12\n- v12 --supports--> v0c\n- v0c --refutes--> v1c\n- v1c --supports--> v04\n- v04 --supports--> v14\n- v14 --refutes--> v1f\n- v1f --supports--> v07\n- v07 --refutes--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v17\n- v17 --refutes--> v24\n- v24 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v1a\n- v1a --supports--> v15\n- v15 --refutes--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v1d\n- v1d --supports--> v11\n- v11 --refutes--> v22\n- v22 --refutes--> v05\n- v05 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v03\n- v07 --supports--> v13\n- v09 --supports--> v06\n- v11 --refutes--> v00\n- v11 --refutes--> v20\n- v11 --supports--> v0a\n- v12 --supports--> v08\n- v14 --supports--> v21\n- v17 --supports--> v02\n- v1c --refutes--> v1b\n- v1f --supports--> v0f\n- v23 --refutes--> v1e\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":"50413bf56df94f37fef05d3ab725763f099b9dd3297550cd60405f7da92d8ddd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --supports--> v13\n- v13 --refutes--> v16\n- v16 --supports--> v1c\n- v1c --refutes--> v10\n- v10 --supports--> v1a\n- v1a --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v19\n- v19 --supports--> v20\n- v20 --supports--> v23\n- v23 --supports--> v1e\n- v1e --refutes--> v06\n- v06 --supports--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v15\n- v15 --supports--> v12\n- v12 --refutes--> v14\n- v14 --refutes--> v0d\n- v0d --refutes--> v1f\n- v1f --supports--> v0a\n- v0a --supports--> v1b\n- v1b --supports--> v00\n- v00 --supports--> v07\n- v07 --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v05 --refutes--> v08\n- v0c --refutes--> v09\n- v10 --refutes--> v22\n- v10 --supports--> v0e\n- v13 --supports--> v1d\n- v15 --supports--> v0b\n- v15 --supports--> v18\n- v17 --refutes--> v02\n- v23 --supports--> v11\n- v24 --refutes--> v01\n- v24 --supports--> v03\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":"2aa386285464a551d5c05a476515605eee719fae4653dd13c0f0466e353e4529","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v0b\n- v0b --refutes--> v1b\n- v1b --refutes--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v1f\n- v1f --refutes--> v1d\n- v1d --supports--> v13\n- v13 --refutes--> v23\n- v23 --supports--> v11\n- v11 --refutes--> v20\n- v20 --refutes--> v1a\n- v1a --refutes--> v04\n- v04 --refutes--> v18\n- v18 --supports--> v00\n- v00 --supports--> v22\n- v22 --supports--> v10\n- v10 --supports--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v03\n- v03 --refutes--> v1e\n- v1e --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v16\n- v03 --refutes--> v19\n- v05 --refutes--> v0e\n- v0a --supports--> v09\n- v0a --supports--> v0c\n- v10 --refutes--> v1c\n- v13 --refutes--> v21\n- v18 --supports--> v01\n- v1a --supports--> v17\n- v1b --supports--> v08\n- v1d --refutes--> v14\n- v1d --refutes--> v24\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":"a8ff9702a2d5c50adfd555dbeddf4f05e8a922a05cf73061da0bdc753800859f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v04\n- v04 --supports--> v10\n- v10 --supports--> v01\n- v01 --supports--> v15\n- v15 --supports--> v24\n- v24 --refutes--> v21\n- v21 --supports--> v17\n- v17 --supports--> v0e\n- v0e --supports--> v18\n- v18 --refutes--> v1e\n- v1e --supports--> v0f\n- v0f --supports--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v16\n- v16 --supports--> v1a\n- v1a --refutes--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v23\n- v23 --supports--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v20\n- v0e --supports--> v1c\n- v10 --refutes--> v19\n- v10 --supports--> v0b\n- v11 --supports--> v00\n- v11 --supports--> v0d\n- v15 --refutes--> v1d\n- v18 --refutes--> v1f\n- v1a --refutes--> v13\n- v1b --refutes--> v08\n- v21 --refutes--> v03\n- v24 --supports--> v22\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":"64c09d269fd1fbcd6916fe787820ed94ce3d661cab210ed315065c7049ba3d2e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --refutes--> v1d\n- v1d --refutes--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v11\n- v11 --refutes--> v24\n- v24 --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --refutes--> v19\n- v19 --refutes--> v14\n- v14 --refutes--> v16\n- v16 --supports--> v1c\n- v1c --refutes--> v08\n- v08 --refutes--> v21\n- v21 --refutes--> v12\n- v12 --refutes--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v20\n- v20 --refutes--> v03\n- v03 --refutes--> v15\n- v15 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v02 --supports--> v10\n- v03 --refutes--> v07\n- v08 --refutes--> v05\n- v0e --supports--> v01\n- v12 --supports--> v1f\n- v14 --refutes--> v22\n- v14 --supports--> v0c\n- v15 --refutes--> v17\n- v19 --refutes--> v1b\n- v19 --refutes--> v23\n- v1a --supports--> v04\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":"c74b298f8e92e7ef87fb9f10144d580debbf9004a9f0cd88db2ffbc186551f1e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v00\n- v00 --supports--> v09\n- v09 --supports--> v01\n- v01 --supports--> v05\n- v05 --supports--> v15\n- v15 --supports--> v0c\n- v0c --supports--> v14\n- v14 --supports--> v16\n- v16 --supports--> v10\n- v10 --refutes--> v19\n- v19 --supports--> v1c\n- v1c --refutes--> v1b\n- v1b --supports--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v03\n- v03 --supports--> v24\n- v24 --refutes--> v0d\n- v0d --refutes--> v21\n- v21 --refutes--> v17\n- v17 --supports--> v18\n- v18 --refutes--> v08\n- v08 --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v1e\n- v0a --supports--> v04\n- v0a --supports--> v07\n- v0c --supports--> v1d\n- v0e --supports--> v06\n- v0e --supports--> v1f\n- v10 --refutes--> v13\n- v10 --supports--> v0b\n- v10 --supports--> v23\n- v14 --supports--> v20\n- v16 --supports--> v11\n- v24 --refutes--> v22\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":"c20352bc526489a813b1b18c5b89c12936a9b74719f8a5329cd3809737f713ac","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v20\n- v20 --supports--> v21\n- v21 --refutes--> v1f\n- v1f --supports--> v0e\n- v0e --supports--> v0b\n- v0b --supports--> v1b\n- v1b --refutes--> v23\n- v23 --supports--> v0c\n- v0c --refutes--> v1a\n- v1a --refutes--> v18\n- v18 --supports--> v02\n- v02 --supports--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v14\n- v14 --supports--> v07\n- v07 --refutes--> v11\n- v11 --refutes--> v1c\n- v1c --refutes--> v16\n- v16 --refutes--> v1d\n- v1d --refutes--> v24\n- v24 --supports--> v05\n- v05 --refutes--> v09\n- v09 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v07 --supports--> v17\n- v0c --refutes--> v15\n- v13 --refutes--> v12\n- v16 --refutes--> v08\n- v18 --supports--> v22\n- v1a --refutes--> v10\n- v1b --supports--> v01\n- v1f --supports--> v0d\n- v20 --refutes--> v04\n- v20 --supports--> v00\n- v23 --supports--> v06\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":"31b2f73cc8d52d82c549e11e777b7058b0f6ddd733bcde9de8f5ee570d7baab9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v0d\n- v0d --supports--> v1a\n- v1a --refutes--> v03\n- v03 --refutes--> v20\n- v20 --refutes--> v01\n- v01 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v14\n- v14 --refutes--> v11\n- v11 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v19\n- v19 --supports--> v1c\n- v1c --supports--> v09\n- v09 --supports--> v1d\n- v1d --refutes--> v24\n- v24 --refutes--> v17\n- v17 --refutes--> v0e\n- v0e --supports--> v13\n- v13 --supports--> v05\n- v05 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v07\n- v08 --supports--> v22\n- v0d --refutes--> v06\n- v0d --supports--> v1f\n- v0e --refutes--> v18\n- v10 --refutes--> v0a\n- v10 --supports--> v23\n- v17 --supports--> v15\n- v19 --supports--> v16\n- v1d --refutes--> v21\n- v1d --supports--> v0f\n- v1d --supports--> v1e\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":"7be906a4e1b45f792ca8d3f7754f500c7477118eb641989a95a28f8c7d6875a7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v19\n- v19 --refutes--> v10\n- v10 --supports--> v1c\n- v1c --supports--> v16\n- v16 --refutes--> v13\n- v13 --refutes--> v03\n- v03 --supports--> v18\n- v18 --supports--> v0e\n- v0e --refutes--> v17\n- v17 --refutes--> v1e\n- v1e --refutes--> v1b\n- v1b --supports--> v07\n- v07 --refutes--> v1d\n- v1d --supports--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v14\n- v14 --refutes--> v12\n- v12 --supports--> v1a\n- v1a --supports--> v11\n- v11 --refutes--> v02\n- v02 --supports--> v1f\n- v1f --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v04\n- v02 --supports--> v23\n- v03 --refutes--> v01\n- v0d --supports--> v21\n- v0e --supports--> v0a\n- v0e --supports--> v24\n- v0f --supports--> v0c\n- v13 --refutes--> v00\n- v13 --refutes--> v20\n- v17 --supports--> v22\n- v1c --refutes--> v06\n- v1f --refutes--> v08\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":"df3351368dba30687a222a50361363ccfe5fa50eb08560dc1e1a734fec759f02","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v1e\n- v1e --supports--> v04\n- v04 --refutes--> v23\n- v23 --supports--> v21\n- v21 --supports--> v03\n- v03 --refutes--> v15\n- v15 --refutes--> v20\n- v20 --refutes--> v1f\n- v1f --supports--> v09\n- v09 --supports--> v18\n- v18 --refutes--> v1a\n- v1a --supports--> v0c\n- v0c --supports--> v0d\n- v0d --refutes--> v05\n- v05 --supports--> v11\n- v11 --supports--> v24\n- v24 --refutes--> v1c\n- v1c --supports--> v17\n- v17 --refutes--> v14\n- v14 --refutes--> v19\n- v19 --supports--> v06\n- v06 --refutes--> v1d\n- v1d --refutes--> v02\n- v02 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v13\n- v0b --supports--> v16\n- v11 --supports--> v10\n- v14 --supports--> v07\n- v15 --supports--> v0e\n- v17 --refutes--> v22\n- v1a --refutes--> v0a\n- v1a --supports--> v0f\n- v1f --refutes--> v01\n- v1f --supports--> v08\n- v1f --supports--> v1b\n- v20 --supports--> v12\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":"7efd6ac2c8a397b7f3a083d5e39919ff53e874ae070abe38367110094eef6b3a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v23\n- v23 --supports--> v13\n- v13 --supports--> v15\n- v15 --refutes--> v1d\n- v1d --refutes--> v20\n- v20 --refutes--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v21\n- v21 --refutes--> v1b\n- v1b --refutes--> v19\n- v19 --refutes--> v07\n- v07 --supports--> v1a\n- v1a --refutes--> v0d\n- v0d --refutes--> v17\n- v17 --supports--> v1e\n- v1e --supports--> v0b\n- v0b --supports--> v1c\n- v1c --supports--> v0e\n- v0e --refutes--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v22\n- v22 --refutes--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v18\n- v07 --supports--> v0c\n- v08 --refutes--> v0a\n- v0b --refutes--> v04\n- v13 --refutes--> v05\n- v13 --supports--> v02\n- v15 --supports--> v11\n- v15 --supports--> v1f\n- v1a --refutes--> v10\n- v22 --supports--> v0f\n- v22 --supports--> v12\n- v23 --supports--> v16\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":"6dae461fc721bcdf8c949ea7f5ddf4374b9540c2f6e8cffbb92663b5cf057cc6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --refutes--> v18\n- v18 --refutes--> v11\n- v11 --refutes--> v1c\n- v1c --supports--> v03\n- v03 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v14\n- v14 --refutes--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v1a\n- v1a --refutes--> v0b\n- v0b --refutes--> v19\n- v19 --refutes--> v17\n- v17 --refutes--> v02\n- v02 --supports--> v20\n- v20 --supports--> v0f\n- v0f --refutes--> v23\n- v23 --refutes--> v1f\n- v1f --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v24\n- v24 --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v01\n- v02 --refutes--> v15\n- v02 --supports--> v09\n- v0a --supports--> v07\n- v0b --refutes--> v08\n- v0f --refutes--> v0e\n- v10 --refutes--> v13\n- v1a --refutes--> v12\n- v1e --refutes--> v22\n- v1e --supports--> v21\n- v24 --refutes--> v16\n- v24 --supports--> v04\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":"9ca4ae7a38aac58738456361489a2bf50b1a49f6f7e3446277e7feba083ac7a4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v1f\n- v1f --refutes--> v16\n- v16 --refutes--> v11\n- v11 --supports--> v02\n- v02 --supports--> v24\n- v24 --supports--> v0c\n- v0c --refutes--> v17\n- v17 --refutes--> v0a\n- v0a --refutes--> v20\n- v20 --refutes--> v0f\n- v0f --refutes--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v14\n- v14 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v21\n- v21 --refutes--> v12\n- v12 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v1e\n- v1e --supports--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v0e\n- v0e --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1a\n- v05 --supports--> v1b\n- v09 --supports--> v19\n- v0b --supports--> v22\n- v0c --supports--> v13\n- v14 --refutes--> v23\n- v18 --refutes--> v08\n- v18 --supports--> v01\n- v1e --supports--> v1c\n- v1f --supports--> v03\n- v21 --supports--> v15\n- v24 --refutes--> v04\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":"25b9f8d7db0f3081ad6207a4add6f4b6d92d85e5bcffb8e6ac62896b03b4699e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v02\n- v02 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v03\n- v03 --supports--> v14\n- v14 --supports--> v12\n- v12 --refutes--> v17\n- v17 --supports--> v00\n- v00 --supports--> v1c\n- v1c --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v22\n- v22 --supports--> v06\n- v06 --refutes--> v23\n- v23 --refutes--> v11\n- v11 --supports--> v20\n- v20 --supports--> v18\n- v18 --supports--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v0a\n- v0a --refutes--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v0f\n- v0f --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0d\n- v01 --supports--> v1f\n- v06 --supports--> v08\n- v0b --supports--> v0c\n- v0b --supports--> v24\n- v0f --supports--> v15\n- v11 --refutes--> v19\n- v14 --refutes--> v1e\n- v17 --supports--> v05\n- v1c --refutes--> v21\n- v20 --supports--> v1a\n- v20 --supports--> v1d\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":"1ed8f5276722c2cce3370604264ee6e8ccecb2952a238801681e175d23e89206","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v0c\n- v0c --refutes--> v1f\n- v1f --supports--> v00\n- v00 --refutes--> v18\n- v18 --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v22\n- v22 --refutes--> v06\n- v06 --refutes--> v24\n- v24 --refutes--> v15\n- v15 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v0f\n- v0f --refutes--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v1e\n- v1e --supports--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v20\n- v20 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v21\n- v03 --supports--> v17\n- v05 --supports--> v10\n- v09 --refutes--> v1b\n- v0a --refutes--> v1a\n- v0c --refutes--> v0d\n- v0c --refutes--> v1d\n- v0c --supports--> v14\n- v0e --refutes--> v08\n- v0f --supports--> v13\n- v22 --refutes--> v16\n- v24 --refutes--> v23\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":"ee89a4fa5dcb04c462230ba84a46cd2b5d5b0545f02efbceaa1bf950cbb47ab4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --refutes--> v20\n- v20 --supports--> v06\n- v06 --refutes--> v10\n- v10 --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v21\n- v21 --supports--> v05\n- v05 --refutes--> v1f\n- v1f --refutes--> v0f\n- v0f --supports--> v13\n- v13 --supports--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --supports--> v07\n- v07 --supports--> v17\n- v17 --refutes--> v12\n- v12 --supports--> v1b\n- v1b --supports--> v0c\n- v0c --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v14\n- v02 --refutes--> v1c\n- v03 --refutes--> v18\n- v05 --supports--> v11\n- v06 --supports--> v04\n- v0c --refutes--> v1e\n- v15 --refutes--> v1d\n- v15 --supports--> v0d\n- v1b --supports--> v0b\n- v1f --supports--> v08\n- v24 --refutes--> v23\n- v24 --supports--> v22\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":"cb72e3bbcefa2bad9d661e3b235169ef19291651a73a0d4fb270260a80a7daa0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --refutes--> v01\n- v01 --supports--> v12\n- v12 --supports--> v1b\n- v1b --refutes--> v02\n- v02 --supports--> v06\n- v06 --supports--> v21\n- v21 --supports--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v14\n- v14 --refutes--> v20\n- v20 --refutes--> v19\n- v19 --supports--> v0d\n- v0d --supports--> v18\n- v18 --refutes--> v1f\n- v1f --supports--> v07\n- v07 --supports--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v17\n- v17 --supports--> v1a\n- v1a --refutes--> v00\n- v00 --refutes--> v1c\n- v1c --refutes--> v24\n- v24 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v01 --supports--> v1d\n- v02 --refutes--> v04\n- v02 --refutes--> v22\n- v05 --refutes--> v13\n- v0d --refutes--> v03\n- v0f --supports--> v23\n- v10 --refutes--> v16\n- v19 --supports--> v0b\n- v1b --refutes--> v0c\n- v1f --supports--> v11\n- v21 --supports--> v15\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":"adee19a22a2b22aee6e601e5cef3600796b0a32349f2b4855d972c5e72ab894d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --supports--> v0d\n- v0d --supports--> v0a\n- v0a --refutes--> v10\n- v10 --supports--> v12\n- v12 --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v24\n- v24 --refutes--> v09\n- v09 --supports--> v1c\n- v1c --supports--> v0f\n- v0f --supports--> v21\n- v21 --refutes--> v04\n- v04 --refutes--> v1a\n- v1a --supports--> v0e\n- v0e --refutes--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v1b\n- v1b --supports--> v1d\n- v1d --refutes--> v23\n- v23 --supports--> v1e\n- v1e --refutes--> v1f\n- v1f --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\n- v05 --supports--> v16\n- v06 --supports--> v18\n- v07 --refutes--> v17\n- v0a --refutes--> v20\n- v0b --supports--> v02\n- v12 --refutes--> v15\n- v1c --supports--> v0c\n- v1e --refutes--> v11\n- v21 --refutes--> v01\n- v23 --supports--> v19\n- v24 --supports--> v00\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":"0ce50449e5368e91935cf445a0287daa241691f703af2f2876c31a8c145148a2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --refutes--> v11\n- v11 --refutes--> v24\n- v24 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v1f\n- v1f --supports--> v1c\n- v1c --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v21\n- v21 --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v17\n- v17 --supports--> v16\n- v16 --supports--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v00\n- v00 --refutes--> v1a\n- v1a --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v1d\n- v06 --refutes--> v18\n- v08 --supports--> v23\n- v0c --supports--> v19\n- v0f --refutes--> v14\n- v10 --supports--> v04\n- v11 --refutes--> v0d\n- v12 --refutes--> v15\n- v17 --refutes--> v03\n- v17 --supports--> v22\n- v21 --refutes--> v1b\n- v24 --refutes--> v1e\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":"12aafa7e0ecf9591197ed11f68950e0a64208b0f54919bdeaa603e3a3afb50c9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v1a\n- v1a --supports--> v10\n- v10 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --refutes--> v03\n- v03 --supports--> v1f\n- v1f --supports--> v1b\n- v1b --supports--> v23\n- v23 --refutes--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v13\n- v13 --supports--> v20\n- v20 --supports--> v04\n- v04 --supports--> v21\n- v21 --supports--> v06\n- v06 --supports--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v19\n- v19 --refutes--> v16\n- v16 --refutes--> v11\n- v11 --refutes--> v0f\n- v0f --refutes--> v0d\n- v0d --refutes--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v15\n- v0a --refutes--> v17\n- v0a --supports--> v18\n- v0c --refutes--> v0e\n- v11 --refutes--> v07\n- v11 --refutes--> v22\n- v12 --refutes--> v1c\n- v13 --supports--> v1d\n- v16 --refutes--> v05\n- v1a --refutes--> v08\n- v20 --refutes--> v09\n- v23 --supports--> v00\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":"a0315d310cf424afa8bf54de3f7330c1a787a31737e713e8241eca494601bab2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --refutes--> v18\n- v18 --refutes--> v21\n- v21 --supports--> v05\n- v05 --refutes--> v1c\n- v1c --supports--> v1e\n- v1e --supports--> v06\n- v06 --supports--> v23\n- v23 --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --refutes--> v12\n- v12 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --refutes--> v14\n- v14 --refutes--> v20\n- v20 --refutes--> v19\n- v19 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --supports--> v16\n- v16 --refutes--> v1f\n- v1f --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v02\n- v09 --refutes--> v0c\n- v0a --refutes--> v13\n- v0b --supports--> v1a\n- v0f --supports--> v04\n- v16 --refutes--> v24\n- v18 --supports--> v17\n- v19 --supports--> v0d\n- v19 --supports--> v1b\n- v1d --refutes--> v10\n- v20 --supports--> v07\n- v21 --supports--> v11\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":"b881ec8f5165c6b74e36ae9da9f96891c0a84e32c11ddf395f46ca263ded6a20","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v14\n- v14 --supports--> v21\n- v21 --refutes--> v0e\n- v0e --refutes--> v20\n- v20 --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v0d\n- v0d --refutes--> v1a\n- v1a --supports--> v1d\n- v1d --refutes--> v17\n- v17 --supports--> v24\n- v24 --supports--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v1b\n- v1b --supports--> v07\n- v07 --refutes--> v23\n- v23 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v18\n- v18 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1e\n- v04 --refutes--> v13\n- v04 --supports--> v0c\n- v06 --supports--> v16\n- v0d --refutes--> v15\n- v14 --refutes--> v0a\n- v17 --supports--> v0f\n- v18 --supports--> v1f\n- v1b --refutes--> v10\n- v1d --supports--> v22\n- v20 --refutes--> v19\n- v23 --refutes--> v05\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":"78cf334f354a34c0dc43eb9248538226dfcdccd763c6fc2cad43044f8a507828","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v02\n- v02 --refutes--> v21\n- v21 --supports--> v0e\n- v0e --supports--> v15\n- v15 --refutes--> v1e\n- v1e --supports--> v17\n- v17 --supports--> v09\n- v09 --supports--> v1a\n- v1a --refutes--> v12\n- v12 --supports--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v20\n- v20 --refutes--> v18\n- v18 --supports--> v24\n- v24 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v1f\n- v1f --supports--> v06\n- v06 --refutes--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v0b\n- v0b --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0f\n- v09 --supports--> v00\n- v0a --supports--> v05\n- v0b --refutes--> v01\n- v0c --refutes--> v1d\n- v0c --refutes--> v23\n- v17 --supports--> v08\n- v18 --refutes--> v1c\n- v1e --supports--> v22\n- v1f --supports--> v1b\n- v20 --refutes--> v16\n- v20 --supports--> v0d\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":"9c6899cb5701f6e31600682ec7ac7cb5ecd363796d194b75b4604698fa7c3f4b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v23\n- v23 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v0c\n- v0c --supports--> v17\n- v17 --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --supports--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v24\n- v24 --refutes--> v1f\n- v1f --refutes--> v1c\n- v1c --refutes--> v11\n- v11 --supports--> v06\n- v06 --refutes--> v19\n- v19 --supports--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v1a\n- v1a --supports--> v04\n- v04 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v20\n- v02 --refutes--> v1e\n- v0a --supports--> v09\n- v0c --refutes--> v21\n- v0e --refutes--> v1b\n- v0e --refutes--> v22\n- v10 --refutes--> v18\n- v13 --refutes--> v0b\n- v13 --supports--> v01\n- v14 --refutes--> v1d\n- v1a --refutes--> v05\n- v1f --refutes--> v15\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":"9cce0c3cdd01ff7284aabc733c5bf5750e4c9db48b8c165d3dbd5edc343b7c85","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --supports--> v13\n- v13 --refutes--> v0c\n- v0c --supports--> v24\n- v24 --refutes--> v21\n- v21 --supports--> v04\n- v04 --supports--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v1e\n- v1e --refutes--> v01\n- v01 --refutes--> v17\n- v17 --supports--> v0d\n- v0d --refutes--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v1a\n- v1a --refutes--> v06\n- v06 --supports--> v1c\n- v1c --refutes--> v0a\n- v0a --supports--> v08\n- v08 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v09\n- v07 --supports--> v1b\n- v0c --refutes--> v19\n- v0d --supports--> v23\n- v12 --supports--> v05\n- v13 --supports--> v1f\n- v17 --refutes--> v14\n- v18 --refutes--> v22\n- v18 --supports--> v20\n- v1c --refutes--> v0b\n- v1d --supports--> v16\n- v24 --refutes--> v0f\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":"4d09b1a657a0810b37b6afe6a5b755a6e2f0f42b8745dc2548d3e07cb0f04eab","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v15\n- v15 --refutes--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v20\n- v20 --supports--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v11\n- v11 --refutes--> v24\n- v24 --supports--> v1d\n- v1d --supports--> v22\n- v22 --refutes--> v0d\n- v0d --refutes--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v18\n- v18 --refutes--> v09\n- v09 --supports--> v1e\n- v1e --refutes--> v1b\n- v1b --refutes--> v02\n- v02 --refutes--> v17\n- v17 --refutes--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v02 --supports--> v08\n- v06 --supports--> v0e\n- v07 --supports--> v23\n- v0a --supports--> v21\n- v10 --refutes--> v1f\n- v13 --refutes--> v0c\n- v17 --refutes--> v0f\n- v1a --supports--> v14\n- v1b --refutes--> v04\n- v20 --refutes--> v03\n- v22 --supports--> v19\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":"e219f17953f7a3fb1335017843f64c646b48fdd43821fb2b48d5d45f27bf9635","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0d\n- v0d --supports--> v19\n- v19 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v23\n- v23 --refutes--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v1d\n- v1d --supports--> v11\n- v11 --refutes--> v00\n- v00 --supports--> v13\n- v13 --refutes--> v17\n- v17 --supports--> v0c\n- v0c --supports--> v1b\n- v1b --refutes--> v1a\n- v1a --supports--> v12\n- v12 --refutes--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v1f\n- v1f --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v0b\n- v08 --supports--> v0f\n- v0c --supports--> v21\n- v0d --supports--> v0a\n- v11 --refutes--> v04\n- v12 --refutes--> v1e\n- v12 --supports--> v10\n- v13 --refutes--> v18\n- v17 --supports--> v22\n- v17 --supports--> v24\n- v1a --refutes--> v05\n- v1d --refutes--> v20\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":"1702cb115c036e280c02b1456e997ea31c08ee9425bee05929121638b83ef495","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v1e\n- v1e --supports--> v11\n- v11 --supports--> v1f\n- v1f --supports--> v15\n- v15 --supports--> v17\n- v17 --supports--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v07\n- v07 --refutes--> v1d\n- v1d --refutes--> v0f\n- v0f --refutes--> v21\n- v21 --supports--> v08\n- v08 --supports--> v10\n- v10 --refutes--> v20\n- v20 --refutes--> v0b\n- v0b --supports--> v0e\n- v0e --refutes--> v14\n- v14 --refutes--> v19\n- v19 --refutes--> v0c\n- v0c --refutes--> v18\n- v18 --refutes--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v01\n- v01 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v03\n- v05 --supports--> v24\n- v06 --supports--> v16\n- v0c --refutes--> v23\n- v0f --supports--> v22\n- v12 --refutes--> v02\n- v12 --refutes--> v0d\n- v12 --supports--> v1b\n- v14 --supports--> v04\n- v14 --supports--> v1a\n- v18 --refutes--> v1c\n- v1f --supports--> v0a\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":"85e5d0085aa901e6b702a3099a3a386038cd2b8ecc3cd6102c440d5d6a7aee99","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v04\n- v04 --supports--> v13\n- v13 --refutes--> v22\n- v22 --refutes--> v17\n- v17 --refutes--> v0b\n- v0b --supports--> v10\n- v10 --refutes--> v19\n- v19 --refutes--> v1d\n- v1d --refutes--> v0e\n- v0e --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v0f\n- v0f --refutes--> v21\n- v21 --supports--> v08\n- v08 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --refutes--> v1c\n- v1c --supports--> v07\n- v07 --refutes--> v23\n- v23 --refutes--> v06\n- v06 --supports--> v00\n- v00 --supports--> v1a\n- v1a --supports--> v20\n- v20 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v24\n- v06 --supports--> v1e\n- v07 --refutes--> v05\n- v09 --supports--> v03\n- v0b --refutes--> v11\n- v0b --refutes--> v16\n- v0b --refutes--> v18\n- v0b --refutes--> v1f\n- v17 --refutes--> v0d\n- v17 --supports--> v1b\n- v1d --supports--> v14\n- v20 --supports--> v12\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":"a034d2481e926effc01e7cb4bc830cad8dedd1e2fac405a9d4c40a50238addec","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v07\n- v07 --supports--> v19\n- v19 --supports--> v21\n- v21 --refutes--> v1f\n- v1f --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v1b\n- v1b --supports--> v14\n- v14 --supports--> v11\n- v11 --supports--> v16\n- v16 --refutes--> v23\n- v23 --refutes--> v09\n- v09 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v0a\n- v0e --refutes--> v02\n- v0e --refutes--> v15\n- v0f --supports--> v03\n- v10 --supports--> v06\n- v14 --refutes--> v20\n- v14 --refutes--> v22\n- v14 --supports--> v18\n- v19 --refutes--> v17\n- v1b --supports--> v1e\n- v1f --refutes--> v24\n- v1f --supports--> v08\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":"f0825194a043b0ac1645181a34cae776790bf596f1f521611d177104a3c88f5c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --refutes--> v12\n- v12 --supports--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v1b\n- v1b --refutes--> v17\n- v17 --refutes--> v16\n- v16 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v13\n- v13 --supports--> v23\n- v23 --supports--> v1f\n- v1f --supports--> v0c\n- v0c --supports--> v0f\n- v0f --refutes--> v21\n- v21 --refutes--> v14\n- v14 --refutes--> v06\n- v06 --supports--> v20\n- v20 --supports--> v10\n- v10 --supports--> v1e\n- v1e --refutes--> v15\n- v15 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v11\n- v06 --refutes--> v01\n- v0c --supports--> v1c\n- v12 --supports--> v03\n- v13 --refutes--> v05\n- v13 --refutes--> v18\n- v20 --refutes--> v1a\n- v20 --supports--> v19\n- v23 --refutes--> v0e\n- v23 --refutes--> v24\n- v23 --supports--> v08\n- v23 --supports--> v1d\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":"49b2d7fd99f966f4e278885eed1b60d62852642b26daf60ca7c986ae3e8aaf0b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v1c\n- v1c --refutes--> v09\n- v09 --supports--> v22\n- v22 --refutes--> v1e\n- v1e --refutes--> v00\n- v00 --refutes--> v17\n- v17 --supports--> v01\n- v01 --supports--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v21\n- v21 --supports--> v0f\n- v0f --refutes--> v24\n- v24 --supports--> v0a\n- v0a --refutes--> v15\n- v15 --supports--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --refutes--> v12\n- v12 --refutes--> v16\n- v16 --refutes--> v23\n- v23 --supports--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v13\n- v08 --supports--> v1a\n- v08 --supports--> v20\n- v10 --refutes--> v05\n- v14 --refutes--> v1f\n- v14 --supports--> v19\n- v16 --supports--> v0e\n- v17 --refutes--> v03\n- v1c --refutes--> v1d\n- v21 --refutes--> v0c\n- v22 --supports--> v11\n- v23 --supports--> v1b\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":"0257c1fdce9b06e7ef145ca73dde1da4f959b14016cb6875ed4f280fec08791c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --supports--> v00\n- v00 --refutes--> v1e\n- v1e --supports--> v24\n- v24 --refutes--> v16\n- v16 --refutes--> v21\n- v21 --supports--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v1d\n- v1d --supports--> v06\n- v06 --refutes--> v20\n- v20 --supports--> v05\n- v05 --supports--> v13\n- v13 --supports--> v11\n- v11 --refutes--> v0d\n- v0d --supports--> v10\n- v10 --supports--> v22\n- v22 --supports--> v14\n- v14 --refutes--> v01\n- v01 --refutes--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v23\n- v23 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v1c\n- v0a --refutes--> v02\n- v0b --refutes--> v1f\n- v10 --supports--> v15\n- v11 --supports--> v09\n- v14 --refutes--> v07\n- v14 --supports--> v1b\n- v16 --refutes--> v08\n- v18 --supports--> v03\n- v1d --supports--> v17\n- v1e --supports--> v0c\n- v23 --supports--> v1a\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":"19e340dfe70e108f1b7e82dd20901f4ddd19f113c3da7a5cbbbcff211a16407e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v08\n- v08 --supports--> v15\n- v15 --refutes--> v1f\n- v1f --refutes--> v04\n- v04 --refutes--> v02\n- v02 --refutes--> v24\n- v24 --supports--> v1a\n- v1a --refutes--> v09\n- v09 --refutes--> v1c\n- v1c --supports--> v21\n- v21 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v23\n- v23 --refutes--> v12\n- v12 --refutes--> v22\n- v22 --supports--> v14\n- v14 --refutes--> v16\n- v16 --refutes--> v00\n- v00 --refutes--> v13\n- v13 --refutes--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v18\n- v18 --refutes--> v0d\n- v0d --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v03\n- v09 --refutes--> v17\n- v15 --refutes--> v19\n- v18 --refutes--> v05\n- v18 --supports--> v1e\n- v1a --refutes--> v07\n- v1a --refutes--> v11\n- v1a --refutes--> v20\n- v1f --refutes--> v10\n- v21 --refutes--> v1d\n- v24 --supports--> v0a\n- v24 --supports--> v1b\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":"0f1b5e2dc8e5acd5c8bbc9f99d610a25f51be54b50dd89bc7cf92835c8583ca0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v07\n- v07 --refutes--> v22\n- v22 --supports--> v10\n- v10 --supports--> v0a\n- v0a --refutes--> v08\n- v08 --supports--> v12\n- v12 --refutes--> v19\n- v19 --supports--> v0e\n- v0e --supports--> v16\n- v16 --supports--> v1d\n- v1d --refutes--> v11\n- v11 --refutes--> v21\n- v21 --refutes--> v13\n- v13 --refutes--> v1e\n- v1e --refutes--> v20\n- v20 --supports--> v1f\n- v1f --supports--> v01\n- v01 --refutes--> v18\n- v18 --supports--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v15\n- v15 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --refutes--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v08 --supports--> v17\n- v0a --refutes--> v04\n- v0e --supports--> v0c\n- v11 --refutes--> v24\n- v12 --refutes--> v03\n- v13 --refutes--> v1a\n- v14 --refutes--> v06\n- v14 --refutes--> v0d\n- v15 --refutes--> v0f\n- v18 --supports--> v09\n- v1b --supports--> v1c\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":"ac6fe14ddf0e2b5ff04953a9280ac8cbc7c07fde32de63118fcdf54935025ccb","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v1d\n- v1d --supports--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v17\n- v17 --refutes--> v1a\n- v1a --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v1f\n- v1f --refutes--> v1b\n- v1b --refutes--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v20\n- v20 --refutes--> v09\n- v09 --supports--> v13\n- v13 --refutes--> v04\n- v04 --refutes--> v16\n- v16 --supports--> v14\n- v14 --supports--> v1c\n- v1c --supports--> v12\n- v12 --refutes--> v00\n- v00 --supports--> v0a\n- v0a --refutes--> v18\n- v18 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v23\n- v0b --supports--> v01\n- v0b --supports--> v19\n- v12 --refutes--> v0e\n- v13 --refutes--> v22\n- v13 --supports--> v05\n- v14 --refutes--> v07\n- v17 --supports--> v0d\n- v1b --supports--> v11\n- v1d --supports--> v21\n- v20 --supports--> v1e\n- v20 --supports--> v24\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":"f6f99e950b1507f4e876d794fd2141cd8b4ffd874eb39f6d6db30f4e0933fe90","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --refutes--> v02\n- v02 --supports--> v1e\n- v1e --refutes--> v19\n- v19 --supports--> v23\n- v23 --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v21\n- v21 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v18\n- v18 --supports--> v20\n- v20 --supports--> v14\n- v14 --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v13\n- v13 --refutes--> v1f\n- v1f --supports--> v08\n- v08 --refutes--> v09\n- v09 --supports--> v24\n- v24 --supports--> v0f\n- v0f --refutes--> v1c\n- v1c --supports--> v0e\n- v0e --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v1d\n- v04 --supports--> v0d\n- v09 --supports--> v03\n- v0e --supports--> v10\n- v11 --supports--> v07\n- v14 --refutes--> v06\n- v14 --supports--> v01\n- v14 --supports--> v15\n- v19 --refutes--> v0a\n- v19 --refutes--> v17\n- v1e --supports--> v00\n- v22 --supports--> v1b\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":"fbbd72c0ca7590da9745fa466ab967b2aad0febdbc686f3f36e8d1f013c0c391","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v18\n- v18 --refutes--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v1c\n- v1c --refutes--> v01\n- v01 --supports--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --supports--> v14\n- v14 --supports--> v19\n- v19 --supports--> v0b\n- v0b --supports--> v09\n- v09 --supports--> v1f\n- v1f --refutes--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v10\n- v10 --supports--> v11\n- v11 --refutes--> v08\n- v08 --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v17\n- v04 --refutes--> v02\n- v04 --supports--> v13\n- v07 --refutes--> v24\n- v07 --supports--> v0e\n- v0d --refutes--> v23\n- v0f --supports--> v22\n- v10 --refutes--> v0a\n- v12 --refutes--> v16\n- v12 --supports--> v21\n- v15 --refutes--> v1d\n- v1b --supports--> v06\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":"4c5996b372374d23bf3aa3cd094db6159d4ec1f0ba90c7681b9aebac837359f8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --refutes--> v13\n- v13 --refutes--> v1e\n- v1e --supports--> v04\n- v04 --supports--> v1d\n- v1d --refutes--> v10\n- v10 --supports--> v20\n- v20 --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v0a\n- v0a --supports--> v05\n- v05 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v23\n- v23 --refutes--> v06\n- v06 --supports--> v16\n- v16 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v0c\n- v0c --refutes--> v14\n- v14 --refutes--> v21\n- v21 --supports--> v24\n- v24 --supports--> v01\n- v01 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v12\n- v02 --supports--> v18\n- v10 --refutes--> v03\n- v14 --supports--> v08\n- v17 --supports--> v00\n- v1a --refutes--> v0f\n- v1c --refutes--> v11\n- v1d --supports--> v1b\n- v1f --refutes--> v0d\n- v1f --refutes--> v22\n- v1f --supports--> v19\n- v23 --supports--> v0e\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":"421848c3a5ea003a8b4c26b1b2030a0eeb02737740cfe16a6ecb3b92a4022c30","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --supports--> v22\n- v22 --supports--> v0e\n- v0e --supports--> v21\n- v21 --refutes--> v13\n- v13 --refutes--> v00\n- v00 --refutes--> v14\n- v14 --refutes--> v15\n- v15 --supports--> v04\n- v04 --supports--> v12\n- v12 --supports--> v24\n- v24 --refutes--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v02\n- v02 --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --refutes--> v1c\n- v1c --supports--> v20\n- v20 --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --supports--> v18\n- v18 --supports--> v23\n- v23 --supports--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v16\n- v07 --supports--> v01\n- v10 --refutes--> v09\n- v10 --refutes--> v11\n- v12 --refutes--> v03\n- v12 --refutes--> v05\n- v13 --supports--> v08\n- v18 --supports--> v1d\n- v18 --supports--> v1f\n- v1c --supports--> v17\n- v21 --supports--> v1e\n- v22 --refutes--> v0f\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":"4de5b84139912dc34d561b650ce47f2d39080ba3f5a35d86b8ba727f1bc5e823","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --supports--> v19\n- v19 --refutes--> v1b\n- v1b --supports--> v24\n- v24 --supports--> v16\n- v16 --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v11\n- v11 --supports--> v07\n- v07 --supports--> v0d\n- v0d --refutes--> v18\n- v18 --refutes--> v22\n- v22 --refutes--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v1e\n- v1e --supports--> v08\n- v08 --refutes--> v21\n- v21 --supports--> v1a\n- v1a --supports--> v0e\n- v0e --refutes--> v0a\n- v0a --refutes--> v1c\n- v1c --refutes--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v00 --refutes--> v10\n- v09 --supports--> v14\n- v0a --supports--> v1d\n- v11 --supports--> v06\n- v11 --supports--> v20\n- v19 --refutes--> v02\n- v19 --refutes--> v04\n- v22 --refutes--> v0b\n- v24 --refutes--> v03\n- v24 --refutes--> v12\n- v24 --supports--> v23\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":"8a26f33bb9c6a9de5f5fcfbb40828e06e60207c1a6b187eb94e97c4a4b774d2b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v08\n- v08 --supports--> v21\n- v21 --supports--> v13\n- v13 --refutes--> v22\n- v22 --supports--> v0a\n- v0a --refutes--> v1f\n- v1f --supports--> v24\n- v24 --supports--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v15\n- v15 --refutes--> v1b\n- v1b --refutes--> v1c\n- v1c --supports--> v16\n- v16 --refutes--> v1d\n- v1d --refutes--> v23\n- v23 --refutes--> v04\n- v04 --supports--> v0f\n- v0f --supports--> v0e\n- v0e --supports--> v17\n- v17 --supports--> v1a\n- v1a --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v12\n- v04 --supports--> v1e\n- v0e --refutes--> v10\n- v0f --refutes--> v0c\n- v0f --supports--> v02\n- v15 --refutes--> v06\n- v15 --supports--> v20\n- v16 --supports--> v00\n- v1a --refutes--> v18\n- v1c --supports--> v01\n- v21 --refutes--> v07\n- v23 --supports--> v19\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":"d83be3b78a4206c39872038b1be4735f1b74c4e24a775ff2f6f355aca8f474d9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v19\n- v19 --supports--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v16\n- v16 --refutes--> v1e\n- v1e --supports--> v0b\n- v0b --supports--> v1f\n- v1f --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v22\n- v22 --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v23\n- v23 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v1b\n- v01 --supports--> v07\n- v09 --supports--> v24\n- v0c --supports--> v0a\n- v0e --refutes--> v1d\n- v0e --refutes--> v21\n- v10 --refutes--> v06\n- v10 --supports--> v20\n- v11 --supports--> v13\n- v15 --supports--> v04\n- v19 --supports--> v14\n- v22 --supports--> v1c\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":"5a00bea948283b582aa1ee01d08557940ce88e09733e97734dccc4928d1f836e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v20\n- v20 --supports--> v04\n- v04 --refutes--> v16\n- v16 --supports--> v0e\n- v0e --supports--> v1e\n- v1e --supports--> v22\n- v22 --refutes--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v0d\n- v0d --supports--> v19\n- v19 --refutes--> v09\n- v09 --refutes--> v12\n- v12 --refutes--> v18\n- v18 --refutes--> v05\n- v05 --refutes--> v24\n- v24 --refutes--> v06\n- v06 --refutes--> v1a\n- v1a --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v13\n- v13 --supports--> v23\n- v23 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v1d\n- v0d --refutes--> v1b\n- v0d --supports--> v07\n- v12 --supports--> v1f\n- v16 --refutes--> v02\n- v17 --refutes--> v21\n- v22 --refutes--> v10\n- v22 --supports--> v0a\n- v22 --supports--> v0b\n- v22 --supports--> v1c\n- v24 --refutes--> v03\n- v24 --refutes--> v11\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":"0c7b5611958b1b4fb3a8b2e085b9578922887cb4b2d17af804fcb2d95f49213e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v16\n- v16 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v17\n- v17 --refutes--> v1a\n- v1a --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --supports--> v04\n- v04 --supports--> v09\n- v09 --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v1c\n- v1c --supports--> v21\n- v21 --refutes--> v18\n- v18 --supports--> v24\n- v24 --refutes--> v13\n- v13 --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v22\n- v07 --refutes--> v06\n- v0a --refutes--> v1d\n- v0a --supports--> v1f\n- v0c --supports--> v1e\n- v0d --refutes--> v0f\n- v0e --supports--> v19\n- v13 --supports--> v20\n- v15 --supports--> v1b\n- v16 --refutes--> v0b\n- v18 --refutes--> v10\n- v1a --supports--> v23\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":"4814cf5f7fb54ee373dfd0d1350f0ab75647bd5231064b7153f4b0b2f8398f7e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v10\n- v10 --supports--> v19\n- v19 --supports--> v18\n- v18 --refutes--> v0f\n- v0f --supports--> v02\n- v02 --supports--> v01\n- v01 --supports--> v15\n- v15 --supports--> v0c\n- v0c --refutes--> v12\n- v12 --refutes--> v1a\n- v1a --supports--> v09\n- v09 --supports--> v1e\n- v1e --refutes--> v1c\n- v1c --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v21\n- v21 --supports--> v20\n- v20 --supports--> v0d\n- v0d --refutes--> v14\n- v14 --refutes--> v23\n- v23 --refutes--> v1d\n- v1d --supports--> v00\n- v00 --refutes--> v16\n- v16 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v02 --supports--> v11\n- v05 --refutes--> v1b\n- v09 --refutes--> v0b\n- v0a --supports--> v17\n- v0c --supports--> v06\n- v0d --supports--> v24\n- v0e --supports--> v08\n- v16 --supports--> v04\n- v18 --refutes--> v1f\n- v1d --refutes--> v07\n- v21 --supports--> v22\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":"92c463ee90e9bbd55409d25223abf245468a03ad7e0befcffa9724459ee5ebb6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --supports--> v1e\n- v1e --supports--> v21\n- v21 --refutes--> v13\n- v13 --supports--> v1f\n- v1f --supports--> v10\n- v10 --supports--> v06\n- v06 --refutes--> v1c\n- v1c --refutes--> v23\n- v23 --refutes--> v0a\n- v0a --refutes--> v17\n- v17 --refutes--> v15\n- v15 --refutes--> v18\n- v18 --refutes--> v0c\n- v0c --refutes--> v19\n- v19 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v12\n- v12 --supports--> v22\n- v22 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v24\n- v24 --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v14\n- v0a --supports--> v05\n- v0b --supports--> v09\n- v0d --supports--> v07\n- v10 --refutes--> v00\n- v10 --refutes--> v11\n- v12 --supports--> v0f\n- v17 --supports--> v02\n- v1c --supports--> v04\n- v1c --supports--> v20\n- v1d --supports--> v08\n- v1f --refutes--> v03\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":"2f252bf3c990fee6d1d4cee7bc427cbde1f0ad1f1fe11d2a8b044b6ab3682ccc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v19\n- v19 --refutes--> v20\n- v20 --supports--> v10\n- v10 --supports--> v11\n- v11 --refutes--> v1e\n- v1e --refutes--> v23\n- v23 --supports--> v00\n- v00 --refutes--> v1d\n- v1d --supports--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v1c\n- v1c --refutes--> v09\n- v09 --supports--> v1a\n- v1a --refutes--> v1f\n- v1f --refutes--> v18\n- v18 --supports--> v15\n- v15 --refutes--> v0a\n- v0a --refutes--> v05\n- v05 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v02\n- v05 --refutes--> v03\n- v07 --refutes--> v12\n- v0f --refutes--> v06\n- v10 --supports--> v1b\n- v11 --refutes--> v22\n- v14 --refutes--> v24\n- v18 --supports--> v16\n- v1c --supports--> v01\n- v1d --refutes--> v21\n- v20 --refutes--> v17\n- v20 --supports--> v13\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":"b66ae5ad55738c71bac4f7357243b8ac2e3d286e57d97859be08189d23e368b4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v23\n- v23 --refutes--> v0b\n- v0b --refutes--> v19\n- v19 --supports--> v09\n- v09 --supports--> v1d\n- v1d --supports--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v22\n- v22 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v21\n- v21 --supports--> v01\n- v01 --supports--> v15\n- v15 --refutes--> v1a\n- v1a --supports--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v18\n- v18 --supports--> v17\n- v17 --refutes--> v1b\n- v1b --supports--> v13\n- v13 --supports--> v0a\n- v0a --refutes--> v0f\n- v0f --refutes--> v1e\n- v1e --refutes--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v03 --supports--> v0e\n- v0a --refutes--> v02\n- v0a --supports--> v00\n- v0f --refutes--> v08\n- v10 --supports--> v16\n- v12 --supports--> v1f\n- v12 --supports--> v20\n- v15 --refutes--> v05\n- v1a --supports--> v11\n- v1a --supports--> v14\n- v1d --refutes--> v24\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":"178e824efcf73e6821e3f49aed5cf39832d9692c68d92e7d956f587bc1749e43","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v23\n- v23 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v0c\n- v0c --supports--> v0f\n- v0f --refutes--> v24\n- v24 --refutes--> v1e\n- v1e --supports--> v16\n- v16 --refutes--> v10\n- v10 --supports--> v04\n- v04 --refutes--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v15\n- v15 --supports--> v1f\n- v1f --supports--> v1b\n- v1b --supports--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v1c\n- v1c --supports--> v1a\n- v1a --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --refutes--> v08\n- v0b --supports--> v20\n- v0f --refutes--> v0a\n- v10 --refutes--> v1d\n- v12 --refutes--> v13\n- v14 --supports--> v11\n- v15 --refutes--> v18\n- v15 --refutes--> v21\n- v1b --supports--> v17\n- v1f --supports--> v05\n- v24 --refutes--> v22\n- v24 --supports--> v02\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":"baa14bfd1c8a863c5774927be89a2b5d3d82e9340b537a4cf59e32574f2f6a6c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v1e\n- v1e --supports--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v00\n- v00 --supports--> v23\n- v23 --refutes--> v13\n- v13 --supports--> v1b\n- v1b --refutes--> v05\n- v05 --supports--> v22\n- v22 --refutes--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v04\n- v04 --refutes--> v24\n- v24 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0b\n- v05 --refutes--> v14\n- v05 --supports--> v12\n- v0e --supports--> v21\n- v0f --refutes--> v19\n- v16 --refutes--> v02\n- v16 --refutes--> v20\n- v1b --supports--> v11\n- v1c --refutes--> v10\n- v1e --supports--> v08\n- v22 --supports--> v1f\n- v23 --supports--> v01\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":"5fc63ef35108a5815a8a0d926fca7b52444273fa5590480913b3b19913cea70a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v1b\n- v1b --supports--> v16\n- v16 --refutes--> v01\n- v01 --refutes--> v1d\n- v1d --supports--> v24\n- v24 --refutes--> v03\n- v03 --supports--> v09\n- v09 --refutes--> v0d\n- v0d --supports--> v1e\n- v1e --refutes--> v1f\n- v1f --refutes--> v00\n- v00 --supports--> v14\n- v14 --refutes--> v20\n- v20 --supports--> v15\n- v15 --supports--> v22\n- v22 --supports--> v06\n- v06 --supports--> v0f\n- v0f --refutes--> v21\n- v21 --supports--> v17\n- v17 --refutes--> v13\n- v13 --refutes--> v12\n- v12 --supports--> v18\n- v18 --supports--> v0b\n- v0b --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v01 --supports--> v19\n- v09 --refutes--> v08\n- v0d --refutes--> v23\n- v0f --supports--> v07\n- v0f --supports--> v0c\n- v0f --supports--> v10\n- v13 --supports--> v1a\n- v18 --refutes--> v04\n- v1f --supports--> v02\n- v21 --supports--> v0a\n- v22 --supports--> v11\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":"4eddfe93ccdb5153b1d3611a40c8b26e1029a7960d242c8a85dbcd1ea6f4aedf","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v16\n- v16 --supports--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v19\n- v19 --supports--> v12\n- v12 --refutes--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v17\n- v17 --supports--> v23\n- v23 --refutes--> v24\n- v24 --supports--> v14\n- v14 --refutes--> v18\n- v18 --refutes--> v0b\n- v0b --refutes--> v1c\n- v1c --supports--> v20\n- v20 --refutes--> v1f\n- v1f --refutes--> v22\n- v22 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v1e\n- v1e --supports--> v0d\n- v0d --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v10\n- v06 --supports--> v04\n- v0b --supports--> v13\n- v0d --supports--> v03\n- v12 --refutes--> v02\n- v14 --supports--> v21\n- v17 --refutes--> v1b\n- v17 --supports--> v09\n- v19 --supports--> v08\n- v1e --supports--> v0f\n- v20 --refutes--> v15\n- v22 --refutes--> v0a\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":"11676397d875e105a3a01a7beb85eb0fc4fe59505c7871c46edb667ef5001fdc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v20\n- v20 --supports--> v02\n- v02 --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --supports--> v21\n- v21 --supports--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v12\n- v12 --refutes--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v17\n- v17 --refutes--> v1a\n- v1a --refutes--> v09\n- v09 --refutes--> v15\n- v15 --supports--> v1c\n- v1c --refutes--> v1f\n- v1f --supports--> v23\n- v23 --refutes--> v1e\n- v1e --refutes--> v01\n- v01 --refutes--> v08\n- v08 --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v06 --refutes--> v05\n- v06 --supports--> v0b\n- v0f --refutes--> v07\n- v12 --supports--> v11\n- v13 --supports--> v24\n- v17 --refutes--> v10\n- v17 --supports--> v18\n- v17 --supports--> v19\n- v1c --supports--> v22\n- v1d --refutes--> v04\n- v1e --supports--> v0d\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":"5d694fdfb446e8de4ec43c7715e4f40eb9b25f286e8a90cb542a888a337a6a7f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v0c\n- v0c --supports--> v16\n- v16 --refutes--> v1f\n- v1f --refutes--> v20\n- v20 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --refutes--> v1b\n- v1b --refutes--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v23\n- v23 --supports--> v04\n- v04 --supports--> v22\n- v22 --supports--> v14\n- v14 --refutes--> v06\n- v06 --supports--> v11\n- v11 --supports--> v05\n- v05 --supports--> v1a\n- v1a --refutes--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v1d\n- v1d --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v19\n- v19 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v04 --refutes--> v15\n- v07 --refutes--> v03\n- v08 --refutes--> v0e\n- v0a --refutes--> v21\n- v0b --supports--> v1c\n- v14 --supports--> v1e\n- v16 --refutes--> v10\n- v19 --refutes--> v24\n- v1b --supports--> v0f\n- v20 --supports--> v18\n- v23 --refutes--> v12\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":"18b38f0fff07ca973b8cc8f0a05c5190552e32f4f1f8dce2a1c45ac8d4181e79","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --refutes--> v1f\n- v1f --supports--> v14\n- v14 --refutes--> v09\n- v09 --supports--> v1a\n- v1a --supports--> v22\n- v22 --refutes--> v18\n- v18 --refutes--> v21\n- v21 --refutes--> v01\n- v01 --supports--> v13\n- v13 --refutes--> v0c\n- v0c --refutes--> v24\n- v24 --supports--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v1c\n- v1c --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v04\n- v04 --supports--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v17\n- v17 --refutes--> v11\n- v11 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v20\n- v0c --refutes--> v08\n- v13 --supports--> v1e\n- v17 --refutes--> v1d\n- v1c --refutes--> v0b\n- v1f --refutes--> v00\n- v1f --supports--> v03\n- v22 --refutes--> v0f\n- v22 --supports--> v05\n- v24 --refutes--> v12\n- v24 --refutes--> v23\n- v24 --supports--> v16\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":"01b309a6a406f31ff4cd28a1f531b6b19a643844fe23c51bbdb6c3ee9db82fe4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v20\n- v20 --refutes--> v22\n- v22 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v14\n- v14 --supports--> v24\n- v24 --supports--> v19\n- v19 --supports--> v06\n- v06 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v1d\n- v1d --refutes--> v16\n- v16 --refutes--> v0a\n- v0a --supports--> v1c\n- v1c --supports--> v13\n- v13 --supports--> v08\n- v08 --supports--> v18\n- v18 --supports--> v1b\n- v1b --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v21\n- v21 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v04 --supports--> v23\n- v07 --refutes--> v10\n- v07 --supports--> v02\n- v0a --supports--> v12\n- v0b --refutes--> v1f\n- v0d --refutes--> v03\n- v0f --refutes--> v09\n- v16 --supports--> v1a\n- v16 --supports--> v1e\n- v1b --refutes--> v17\n- v22 --supports--> v0c\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":"89a1507b0abf898f94c9493601a3d241f663dfb8114af493ae464a5a21afcf4d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v17\n- v17 --refutes--> v00\n- v00 --supports--> v01\n- v01 --supports--> v1a\n- v1a --supports--> v24\n- v24 --supports--> v19\n- v19 --supports--> v1c\n- v1c --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --supports--> v03\n- v03 --refutes--> v1d\n- v1d --supports--> v0a\n- v0a --supports--> v14\n- v14 --supports--> v09\n- v09 --supports--> v11\n- v11 --refutes--> v0c\n- v0c --refutes--> v1e\n- v1e --refutes--> v06\n- v06 --supports--> v22\n- v22 --supports--> v21\n- v21 --refutes--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v00 --refutes--> v10\n- v00 --supports--> v23\n- v01 --refutes--> v20\n- v01 --supports--> v13\n- v07 --refutes--> v1f\n- v0b --supports--> v08\n- v0f --supports--> v15\n- v11 --refutes--> v18\n- v16 --refutes--> v04\n- v1a --refutes--> v12\n- v24 --refutes--> v02\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":"301222568c1a2209686e62b4428ea251482bb7dc39c3cba96ddc80815a60a1c7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --refutes--> v1f\n- v1f --supports--> v0b\n- v0b --supports--> v1d\n- v1d --refutes--> v15\n- v15 --supports--> v1b\n- v1b --refutes--> v24\n- v24 --refutes--> v12\n- v12 --supports--> v22\n- v22 --refutes--> v0d\n- v0d --refutes--> v18\n- v18 --supports--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v20\n- v20 --supports--> v14\n- v14 --supports--> v16\n- v16 --supports--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v0e\n- v0e --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v23\n- v23 --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v00\n- v08 --refutes--> v03\n- v09 --refutes--> v19\n- v0b --refutes--> v0f\n- v0e --supports--> v17\n- v13 --refutes--> v1c\n- v15 --supports--> v1e\n- v18 --supports--> v05\n- v1b --supports--> v1a\n- v1d --refutes--> v11\n- v22 --refutes--> v10\n- v23 --supports--> v04\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":"047df488c128198266926175d911d752b40147da02b369970f273df5865516f9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v1b\n- v1b --supports--> v03\n- v03 --refutes--> v1f\n- v1f --supports--> v17\n- v17 --refutes--> v12\n- v12 --supports--> v14\n- v14 --refutes--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v21\n- v21 --refutes--> v24\n- v24 --supports--> v00\n- v00 --refutes--> v04\n- v04 --refutes--> v11\n- v11 --refutes--> v05\n- v05 --refutes--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v08\n- v08 --supports--> v1d\n- v1d --refutes--> v1e\n- v1e --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0a\n- v03 --refutes--> v06\n- v03 --refutes--> v07\n- v05 --refutes--> v23\n- v10 --refutes--> v15\n- v16 --supports--> v13\n- v17 --refutes--> v22\n- v17 --supports--> v1a\n- v1e --refutes--> v02\n- v21 --refutes--> v19\n- v24 --supports--> v0b\n- v24 --supports--> v20\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":"0bb0dcf4b7c0d7766b9be216b7e1761d7bba8cf587d2407a8bb33af06005ca5c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --supports--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v15\n- v15 --supports--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v20\n- v20 --supports--> v00\n- v00 --supports--> v12\n- v12 --supports--> v16\n- v16 --supports--> v1a\n- v1a --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v02\n- v02 --refutes--> v0d\n- v0d --refutes--> v10\n- v10 --refutes--> v1b\n- v1b --supports--> v1e\n- v1e --supports--> v22\n- v22 --refutes--> v1d\n- v1d --supports--> v0f\n- v0f --refutes--> v0a\n- v0a --supports--> v1c\n- v1c --supports--> v17\n- v17 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v18\n- v04 --supports--> v21\n- v07 --refutes--> v11\n- v08 --supports--> v0b\n- v13 --supports--> v0c\n- v16 --supports--> v14\n- v17 --supports--> v24\n- v1a --refutes--> v19\n- v1b --refutes--> v23\n- v1d --refutes--> v01\n- v1d --refutes--> v09\n- v1d --supports--> v1f\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":"809703ea0dbc8fe29a6a1be6bd893f74d8517522ecca60a5d5d8623dc4d84045","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v18\n- v18 --supports--> v0e\n- v0e --supports--> v17\n- v17 --refutes--> v1d\n- v1d --refutes--> v03\n- v03 --supports--> v1b\n- v1b --refutes--> v00\n- v00 --supports--> v1f\n- v1f --refutes--> v11\n- v11 --supports--> v06\n- v06 --refutes--> v16\n- v16 --supports--> v15\n- v15 --supports--> v07\n- v07 --supports--> v0f\n- v0f --refutes--> v14\n- v14 --refutes--> v1e\n- v1e --refutes--> v13\n- v13 --supports--> v24\n- v24 --supports--> v0c\n- v0c --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1c\n- v05 --refutes--> v20\n- v06 --supports--> v10\n- v08 --supports--> v1a\n- v08 --supports--> v23\n- v0c --refutes--> v12\n- v0d --supports--> v09\n- v0f --refutes--> v02\n- v15 --refutes--> v21\n- v16 --refutes--> v19\n- v17 --supports--> v01\n- v1d --refutes--> v22\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":"914ba88ce255cdd150d93dd259dbb47e7502d86e09e4defbef3bd329bc9ceaae","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v14\n- v14 --supports--> v1f\n- v1f --supports--> v1d\n- v1d --supports--> v16\n- v16 --refutes--> v12\n- v12 --supports--> v17\n- v17 --supports--> v06\n- v06 --supports--> v00\n- v00 --supports--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v11\n- v11 --refutes--> v18\n- v18 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v22\n- v22 --supports--> v01\n- v01 --refutes--> v24\n- v24 --supports--> v0d\n- v0d --supports--> v1a\n- v1a --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v10\n- v09 --refutes--> v0e\n- v0c --refutes--> v21\n- v0d --refutes--> v0b\n- v13 --supports--> v04\n- v14 --refutes--> v1e\n- v14 --refutes--> v23\n- v18 --refutes--> v07\n- v18 --supports--> v02\n- v1d --refutes--> v20\n- v1d --supports--> v19\n- v22 --refutes--> v1c\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":"27fdbe0da1a23a398e4e6c502da2cbf2ac91988dc5558ec4a80a82c981cc1af2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v09\n- v09 --refutes--> v19\n- v19 --refutes--> v00\n- v00 --supports--> v18\n- v18 --refutes--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v24\n- v24 --refutes--> v1e\n- v1e --refutes--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v23\n- v23 --supports--> v01\n- v01 --refutes--> v1b\n- v1b --refutes--> v0e\n- v0e --refutes--> v08\n- v08 --supports--> v21\n- v21 --refutes--> v03\n- v03 --supports--> v1f\n- v1f --supports--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v13\n- v01 --refutes--> v06\n- v01 --supports--> v10\n- v05 --supports--> v02\n- v07 --refutes--> v17\n- v08 --supports--> v1a\n- v09 --refutes--> v20\n- v12 --refutes--> v1d\n- v18 --refutes--> v14\n- v19 --supports--> v15\n- v1b --refutes--> v0d\n- v1e --supports--> v0a\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":"9042211e27dac1af4117e091302b10da40464c87dc607ddd10118d0d76e6f13e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v1f\n- v1f --supports--> v05\n- v05 --refutes--> v16\n- v16 --refutes--> v03\n- v03 --supports--> v24\n- v24 --refutes--> v12\n- v12 --refutes--> v01\n- v01 --refutes--> v13\n- v13 --refutes--> v1e\n- v1e --refutes--> v06\n- v06 --supports--> v22\n- v22 --supports--> v1d\n- v1d --supports--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v1c\n- v08 --supports--> v23\n- v0a --refutes--> v00\n- v0a --supports--> v1a\n- v0e --supports--> v02\n- v10 --refutes--> v0b\n- v11 --supports--> v14\n- v12 --supports--> v20\n- v13 --supports--> v15\n- v13 --supports--> v21\n- v18 --refutes--> v17\n- v1e --supports--> v1b\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":"4326b09d461762e732be45e871744bbc31f2eed2a8bc72389aaf30e2bc53018a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0a\n- v0a --refutes--> v14\n- v14 --refutes--> v21\n- v21 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v24\n- v24 --refutes--> v0b\n- v0b --supports--> v1d\n- v1d --supports--> v18\n- v18 --refutes--> v1e\n- v1e --supports--> v1f\n- v1f --refutes--> v02\n- v02 --supports--> v1b\n- v1b --refutes--> v23\n- v23 --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v17\n- v17 --refutes--> v08\n- v08 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v16\n- v16 --refutes--> v12\n- v12 --refutes--> v1a\n- v1a --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v20\n- v02 --supports--> v1c\n- v04 --refutes--> v01\n- v06 --refutes--> v15\n- v08 --supports--> v0d\n- v0a --refutes--> v0f\n- v0b --refutes--> v09\n- v0b --refutes--> v22\n- v1a --supports--> v11\n- v1e --supports--> v10\n- v1f --supports--> v19\n- v21 --supports--> v00\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":"8a1380c3078825d121f1357b0389af63cba8d46ef8f5edaa0fa0bed42f06c31b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v23\n- v23 --supports--> v1a\n- v1a --refutes--> v1d\n- v1d --supports--> v21\n- v21 --supports--> v12\n- v12 --refutes--> v1b\n- v1b --supports--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v1e\n- v1e --supports--> v06\n- v06 --refutes--> v05\n- v05 --refutes--> v20\n- v20 --refutes--> v11\n- v11 --supports--> v00\n- v00 --refutes--> v16\n- v16 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v1f\n- v1f --refutes--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v02 --supports--> v13\n- v04 --refutes--> v0a\n- v04 --refutes--> v18\n- v05 --supports--> v14\n- v0d --refutes--> v0c\n- v10 --supports--> v15\n- v17 --refutes--> v22\n- v19 --supports--> v07\n- v1a --supports--> v09\n- v1b --refutes--> v1c\n- v1f --refutes--> v08\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":"2de15f51cbe0f98b54dda65904d252979df86b5dc7c98b828293cc41b133ce40","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v07\n- v07 --supports--> v1d\n- v1d --supports--> v1e\n- v1e --supports--> v1f\n- v1f --supports--> v11\n- v11 --refutes--> v0c\n- v0c --supports--> v1c\n- v1c --refutes--> v23\n- v23 --supports--> v24\n- v24 --refutes--> v22\n- v22 --refutes--> v1a\n- v1a --refutes--> v21\n- v21 --supports--> v0d\n- v0d --supports--> v14\n- v14 --refutes--> v12\n- v12 --supports--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v20\n- v20 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v03\n- v03 --refutes--> v19\n- v19 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v02\n- v0c --refutes--> v05\n- v1a --refutes--> v10\n- v1c --refutes--> v17\n- v1d --refutes--> v09\n- v1d --supports--> v01\n- v1e --refutes--> v18\n- v1e --supports--> v16\n- v20 --refutes--> v1b\n- v22 --supports--> v00\n- v23 --refutes--> v13\n- v24 --supports--> v15\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":"d41abb84a03581fbba49359c9eb28fe6a493646cf4376bfea84e5f050c2db837","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --refutes--> v1b\n- v1b --supports--> v05\n- v05 --refutes--> v0f\n- v0f --refutes--> v20\n- v20 --refutes--> v0c\n- v0c --refutes--> v24\n- v24 --supports--> v16\n- v16 --refutes--> v1c\n- v1c --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v18\n- v18 --refutes--> v0b\n- v0b --refutes--> v10\n- v10 --supports--> v23\n- v23 --refutes--> v1d\n- v1d --refutes--> v01\n- v01 --supports--> v03\n- v03 --supports--> v0e\n- v0e --refutes--> v1f\n- v1f --refutes--> v0d\n- v0d --supports--> v08\n- v08 --supports--> v19\n- v19 --refutes--> v21\n- v21 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v00\n- v05 --refutes--> v02\n- v0d --refutes--> v07\n- v0d --refutes--> v0a\n- v10 --supports--> v1e\n- v18 --supports--> v09\n- v18 --supports--> v14\n- v18 --supports--> v17\n- v1f --supports--> v04\n- v1f --supports--> v06\n- v21 --refutes--> v1a\n- v24 --refutes--> v13\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":"942568c821deb1b6deb905cae0b98221f744454d4fb2a5887e86b66a41ce3e61","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v0f\n- v0f --refutes--> v1e\n- v1e --refutes--> v09\n- v09 --supports--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v16\n- v16 --refutes--> v05\n- v05 --refutes--> v1b\n- v1b --refutes--> v20\n- v20 --refutes--> v22\n- v22 --supports--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v0c\n- v0c --refutes--> v1c\n- v1c --refutes--> v01\n- v01 --refutes--> v17\n- v17 --refutes--> v0a\n- v0a --supports--> v23\n- v23 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v1d\n- v05 --refutes--> v1f\n- v05 --refutes--> v24\n- v07 --refutes--> v1a\n- v0b --supports--> v21\n- v12 --supports--> v02\n- v17 --supports--> v0d\n- v18 --supports--> v15\n- v1b --refutes--> v03\n- v1c --supports--> v11\n- v20 --supports--> v13\n- v23 --refutes--> v0e\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":"b2aca722a1e2a6913ccb66e2303e1b03c3d00fba60076b3a15b7b125be78f14a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v13\n- v13 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v10\n- v10 --refutes--> v07\n- v07 --refutes--> v09\n- v09 --refutes--> v1d\n- v1d --supports--> v1a\n- v1a --supports--> v14\n- v14 --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --refutes--> v0a\n- v0a --refutes--> v1e\n- v1e --refutes--> v22\n- v22 --refutes--> v23\n- v23 --supports--> v1f\n- v1f --supports--> v21\n- v21 --supports--> v17\n- v17 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v18\n- v07 --supports--> v16\n- v0a --supports--> v02\n- v0a --supports--> v12\n- v0d --refutes--> v03\n- v1b --refutes--> v06\n- v1b --refutes--> v11\n- v1b --supports--> v05\n- v1e --supports--> v1c\n- v1e --supports--> v24\n- v22 --supports--> v19\n- v23 --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":"77a7115b762c36a1d34cc7b48b7cc88e60d0d7b00ed378e73fa5098acdd28c25","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v06\n- v06 --refutes--> v1c\n- v1c --supports--> v05\n- v05 --supports--> v17\n- v17 --supports--> v1f\n- v1f --refutes--> v21\n- v21 --supports--> v16\n- v16 --refutes--> v0b\n- v0b --refutes--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v24\n- v24 --supports--> v20\n- v20 --supports--> v1b\n- v1b --refutes--> v03\n- v03 --refutes--> v18\n- v18 --supports--> v02\n- v02 --refutes--> v22\n- v22 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --supports--> v10\n- v10 --supports--> v12\n- v12 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v11\n- v03 --supports--> v0a\n- v03 --supports--> v1a\n- v04 --supports--> v08\n- v0b --refutes--> v14\n- v16 --supports--> v19\n- v17 --refutes--> v09\n- v18 --refutes--> v1d\n- v1b --refutes--> v01\n- v1c --supports--> v0d\n- v1f --refutes--> v0c\n- v1f --refutes--> v1e\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":"ec41cdad1da1fa4101e657f116b06ea3862d94592810ddbb320e00104d0263d6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v06\n- v06 --supports--> v20\n- v20 --refutes--> v0c\n- v0c --supports--> v18\n- v18 --supports--> v00\n- v00 --supports--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v1f\n- v1f --refutes--> v05\n- v05 --supports--> v13\n- v13 --refutes--> v16\n- v16 --supports--> v21\n- v21 --refutes--> v19\n- v19 --refutes--> v01\n- v01 --refutes--> v1d\n- v1d --refutes--> v04\n- v04 --refutes--> v0e\n- v0e --refutes--> v1c\n- v1c --refutes--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --supports--> v1a\n- v1a --supports--> v0a\n- v0a --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v05 --supports--> v02\n- v05 --supports--> v14\n- v0a --supports--> v1e\n- v0b --supports--> v08\n- v0c --refutes--> v0f\n- v12 --refutes--> v23\n- v12 --refutes--> v24\n- v15 --supports--> v17\n- v1a --supports--> v22\n- v1b --supports--> v09\n- v1f --refutes--> v10\n\n[VERIFIED ROOT]\nv15 = TRUE\n\n[TARGET]\nv0d\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"f9396396c692e3f0785f92fa0e3afff8789d45ee189ad6228ba45c787dd4d727","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v13\n- v13 --refutes--> v00\n- v00 --refutes--> v20\n- v20 --supports--> v17\n- v17 --refutes--> v1b\n- v1b --supports--> v02\n- v02 --supports--> v08\n- v08 --refutes--> v12\n- v12 --refutes--> v06\n- v06 --supports--> v1d\n- v1d --supports--> v24\n- v24 --supports--> v22\n- v22 --refutes--> v0e\n- v0e --refutes--> v23\n- v23 --supports--> v0f\n- v0f --refutes--> v0a\n- v0a --supports--> v07\n- v07 --refutes--> v1e\n- v1e --refutes--> v11\n- v11 --supports--> v1a\n- v1a --refutes--> v1f\n- v1f --supports--> v04\n- v04 --supports--> v21\n- v21 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v0b\n- v06 --supports--> v0c\n- v08 --refutes--> v19\n- v09 --refutes--> v15\n- v11 --supports--> v10\n- v1d --supports--> v16\n- v1d --supports--> v1c\n- v1e --refutes--> v05\n- v1f --refutes--> v03\n- v1f --supports--> v01\n- v20 --refutes--> v0d\n- v23 --supports--> v14\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":"0de79f51a4d69fab595bad38ac48b8454e5d9c24ebd94f17266dd4ebb7423f86","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v0f\n- v0f --supports--> v15\n- v15 --refutes--> v24\n- v24 --refutes--> v06\n- v06 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v1d\n- v1d --refutes--> v17\n- v17 --refutes--> v1b\n- v1b --supports--> v1c\n- v1c --refutes--> v0c\n- v0c --supports--> v02\n- v02 --supports--> v16\n- v16 --supports--> v20\n- v20 --supports--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v1f\n- v1f --supports--> v08\n- v08 --supports--> v10\n- v10 --supports--> v22\n- v22 --supports--> v0a\n- v0a --supports--> v11\n- v11 --refutes--> v13\n- v13 --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v0b\n- v08 --supports--> v14\n- v0c --refutes--> v03\n- v0c --refutes--> v1e\n- v0f --refutes--> v01\n- v11 --refutes--> v09\n- v11 --refutes--> v0d\n- v15 --refutes--> v23\n- v17 --refutes--> v12\n- v1c --refutes--> v18\n- v1c --supports--> v21\n- v20 --refutes--> v0e\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":"deb6086d4b589b88e5ed4244354fff2221333eca7f60d8c810ad3e7f62c478c0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v07\n- v07 --supports--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v21\n- v21 --supports--> v09\n- v09 --refutes--> v15\n- v15 --supports--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v1a\n- v1a --supports--> v02\n- v02 --supports--> v1e\n- v1e --refutes--> v1d\n- v1d --refutes--> v12\n- v12 --supports--> v06\n- v06 --supports--> v20\n- v20 --supports--> v0e\n- v0e --supports--> v1b\n- v1b --refutes--> v24\n- v24 --refutes--> v1f\n- v1f --supports--> v14\n- v14 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v22\n- v0b --refutes--> v0a\n- v0d --supports--> v23\n- v13 --supports--> v1c\n- v14 --supports--> v04\n- v15 --refutes--> v10\n- v18 --refutes--> v19\n- v1a --refutes--> v0c\n- v1b --supports--> v05\n- v1b --supports--> v16\n- v1d --supports--> v08\n- v24 --refutes--> v00\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":"f1a6f238c8c84e3d73d353b9b6989d4ebdf7761992f162ec936807b92d1d64e3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --refutes--> v22\n- v22 --refutes--> v01\n- v01 --supports--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v1d\n- v1d --supports--> v24\n- v24 --supports--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v20\n- v20 --supports--> v10\n- v10 --supports--> v13\n- v13 --supports--> v21\n- v21 --supports--> v1f\n- v1f --refutes--> v08\n- v08 --refutes--> v06\n- v06 --refutes--> v1a\n- v1a --supports--> v1c\n- v1c --supports--> v0e\n- v0e --supports--> v23\n- v23 --refutes--> v0b\n- v0b --refutes--> v19\n- v19 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0d\n- v03 --refutes--> v02\n- v03 --refutes--> v12\n- v0c --supports--> v09\n- v18 --supports--> v17\n- v19 --supports--> v05\n- v1a --supports--> v07\n- v1b --supports--> v0a\n- v1f --supports--> v14\n- v20 --refutes--> v16\n- v21 --refutes--> v00\n- v23 --refutes--> v11\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":"6158608e3aedbaf91184493748598d8e4695272a012dcf6bb4848be2eaa3d118","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v20\n- v20 --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v23\n- v23 --supports--> v21\n- v21 --supports--> v12\n- v12 --supports--> v18\n- v18 --supports--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v22\n- v22 --refutes--> v1c\n- v1c --refutes--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v07\n- v07 --supports--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --refutes--> v1a\n- v1a --supports--> v09\n- v09 --refutes--> v0d\n- v0d --refutes--> v1b\n- v1b --supports--> v13\n- v13 --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v19\n- v05 --supports--> v17\n- v0a --refutes--> v0c\n- v0a --supports--> v04\n- v0e --supports--> v0b\n- v13 --refutes--> v10\n- v18 --refutes--> v1f\n- v1a --supports--> v1e\n- v1b --refutes--> v0f\n- v1c --supports--> v24\n- v22 --supports--> v15\n- v23 --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":"fa873932a15f48f7b89d89894fe4efbb60fd464827ca698fa3b81811c2e17dc0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v22\n- v22 --refutes--> v24\n- v24 --refutes--> v1c\n- v1c --supports--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v18\n- v18 --supports--> v1a\n- v1a --supports--> v0c\n- v0c --refutes--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v15\n- v15 --refutes--> v23\n- v23 --refutes--> v1d\n- v1d --refutes--> v19\n- v19 --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v0b\n- v0b --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v1f\n- v05 --refutes--> v01\n- v05 --supports--> v13\n- v06 --supports--> v0d\n- v0b --refutes--> v10\n- v0b --refutes--> v16\n- v0c --refutes--> v0e\n- v11 --refutes--> v1e\n- v15 --refutes--> v17\n- v18 --refutes--> v21\n- v1b --supports--> v02\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":"48b94812b77769f1eb96aa30b2114d5140237478542a914a9d0180833ce0f0a3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v06\n- v06 --refutes--> v23\n- v23 --supports--> v04\n- v04 --refutes--> v10\n- v10 --supports--> v09\n- v09 --supports--> v1f\n- v1f --refutes--> v03\n- v03 --supports--> v18\n- v18 --refutes--> v14\n- v14 --supports--> v00\n- v00 --refutes--> v21\n- v21 --refutes--> v20\n- v20 --refutes--> v11\n- v11 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v1b\n- v1b --supports--> v0c\n- v0c --supports--> v05\n- v05 --supports--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v1c\n- v1c --refutes--> v07\n- v07 --supports--> v24\n- v24 --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v1a\n- v04 --refutes--> v22\n- v09 --supports--> v01\n- v0b --refutes--> v1e\n- v0b --supports--> v15\n- v10 --refutes--> v08\n- v12 --supports--> v0d\n- v14 --supports--> v0f\n- v20 --refutes--> v02\n- v20 --refutes--> v19\n- v24 --refutes--> v17\n- v24 --supports--> v13\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":"f4f58df9b27de986ace261cd4f69a2e4c3f755c2905f876875f88b18c09b2e17","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v20\n- v20 --supports--> v01\n- v01 --refutes--> v14\n- v14 --refutes--> v24\n- v24 --refutes--> v1d\n- v1d --supports--> v1e\n- v1e --supports--> v10\n- v10 --refutes--> v0e\n- v0e --refutes--> v19\n- v19 --supports--> v0b\n- v0b --supports--> v1a\n- v1a --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --refutes--> v09\n- v09 --refutes--> v05\n- v05 --refutes--> v1f\n- v1f --supports--> v23\n- v23 --refutes--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v13\n- v13 --supports--> v22\n- v22 --supports--> v00\n- v00 --supports--> v16\n- v16 --refutes--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v18\n- v09 --refutes--> v17\n- v13 --supports--> v06\n- v13 --supports--> v11\n- v16 --supports--> v08\n- v19 --refutes--> v02\n- v1d --supports--> v0f\n- v1d --supports--> v15\n- v1e --refutes--> v12\n- v1e --supports--> v1c\n- v20 --supports--> v04\n- v24 --supports--> v1b\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":"fd7d130f83ac2c5b2cdf9bfe51407cf15d6f27fbcbd0b71dd8430496fa42fc59","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v02\n- v02 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v15\n- v15 --supports--> v18\n- v18 --supports--> v0b\n- v0b --refutes--> v1d\n- v1d --supports--> v08\n- v08 --supports--> v06\n- v06 --refutes--> v20\n- v20 --refutes--> v1b\n- v1b --refutes--> v1f\n- v1f --refutes--> v16\n- v16 --refutes--> v21\n- v21 --refutes--> v03\n- v03 --supports--> v22\n- v22 --refutes--> v24\n- v24 --supports--> v1a\n- v1a --supports--> v04\n- v04 --supports--> v19\n- v19 --refutes--> v0a\n- v0a --refutes--> v17\n- v17 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0c\n- v02 --refutes--> v0d\n- v02 --supports--> v11\n- v03 --refutes--> v10\n- v08 --refutes--> v23\n- v0a --refutes--> v09\n- v0a --supports--> v07\n- v14 --supports--> v13\n- v19 --supports--> v0e\n- v1a --supports--> v12\n- v1f --refutes--> v1e\n- v20 --supports--> v0f\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":"0fe7ae4277237e3de1b7dfbc0160cb0978496240a59e846d3e8594c7fe19081c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v15\n- v15 --refutes--> v1c\n- v1c --refutes--> v1e\n- v1e --refutes--> v17\n- v17 --supports--> v02\n- v02 --supports--> v16\n- v16 --refutes--> v1f\n- v1f --supports--> v0b\n- v0b --refutes--> v20\n- v20 --refutes--> v1d\n- v1d --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v1b\n- v1b --refutes--> v18\n- v18 --refutes--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v19\n- v19 --refutes--> v06\n- v06 --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v05 --refutes--> v13\n- v05 --supports--> v23\n- v10 --refutes--> v09\n- v10 --refutes--> v12\n- v16 --supports--> v0f\n- v17 --supports--> v21\n- v19 --refutes--> v22\n- v1c --refutes--> v04\n- v1d --supports--> v03\n- v1e --supports--> v24\n- v20 --supports--> v0e\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":"d0eb6b481aaef48a1512bd2625dca312f4b1eccbecc69e8ece495ecb972e8804","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v0e\n- v0e --refutes--> v18\n- v18 --supports--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v02\n- v02 --refutes--> v23\n- v23 --refutes--> v0a\n- v0a --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --supports--> v10\n- v10 --refutes--> v24\n- v24 --refutes--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v1f\n- v1f --supports--> v1c\n- v1c --refutes--> v20\n- v20 --supports--> v1d\n- v1d --supports--> v1b\n- v1b --refutes--> v1a\n- v1a --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v08\n- v02 --supports--> v17\n- v09 --supports--> v04\n- v0d --supports--> v06\n- v0f --supports--> v07\n- v10 --supports--> v21\n- v12 --refutes--> v22\n- v19 --supports--> v0c\n- v1d --refutes--> v03\n- v1f --refutes--> v15\n- v20 --refutes--> v13\n- v24 --supports--> v1e\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":"7fa98be72a6d19142b7430d5c6553021516ba30cbe89db1ea4f58c4966652496","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v17\n- v17 --supports--> v20\n- v20 --supports--> v18\n- v18 --supports--> v13\n- v13 --supports--> v09\n- v09 --supports--> v11\n- v11 --supports--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v14\n- v14 --supports--> v07\n- v07 --refutes--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v1c\n- v1c --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v21\n- v21 --refutes--> v24\n- v24 --supports--> v0f\n- v0f --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v02 --supports--> v00\n- v06 --refutes--> v15\n- v09 --refutes--> v12\n- v09 --refutes--> v1e\n- v0b --supports--> v1a\n- v0c --supports--> v22\n- v0f --refutes--> v04\n- v0f --supports--> v23\n- v10 --supports--> v19\n- v11 --supports--> v1f\n- v17 --refutes--> v1d\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":"7e19824e487692f500ff58e2d7dd7fb7e2d97eda74b5dfaf65414f5695f1cc46","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v16\n- v16 --refutes--> v24\n- v24 --supports--> v08\n- v08 --supports--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v20\n- v20 --supports--> v1b\n- v1b --supports--> v1c\n- v1c --refutes--> v1e\n- v1e --supports--> v13\n- v13 --supports--> v14\n- v14 --supports--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v0b\n- v0b --supports--> v19\n- v19 --supports--> v01\n- v01 --refutes--> v23\n- v23 --supports--> v0a\n- v0a --supports--> v1a\n- v1a --refutes--> v03\n- v03 --refutes--> v22\n- v22 --supports--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v05\n- v03 --supports--> v1f\n- v08 --supports--> v1d\n- v0f --supports--> v06\n- v10 --refutes--> v09\n- v13 --refutes--> v0d\n- v14 --supports--> v0e\n- v19 --supports--> v0c\n- v1b --refutes--> v17\n- v1e --refutes--> v11\n- v20 --supports--> v21\n- v22 --refutes--> v07\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":"2551b719278dd4d403e549039430675c6983bd83b47da2fcbcad2de9639ce6da","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v12\n- v12 --supports--> v06\n- v06 --supports--> v02\n- v02 --refutes--> v11\n- v11 --supports--> v0a\n- v0a --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v03\n- v03 --supports--> v04\n- v04 --refutes--> v13\n- v13 --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v0d\n- v0d --supports--> v0f\n- v0f --supports--> v23\n- v23 --supports--> v17\n- v17 --refutes--> v1f\n- v1f --refutes--> v21\n- v21 --supports--> v18\n- v18 --refutes--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v0e\n- v0e --supports--> v1a\n- v1a --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v22\n- v06 --refutes--> v0c\n- v06 --refutes--> v24\n- v0d --refutes--> v05\n- v0d --supports--> v10\n- v11 --refutes--> v15\n- v13 --refutes--> v1c\n- v13 --supports--> v19\n- v14 --supports--> v01\n- v1a --supports--> v1d\n- v1b --refutes--> v16\n- v1f --refutes--> v1e\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":"630e73352795e96c0bf16795ea3cf86b3fb1bac6fa7568337d0d5f3a8e74a346","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v1d\n- v1d --refutes--> v09\n- v09 --supports--> v0d\n- v0d --supports--> v19\n- v19 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v22\n- v22 --refutes--> v20\n- v20 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v24\n- v24 --supports--> v0a\n- v0a --supports--> v21\n- v21 --refutes--> v11\n- v11 --refutes--> v17\n- v17 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --refutes--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v12\n- v03 --refutes--> v14\n- v04 --supports--> v23\n- v05 --refutes--> v13\n- v09 --refutes--> v1c\n- v0a --refutes--> v1b\n- v0b --supports--> v07\n- v0d --refutes--> v15\n- v0f --supports--> v02\n- v19 --refutes--> v1a\n- v21 --supports--> v18\n- v22 --supports--> v1e\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":"9e1e792d5a7c7531f61652622994d9d8cfdc4b7284d6d4677d3f8b8a30ae0375","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v24\n- v24 --refutes--> v18\n- v18 --refutes--> v10\n- v10 --refutes--> v16\n- v16 --refutes--> v0a\n- v0a --refutes--> v21\n- v21 --refutes--> v1c\n- v1c --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v1b\n- v1b --refutes--> v1d\n- v1d --refutes--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v1f\n- v1f --refutes--> v22\n- v22 --refutes--> v00\n- v00 --refutes--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v01\n- v01 --refutes--> v17\n- v17 --refutes--> v23\n- v23 --supports--> v04\n- v04 --refutes--> v12\n- v12 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v05 --supports--> v03\n- v06 --refutes--> v08\n- v06 --supports--> v11\n- v0a --refutes--> v09\n- v15 --refutes--> v20\n- v16 --supports--> v0c\n- v17 --refutes--> v02\n- v1c --refutes--> v13\n- v1c --refutes--> v19\n- v22 --supports--> v0d\n- v23 --refutes--> v1e\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":"765aa03bdfa01cd27b809cbb5211111234ec6e61b25f85e0eadae293eae171d0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v18\n- v18 --supports--> v23\n- v23 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --refutes--> v12\n- v12 --supports--> v11\n- v11 --supports--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --supports--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v19\n- v19 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v21\n- v21 --refutes--> v17\n- v17 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v15\n- v07 --refutes--> v09\n- v07 --supports--> v1c\n- v0f --refutes--> v1d\n- v0f --refutes--> v24\n- v0f --supports--> v20\n- v10 --supports--> v1f\n- v10 --supports--> v22\n- v19 --refutes--> v02\n- v1a --refutes--> v16\n- v1e --supports--> v0d\n- v21 --supports--> v0c\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":"a6a3899d7747b2994ce2a916d04ac6b46674d8509ac421c660e11318d34dbdc6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v10\n- v10 --refutes--> v18\n- v18 --supports--> v11\n- v11 --supports--> v22\n- v22 --refutes--> v00\n- v00 --supports--> v24\n- v24 --supports--> v17\n- v17 --refutes--> v16\n- v16 --refutes--> v12\n- v12 --supports--> v21\n- v21 --refutes--> v04\n- v04 --refutes--> v23\n- v23 --refutes--> v0c\n- v0c --supports--> v1e\n- v1e --refutes--> v1a\n- v1a --supports--> v14\n- v14 --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --supports--> v07\n- v07 --supports--> v1b\n- v1b --refutes--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v1c\n- v1c --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1f\n- v07 --supports--> v06\n- v11 --supports--> v20\n- v12 --supports--> v0e\n- v17 --refutes--> v1d\n- v18 --supports--> v13\n- v18 --supports--> v15\n- v1b --refutes--> v03\n- v1b --supports--> v08\n- v1e --refutes--> v05\n- v22 --supports--> v02\n- v23 --supports--> v0b\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":"66b28ac602cf02235a4ad0eca034acacca5983fbda88159ff7adf0763ff2a6a1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v04\n- v04 --supports--> v0d\n- v0d --supports--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v14\n- v14 --supports--> v06\n- v06 --refutes--> v13\n- v13 --supports--> v24\n- v24 --supports--> v1e\n- v1e --supports--> v0e\n- v0e --refutes--> v21\n- v21 --supports--> v09\n- v09 --supports--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v08\n- v08 --supports--> v17\n- v17 --refutes--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v1d\n- v1d --supports--> v16\n- v16 --refutes--> v23\n- v23 --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1b\n- v04 --refutes--> v15\n- v06 --refutes--> v03\n- v0b --refutes--> v1a\n- v0b --supports--> v05\n- v0c --supports--> v1f\n- v0f --supports--> v02\n- v1d --supports--> v10\n- v1e --supports--> v1c\n- v1e --supports--> v22\n- v21 --refutes--> v19\n- v24 --refutes--> v18\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":"1819ffc6ea649e2ee273206924bf6e2a9f62e72ef21cc661ac53691b43b1b34b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v15\n- v15 --refutes--> v0d\n- v0d --refutes--> v06\n- v06 --refutes--> v1e\n- v1e --refutes--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v21\n- v21 --refutes--> v20\n- v20 --refutes--> v0f\n- v0f --supports--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v14\n- v14 --refutes--> v23\n- v23 --refutes--> v22\n- v22 --refutes--> v01\n- v01 --refutes--> v0b\n- v0b --supports--> v24\n- v24 --refutes--> v1f\n- v1f --supports--> v00\n- v00 --supports--> v1c\n- v1c --supports--> v18\n- v18 --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v13\n- v06 --supports--> v1d\n- v0b --supports--> v09\n- v18 --supports--> v11\n- v1b --refutes--> v07\n- v1b --refutes--> v0c\n- v1b --supports--> v08\n- v1b --supports--> v19\n- v1f --refutes--> v12\n- v21 --supports--> v16\n- v22 --supports--> v04\n- v24 --supports--> v17\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":"bcb5ca62a2a350b53864d8f8d40ce3465bfa3e24fa936a671ec854a6f961ee2c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --refutes--> v05\n- v05 --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v0b\n- v0b --supports--> v15\n- v15 --supports--> v13\n- v13 --supports--> v20\n- v20 --refutes--> v19\n- v19 --refutes--> v14\n- v14 --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v11\n- v11 --supports--> v18\n- v18 --refutes--> v06\n- v06 --supports--> v1e\n- v1e --supports--> v24\n- v24 --supports--> v1d\n- v1d --refutes--> v0a\n- v0a --refutes--> v22\n- v22 --supports--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v23\n- v03 --refutes--> v0c\n- v07 --refutes--> v04\n- v0a --supports--> v12\n- v0b --refutes--> v0d\n- v0b --supports--> v17\n- v0e --refutes--> v1c\n- v11 --refutes--> v1b\n- v13 --refutes--> v10\n- v13 --supports--> v08\n- v18 --refutes--> v1a\n- v1e --refutes--> v16\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":"e783fa83965164c0ff98f74793dc6926ee887818341490f2e034bda666fe1adf","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v18\n- v18 --refutes--> v02\n- v02 --refutes--> v0d\n- v0d --supports--> v10\n- v10 --refutes--> v20\n- v20 --refutes--> v12\n- v12 --supports--> v16\n- v16 --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v1a\n- v1a --refutes--> v00\n- v00 --refutes--> v1e\n- v1e --supports--> v07\n- v07 --supports--> v22\n- v22 --refutes--> v24\n- v24 --supports--> v17\n- v17 --refutes--> v05\n- v05 --refutes--> v21\n- v21 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v23\n- v01 --refutes--> v04\n- v02 --refutes--> v1f\n- v05 --refutes--> v15\n- v05 --supports--> v1d\n- v0b --refutes--> v14\n- v11 --supports--> v0e\n- v12 --refutes--> v09\n- v12 --supports--> v0a\n- v16 --refutes--> v1b\n- v18 --refutes--> v0c\n- v21 --refutes--> v1c\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":"81c6ceab0e491f8c610a7af3145022558d3183e0d635f64bb7ef95884495b62d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v1e\n- v1e --refutes--> v1c\n- v1c --refutes--> v15\n- v15 --refutes--> v02\n- v02 --supports--> v1a\n- v1a --refutes--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v1b\n- v1b --refutes--> v00\n- v00 --refutes--> v16\n- v16 --supports--> v18\n- v18 --supports--> v24\n- v24 --supports--> v23\n- v23 --refutes--> v0e\n- v0e --refutes--> v17\n- v17 --refutes--> v1d\n- v1d --supports--> v03\n- v03 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v13\n- v13 --supports--> v05\n- v05 --supports--> v1f\n- v1f --supports--> v19\n- v19 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --refutes--> v21\n- v0e --supports--> v20\n- v13 --refutes--> v04\n- v13 --refutes--> v11\n- v1a --supports--> v22\n- v1b --supports--> v08\n- v1c --supports--> v0d\n- v1d --refutes--> v09\n- v1d --refutes--> v14\n- v1d --supports--> v12\n- v23 --supports--> v0f\n- v24 --refutes--> v07\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":"220d2a7c15ecc180141d2315b036e56dacdcfb8def2114834938e4e34594b0e3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v10\n- v10 --refutes--> v18\n- v18 --supports--> v0e\n- v0e --supports--> v02\n- v02 --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v1c\n- v1c --supports--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v11\n- v11 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --supports--> v1f\n- v1f --supports--> v0b\n- v0b --supports--> v24\n- v24 --supports--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v16\n- v16 --refutes--> v09\n- v09 --refutes--> v08\n- v08 --refutes--> v22\n- v22 --refutes--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v02 --supports--> v1b\n- v08 --supports--> v19\n- v0b --refutes--> v06\n- v0e --supports--> v1d\n- v10 --refutes--> v23\n- v12 --refutes--> v04\n- v12 --supports--> v1e\n- v13 --supports--> v17\n- v13 --supports--> v21\n- v18 --refutes--> v1a\n- v18 --supports--> v14\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":"3a90fa12faa69808d841adad2bc80bc02b2e228acdec37ccbabbcb61265f0621","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v12\n- v12 --refutes--> v07\n- v07 --supports--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v17\n- v17 --refutes--> v0a\n- v0a --supports--> v21\n- v21 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v1e\n- v1e --refutes--> v18\n- v18 --refutes--> v13\n- v13 --refutes--> v0f\n- v0f --refutes--> v20\n- v20 --supports--> v1f\n- v1f --supports--> v14\n- v14 --refutes--> v1b\n- v1b --supports--> v22\n- v22 --refutes--> v10\n- v10 --refutes--> v23\n- v23 --refutes--> v1a\n- v1a --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v11\n- v03 --refutes--> v05\n- v07 --supports--> v15\n- v12 --refutes--> v1d\n- v13 --refutes--> v16\n- v17 --refutes--> v06\n- v18 --refutes--> v0d\n- v1a --refutes--> v24\n- v1b --refutes--> v08\n- v1f --supports--> v19\n- v20 --supports--> v0e\n- v22 --supports--> v1c\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":"eddb4dac17ee0cf76bd725481d6ff51e1e3f1a9e9d2ab3f53d2003d814d1587f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v01\n- v01 --refutes--> v23\n- v23 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v00\n- v00 --refutes--> v1f\n- v1f --supports--> v0d\n- v0d --refutes--> v18\n- v18 --supports--> v1a\n- v1a --supports--> v10\n- v10 --refutes--> v1b\n- v1b --supports--> v05\n- v05 --refutes--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v21\n- v21 --supports--> v24\n- v24 --supports--> v08\n- v08 --refutes--> v22\n- v22 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0b\n- v06 --supports--> v14\n- v06 --supports--> v17\n- v0c --refutes--> v03\n- v0d --refutes--> v07\n- v0f --refutes--> v16\n- v0f --refutes--> v1d\n- v13 --supports--> v11\n- v19 --refutes--> v20\n- v1a --refutes--> v0e\n- v1b --supports--> v15\n- v1f --refutes--> v12\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":"bffe8c37358e54e7e473e587b7ef1648a5ac598483dc048bf5d0300ad18c2cb7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v1f\n- v1f --refutes--> v1a\n- v1a --supports--> v10\n- v10 --supports--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v0d\n- v0d --refutes--> v1b\n- v1b --refutes--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v02\n- v02 --supports--> v07\n- v07 --refutes--> v19\n- v19 --supports--> v1e\n- v1e --refutes--> v17\n- v17 --supports--> v24\n- v24 --supports--> v05\n- v05 --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v0e\n- v0e --refutes--> v0a\n- v0a --supports--> v18\n- v18 --supports--> v0f\n- v0f --refutes--> v16\n- v16 --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v21\n- v04 --supports--> v01\n- v05 --refutes--> v23\n- v05 --supports--> v0c\n- v06 --supports--> v09\n- v07 --refutes--> v14\n- v0b --refutes--> v1d\n- v0b --refutes--> v20\n- v0f --refutes--> v11\n- v0f --supports--> v03\n- v16 --supports--> v15\n- v18 --refutes--> v1c\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":"c13d613b8d3de2b0bb56b497dc00b4db6dbd05499f6a8947a99d2e02bc9ee085","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v04\n- v04 --refutes--> v21\n- v21 --refutes--> v16\n- v16 --supports--> v1d\n- v1d --supports--> v06\n- v06 --refutes--> v15\n- v15 --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --supports--> v0a\n- v0a --supports--> v13\n- v13 --supports--> v1e\n- v1e --supports--> v08\n- v08 --refutes--> v18\n- v18 --supports--> v20\n- v20 --supports--> v03\n- v03 --supports--> v07\n- v07 --supports--> v0d\n- v0d --refutes--> v11\n- v11 --refutes--> v1f\n- v1f --supports--> v17\n- v17 --refutes--> v1b\n- v1b --supports--> v14\n- v14 --supports--> v19\n- v19 --refutes--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v04 --refutes--> v22\n- v06 --supports--> v0e\n- v07 --supports--> v12\n- v09 --supports--> v0f\n- v11 --supports--> v0c\n- v15 --supports--> v23\n- v16 --supports--> v10\n- v17 --refutes--> v0b\n- v17 --supports--> v01\n- v19 --refutes--> v1a\n- v20 --refutes--> v02\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":"1e1a7f248e24d93ef1df869609caf6a4db8268ac4cc15838b7032214c5fbc759","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v23\n- v23 --refutes--> v1b\n- v1b --refutes--> v01\n- v01 --refutes--> v18\n- v18 --refutes--> v09\n- v09 --supports--> v13\n- v13 --refutes--> v00\n- v00 --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v11\n- v11 --refutes--> v24\n- v24 --supports--> v10\n- v10 --supports--> v15\n- v15 --supports--> v1f\n- v1f --supports--> v16\n- v16 --refutes--> v21\n- v21 --supports--> v07\n- v07 --refutes--> v20\n- v20 --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v03 --supports--> v02\n- v09 --refutes--> v1c\n- v0e --refutes--> v17\n- v0e --supports--> v0d\n- v15 --supports--> v1e\n- v1b --refutes--> v0b\n- v1f --refutes--> v19\n- v1f --refutes--> v1a\n- v1f --supports--> v0a\n- v21 --supports--> v14\n- v23 --refutes--> v22\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":"d743fecfc2a7002d59776a0c63d69b00d4abb535841bf2c4b5f5e72e4775cbe1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v04\n- v04 --supports--> v1f\n- v1f --refutes--> v24\n- v24 --refutes--> v23\n- v23 --refutes--> v22\n- v22 --supports--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v16\n- v16 --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v0f\n- v0f --refutes--> v1b\n- v1b --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v19\n- v19 --refutes--> v14\n- v14 --supports--> v13\n- v13 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v01 --supports--> v18\n- v07 --supports--> v17\n- v0e --refutes--> v1d\n- v0f --refutes--> v21\n- v15 --refutes--> v0b\n- v15 --supports--> v20\n- v1a --supports--> v1e\n- v1b --refutes--> v05\n- v1b --refutes--> v0c\n- v23 --refutes--> v11\n- v23 --supports--> v02\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":"4ad5a895ddb950d7cc8fe9bc4e5a800dfbba4e1a327420b04c38c4195dd95aac","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v16\n- v16 --supports--> v17\n- v17 --refutes--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v1d\n- v1d --supports--> v0a\n- v0a --refutes--> v22\n- v22 --supports--> v18\n- v18 --refutes--> v24\n- v24 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v1e\n- v1e --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v20\n- v20 --supports--> v0e\n- v0e --refutes--> v05\n- v05 --refutes--> v19\n- v19 --supports--> v10\n- v10 --supports--> v21\n- v21 --refutes--> v1f\n- v1f --refutes--> v23\n- v23 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v06 --supports--> v1b\n- v08 --supports--> v1a\n- v0e --refutes--> v11\n- v0e --supports--> v0d\n- v17 --refutes--> v02\n- v1d --refutes--> v15\n- v1d --supports--> v04\n- v20 --supports--> v01\n- v21 --refutes--> v0f\n- v22 --supports--> v1c\n- v23 --refutes--> v07\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":"6d38c07f68bd87f9a7028fef8571aa927997cb978a2cb66f9d2d56fc9260c912","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --refutes--> v26\n- v26 --supports--> v16\n- v16 --refutes--> v14\n- v14 --refutes--> v15\n- v15 --supports--> v24\n- v24 --supports--> v23\n- v23 --supports--> v0e\n- v0e --refutes--> v29\n- v29 --supports--> v12\n- v12 --refutes--> v0d\n- v0d --refutes--> v27\n- v27 --supports--> v28\n- v28 --supports--> v10\n- v10 --supports--> v1b\n- v1b --supports--> v19\n- v19 --refutes--> v1e\n- v1e --supports--> v0f\n- v0f --refutes--> v09\n- v09 --refutes--> v13\n- v13 --supports--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v1f\n- v1f --refutes--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v1c\n- v1c --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v21\n- v05 --refutes--> v01\n- v09 --refutes--> v1d\n- v09 --supports--> v02\n- v0b --supports--> v18\n- v11 --refutes--> v22\n- v15 --refutes--> v0a\n- v1b --supports--> v08\n- v1f --refutes--> v04\n- v24 --supports--> v17\n- v25 --refutes--> v03\n- v25 --supports--> v06\n- v26 --supports--> v20\n- v26 --supports--> v2a\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":"334915413ea06cee2624b04223caec1dc69cf9b36dfbceffb3f38fc1145916b7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --supports--> v11\n- v11 --refutes--> v01\n- v01 --supports--> v29\n- v29 --supports--> v21\n- v21 --supports--> v24\n- v24 --supports--> v04\n- v04 --supports--> v14\n- v14 --supports--> v0c\n- v0c --refutes--> v15\n- v15 --refutes--> v19\n- v19 --refutes--> v0a\n- v0a --supports--> v1a\n- v1a --supports--> v06\n- v06 --refutes--> v23\n- v23 --refutes--> v02\n- v02 --supports--> v13\n- v13 --refutes--> v08\n- v08 --refutes--> v1c\n- v1c --refutes--> v09\n- v09 --refutes--> v16\n- v16 --refutes--> v05\n- v05 --supports--> v27\n- v27 --supports--> v00\n- v00 --supports--> v28\n- v28 --refutes--> v22\n- v22 --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v25\n- v03 --supports--> v0d\n- v07 --refutes--> v18\n- v07 --refutes--> v2a\n- v08 --refutes--> v10\n- v09 --refutes--> v20\n- v0c --refutes--> v1b\n- v12 --supports--> v1f\n- v14 --supports--> v26\n- v16 --refutes--> v1e\n- v1c --refutes--> v1d\n- v1c --supports--> v17\n- v23 --refutes--> v0e\n- v27 --supports--> v0b\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":"9e0b3f3b84c6154a3ad0676488b5448a94f478dbae74b00c1dd9bec9812027a8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v19\n- v19 --refutes--> v02\n- v02 --supports--> v1f\n- v1f --refutes--> v12\n- v12 --refutes--> v01\n- v01 --supports--> v16\n- v16 --supports--> v00\n- v00 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v27\n- v27 --refutes--> v25\n- v25 --refutes--> v1d\n- v1d --refutes--> v08\n- v08 --refutes--> v07\n- v07 --supports--> v21\n- v21 --supports--> v14\n- v14 --refutes--> v18\n- v18 --refutes--> v24\n- v24 --supports--> v2a\n- v2a --supports--> v1e\n- v1e --refutes--> v1b\n- v1b --refutes--> v10\n- v10 --supports--> v26\n- v26 --refutes--> v1c\n- v1c --supports--> v20\n- v20 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v00 --refutes--> v13\n- v00 --supports--> v28\n- v06 --supports--> v15\n- v0a --refutes--> v1a\n- v10 --supports--> v04\n- v11 --refutes--> v22\n- v16 --refutes--> v0d\n- v1d --supports--> v0b\n- v20 --refutes--> v09\n- v20 --refutes--> v29\n- v20 --supports--> v0e\n- v24 --refutes--> v17\n- v2a --supports--> v03\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":"20a35766dd7c14ae40dbd676de96ad0d10535e0d448b0264194ad122709992ee","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v24\n- v24 --refutes--> v1b\n- v1b --supports--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v21\n- v21 --supports--> v1c\n- v1c --refutes--> v10\n- v10 --refutes--> v02\n- v02 --refutes--> v17\n- v17 --refutes--> v26\n- v26 --refutes--> v00\n- v00 --supports--> v13\n- v13 --refutes--> v29\n- v29 --refutes--> v2a\n- v2a --supports--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v01\n- v01 --refutes--> v0d\n- v0d --refutes--> v28\n- v28 --supports--> v0f\n- v0f --refutes--> v14\n- v14 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v1a\n- v1a --supports--> v19\n- v19 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v20\n- v0a --refutes--> v0e\n- v0c --refutes--> v22\n- v0c --supports--> v23\n- v0f --refutes--> v1f\n- v13 --refutes--> v25\n- v14 --refutes--> v11\n- v17 --supports--> v18\n- v21 --refutes--> v1e\n- v26 --supports--> v1d\n- v27 --supports--> v08\n- v29 --supports--> v15\n- v2a --supports--> v03\n- v2a --supports--> v12\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":"1f7584e939f55030c6aeac725d4e8490e1c62c5e136a191f08cc059860208cea","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0a\n- v0a --supports--> v20\n- v20 --supports--> v09\n- v09 --supports--> v22\n- v22 --refutes--> v1b\n- v1b --refutes--> v1a\n- v1a --supports--> v28\n- v28 --supports--> v11\n- v11 --supports--> v24\n- v24 --supports--> v17\n- v17 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v15\n- v15 --supports--> v08\n- v08 --refutes--> v2a\n- v2a --refutes--> v0c\n- v0c --supports--> v05\n- v05 --refutes--> v18\n- v18 --supports--> v1e\n- v1e --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v10\n- v10 --supports--> v27\n- v27 --refutes--> v16\n- v16 --refutes--> v29\n- v29 --refutes--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v25\n- v09 --refutes--> v19\n- v09 --supports--> v21\n- v0d --supports--> v01\n- v10 --supports--> v03\n- v11 --supports--> v06\n- v1a --supports--> v1f\n- v20 --refutes--> v04\n- v20 --supports--> v07\n- v22 --refutes--> v14\n- v22 --supports--> v13\n- v27 --supports--> v0e\n- v27 --supports--> v23\n- v2a --supports--> v0b\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":"05eb608acf8443c968ea8e8d21b589344e048d8371c149ece413feca944a708c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v27\n- v27 --supports--> v28\n- v28 --supports--> v11\n- v11 --supports--> v14\n- v14 --refutes--> v23\n- v23 --supports--> v16\n- v16 --refutes--> v0c\n- v0c --supports--> v06\n- v06 --refutes--> v09\n- v09 --refutes--> v17\n- v17 --refutes--> v1e\n- v1e --supports--> v1d\n- v1d --refutes--> v1f\n- v1f --supports--> v22\n- v22 --supports--> v10\n- v10 --refutes--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v15\n- v15 --supports--> v21\n- v21 --supports--> v03\n- v03 --supports--> v25\n- v25 --supports--> v07\n- v07 --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --supports--> v02\n- v02 --supports--> v12\n- v12 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v13\n- v02 --refutes--> v18\n- v06 --supports--> v26\n- v09 --supports--> v20\n- v0e --refutes--> v29\n- v0e --supports--> v1b\n- v10 --supports--> v24\n- v11 --supports--> v0f\n- v14 --supports--> v01\n- v1d --supports--> v2a\n- v1f --refutes--> v08\n- v1f --supports--> v1c\n- v21 --refutes--> v19\n- v22 --refutes--> v0a\n\n[VERIFIED ROOT]\nv05 = TRUE\n\n[TARGET]\nv0b\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"c93d583736c006d5531e4f4b66ad224450671699e2db3ba6bfe58a72d74d2d71","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v1a\n- v1a --refutes--> v12\n- v12 --supports--> v15\n- v15 --refutes--> v0c\n- v0c --supports--> v0e\n- v0e --supports--> v1d\n- v1d --supports--> v10\n- v10 --supports--> v17\n- v17 --refutes--> v28\n- v28 --supports--> v07\n- v07 --supports--> v08\n- v08 --refutes--> v24\n- v24 --refutes--> v11\n- v11 --refutes--> v29\n- v29 --refutes--> v1b\n- v1b --refutes--> v16\n- v16 --supports--> v20\n- v20 --supports--> v14\n- v14 --refutes--> v13\n- v13 --supports--> v1c\n- v1c --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --refutes--> v22\n- v22 --supports--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v03\n- v03 --supports--> v0d\n- v0d --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v25\n- v08 --refutes--> v26\n- v0a --refutes--> v21\n- v0e --refutes--> v06\n- v0e --refutes--> v1f\n- v10 --refutes--> v0b\n- v11 --refutes--> v2a\n- v14 --refutes--> v01\n- v14 --refutes--> v09\n- v14 --supports--> v00\n- v1c --supports--> v19\n- v20 --refutes--> v18\n- v20 --supports--> v04\n- v23 --supports--> v27\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":"29ba597b9e5cb2c0e9870f0f612e10755535219d442f7abd4b01ae97662218f5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v03\n- v03 --supports--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v13\n- v13 --refutes--> v1e\n- v1e --supports--> v21\n- v21 --refutes--> v24\n- v24 --refutes--> v26\n- v26 --supports--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v1c\n- v1c --supports--> v0f\n- v0f --refutes--> v0e\n- v0e --supports--> v22\n- v22 --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v1d\n- v1d --supports--> v06\n- v06 --supports--> v1f\n- v1f --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --supports--> v2a\n- v2a --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v05\n- v05 --refutes--> v20\n- v20 --refutes--> v12\n- v12 --supports--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v28\n- v04 --refutes--> v29\n- v06 --refutes--> v16\n- v07 --refutes--> v18\n- v0b --supports--> v09\n- v0f --supports--> v25\n- v10 --refutes--> v15\n- v13 --supports--> v27\n- v17 --supports--> v01\n- v17 --supports--> v0c\n- v1a --refutes--> v23\n- v1d --refutes--> v02\n- v1d --refutes--> v19\n- v26 --supports--> v1b\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":"d85b4328669b0b6a116893c8def74743241b105bcbc566da195d70a776e198f1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v10\n- v10 --refutes--> v20\n- v20 --supports--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v0c\n- v0c --refutes--> v1e\n- v1e --refutes--> v00\n- v00 --refutes--> v0b\n- v0b --refutes--> v22\n- v22 --refutes--> v0d\n- v0d --supports--> v19\n- v19 --refutes--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v24\n- v24 --supports--> v29\n- v29 --supports--> v23\n- v23 --refutes--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v05\n- v05 --supports--> v18\n- v18 --refutes--> v06\n- v06 --refutes--> v26\n- v26 --supports--> v13\n- v13 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0a\n- v06 --supports--> v07\n- v0b --supports--> v01\n- v0e --refutes--> v08\n- v0e --supports--> v09\n- v10 --refutes--> v21\n- v10 --refutes--> v25\n- v11 --refutes--> v27\n- v11 --refutes--> v2a\n- v11 --supports--> v28\n- v16 --refutes--> v14\n- v16 --supports--> v1f\n- v1a --refutes--> v03\n- v1c --refutes--> v12\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":"26b4f4716d90a2aa89bad90d66a00df0ae27a6377e6447c2fcbded84237b1ab2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v17\n- v17 --refutes--> v08\n- v08 --supports--> v13\n- v13 --refutes--> v1c\n- v1c --supports--> v28\n- v28 --refutes--> v09\n- v09 --refutes--> v23\n- v23 --refutes--> v15\n- v15 --refutes--> v10\n- v10 --supports--> v21\n- v21 --supports--> v1a\n- v1a --supports--> v04\n- v04 --supports--> v05\n- v05 --supports--> v20\n- v20 --refutes--> v11\n- v11 --refutes--> v0e\n- v0e --refutes--> v27\n- v27 --supports--> v22\n- v22 --refutes--> v29\n- v29 --refutes--> v0f\n- v0f --supports--> v25\n- v25 --supports--> v2a\n- v2a --refutes--> v01\n- v01 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v1f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v12\n- v0b --refutes--> v18\n- v0e --refutes--> v24\n- v10 --refutes--> v26\n- v13 --supports--> v14\n- v15 --refutes--> v1e\n- v20 --supports--> v1b\n- v21 --refutes--> v0d\n- v21 --supports--> v19\n- v22 --refutes--> v07\n- v22 --supports--> v0a\n- v23 --refutes--> v1d\n- v23 --supports--> v06\n- v25 --refutes--> v0c\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":"41cb3f2c949d549d1bbf05f382dcf5e73a74d26d6ebd9a1f74859e7f163e4049","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v2a\n- v2a --supports--> v24\n- v24 --refutes--> v06\n- v06 --refutes--> v01\n- v01 --refutes--> v27\n- v27 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v1c\n- v1c --refutes--> v1b\n- v1b --refutes--> v1d\n- v1d --refutes--> v11\n- v11 --supports--> v19\n- v19 --supports--> v29\n- v29 --refutes--> v26\n- v26 --supports--> v15\n- v15 --supports--> v16\n- v16 --supports--> v18\n- v18 --supports--> v28\n- v28 --refutes--> v14\n- v14 --supports--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v13\n- v13 --refutes--> v1f\n- v1f --supports--> v0b\n- v0b --refutes--> v1e\n- v1e --supports--> v1a\n- v1a --refutes--> v08\n- v08 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v22\n- v06 --refutes--> v0c\n- v06 --supports--> v0d\n- v06 --supports--> v21\n- v11 --refutes--> v20\n- v13 --supports--> v25\n- v14 --supports--> v0e\n- v1c --refutes--> v02\n- v1e --refutes--> v07\n- v23 --refutes--> v03\n- v26 --refutes--> v04\n- v27 --refutes--> v05\n- v28 --supports--> v0a\n- v29 --supports--> v17\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":"c3fa6a8557d4a045b1a2fc9aeb86ba599975e66ce1bf4f0cce64e1653e4c693c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --refutes--> v01\n- v01 --supports--> v00\n- v00 --refutes--> v15\n- v15 --supports--> v1c\n- v1c --refutes--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v28\n- v28 --supports--> v1d\n- v1d --supports--> v22\n- v22 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v12\n- v12 --supports--> v14\n- v14 --refutes--> v2a\n- v2a --supports--> v13\n- v13 --refutes--> v0f\n- v0f --refutes--> v26\n- v26 --refutes--> v11\n- v11 --supports--> v1b\n- v1b --refutes--> v27\n- v27 --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v07\n- v07 --supports--> v20\n- v20 --refutes--> v17\n- v17 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v05\n- v0f --supports--> v09\n- v10 --refutes--> v02\n- v12 --refutes--> v23\n- v16 --refutes--> v19\n- v17 --supports--> v0e\n- v17 --supports--> v29\n- v1c --supports--> v1e\n- v20 --refutes--> v04\n- v20 --supports--> v1a\n- v20 --supports--> v21\n- v22 --supports--> v18\n- v26 --supports--> v25\n- v27 --supports--> v06\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":"7bb1066d1e242c9614365e472e0f0af5c8c1ddb9741a3668eb07f9a80063ad6b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v20\n- v20 --refutes--> v05\n- v05 --refutes--> v1b\n- v1b --refutes--> v17\n- v17 --supports--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v12\n- v12 --refutes--> v2a\n- v2a --refutes--> v26\n- v26 --supports--> v00\n- v00 --supports--> v24\n- v24 --supports--> v1f\n- v1f --refutes--> v10\n- v10 --refutes--> v27\n- v27 --refutes--> v28\n- v28 --refutes--> v0a\n- v0a --refutes--> v1e\n- v1e --refutes--> v1c\n- v1c --refutes--> v21\n- v21 --supports--> v0d\n- v0d --supports--> v22\n- v22 --supports--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v1d\n- v1d --refutes--> v15\n- v15 --supports--> v11\n- v11 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v13\n- v0b --supports--> v03\n- v10 --refutes--> v16\n- v12 --refutes--> v23\n- v15 --refutes--> v19\n- v15 --supports--> v02\n- v17 --supports--> v18\n- v1a --supports--> v0e\n- v1a --supports--> v29\n- v22 --refutes--> v25\n- v22 --supports--> v01\n- v26 --refutes--> v04\n- v27 --refutes--> v08\n- v2a --refutes--> v0f\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":"225ec58e77a2ff3b1bf873a5ec919afbd05ab8496ac666e7ff68953f18e22667","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v16\n- v16 --refutes--> v1f\n- v1f --refutes--> v08\n- v08 --supports--> v20\n- v20 --refutes--> v26\n- v26 --refutes--> v00\n- v00 --supports--> v25\n- v25 --supports--> v0b\n- v0b --supports--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v1d\n- v1d --refutes--> v22\n- v22 --refutes--> v15\n- v15 --refutes--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v06\n- v06 --refutes--> v0e\n- v0e --supports--> v2a\n- v2a --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --supports--> v0d\n- v0d --supports--> v1a\n- v1a --supports--> v11\n- v11 --supports--> v0f\n- v0f --refutes--> v1b\n- v1b --refutes--> v1c\n- v1c --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v19\n- v0b --refutes--> v0c\n- v0b --refutes--> v29\n- v11 --refutes--> v27\n- v11 --supports--> v12\n- v1b --refutes--> v17\n- v1f --refutes--> v02\n- v1f --refutes--> v18\n- v1f --refutes--> v28\n- v1f --supports--> v04\n- v20 --supports--> v14\n- v22 --refutes--> v09\n- v23 --supports--> v24\n- v26 --refutes--> v21\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":"1412cbf83e9e7403d198265996f040ecd2c1322cd559f45a18b89f4708b30553","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v12\n- v12 --refutes--> v18\n- v18 --supports--> v09\n- v09 --refutes--> v22\n- v22 --refutes--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v1b\n- v1b --refutes--> v28\n- v28 --supports--> v19\n- v19 --supports--> v05\n- v05 --supports--> v00\n- v00 --supports--> v27\n- v27 --supports--> v10\n- v10 --refutes--> v1c\n- v1c --refutes--> v0f\n- v0f --refutes--> v2a\n- v2a --supports--> v13\n- v13 --supports--> v01\n- v01 --supports--> v1e\n- v1e --refutes--> v15\n- v15 --refutes--> v25\n- v25 --refutes--> v1d\n- v1d --supports--> v14\n- v14 --supports--> v23\n- v23 --supports--> v17\n- v17 --refutes--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v08\n- v01 --refutes--> v0d\n- v04 --refutes--> v29\n- v05 --supports--> v0c\n- v06 --refutes--> v1f\n- v11 --refutes--> v0b\n- v11 --supports--> v21\n- v13 --refutes--> v03\n- v14 --supports--> v07\n- v15 --refutes--> v02\n- v1c --refutes--> v24\n- v23 --supports--> v20\n- v27 --supports--> v1a\n- v28 --supports--> v16\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":"896116814a5b4a36d48619a25c3a1ba581a4d268760646b40bf1b83c02f1fae6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v26\n- v26 --refutes--> v0e\n- v0e --supports--> v20\n- v20 --supports--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v24\n- v24 --refutes--> v0d\n- v0d --refutes--> v06\n- v06 --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --refutes--> v1c\n- v1c --refutes--> v17\n- v17 --refutes--> v11\n- v11 --refutes--> v1a\n- v1a --supports--> v05\n- v05 --refutes--> v15\n- v15 --refutes--> v21\n- v21 --supports--> v0f\n- v0f --refutes--> v16\n- v16 --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v25\n- v25 --supports--> v27\n- v27 --supports--> v14\n- v14 --supports--> v09\n- v09 --supports--> v03\n- v03 --refutes--> v2a\n- v2a --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v02 --supports--> v28\n- v03 --refutes--> v22\n- v07 --supports--> v10\n- v0d --supports--> v18\n- v0e --refutes--> v0b\n- v0e --refutes--> v29\n- v1c --refutes--> v1b\n- v1e --supports--> v04\n- v1e --supports--> v1f\n- v20 --refutes--> v0c\n- v21 --refutes--> v23\n- v21 --supports--> v19\n- v24 --supports--> v1d\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":"5531c9d98b911ed872cf5479d47d6b9095d3e7395df5a90af00dbb70e0b85830","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --supports--> v2a\n- v2a --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v26\n- v26 --supports--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v1b\n- v1b --supports--> v17\n- v17 --supports--> v28\n- v28 --refutes--> v0a\n- v0a --supports--> v1a\n- v1a --refutes--> v05\n- v05 --supports--> v0c\n- v0c --supports--> v10\n- v10 --refutes--> v14\n- v14 --supports--> v29\n- v29 --supports--> v03\n- v03 --supports--> v1f\n- v1f --refutes--> v20\n- v20 --supports--> v1d\n- v1d --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --refutes--> v00\n- v00 --refutes--> v13\n- v13 --supports--> v09\n- v09 --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v24\n- v03 --supports--> v19\n- v05 --refutes--> v22\n- v05 --supports--> v06\n- v0c --supports--> v21\n- v10 --refutes--> v0d\n- v12 --refutes--> v0f\n- v14 --refutes--> v07\n- v14 --supports--> v18\n- v1e --refutes--> v15\n- v1e --supports--> v27\n- v20 --supports--> v23\n- v2a --refutes--> v0b\n- v2a --refutes--> v25\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":"0fc9341cc35b36a0ba3097746d058a89dc22181e99592339c4f75b45e20825ef","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v22\n- v22 --supports--> v04\n- v04 --supports--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v19\n- v19 --refutes--> v02\n- v02 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v2a\n- v2a --refutes--> v20\n- v20 --supports--> v08\n- v08 --refutes--> v14\n- v14 --refutes--> v1e\n- v1e --refutes--> v13\n- v13 --supports--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v09\n- v09 --supports--> v26\n- v26 --supports--> v27\n- v27 --supports--> v0a\n- v0a --supports--> v15\n- v15 --supports--> v12\n- v12 --refutes--> v23\n- v23 --refutes--> v24\n- v24 --supports--> v16\n- v16 --supports--> v17\n- v17 --supports--> v06\n- v06 --supports--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1b\n- v01 --supports--> v29\n- v06 --refutes--> v10\n- v08 --refutes--> v0b\n- v09 --refutes--> v1a\n- v0e --refutes--> v07\n- v14 --supports--> v21\n- v16 --supports--> v18\n- v16 --supports--> v1f\n- v1c --refutes--> v0f\n- v1c --supports--> v05\n- v22 --refutes--> v1d\n- v23 --supports--> v28\n- v27 --supports--> v0d\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":"f43e7d19be03a343bf81570f5c0c17408b6f71831eeba576ae8572e48eef8bd6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --refutes--> v03\n- v03 --supports--> v0d\n- v0d --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v1f\n- v1f --supports--> v1c\n- v1c --refutes--> v19\n- v19 --refutes--> v1b\n- v1b --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v28\n- v28 --supports--> v1a\n- v1a --supports--> v12\n- v12 --refutes--> v22\n- v22 --refutes--> v17\n- v17 --supports--> v2a\n- v2a --supports--> v27\n- v27 --supports--> v25\n- v25 --supports--> v0e\n- v0e --refutes--> v18\n- v18 --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --refutes--> v26\n- v26 --supports--> v20\n- v20 --refutes--> v29\n- v29 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v16\n- v08 --supports--> v04\n- v0b --refutes--> v1d\n- v0d --refutes--> v06\n- v0f --supports--> v00\n- v17 --refutes--> v14\n- v19 --supports--> v15\n- v1a --refutes--> v21\n- v1e --refutes--> v0a\n- v20 --refutes--> v10\n- v20 --supports--> v11\n- v20 --supports--> v13\n- v26 --refutes--> v24\n- v29 --supports--> v09\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":"a36048e640756bbd2f4d2585799d242cdf3a2ee391657f6dba97418388e839ed","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v19\n- v19 --supports--> v16\n- v16 --refutes--> v1b\n- v1b --supports--> v05\n- v05 --supports--> v02\n- v02 --supports--> v0d\n- v0d --refutes--> v29\n- v29 --refutes--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v1d\n- v1d --refutes--> v27\n- v27 --refutes--> v26\n- v26 --supports--> v25\n- v25 --supports--> v28\n- v28 --supports--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v1a\n- v1a --supports--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v22\n- v22 --supports--> v1e\n- v1e --refutes--> v15\n- v15 --supports--> v07\n- v07 --supports--> v03\n- v03 --refutes--> v24\n- v24 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v09\n- v07 --supports--> v17\n- v0a --supports--> v00\n- v0c --refutes--> v0b\n- v0f --refutes--> v0e\n- v13 --supports--> v21\n- v14 --refutes--> v2a\n- v15 --refutes--> v1f\n- v16 --refutes--> v20\n- v19 --refutes--> v23\n- v1d --refutes--> v12\n- v25 --supports--> v10\n- v27 --refutes--> v01\n- v28 --supports--> v18\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":"8ea5b2f3cb48d9cbecd8c338602e6a878dfa9a3c277bb8dd15c01f7a733abbff","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v24\n- v24 --supports--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v21\n- v21 --refutes--> v1c\n- v1c --supports--> v23\n- v23 --supports--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v2a\n- v2a --supports--> v0e\n- v0e --supports--> v13\n- v13 --supports--> v1b\n- v1b --supports--> v00\n- v00 --refutes--> v29\n- v29 --refutes--> v20\n- v20 --refutes--> v17\n- v17 --refutes--> v06\n- v06 --refutes--> v1d\n- v1d --refutes--> v25\n- v25 --refutes--> v26\n- v26 --supports--> v10\n- v10 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v1a\n- v1a --supports--> v1f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v19\n- v0a --supports--> v03\n- v0a --supports--> v27\n- v0f --supports--> v22\n- v13 --supports--> v28\n- v14 --refutes--> v0c\n- v1b --supports--> v15\n- v1b --supports--> v1e\n- v1d --supports--> v02\n- v21 --supports--> v18\n- v26 --refutes--> v04\n- v26 --refutes--> v09\n- v29 --supports--> v16\n- v2a --refutes--> v07\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":"f4816dd1659309621f380d8cdb99b67c2f51821ef09021afc75305cae07fd4f4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v1f\n- v1f --supports--> v1d\n- v1d --refutes--> v19\n- v19 --refutes--> v17\n- v17 --supports--> v05\n- v05 --supports--> v20\n- v20 --supports--> v08\n- v08 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v24\n- v24 --refutes--> v21\n- v21 --supports--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v27\n- v27 --supports--> v29\n- v29 --refutes--> v15\n- v15 --refutes--> v1a\n- v1a --supports--> v23\n- v23 --supports--> v1c\n- v1c --refutes--> v0e\n- v0e --refutes--> v04\n- v04 --supports--> v01\n- v01 --supports--> v2a\n- v2a --refutes--> v14\n- v14 --supports--> v0c\n- v0c --refutes--> v18\n- v18 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v12\n- v03 --refutes--> v07\n- v03 --supports--> v26\n- v05 --refutes--> v06\n- v14 --supports--> v28\n- v17 --refutes--> v1e\n- v1f --refutes--> v1b\n- v1f --supports--> v10\n- v20 --supports--> v13\n- v20 --supports--> v25\n- v21 --supports--> v16\n- v27 --refutes--> v0a\n- v27 --refutes--> v0d\n- v29 --refutes--> v22\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":"4d8bc7618ca5f4e7f4e0b2b4b2d05c0aed6eb57793eab09c7bf5f5ddb0c476df","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --supports--> v1b\n- v1b --refutes--> v19\n- v19 --refutes--> v21\n- v21 --supports--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v16\n- v16 --refutes--> v18\n- v18 --supports--> v04\n- v04 --supports--> v14\n- v14 --supports--> v12\n- v12 --refutes--> v1c\n- v1c --refutes--> v02\n- v02 --refutes--> v25\n- v25 --refutes--> v0d\n- v0d --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v1d\n- v1d --refutes--> v22\n- v22 --supports--> v29\n- v29 --supports--> v1a\n- v1a --refutes--> v03\n- v03 --supports--> v11\n- v11 --supports--> v13\n- v13 --refutes--> v17\n- v17 --supports--> v10\n- v10 --refutes--> v15\n- v15 --supports--> v09\n- v09 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v23\n- v02 --supports--> v05\n- v09 --supports--> v0a\n- v13 --supports--> v0e\n- v16 --refutes--> v28\n- v17 --refutes--> v0f\n- v1a --refutes--> v00\n- v1a --supports--> v26\n- v1d --refutes--> v24\n- v1d --supports--> v0c\n- v1d --supports--> v2a\n- v25 --supports--> v20\n- v27 --refutes--> v1e\n- v29 --refutes--> v1f\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":"29585c27a0292441232b6bec1a5414a7daea7e41fe2cbe3b426a1ccf28fabc5b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --supports--> v28\n- v28 --supports--> v29\n- v29 --refutes--> v13\n- v13 --supports--> v03\n- v03 --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --supports--> v25\n- v25 --supports--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v04\n- v04 --supports--> v16\n- v16 --supports--> v1f\n- v1f --supports--> v1a\n- v1a --supports--> v09\n- v09 --supports--> v1c\n- v1c --refutes--> v06\n- v06 --refutes--> v15\n- v15 --supports--> v20\n- v20 --refutes--> v19\n- v19 --refutes--> v0d\n- v0d --supports--> v12\n- v12 --supports--> v0b\n- v0b --supports--> v0f\n- v0f --refutes--> v10\n- v10 --supports--> v1b\n- v1b --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v07\n- v0b --refutes--> v1e\n- v10 --supports--> v24\n- v11 --supports--> v01\n- v13 --supports--> v17\n- v15 --supports--> v14\n- v15 --supports--> v21\n- v1b --refutes--> v2a\n- v1d --supports--> v26\n- v1f --refutes--> v23\n- v20 --supports--> v22\n- v29 --refutes--> v18\n- v29 --supports--> v08\n- v29 --supports--> v0e\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":"c80a7c7fcccbc87aa73f44da1befa3bd91ced31f1cc4dd1b9ef681991ce14a12","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v06\n- v06 --supports--> v1b\n- v1b --refutes--> v14\n- v14 --supports--> v04\n- v04 --refutes--> v03\n- v03 --supports--> v02\n- v02 --supports--> v27\n- v27 --refutes--> v25\n- v25 --refutes--> v1f\n- v1f --supports--> v10\n- v10 --supports--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v07\n- v07 --supports--> v13\n- v13 --supports--> v18\n- v18 --refutes--> v12\n- v12 --refutes--> v01\n- v01 --refutes--> v1d\n- v1d --refutes--> v0f\n- v0f --refutes--> v09\n- v09 --supports--> v17\n- v17 --supports--> v0c\n- v0c --supports--> v29\n- v29 --supports--> v21\n- v21 --supports--> v19\n- v19 --supports--> v1a\n- v1a --supports--> v0e\n- v0e --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v2a\n- v06 --supports--> v28\n- v07 --refutes--> v05\n- v07 --refutes--> v1c\n- v09 --supports--> v26\n- v11 --supports--> v24\n- v15 --refutes--> v20\n- v17 --refutes--> v08\n- v17 --refutes--> v1e\n- v1a --refutes--> v00\n- v1b --refutes--> v0d\n- v1b --refutes--> v22\n- v27 --supports--> v16\n- v29 --supports--> v23\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":"41a17035e64d373060a732d4765324d8384fd1ffec5943ca37aa4df7ac69d4bf","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v0d\n- v0d --refutes--> v1c\n- v1c --refutes--> v25\n- v25 --supports--> v19\n- v19 --supports--> v0a\n- v0a --refutes--> v20\n- v20 --refutes--> v1b\n- v1b --supports--> v0f\n- v0f --supports--> v26\n- v26 --supports--> v01\n- v01 --supports--> v24\n- v24 --refutes--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v1f\n- v1f --refutes--> v16\n- v16 --refutes--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v22\n- v22 --refutes--> v08\n- v08 --supports--> v04\n- v04 --supports--> v2a\n- v2a --supports--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v27\n- v27 --supports--> v0c\n- v0c --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v02\n- v01 --refutes--> v0e\n- v01 --supports--> v18\n- v03 --supports--> v17\n- v05 --refutes--> v28\n- v06 --refutes--> v23\n- v0d --supports--> v1a\n- v10 --refutes--> v1d\n- v1f --supports--> v09\n- v20 --refutes--> v13\n- v22 --supports--> v15\n- v22 --supports--> v1e\n- v25 --refutes--> v29\n- v27 --refutes--> v21\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":"9772623a220e5f6f11f7ffa371670f4c39f88df26ef8e1f642b653c152391d09","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v0c\n- v0c --refutes--> v02\n- v02 --refutes--> v23\n- v23 --supports--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v18\n- v18 --refutes--> v29\n- v29 --refutes--> v05\n- v05 --refutes--> v20\n- v20 --refutes--> v1f\n- v1f --refutes--> v08\n- v08 --refutes--> v1c\n- v1c --supports--> v17\n- v17 --refutes--> v2a\n- v2a --refutes--> v01\n- v01 --refutes--> v04\n- v04 --refutes--> v25\n- v25 --supports--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v22\n- v22 --supports--> v03\n- v03 --refutes--> v16\n- v16 --refutes--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v27\n- v27 --refutes--> v13\n- v13 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v28\n- v03 --supports--> v0e\n- v10 --refutes--> v1e\n- v14 --refutes--> v06\n- v14 --supports--> v1d\n- v15 --supports--> v19\n- v15 --supports--> v21\n- v17 --refutes--> v00\n- v18 --refutes--> v0f\n- v1c --supports--> v1b\n- v23 --refutes--> v0b\n- v29 --refutes--> v24\n- v2a --refutes--> v26\n- v2a --supports--> v11\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":"9d366dd4bbb50750ce793820a82ccd16e7e9fda76173036427a9e5ad06c68b6b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v26\n- v26 --supports--> v15\n- v15 --supports--> v1e\n- v1e --refutes--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v20\n- v20 --supports--> v13\n- v13 --refutes--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v19\n- v19 --refutes--> v07\n- v07 --supports--> v1d\n- v1d --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v22\n- v22 --refutes--> v14\n- v14 --refutes--> v27\n- v27 --supports--> v0a\n- v0a --supports--> v25\n- v25 --supports--> v02\n- v02 --supports--> v00\n- v00 --supports--> v01\n- v01 --supports--> v16\n- v16 --supports--> v04\n- v04 --supports--> v21\n- v21 --supports--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v28\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v24\n- v04 --refutes--> v23\n- v05 --supports--> v1c\n- v07 --supports--> v03\n- v08 --refutes--> v06\n- v08 --refutes--> v17\n- v08 --supports--> v11\n- v0a --supports--> v10\n- v0a --supports--> v29\n- v16 --refutes--> v2a\n- v18 --supports--> v0b\n- v19 --supports--> v1f\n- v1d --refutes--> v09\n- v22 --refutes--> v12\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":"27446c2a9aa8db616eff6f7de9a9cf64e00b000141ced6d27ba0363b5e8c2be3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v25\n- v25 --refutes--> v29\n- v29 --refutes--> v12\n- v12 --supports--> v13\n- v13 --supports--> v18\n- v18 --supports--> v0c\n- v0c --supports--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v1d\n- v1d --refutes--> v24\n- v24 --refutes--> v0e\n- v0e --supports--> v01\n- v01 --refutes--> v1a\n- v1a --refutes--> v11\n- v11 --supports--> v16\n- v16 --refutes--> v0b\n- v0b --supports--> v00\n- v00 --refutes--> v09\n- v09 --supports--> v14\n- v14 --supports--> v23\n- v23 --refutes--> v1b\n- v1b --refutes--> v1e\n- v1e --refutes--> v27\n- v27 --refutes--> v28\n- v28 --supports--> v2a\n- v2a --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v21\n- v03 --supports--> v26\n- v0b --supports--> v20\n- v11 --supports--> v1f\n- v14 --supports--> v1c\n- v1b --refutes--> v22\n- v1b --supports--> v15\n- v1d --supports--> v0a\n- v1e --refutes--> v0d\n- v23 --supports--> v08\n- v24 --supports--> v07\n- v27 --refutes--> v19\n- v27 --supports--> v0f\n- v29 --supports--> v02\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":"beb1f9e02b609b71f9bd604cd644ec4d87fb202848bc9b077fa6bf9f29fff8ee","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --supports--> v21\n- v21 --refutes--> v10\n- v10 --supports--> v1a\n- v1a --supports--> v1d\n- v1d --refutes--> v01\n- v01 --supports--> v07\n- v07 --refutes--> v27\n- v27 --refutes--> v19\n- v19 --supports--> v15\n- v15 --refutes--> v23\n- v23 --refutes--> v1c\n- v1c --supports--> v20\n- v20 --refutes--> v22\n- v22 --refutes--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v03\n- v03 --refutes--> v13\n- v13 --supports--> v11\n- v11 --refutes--> v0c\n- v0c --refutes--> v2a\n- v2a --supports--> v29\n- v29 --refutes--> v14\n- v14 --supports--> v26\n- v26 --supports--> v0e\n- v0e --supports--> v1f\n- v1f --refutes--> v02\n- v02 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v1b\n- v02 --supports--> v28\n- v11 --refutes--> v1e\n- v13 --supports--> v18\n- v15 --refutes--> v08\n- v15 --supports--> v0a\n- v15 --supports--> v0d\n- v17 --supports--> v09\n- v1c --refutes--> v12\n- v1c --supports--> v0b\n- v1d --supports--> v06\n- v1f --refutes--> v16\n- v23 --supports--> v24\n- v27 --supports--> v0f\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":"a902a13ed468658a8ca424cd4454793aeb0c82f303ee54815f331c589b2b7792","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --supports--> v0b\n- v0b --refutes--> v0e\n- v0e --refutes--> v2a\n- v2a --refutes--> v20\n- v20 --refutes--> v16\n- v16 --supports--> v12\n- v12 --refutes--> v21\n- v21 --supports--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v10\n- v10 --supports--> v07\n- v07 --supports--> v24\n- v24 --supports--> v14\n- v14 --supports--> v1c\n- v1c --supports--> v1a\n- v1a --refutes--> v00\n- v00 --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v29\n- v29 --refutes--> v03\n- v03 --supports--> v18\n- v18 --refutes--> v23\n- v23 --refutes--> v25\n- v25 --refutes--> v22\n- v22 --supports--> v01\n- v01 --refutes--> v09\n- v09 --supports--> v11\n- v11 --supports--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v06 --supports--> v0f\n- v0b --supports--> v13\n- v0e --refutes--> v17\n- v0e --refutes--> v1b\n- v0e --refutes--> v1d\n- v0e --supports--> v05\n- v12 --supports--> v0c\n- v12 --supports--> v1f\n- v15 --refutes--> v1e\n- v19 --supports--> v0d\n- v19 --supports--> v28\n- v21 --supports--> v0a\n- v29 --refutes--> v27\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":"99da8a59e3e67c92f5fa495768b2e7390bf67447fd925f7bb968cb4fb2d6f920","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v21\n- v21 --supports--> v22\n- v22 --refutes--> v1a\n- v1a --refutes--> v2a\n- v2a --refutes--> v17\n- v17 --refutes--> v08\n- v08 --refutes--> v13\n- v13 --refutes--> v1b\n- v1b --refutes--> v0b\n- v0b --supports--> v1e\n- v1e --supports--> v16\n- v16 --refutes--> v25\n- v25 --supports--> v06\n- v06 --refutes--> v1f\n- v1f --supports--> v15\n- v15 --supports--> v00\n- v00 --supports--> v12\n- v12 --supports--> v28\n- v28 --refutes--> v14\n- v14 --supports--> v10\n- v10 --refutes--> v27\n- v27 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v05 --supports--> v1d\n- v0c --supports--> v01\n- v0d --supports--> v07\n- v0d --supports--> v0a\n- v0d --supports--> v0f\n- v12 --refutes--> v29\n- v13 --supports--> v24\n- v15 --supports--> v11\n- v1a --refutes--> v23\n- v1b --supports--> v20\n- v1f --refutes--> v18\n- v21 --supports--> v26\n- v28 --supports--> v1c\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":"d2cf89795b1c15b685b956d8b0a8571228c5224e061a2399d341ceb064e1f259","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2a --supports--> v28\n- v28 --refutes--> v1b\n- v1b --supports--> v20\n- v20 --supports--> v1e\n- v1e --supports--> v0e\n- v0e --supports--> v03\n- v03 --refutes--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v26\n- v26 --refutes--> v11\n- v11 --supports--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v29\n- v29 --supports--> v15\n- v15 --supports--> v14\n- v14 --refutes--> v1f\n- v1f --refutes--> v17\n- v17 --supports--> v1a\n- v1a --supports--> v0f\n- v0f --refutes--> v02\n- v02 --supports--> v21\n- v21 --supports--> v12\n- v12 --refutes--> v1d\n- v1d --refutes--> v07\n- v07 --supports--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v19\n- v04 --refutes--> v1c\n- v07 --refutes--> v10\n- v0b --supports--> v18\n- v0d --supports--> v24\n- v0e --supports--> v27\n- v0f --refutes--> v13\n- v12 --refutes--> v0a\n- v12 --supports--> v25\n- v14 --supports--> v06\n- v14 --supports--> v22\n- v15 --refutes--> v23\n- v1d --supports--> v00\n- v29 --refutes--> v08\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":"e0dadd5ed01e4baebe5efa9369ab12b35b78b642d358c12f81807e327fb84c80","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2a --supports--> v24\n- v24 --supports--> v06\n- v06 --refutes--> v13\n- v13 --refutes--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v0e\n- v0e --refutes--> v22\n- v22 --refutes--> v15\n- v15 --supports--> v05\n- v05 --supports--> v27\n- v27 --supports--> v20\n- v20 --supports--> v14\n- v14 --refutes--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --refutes--> v09\n- v09 --supports--> v1c\n- v1c --supports--> v0d\n- v0d --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v23\n- v23 --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v18\n- v18 --refutes--> v16\n- v16 --refutes--> v04\n- v04 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v21\n- v04 --supports--> v12\n- v04 --supports--> v25\n- v06 --refutes--> v10\n- v09 --refutes--> v0f\n- v09 --supports--> v19\n- v13 --refutes--> v28\n- v18 --refutes--> v29\n- v18 --supports--> v26\n- v1a --refutes--> v17\n- v1b --supports--> v0c\n- v24 --refutes--> v02\n- v27 --refutes--> v08\n- v27 --supports--> v1f\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":"7440e7ac3909294927281fce96cd90502fffe84c681b0e28eb46e73d4cdadd84","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --supports--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v25\n- v25 --refutes--> v03\n- v03 --refutes--> v23\n- v23 --supports--> v16\n- v16 --supports--> v24\n- v24 --refutes--> v1e\n- v1e --supports--> v0c\n- v0c --refutes--> v17\n- v17 --refutes--> v11\n- v11 --refutes--> v14\n- v14 --refutes--> v0d\n- v0d --refutes--> v1d\n- v1d --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v27\n- v27 --refutes--> v18\n- v18 --supports--> v1a\n- v1a --supports--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --refutes--> v22\n- v22 --supports--> v04\n- v04 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v05\n- v03 --refutes--> v2a\n- v06 --refutes--> v07\n- v0c --supports--> v1f\n- v0e --supports--> v10\n- v11 --supports--> v1c\n- v15 --refutes--> v21\n- v18 --supports--> v01\n- v1a --supports--> v0a\n- v1d --refutes--> v26\n- v22 --supports--> v28\n- v22 --supports--> v29\n- v24 --refutes--> v19\n- v27 --supports--> v02\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":"27d2faddfbf2b792210dcce8a8351a2f48b2b5dfdc341b550e897d0fb3278c12","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v04\n- v04 --refutes--> v26\n- v26 --supports--> v27\n- v27 --supports--> v0a\n- v0a --refutes--> v17\n- v17 --refutes--> v24\n- v24 --refutes--> v02\n- v02 --refutes--> v29\n- v29 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v1d\n- v1d --refutes--> v2a\n- v2a --refutes--> v00\n- v00 --supports--> v10\n- v10 --supports--> v1a\n- v1a --supports--> v05\n- v05 --supports--> v09\n- v09 --refutes--> v1b\n- v1b --refutes--> v01\n- v01 --refutes--> v20\n- v20 --refutes--> v0c\n- v0c --supports--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v21\n- v21 --refutes--> v18\n- v18 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0b\n- v01 --supports--> v23\n- v02 --refutes--> v0d\n- v04 --refutes--> v28\n- v07 --supports--> v1e\n- v08 --refutes--> v19\n- v0a --refutes--> v16\n- v10 --refutes--> v06\n- v18 --refutes--> v22\n- v1b --refutes--> v1f\n- v1b --supports--> v13\n- v1c --supports--> v25\n- v20 --supports--> v15\n- v24 --supports--> v14\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":"ede4030159411cb0433963938e3f59831c85e1cde4b23c8b2ea317d7534c2c7e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v19\n- v19 --refutes--> v26\n- v26 --supports--> v09\n- v09 --refutes--> v13\n- v13 --supports--> v24\n- v24 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --refutes--> v02\n- v02 --refutes--> v1f\n- v1f --refutes--> v1e\n- v1e --refutes--> v29\n- v29 --refutes--> v18\n- v18 --supports--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v21\n- v21 --refutes--> v0f\n- v0f --supports--> v23\n- v23 --refutes--> v08\n- v08 --refutes--> v2a\n- v2a --supports--> v04\n- v04 --supports--> v1d\n- v1d --supports--> v10\n- v10 --refutes--> v1c\n- v1c --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v14\n- v00 --supports--> v25\n- v0b --refutes--> v11\n- v0b --supports--> v05\n- v0d --refutes--> v03\n- v0e --refutes--> v0a\n- v0e --refutes--> v1b\n- v0e --supports--> v22\n- v15 --supports--> v28\n- v18 --supports--> v01\n- v18 --supports--> v16\n- v19 --supports--> v20\n- v1d --refutes--> v07\n- v21 --refutes--> v27\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":"d7b23182321b8f5fab1a933f48df1759e58bd8475f3cafc06664665e0df34d74","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v08\n- v08 --supports--> v13\n- v13 --refutes--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v16\n- v16 --refutes--> v19\n- v19 --refutes--> v29\n- v29 --supports--> v0d\n- v0d --supports--> v09\n- v09 --supports--> v11\n- v11 --supports--> v04\n- v04 --supports--> v25\n- v25 --supports--> v0e\n- v0e --refutes--> v14\n- v14 --supports--> v0c\n- v0c --supports--> v01\n- v01 --refutes--> v00\n- v00 --supports--> v21\n- v21 --refutes--> v20\n- v20 --supports--> v06\n- v06 --refutes--> v28\n- v28 --refutes--> v22\n- v22 --refutes--> v1d\n- v1d --supports--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v15\n- v15 --supports--> v24\n- v24 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v01 --refutes--> v1b\n- v04 --supports--> v26\n- v05 --supports--> v03\n- v05 --supports--> v10\n- v0d --refutes--> v0b\n- v11 --supports--> v07\n- v14 --supports--> v23\n- v16 --supports--> v1f\n- v19 --refutes--> v17\n- v19 --supports--> v27\n- v1d --refutes--> v18\n- v1e --supports--> v0f\n- v24 --refutes--> v2a\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":"469f7159474d0e4abeca1f5bb1d20a0502d8ac77cc410ed21fdcac1224e5f39d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v26 --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v01\n- v01 --refutes--> v27\n- v27 --refutes--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v21\n- v21 --refutes--> v24\n- v24 --supports--> v19\n- v19 --refutes--> v02\n- v02 --supports--> v1a\n- v1a --refutes--> v18\n- v18 --refutes--> v1e\n- v1e --refutes--> v08\n- v08 --refutes--> v1f\n- v1f --refutes--> v04\n- v04 --refutes--> v23\n- v23 --refutes--> v14\n- v14 --supports--> v28\n- v28 --supports--> v22\n- v22 --refutes--> v20\n- v20 --refutes--> v29\n- v29 --supports--> v03\n- v03 --supports--> v16\n- v16 --refutes--> v00\n- v00 --refutes--> v0e\n- v0e --supports--> v0f\n- v0f --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1c\n- v03 --supports--> v15\n- v04 --supports--> v0c\n- v0b --refutes--> v25\n- v0e --refutes--> v06\n- v0f --refutes--> v12\n- v0f --refutes--> v13\n- v14 --supports--> v09\n- v19 --refutes--> v2a\n- v20 --refutes--> v0d\n- v22 --supports--> v1b\n- v22 --supports--> v1d\n- v24 --supports--> v17\n- v28 --supports--> v0a\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":"2df5d3048d8355914931b0a5141003974f97fb19d36dd694fb456b4b6bd21f51","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v1d\n- v1d --refutes--> v28\n- v28 --supports--> v29\n- v29 --supports--> v02\n- v02 --supports--> v05\n- v05 --supports--> v16\n- v16 --supports--> v15\n- v15 --refutes--> v0f\n- v0f --supports--> v1e\n- v1e --refutes--> v17\n- v17 --refutes--> v2a\n- v2a --supports--> v1a\n- v1a --refutes--> v10\n- v10 --refutes--> v06\n- v06 --refutes--> v24\n- v24 --refutes--> v14\n- v14 --refutes--> v1c\n- v1c --supports--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v21\n- v21 --supports--> v04\n- v04 --supports--> v1f\n- v1f --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v27\n- v27 --refutes--> v0a\n- v0a --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v00 --supports--> v09\n- v00 --supports--> v23\n- v05 --refutes--> v18\n- v05 --refutes--> v22\n- v12 --refutes--> v26\n- v12 --supports--> v11\n- v16 --refutes--> v01\n- v16 --refutes--> v20\n- v17 --supports--> v1b\n- v1c --refutes--> v25\n- v1d --refutes--> v07\n- v1e --supports--> v13\n- v2a --refutes--> v08\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":"890dc85490697ff407dfcbcee229d0ee729b83784b2bfdfe64108816fb1760ee","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v03\n- v03 --refutes--> v1f\n- v1f --refutes--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v22\n- v22 --refutes--> v16\n- v16 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --refutes--> v17\n- v17 --supports--> v1e\n- v1e --refutes--> v29\n- v29 --supports--> v08\n- v08 --supports--> v01\n- v01 --refutes--> v13\n- v13 --supports--> v21\n- v21 --supports--> v10\n- v10 --supports--> v27\n- v27 --supports--> v0e\n- v0e --refutes--> v1b\n- v1b --supports--> v0a\n- v0a --supports--> v25\n- v25 --supports--> v26\n- v26 --supports--> v0c\n- v0c --refutes--> v18\n- v18 --supports--> v11\n- v11 --refutes--> v12\n- v12 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0b\n- v04 --refutes--> v20\n- v06 --refutes--> v2a\n- v08 --refutes--> v14\n- v0a --refutes--> v24\n- v0e --refutes--> v09\n- v16 --supports--> v19\n- v18 --refutes--> v02\n- v18 --supports--> v15\n- v21 --refutes--> v23\n- v21 --supports--> v0f\n- v26 --refutes--> v1a\n- v26 --supports--> v05\n- v27 --supports--> v28\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":"ddf53f4913842a401a55d24a1c6927966f277ec7cabae60b2b5ca8658b1256d5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --refutes--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v29\n- v29 --refutes--> v21\n- v21 --supports--> v23\n- v23 --supports--> v1e\n- v1e --supports--> v1a\n- v1a --refutes--> v25\n- v25 --refutes--> v24\n- v24 --refutes--> v0c\n- v0c --refutes--> v17\n- v17 --refutes--> v1f\n- v1f --supports--> v08\n- v08 --refutes--> v1c\n- v1c --refutes--> v2a\n- v2a --supports--> v06\n- v06 --refutes--> v0d\n- v0d --refutes--> v14\n- v14 --refutes--> v28\n- v28 --supports--> v02\n- v02 --supports--> v20\n- v20 --refutes--> v22\n- v22 --refutes--> v10\n- v10 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v16\n- v0c --supports--> v0a\n- v0f --supports--> v13\n- v10 --refutes--> v0e\n- v11 --refutes--> v19\n- v1b --supports--> v03\n- v1c --supports--> v01\n- v1c --supports--> v26\n- v1e --refutes--> v07\n- v24 --supports--> v15\n- v25 --refutes--> v00\n- v25 --refutes--> v18\n- v27 --refutes--> v09\n- v2a --supports--> v12\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":"60c393994c4472b53045b725b698c73e67f7a187cf0b52282f21e4663e809563","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --supports--> v1f\n- v1f --supports--> v06\n- v06 --refutes--> v21\n- v21 --supports--> v1c\n- v1c --supports--> v07\n- v07 --refutes--> v2a\n- v2a --refutes--> v0e\n- v0e --supports--> v27\n- v27 --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v22\n- v22 --refutes--> v26\n- v26 --refutes--> v14\n- v14 --supports--> v10\n- v10 --supports--> v20\n- v20 --refutes--> v0b\n- v0b --supports--> v23\n- v23 --supports--> v16\n- v16 --supports--> v0f\n- v0f --supports--> v1a\n- v1a --supports--> v1b\n- v1b --refutes--> v24\n- v24 --refutes--> v01\n- v01 --refutes--> v15\n- v15 --supports--> v1d\n- v1d --supports--> v19\n- v19 --supports--> v17\n- v17 --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v12\n- v01 --supports--> v13\n- v06 --refutes--> v29\n- v07 --supports--> v0d\n- v0b --refutes--> v02\n- v0b --refutes--> v11\n- v0e --refutes--> v28\n- v10 --supports--> v09\n- v14 --refutes--> v08\n- v17 --supports--> v04\n- v17 --supports--> v0a\n- v1c --supports--> v03\n- v1f --supports--> v05\n- v26 --supports--> v18\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":"e00ec400d21e03699eba27b28bb636c14ed3928f099d4980b984121d175a2c14","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v11\n- v11 --refutes--> v19\n- v19 --refutes--> v10\n- v10 --refutes--> v07\n- v07 --supports--> v1b\n- v1b --supports--> v06\n- v06 --refutes--> v29\n- v29 --refutes--> v0a\n- v0a --supports--> v1e\n- v1e --supports--> v1d\n- v1d --supports--> v13\n- v13 --supports--> v09\n- v09 --supports--> v00\n- v00 --refutes--> v21\n- v21 --supports--> v24\n- v24 --supports--> v0b\n- v0b --refutes--> v26\n- v26 --refutes--> v17\n- v17 --refutes--> v15\n- v15 --supports--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v0c\n- v0c --supports--> v05\n- v05 --supports--> v1f\n- v1f --refutes--> v16\n- v16 --supports--> v18\n- v18 --supports--> v23\n- v23 --refutes--> v28\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v22\n- v07 --refutes--> v27\n- v07 --supports--> v2a\n- v09 --refutes--> v25\n- v11 --supports--> v14\n- v17 --refutes--> v20\n- v17 --supports--> v0e\n- v17 --supports--> v12\n- v1a --refutes--> v0f\n- v1b --supports--> v02\n- v1d --supports--> v01\n- v1e --refutes--> v04\n- v1e --supports--> v08\n- v23 --refutes--> v1c\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":"3a1dab79298552b6eb4928370f19829ece6367132ec69bc27605cdd389060855","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v17\n- v17 --supports--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v1a\n- v1a --supports--> v18\n- v18 --supports--> v0f\n- v0f --supports--> v19\n- v19 --supports--> v1d\n- v1d --supports--> v20\n- v20 --refutes--> v01\n- v01 --supports--> v29\n- v29 --supports--> v15\n- v15 --supports--> v1c\n- v1c --refutes--> v1b\n- v1b --refutes--> v16\n- v16 --refutes--> v23\n- v23 --supports--> v28\n- v28 --supports--> v0d\n- v0d --supports--> v21\n- v21 --supports--> v06\n- v06 --supports--> v24\n- v24 --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v2a\n- v2a --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v10\n- v09 --supports--> v25\n- v0a --refutes--> v1f\n- v0a --supports--> v02\n- v0a --supports--> v07\n- v0d --supports--> v13\n- v0d --supports--> v27\n- v16 --refutes--> v14\n- v19 --refutes--> v00\n- v1d --supports--> v11\n- v1d --supports--> v26\n- v1e --refutes--> v12\n- v21 --refutes--> v0e\n- v28 --refutes--> v04\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":"b6955ef388ebd348636e951ee713e35fb2dc4c9932cedcfb80ac3ef4b0837f0c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v26\n- v26 --refutes--> v23\n- v23 --refutes--> v09\n- v09 --supports--> v27\n- v27 --supports--> v05\n- v05 --refutes--> v1f\n- v1f --supports--> v22\n- v22 --refutes--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v0c\n- v0c --refutes--> v28\n- v28 --refutes--> v14\n- v14 --refutes--> v29\n- v29 --supports--> v04\n- v04 --supports--> v0d\n- v0d --supports--> v17\n- v17 --supports--> v0f\n- v0f --refutes--> v08\n- v08 --supports--> v07\n- v07 --refutes--> v15\n- v15 --supports--> v00\n- v00 --supports--> v2a\n- v2a --refutes--> v21\n- v21 --supports--> v06\n- v06 --supports--> v18\n- v18 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v00 --supports--> v25\n- v04 --refutes--> v02\n- v07 --supports--> v12\n- v08 --supports--> v0e\n- v08 --supports--> v24\n- v0a --refutes--> v19\n- v0b --refutes--> v1d\n- v0d --refutes--> v10\n- v1a --refutes--> v1e\n- v1f --refutes--> v11\n- v26 --supports--> v20\n- v28 --refutes--> v1b\n- v29 --refutes--> v16\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":"92fbe4c49bf10b1a56a7fdddefc88216fb8fc6966b711834a00544833148bb74","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v13\n- v13 --refutes--> v12\n- v12 --refutes--> v1f\n- v1f --refutes--> v02\n- v02 --supports--> v24\n- v24 --refutes--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v29\n- v29 --supports--> v17\n- v17 --refutes--> v00\n- v00 --supports--> v20\n- v20 --refutes--> v21\n- v21 --supports--> v15\n- v15 --refutes--> v2a\n- v2a --refutes--> v1a\n- v1a --refutes--> v01\n- v01 --refutes--> v26\n- v26 --supports--> v28\n- v28 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --refutes--> v27\n- v27 --refutes--> v07\n- v07 --supports--> v19\n- v19 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0a\n- v01 --supports--> v10\n- v02 --supports--> v22\n- v03 --refutes--> v0b\n- v05 --supports--> v25\n- v07 --refutes--> v1b\n- v17 --supports--> v0f\n- v19 --refutes--> v0d\n- v19 --supports--> v08\n- v1c --supports--> v1e\n- v20 --refutes--> v14\n- v24 --supports--> v06\n- v24 --supports--> v18\n- v29 --supports--> v11\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":"45a802a2ebef4bc0e866247b884ae4f166e58dea7fe3480bc9d9fcc132e134ad","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v04\n- v04 --supports--> v23\n- v23 --supports--> v0f\n- v0f --supports--> v12\n- v12 --refutes--> v06\n- v06 --refutes--> v17\n- v17 --supports--> v24\n- v24 --refutes--> v28\n- v28 --refutes--> v16\n- v16 --supports--> v20\n- v20 --supports--> v1f\n- v1f --supports--> v21\n- v21 --refutes--> v15\n- v15 --supports--> v0a\n- v0a --supports--> v18\n- v18 --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --supports--> v02\n- v02 --supports--> v05\n- v05 --refutes--> v22\n- v22 --supports--> v03\n- v03 --supports--> v13\n- v13 --refutes--> v14\n- v14 --refutes--> v10\n- v10 --refutes--> v1d\n- v1d --refutes--> v0b\n- v0b --refutes--> v25\n- v25 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v07\n- v0a --refutes--> v0e\n- v0a --refutes--> v27\n- v13 --refutes--> v1b\n- v13 --refutes--> v26\n- v14 --refutes--> v2a\n- v15 --supports--> v1e\n- v16 --supports--> v1c\n- v1a --refutes--> v00\n- v1a --supports--> v29\n- v1f --refutes--> v11\n- v1f --supports--> v19\n- v21 --refutes--> v08\n- v25 --supports--> v0d\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":"8ee1cd1e423af02cc8c133f4060a590c3c8d1081b4c0d6d2bd70d4318ed5e0e5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v18\n- v18 --refutes--> v28\n- v28 --supports--> v01\n- v01 --refutes--> v1c\n- v1c --supports--> v09\n- v09 --refutes--> v02\n- v02 --supports--> v17\n- v17 --refutes--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v05\n- v05 --supports--> v24\n- v24 --refutes--> v1a\n- v1a --refutes--> v07\n- v07 --supports--> v22\n- v22 --refutes--> v19\n- v19 --supports--> v29\n- v29 --refutes--> v14\n- v14 --supports--> v26\n- v26 --refutes--> v10\n- v10 --supports--> v20\n- v20 --refutes--> v1e\n- v1e --supports--> v04\n- v04 --refutes--> v1b\n- v1b --refutes--> v12\n- v12 --refutes--> v25\n- v25 --supports--> v0f\n- v0f --refutes--> v23\n- v23 --refutes--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v08\n- v05 --supports--> v16\n- v0f --refutes--> v15\n- v10 --refutes--> v27\n- v12 --supports--> v1d\n- v17 --refutes--> v2a\n- v19 --refutes--> v0e\n- v19 --supports--> v0d\n- v1e --refutes--> v1f\n- v1e --supports--> v21\n- v22 --supports--> v06\n- v23 --supports--> v00\n- v24 --refutes--> v13\n- v29 --refutes--> v0a\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":"a1e50c3a443ddad0dafafcc900917be794e8462dfeca6dc4a7c5d1f19a7991c6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v20\n- v20 --refutes--> v1e\n- v1e --supports--> v08\n- v08 --supports--> v29\n- v29 --supports--> v00\n- v00 --supports--> v04\n- v04 --refutes--> v1c\n- v1c --supports--> v25\n- v25 --refutes--> v28\n- v28 --refutes--> v0c\n- v0c --supports--> v26\n- v26 --refutes--> v22\n- v22 --supports--> v14\n- v14 --refutes--> v01\n- v01 --refutes--> v24\n- v24 --refutes--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v23\n- v23 --refutes--> v10\n- v10 --supports--> v1b\n- v1b --supports--> v1f\n- v1f --supports--> v17\n- v17 --refutes--> v05\n- v05 --refutes--> v0f\n- v0f --supports--> v0a\n- v0a --refutes--> v12\n- v12 --supports--> v1a\n- v1a --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v09\n- v0b --refutes--> v2a\n- v0b --supports--> v18\n- v0b --supports--> v27\n- v0f --refutes--> v0e\n- v0f --refutes--> v13\n- v10 --supports--> v11\n- v15 --supports--> v07\n- v1b --supports--> v03\n- v1c --supports--> v06\n- v1f --refutes--> v1d\n- v25 --supports--> v21\n- v28 --supports--> v16\n- v29 --supports--> v02\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":"4db5dd6c468e20236efbb4e913dd0a845a77322476738e9236f7d63620eb5f3e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --supports--> v09\n- v09 --supports--> v1d\n- v1d --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --refutes--> v16\n- v16 --refutes--> v10\n- v10 --supports--> v0a\n- v0a --supports--> v0b\n- v0b --refutes--> v1a\n- v1a --supports--> v29\n- v29 --supports--> v18\n- v18 --supports--> v1f\n- v1f --supports--> v1c\n- v1c --refutes--> v20\n- v20 --refutes--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v11\n- v11 --refutes--> v12\n- v12 --refutes--> v15\n- v15 --supports--> v04\n- v04 --refutes--> v26\n- v26 --supports--> v23\n- v23 --supports--> v19\n- v19 --supports--> v28\n- v28 --refutes--> v2a\n- v2a --refutes--> v0e\n- v0e --supports--> v21\n- v21 --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v1e\n- v0d --refutes--> v02\n- v12 --refutes--> v03\n- v12 --supports--> v0c\n- v14 --supports--> v00\n- v16 --refutes--> v06\n- v19 --refutes--> v05\n- v1d --refutes--> v1b\n- v1d --refutes--> v24\n- v1d --supports--> v25\n- v1f --supports--> v27\n- v29 --refutes--> v22\n- v2a --refutes--> v08\n- v2a --supports--> v01\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":"9060fbef5d1615f14a11981eec6040c33957c636ca8b59185ca821de454bd486","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v02\n- v02 --supports--> v23\n- v23 --refutes--> v10\n- v10 --refutes--> v17\n- v17 --supports--> v0d\n- v0d --supports--> v16\n- v16 --refutes--> v03\n- v03 --refutes--> v1d\n- v1d --refutes--> v29\n- v29 --refutes--> v1c\n- v1c --supports--> v04\n- v04 --supports--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v24\n- v24 --refutes--> v09\n- v09 --refutes--> v20\n- v20 --supports--> v26\n- v26 --refutes--> v0b\n- v0b --refutes--> v2a\n- v2a --refutes--> v1a\n- v1a --refutes--> v1e\n- v1e --supports--> v18\n- v18 --refutes--> v1f\n- v1f --supports--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v28\n- v28 --refutes--> v05\n- v05 --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v02 --refutes--> v12\n- v05 --supports--> v19\n- v07 --supports--> v25\n- v09 --refutes--> v01\n- v09 --refutes--> v27\n- v0c --refutes--> v11\n- v0f --supports--> v0a\n- v1f --supports--> v08\n- v23 --supports--> v13\n- v23 --supports--> v1b\n- v26 --refutes--> v14\n- v26 --refutes--> v21\n- v29 --refutes--> v22\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":"cc6d14ae534a1950dfa48644b2b061667607d15d7114ede155acb690bbcd1083","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v04\n- v04 --supports--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v26\n- v26 --refutes--> v27\n- v27 --refutes--> v02\n- v02 --refutes--> v06\n- v06 --refutes--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v03\n- v03 --refutes--> v1a\n- v1a --refutes--> v1f\n- v1f --supports--> v23\n- v23 --supports--> v08\n- v08 --supports--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v0d\n- v0d --refutes--> v29\n- v29 --supports--> v25\n- v25 --supports--> v28\n- v28 --refutes--> v19\n- v19 --refutes--> v14\n- v14 --refutes--> v18\n- v18 --refutes--> v07\n- v07 --refutes--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v15\n- v15 --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v1d\n- v0c --refutes--> v10\n- v0c --supports--> v22\n- v13 --supports--> v24\n- v14 --supports--> v01\n- v15 --supports--> v05\n- v16 --supports--> v20\n- v18 --supports--> v0a\n- v19 --supports--> v1b\n- v1f --refutes--> v00\n- v27 --refutes--> v11\n- v28 --refutes--> v1e\n- v28 --supports--> v21\n- v28 --supports--> v2a\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":"87eed803035dba91cb46a621291b85c5c867c27f2b432e84847135e17e66a7f0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v26 --refutes--> v14\n- v14 --refutes--> v21\n- v21 --refutes--> v0e\n- v0e --refutes--> v1e\n- v1e --supports--> v20\n- v20 --supports--> v22\n- v22 --supports--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v2a\n- v2a --refutes--> v0c\n- v0c --supports--> v25\n- v25 --refutes--> v11\n- v11 --supports--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v23\n- v23 --refutes--> v19\n- v19 --refutes--> v24\n- v24 --supports--> v18\n- v18 --refutes--> v01\n- v01 --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v29\n- v29 --supports--> v13\n- v13 --supports--> v03\n- v03 --supports--> v27\n- v27 --supports--> v1c\n- v1c --supports--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v02 --refutes--> v1d\n- v05 --refutes--> v07\n- v09 --supports--> v15\n- v0e --refutes--> v06\n- v11 --supports--> v10\n- v11 --supports--> v1f\n- v18 --refutes--> v16\n- v19 --supports--> v17\n- v24 --refutes--> v1a\n- v24 --refutes--> v28\n- v24 --supports--> v0b\n- v26 --refutes--> v08\n- v29 --supports--> v0d\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":"4297a3fdfce15ef3bb4e644ff52b66fad51be4ec6b42d5d7209986073bbfff82","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v01 --supports--> v11\n- v11 --supports--> v04\n- v04 --supports--> v1b\n- v1b --refutes--> v03\n- v03 --refutes--> v1c\n- v1c --supports--> v12\n- v12 --refutes--> v02\n- v02 --supports--> v19\n- v19 --supports--> v0c\n- v0c --refutes--> v25\n- v25 --supports--> v1d\n- v1d --supports--> v1e\n- v1e --refutes--> v17\n- v17 --supports--> v24\n- v24 --supports--> v05\n- v05 --supports--> v18\n- v18 --refutes--> v00\n- v00 --supports--> v21\n- v21 --supports--> v26\n- v26 --refutes--> v0f\n- v0f --refutes--> v10\n- v10 --refutes--> v20\n- v20 --supports--> v15\n- v15 --supports--> v27\n- v27 --supports--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v1f\n- v1f --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v08\n- v03 --supports--> v0a\n- v04 --supports--> v23\n- v05 --refutes--> v2a\n- v05 --supports--> v0d\n- v0f --supports--> v29\n- v10 --refutes--> v1a\n- v19 --refutes--> v28\n- v1b --refutes--> v22\n- v1d --refutes--> v06\n- v1e --supports--> v16\n- v20 --refutes--> v07\n- v25 --supports--> v09\n- v26 --refutes--> v0b\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":"c78c1d6e065d19006c497a2b6f0724576faf501d2060721c1284a0348a34cd2a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v04 --refutes--> v10\n- v10 --supports--> v2a\n- v2a --supports--> v0c\n- v0c --refutes--> v07\n- v07 --refutes--> v21\n- v21 --supports--> v29\n- v29 --supports--> v12\n- v12 --supports--> v01\n- v01 --supports--> v1e\n- v1e --supports--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --supports--> v08\n- v08 --refutes--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v20\n- v20 --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v22\n- v22 --supports--> v1c\n- v1c --refutes--> v0a\n- v0a --refutes--> v28\n- v28 --refutes--> v25\n- v25 --refutes--> v1d\n- v1d --supports--> v14\n- v14 --refutes--> v05\n- v05 --supports--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v18\n- v05 --supports--> v1a\n- v07 --supports--> v03\n- v0a --refutes--> v1f\n- v0c --supports--> v26\n- v0f --supports--> v06\n- v10 --supports--> v0d\n- v11 --supports--> v23\n- v14 --refutes--> v00\n- v19 --refutes--> v0e\n- v25 --supports--> v24\n- v28 --supports--> v17\n- v29 --refutes--> v0b\n- v2a --supports--> v1b\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":"a7215b54707039c796bf2eb253b0cca3e9118b031cc3dfe43446626a23320851","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v26\n- v26 --refutes--> v1d\n- v1d --supports--> v21\n- v21 --supports--> v10\n- v10 --supports--> v25\n- v25 --supports--> v00\n- v00 --supports--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v29\n- v29 --supports--> v08\n- v08 --supports--> v15\n- v15 --supports--> v1c\n- v1c --refutes--> v01\n- v01 --refutes--> v19\n- v19 --supports--> v05\n- v05 --supports--> v1a\n- v1a --supports--> v0c\n- v0c --supports--> v1f\n- v1f --refutes--> v27\n- v27 --refutes--> v1e\n- v1e --refutes--> v18\n- v18 --refutes--> v16\n- v16 --supports--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v22\n- v22 --refutes--> v24\n- v24 --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v17\n- v0c --refutes--> v23\n- v0e --refutes--> v20\n- v0e --refutes--> v2a\n- v10 --supports--> v06\n- v16 --refutes--> v04\n- v18 --refutes--> v03\n- v1c --refutes--> v14\n- v1e --supports--> v1b\n- v1e --supports--> v28\n- v21 --supports--> v07\n- v22 --supports--> v11\n- v25 --refutes--> v12\n- v25 --supports--> v0d\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":"72e1850356301ed077b7d50130ac7382bb8cb69238e978e0c3904e2d5427fbd7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v07\n- v07 --supports--> v26\n- v26 --supports--> v1e\n- v1e --refutes--> v06\n- v06 --supports--> v1b\n- v1b --supports--> v09\n- v09 --supports--> v25\n- v25 --supports--> v24\n- v24 --refutes--> v22\n- v22 --supports--> v10\n- v10 --refutes--> v05\n- v05 --refutes--> v2a\n- v2a --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v13\n- v13 --refutes--> v1d\n- v1d --refutes--> v29\n- v29 --supports--> v17\n- v17 --supports--> v04\n- v04 --supports--> v08\n- v08 --supports--> v12\n- v12 --supports--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v16\n- v16 --refutes--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v18\n- v04 --refutes--> v00\n- v06 --supports--> v23\n- v0d --supports--> v02\n- v0d --supports--> v1f\n- v0e --refutes--> v15\n- v0f --supports--> v0c\n- v14 --supports--> v0b\n- v16 --refutes--> v0a\n- v16 --refutes--> v1a\n- v16 --supports--> v20\n- v19 --refutes--> v21\n- v1e --refutes--> v01\n- v25 --refutes--> v28\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":"f063c0cc4a89da58782d5b69e24afd38cab34f37e93413a67deae324c4d0c73f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v12\n- v12 --refutes--> v1d\n- v1d --supports--> v24\n- v24 --supports--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v23\n- v23 --refutes--> v13\n- v13 --supports--> v18\n- v18 --refutes--> v0f\n- v0f --supports--> v2a\n- v2a --supports--> v1b\n- v1b --supports--> v27\n- v27 --supports--> v0a\n- v0a --supports--> v11\n- v11 --supports--> v29\n- v29 --supports--> v17\n- v17 --supports--> v26\n- v26 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v19\n- v19 --supports--> v1c\n- v1c --supports--> v20\n- v20 --refutes--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v09 --refutes--> v15\n- v0a --refutes--> v21\n- v0d --refutes--> v1e\n- v0f --supports--> v07\n- v10 --supports--> v0e\n- v10 --supports--> v1f\n- v17 --supports--> v1a\n- v1c --supports--> v22\n- v1d --refutes--> v25\n- v24 --refutes--> v28\n- v26 --refutes--> v00\n- v27 --refutes--> v0c\n- v2a --refutes--> v01\n- v2a --supports--> v14\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":"0fe7ae2e7feb02f18b7d481b8edf95d9b84103c90cedc89a5bbebbc9922e9389","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v17\n- v17 --refutes--> v18\n- v18 --supports--> v08\n- v08 --refutes--> v03\n- v03 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --supports--> v1d\n- v1d --refutes--> v20\n- v20 --supports--> v01\n- v01 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v2a\n- v2a --refutes--> v24\n- v24 --supports--> v12\n- v12 --refutes--> v21\n- v21 --supports--> v19\n- v19 --supports--> v28\n- v28 --refutes--> v15\n- v15 --supports--> v27\n- v27 --supports--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v00\n- v00 --refutes--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v29\n- v01 --refutes--> v22\n- v02 --supports--> v10\n- v05 --supports--> v26\n- v0b --refutes--> v1e\n- v0b --supports--> v1c\n- v0d --refutes--> v07\n- v0e --supports--> v11\n- v12 --supports--> v25\n- v13 --refutes--> v1f\n- v17 --refutes--> v0a\n- v17 --supports--> v04\n- v1a --supports--> v16\n- v1d --refutes--> v0f\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":"aacc459cc13f0eb6913890d62b5f33b9653ad849b07b9d169f202390d8261656","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v21 --supports--> v0a\n- v0a --refutes--> v03\n- v03 --supports--> v17\n- v17 --supports--> v04\n- v04 --refutes--> v15\n- v15 --supports--> v22\n- v22 --refutes--> v06\n- v06 --supports--> v01\n- v01 --refutes--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v02\n- v02 --refutes--> v24\n- v24 --supports--> v27\n- v27 --supports--> v29\n- v29 --supports--> v11\n- v11 --refutes--> v16\n- v16 --refutes--> v28\n- v28 --supports--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v1c\n- v1c --supports--> v12\n- v12 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v1f\n- v1f --refutes--> v20\n- v20 --refutes--> v26\n- v26 --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0c\n- v02 --refutes--> v1e\n- v06 --refutes--> v18\n- v08 --supports--> v10\n- v0d --refutes--> v0b\n- v0e --supports--> v07\n- v0e --supports--> v23\n- v11 --refutes--> v1d\n- v14 --refutes--> v2a\n- v17 --supports--> v00\n- v20 --refutes--> v05\n- v22 --refutes--> v19\n- v22 --supports--> v1a\n- v28 --refutes--> v1b\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":"13e0c2525dc01679b336c99034067b8079707ddbe299ce8773c8f6d7357bc21a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v13\n- v13 --refutes--> v16\n- v16 --refutes--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v21\n- v21 --refutes--> v12\n- v12 --refutes--> v02\n- v02 --supports--> v0c\n- v0c --refutes--> v1a\n- v1a --refutes--> v10\n- v10 --refutes--> v27\n- v27 --supports--> v1d\n- v1d --refutes--> v20\n- v20 --refutes--> v14\n- v14 --supports--> v0f\n- v0f --supports--> v0a\n- v0a --supports--> v1b\n- v1b --supports--> v1f\n- v1f --refutes--> v28\n- v28 --supports--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v22\n- v22 --supports--> v03\n- v03 --supports--> v17\n- v17 --refutes--> v19\n- v19 --refutes--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v26\n- v04 --supports--> v01\n- v06 --refutes--> v0b\n- v06 --supports--> v25\n- v0a --refutes--> v11\n- v0a --supports--> v2a\n- v0f --supports--> v1c\n- v10 --refutes--> v05\n- v16 --refutes--> v08\n- v19 --supports--> v29\n- v1f --refutes--> v07\n- v20 --refutes--> v18\n- v21 --refutes--> v00\n- v22 --supports--> v24\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":"2e6ca80e048dbec073cf84a48b464ae7138de9c333d51ec9538938842a0a7a45","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v10\n- v10 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v01\n- v01 --refutes--> v17\n- v17 --supports--> v06\n- v06 --refutes--> v08\n- v08 --supports--> v22\n- v22 --refutes--> v1b\n- v1b --refutes--> v1d\n- v1d --supports--> v07\n- v07 --supports--> v0a\n- v0a --supports--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v25\n- v25 --refutes--> v12\n- v12 --supports--> v26\n- v26 --refutes--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v15\n- v15 --supports--> v27\n- v27 --refutes--> v18\n- v18 --supports--> v28\n- v28 --supports--> v19\n- v19 --refutes--> v11\n- v11 --refutes--> v24\n- v24 --refutes--> v09\n- v09 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v1c\n- v03 --supports--> v1f\n- v06 --refutes--> v23\n- v06 --supports--> v2a\n- v0e --refutes--> v0b\n- v0e --refutes--> v13\n- v0e --refutes--> v1e\n- v0e --refutes--> v21\n- v12 --refutes--> v20\n- v19 --supports--> v0f\n- v25 --refutes--> v1a\n- v25 --refutes--> v29\n- v28 --refutes--> v16\n- v28 --supports--> v0c\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":"19bb4e50e0d8e9f4ca56e2dd9e0c0f8f8524ad3715d398cf76e94ecd253534f8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v01\n- v01 --supports--> v27\n- v27 --supports--> v28\n- v28 --supports--> v1a\n- v1a --supports--> v02\n- v02 --supports--> v13\n- v13 --supports--> v0b\n- v0b --refutes--> v03\n- v03 --supports--> v29\n- v29 --supports--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v1d\n- v1d --refutes--> v21\n- v21 --refutes--> v11\n- v11 --refutes--> v20\n- v20 --supports--> v04\n- v04 --refutes--> v1b\n- v1b --supports--> v2a\n- v2a --refutes--> v1e\n- v1e --refutes--> v17\n- v17 --supports--> v25\n- v25 --supports--> v0e\n- v0e --refutes--> v22\n- v22 --refutes--> v19\n- v19 --refutes--> v18\n- v18 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v1f\n- v0b --refutes--> v06\n- v0b --supports--> v07\n- v0e --refutes--> v0f\n- v19 --refutes--> v05\n- v20 --refutes--> v26\n- v21 --supports--> v08\n- v22 --refutes--> v0d\n- v27 --supports--> v0a\n- v28 --supports--> v15\n- v29 --refutes--> v00\n- v2a --refutes--> v14\n- v2a --refutes--> v1c\n- v2a --supports--> v24\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":"a6175c32ba147ed9f50a052261840ba4b634909a19fd2b5d8ef37e4fe859f62d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v0e\n- v0e --supports--> v2a\n- v2a --refutes--> v0d\n- v0d --refutes--> v23\n- v23 --refutes--> v1d\n- v1d --supports--> v20\n- v20 --supports--> v1c\n- v1c --refutes--> v02\n- v02 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v29\n- v29 --supports--> v17\n- v17 --refutes--> v08\n- v08 --refutes--> v1f\n- v1f --refutes--> v07\n- v07 --refutes--> v05\n- v05 --refutes--> v00\n- v00 --supports--> v24\n- v24 --supports--> v11\n- v11 --supports--> v15\n- v15 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v25\n- v25 --supports--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0f\n- v01 --supports--> v28\n- v02 --refutes--> v19\n- v03 --refutes--> v26\n- v03 --supports--> v21\n- v07 --supports--> v09\n- v0d --supports--> v27\n- v11 --refutes--> v12\n- v13 --refutes--> v14\n- v13 --supports--> v18\n- v15 --supports--> v1b\n- v16 --supports--> v10\n- v1f --refutes--> v1e\n- v24 --refutes--> v0b\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":"cbf6ce3c070fd01305349b9a39be397008f92b81b0f21f4311eff856538fe50d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v0e\n- v0e --refutes--> v28\n- v28 --refutes--> v0c\n- v0c --refutes--> v22\n- v22 --refutes--> v23\n- v23 --supports--> v0f\n- v0f --supports--> v01\n- v01 --refutes--> v16\n- v16 --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v02\n- v02 --supports--> v19\n- v19 --supports--> v12\n- v12 --supports--> v05\n- v05 --refutes--> v26\n- v26 --supports--> v2a\n- v2a --refutes--> v1e\n- v1e --refutes--> v14\n- v14 --refutes--> v1b\n- v1b --supports--> v24\n- v24 --refutes--> v1f\n- v1f --refutes--> v20\n- v20 --supports--> v00\n- v00 --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0d\n- v01 --supports--> v29\n- v09 --supports--> v13\n- v0b --refutes--> v17\n- v0c --refutes--> v07\n- v11 --refutes--> v27\n- v14 --refutes--> v08\n- v14 --refutes--> v1d\n- v15 --refutes--> v03\n- v1f --refutes--> v25\n- v20 --supports--> v10\n- v20 --supports--> v18\n- v23 --supports--> v21\n- v28 --refutes--> v06\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":"8b5ca2511d550e3b2ba493d393d753c182798576a5c1ce8c11733ff9d7b4b1ca","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v28\n- v28 --supports--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v02\n- v02 --supports--> v0f\n- v0f --refutes--> v21\n- v21 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --refutes--> v1b\n- v1b --refutes--> v04\n- v04 --supports--> v1a\n- v1a --refutes--> v1d\n- v1d --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v10\n- v10 --supports--> v23\n- v23 --refutes--> v1c\n- v1c --supports--> v25\n- v25 --refutes--> v2a\n- v2a --supports--> v15\n- v15 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v06\n- v06 --supports--> v1f\n- v1f --refutes--> v12\n- v12 --refutes--> v16\n- v16 --supports--> v03\n- v03 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v29\n- v08 --supports--> v0e\n- v09 --supports--> v05\n- v0b --refutes--> v0d\n- v0b --refutes--> v13\n- v0b --supports--> v11\n- v0c --refutes--> v20\n- v0c --supports--> v22\n- v12 --refutes--> v26\n- v12 --supports--> v17\n- v15 --refutes--> v00\n- v16 --refutes--> v27\n- v23 --refutes--> v1e\n- v2a --refutes--> v24\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":"1441bb5a1eb077af1d6388a4e46e529ca0ddb1619f40c330ef75fbb45b8a66f7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v07\n- v07 --supports--> v1a\n- v1a --supports--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v1c\n- v1c --refutes--> v10\n- v10 --supports--> v0c\n- v0c --supports--> v0e\n- v0e --refutes--> v1f\n- v1f --refutes--> v04\n- v04 --refutes--> v16\n- v16 --refutes--> v00\n- v00 --refutes--> v22\n- v22 --refutes--> v18\n- v18 --refutes--> v0f\n- v0f --supports--> v17\n- v17 --supports--> v09\n- v09 --supports--> v21\n- v21 --supports--> v11\n- v11 --refutes--> v05\n- v05 --refutes--> v25\n- v25 --supports--> v14\n- v14 --supports--> v1d\n- v1d --supports--> v01\n- v01 --refutes--> v06\n- v06 --supports--> v24\n- v24 --refutes--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v29\n- v06 --refutes--> v08\n- v09 --supports--> v27\n- v0c --refutes--> v15\n- v0e --refutes--> v20\n- v10 --supports--> v26\n- v14 --refutes--> v1b\n- v16 --supports--> v1e\n- v21 --refutes--> v02\n- v23 --refutes--> v0b\n- v23 --supports--> v2a\n- v24 --refutes--> v13\n- v24 --supports--> v03\n- v24 --supports--> v28\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":"b24f12580db896463dfce3a3f0e45c9dd255c69942e6556edbbe02d53d1ade62","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v1f\n- v1f --refutes--> v26\n- v26 --supports--> v10\n- v10 --refutes--> v17\n- v17 --supports--> v2a\n- v2a --supports--> v07\n- v07 --supports--> v19\n- v19 --refutes--> v1b\n- v1b --refutes--> v22\n- v22 --supports--> v16\n- v16 --supports--> v0f\n- v0f --supports--> v01\n- v01 --supports--> v29\n- v29 --supports--> v27\n- v27 --refutes--> v0b\n- v0b --refutes--> v1a\n- v1a --supports--> v1d\n- v1d --refutes--> v21\n- v21 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --supports--> v13\n- v13 --supports--> v11\n- v11 --refutes--> v00\n- v00 --refutes--> v0a\n- v0a --supports--> v20\n- v20 --refutes--> v0d\n- v0d --supports--> v06\n- v06 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v18\n- v09 --supports--> v0c\n- v0d --refutes--> v02\n- v0d --refutes--> v03\n- v11 --refutes--> v15\n- v14 --supports--> v24\n- v17 --supports--> v12\n- v1d --refutes--> v1e\n- v1d --supports--> v05\n- v22 --refutes--> v04\n- v27 --refutes--> v0e\n- v27 --refutes--> v25\n- v2a --refutes--> v1c\n- v2a --supports--> v28\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":"d8c426c90b78e0d1232a8a37096cd9e202273beb846c9804c3ca477969ba9ec6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --refutes--> v26\n- v26 --refutes--> v04\n- v04 --supports--> v07\n- v07 --supports--> v0b\n- v0b --refutes--> v24\n- v24 --refutes--> v1f\n- v1f --supports--> v0e\n- v0e --supports--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v2a\n- v2a --supports--> v1e\n- v1e --supports--> v19\n- v19 --refutes--> v06\n- v06 --supports--> v10\n- v10 --supports--> v29\n- v29 --refutes--> v15\n- v15 --refutes--> v28\n- v28 --refutes--> v0f\n- v0f --refutes--> v01\n- v01 --refutes--> v27\n- v27 --supports--> v0d\n- v0d --refutes--> v11\n- v11 --supports--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v13\n- v13 --refutes--> v1d\n- v1d --refutes--> v20\n- v20 --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v18\n- v06 --refutes--> v1a\n- v06 --refutes--> v1b\n- v09 --refutes--> v17\n- v0b --supports--> v12\n- v0d --refutes--> v16\n- v10 --supports--> v0a\n- v11 --supports--> v21\n- v15 --refutes--> v14\n- v19 --refutes--> v08\n- v1d --supports--> v25\n- v1e --supports--> v23\n- v22 --supports--> v02\n- v26 --supports--> v1c\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":"18522ac18e41a8662de6ecca63d1a020fd02fceee2ca79c789d31062deeae3cd","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v28 --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v14\n- v14 --supports--> v11\n- v11 --supports--> v22\n- v22 --refutes--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v25\n- v25 --refutes--> v08\n- v08 --supports--> v12\n- v12 --supports--> v06\n- v06 --supports--> v01\n- v01 --supports--> v29\n- v29 --refutes--> v0c\n- v0c --supports--> v27\n- v27 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v1c\n- v1c --supports--> v26\n- v26 --refutes--> v05\n- v05 --refutes--> v18\n- v18 --refutes--> v0b\n- v0b --supports--> v1d\n- v1d --supports--> v23\n- v23 --supports--> v0d\n- v0d --refutes--> v13\n- v13 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v20\n- v03 --refutes--> v1e\n- v06 --supports--> v02\n- v0b --supports--> v0a\n- v0b --supports--> v15\n- v0d --supports--> v1f\n- v12 --supports--> v19\n- v14 --refutes--> v07\n- v1a --refutes--> v24\n- v22 --refutes--> v16\n- v22 --supports--> v2a\n- v27 --supports--> v0f\n- v27 --supports--> v1b\n- v28 --refutes--> v21\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":"7d7415c25f385251185044eb37f23ee79758870dbcdaedac196fc8257ce12c81","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v00\n- v00 --supports--> v2a\n- v2a --supports--> v19\n- v19 --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v20\n- v20 --supports--> v13\n- v13 --supports--> v15\n- v15 --refutes--> v07\n- v07 --supports--> v1b\n- v1b --supports--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v1e\n- v1e --refutes--> v05\n- v05 --refutes--> v26\n- v26 --refutes--> v29\n- v29 --refutes--> v10\n- v10 --supports--> v1a\n- v1a --refutes--> v22\n- v22 --supports--> v25\n- v25 --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v23\n- v23 --supports--> v17\n- v17 --refutes--> v16\n- v16 --supports--> v08\n- v08 --supports--> v1c\n- v1c --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v04 --refutes--> v14\n- v05 --refutes--> v0d\n- v10 --refutes--> v1f\n- v15 --refutes--> v09\n- v17 --supports--> v24\n- v19 --refutes--> v27\n- v19 --refutes--> v28\n- v22 --refutes--> v18\n- v23 --refutes--> v0e\n- v25 --refutes--> v03\n- v25 --refutes--> v1d\n- v25 --supports--> v21\n- v26 --refutes--> v02\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":"3b0294812820c47676dc5a90f4878bcf03248111423906a1f74361e735541fc4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v16\n- v16 --refutes--> v0f\n- v0f --refutes--> v17\n- v17 --supports--> v08\n- v08 --supports--> v21\n- v21 --refutes--> v18\n- v18 --supports--> v1a\n- v1a --refutes--> v28\n- v28 --supports--> v1e\n- v1e --supports--> v14\n- v14 --supports--> v23\n- v23 --refutes--> v02\n- v02 --supports--> v1c\n- v1c --refutes--> v0a\n- v0a --refutes--> v29\n- v29 --supports--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v06\n- v06 --supports--> v0c\n- v0c --supports--> v22\n- v22 --refutes--> v11\n- v11 --supports--> v0d\n- v0d --refutes--> v27\n- v27 --refutes--> v1b\n- v1b --refutes--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v0e\n- v0e --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v19\n- v05 --supports--> v20\n- v06 --supports--> v24\n- v08 --refutes--> v26\n- v08 --supports--> v10\n- v0a --supports--> v2a\n- v0e --refutes--> v1f\n- v11 --supports--> v00\n- v16 --supports--> v12\n- v18 --refutes--> v09\n- v1d --supports--> v15\n- v1e --refutes--> v13\n- v1e --supports--> v03\n- v23 --supports--> v01\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":"2dacd1908c4fee6d88e7c5caa56fcd137fce9af666bc4d75fcd76095e180ea39","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --supports--> v0c\n- v0c --supports--> v16\n- v16 --refutes--> v29\n- v29 --refutes--> v08\n- v08 --refutes--> v10\n- v10 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --supports--> v04\n- v04 --refutes--> v14\n- v14 --refutes--> v18\n- v18 --supports--> v19\n- v19 --supports--> v2a\n- v2a --supports--> v0d\n- v0d --refutes--> v13\n- v13 --supports--> v22\n- v22 --refutes--> v07\n- v07 --supports--> v00\n- v00 --supports--> v09\n- v09 --supports--> v21\n- v21 --refutes--> v1c\n- v1c --supports--> v02\n- v02 --supports--> v06\n- v06 --supports--> v17\n- v17 --supports--> v20\n- v20 --supports--> v0e\n- v0e --refutes--> v25\n- v25 --refutes--> v05\n- v05 --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v15\n- v05 --refutes--> v27\n- v09 --refutes--> v03\n- v09 --refutes--> v12\n- v09 --refutes--> v28\n- v09 --supports--> v1b\n- v0b --refutes--> v01\n- v0d --supports--> v0f\n- v13 --supports--> v0a\n- v13 --supports--> v11\n- v19 --supports--> v1e\n- v1c --supports--> v24\n- v20 --refutes--> v26\n- v2a --refutes--> v1d\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":"b27696308c1af6b66936201c0900f052836b3bdadca398dab65e25f1db102704","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --refutes--> v05\n- v05 --refutes--> v23\n- v23 --supports--> v21\n- v21 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v28\n- v28 --supports--> v17\n- v17 --refutes--> v1f\n- v1f --refutes--> v22\n- v22 --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v27\n- v27 --refutes--> v06\n- v06 --refutes--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v14\n- v14 --refutes--> v03\n- v03 --refutes--> v2a\n- v2a --refutes--> v18\n- v18 --supports--> v02\n- v02 --supports--> v1a\n- v1a --refutes--> v15\n- v15 --supports--> v0e\n- v0e --supports--> v26\n- v26 --refutes--> v16\n- v16 --supports--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v04\n- v0a --refutes--> v1e\n- v0b --refutes--> v25\n- v11 --refutes--> v1b\n- v15 --refutes--> v07\n- v16 --refutes--> v1c\n- v17 --refutes--> v29\n- v1a --supports--> v09\n- v22 --supports--> v0c\n- v23 --supports--> v1d\n- v26 --refutes--> v10\n- v27 --refutes--> v12\n- v28 --supports--> v0d\n- v28 --supports--> v20\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":"4887b4a674fd18afa7d65260ab597bf70ce81e059e14be39a1c7041597156ce8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --supports--> v1d\n- v1d --refutes--> v02\n- v02 --supports--> v14\n- v14 --refutes--> v12\n- v12 --refutes--> v11\n- v11 --refutes--> v21\n- v21 --supports--> v03\n- v03 --refutes--> v24\n- v24 --supports--> v29\n- v29 --supports--> v0e\n- v0e --refutes--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v2a\n- v2a --supports--> v0c\n- v0c --refutes--> v23\n- v23 --supports--> v00\n- v00 --supports--> v1c\n- v1c --supports--> v22\n- v22 --supports--> v07\n- v07 --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v28\n- v28 --refutes--> v16\n- v16 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v13\n- v13 --supports--> v26\n- v26 --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v09\n- v04 --supports--> v08\n- v07 --refutes--> v01\n- v11 --supports--> v27\n- v12 --supports--> v0a\n- v13 --supports--> v1a\n- v14 --supports--> v0f\n- v16 --refutes--> v19\n- v16 --supports--> v0d\n- v1d --refutes--> v15\n- v24 --supports--> v1e\n- v26 --refutes--> v18\n- v28 --supports--> v1b\n- v2a --refutes--> v25\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":"6e6988209476e7dd788dcc7435a93c8c294fbeaaf67bff7508f7cb9afda9bb15","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v14\n- v14 --refutes--> v03\n- v03 --refutes--> v2a\n- v2a --supports--> v13\n- v13 --refutes--> v1b\n- v1b --refutes--> v28\n- v28 --refutes--> v0c\n- v0c --refutes--> v23\n- v23 --refutes--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v17\n- v17 --supports--> v1c\n- v1c --refutes--> v05\n- v05 --supports--> v22\n- v22 --supports--> v20\n- v20 --refutes--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v1a\n- v1a --supports--> v1f\n- v1f --refutes--> v04\n- v04 --supports--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v24\n- v24 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v19\n- v00 --refutes--> v25\n- v01 --supports--> v1e\n- v02 --refutes--> v0d\n- v04 --refutes--> v29\n- v07 --supports--> v0b\n- v0f --refutes--> v26\n- v0f --supports--> v10\n- v17 --supports--> v1d\n- v1a --refutes--> v11\n- v1c --supports--> v18\n- v1f --supports--> v21\n- v24 --refutes--> v08\n- v28 --supports--> v15\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":"1ad44f27ba0f83c0fea1f55c204b0abe46b9811830a2629af07186db80ed2b57","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v0a\n- v0a --refutes--> v00\n- v00 --refutes--> v29\n- v29 --supports--> v13\n- v13 --refutes--> v20\n- v20 --refutes--> v24\n- v24 --supports--> v14\n- v14 --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v25\n- v25 --refutes--> v09\n- v09 --refutes--> v0f\n- v0f --supports--> v1f\n- v1f --supports--> v21\n- v21 --supports--> v12\n- v12 --refutes--> v0e\n- v0e --supports--> v15\n- v15 --refutes--> v18\n- v18 --refutes--> v1b\n- v1b --refutes--> v22\n- v22 --refutes--> v1e\n- v1e --supports--> v0c\n- v0c --supports--> v03\n- v03 --refutes--> v10\n- v10 --supports--> v17\n- v17 --refutes--> v23\n- v23 --supports--> v27\n- v27 --refutes--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v06\n- v03 --supports--> v04\n- v0e --refutes--> v26\n- v0f --refutes--> v1a\n- v10 --supports--> v11\n- v12 --refutes--> v2a\n- v15 --supports--> v28\n- v17 --refutes--> v01\n- v17 --supports--> v02\n- v1c --supports--> v1d\n- v21 --refutes--> v0d\n- v23 --supports--> v07\n- v27 --supports--> v08\n- v29 --supports--> v19\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":"ed3236f13e48ee6758d6da64e9abd32afab615fd5d3515b9dd5ef3b9a32f9cdc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v24\n- v24 --refutes--> v03\n- v03 --supports--> v1e\n- v1e --supports--> v23\n- v23 --refutes--> v08\n- v08 --supports--> v16\n- v16 --supports--> v20\n- v20 --supports--> v22\n- v22 --supports--> v1f\n- v1f --supports--> v19\n- v19 --supports--> v21\n- v21 --supports--> v25\n- v25 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v2a\n- v2a --refutes--> v0c\n- v0c --refutes--> v15\n- v15 --refutes--> v1d\n- v1d --supports--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v27\n- v27 --refutes--> v28\n- v28 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0b\n- v03 --supports--> v29\n- v05 --supports--> v01\n- v0a --supports--> v10\n- v0c --refutes--> v0f\n- v0d --supports--> v02\n- v14 --supports--> v1a\n- v16 --supports--> v13\n- v16 --supports--> v17\n- v19 --supports--> v26\n- v1b --supports--> v1c\n- v20 --refutes--> v12\n- v22 --refutes--> v06\n- v28 --supports--> v0e\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":"3d8eae38fd155e14171ccbdad26ede62a07fabe0d567e98101c9612c05f4b065","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --supports--> v17\n- v17 --supports--> v2a\n- v2a --refutes--> v23\n- v23 --supports--> v1d\n- v1d --refutes--> v12\n- v12 --refutes--> v19\n- v19 --supports--> v22\n- v22 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v0d\n- v0d --supports--> v1c\n- v1c --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v1f\n- v1f --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v26\n- v26 --refutes--> v0e\n- v0e --supports--> v14\n- v14 --supports--> v27\n- v27 --supports--> v1a\n- v1a --supports--> v05\n- v05 --refutes--> v21\n- v21 --refutes--> v1e\n- v1e --refutes--> v20\n- v20 --supports--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v07\n- v04 --supports--> v25\n- v06 --refutes--> v24\n- v0d --refutes--> v16\n- v0d --supports--> v15\n- v17 --supports--> v18\n- v17 --supports--> v1b\n- v1e --refutes--> v28\n- v21 --supports--> v29\n- v22 --supports--> v10\n- v26 --refutes--> v03\n- v26 --refutes--> v08\n- v26 --supports--> v01\n- v2a --supports--> v0c\n\n[VERIFIED ROOT]\nv06 = FALSE\n\n[TARGET]\nv11\n\n[QUERY]\nPropagate truth from the verified root to the target. Output exactly TRUE or FALSE.","role":"user"}],"prompt_hash":"d4f754db7845586668252f39860d42d3cfc06e89a60cc813c345f018706e3455","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v09\n- v09 --refutes--> v19\n- v19 --supports--> v1c\n- v1c --refutes--> v24\n- v24 --supports--> v13\n- v13 --refutes--> v1b\n- v1b --refutes--> v07\n- v07 --supports--> v0f\n- v0f --supports--> v23\n- v23 --supports--> v05\n- v05 --supports--> v27\n- v27 --refutes--> v16\n- v16 --supports--> v20\n- v20 --supports--> v0d\n- v0d --supports--> v22\n- v22 --supports--> v11\n- v11 --refutes--> v00\n- v00 --supports--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v0c\n- v0c --supports--> v2a\n- v2a --supports--> v18\n- v18 --supports--> v1f\n- v1f --refutes--> v08\n- v08 --supports--> v03\n- v03 --refutes--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0b\n- v00 --refutes--> v14\n- v05 --refutes--> v0e\n- v07 --supports--> v25\n- v07 --supports--> v29\n- v0c --supports--> v04\n- v16 --refutes--> v02\n- v16 --refutes--> v26\n- v18 --supports--> v10\n- v1b --supports--> v28\n- v24 --supports--> v06\n- v24 --supports--> v1a\n- v27 --refutes--> v21\n- v2a --refutes--> v1d\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":"f3649bc1025cb55a1e90bcb3edc4937edb7a06b70cdf6722e2dbe458c2d8d593","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v09\n- v09 --supports--> v2a\n- v2a --refutes--> v28\n- v28 --refutes--> v19\n- v19 --refutes--> v0f\n- v0f --refutes--> v04\n- v04 --supports--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v1f\n- v1f --refutes--> v23\n- v23 --refutes--> v21\n- v21 --supports--> v27\n- v27 --refutes--> v1e\n- v1e --supports--> v25\n- v25 --refutes--> v06\n- v06 --refutes--> v22\n- v22 --supports--> v0d\n- v0d --refutes--> v0b\n- v0b --supports--> v15\n- v15 --refutes--> v26\n- v26 --refutes--> v11\n- v11 --refutes--> v00\n- v00 --supports--> v12\n- v12 --refutes--> v05\n- v05 --refutes--> v03\n- v03 --supports--> v13\n- v13 --refutes--> v24\n- v24 --refutes--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v14\n- v03 --supports--> v20\n- v04 --supports--> v18\n- v05 --supports--> v1c\n- v0b --refutes--> v1a\n- v0d --supports--> v07\n- v0f --supports--> v08\n- v19 --refutes--> v29\n- v1d --refutes--> v0e\n- v1e --refutes--> v0a\n- v23 --refutes--> v02\n- v23 --refutes--> v10\n- v24 --supports--> v16\n- v25 --refutes--> v1b\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":"866de0f3cfa1522cb6c52d4a0e9b43a6c00a4862ca6d8d33607a46454e0413a5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v22\n- v22 --supports--> v00\n- v00 --supports--> v27\n- v27 --refutes--> v1f\n- v1f --refutes--> v17\n- v17 --refutes--> v24\n- v24 --supports--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v1b\n- v1b --supports--> v01\n- v01 --supports--> v2a\n- v2a --supports--> v13\n- v13 --supports--> v1e\n- v1e --refutes--> v11\n- v11 --supports--> v23\n- v23 --refutes--> v0b\n- v0b --refutes--> v1d\n- v1d --refutes--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v18\n- v18 --refutes--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v1a\n- v1a --supports--> v29\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v05 --supports--> v21\n- v0a --supports--> v19\n- v13 --refutes--> v04\n- v16 --supports--> v09\n- v17 --supports--> v07\n- v18 --refutes--> v25\n- v1a --refutes--> v0c\n- v1d --supports--> v28\n- v1f --refutes--> v26\n- v24 --refutes--> v10\n- v27 --refutes--> v14\n- v27 --supports--> v02\n- v2a --supports--> v20\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":"bd06cec79de8ca1a1f98df904bc00d3831de284a28a916c6581ff4705f725008","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --refutes--> v26\n- v26 --supports--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v22\n- v22 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v0f\n- v0f --refutes--> v28\n- v28 --supports--> v19\n- v19 --supports--> v1d\n- v1d --supports--> v25\n- v25 --supports--> v04\n- v04 --refutes--> v27\n- v27 --supports--> v18\n- v18 --refutes--> v23\n- v23 --refutes--> v08\n- v08 --refutes--> v05\n- v05 --refutes--> v2a\n- v2a --supports--> v1e\n- v1e --refutes--> v1b\n- v1b --refutes--> v1c\n- v1c --refutes--> v21\n- v21 --refutes--> v01\n- v01 --supports--> v20\n- v20 --refutes--> v15\n- v15 --supports--> v0c\n- v0c --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v29\n- v05 --refutes--> v12\n- v06 --refutes--> v16\n- v0c --supports--> v13\n- v15 --refutes--> v1f\n- v1b --refutes--> v0b\n- v1b --refutes--> v17\n- v1c --refutes--> v09\n- v1c --supports--> v0d\n- v1e --refutes--> v24\n- v20 --supports--> v10\n- v22 --refutes--> v14\n- v23 --refutes--> v0a\n- v26 --supports--> v03\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":"47b838d20a0c1680cdded77c0409e330f3f0f333d462a0e302547e27a6654f0b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --supports--> v29\n- v29 --supports--> v08\n- v08 --refutes--> v24\n- v24 --supports--> v10\n- v10 --supports--> v1e\n- v1e --supports--> v22\n- v22 --refutes--> v20\n- v20 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v05\n- v05 --supports--> v19\n- v19 --refutes--> v0d\n- v0d --supports--> v28\n- v28 --supports--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v1c\n- v1c --supports--> v23\n- v23 --refutes--> v00\n- v00 --supports--> v27\n- v27 --supports--> v25\n- v25 --refutes--> v0e\n- v0e --supports--> v2a\n- v2a --refutes--> v03\n- v03 --refutes--> v0c\n- v0c --refutes--> v1a\n- v1a --supports--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v14\n- v02 --supports--> v0a\n- v02 --supports--> v18\n- v05 --refutes--> v1d\n- v05 --refutes--> v26\n- v05 --supports--> v16\n- v08 --refutes--> v15\n- v0c --refutes--> v21\n- v11 --supports--> v12\n- v13 --refutes--> v06\n- v19 --refutes--> v1f\n- v1c --refutes--> v17\n- v20 --refutes--> v07\n- v23 --refutes--> v04\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":"009d8fd37852d97a3affb72d683e8e635dacdbcf6005b7c534c821b93e4d2b77","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v1e\n- v1e --refutes--> v23\n- v23 --refutes--> v27\n- v27 --supports--> v1b\n- v1b --refutes--> v1d\n- v1d --supports--> v25\n- v25 --supports--> v01\n- v01 --refutes--> v18\n- v18 --refutes--> v1f\n- v1f --supports--> v02\n- v02 --supports--> v06\n- v06 --refutes--> v12\n- v12 --supports--> v10\n- v10 --supports--> v11\n- v11 --supports--> v00\n- v00 --supports--> v08\n- v08 --supports--> v21\n- v21 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v29\n- v29 --refutes--> v26\n- v26 --supports--> v16\n- v16 --refutes--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v07\n- v07 --refutes--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v04\n- v08 --supports--> v09\n- v0a --refutes--> v22\n- v0a --refutes--> v24\n- v0c --refutes--> v05\n- v0f --supports--> v2a\n- v11 --supports--> v0b\n- v11 --supports--> v20\n- v13 --supports--> v19\n- v16 --refutes--> v1c\n- v1b --refutes--> v28\n- v1f --refutes--> v03\n- v26 --refutes--> v0e\n- v27 --refutes--> v1a\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":"59f97448318dc0dd2cde40daa75bbeb5462042371263301286961f59a4389b39","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --refutes--> v02\n- v02 --supports--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v1e\n- v1e --supports--> v29\n- v29 --supports--> v22\n- v22 --refutes--> v10\n- v10 --supports--> v1c\n- v1c --supports--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v27\n- v27 --supports--> v01\n- v01 --refutes--> v18\n- v18 --refutes--> v13\n- v13 --supports--> v0c\n- v0c --refutes--> v1d\n- v1d --refutes--> v28\n- v28 --supports--> v1a\n- v1a --refutes--> v19\n- v19 --supports--> v00\n- v00 --refutes--> v03\n- v03 --refutes--> v09\n- v09 --refutes--> v24\n- v24 --supports--> v06\n- v06 --refutes--> v21\n- v21 --refutes--> v26\n- v26 --supports--> v1b\n- v1b --refutes--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v12\n- v03 --refutes--> v17\n- v04 --refutes--> v0f\n- v06 --refutes--> v11\n- v09 --refutes--> v08\n- v0a --supports--> v0d\n- v0b --refutes--> v2a\n- v0c --refutes--> v07\n- v13 --supports--> v1f\n- v19 --supports--> v16\n- v1c --supports--> v15\n- v1e --refutes--> v14\n- v21 --refutes--> v0e\n- v21 --refutes--> v20\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":"8b86da894c5085716992b42ce385209c98bf04c81baf49d51ecb6ced6477e623","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v23\n- v23 --refutes--> v20\n- v20 --refutes--> v04\n- v04 --supports--> v16\n- v16 --refutes--> v25\n- v25 --refutes--> v01\n- v01 --refutes--> v13\n- v13 --supports--> v17\n- v17 --supports--> v02\n- v02 --refutes--> v2a\n- v2a --refutes--> v11\n- v11 --refutes--> v07\n- v07 --refutes--> v10\n- v10 --refutes--> v05\n- v05 --refutes--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v0d\n- v0d --refutes--> v1d\n- v1d --refutes--> v1a\n- v1a --refutes--> v29\n- v29 --supports--> v22\n- v22 --refutes--> v18\n- v18 --supports--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v27\n- v27 --refutes--> v00\n- v00 --supports--> v0c\n- v0c --refutes--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v03\n- v04 --refutes--> v26\n- v05 --refutes--> v06\n- v0c --refutes--> v08\n- v0c --supports--> v28\n- v10 --refutes--> v24\n- v11 --refutes--> v19\n- v11 --refutes--> v1b\n- v18 --refutes--> v21\n- v1e --supports--> v09\n- v1e --supports--> v0e\n- v20 --supports--> v0f\n- v22 --refutes--> v1f\n- v2a --refutes--> v0b\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":"13be4c783217a8582e1660195677e6ed319a8bb4c9a0b03da3dfc2d1bed74574","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --refutes--> v18\n- v18 --refutes--> v13\n- v13 --supports--> v16\n- v16 --refutes--> v09\n- v09 --supports--> v22\n- v22 --refutes--> v1e\n- v1e --refutes--> v1b\n- v1b --refutes--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v24\n- v24 --refutes--> v17\n- v17 --refutes--> v12\n- v12 --refutes--> v27\n- v27 --refutes--> v25\n- v25 --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v0f\n- v0f --refutes--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v1d\n- v1d --refutes--> v14\n- v14 --refutes--> v29\n- v29 --refutes--> v20\n- v20 --refutes--> v0a\n- v0a --supports--> v28\n- v28 --supports--> v2a\n- v2a --supports--> v04\n- v04 --supports--> v0c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0e\n- v03 --refutes--> v01\n- v03 --supports--> v15\n- v09 --refutes--> v02\n- v09 --refutes--> v05\n- v0a --supports--> v0b\n- v10 --supports--> v11\n- v17 --supports--> v07\n- v17 --supports--> v1f\n- v1a --refutes--> v19\n- v1b --supports--> v23\n- v1b --supports--> v26\n- v25 --refutes--> v21\n- v2a --refutes--> v1c\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":"e10ee2f769beb908161f02794fd391ae8695bce1d6d382d49f1a2ea87e041a28","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --refutes--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v27\n- v27 --supports--> v02\n- v02 --supports--> v1d\n- v1d --supports--> v2a\n- v2a --supports--> v11\n- v11 --refutes--> v25\n- v25 --refutes--> v09\n- v09 --refutes--> v1a\n- v1a --refutes--> v22\n- v22 --supports--> v01\n- v01 --refutes--> v29\n- v29 --refutes--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v26\n- v26 --refutes--> v0e\n- v0e --supports--> v1b\n- v1b --supports--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v28\n- v28 --refutes--> v13\n- v13 --refutes--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v05\n- v0b --supports--> v12\n- v0f --refutes--> v00\n- v0f --supports--> v1c\n- v11 --supports--> v23\n- v1d --refutes--> v0c\n- v1f --refutes--> v19\n- v1f --supports--> v18\n- v22 --supports--> v16\n- v25 --refutes--> v21\n- v26 --refutes--> v07\n- v27 --refutes--> v20\n- v27 --refutes--> v24\n- v29 --supports--> v06\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":"0940ff9e9fcb5fa38f7b6f2eff2a81102942f00a44872ccd3132ad18cee6a48d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v29 --refutes--> v0d\n- v0d --supports--> v13\n- v13 --supports--> v19\n- v19 --refutes--> v1f\n- v1f --supports--> v18\n- v18 --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --supports--> v28\n- v28 --supports--> v16\n- v16 --refutes--> v08\n- v08 --supports--> v03\n- v03 --supports--> v1a\n- v1a --refutes--> v25\n- v25 --refutes--> v00\n- v00 --supports--> v1b\n- v1b --supports--> v0f\n- v0f --refutes--> v1c\n- v1c --supports--> v24\n- v24 --supports--> v05\n- v05 --refutes--> v1e\n- v1e --supports--> v2a\n- v2a --supports--> v27\n- v27 --supports--> v1d\n- v1d --supports--> v10\n- v10 --supports--> v06\n- v06 --supports--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v01\n- v0a --supports--> v09\n- v0e --supports--> v20\n- v10 --supports--> v12\n- v18 --supports--> v04\n- v19 --refutes--> v22\n- v19 --supports--> v11\n- v19 --supports--> v17\n- v1a --refutes--> v14\n- v1a --supports--> v21\n- v1a --supports--> v26\n- v1b --refutes--> v07\n- v1d --refutes--> v0b\n- v29 --supports--> v0c\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":"52a3dca45885375c8dbff33d1299bf4e33631a5c8d00fe2c02b47fee40d4f7ca","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v1d\n- v1d --supports--> v19\n- v19 --supports--> v1e\n- v1e --supports--> v04\n- v04 --supports--> v10\n- v10 --supports--> v1b\n- v1b --supports--> v0e\n- v0e --supports--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v12\n- v12 --supports--> v0b\n- v0b --refutes--> v1a\n- v1a --refutes--> v07\n- v07 --refutes--> v00\n- v00 --supports--> v27\n- v27 --refutes--> v0f\n- v0f --refutes--> v1f\n- v1f --refutes--> v03\n- v03 --refutes--> v0a\n- v0a --refutes--> v22\n- v22 --supports--> v15\n- v15 --refutes--> v08\n- v08 --refutes--> v21\n- v21 --supports--> v0d\n- v0d --supports--> v09\n- v09 --supports--> v25\n- v25 --supports--> v26\n- v26 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v16\n- v07 --supports--> v1c\n- v0f --refutes--> v02\n- v10 --supports--> v11\n- v14 --refutes--> v05\n- v15 --refutes--> v24\n- v19 --refutes--> v28\n- v1b --refutes--> v18\n- v1e --refutes--> v29\n- v1f --refutes--> v23\n- v1f --supports--> v06\n- v21 --supports--> v0c\n- v22 --supports--> v20\n- v25 --supports--> v2a\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":"97c7591b0d786b737bf533a89dca77492e466036cefd966bb71be3ef7bc9eeff","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v0b\n- v0b --supports--> v1e\n- v1e --refutes--> v11\n- v11 --refutes--> v1b\n- v1b --supports--> v28\n- v28 --refutes--> v22\n- v22 --supports--> v25\n- v25 --refutes--> v18\n- v18 --supports--> v0f\n- v0f --refutes--> v2a\n- v2a --supports--> v16\n- v16 --supports--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v1c\n- v1c --supports--> v29\n- v29 --refutes--> v12\n- v12 --refutes--> v21\n- v21 --refutes--> v0c\n- v0c --supports--> v19\n- v19 --refutes--> v26\n- v26 --refutes--> v13\n- v13 --refutes--> v01\n- v01 --refutes--> v27\n- v27 --supports--> v05\n- v05 --refutes--> v10\n- v10 --refutes--> v03\n- v03 --refutes--> v17\n- v17 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0e\n- v0f --refutes--> v1d\n- v10 --refutes--> v24\n- v13 --refutes--> v14\n- v18 --supports--> v1f\n- v19 --refutes--> v07\n- v1b --supports--> v08\n- v1c --supports--> v1a\n- v1e --supports--> v0a\n- v22 --refutes--> v04\n- v25 --supports--> v02\n- v26 --supports--> v00\n- v29 --refutes--> v20\n- v29 --supports--> v15\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":"16a3ca0c7e7e9a5b90299f2ce10ba85e2d5a292b7bd43927ee8b977cd89bf4fe","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v17\n- v17 --refutes--> v12\n- v12 --refutes--> v22\n- v22 --refutes--> v0d\n- v0d --supports--> v0c\n- v0c --refutes--> v21\n- v21 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v11\n- v11 --supports--> v20\n- v20 --supports--> v0e\n- v0e --refutes--> v1f\n- v1f --supports--> v24\n- v24 --refutes--> v1a\n- v1a --refutes--> v29\n- v29 --refutes--> v26\n- v26 --supports--> v0f\n- v0f --supports--> v1d\n- v1d --refutes--> v27\n- v27 --supports--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v2a\n- v2a --refutes--> v15\n- v15 --supports--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v1c\n- v1c --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v16\n- v07 --refutes--> v00\n- v0a --supports--> v13\n- v0c --refutes--> v0b\n- v0c --supports--> v02\n- v0c --supports--> v19\n- v0d --refutes--> v1e\n- v12 --supports--> v05\n- v14 --supports--> v03\n- v17 --refutes--> v23\n- v1c --refutes--> v25\n- v1d --refutes--> v08\n- v22 --refutes--> v09\n- v22 --supports--> v28\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":"d8aa192868faba4f85a6f7429dbc8549a9847a0c1e1ebe9bbf180e119d93a074","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --supports--> v16\n- v16 --refutes--> v07\n- v07 --supports--> v24\n- v24 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --refutes--> v0a\n- v0a --supports--> v23\n- v23 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --refutes--> v1c\n- v1c --supports--> v11\n- v11 --refutes--> v13\n- v13 --supports--> v0b\n- v0b --refutes--> v09\n- v09 --supports--> v18\n- v18 --refutes--> v20\n- v20 --refutes--> v21\n- v21 --supports--> v1f\n- v1f --refutes--> v28\n- v28 --supports--> v04\n- v04 --supports--> v05\n- v05 --refutes--> v2a\n- v2a --supports--> v1e\n- v1e --refutes--> v14\n- v14 --supports--> v1d\n- v1d --supports--> v03\n- v03 --supports--> v10\n- v10 --supports--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v02\n- v04 --supports--> v1a\n- v05 --supports--> v12\n- v0b --supports--> v26\n- v0e --refutes--> v01\n- v0e --supports--> v17\n- v18 --supports--> v08\n- v18 --supports--> v0c\n- v1c --supports--> v06\n- v1c --supports--> v15\n- v1d --supports--> v22\n- v1f --refutes--> v1b\n- v23 --supports--> v19\n- v24 --refutes--> v29\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":"a2485967329fb6e78d84bf8d0c3c3ac97f4f9a284c6fa5ea824a3b2a3751e3b4","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v11\n- v11 --supports--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v23\n- v23 --refutes--> v18\n- v18 --supports--> v1e\n- v1e --supports--> v1b\n- v1b --refutes--> v17\n- v17 --refutes--> v1f\n- v1f --supports--> v20\n- v20 --refutes--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v16\n- v16 --supports--> v0b\n- v0b --supports--> v08\n- v08 --supports--> v1c\n- v1c --refutes--> v14\n- v14 --refutes--> v29\n- v29 --supports--> v0c\n- v0c --refutes--> v2a\n- v2a --supports--> v15\n- v15 --refutes--> v19\n- v19 --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --refutes--> v24\n- v24 --refutes--> v21\n- v21 --refutes--> v28\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v22\n- v0d --refutes--> v27\n- v13 --supports--> v09\n- v15 --refutes--> v1a\n- v16 --supports--> v10\n- v1b --refutes--> v12\n- v1b --refutes--> v26\n- v1b --supports--> v25\n- v1d --refutes--> v07\n- v20 --refutes--> v06\n- v20 --supports--> v03\n- v23 --refutes--> v01\n- v23 --supports--> v0a\n- v24 --refutes--> v00\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":"a2cbead6db37d2595212150d807931e927998e018fcc5da98c73030ef7e3949c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0f --refutes--> v00\n- v00 --refutes--> v27\n- v27 --refutes--> v0b\n- v0b --refutes--> v2a\n- v2a --refutes--> v24\n- v24 --supports--> v1f\n- v1f --supports--> v28\n- v28 --refutes--> v1d\n- v1d --refutes--> v25\n- v25 --supports--> v18\n- v18 --refutes--> v1e\n- v1e --supports--> v0a\n- v0a --refutes--> v01\n- v01 --refutes--> v15\n- v15 --refutes--> v23\n- v23 --refutes--> v11\n- v11 --supports--> v22\n- v22 --supports--> v19\n- v19 --refutes--> v17\n- v17 --supports--> v08\n- v08 --supports--> v1a\n- v1a --supports--> v20\n- v20 --supports--> v1b\n- v1b --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v12\n- v12 --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v1c\n- v00 --supports--> v13\n- v08 --supports--> v29\n- v11 --supports--> v0e\n- v12 --refutes--> v05\n- v18 --supports--> v09\n- v19 --supports--> v07\n- v1b --refutes--> v02\n- v1f --refutes--> v10\n- v20 --supports--> v26\n- v22 --refutes--> v14\n- v23 --refutes--> v16\n- v28 --supports--> v21\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":"57e182006aee56a798a2d454c589991ac4151129f544a94f166c3b0eb44a119f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v19\n- v19 --supports--> v10\n- v10 --supports--> v05\n- v05 --supports--> v21\n- v21 --supports--> v1b\n- v1b --refutes--> v0c\n- v0c --refutes--> v18\n- v18 --supports--> v22\n- v22 --refutes--> v17\n- v17 --supports--> v14\n- v14 --supports--> v29\n- v29 --supports--> v12\n- v12 --refutes--> v02\n- v02 --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v26\n- v26 --refutes--> v0e\n- v0e --supports--> v0a\n- v0a --supports--> v1c\n- v1c --refutes--> v20\n- v20 --refutes--> v13\n- v13 --supports--> v0f\n- v0f --refutes--> v16\n- v16 --supports--> v24\n- v24 --supports--> v1a\n- v1a --refutes--> v28\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0d\n- v00 --supports--> v27\n- v0a --supports--> v01\n- v0a --supports--> v06\n- v0c --refutes--> v25\n- v0c --supports--> v09\n- v0c --supports--> v2a\n- v0e --supports--> v11\n- v0f --supports--> v15\n- v13 --refutes--> v1d\n- v18 --supports--> v04\n- v19 --refutes--> v1f\n- v19 --refutes--> v23\n- v26 --supports--> v1e\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":"25cdc9079033f127afb1d3d53ce5499932ec2f19ea20f3edddcca27b92ca284e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v1e\n- v1e --supports--> v04\n- v04 --refutes--> v0c\n- v0c --refutes--> v24\n- v24 --supports--> v03\n- v03 --supports--> v08\n- v08 --refutes--> v1a\n- v1a --refutes--> v11\n- v11 --supports--> v17\n- v17 --refutes--> v1f\n- v1f --refutes--> v1c\n- v1c --supports--> v16\n- v16 --supports--> v25\n- v25 --refutes--> v12\n- v12 --refutes--> v22\n- v22 --supports--> v09\n- v09 --supports--> v2a\n- v2a --supports--> v0d\n- v0d --refutes--> v21\n- v21 --supports--> v15\n- v15 --supports--> v05\n- v05 --supports--> v02\n- v02 --supports--> v20\n- v20 --refutes--> v29\n- v29 --supports--> v28\n- v28 --supports--> v18\n- v18 --refutes--> v10\n- v10 --refutes--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0f\n- v0c --supports--> v0a\n- v0d --refutes--> v1d\n- v12 --supports--> v00\n- v17 --refutes--> v07\n- v17 --refutes--> v19\n- v1a --supports--> v06\n- v1f --supports--> v0e\n- v20 --refutes--> v27\n- v21 --supports--> v01\n- v22 --refutes--> v13\n- v28 --refutes--> v14\n- v28 --supports--> v0b\n- v2a --supports--> v1b\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":"59b3ef9ec8e5ba47e5ec7d48fbb46c8d62dde6f6ef51ea3f549d053c50939b14","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v2a\n- v2a --supports--> v19\n- v19 --supports--> v0d\n- v0d --refutes--> v27\n- v27 --refutes--> v1e\n- v1e --refutes--> v16\n- v16 --supports--> v28\n- v28 --refutes--> v0e\n- v0e --supports--> v24\n- v24 --refutes--> v1d\n- v1d --refutes--> v10\n- v10 --refutes--> v0a\n- v0a --supports--> v21\n- v21 --refutes--> v22\n- v22 --supports--> v26\n- v26 --refutes--> v15\n- v15 --refutes--> v02\n- v02 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v25\n- v25 --refutes--> v09\n- v09 --supports--> v23\n- v23 --refutes--> v1a\n- v1a --supports--> v07\n- v07 --refutes--> v00\n- v00 --refutes--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v01 --supports--> v05\n- v01 --supports--> v14\n- v08 --supports--> v17\n- v0c --supports--> v0b\n- v0e --supports--> v1f\n- v10 --supports--> v13\n- v1a --refutes--> v18\n- v1c --refutes--> v04\n- v1c --supports--> v11\n- v1e --refutes--> v06\n- v21 --supports--> v29\n- v23 --refutes--> v20\n- v28 --refutes--> v1b\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":"f2018094134f9a88f9ea0f4fdbc9a49a257ac42d62af47733efaaf96dd209927","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v26\n- v26 --supports--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v1e\n- v1e --refutes--> v2a\n- v2a --refutes--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v1f\n- v1f --supports--> v19\n- v19 --refutes--> v04\n- v04 --supports--> v14\n- v14 --refutes--> v29\n- v29 --supports--> v08\n- v08 --supports--> v28\n- v28 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --refutes--> v16\n- v16 --refutes--> v24\n- v24 --refutes--> v15\n- v15 --supports--> v20\n- v20 --supports--> v06\n- v06 --supports--> v22\n- v22 --supports--> v1d\n- v1d --supports--> v03\n- v03 --refutes--> v1a\n- v1a --refutes--> v13\n- v13 --supports--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v10\n- v03 --supports--> v25\n- v04 --supports--> v18\n- v09 --refutes--> v12\n- v0b --refutes--> v21\n- v13 --supports--> v02\n- v1d --refutes--> v00\n- v1e --refutes--> v0a\n- v1e --supports--> v0d\n- v1f --supports--> v11\n- v24 --supports--> v1c\n- v27 --refutes--> v01\n- v27 --supports--> v05\n- v28 --supports--> v17\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":"4a6d48ec6792d5d09f66fb300dc973ae1bd225fba1b61368ca1c629a3221dc12","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --refutes--> v0f\n- v0f --supports--> v15\n- v15 --supports--> v00\n- v00 --refutes--> v11\n- v11 --refutes--> v27\n- v27 --refutes--> v1c\n- v1c --refutes--> v1b\n- v1b --refutes--> v20\n- v20 --refutes--> v02\n- v02 --supports--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v21\n- v21 --refutes--> v04\n- v04 --refutes--> v07\n- v07 --supports--> v0e\n- v0e --supports--> v13\n- v13 --supports--> v03\n- v03 --refutes--> v16\n- v16 --supports--> v12\n- v12 --refutes--> v1e\n- v1e --supports--> v18\n- v18 --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v25\n- v25 --supports--> v1d\n- v1d --supports--> v14\n- v14 --refutes--> v0a\n- v0a --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v19\n- v01 --supports--> v1a\n- v05 --refutes--> v26\n- v07 --supports--> v08\n- v07 --supports--> v29\n- v0d --refutes--> v1f\n- v0d --refutes--> v28\n- v0e --refutes--> v23\n- v11 --refutes--> v0b\n- v15 --refutes--> v0c\n- v18 --refutes--> v2a\n- v1b --supports--> v22\n- v25 --refutes--> v09\n- v27 --refutes--> v06\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":"8658565802d6415cdb5c595f80ffb110a0c5f797a3c45fb05f27b85797421230","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --refutes--> v18\n- v18 --supports--> v24\n- v24 --supports--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v1e\n- v1e --supports--> v1b\n- v1b --supports--> v2a\n- v2a --refutes--> v0a\n- v0a --supports--> v15\n- v15 --supports--> v04\n- v04 --supports--> v26\n- v26 --refutes--> v14\n- v14 --supports--> v0c\n- v0c --supports--> v1f\n- v1f --refutes--> v1d\n- v1d --supports--> v29\n- v29 --refutes--> v28\n- v28 --refutes--> v05\n- v05 --supports--> v25\n- v25 --refutes--> v13\n- v13 --supports--> v1c\n- v1c --refutes--> v08\n- v08 --supports--> v10\n- v10 --refutes--> v20\n- v20 --refutes--> v12\n- v12 --supports--> v19\n- v19 --refutes--> v09\n- v09 --refutes--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v22\n- v00 --refutes--> v23\n- v04 --refutes--> v27\n- v05 --supports--> v0d\n- v13 --refutes--> v21\n- v16 --refutes--> v01\n- v18 --supports--> v17\n- v1b --refutes--> v02\n- v1f --refutes--> v07\n- v20 --refutes--> v1a\n- v24 --refutes--> v03\n- v24 --supports--> v0e\n- v26 --refutes--> v11\n- v2a --refutes--> v0f\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":"306743209b77ab9e147c1b53490bf459431bcc8885c65acd315e8b7991d8a831","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v22\n- v22 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v1d\n- v1d --refutes--> v29\n- v29 --supports--> v0d\n- v0d --supports--> v0c\n- v0c --supports--> v07\n- v07 --supports--> v01\n- v01 --supports--> v12\n- v12 --supports--> v0a\n- v0a --supports--> v18\n- v18 --refutes--> v09\n- v09 --supports--> v25\n- v25 --refutes--> v20\n- v20 --supports--> v00\n- v00 --refutes--> v13\n- v13 --supports--> v23\n- v23 --refutes--> v27\n- v27 --supports--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v1f\n- v1f --supports--> v15\n- v15 --refutes--> v06\n- v06 --supports--> v05\n- v05 --supports--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0f\n- v06 --refutes--> v02\n- v0d --refutes--> v21\n- v0e --supports--> v1c\n- v0e --supports--> v1e\n- v11 --supports--> v14\n- v12 --refutes--> v04\n- v12 --refutes--> v16\n- v20 --refutes--> v28\n- v20 --supports--> v2a\n- v25 --supports--> v0b\n- v27 --refutes--> v17\n- v27 --supports--> v24\n- v29 --supports--> v1b\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":"bcedde3ce4548ec3e2b265959b29050a1d34c2703da4b15ed37c24eed9d0d693","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v09\n- v09 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v16\n- v16 --supports--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v03\n- v03 --supports--> v10\n- v10 --supports--> v1f\n- v1f --refutes--> v1c\n- v1c --supports--> v1a\n- v1a --refutes--> v21\n- v21 --supports--> v18\n- v18 --refutes--> v04\n- v04 --refutes--> v17\n- v17 --refutes--> v20\n- v20 --supports--> v02\n- v02 --refutes--> v2a\n- v2a --supports--> v00\n- v00 --supports--> v23\n- v23 --refutes--> v29\n- v29 --supports--> v08\n- v08 --supports--> v11\n- v11 --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v1e\n- v04 --refutes--> v1d\n- v09 --refutes--> v24\n- v0a --refutes--> v14\n- v0a --supports--> v1b\n- v0c --refutes--> v01\n- v0c --supports--> v26\n- v10 --supports--> v28\n- v11 --supports--> v15\n- v17 --refutes--> v12\n- v1f --refutes--> v13\n- v29 --refutes--> v19\n- v2a --refutes--> v27\n- v2a --supports--> v25\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":"d0202b01caf6d5d90be48b9f4f0b0da3775bcfac2505e787f13351dd42a15e16","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v12\n- v12 --supports--> v23\n- v23 --supports--> v22\n- v22 --refutes--> v1e\n- v1e --supports--> v07\n- v07 --refutes--> v26\n- v26 --supports--> v0b\n- v0b --supports--> v18\n- v18 --refutes--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v16\n- v16 --supports--> v09\n- v09 --supports--> v13\n- v13 --refutes--> v1f\n- v1f --supports--> v0c\n- v0c --refutes--> v24\n- v24 --supports--> v2a\n- v2a --supports--> v21\n- v21 --supports--> v29\n- v29 --refutes--> v01\n- v01 --supports--> v27\n- v27 --supports--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v02\n- v02 --supports--> v14\n- v14 --supports--> v20\n- v20 --supports--> v03\n- v03 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v1c\n- v01 --supports--> v0a\n- v01 --supports--> v0f\n- v02 --refutes--> v05\n- v03 --refutes--> v1d\n- v03 --supports--> v06\n- v03 --supports--> v1b\n- v15 --refutes--> v04\n- v15 --refutes--> v10\n- v18 --supports--> v19\n- v1f --refutes--> v28\n- v20 --refutes--> v17\n- v22 --supports--> v25\n- v24 --refutes--> v0e\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":"77ad146c7324e9760503c19d77fd9c8da25329bf4d6a48ba78ebb04e5fd96516","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v28 --refutes--> v20\n- v20 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v21\n- v21 --refutes--> v08\n- v08 --refutes--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v05\n- v05 --supports--> v0a\n- v0a --refutes--> v18\n- v18 --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v23\n- v23 --refutes--> v1c\n- v1c --supports--> v14\n- v14 --refutes--> v25\n- v25 --refutes--> v19\n- v19 --supports--> v0c\n- v0c --supports--> v22\n- v22 --refutes--> v1b\n- v1b --refutes--> v2a\n- v2a --supports--> v27\n- v27 --supports--> v00\n- v00 --refutes--> v1a\n- v1a --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v26\n- v06 --refutes--> v0f\n- v07 --supports--> v1e\n- v0c --refutes--> v04\n- v15 --refutes--> v0b\n- v15 --supports--> v1f\n- v15 --supports--> v29\n- v19 --supports--> v11\n- v1b --supports--> v09\n- v1c --refutes--> v17\n- v22 --refutes--> v01\n- v27 --supports--> v24\n- v28 --supports--> v13\n- v2a --refutes--> v16\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":"e2d5cfab612aaac9dbce0eb1e5a65b9792f14894c0f06da048739c761a033cfb","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v10\n- v10 --supports--> v16\n- v16 --supports--> v19\n- v19 --refutes--> v17\n- v17 --supports--> v09\n- v09 --refutes--> v26\n- v26 --supports--> v00\n- v00 --refutes--> v22\n- v22 --supports--> v04\n- v04 --supports--> v12\n- v12 --supports--> v1d\n- v1d --supports--> v15\n- v15 --supports--> v28\n- v28 --refutes--> v05\n- v05 --refutes--> v29\n- v29 --supports--> v14\n- v14 --supports--> v1a\n- v1a --refutes--> v24\n- v24 --supports--> v1e\n- v1e --supports--> v11\n- v11 --refutes--> v21\n- v21 --refutes--> v0f\n- v0f --supports--> v20\n- v20 --supports--> v07\n- v07 --refutes--> v2a\n- v2a --refutes--> v08\n- v08 --supports--> v27\n- v27 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v01\n- v05 --supports--> v06\n- v07 --refutes--> v1c\n- v08 --refutes--> v02\n- v0f --refutes--> v0a\n- v12 --supports--> v0d\n- v17 --supports--> v1f\n- v1d --supports--> v03\n- v1e --refutes--> v0c\n- v1e --refutes--> v23\n- v1e --supports--> v13\n- v20 --supports--> v0b\n- v24 --supports--> v1b\n- v29 --supports--> v25\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":"02ac8e202269b8843484e6e110d41ee6cdbcb61396de0e1d9763f74333e45c52","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v27\n- v27 --supports--> v06\n- v06 --supports--> v1b\n- v1b --refutes--> v11\n- v11 --supports--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v21\n- v21 --refutes--> v1a\n- v1a --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v0c\n- v0c --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v29\n- v29 --refutes--> v28\n- v28 --refutes--> v26\n- v26 --refutes--> v0b\n- v0b --supports--> v15\n- v15 --supports--> v0a\n- v0a --supports--> v02\n- v02 --refutes--> v23\n- v23 --refutes--> v17\n- v17 --refutes--> v1e\n- v1e --supports--> v00\n- v00 --supports--> v12\n- v12 --refutes--> v08\n- v08 --supports--> v22\n- v22 --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v02 --supports--> v01\n- v0b --refutes--> v13\n- v0b --supports--> v03\n- v0b --supports--> v0d\n- v12 --refutes--> v1d\n- v12 --supports--> v07\n- v17 --refutes--> v25\n- v19 --supports--> v2a\n- v1b --supports--> v0e\n- v1b --supports--> v18\n- v1b --supports--> v1f\n- v27 --refutes--> v10\n- v29 --refutes--> v24\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":"e415f1f317bf82360410be6644bfdc67913a4136d6b5bc27a0e129663277c30e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v06\n- v06 --supports--> v2a\n- v2a --refutes--> v1e\n- v1e --refutes--> v00\n- v00 --supports--> v0e\n- v0e --refutes--> v20\n- v20 --refutes--> v25\n- v25 --supports--> v1d\n- v1d --refutes--> v07\n- v07 --refutes--> v01\n- v01 --supports--> v29\n- v29 --supports--> v24\n- v24 --supports--> v27\n- v27 --refutes--> v19\n- v19 --refutes--> v0b\n- v0b --refutes--> v02\n- v02 --refutes--> v22\n- v22 --supports--> v10\n- v10 --refutes--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v08\n- v08 --supports--> v21\n- v21 --refutes--> v03\n- v03 --supports--> v1a\n- v1a --refutes--> v0a\n- v0a --supports--> v04\n- v04 --refutes--> v28\n- v28 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v23\n- v01 --refutes--> v1c\n- v02 --refutes--> v1f\n- v02 --supports--> v1b\n- v07 --supports--> v05\n- v0b --refutes--> v09\n- v0b --refutes--> v13\n- v1d --supports--> v14\n- v21 --refutes--> v11\n- v25 --supports--> v26\n- v28 --refutes--> v12\n- v29 --supports--> v0f\n- v2a --refutes--> v16\n- v2a --refutes--> v18\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":"ac7e07309340624f02356688f62ad8fe93a9c8dba6c240b241de8139314efc45","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v23\n- v23 --supports--> v24\n- v24 --supports--> v1b\n- v1b --refutes--> v0d\n- v0d --supports--> v16\n- v16 --refutes--> v19\n- v19 --supports--> v25\n- v25 --supports--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v21\n- v21 --supports--> v27\n- v27 --refutes--> v05\n- v05 --refutes--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v26\n- v26 --refutes--> v07\n- v07 --supports--> v2a\n- v2a --supports--> v0b\n- v0b --refutes--> v22\n- v22 --refutes--> v0a\n- v0a --refutes--> v1c\n- v1c --supports--> v15\n- v15 --supports--> v13\n- v13 --supports--> v20\n- v20 --supports--> v04\n- v04 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v00\n- v04 --supports--> v14\n- v07 --supports--> v1a\n- v0a --supports--> v08\n- v12 --refutes--> v18\n- v12 --supports--> v1d\n- v19 --refutes--> v06\n- v1b --supports--> v0f\n- v1c --refutes--> v1f\n- v1c --refutes--> v28\n- v20 --refutes--> v29\n- v21 --supports--> v10\n- v23 --supports--> v02\n- v23 --supports--> v17\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":"7a2cdb8e88ae0dbbbef7cdd1c0a811a3e64b22943cc3c64e1649dbada71d071e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v26\n- v26 --refutes--> v05\n- v05 --supports--> v25\n- v25 --refutes--> v06\n- v06 --refutes--> v0b\n- v0b --refutes--> v1d\n- v1d --refutes--> v0f\n- v0f --supports--> v02\n- v02 --refutes--> v1a\n- v1a --supports--> v27\n- v27 --supports--> v24\n- v24 --refutes--> v21\n- v21 --refutes--> v01\n- v01 --refutes--> v14\n- v14 --supports--> v10\n- v10 --supports--> v1e\n- v1e --refutes--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v22\n- v22 --supports--> v0a\n- v0a --supports--> v1c\n- v1c --refutes--> v1b\n- v1b --supports--> v29\n- v29 --refutes--> v15\n- v15 --refutes--> v18\n- v18 --refutes--> v11\n- v11 --supports--> v03\n- v03 --supports--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v04\n- v01 --refutes--> v28\n- v03 --supports--> v16\n- v0b --refutes--> v07\n- v10 --supports--> v1f\n- v14 --supports--> v0c\n- v15 --supports--> v00\n- v18 --supports--> v08\n- v1a --refutes--> v12\n- v1b --supports--> v17\n- v1c --supports--> v20\n- v21 --supports--> v19\n- v25 --refutes--> v0e\n- v25 --refutes--> v13\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":"81c335fa8bcd9f03c3474f0cdde5284830fd26facef24f65a6a5f7397decbaba","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2a --supports--> v28\n- v28 --refutes--> v01\n- v01 --supports--> v22\n- v22 --supports--> v1d\n- v1d --refutes--> v1c\n- v1c --refutes--> v19\n- v19 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v16\n- v16 --refutes--> v25\n- v25 --supports--> v09\n- v09 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v07\n- v07 --supports--> v1b\n- v1b --refutes--> v00\n- v00 --refutes--> v27\n- v27 --refutes--> v24\n- v24 --refutes--> v13\n- v13 --refutes--> v0b\n- v0b --supports--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v0c\n- v0c --refutes--> v11\n- v11 --refutes--> v29\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v14\n- v00 --refutes--> v21\n- v02 --supports--> v0d\n- v04 --supports--> v1a\n- v04 --supports--> v23\n- v09 --refutes--> v1f\n- v0e --refutes--> v15\n- v0e --supports--> v17\n- v12 --refutes--> v20\n- v12 --supports--> v06\n- v18 --refutes--> v03\n- v1d --supports--> v1e\n- v27 --refutes--> v10\n- v2a --refutes--> v26\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":"30d960a7af4b92cb9fee08357331ce4a30cd692cf9dd5a466ad062781097b9c7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v21 --supports--> v1b\n- v1b --supports--> v22\n- v22 --supports--> v00\n- v00 --refutes--> v1f\n- v1f --refutes--> v16\n- v16 --refutes--> v02\n- v02 --supports--> v24\n- v24 --supports--> v0a\n- v0a --supports--> v12\n- v12 --supports--> v17\n- v17 --supports--> v1c\n- v1c --supports--> v06\n- v06 --supports--> v14\n- v14 --supports--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v26\n- v26 --refutes--> v19\n- v19 --supports--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v18\n- v18 --refutes--> v03\n- v03 --refutes--> v27\n- v27 --refutes--> v25\n- v25 --supports--> v01\n- v01 --supports--> v10\n- v10 --refutes--> v1e\n- v1e --refutes--> v0c\n- v0c --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0b\n- v00 --supports--> v0d\n- v01 --supports--> v29\n- v02 --supports--> v28\n- v03 --refutes--> v2a\n- v09 --refutes--> v05\n- v09 --supports--> v0f\n- v09 --supports--> v15\n- v10 --supports--> v07\n- v12 --refutes--> v1d\n- v16 --supports--> v13\n- v16 --supports--> v20\n- v1f --supports--> v23\n- v22 --supports--> v1a\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":"e4bbf35ede8d342b74629fe09c132acc3869c2d2e0d963172cb685e43ad699f5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --supports--> v1a\n- v1a --refutes--> v24\n- v24 --supports--> v08\n- v08 --supports--> v19\n- v19 --refutes--> v15\n- v15 --refutes--> v25\n- v25 --supports--> v0d\n- v0d --supports--> v00\n- v00 --refutes--> v18\n- v18 --supports--> v27\n- v27 --refutes--> v0e\n- v0e --supports--> v17\n- v17 --refutes--> v21\n- v21 --supports--> v16\n- v16 --refutes--> v0c\n- v0c --supports--> v26\n- v26 --refutes--> v04\n- v04 --supports--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v28\n- v28 --supports--> v22\n- v22 --refutes--> v23\n- v23 --refutes--> v0b\n- v0b --supports--> v2a\n- v2a --supports--> v1f\n- v1f --refutes--> v06\n- v06 --refutes--> v02\n- v02 --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v11\n- v0e --refutes--> v0a\n- v10 --refutes--> v05\n- v10 --supports--> v0f\n- v18 --supports--> v09\n- v19 --refutes--> v12\n- v19 --refutes--> v13\n- v22 --supports--> v1e\n- v25 --refutes--> v14\n- v25 --refutes--> v1d\n- v27 --supports--> v01\n- v27 --supports--> v07\n- v28 --refutes--> v29\n- v28 --supports--> v03\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":"749524bfb084d3ec164320a640ba9b041877e86a6a888a13340a4cf5635db719","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v29\n- v29 --supports--> v09\n- v09 --supports--> v1d\n- v1d --supports--> v0a\n- v0a --refutes--> v20\n- v20 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --refutes--> v12\n- v12 --refutes--> v02\n- v02 --supports--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v22\n- v22 --refutes--> v26\n- v26 --supports--> v05\n- v05 --supports--> v04\n- v04 --supports--> v13\n- v13 --supports--> v15\n- v15 --supports--> v27\n- v27 --refutes--> v0f\n- v0f --supports--> v18\n- v18 --supports--> v24\n- v24 --supports--> v1c\n- v1c --refutes--> v16\n- v16 --supports--> v07\n- v07 --refutes--> v0c\n- v0c --refutes--> v1a\n- v1a --refutes--> v0b\n- v0b --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v00\n- v05 --refutes--> v2a\n- v07 --refutes--> v06\n- v08 --supports--> v19\n- v0b --refutes--> v1e\n- v0b --supports--> v0d\n- v0e --supports--> v14\n- v0e --supports--> v1b\n- v12 --supports--> v01\n- v16 --refutes--> v28\n- v1a --supports--> v21\n- v22 --supports--> v1f\n- v27 --refutes--> v11\n- v29 --supports--> v23\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":"9f670e0724fa0bd4353dea063ea90fd92a305a993d599d1d624ec855cd175c2e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --supports--> v26\n- v26 --supports--> v05\n- v05 --refutes--> v04\n- v04 --supports--> v24\n- v24 --refutes--> v1b\n- v1b --supports--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v0b\n- v0b --refutes--> v12\n- v12 --supports--> v2a\n- v2a --supports--> v29\n- v29 --supports--> v21\n- v21 --refutes--> v0a\n- v0a --refutes--> v1d\n- v1d --refutes--> v14\n- v14 --refutes--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v27\n- v27 --supports--> v00\n- v00 --refutes--> v23\n- v23 --supports--> v0f\n- v0f --supports--> v01\n- v01 --supports--> v02\n- v02 --supports--> v13\n- v13 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v1f\n- v1f --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v22\n- v03 --refutes--> v11\n- v03 --supports--> v16\n- v05 --supports--> v18\n- v06 --supports--> v1c\n- v07 --supports--> v1a\n- v0b --supports--> v10\n- v14 --supports--> v0e\n- v1d --supports--> v20\n- v1d --supports--> v28\n- v1f --supports--> v0d\n- v24 --supports--> v09\n- v27 --refutes--> v19\n- v27 --supports--> v08\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":"70278ac8f8658b83e9ee71e44d76865e8677c3eefdcecab1733eb989b5a75640","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1a --supports--> v27\n- v27 --supports--> v1b\n- v1b --refutes--> v1c\n- v1c --refutes--> v21\n- v21 --supports--> v08\n- v08 --refutes--> v15\n- v15 --refutes--> v1f\n- v1f --supports--> v28\n- v28 --refutes--> v0d\n- v0d --refutes--> v17\n- v17 --refutes--> v23\n- v23 --refutes--> v26\n- v26 --refutes--> v04\n- v04 --refutes--> v11\n- v11 --refutes--> v19\n- v19 --refutes--> v1d\n- v1d --supports--> v03\n- v03 --refutes--> v22\n- v22 --refutes--> v05\n- v05 --refutes--> v20\n- v20 --refutes--> v09\n- v09 --refutes--> v00\n- v00 --refutes--> v18\n- v18 --refutes--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v25\n- v25 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1e\n- v05 --refutes--> v01\n- v08 --supports--> v16\n- v11 --refutes--> v2a\n- v11 --supports--> v29\n- v15 --supports--> v02\n- v19 --supports--> v0e\n- v1c --refutes--> v0c\n- v1c --supports--> v10\n- v1d --supports--> v0f\n- v1d --supports--> v13\n- v1f --supports--> v24\n- v20 --supports--> v0a\n- v21 --refutes--> v14\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":"c1f8baa94969b4f9f798a49bd15a232b6fa9c4796ce48a80aeb0c8dcca1bc247","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --supports--> v03\n- v03 --refutes--> v02\n- v02 --refutes--> v22\n- v22 --supports--> v0d\n- v0d --supports--> v1e\n- v1e --supports--> v19\n- v19 --supports--> v0e\n- v0e --refutes--> v28\n- v28 --refutes--> v11\n- v11 --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v0a\n- v0a --refutes--> v23\n- v23 --refutes--> v27\n- v27 --refutes--> v00\n- v00 --supports--> v17\n- v17 --supports--> v12\n- v12 --refutes--> v0b\n- v0b --supports--> v18\n- v18 --supports--> v26\n- v26 --supports--> v1c\n- v1c --refutes--> v1f\n- v1f --supports--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v14\n- v14 --refutes--> v2a\n- v2a --refutes--> v20\n- v20 --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0f\n- v03 --supports--> v0c\n- v07 --refutes--> v1a\n- v17 --supports--> v29\n- v19 --refutes--> v10\n- v1c --refutes--> v21\n- v1f --supports--> v04\n- v22 --refutes--> v05\n- v22 --supports--> v13\n- v23 --supports--> v16\n- v26 --refutes--> v09\n- v27 --refutes--> v25\n- v28 --supports--> v1b\n- v2a --refutes--> v08\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":"135135e2205a802756c1ec747f0a9f5bd5f736c47226200d63a05f608aadd816","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v19\n- v19 --refutes--> v14\n- v14 --supports--> v1d\n- v1d --refutes--> v0b\n- v0b --refutes--> v09\n- v09 --refutes--> v28\n- v28 --refutes--> v16\n- v16 --refutes--> v22\n- v22 --refutes--> v1c\n- v1c --supports--> v0a\n- v0a --supports--> v23\n- v23 --supports--> v06\n- v06 --refutes--> v25\n- v25 --refutes--> v02\n- v02 --supports--> v0e\n- v0e --supports--> v12\n- v12 --supports--> v03\n- v03 --supports--> v13\n- v13 --refutes--> v20\n- v20 --refutes--> v1a\n- v1a --supports--> v17\n- v17 --supports--> v0d\n- v0d --supports--> v10\n- v10 --supports--> v21\n- v21 --supports--> v15\n- v15 --supports--> v11\n- v11 --refutes--> v27\n- v27 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v04\n- v0d --supports--> v01\n- v0e --refutes--> v07\n- v11 --supports--> v1f\n- v12 --refutes--> v29\n- v15 --supports--> v00\n- v17 --supports--> v24\n- v21 --refutes--> v0c\n- v22 --refutes--> v2a\n- v23 --refutes--> v05\n- v23 --refutes--> v1e\n- v23 --supports--> v26\n- v25 --refutes--> v0f\n- v27 --refutes--> v1b\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":"646cc2d53b4015c57bd3b8995075876305120fdd29383167583a5a6f6ffe0b2d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v21 --supports--> v01\n- v01 --supports--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v24\n- v24 --refutes--> v28\n- v28 --supports--> v23\n- v23 --supports--> v13\n- v13 --supports--> v03\n- v03 --refutes--> v0d\n- v0d --refutes--> v07\n- v07 --supports--> v15\n- v15 --refutes--> v1f\n- v1f --refutes--> v14\n- v14 --supports--> v17\n- v17 --supports--> v08\n- v08 --supports--> v27\n- v27 --refutes--> v25\n- v25 --supports--> v02\n- v02 --supports--> v0b\n- v0b --supports--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v1d\n- v1d --refutes--> v0a\n- v0a --refutes--> v19\n- v19 --supports--> v0c\n- v0c --refutes--> v22\n- v22 --supports--> v00\n- v00 --supports--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1c\n- v01 --supports--> v1b\n- v07 --refutes--> v1a\n- v0a --refutes--> v20\n- v0b --refutes--> v05\n- v0b --supports--> v1e\n- v10 --refutes--> v29\n- v13 --refutes--> v11\n- v17 --supports--> v06\n- v21 --refutes--> v18\n- v22 --supports--> v09\n- v24 --supports--> v0f\n- v25 --supports--> v26\n- v28 --supports--> v16\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":"901b1a2a4bd512a0529213efe5e3edd2f790c29c0de29f09a7633504bc6dd4e0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v26\n- v26 --supports--> v03\n- v03 --supports--> v1c\n- v1c --supports--> v09\n- v09 --supports--> v27\n- v27 --supports--> v12\n- v12 --refutes--> v20\n- v20 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --supports--> v0d\n- v0d --supports--> v15\n- v15 --refutes--> v1e\n- v1e --supports--> v19\n- v19 --supports--> v22\n- v22 --refutes--> v28\n- v28 --supports--> v25\n- v25 --refutes--> v04\n- v04 --refutes--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v2a\n- v2a --supports--> v06\n- v06 --refutes--> v23\n- v23 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v24\n- v24 --refutes--> v1a\n- v1a --supports--> v0e\n- v0e --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v16\n- v01 --refutes--> v1f\n- v01 --supports--> v1b\n- v03 --refutes--> v14\n- v04 --supports--> v13\n- v07 --refutes--> v0a\n- v0d --refutes--> v0c\n- v0d --refutes--> v29\n- v10 --refutes--> v17\n- v18 --supports--> v0f\n- v1a --refutes--> v08\n- v23 --refutes--> v05\n- v27 --refutes--> v11\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":"68d0b9b6d8d13039a8cd9df95e9cd6a7f564440479f9280c680f892457914490","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v0d\n- v0d --refutes--> v03\n- v03 --supports--> v1f\n- v1f --supports--> v29\n- v29 --refutes--> v25\n- v25 --supports--> v21\n- v21 --supports--> v28\n- v28 --refutes--> v1d\n- v1d --supports--> v27\n- v27 --refutes--> v18\n- v18 --refutes--> v00\n- v00 --supports--> v16\n- v16 --supports--> v07\n- v07 --supports--> v12\n- v12 --supports--> v17\n- v17 --supports--> v20\n- v20 --refutes--> v09\n- v09 --refutes--> v1e\n- v1e --supports--> v24\n- v24 --refutes--> v22\n- v22 --refutes--> v14\n- v14 --supports--> v19\n- v19 --supports--> v0b\n- v0b --refutes--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v1b\n- v1b --refutes--> v0a\n- v0a --supports--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --supports--> v10\n- v09 --supports--> v1c\n- v0b --refutes--> v05\n- v0d --supports--> v13\n- v0d --supports--> v1a\n- v12 --supports--> v08\n- v14 --supports--> v11\n- v15 --refutes--> v23\n- v16 --refutes--> v06\n- v18 --refutes--> v26\n- v19 --supports--> v0c\n- v19 --supports--> v0f\n- v1d --supports--> v0e\n- v29 --supports--> v02\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":"d767212b60b20587fa1161ba763ee66e4cf5c7ee4ee55c2b90f1af05b295d4d3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --supports--> v13\n- v13 --supports--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v21\n- v21 --supports--> v10\n- v10 --refutes--> v2a\n- v2a --refutes--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v1c\n- v1c --supports--> v1a\n- v1a --supports--> v17\n- v17 --refutes--> v15\n- v15 --refutes--> v01\n- v01 --refutes--> v23\n- v23 --supports--> v0c\n- v0c --supports--> v18\n- v18 --refutes--> v0e\n- v0e --supports--> v07\n- v07 --refutes--> v1d\n- v1d --refutes--> v25\n- v25 --refutes--> v08\n- v08 --refutes--> v29\n- v29 --supports--> v1e\n- v1e --supports--> v0a\n- v0a --supports--> v1f\n- v1f --refutes--> v11\n- v11 --supports--> v12\n- v12 --supports--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v09\n- v0a --refutes--> v0b\n- v0a --refutes--> v28\n- v10 --supports--> v26\n- v11 --refutes--> v1b\n- v13 --refutes--> v22\n- v14 --supports--> v16\n- v18 --supports--> v02\n- v1f --supports--> v00\n- v21 --supports--> v20\n- v23 --refutes--> v24\n- v25 --supports--> v06\n- v29 --supports--> v0d\n- v2a --refutes--> v05\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":"a74b9214d3a21ed74f9cd83e459ca19901bd63e7178715155dee64423c7e0a49","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v29\n- v29 --supports--> v00\n- v00 --refutes--> v1f\n- v1f --supports--> v28\n- v28 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v13\n- v13 --supports--> v08\n- v08 --supports--> v19\n- v19 --supports--> v0b\n- v0b --refutes--> v02\n- v02 --supports--> v16\n- v16 --refutes--> v03\n- v03 --supports--> v1b\n- v1b --supports--> v04\n- v04 --refutes--> v21\n- v21 --supports--> v22\n- v22 --supports--> v23\n- v23 --refutes--> v25\n- v25 --supports--> v01\n- v01 --supports--> v17\n- v17 --supports--> v1d\n- v1d --refutes--> v09\n- v09 --supports--> v20\n- v20 --supports--> v0c\n- v0c --supports--> v1e\n- v1e --supports--> v2a\n- v2a --supports--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0d\n- v01 --supports--> v0f\n- v03 --supports--> v05\n- v09 --supports--> v1c\n- v0a --refutes--> v15\n- v0a --supports--> v06\n- v0b --supports--> v27\n- v0e --refutes--> v11\n- v0e --refutes--> v24\n- v17 --refutes--> v18\n- v1b --supports--> v1a\n- v1f --refutes--> v07\n- v23 --supports--> v10\n- v25 --refutes--> v14\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":"36b2e1cea65473f96841b6feeb1cf014371ea93ddb7a4b7852bad81c499a4f0b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v21\n- v21 --refutes--> v08\n- v08 --supports--> v02\n- v02 --supports--> v1b\n- v1b --supports--> v25\n- v25 --supports--> v00\n- v00 --supports--> v1a\n- v1a --refutes--> v06\n- v06 --supports--> v17\n- v17 --supports--> v13\n- v13 --supports--> v1f\n- v1f --supports--> v0e\n- v0e --refutes--> v28\n- v28 --refutes--> v1e\n- v1e --supports--> v20\n- v20 --refutes--> v26\n- v26 --refutes--> v24\n- v24 --refutes--> v27\n- v27 --refutes--> v2a\n- v2a --supports--> v0d\n- v0d --supports--> v16\n- v16 --supports--> v0b\n- v0b --supports--> v10\n- v10 --refutes--> v09\n- v09 --refutes--> v12\n- v12 --supports--> v1c\n- v1c --supports--> v15\n- v15 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --refutes--> v11\n- v0e --supports--> v0f\n- v10 --refutes--> v01\n- v12 --supports--> v19\n- v12 --supports--> v1d\n- v13 --supports--> v22\n- v13 --supports--> v29\n- v16 --supports--> v0c\n- v1a --supports--> v0a\n- v1e --refutes--> v04\n- v1e --supports--> v23\n- v24 --refutes--> v07\n- v24 --supports--> v18\n- v27 --refutes--> v14\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":"0dae651560f11586115c64c578a40012c0f046861f8ee4dfbd5e46c14d15aa43","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v0f\n- v0f --refutes--> v03\n- v03 --refutes--> v23\n- v23 --refutes--> v2a\n- v2a --supports--> v06\n- v06 --supports--> v24\n- v24 --refutes--> v22\n- v22 --refutes--> v00\n- v00 --refutes--> v25\n- v25 --supports--> v04\n- v04 --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v0d\n- v0d --supports--> v1e\n- v1e --refutes--> v16\n- v16 --refutes--> v27\n- v27 --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --refutes--> v0c\n- v0c --refutes--> v26\n- v26 --refutes--> v15\n- v15 --supports--> v0e\n- v0e --refutes--> v13\n- v13 --refutes--> v14\n- v14 --refutes--> v1a\n- v1a --supports--> v18\n- v18 --supports--> v09\n- v09 --refutes--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1b\n- v00 --supports--> v1c\n- v03 --refutes--> v02\n- v04 --refutes--> v0b\n- v04 --refutes--> v29\n- v0f --refutes--> v11\n- v14 --refutes--> v28\n- v18 --refutes--> v05\n- v18 --refutes--> v21\n- v1a --supports--> v12\n- v1a --supports--> v19\n- v1e --refutes--> v20\n- v1f --refutes--> v1d\n- v24 --supports--> v0a\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":"365e3cbdfd9e12f391d0c675c83027fad229d5abd830e9ec11bb44118ef3d961","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v18\n- v18 --supports--> v00\n- v00 --supports--> v09\n- v09 --supports--> v2a\n- v2a --supports--> v1c\n- v1c --refutes--> v04\n- v04 --refutes--> v0d\n- v0d --supports--> v06\n- v06 --refutes--> v0f\n- v0f --supports--> v07\n- v07 --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --supports--> v1b\n- v1b --refutes--> v28\n- v28 --supports--> v15\n- v15 --refutes--> v0a\n- v0a --supports--> v14\n- v14 --supports--> v1a\n- v1a --refutes--> v01\n- v01 --refutes--> v21\n- v21 --supports--> v05\n- v05 --refutes--> v29\n- v29 --supports--> v1f\n- v1f --supports--> v1e\n- v1e --supports--> v24\n- v24 --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v05 --refutes--> v10\n- v07 --supports--> v13\n- v0e --supports--> v12\n- v0f --refutes--> v03\n- v14 --supports--> v19\n- v18 --refutes--> v0b\n- v1b --refutes--> v16\n- v1b --refutes--> v27\n- v1c --refutes--> v23\n- v1d --supports--> v26\n- v1e --refutes--> v20\n- v21 --refutes--> v17\n- v24 --supports--> v22\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":"b79f8b100c3d078b9dd280bc0645e0e8285a4ea1fffd2b48976b3c84263f030a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --supports--> v2f\n- v2f --refutes--> v22\n- v22 --supports--> v0a\n- v0a --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v12\n- v12 --refutes--> v05\n- v05 --supports--> v17\n- v17 --refutes--> v04\n- v04 --refutes--> v1e\n- v1e --supports--> v1f\n- v1f --supports--> v0e\n- v0e --refutes--> v2c\n- v2c --supports--> v20\n- v20 --refutes--> v13\n- v13 --refutes--> v10\n- v10 --supports--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v1a\n- v1a --supports--> v28\n- v28 --refutes--> v02\n- v02 --refutes--> v26\n- v26 --refutes--> v1c\n- v1c --supports--> v2a\n- v2a --refutes--> v03\n- v03 --supports--> v2e\n- v2e --refutes--> v00\n- v00 --refutes--> v2b\n- v2b --supports--> v27\n- v27 --supports--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v25\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0d\n- v05 --supports--> v24\n- v0a --supports--> v29\n- v13 --supports--> v11\n- v17 --supports--> v0f\n- v1a --refutes--> v09\n- v1c --refutes--> v07\n- v1f --refutes--> v16\n- v1f --refutes--> v18\n- v20 --supports--> v2d\n- v26 --refutes--> v0b\n- v26 --refutes--> v30\n- v28 --refutes--> v14\n- v2b --refutes--> v19\n- v2b --refutes--> v23\n- v2c --refutes--> v21\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":"0e4ff78e04609ca3cdd01e69103f56bdd92a5db9d16cff45bf43649d3baacae3","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v04\n- v04 --supports--> v25\n- v25 --supports--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v26\n- v26 --supports--> v10\n- v10 --supports--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v0d\n- v0d --refutes--> v2b\n- v2b --supports--> v0e\n- v0e --refutes--> v2e\n- v2e --refutes--> v29\n- v29 --supports--> v08\n- v08 --refutes--> v19\n- v19 --supports--> v2c\n- v2c --refutes--> v13\n- v13 --refutes--> v14\n- v14 --refutes--> v2f\n- v2f --supports--> v02\n- v02 --supports--> v21\n- v21 --refutes--> v2d\n- v2d --supports--> v16\n- v16 --refutes--> v2a\n- v2a --supports--> v09\n- v09 --refutes--> v24\n- v24 --supports--> v0c\n- v0c --supports--> v01\n- v01 --supports--> v30\n- v30 --refutes--> v1c\n- v1c --refutes--> v05\n- v05 --supports--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v17\n- v02 --supports--> v0a\n- v03 --refutes--> v22\n- v06 --refutes--> v27\n- v06 --supports--> v1a\n- v07 --refutes--> v1f\n- v08 --refutes--> v11\n- v0c --refutes--> v0b\n- v0c --supports--> v28\n- v0d --refutes--> v1e\n- v12 --refutes--> v0f\n- v12 --supports--> v00\n- v1b --refutes--> v23\n- v24 --supports--> v15\n- v25 --supports--> v20\n- v2c --supports--> v18\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":"c77ec2c988417a30e46193bfa903643394c2b028a3fc2fa670557dadc171b63d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --refutes--> v22\n- v22 --refutes--> v05\n- v05 --refutes--> v1a\n- v1a --refutes--> v2e\n- v2e --supports--> v1e\n- v1e --refutes--> v30\n- v30 --refutes--> v07\n- v07 --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --refutes--> v28\n- v28 --supports--> v2a\n- v2a --supports--> v2c\n- v2c --supports--> v25\n- v25 --supports--> v01\n- v01 --supports--> v2f\n- v2f --supports--> v0c\n- v0c --supports--> v13\n- v13 --supports--> v2b\n- v2b --supports--> v03\n- v03 --supports--> v06\n- v06 --refutes--> v20\n- v20 --supports--> v1d\n- v1d --supports--> v1c\n- v1c --supports--> v29\n- v29 --supports--> v26\n- v26 --refutes--> v1f\n- v1f --refutes--> v02\n- v02 --refutes--> v10\n- v10 --supports--> v15\n- v15 --refutes--> v11\n- v11 --refutes--> v0e\n- v0e --supports--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v06 --refutes--> v0b\n- v07 --refutes--> v0d\n- v0e --refutes--> v16\n- v11 --refutes--> v23\n- v13 --supports--> v21\n- v1b --refutes--> v27\n- v1c --refutes--> v19\n- v1e --supports--> v0a\n- v1e --supports--> v12\n- v20 --refutes--> v08\n- v22 --supports--> v24\n- v28 --refutes--> v17\n- v2b --refutes--> v04\n- v2c --supports--> v2d\n- v30 --refutes--> v09\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":"7cf6349014ef23c6b27ebe38211e00b2b60a681f2c5d9b789fcb052a25b9c65c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v2e\n- v2e --refutes--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v15\n- v15 --refutes--> v25\n- v25 --refutes--> v07\n- v07 --supports--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v10\n- v10 --refutes--> v0b\n- v0b --supports--> v1d\n- v1d --refutes--> v26\n- v26 --supports--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v18\n- v18 --refutes--> v22\n- v22 --supports--> v0f\n- v0f --supports--> v2f\n- v2f --refutes--> v1e\n- v1e --supports--> v13\n- v13 --supports--> v20\n- v20 --supports--> v02\n- v02 --refutes--> v1b\n- v1b --refutes--> v1a\n- v1a --supports--> v01\n- v01 --supports--> v08\n- v08 --supports--> v0c\n- v0c --supports--> v19\n- v19 --refutes--> v1c\n- v1c --refutes--> v29\n- v29 --supports--> v2d\n- v2d --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v00\n- v07 --refutes--> v16\n- v08 --refutes--> v2b\n- v0b --supports--> v1f\n- v0c --refutes--> v0a\n- v0f --supports--> v24\n- v12 --refutes--> v27\n- v12 --refutes--> v2a\n- v19 --refutes--> v05\n- v19 --refutes--> v30\n- v1b --refutes--> v2c\n- v1d --refutes--> v17\n- v22 --refutes--> v09\n- v22 --supports--> v11\n- v26 --refutes--> v28\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":"4a52b638ad9d7547230036744a1a397d04e3e449aadcb69673ac0ed4a0e9cbf0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v2f\n- v2f --supports--> v2d\n- v2d --refutes--> v21\n- v21 --refutes--> v1c\n- v1c --supports--> v17\n- v17 --supports--> v24\n- v24 --supports--> v28\n- v28 --refutes--> v00\n- v00 --supports--> v2b\n- v2b --refutes--> v01\n- v01 --supports--> v23\n- v23 --supports--> v26\n- v26 --refutes--> v07\n- v07 --supports--> v30\n- v30 --supports--> v0b\n- v0b --supports--> v1e\n- v1e --refutes--> v18\n- v18 --supports--> v04\n- v04 --refutes--> v20\n- v20 --refutes--> v1b\n- v1b --supports--> v11\n- v11 --supports--> v1d\n- v1d --supports--> v29\n- v29 --supports--> v25\n- v25 --refutes--> v0a\n- v0a --supports--> v16\n- v16 --refutes--> v2c\n- v2c --supports--> v12\n- v12 --supports--> v0e\n- v0e --refutes--> v2a\n- v2a --supports--> v1a\n- v1a --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v03\n- v07 --supports--> v08\n- v11 --supports--> v0f\n- v16 --refutes--> v06\n- v16 --supports--> v05\n- v17 --supports--> v1f\n- v1c --refutes--> v2e\n- v23 --supports--> v02\n- v24 --refutes--> v09\n- v24 --supports--> v10\n- v29 --refutes--> v14\n- v2c --refutes--> v0d\n- v2c --refutes--> v27\n- v2d --refutes--> v19\n- v2f --supports--> v22\n- v30 --refutes--> v15\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":"1187c5c7dd164ace84b99e69b9a5f957e2565c915cb09710eff5ea49541317f7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v2a\n- v2a --refutes--> v1f\n- v1f --refutes--> v2d\n- v2d --supports--> v15\n- v15 --supports--> v24\n- v24 --supports--> v07\n- v07 --refutes--> v28\n- v28 --supports--> v27\n- v27 --refutes--> v0c\n- v0c --supports--> v18\n- v18 --refutes--> v2f\n- v2f --refutes--> v1a\n- v1a --supports--> v08\n- v08 --supports--> v13\n- v13 --refutes--> v2e\n- v2e --supports--> v29\n- v29 --refutes--> v14\n- v14 --refutes--> v02\n- v02 --supports--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v21\n- v21 --refutes--> v22\n- v22 --supports--> v26\n- v26 --refutes--> v00\n- v00 --refutes--> v20\n- v20 --refutes--> v10\n- v10 --supports--> v05\n- v05 --supports--> v19\n- v19 --refutes--> v0f\n- v0f --supports--> v0d\n- v0d --supports--> v16\n- v16 --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v00 --supports--> v25\n- v08 --supports--> v1d\n- v0d --refutes--> v0b\n- v0d --refutes--> v17\n- v0f --refutes--> v1e\n- v0f --supports--> v0a\n- v12 --supports--> v1b\n- v15 --supports--> v0e\n- v16 --supports--> v1c\n- v20 --refutes--> v11\n- v20 --refutes--> v2c\n- v22 --supports--> v2b\n- v26 --supports--> v04\n- v26 --supports--> v30\n- v27 --refutes--> v01\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":"2dcf667261b14e7eea4c044dc51d695cff89189cd01a27bea6637c8f6629f451","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v28 --supports--> v03\n- v03 --refutes--> v1e\n- v1e --refutes--> v1c\n- v1c --refutes--> v15\n- v15 --refutes--> v13\n- v13 --refutes--> v0f\n- v0f --supports--> v09\n- v09 --refutes--> v18\n- v18 --refutes--> v10\n- v10 --refutes--> v22\n- v22 --refutes--> v11\n- v11 --supports--> v1d\n- v1d --refutes--> v02\n- v02 --supports--> v20\n- v20 --supports--> v1a\n- v1a --refutes--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v2f\n- v2f --supports--> v12\n- v12 --supports--> v30\n- v30 --refutes--> v0d\n- v0d --supports--> v0e\n- v0e --refutes--> v24\n- v24 --supports--> v29\n- v29 --refutes--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v14\n- v14 --refutes--> v0c\n- v0c --refutes--> v2e\n- v2e --refutes--> v26\n- v26 --refutes--> v0b\n- v0b --supports--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v17\n- v09 --refutes--> v2d\n- v0b --refutes--> v0a\n- v0f --supports--> v07\n- v0f --supports--> v27\n- v10 --refutes--> v1f\n- v11 --refutes--> v2b\n- v19 --refutes--> v05\n- v1a --refutes--> v04\n- v1c --supports--> v01\n- v1e --refutes--> v25\n- v22 --refutes--> v21\n- v22 --refutes--> v2c\n- v26 --supports--> v1b\n- v2f --refutes--> v2a\n- v30 --supports--> v00\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":"0099b14da415f57d7441de8f5839e85a4494464aa80aa99c7ec0f9eaeb4e58f2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v10 --refutes--> v18\n- v18 --refutes--> v14\n- v14 --supports--> v1f\n- v1f --refutes--> v02\n- v02 --refutes--> v2e\n- v2e --supports--> v24\n- v24 --refutes--> v1c\n- v1c --supports--> v29\n- v29 --supports--> v2b\n- v2b --refutes--> v15\n- v15 --supports--> v21\n- v21 --refutes--> v20\n- v20 --supports--> v0c\n- v0c --supports--> v07\n- v07 --refutes--> v23\n- v23 --supports--> v22\n- v22 --refutes--> v2f\n- v2f --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v1b\n- v1b --refutes--> v19\n- v19 --supports--> v09\n- v09 --supports--> v12\n- v12 --refutes--> v30\n- v30 --supports--> v1e\n- v1e --refutes--> v05\n- v05 --refutes--> v27\n- v27 --refutes--> v2c\n- v2c --supports--> v00\n- v00 --refutes--> v2d\n- v2d --supports--> v28\n- v28 --refutes--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v1a\n- v02 --refutes--> v26\n- v02 --supports--> v13\n- v05 --supports--> v03\n- v07 --supports--> v0d\n- v0c --refutes--> v08\n- v10 --refutes--> v0f\n- v10 --refutes--> v25\n- v18 --supports--> v11\n- v1e --refutes--> v06\n- v20 --refutes--> v17\n- v21 --supports--> v1d\n- v28 --refutes--> v0e\n- v2b --refutes--> v01\n- v2d --supports--> v0a\n- v30 --refutes--> v16\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":"d202031359b09e7aabc565edb446625cbb462f1e693fac87155ba0adfe4d4a3d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --refutes--> v0b\n- v0b --refutes--> v1a\n- v1a --supports--> v17\n- v17 --supports--> v20\n- v20 --refutes--> v22\n- v22 --supports--> v07\n- v07 --refutes--> v02\n- v02 --refutes--> v23\n- v23 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v2e\n- v2e --supports--> v2c\n- v2c --supports--> v1e\n- v1e --supports--> v00\n- v00 --supports--> v30\n- v30 --refutes--> v12\n- v12 --refutes--> v1c\n- v1c --supports--> v19\n- v19 --supports--> v1f\n- v1f --refutes--> v29\n- v29 --refutes--> v21\n- v21 --supports--> v09\n- v09 --refutes--> v01\n- v01 --refutes--> v03\n- v03 --refutes--> v25\n- v25 --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v05\n- v05 --supports--> v13\n- v13 --refutes--> v24\n- v24 --supports--> v16\n- v16 --refutes--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v18\n- v09 --refutes--> v04\n- v09 --supports--> v2f\n- v0c --supports--> v27\n- v13 --refutes--> v2d\n- v17 --refutes--> v26\n- v1c --supports--> v15\n- v1e --refutes--> v14\n- v20 --supports--> v2a\n- v21 --refutes--> v11\n- v24 --refutes--> v28\n- v25 --refutes--> v10\n- v2c --refutes--> v0a\n- v2c --refutes--> v0e\n- v2c --supports--> v1b\n- v2e --supports--> v0d\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":"dda7c144992f2e8c1e945f4ee2a36af95957f321c6ac43a27f171931c0198d35","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --supports--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v25\n- v25 --refutes--> v06\n- v06 --supports--> v21\n- v21 --supports--> v27\n- v27 --refutes--> v0b\n- v0b --refutes--> v2d\n- v2d --supports--> v03\n- v03 --refutes--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v1a\n- v1a --refutes--> v26\n- v26 --refutes--> v28\n- v28 --supports--> v1d\n- v1d --supports--> v1b\n- v1b --supports--> v2b\n- v2b --refutes--> v30\n- v30 --refutes--> v16\n- v16 --supports--> v12\n- v12 --refutes--> v0c\n- v0c --supports--> v0d\n- v0d --supports--> v2c\n- v2c --refutes--> v29\n- v29 --refutes--> v1c\n- v1c --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v1e\n- v1e --refutes--> v0f\n- v0f --supports--> v2f\n- v2f --supports--> v02\n- v02 --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v15\n- v03 --supports--> v22\n- v0b --supports--> v2e\n- v11 --refutes--> v07\n- v12 --refutes--> v14\n- v16 --refutes--> v00\n- v16 --refutes--> v0e\n- v16 --refutes--> v2a\n- v16 --supports--> v1f\n- v17 --refutes--> v05\n- v1b --refutes--> v18\n- v1e --refutes--> v10\n- v23 --supports--> v0a\n- v25 --refutes--> v04\n- v26 --refutes--> v19\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":"d34c8a48946e6691da0bc36732a9e08bab2f4a0b2cbbd6ce2f777e612439465d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --supports--> v23\n- v23 --refutes--> v07\n- v07 --supports--> v01\n- v01 --supports--> v1d\n- v1d --refutes--> v10\n- v10 --supports--> v0b\n- v0b --supports--> v03\n- v03 --supports--> v12\n- v12 --refutes--> v2c\n- v2c --refutes--> v0d\n- v0d --supports--> v14\n- v14 --refutes--> v00\n- v00 --supports--> v1e\n- v1e --supports--> v28\n- v28 --supports--> v2e\n- v2e --supports--> v21\n- v21 --supports--> v29\n- v29 --supports--> v22\n- v22 --refutes--> v26\n- v26 --supports--> v27\n- v27 --supports--> v19\n- v19 --supports--> v20\n- v20 --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v09\n- v09 --refutes--> v15\n- v15 --supports--> v2d\n- v2d --supports--> v30\n- v30 --refutes--> v1b\n- v1b --refutes--> v1a\n- v1a --refutes--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v13\n- v07 --supports--> v05\n- v0c --supports--> v11\n- v10 --supports--> v2f\n- v14 --refutes--> v1c\n- v14 --supports--> v16\n- v15 --supports--> v18\n- v19 --refutes--> v2b\n- v1a --supports--> v25\n- v1b --refutes--> v0f\n- v1d --supports--> v04\n- v1e --refutes--> v0e\n- v22 --supports--> v2a\n- v27 --refutes--> v24\n- v29 --supports--> v06\n- v2e --supports--> v1f\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":"4e54a4c080d84006998896ec936e6b6066895c27d0b1f685b4163894cda2e1f7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v19\n- v19 --refutes--> v30\n- v30 --supports--> v2e\n- v2e --supports--> v16\n- v16 --refutes--> v26\n- v26 --refutes--> v0b\n- v0b --refutes--> v0c\n- v0c --supports--> v1c\n- v1c --refutes--> v2b\n- v2b --refutes--> v27\n- v27 --refutes--> v23\n- v23 --refutes--> v24\n- v24 --refutes--> v12\n- v12 --refutes--> v1a\n- v1a --supports--> v00\n- v00 --supports--> v2a\n- v2a --supports--> v29\n- v29 --supports--> v2c\n- v2c --supports--> v06\n- v06 --refutes--> v08\n- v08 --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v1d\n- v1d --supports--> v21\n- v21 --refutes--> v09\n- v09 --supports--> v17\n- v17 --refutes--> v28\n- v28 --supports--> v07\n- v07 --supports--> v0d\n- v0d --supports--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v1b\n- v04 --refutes--> v22\n- v05 --supports--> v01\n- v06 --refutes--> v15\n- v06 --supports--> v02\n- v09 --refutes--> v10\n- v09 --refutes--> v11\n- v0a --supports--> v20\n- v0b --supports--> v18\n- v0f --refutes--> v2f\n- v17 --supports--> v1e\n- v1c --supports--> v0e\n- v1d --supports--> v1f\n- v26 --refutes--> v25\n- v28 --refutes--> v14\n- v2c --supports--> v2d\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":"3faff360fae564fff105a99a970c7ca87e69b4e96b70ddcff61078ddc1f375e9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v2b\n- v2b --supports--> v19\n- v19 --refutes--> v12\n- v12 --refutes--> v1f\n- v1f --refutes--> v1a\n- v1a --supports--> v16\n- v16 --supports--> v23\n- v23 --refutes--> v1c\n- v1c --refutes--> v11\n- v11 --refutes--> v26\n- v26 --supports--> v08\n- v08 --supports--> v1b\n- v1b --refutes--> v0d\n- v0d --refutes--> v28\n- v28 --supports--> v1e\n- v1e --supports--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --supports--> v20\n- v20 --supports--> v1d\n- v1d --refutes--> v00\n- v00 --supports--> v0e\n- v0e --refutes--> v2f\n- v2f --supports--> v18\n- v18 --refutes--> v25\n- v25 --supports--> v17\n- v17 --supports--> v14\n- v14 --refutes--> v30\n- v30 --refutes--> v03\n- v03 --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v01 --refutes--> v27\n- v0d --supports--> v02\n- v11 --refutes--> v2d\n- v14 --refutes--> v13\n- v14 --supports--> v2e\n- v17 --refutes--> v05\n- v18 --refutes--> v0b\n- v19 --refutes--> v29\n- v19 --supports--> v2c\n- v1b --supports--> v24\n- v1c --supports--> v22\n- v1d --supports--> v2a\n- v20 --supports--> v21\n- v28 --refutes--> v06\n- v30 --supports--> v15\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":"171e6ee6962e43f4eda0b8b176aa2b7082274c52affc95f9e8f4151f4f600a62","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --refutes--> v1e\n- v1e --supports--> v01\n- v01 --refutes--> v02\n- v02 --refutes--> v2b\n- v2b --refutes--> v2e\n- v2e --supports--> v27\n- v27 --refutes--> v17\n- v17 --supports--> v14\n- v14 --supports--> v09\n- v09 --refutes--> v05\n- v05 --supports--> v12\n- v12 --refutes--> v30\n- v30 --supports--> v1d\n- v1d --supports--> v16\n- v16 --supports--> v28\n- v28 --refutes--> v15\n- v15 --refutes--> v03\n- v03 --refutes--> v21\n- v21 --supports--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v0d\n- v0d --refutes--> v29\n- v29 --supports--> v23\n- v23 --refutes--> v1a\n- v1a --refutes--> v1b\n- v1b --refutes--> v0f\n- v0f --supports--> v2a\n- v2a --refutes--> v2c\n- v2c --supports--> v11\n- v11 --supports--> v1c\n- v1c --refutes--> v1f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v20\n- v03 --supports--> v13\n- v05 --supports--> v08\n- v07 --refutes--> v26\n- v09 --refutes--> v0a\n- v0f --refutes--> v2d\n- v17 --supports--> v19\n- v17 --supports--> v22\n- v1b --refutes--> v04\n- v1b --refutes--> v0e\n- v1c --refutes--> v0b\n- v1c --supports--> v24\n- v1e --refutes--> v10\n- v29 --refutes--> v2f\n- v2b --refutes--> v06\n- v2e --refutes--> v18\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":"2f3e35227d6a7200bb20dd93c92a9903c2c006e10346875467aa8733d0a96aa0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --supports--> v29\n- v29 --refutes--> v07\n- v07 --supports--> v1c\n- v1c --supports--> v02\n- v02 --refutes--> v2d\n- v2d --refutes--> v28\n- v28 --supports--> v0c\n- v0c --supports--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v10\n- v10 --refutes--> v18\n- v18 --supports--> v00\n- v00 --refutes--> v13\n- v13 --supports--> v23\n- v23 --supports--> v2f\n- v2f --supports--> v12\n- v12 --refutes--> v16\n- v16 --refutes--> v0e\n- v0e --refutes--> v0a\n- v0a --refutes--> v0d\n- v0d --supports--> v26\n- v26 --supports--> v2e\n- v2e --supports--> v05\n- v05 --refutes--> v22\n- v22 --refutes--> v06\n- v06 --refutes--> v1d\n- v1d --supports--> v17\n- v17 --refutes--> v21\n- v21 --supports--> v0f\n- v0f --refutes--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v2a\n- v04 --supports--> v03\n- v06 --supports--> v01\n- v07 --refutes--> v27\n- v0e --supports--> v1b\n- v0f --refutes--> v14\n- v0f --supports--> v24\n- v16 --refutes--> v1f\n- v19 --supports--> v30\n- v1c --refutes--> v1a\n- v1d --supports--> v2c\n- v23 --refutes--> v08\n- v25 --refutes--> v1e\n- v29 --refutes--> v20\n- v2f --refutes--> v0b\n- v2f --supports--> v2b\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":"172f2d1ed1fdb30bbff13ebfa5fad3f6adf4ab9529c04f9abe52f51a46dbbb58","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v26\n- v26 --refutes--> v2c\n- v2c --refutes--> v2e\n- v2e --refutes--> v0f\n- v0f --supports--> v20\n- v20 --refutes--> v17\n- v17 --refutes--> v02\n- v02 --refutes--> v11\n- v11 --refutes--> v04\n- v04 --supports--> v30\n- v30 --supports--> v1c\n- v1c --refutes--> v00\n- v00 --refutes--> v06\n- v06 --supports--> v1a\n- v1a --refutes--> v23\n- v23 --supports--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v25\n- v25 --refutes--> v12\n- v12 --supports--> v1f\n- v1f --supports--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v0a\n- v0a --refutes--> v2d\n- v2d --refutes--> v1d\n- v1d --refutes--> v21\n- v21 --refutes--> v2f\n- v2f --refutes--> v1b\n- v1b --supports--> v27\n- v27 --supports--> v10\n- v10 --supports--> v01\n- v01 --refutes--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0b\n- v01 --supports--> v0c\n- v03 --refutes--> v09\n- v10 --refutes--> v13\n- v12 --refutes--> v14\n- v18 --supports--> v08\n- v19 --refutes--> v2a\n- v19 --supports--> v0e\n- v1a --supports--> v16\n- v1e --supports--> v05\n- v1e --supports--> v29\n- v1f --refutes--> v22\n- v1f --supports--> v2b\n- v23 --supports--> v28\n- v2f --supports--> v0d\n- v30 --supports--> v07\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":"05a169828af69e613a851385ccc2132796c8f7c5e2dec80469cf83a676b83272","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2b --refutes--> v16\n- v16 --refutes--> v0e\n- v0e --refutes--> v06\n- v06 --supports--> v09\n- v09 --refutes--> v2d\n- v2d --supports--> v23\n- v23 --supports--> v18\n- v18 --supports--> v2a\n- v2a --refutes--> v00\n- v00 --supports--> v20\n- v20 --supports--> v13\n- v13 --supports--> v28\n- v28 --supports--> v02\n- v02 --supports--> v14\n- v14 --supports--> v17\n- v17 --refutes--> v1f\n- v1f --supports--> v19\n- v19 --refutes--> v26\n- v26 --refutes--> v15\n- v15 --supports--> v22\n- v22 --refutes--> v21\n- v21 --supports--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v27\n- v27 --supports--> v0d\n- v0d --refutes--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v30\n- v30 --supports--> v2f\n- v2f --refutes--> v1d\n- v1d --refutes--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v29\n- v05 --refutes--> v24\n- v06 --supports--> v07\n- v14 --refutes--> v2e\n- v17 --refutes--> v1a\n- v19 --refutes--> v1e\n- v20 --supports--> v08\n- v21 --refutes--> v12\n- v21 --refutes--> v1b\n- v21 --supports--> v0b\n- v23 --supports--> v1c\n- v27 --refutes--> v2c\n- v28 --supports--> v25\n- v2f --supports--> v0f\n- v30 --refutes--> v0a\n- v30 --supports--> v04\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":"be75a06badd3bdb1cbe3a2e9932c9d52d1397f531e78820d7c46b338b9e8b48c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --refutes--> v1c\n- v1c --refutes--> v22\n- v22 --refutes--> v01\n- v01 --supports--> v1b\n- v1b --supports--> v06\n- v06 --supports--> v14\n- v14 --refutes--> v2b\n- v2b --supports--> v02\n- v02 --refutes--> v20\n- v20 --supports--> v21\n- v21 --supports--> v26\n- v26 --refutes--> v0f\n- v0f --supports--> v1f\n- v1f --supports--> v0c\n- v0c --refutes--> v15\n- v15 --supports--> v1d\n- v1d --refutes--> v2a\n- v2a --refutes--> v03\n- v03 --refutes--> v1e\n- v1e --refutes--> v12\n- v12 --supports--> v07\n- v07 --refutes--> v30\n- v30 --refutes--> v2c\n- v2c --refutes--> v08\n- v08 --supports--> v24\n- v24 --supports--> v09\n- v09 --supports--> v04\n- v04 --supports--> v10\n- v10 --refutes--> v17\n- v17 --refutes--> v13\n- v13 --supports--> v0b\n- v0b --refutes--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v29\n- v04 --supports--> v0d\n- v09 --refutes--> v0e\n- v0c --refutes--> v1a\n- v0f --supports--> v23\n- v12 --supports--> v19\n- v13 --refutes--> v16\n- v14 --refutes--> v00\n- v1c --refutes--> v2e\n- v1d --refutes--> v05\n- v1d --supports--> v28\n- v21 --refutes--> v11\n- v22 --supports--> v0a\n- v25 --refutes--> v18\n- v25 --refutes--> v2f\n- v30 --supports--> v2d\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":"57543b0b1700d60712a8012ef0f4e427f23b48ebc0f7faae7d51421654dfdba0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --supports--> v00\n- v00 --supports--> v13\n- v13 --refutes--> v16\n- v16 --refutes--> v04\n- v04 --supports--> v0f\n- v0f --supports--> v1d\n- v1d --refutes--> v05\n- v05 --supports--> v2f\n- v2f --supports--> v2a\n- v2a --supports--> v07\n- v07 --supports--> v18\n- v18 --supports--> v01\n- v01 --refutes--> v25\n- v25 --supports--> v1a\n- v1a --supports--> v0d\n- v0d --refutes--> v1e\n- v1e --supports--> v0e\n- v0e --refutes--> v2e\n- v2e --refutes--> v0a\n- v0a --refutes--> v17\n- v17 --refutes--> v23\n- v23 --supports--> v2c\n- v2c --refutes--> v29\n- v29 --refutes--> v24\n- v24 --supports--> v0c\n- v0c --supports--> v21\n- v21 --supports--> v30\n- v30 --supports--> v06\n- v06 --refutes--> v12\n- v12 --refutes--> v26\n- v26 --supports--> v14\n- v14 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v08\n- v05 --refutes--> v1f\n- v0f --refutes--> v15\n- v12 --supports--> v2b\n- v14 --supports--> v03\n- v14 --supports--> v0b\n- v1a --refutes--> v11\n- v1d --supports--> v20\n- v22 --refutes--> v02\n- v22 --supports--> v09\n- v24 --refutes--> v19\n- v24 --supports--> v28\n- v24 --supports--> v2d\n- v26 --supports--> v27\n- v29 --supports--> v1b\n- v2e --supports--> v10\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":"1bde9f2841522ff49bf695224507228a3ee7078c1642cb3b08fc1df6a0c10225","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --supports--> v19\n- v19 --supports--> v03\n- v03 --refutes--> v1a\n- v1a --refutes--> v1f\n- v1f --supports--> v29\n- v29 --supports--> v06\n- v06 --supports--> v07\n- v07 --refutes--> v26\n- v26 --supports--> v16\n- v16 --supports--> v0b\n- v0b --supports--> v11\n- v11 --supports--> v08\n- v08 --refutes--> v18\n- v18 --supports--> v2f\n- v2f --supports--> v2d\n- v2d --supports--> v24\n- v24 --refutes--> v1d\n- v1d --supports--> v28\n- v28 --refutes--> v21\n- v21 --supports--> v13\n- v13 --refutes--> v0f\n- v0f --refutes--> v15\n- v15 --supports--> v0c\n- v0c --refutes--> v02\n- v02 --supports--> v09\n- v09 --refutes--> v20\n- v20 --refutes--> v25\n- v25 --supports--> v1e\n- v1e --supports--> v2e\n- v2e --refutes--> v30\n- v30 --supports--> v2a\n- v2a --refutes--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v1b\n- v03 --supports--> v27\n- v08 --supports--> v10\n- v0c --refutes--> v0e\n- v0f --refutes--> v04\n- v1a --refutes--> v14\n- v1f --supports--> v01\n- v20 --refutes--> v12\n- v22 --refutes--> v00\n- v22 --refutes--> v1c\n- v24 --refutes--> v0d\n- v26 --refutes--> v17\n- v29 --refutes--> v05\n- v2a --refutes--> v2b\n- v2a --refutes--> v2c\n- v2e --supports--> v23\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":"743bb64e11b0e2f4336bf0a16371c7c81f84ca2f22958d93cf92ecb3c238ad8f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2a --supports--> v2f\n- v2f --refutes--> v05\n- v05 --refutes--> v19\n- v19 --refutes--> v14\n- v14 --supports--> v04\n- v04 --refutes--> v27\n- v27 --refutes--> v07\n- v07 --supports--> v2e\n- v2e --refutes--> v00\n- v00 --refutes--> v30\n- v30 --supports--> v0c\n- v0c --refutes--> v16\n- v16 --supports--> v24\n- v24 --supports--> v22\n- v22 --refutes--> v2b\n- v2b --refutes--> v18\n- v18 --supports--> v03\n- v03 --refutes--> v08\n- v08 --supports--> v1b\n- v1b --refutes--> v26\n- v26 --supports--> v17\n- v17 --supports--> v09\n- v09 --supports--> v1f\n- v1f --refutes--> v01\n- v01 --refutes--> v1e\n- v1e --supports--> v2d\n- v2d --refutes--> v1d\n- v1d --refutes--> v29\n- v29 --refutes--> v10\n- v10 --refutes--> v21\n- v21 --refutes--> v1c\n- v1c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v15\n- v03 --refutes--> v0b\n- v04 --supports--> v0e\n- v05 --refutes--> v0d\n- v0c --supports--> v28\n- v16 --refutes--> v2c\n- v1b --refutes--> v20\n- v1b --supports--> v11\n- v1b --supports--> v1a\n- v1b --supports--> v25\n- v1e --supports--> v02\n- v1f --refutes--> v23\n- v21 --supports--> v12\n- v2f --supports--> v0a\n- v30 --refutes--> v0f\n- v30 --supports--> v13\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":"d678b30eae1a6597f1fa04841a24af5c04e1752a0aad5a5481087ee723a64042","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v1d\n- v1d --supports--> v2b\n- v2b --supports--> v1e\n- v1e --refutes--> v16\n- v16 --supports--> v1b\n- v1b --refutes--> v12\n- v12 --supports--> v22\n- v22 --refutes--> v30\n- v30 --refutes--> v01\n- v01 --supports--> v11\n- v11 --supports--> v0e\n- v0e --supports--> v29\n- v29 --supports--> v02\n- v02 --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v28\n- v28 --supports--> v17\n- v17 --supports--> v18\n- v18 --supports--> v24\n- v24 --supports--> v13\n- v13 --supports--> v23\n- v23 --supports--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v03\n- v03 --refutes--> v20\n- v20 --supports--> v14\n- v14 --refutes--> v2c\n- v2c --refutes--> v08\n- v08 --refutes--> v25\n- v25 --supports--> v09\n- v09 --supports--> v21\n- v21 --supports--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v1f\n- v0d --refutes--> v2d\n- v10 --supports--> v06\n- v12 --refutes--> v27\n- v12 --supports--> v2f\n- v13 --refutes--> v0c\n- v17 --refutes--> v00\n- v17 --supports--> v0a\n- v18 --supports--> v05\n- v1d --refutes--> v2e\n- v1e --supports--> v0f\n- v22 --refutes--> v2a\n- v23 --supports--> v04\n- v24 --supports--> v19\n- v2b --refutes--> v1c\n- v30 --supports--> v26\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":"e61c2fe197f362b7c858793acdc950e25266f1cee2d1e8beab59ace154c72272","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v26 --refutes--> v16\n- v16 --refutes--> v2a\n- v2a --supports--> v30\n- v30 --supports--> v09\n- v09 --refutes--> v20\n- v20 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v2e\n- v2e --supports--> v04\n- v04 --refutes--> v1c\n- v1c --supports--> v07\n- v07 --refutes--> v06\n- v06 --refutes--> v14\n- v14 --supports--> v02\n- v02 --supports--> v27\n- v27 --refutes--> v2d\n- v2d --supports--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v18\n- v18 --refutes--> v11\n- v11 --refutes--> v10\n- v10 --refutes--> v15\n- v15 --refutes--> v05\n- v05 --refutes--> v2f\n- v2f --refutes--> v13\n- v13 --supports--> v23\n- v23 --supports--> v1d\n- v1d --supports--> v0e\n- v0e --supports--> v08\n- v08 --refutes--> v24\n- v24 --supports--> v0b\n- v0b --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1b\n- v06 --supports--> v25\n- v07 --supports--> v17\n- v0b --supports--> v19\n- v13 --supports--> v0d\n- v13 --supports--> v2b\n- v14 --supports--> v21\n- v14 --supports--> v28\n- v16 --supports--> v22\n- v18 --refutes--> v2c\n- v1c --refutes--> v1a\n- v1c --supports--> v0f\n- v1c --supports--> v1e\n- v1d --refutes--> v01\n- v1d --refutes--> v1f\n- v23 --supports--> v29\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":"ee6e679ed2749d1f2da13005c54fde0fcf35d7e368e0faba48f2741aab037f82","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --supports--> v0b\n- v0b --refutes--> v13\n- v13 --supports--> v26\n- v26 --refutes--> v06\n- v06 --refutes--> v10\n- v10 --supports--> v17\n- v17 --supports--> v22\n- v22 --supports--> v07\n- v07 --supports--> v21\n- v21 --refutes--> v0e\n- v0e --refutes--> v30\n- v30 --refutes--> v24\n- v24 --refutes--> v1d\n- v1d --supports--> v1a\n- v1a --supports--> v1f\n- v1f --supports--> v28\n- v28 --supports--> v1c\n- v1c --refutes--> v18\n- v18 --refutes--> v25\n- v25 --refutes--> v0c\n- v0c --refutes--> v2a\n- v2a --supports--> v01\n- v01 --supports--> v04\n- v04 --supports--> v2d\n- v2d --refutes--> v0a\n- v0a --supports--> v0f\n- v0f --supports--> v2f\n- v2f --refutes--> v15\n- v15 --refutes--> v1e\n- v1e --supports--> v00\n- v00 --supports--> v27\n- v27 --supports--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v14\n- v06 --refutes--> v16\n- v06 --supports--> v11\n- v07 --refutes--> v2c\n- v0e --refutes--> v2b\n- v15 --refutes--> v09\n- v17 --refutes--> v0d\n- v1a --supports--> v20\n- v1f --supports--> v05\n- v21 --supports--> v19\n- v24 --supports--> v03\n- v25 --supports--> v12\n- v26 --supports--> v1b\n- v28 --refutes--> v02\n- v30 --refutes--> v23\n- v30 --refutes--> v29\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":"1fb1583399beb19928db496b4057ce6bbdb659a859c158b0edf420a56b1e8f7d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1b --refutes--> v22\n- v22 --refutes--> v15\n- v15 --supports--> v03\n- v03 --refutes--> v21\n- v21 --refutes--> v14\n- v14 --refutes--> v26\n- v26 --refutes--> v06\n- v06 --supports--> v0b\n- v0b --refutes--> v2b\n- v2b --refutes--> v09\n- v09 --supports--> v2f\n- v2f --supports--> v19\n- v19 --supports--> v0e\n- v0e --refutes--> v25\n- v25 --supports--> v11\n- v11 --refutes--> v16\n- v16 --supports--> v0d\n- v0d --supports--> v18\n- v18 --supports--> v28\n- v28 --supports--> v04\n- v04 --refutes--> v1e\n- v1e --supports--> v13\n- v13 --supports--> v17\n- v17 --refutes--> v0c\n- v0c --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v2d\n- v2d --refutes--> v30\n- v30 --refutes--> v1c\n- v1c --refutes--> v10\n- v10 --refutes--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v1f\n- v06 --supports--> v2a\n- v0c --supports--> v29\n- v11 --refutes--> v2c\n- v12 --refutes--> v1a\n- v12 --refutes--> v23\n- v12 --supports--> v01\n- v13 --supports--> v08\n- v15 --refutes--> v24\n- v15 --supports--> v05\n- v1e --refutes--> v1d\n- v21 --refutes--> v02\n- v21 --supports--> v20\n- v22 --supports--> v2e\n- v28 --supports--> v07\n- v30 --refutes--> v0f\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":"55bc7791ff7079943c086facbb843cc98953e7837b015d31f31e54344ff019b8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --refutes--> v24\n- v24 --supports--> v25\n- v25 --refutes--> v2c\n- v2c --supports--> v0c\n- v0c --supports--> v2d\n- v2d --refutes--> v04\n- v04 --refutes--> v0f\n- v0f --supports--> v13\n- v13 --refutes--> v28\n- v28 --supports--> v15\n- v15 --supports--> v18\n- v18 --supports--> v19\n- v19 --refutes--> v01\n- v01 --supports--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v0b\n- v0b --refutes--> v30\n- v30 --supports--> v09\n- v09 --supports--> v1b\n- v1b --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v1f\n- v1f --supports--> v00\n- v00 --supports--> v2b\n- v2b --refutes--> v16\n- v16 --supports--> v20\n- v20 --refutes--> v0a\n- v0a --refutes--> v2f\n- v2f --refutes--> v23\n- v23 --refutes--> v06\n- v06 --refutes--> v1e\n- v1e --supports--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v14\n- v02 --supports--> v17\n- v02 --supports--> v22\n- v09 --refutes--> v1d\n- v0a --supports--> v1a\n- v0b --refutes--> v1c\n- v0e --supports--> v26\n- v0f --supports--> v29\n- v18 --refutes--> v10\n- v19 --supports--> v05\n- v1f --refutes--> v27\n- v28 --supports--> v2a\n- v2b --refutes--> v0d\n- v2b --supports--> v08\n- v2c --refutes--> v03\n- v2f --supports--> v21\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":"a015da2b9ab04d114a5f440560522d79e72183be06f16465cf5c396b6f2ba016","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v26 --supports--> v10\n- v10 --supports--> v2c\n- v2c --supports--> v0d\n- v0d --supports--> v23\n- v23 --supports--> v16\n- v16 --refutes--> v1a\n- v1a --supports--> v27\n- v27 --supports--> v05\n- v05 --refutes--> v04\n- v04 --refutes--> v21\n- v21 --supports--> v29\n- v29 --refutes--> v0e\n- v0e --refutes--> v0b\n- v0b --refutes--> v1d\n- v1d --supports--> v01\n- v01 --refutes--> v08\n- v08 --supports--> v25\n- v25 --refutes--> v06\n- v06 --refutes--> v2a\n- v2a --supports--> v00\n- v00 --refutes--> v1c\n- v1c --supports--> v13\n- v13 --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --supports--> v2b\n- v2b --refutes--> v1f\n- v1f --refutes--> v17\n- v17 --refutes--> v1e\n- v1e --refutes--> v03\n- v03 --refutes--> v0c\n- v0c --supports--> v2f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --supports--> v0f\n- v0b --supports--> v18\n- v0c --refutes--> v22\n- v0c --supports--> v09\n- v0e --supports--> v15\n- v10 --refutes--> v02\n- v11 --refutes--> v20\n- v13 --supports--> v1b\n- v13 --supports--> v2d\n- v17 --refutes--> v19\n- v17 --refutes--> v28\n- v1f --refutes--> v14\n- v26 --refutes--> v24\n- v27 --refutes--> v30\n- v27 --supports--> v2e\n- v2a --refutes--> v12\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":"02c20d27c41745a6a6e98858213ee68ea94c72661a3fbdb4baefeb5e284bdde7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2d --refutes--> v22\n- v22 --supports--> v0f\n- v0f --supports--> v11\n- v11 --refutes--> v26\n- v26 --refutes--> v16\n- v16 --supports--> v1b\n- v1b --refutes--> v2e\n- v2e --supports--> v2c\n- v2c --refutes--> v1c\n- v1c --refutes--> v10\n- v10 --refutes--> v13\n- v13 --refutes--> v04\n- v04 --supports--> v12\n- v12 --refutes--> v21\n- v21 --supports--> v1a\n- v1a --refutes--> v1e\n- v1e --refutes--> v2b\n- v2b --supports--> v08\n- v08 --supports--> v00\n- v00 --supports--> v0d\n- v0d --refutes--> v09\n- v09 --refutes--> v18\n- v18 --refutes--> v07\n- v07 --supports--> v17\n- v17 --supports--> v1f\n- v1f --refutes--> v0a\n- v0a --supports--> v05\n- v05 --supports--> v30\n- v30 --refutes--> v2f\n- v2f --refutes--> v2a\n- v2a --refutes--> v0c\n- v0c --refutes--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v1d\n- v09 --supports--> v06\n- v0a --refutes--> v19\n- v0f --supports--> v01\n- v13 --supports--> v20\n- v17 --refutes--> v0b\n- v17 --refutes--> v23\n- v1a --supports--> v02\n- v1b --supports--> v28\n- v21 --supports--> v14\n- v26 --supports--> v27\n- v2a --refutes--> v03\n- v2b --refutes--> v24\n- v2b --supports--> v29\n- v2f --refutes--> v25\n- v2f --supports--> v15\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":"ae3b1931863a00c0caaf7ad0881319937748b84052d8913af37f580fb2986a5d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --supports--> v28\n- v28 --refutes--> v13\n- v13 --supports--> v1c\n- v1c --refutes--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v1f\n- v1f --refutes--> v1b\n- v1b --supports--> v10\n- v10 --supports--> v29\n- v29 --supports--> v2a\n- v2a --supports--> v02\n- v02 --refutes--> v0e\n- v0e --refutes--> v16\n- v16 --supports--> v21\n- v21 --refutes--> v2b\n- v2b --refutes--> v00\n- v00 --supports--> v0b\n- v0b --supports--> v14\n- v14 --supports--> v26\n- v26 --supports--> v19\n- v19 --refutes--> v20\n- v20 --refutes--> v17\n- v17 --refutes--> v1a\n- v1a --supports--> v0a\n- v0a --supports--> v03\n- v03 --supports--> v1d\n- v1d --refutes--> v09\n- v09 --refutes--> v24\n- v24 --refutes--> v08\n- v08 --supports--> v22\n- v22 --refutes--> v25\n- v25 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v05\n- v08 --refutes--> v27\n- v09 --refutes--> v2f\n- v0a --refutes--> v11\n- v0b --supports--> v0f\n- v0c --refutes--> v12\n- v10 --refutes--> v01\n- v10 --supports--> v2d\n- v10 --supports--> v2e\n- v13 --supports--> v0d\n- v1a --refutes--> v07\n- v21 --supports--> v04\n- v21 --supports--> v23\n- v24 --refutes--> v30\n- v28 --supports--> v2c\n- v29 --refutes--> v15\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":"8682b803b4f2287304a6ca61feebb21faef7d61b768437223cf1118930b5a909","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v25\n- v25 --refutes--> v07\n- v07 --refutes--> v18\n- v18 --refutes--> v23\n- v23 --supports--> v0e\n- v0e --refutes--> v27\n- v27 --refutes--> v1f\n- v1f --refutes--> v0c\n- v0c --supports--> v2f\n- v2f --supports--> v29\n- v29 --supports--> v1a\n- v1a --supports--> v01\n- v01 --refutes--> v10\n- v10 --supports--> v26\n- v26 --refutes--> v05\n- v05 --supports--> v2d\n- v2d --refutes--> v17\n- v17 --supports--> v08\n- v08 --refutes--> v12\n- v12 --supports--> v2a\n- v2a --supports--> v30\n- v30 --refutes--> v0a\n- v0a --refutes--> v19\n- v19 --refutes--> v00\n- v00 --refutes--> v2b\n- v2b --supports--> v02\n- v02 --refutes--> v24\n- v24 --supports--> v2e\n- v2e --refutes--> v03\n- v03 --supports--> v21\n- v21 --refutes--> v20\n- v20 --refutes--> v13\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v03 --refutes--> v06\n- v05 --supports--> v16\n- v05 --supports--> v22\n- v08 --refutes--> v1e\n- v08 --supports--> v11\n- v0e --refutes--> v09\n- v10 --refutes--> v14\n- v15 --refutes--> v1b\n- v19 --supports--> v0d\n- v20 --refutes--> v1c\n- v26 --supports--> v28\n- v27 --refutes--> v0b\n- v2a --refutes--> v1d\n- v2a --refutes--> v2c\n- v2b --supports--> v0f\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":"3cc55b4648fcef40e359ca9dfd4948ff636c9a81afe444ccfe6eec2400f23b3a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v22\n- v22 --supports--> v0e\n- v0e --refutes--> v15\n- v15 --supports--> v25\n- v25 --refutes--> v2a\n- v2a --supports--> v28\n- v28 --refutes--> v2d\n- v2d --supports--> v04\n- v04 --supports--> v21\n- v21 --refutes--> v16\n- v16 --supports--> v00\n- v00 --supports--> v05\n- v05 --supports--> v0c\n- v0c --refutes--> v14\n- v14 --refutes--> v13\n- v13 --supports--> v06\n- v06 --supports--> v1a\n- v1a --supports--> v2b\n- v2b --refutes--> v2c\n- v2c --refutes--> v2f\n- v2f --supports--> v18\n- v18 --supports--> v12\n- v12 --supports--> v01\n- v01 --refutes--> v07\n- v07 --refutes--> v26\n- v26 --supports--> v03\n- v03 --supports--> v0f\n- v0f --supports--> v29\n- v29 --supports--> v10\n- v10 --supports--> v24\n- v24 --refutes--> v1c\n- v1c --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v2e\n- v01 --refutes--> v20\n- v01 --refutes--> v27\n- v08 --refutes--> v1d\n- v0e --refutes--> v30\n- v12 --supports--> v11\n- v13 --refutes--> v1b\n- v14 --supports--> v17\n- v1a --supports--> v0a\n- v2a --refutes--> v1f\n- v2c --refutes--> v09\n- v2c --supports--> v0d\n- v2d --supports--> v02\n- v2d --supports--> v23\n- v2f --supports--> v19\n- v2f --supports--> v1e\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":"b7c80fa9e5252ae13540b753d5b50f1d35eb2c6644395476a6542f4bf4ce1e69","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v08\n- v08 --supports--> v22\n- v22 --supports--> v2e\n- v2e --refutes--> v0b\n- v0b --refutes--> v24\n- v24 --supports--> v26\n- v26 --refutes--> v01\n- v01 --refutes--> v1c\n- v1c --supports--> v06\n- v06 --refutes--> v29\n- v29 --supports--> v13\n- v13 --refutes--> v11\n- v11 --refutes--> v02\n- v02 --refutes--> v15\n- v15 --supports--> v16\n- v16 --refutes--> v2f\n- v2f --refutes--> v28\n- v28 --supports--> v0f\n- v0f --supports--> v1d\n- v1d --refutes--> v21\n- v21 --refutes--> v10\n- v10 --refutes--> v1b\n- v1b --supports--> v30\n- v30 --supports--> v0c\n- v0c --supports--> v04\n- v04 --supports--> v07\n- v07 --refutes--> v0e\n- v0e --supports--> v12\n- v12 --supports--> v09\n- v09 --refutes--> v1f\n- v1f --supports--> v2d\n- v2d --refutes--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v03\n- v04 --refutes--> v25\n- v07 --supports--> v17\n- v09 --supports--> v0a\n- v0c --supports--> v18\n- v0e --supports--> v19\n- v0f --refutes--> v14\n- v0f --refutes--> v23\n- v0f --supports--> v2a\n- v12 --supports--> v1a\n- v1b --supports--> v0d\n- v21 --refutes--> v27\n- v28 --supports--> v2c\n- v2f --refutes--> v1e\n- v2f --refutes--> v2b\n- v30 --refutes--> v05\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":"073420a1ad957e723fc2fad6a90fd035dfbf69d1e5d819aa93d04a2b29b825d8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2d --supports--> v24\n- v24 --refutes--> v1e\n- v1e --refutes--> v25\n- v25 --refutes--> v30\n- v30 --refutes--> v05\n- v05 --refutes--> v08\n- v08 --supports--> v0d\n- v0d --refutes--> v27\n- v27 --supports--> v28\n- v28 --refutes--> v19\n- v19 --supports--> v2e\n- v2e --refutes--> v1f\n- v1f --supports--> v20\n- v20 --supports--> v0a\n- v0a --refutes--> v26\n- v26 --refutes--> v0c\n- v0c --refutes--> v2a\n- v2a --supports--> v00\n- v00 --refutes--> v1b\n- v1b --refutes--> v14\n- v14 --supports--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v13\n- v13 --supports--> v01\n- v01 --supports--> v04\n- v04 --refutes--> v02\n- v02 --supports--> v1c\n- v1c --supports--> v22\n- v22 --refutes--> v06\n- v06 --supports--> v10\n- v10 --supports--> v2c\n- v2c --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v15\n- v06 --supports--> v21\n- v0a --refutes--> v03\n- v12 --supports--> v0f\n- v12 --supports--> v2f\n- v19 --refutes--> v29\n- v1f --supports--> v11\n- v22 --refutes--> v18\n- v25 --supports--> v2b\n- v28 --supports--> v0b\n- v28 --supports--> v0e\n- v2a --refutes--> v1a\n- v2a --supports--> v23\n- v2c --supports--> v17\n- v30 --supports--> v16\n- v30 --supports--> v1d\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":"8d3584b2b2877db2b306a6b29e16b39030f28c7cc9185670eccc0e878bf77a4b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v1d\n- v1d --supports--> v0f\n- v0f --refutes--> v0e\n- v0e --refutes--> v19\n- v19 --refutes--> v11\n- v11 --supports--> v2e\n- v2e --supports--> v1c\n- v1c --refutes--> v08\n- v08 --supports--> v23\n- v23 --refutes--> v1f\n- v1f --supports--> v04\n- v04 --supports--> v0c\n- v0c --supports--> v0b\n- v0b --refutes--> v20\n- v20 --refutes--> v07\n- v07 --supports--> v10\n- v10 --refutes--> v02\n- v02 --supports--> v21\n- v21 --supports--> v05\n- v05 --refutes--> v26\n- v26 --refutes--> v24\n- v24 --refutes--> v09\n- v09 --refutes--> v28\n- v28 --supports--> v2c\n- v2c --refutes--> v2d\n- v2d --refutes--> v1a\n- v1a --refutes--> v06\n- v06 --refutes--> v30\n- v30 --refutes--> v1b\n- v1b --refutes--> v2b\n- v2b --refutes--> v01\n- v01 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v1e\n- v0c --refutes--> v25\n- v0c --supports--> v2a\n- v0f --supports--> v22\n- v13 --supports--> v03\n- v1a --supports--> v29\n- v1c --refutes--> v0a\n- v1c --supports--> v0d\n- v1c --supports--> v12\n- v1d --supports--> v16\n- v1f --refutes--> v15\n- v1f --supports--> v18\n- v24 --refutes--> v00\n- v24 --supports--> v17\n- v2d --supports--> v2f\n- v30 --refutes--> v27\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":"4e2b86922a89ff4932347ec49d21595b8c9e49d21d213d873e6e84074e5e8ec9","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v04\n- v04 --refutes--> v1c\n- v1c --supports--> v16\n- v16 --supports--> v11\n- v11 --refutes--> v0b\n- v0b --supports--> v22\n- v22 --refutes--> v10\n- v10 --refutes--> v03\n- v03 --refutes--> v15\n- v15 --refutes--> v1e\n- v1e --supports--> v24\n- v24 --refutes--> v2d\n- v2d --refutes--> v18\n- v18 --supports--> v26\n- v26 --supports--> v21\n- v21 --refutes--> v20\n- v20 --supports--> v01\n- v01 --supports--> v27\n- v27 --refutes--> v2b\n- v2b --supports--> v0e\n- v0e --supports--> v12\n- v12 --supports--> v1d\n- v1d --supports--> v25\n- v25 --supports--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v2f\n- v2f --supports--> v2c\n- v2c --refutes--> v2a\n- v2a --refutes--> v23\n- v23 --supports--> v28\n- v28 --supports--> v19\n- v19 --refutes--> v1f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v0d\n- v02 --refutes--> v17\n- v02 --refutes--> v29\n- v07 --refutes--> v08\n- v07 --supports--> v2e\n- v0b --refutes--> v00\n- v0b --refutes--> v14\n- v18 --refutes--> v0a\n- v19 --refutes--> v0f\n- v22 --refutes--> v13\n- v22 --supports--> v06\n- v26 --refutes--> v0c\n- v28 --refutes--> v1a\n- v28 --supports--> v09\n- v28 --supports--> v30\n- v2c --refutes--> v1b\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":"97ccc25a1b5c4ab233ffa2e431f7711c16c31a501e0950355bfa6316c5df55cb","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v05\n- v05 --supports--> v2e\n- v2e --supports--> v0c\n- v0c --supports--> v2a\n- v2a --refutes--> v11\n- v11 --supports--> v0f\n- v0f --supports--> v1e\n- v1e --refutes--> v0a\n- v0a --supports--> v2d\n- v2d --refutes--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v26\n- v26 --refutes--> v20\n- v20 --supports--> v01\n- v01 --supports--> v0b\n- v0b --refutes--> v24\n- v24 --supports--> v22\n- v22 --supports--> v17\n- v17 --refutes--> v25\n- v25 --supports--> v23\n- v23 --supports--> v30\n- v30 --refutes--> v0d\n- v0d --supports--> v28\n- v28 --refutes--> v21\n- v21 --supports--> v10\n- v10 --refutes--> v1b\n- v1b --supports--> v1a\n- v1a --refutes--> v12\n- v12 --refutes--> v16\n- v16 --refutes--> v09\n- v09 --refutes--> v1c\n- v1c --refutes--> v2f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0e --refutes--> v00\n- v16 --supports--> v1f\n- v17 --supports--> v29\n- v1b --refutes--> v07\n- v1b --supports--> v27\n- v20 --refutes--> v03\n- v20 --supports--> v2c\n- v21 --refutes--> v19\n- v21 --supports--> v13\n- v22 --refutes--> v1d\n- v24 --supports--> v02\n- v28 --refutes--> v2b\n- v28 --supports--> v18\n- v2a --refutes--> v15\n- v2e --refutes--> v04\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":"f6a3faa9651a744ec1b2eeaa7483ca39835b60610335bd0013dd49eb38ed35d0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --supports--> v24\n- v24 --refutes--> v04\n- v04 --refutes--> v1a\n- v1a --supports--> v17\n- v17 --supports--> v1c\n- v1c --refutes--> v0e\n- v0e --supports--> v12\n- v12 --refutes--> v1d\n- v1d --supports--> v30\n- v30 --supports--> v2b\n- v2b --refutes--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v0f\n- v0f --refutes--> v25\n- v25 --refutes--> v2a\n- v2a --supports--> v23\n- v23 --refutes--> v02\n- v02 --supports--> v21\n- v21 --supports--> v20\n- v20 --supports--> v18\n- v18 --supports--> v06\n- v06 --supports--> v03\n- v03 --refutes--> v22\n- v22 --supports--> v2e\n- v2e --refutes--> v2f\n- v2f --refutes--> v10\n- v10 --refutes--> v05\n- v05 --supports--> v19\n- v19 --supports--> v0b\n- v0b --supports--> v15\n- v15 --refutes--> v09\n- v09 --refutes--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v03 --refutes--> v2d\n- v03 --supports--> v26\n- v09 --supports--> v16\n- v0b --refutes--> v0c\n- v12 --supports--> v08\n- v15 --supports--> v29\n- v15 --supports--> v2c\n- v19 --supports--> v28\n- v1a --refutes--> v1f\n- v1d --supports--> v14\n- v22 --refutes--> v13\n- v2b --supports--> v11\n- v2b --supports--> v27\n- v2e --supports--> v1e\n- v2f --supports--> v1b\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":"bf6794b031383920b7487f7d2d72d515539986050afd3db2f4437b14516fc565","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v2c\n- v2c --refutes--> v20\n- v20 --refutes--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v1e\n- v1e --refutes--> v03\n- v03 --supports--> v2b\n- v2b --refutes--> v07\n- v07 --refutes--> v15\n- v15 --supports--> v00\n- v00 --refutes--> v0f\n- v0f --refutes--> v18\n- v18 --supports--> v16\n- v16 --supports--> v2f\n- v2f --refutes--> v06\n- v06 --refutes--> v24\n- v24 --supports--> v12\n- v12 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v1a\n- v1a --supports--> v0e\n- v0e --supports--> v29\n- v29 --supports--> v1b\n- v1b --refutes--> v23\n- v23 --refutes--> v10\n- v10 --refutes--> v1c\n- v1c --refutes--> v25\n- v25 --supports--> v11\n- v11 --supports--> v30\n- v30 --supports--> v27\n- v27 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --supports--> v17\n- v06 --supports--> v0d\n- v0e --refutes--> v2d\n- v0e --supports--> v28\n- v11 --refutes--> v09\n- v12 --supports--> v14\n- v15 --refutes--> v2e\n- v1a --supports--> v22\n- v1b --refutes--> v26\n- v1b --supports--> v19\n- v20 --refutes--> v0c\n- v23 --refutes--> v2a\n- v25 --refutes--> v0b\n- v2b --supports--> v1d\n- v2b --supports--> v1f\n- v30 --refutes--> v21\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":"52bc8ed8253d8ab5456317aadb773fdb4766188e5fbc83e70c580ff19816daab","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v29 --refutes--> v13\n- v13 --supports--> v1f\n- v1f --refutes--> v23\n- v23 --refutes--> v01\n- v01 --refutes--> v0d\n- v0d --refutes--> v02\n- v02 --refutes--> v30\n- v30 --refutes--> v2f\n- v2f --supports--> v2b\n- v2b --refutes--> v15\n- v15 --supports--> v19\n- v19 --supports--> v25\n- v25 --supports--> v20\n- v20 --refutes--> v17\n- v17 --supports--> v24\n- v24 --supports--> v11\n- v11 --supports--> v26\n- v26 --refutes--> v00\n- v00 --refutes--> v1c\n- v1c --refutes--> v28\n- v28 --refutes--> v0a\n- v0a --supports--> v2c\n- v2c --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v1a\n- v1a --supports--> v0b\n- v0b --refutes--> v0c\n- v0c --refutes--> v10\n- v10 --refutes--> v2a\n- v2a --refutes--> v0f\n- v0f --supports--> v2e\n- v2e --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1b\n- v07 --refutes--> v1e\n- v0a --refutes--> v1d\n- v0c --supports--> v0e\n- v0c --supports--> v22\n- v0d --refutes--> v04\n- v12 --supports--> v16\n- v15 --refutes--> v27\n- v1f --refutes--> v18\n- v23 --refutes--> v06\n- v24 --refutes--> v21\n- v25 --refutes--> v05\n- v25 --supports--> v14\n- v29 --refutes--> v03\n- v2e --refutes--> v2d\n- v30 --supports--> v08\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":"a814bcc978a41da25655ca0f364cde7dd4fc545095ac0f83b1b80176dc0feafc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v02\n- v02 --supports--> v26\n- v26 --refutes--> v19\n- v19 --refutes--> v1f\n- v1f --refutes--> v2a\n- v2a --refutes--> v25\n- v25 --supports--> v00\n- v00 --refutes--> v23\n- v23 --refutes--> v1c\n- v1c --refutes--> v0c\n- v0c --refutes--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v22\n- v22 --refutes--> v17\n- v17 --refutes--> v28\n- v28 --supports--> v2c\n- v2c --refutes--> v1a\n- v1a --supports--> v16\n- v16 --refutes--> v2f\n- v2f --supports--> v27\n- v27 --supports--> v13\n- v13 --supports--> v0a\n- v0a --refutes--> v10\n- v10 --refutes--> v04\n- v04 --refutes--> v20\n- v20 --supports--> v2e\n- v2e --refutes--> v1b\n- v1b --refutes--> v2d\n- v2d --supports--> v24\n- v24 --refutes--> v29\n- v29 --supports--> v1d\n- v1d --refutes--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v05\n- v03 --supports--> v0f\n- v0c --refutes--> v14\n- v0c --refutes--> v18\n- v0c --supports--> v2b\n- v10 --refutes--> v0e\n- v1b --supports--> v1e\n- v1c --refutes--> v08\n- v20 --refutes--> v11\n- v20 --refutes--> v30\n- v20 --supports--> v09\n- v22 --refutes--> v0d\n- v25 --supports--> v01\n- v28 --supports--> v06\n- v29 --supports--> v12\n- v2a --supports--> v07\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":"2cdd5e1f54ca1549e96c7ef2599d212aa0160c55cc10f65811f15cbe6aa7b7a1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2b --refutes--> v07\n- v07 --refutes--> v10\n- v10 --supports--> v22\n- v22 --refutes--> v28\n- v28 --supports--> v21\n- v21 --refutes--> v2c\n- v2c --refutes--> v2e\n- v2e --refutes--> v2f\n- v2f --supports--> v12\n- v12 --refutes--> v30\n- v30 --refutes--> v08\n- v08 --refutes--> v18\n- v18 --supports--> v00\n- v00 --supports--> v0f\n- v0f --supports--> v25\n- v25 --refutes--> v01\n- v01 --refutes--> v17\n- v17 --refutes--> v0d\n- v0d --refutes--> v1b\n- v1b --supports--> v0e\n- v0e --supports--> v1d\n- v1d --supports--> v11\n- v11 --supports--> v02\n- v02 --refutes--> v0a\n- v0a --supports--> v16\n- v16 --refutes--> v1e\n- v1e --supports--> v2a\n- v2a --supports--> v27\n- v27 --supports--> v26\n- v26 --refutes--> v0c\n- v0c --supports--> v29\n- v29 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v20\n- v07 --supports--> v1f\n- v07 --supports--> v24\n- v0e --supports--> v15\n- v10 --supports--> v06\n- v12 --refutes--> v1a\n- v17 --refutes--> v03\n- v1e --supports--> v19\n- v22 --refutes--> v13\n- v25 --supports--> v1c\n- v27 --refutes--> v2d\n- v28 --supports--> v14\n- v2b --refutes--> v09\n- v2c --refutes--> v0b\n- v2f --refutes--> v23\n- v30 --supports--> v04\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":"7ec40d4eaca096cd7ae0ae8f62f2a72835f2f57b565aad5a13c2b157e1c2949c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v30 --refutes--> v12\n- v12 --supports--> v16\n- v16 --refutes--> v25\n- v25 --supports--> v2b\n- v2b --refutes--> v2f\n- v2f --supports--> v1c\n- v1c --supports--> v05\n- v05 --supports--> v15\n- v15 --refutes--> v2d\n- v2d --refutes--> v17\n- v17 --supports--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v1d\n- v1d --supports--> v0a\n- v0a --supports--> v19\n- v19 --supports--> v07\n- v07 --supports--> v13\n- v13 --supports--> v27\n- v27 --supports--> v29\n- v29 --refutes--> v0d\n- v0d --supports--> v10\n- v10 --refutes--> v2c\n- v2c --supports--> v1f\n- v1f --refutes--> v28\n- v28 --refutes--> v2a\n- v2a --supports--> v03\n- v03 --refutes--> v02\n- v02 --supports--> v20\n- v20 --refutes--> v0e\n- v0e --refutes--> v0c\n- v0c --supports--> v0f\n- v0f --supports--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v1a\n- v05 --refutes--> v14\n- v07 --supports--> v21\n- v0a --supports--> v01\n- v0b --refutes--> v23\n- v0c --refutes--> v2e\n- v0c --supports--> v24\n- v0e --refutes--> v11\n- v0e --supports--> v1e\n- v12 --refutes--> v26\n- v13 --refutes--> v08\n- v1d --supports--> v06\n- v29 --supports--> v1b\n- v2b --supports--> v09\n- v2c --supports--> v04\n- v2f --supports--> v18\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":"eb42ece08643d029731d488b89233910e73322f4fad29a83422d29073392a295","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --supports--> v27\n- v27 --supports--> v2f\n- v2f --refutes--> v2b\n- v2b --supports--> v01\n- v01 --refutes--> v0b\n- v0b --refutes--> v20\n- v20 --refutes--> v22\n- v22 --supports--> v08\n- v08 --supports--> v07\n- v07 --supports--> v1f\n- v1f --supports--> v0c\n- v0c --supports--> v00\n- v00 --refutes--> v1c\n- v1c --refutes--> v11\n- v11 --supports--> v24\n- v24 --refutes--> v21\n- v21 --supports--> v04\n- v04 --refutes--> v1a\n- v1a --supports--> v28\n- v28 --supports--> v0e\n- v0e --supports--> v0a\n- v0a --refutes--> v2d\n- v2d --refutes--> v29\n- v29 --supports--> v2a\n- v2a --refutes--> v15\n- v15 --refutes--> v25\n- v25 --supports--> v1e\n- v1e --supports--> v1d\n- v1d --supports--> v18\n- v18 --supports--> v03\n- v03 --refutes--> v12\n- v12 --supports--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v10\n- v00 --refutes--> v1b\n- v01 --supports--> v26\n- v04 --refutes--> v0f\n- v0d --supports--> v16\n- v0e --supports--> v17\n- v0e --supports--> v30\n- v12 --supports--> v2c\n- v1e --refutes--> v05\n- v1f --refutes--> v02\n- v21 --refutes--> v14\n- v25 --refutes--> v19\n- v27 --supports--> v13\n- v29 --refutes--> v09\n- v2b --refutes--> v23\n- v2f --supports--> v06\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":"a957199427dea74f533e5574271de1757f25f66c7a8252ae66525ae993ee6c27","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --supports--> v04\n- v04 --supports--> v2b\n- v2b --supports--> v23\n- v23 --supports--> v1b\n- v1b --refutes--> v13\n- v13 --refutes--> v07\n- v07 --refutes--> v25\n- v25 --refutes--> v1a\n- v1a --refutes--> v02\n- v02 --refutes--> v30\n- v30 --supports--> v11\n- v11 --supports--> v15\n- v15 --refutes--> v0e\n- v0e --supports--> v10\n- v10 --supports--> v00\n- v00 --refutes--> v2a\n- v2a --refutes--> v17\n- v17 --refutes--> v1c\n- v1c --refutes--> v0b\n- v0b --supports--> v2e\n- v2e --refutes--> v08\n- v08 --refutes--> v24\n- v24 --supports--> v01\n- v01 --refutes--> v18\n- v18 --supports--> v19\n- v19 --refutes--> v1e\n- v1e --supports--> v2c\n- v2c --supports--> v2f\n- v2f --refutes--> v12\n- v12 --supports--> v29\n- v29 --refutes--> v0c\n- v0c --supports--> v06\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v14\n- v11 --refutes--> v0a\n- v13 --refutes--> v28\n- v13 --supports--> v09\n- v15 --refutes--> v27\n- v1b --refutes--> v20\n- v1b --supports--> v2d\n- v1e --refutes--> v03\n- v1e --refutes--> v0d\n- v1e --supports--> v05\n- v25 --refutes--> v26\n- v25 --supports--> v1f\n- v29 --refutes--> v21\n- v2a --refutes--> v22\n- v2e --refutes--> v0f\n- v2e --supports--> v1d\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":"92754c951b86901bc1cdd26eb96805fee6265ef8fe096a23d3c1176656e93616","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v23\n- v23 --supports--> v2a\n- v2a --refutes--> v2d\n- v2d --refutes--> v26\n- v26 --refutes--> v04\n- v04 --refutes--> v27\n- v27 --supports--> v2f\n- v2f --refutes--> v18\n- v18 --refutes--> v19\n- v19 --supports--> v0f\n- v0f --refutes--> v16\n- v16 --refutes--> v20\n- v20 --supports--> v2b\n- v2b --refutes--> v15\n- v15 --supports--> v1b\n- v1b --refutes--> v21\n- v21 --refutes--> v08\n- v08 --supports--> v03\n- v03 --supports--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v14\n- v14 --refutes--> v13\n- v13 --supports--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v1f\n- v1f --refutes--> v00\n- v00 --supports--> v09\n- v09 --supports--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v24\n- v24 --refutes--> v1e\n- v1e --refutes--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v07\n- v04 --supports--> v06\n- v09 --supports--> v0a\n- v0c --supports--> v17\n- v0d --refutes--> v1d\n- v0d --supports--> v2c\n- v0f --refutes--> v30\n- v18 --supports--> v10\n- v19 --supports--> v0e\n- v1c --refutes--> v28\n- v1e --supports--> v22\n- v23 --refutes--> v0b\n- v24 --refutes--> v29\n- v27 --supports--> v02\n- v2a --refutes--> v25\n- v2a --supports--> v1a\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":"15151c5dc7844ede5f29dc032aa44a1d4f5116ae97118f6c520f93b258c4dede","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --supports--> v28\n- v28 --supports--> v0f\n- v0f --refutes--> v05\n- v05 --supports--> v07\n- v07 --refutes--> v14\n- v14 --refutes--> v10\n- v10 --supports--> v0b\n- v0b --refutes--> v1c\n- v1c --supports--> v03\n- v03 --refutes--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v22\n- v22 --refutes--> v1e\n- v1e --supports--> v2e\n- v2e --refutes--> v02\n- v02 --refutes--> v2c\n- v2c --supports--> v30\n- v30 --supports--> v1a\n- v1a --refutes--> v1d\n- v1d --supports--> v2f\n- v2f --supports--> v21\n- v21 --supports--> v16\n- v16 --refutes--> v17\n- v17 --refutes--> v13\n- v13 --refutes--> v00\n- v00 --refutes--> v26\n- v26 --supports--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v20\n- v20 --refutes--> v18\n- v18 --supports--> v25\n- v25 --refutes--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v03 --refutes--> v01\n- v03 --refutes--> v2a\n- v05 --refutes--> v23\n- v07 --refutes--> v2d\n- v09 --supports--> v24\n- v10 --refutes--> v1f\n- v13 --supports--> v0e\n- v17 --refutes--> v04\n- v1d --refutes--> v15\n- v1d --supports--> v12\n- v1d --supports--> v29\n- v1e --refutes--> v08\n- v2e --refutes--> v2b\n- v2e --supports--> v0d\n- v2e --supports--> v19\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":"c1d86cc306b61092b3d79bef317cb5c5cc667f3a9d96fdd5722c9d185a4570da","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v15\n- v15 --refutes--> v2d\n- v2d --refutes--> v26\n- v26 --refutes--> v1a\n- v1a --refutes--> v06\n- v06 --refutes--> v28\n- v28 --supports--> v02\n- v02 --supports--> v0a\n- v0a --refutes--> v12\n- v12 --refutes--> v27\n- v27 --refutes--> v20\n- v20 --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v21\n- v21 --refutes--> v0b\n- v0b --supports--> v17\n- v17 --refutes--> v2e\n- v2e --supports--> v18\n- v18 --supports--> v2b\n- v2b --supports--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v24\n- v24 --supports--> v1e\n- v1e --refutes--> v30\n- v30 --supports--> v09\n- v09 --supports--> v05\n- v05 --refutes--> v14\n- v14 --supports--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v11\n- v05 --refutes--> v08\n- v05 --refutes--> v2f\n- v09 --refutes--> v2c\n- v0c --refutes--> v0f\n- v0c --refutes--> v10\n- v12 --refutes--> v1b\n- v13 --refutes--> v0e\n- v13 --supports--> v29\n- v15 --refutes--> v23\n- v18 --supports--> v19\n- v27 --refutes--> v1f\n- v2e --refutes--> v00\n- v2e --refutes--> v22\n- v2e --supports--> v0d\n- v2e --supports--> v1d\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":"b31b8fcb5b46aa5d0900b112472c17e862df8ad023a3213905261aef40a27efc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v00 --supports--> v16\n- v16 --refutes--> v22\n- v22 --refutes--> v0e\n- v0e --supports--> v1d\n- v1d --supports--> v0a\n- v0a --supports--> v2e\n- v2e --supports--> v21\n- v21 --supports--> v1c\n- v1c --refutes--> v11\n- v11 --refutes--> v1b\n- v1b --supports--> v09\n- v09 --supports--> v06\n- v06 --refutes--> v03\n- v03 --supports--> v2c\n- v2c --refutes--> v0c\n- v0c --supports--> v27\n- v27 --refutes--> v28\n- v28 --refutes--> v13\n- v13 --refutes--> v2d\n- v2d --refutes--> v02\n- v02 --refutes--> v25\n- v25 --supports--> v2f\n- v2f --refutes--> v24\n- v24 --supports--> v1f\n- v1f --refutes--> v23\n- v23 --refutes--> v12\n- v12 --supports--> v01\n- v01 --supports--> v18\n- v18 --supports--> v05\n- v05 --refutes--> v2b\n- v2b --supports--> v17\n- v17 --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v15\n- v02 --refutes--> v0d\n- v02 --refutes--> v30\n- v06 --supports--> v10\n- v09 --supports--> v20\n- v12 --supports--> v0b\n- v17 --refutes--> v0f\n- v1d --supports--> v08\n- v21 --refutes--> v1a\n- v22 --refutes--> v14\n- v22 --refutes--> v26\n- v24 --supports--> v29\n- v27 --refutes--> v07\n- v2d --supports--> v04\n- v2e --refutes--> v1e\n- v2e --refutes--> v2a\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":"2768e8101b98fc6eeddaa79318cfc6478c4cd13edbd6f44cacd98ecbd377ad66","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v23\n- v23 --supports--> v2e\n- v2e --supports--> v10\n- v10 --supports--> v06\n- v06 --supports--> v00\n- v00 --refutes--> v30\n- v30 --supports--> v0d\n- v0d --supports--> v01\n- v01 --refutes--> v12\n- v12 --supports--> v0c\n- v0c --supports--> v16\n- v16 --supports--> v13\n- v13 --supports--> v05\n- v05 --supports--> v14\n- v14 --refutes--> v1d\n- v1d --refutes--> v03\n- v03 --supports--> v02\n- v02 --refutes--> v1c\n- v1c --refutes--> v26\n- v26 --refutes--> v27\n- v27 --refutes--> v0a\n- v0a --refutes--> v0e\n- v0e --supports--> v2a\n- v2a --refutes--> v25\n- v25 --supports--> v19\n- v19 --refutes--> v17\n- v17 --supports--> v2d\n- v2d --supports--> v2c\n- v2c --refutes--> v04\n- v04 --supports--> v1f\n- v1f --refutes--> v20\n- v20 --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v11\n- v01 --supports--> v1a\n- v02 --supports--> v21\n- v03 --supports--> v1b\n- v05 --refutes--> v29\n- v06 --supports--> v09\n- v0c --refutes--> v18\n- v13 --refutes--> v2f\n- v13 --supports--> v2b\n- v1c --refutes--> v15\n- v1c --supports--> v28\n- v1d --supports--> v08\n- v26 --refutes--> v22\n- v2a --supports--> v24\n- v2d --refutes--> v0b\n- v30 --supports--> v1e\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":"3c1ee206850979bcdc7b08f493168f140b76e101009458f408af5a21d8b54ff7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v25\n- v25 --refutes--> v29\n- v29 --refutes--> v30\n- v30 --refutes--> v07\n- v07 --refutes--> v1c\n- v1c --supports--> v1b\n- v1b --supports--> v08\n- v08 --refutes--> v1f\n- v1f --refutes--> v2f\n- v2f --supports--> v2d\n- v2d --supports--> v06\n- v06 --refutes--> v0d\n- v0d --supports--> v1d\n- v1d --supports--> v17\n- v17 --refutes--> v20\n- v20 --refutes--> v21\n- v21 --refutes--> v04\n- v04 --refutes--> v23\n- v23 --supports--> v19\n- v19 --refutes--> v27\n- v27 --supports--> v18\n- v18 --refutes--> v28\n- v28 --supports--> v16\n- v16 --supports--> v14\n- v14 --supports--> v2a\n- v2a --refutes--> v13\n- v13 --refutes--> v0e\n- v0e --supports--> v24\n- v24 --refutes--> v02\n- v02 --supports--> v0f\n- v0f --supports--> v2c\n- v2c --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v11\n- v06 --refutes--> v2e\n- v08 --supports--> v15\n- v0e --refutes--> v26\n- v0e --refutes--> v2b\n- v13 --refutes--> v10\n- v19 --refutes--> v05\n- v1d --refutes--> v01\n- v21 --supports--> v1a\n- v24 --supports--> v00\n- v25 --refutes--> v03\n- v27 --refutes--> v0a\n- v29 --supports--> v22\n- v2a --refutes--> v0b\n- v2d --supports--> v09\n- v2f --supports--> v12\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":"3a46a44de2935e5b357f189a53da6654049161e977ab224f7c77cc4f28b63155","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v1a\n- v1a --supports--> v04\n- v04 --refutes--> v00\n- v00 --supports--> v07\n- v07 --supports--> v23\n- v23 --supports--> v2e\n- v2e --supports--> v0f\n- v0f --supports--> v2b\n- v2b --supports--> v0b\n- v0b --supports--> v26\n- v26 --refutes--> v19\n- v19 --refutes--> v1b\n- v1b --refutes--> v12\n- v12 --supports--> v2d\n- v2d --supports--> v20\n- v20 --supports--> v16\n- v16 --refutes--> v0c\n- v0c --supports--> v10\n- v10 --supports--> v24\n- v24 --supports--> v30\n- v30 --refutes--> v2a\n- v2a --supports--> v09\n- v09 --supports--> v2c\n- v2c --supports--> v28\n- v28 --supports--> v14\n- v14 --refutes--> v08\n- v08 --refutes--> v29\n- v29 --supports--> v13\n- v13 --refutes--> v01\n- v01 --supports--> v06\n- v06 --supports--> v17\n- v17 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v1c\n- v08 --supports--> v0a\n- v09 --refutes--> v0e\n- v12 --supports--> v1f\n- v16 --refutes--> v02\n- v16 --refutes--> v15\n- v17 --refutes--> v0d\n- v17 --refutes--> v21\n- v20 --refutes--> v18\n- v26 --refutes--> v05\n- v26 --supports--> v25\n- v27 --refutes--> v22\n- v28 --refutes--> v2f\n- v29 --refutes--> v1d\n- v2d --refutes--> v03\n- v2e --supports--> v1e\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":"d4185cb8c5eb3f8c778e5e9f70136504ee3aac8619fd5397a9791014765c83a5","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0d --refutes--> v27\n- v27 --supports--> v0a\n- v0a --supports--> v12\n- v12 --supports--> v24\n- v24 --refutes--> v2e\n- v2e --refutes--> v1c\n- v1c --refutes--> v08\n- v08 --supports--> v29\n- v29 --supports--> v03\n- v03 --supports--> v13\n- v13 --refutes--> v26\n- v26 --refutes--> v0f\n- v0f --supports--> v10\n- v10 --refutes--> v1a\n- v1a --refutes--> v05\n- v05 --supports--> v04\n- v04 --refutes--> v1b\n- v1b --refutes--> v1f\n- v1f --supports--> v17\n- v17 --supports--> v02\n- v02 --supports--> v2f\n- v2f --supports--> v25\n- v25 --refutes--> v22\n- v22 --supports--> v1d\n- v1d --refutes--> v2a\n- v2a --refutes--> v0c\n- v0c --supports--> v15\n- v15 --supports--> v14\n- v14 --supports--> v01\n- v01 --refutes--> v16\n- v16 --supports--> v28\n- v28 --refutes--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v11\n- v02 --refutes--> v2c\n- v05 --supports--> v21\n- v0c --supports--> v09\n- v0f --supports--> v20\n- v12 --supports--> v0e\n- v1c --supports--> v00\n- v1c --supports--> v19\n- v1d --refutes--> v18\n- v24 --supports--> v1e\n- v28 --supports--> v2d\n- v29 --refutes--> v06\n- v29 --refutes--> v07\n- v2a --refutes--> v30\n- v2a --supports--> v23\n- v2e --supports--> v0b\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":"00d514b8ececd4238b9f8e49040f4656fd548690cc1b26f2a579a309ca5e993a","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v12\n- v12 --refutes--> v1e\n- v1e --supports--> v2b\n- v2b --supports--> v2a\n- v2a --refutes--> v18\n- v18 --refutes--> v25\n- v25 --refutes--> v29\n- v29 --refutes--> v30\n- v30 --refutes--> v0b\n- v0b --refutes--> v0a\n- v0a --refutes--> v26\n- v26 --supports--> v0d\n- v0d --supports--> v1a\n- v1a --refutes--> v13\n- v13 --refutes--> v01\n- v01 --refutes--> v00\n- v00 --refutes--> v09\n- v09 --supports--> v19\n- v19 --refutes--> v16\n- v16 --refutes--> v10\n- v10 --refutes--> v1b\n- v1b --refutes--> v28\n- v28 --supports--> v23\n- v23 --refutes--> v24\n- v24 --supports--> v2d\n- v2d --refutes--> v0e\n- v0e --refutes--> v1c\n- v1c --supports--> v02\n- v02 --supports--> v1d\n- v1d --refutes--> v07\n- v07 --supports--> v27\n- v27 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0f\n- v0a --refutes--> v0c\n- v0e --supports--> v17\n- v14 --refutes--> v11\n- v18 --supports--> v21\n- v18 --supports--> v22\n- v1a --refutes--> v04\n- v1b --supports--> v06\n- v26 --refutes--> v08\n- v26 --refutes--> v15\n- v29 --refutes--> v1f\n- v29 --supports--> v2f\n- v2a --refutes--> v2e\n- v2b --refutes--> v05\n- v2b --refutes--> v2c\n- v2d --refutes--> v20\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":"c1a84dab2d7815f22bb787da7f0e17a67d1b1d00d631c5485f2397cbabd84744","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --supports--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v1e\n- v1e --supports--> v1b\n- v1b --refutes--> v25\n- v25 --supports--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v2c\n- v2c --supports--> v2a\n- v2a --supports--> v21\n- v21 --supports--> v0e\n- v0e --refutes--> v01\n- v01 --refutes--> v14\n- v14 --supports--> v29\n- v29 --supports--> v0b\n- v0b --supports--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v11\n- v11 --supports--> v2d\n- v2d --refutes--> v09\n- v09 --supports--> v2b\n- v2b --supports--> v15\n- v15 --refutes--> v1a\n- v1a --refutes--> v0d\n- v0d --supports--> v0a\n- v0a --supports--> v16\n- v16 --supports--> v06\n- v06 --refutes--> v23\n- v23 --supports--> v17\n- v17 --refutes--> v19\n- v19 --refutes--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1d\n- v04 --refutes--> v22\n- v05 --supports--> v2f\n- v06 --supports--> v0c\n- v0e --supports--> v03\n- v14 --refutes--> v24\n- v16 --refutes--> v27\n- v1b --refutes--> v1f\n- v1c --refutes--> v18\n- v1e --refutes--> v30\n- v20 --supports--> v28\n- v23 --supports--> v13\n- v23 --supports--> v26\n- v29 --supports--> v00\n- v2d --refutes--> v02\n- v2d --refutes--> v07\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":"0f09e627be7b4954974a4539caddd67713f3d4fbebb326f3f9f1480e2142443b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --supports--> v30\n- v30 --refutes--> v20\n- v20 --supports--> v24\n- v24 --supports--> v2f\n- v2f --refutes--> v26\n- v26 --refutes--> v1c\n- v1c --supports--> v0d\n- v0d --supports--> v22\n- v22 --refutes--> v27\n- v27 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v2a\n- v2a --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v13\n- v13 --supports--> v11\n- v11 --refutes--> v0b\n- v0b --supports--> v14\n- v14 --supports--> v03\n- v03 --refutes--> v2b\n- v2b --refutes--> v00\n- v00 --supports--> v05\n- v05 --refutes--> v2e\n- v2e --refutes--> v2d\n- v2d --refutes--> v19\n- v19 --refutes--> v06\n- v06 --refutes--> v1a\n- v1a --supports--> v07\n- v07 --supports--> v17\n- v17 --supports--> v18\n- v18 --refutes--> v21\n- v21 --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v28\n- v04 --supports--> v0c\n- v08 --refutes--> v23\n- v08 --supports--> v1e\n- v0b --refutes--> v10\n- v13 --refutes--> v29\n- v17 --refutes--> v25\n- v18 --supports--> v12\n- v18 --supports--> v1d\n- v18 --supports--> v2c\n- v1a --supports--> v16\n- v24 --refutes--> v15\n- v2a --refutes--> v0f\n- v2b --supports--> v01\n- v2f --refutes--> v02\n- v2f --refutes--> v1f\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":"2a611e3da8e335a341414b5763d1afda38674116df83a0232f4028f4f59b5047","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v08 --refutes--> v14\n- v14 --supports--> v12\n- v12 --supports--> v25\n- v25 --supports--> v02\n- v02 --supports--> v24\n- v24 --supports--> v20\n- v20 --refutes--> v0d\n- v0d --refutes--> v1b\n- v1b --refutes--> v17\n- v17 --refutes--> v10\n- v10 --refutes--> v09\n- v09 --supports--> v0c\n- v0c --supports--> v1e\n- v1e --refutes--> v30\n- v30 --supports--> v1c\n- v1c --refutes--> v1a\n- v1a --supports--> v2a\n- v2a --supports--> v11\n- v11 --refutes--> v18\n- v18 --supports--> v2e\n- v2e --supports--> v29\n- v29 --refutes--> v0a\n- v0a --supports--> v0b\n- v0b --supports--> v03\n- v03 --refutes--> v2c\n- v2c --supports--> v21\n- v21 --supports--> v00\n- v00 --supports--> v2f\n- v2f --refutes--> v2d\n- v2d --refutes--> v1d\n- v1d --supports--> v23\n- v23 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v15\n- v02 --refutes--> v0f\n- v03 --supports--> v05\n- v0b --supports--> v1f\n- v0d --refutes--> v07\n- v11 --refutes--> v26\n- v1e --supports--> v04\n- v20 --refutes--> v01\n- v20 --supports--> v2b\n- v21 --supports--> v28\n- v25 --refutes--> v27\n- v2d --supports--> v13\n- v2e --refutes--> v19\n- v2e --supports--> v0e\n- v2f --refutes--> v22\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":"352778e2d61374de915aa1dcb170ee0fbe5acb5f8a7f881e975726057825ff00","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --supports--> v0f\n- v0f --refutes--> v05\n- v05 --supports--> v2a\n- v2a --supports--> v1d\n- v1d --supports--> v13\n- v13 --supports--> v16\n- v16 --supports--> v1e\n- v1e --supports--> v27\n- v27 --supports--> v09\n- v09 --supports--> v1b\n- v1b --supports--> v28\n- v28 --refutes--> v19\n- v19 --refutes--> v2d\n- v2d --refutes--> v0b\n- v0b --refutes--> v15\n- v15 --refutes--> v0e\n- v0e --refutes--> v21\n- v21 --supports--> v1f\n- v1f --refutes--> v14\n- v14 --supports--> v04\n- v04 --supports--> v02\n- v02 --refutes--> v1a\n- v1a --supports--> v25\n- v25 --refutes--> v12\n- v12 --refutes--> v0a\n- v0a --refutes--> v11\n- v11 --supports--> v07\n- v07 --supports--> v03\n- v03 --supports--> v26\n- v26 --refutes--> v06\n- v06 --supports--> v08\n- v08 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v2c\n- v02 --supports--> v30\n- v0b --refutes--> v2f\n- v0e --supports--> v1c\n- v13 --supports--> v22\n- v16 --refutes--> v23\n- v16 --refutes--> v2e\n- v19 --supports--> v10\n- v1d --supports--> v0c\n- v1e --refutes--> v20\n- v1e --supports--> v00\n- v21 --refutes--> v17\n- v24 --refutes--> v01\n- v24 --refutes--> v0d\n- v27 --refutes--> v29\n- v2d --supports--> v2b\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":"0cf09e9db0c99464e25f4010f38e8adc256230207142ff3de3cbf53180ec9830","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v1d\n- v1d --refutes--> v06\n- v06 --refutes--> v20\n- v20 --refutes--> v2a\n- v2a --refutes--> v2e\n- v2e --supports--> v1e\n- v1e --supports--> v27\n- v27 --refutes--> v0e\n- v0e --supports--> v15\n- v15 --supports--> v17\n- v17 --refutes--> v04\n- v04 --refutes--> v30\n- v30 --supports--> v1a\n- v1a --refutes--> v29\n- v29 --refutes--> v22\n- v22 --supports--> v14\n- v14 --refutes--> v1c\n- v1c --refutes--> v1f\n- v1f --refutes--> v0a\n- v0a --refutes--> v28\n- v28 --supports--> v11\n- v11 --refutes--> v2f\n- v2f --supports--> v26\n- v26 --refutes--> v18\n- v18 --supports--> v10\n- v10 --refutes--> v0f\n- v0f --refutes--> v1b\n- v1b --supports--> v09\n- v09 --refutes--> v03\n- v03 --refutes--> v21\n- v21 --refutes--> v07\n- v07 --refutes--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v2d\n- v09 --supports--> v05\n- v0f --refutes--> v0c\n- v11 --supports--> v00\n- v11 --supports--> v08\n- v15 --supports--> v2c\n- v1a --supports--> v02\n- v1c --refutes--> v13\n- v1d --refutes--> v12\n- v1d --supports--> v01\n- v22 --supports--> v0d\n- v23 --supports--> v19\n- v2e --refutes--> v0b\n- v2e --refutes--> v24\n- v2e --supports--> v25\n- v2f --supports--> v2b\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":"b356aaeb541ddd67670ba19f5decd91408ff84f179e99c70fa6619bba2de4481","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --refutes--> v08\n- v08 --supports--> v2f\n- v2f --supports--> v28\n- v28 --supports--> v30\n- v30 --supports--> v1d\n- v1d --refutes--> v0e\n- v0e --refutes--> v18\n- v18 --supports--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v06\n- v06 --refutes--> v16\n- v16 --supports--> v0d\n- v0d --refutes--> v26\n- v26 --refutes--> v1e\n- v1e --supports--> v11\n- v11 --refutes--> v23\n- v23 --refutes--> v21\n- v21 --refutes--> v17\n- v17 --refutes--> v2d\n- v2d --supports--> v12\n- v12 --refutes--> v09\n- v09 --refutes--> v0b\n- v0b --supports--> v19\n- v19 --supports--> v0c\n- v0c --supports--> v2e\n- v2e --supports--> v02\n- v02 --refutes--> v20\n- v20 --refutes--> v2a\n- v2a --refutes--> v13\n- v13 --refutes--> v14\n- v14 --refutes--> v2c\n- v2c --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v05\n- v0c --supports--> v1c\n- v13 --refutes--> v07\n- v17 --supports--> v2b\n- v18 --refutes--> v15\n- v19 --supports--> v03\n- v1d --supports--> v25\n- v1e --refutes--> v0f\n- v1e --refutes--> v24\n- v1e --supports--> v1b\n- v20 --supports--> v00\n- v21 --refutes--> v01\n- v26 --refutes--> v04\n- v26 --refutes--> v27\n- v2c --refutes--> v29\n- v30 --supports--> v10\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":"0b47fb9ba6b07a0a535f81c55cb5b4f62496520c19e15dc5e57a87215e179c3e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v09\n- v09 --refutes--> v11\n- v11 --refutes--> v04\n- v04 --refutes--> v0b\n- v0b --supports--> v23\n- v23 --supports--> v2f\n- v2f --supports--> v07\n- v07 --refutes--> v0d\n- v0d --refutes--> v20\n- v20 --supports--> v05\n- v05 --refutes--> v16\n- v16 --supports--> v10\n- v10 --refutes--> v13\n- v13 --supports--> v14\n- v14 --supports--> v22\n- v22 --supports--> v1a\n- v1a --refutes--> v0e\n- v0e --refutes--> v01\n- v01 --supports--> v08\n- v08 --supports--> v28\n- v28 --supports--> v1b\n- v1b --refutes--> v18\n- v18 --refutes--> v03\n- v03 --refutes--> v1f\n- v1f --supports--> v06\n- v06 --refutes--> v02\n- v02 --supports--> v2d\n- v2d --supports--> v29\n- v29 --refutes--> v30\n- v30 --refutes--> v26\n- v26 --supports--> v25\n- v25 --supports--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v27\n- v06 --refutes--> v0a\n- v07 --refutes--> v0c\n- v10 --supports--> v2e\n- v13 --supports--> v15\n- v14 --refutes--> v1e\n- v16 --supports--> v0f\n- v17 --refutes--> v2a\n- v1a --refutes--> v19\n- v23 --refutes--> v1d\n- v23 --supports--> v21\n- v25 --refutes--> v2b\n- v28 --supports--> v00\n- v29 --refutes--> v24\n- v30 --refutes--> v1c\n- v30 --refutes--> v2c\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":"873f58013210ca54d81e7a6ddea39f178beb66fb09e349450ff96f802f0e7b36","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --supports--> v2d\n- v2d --refutes--> v09\n- v09 --refutes--> v1b\n- v1b --refutes--> v2c\n- v2c --refutes--> v00\n- v00 --supports--> v12\n- v12 --supports--> v18\n- v18 --refutes--> v01\n- v01 --supports--> v0c\n- v0c --refutes--> v19\n- v19 --supports--> v1e\n- v1e --refutes--> v1f\n- v1f --refutes--> v0a\n- v0a --refutes--> v21\n- v21 --refutes--> v1c\n- v1c --refutes--> v2f\n- v2f --supports--> v07\n- v07 --refutes--> v26\n- v26 --supports--> v1a\n- v1a --refutes--> v08\n- v08 --supports--> v24\n- v24 --supports--> v1d\n- v1d --supports--> v02\n- v02 --refutes--> v14\n- v14 --refutes--> v0e\n- v0e --supports--> v06\n- v06 --refutes--> v2e\n- v2e --refutes--> v04\n- v04 --refutes--> v13\n- v13 --refutes--> v0b\n- v0b --refutes--> v29\n- v29 --supports--> v20\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0f\n- v02 --supports--> v23\n- v02 --supports--> v2b\n- v05 --refutes--> v0d\n- v06 --supports--> v2a\n- v07 --refutes--> v15\n- v07 --supports--> v10\n- v08 --supports--> v16\n- v0e --supports--> v22\n- v0e --supports--> v28\n- v1d --refutes--> v30\n- v1d --supports--> v03\n- v24 --refutes--> v11\n- v24 --supports--> v17\n- v29 --supports--> v27\n- v2f --refutes--> v25\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":"2be41e203f3a40751c944228f3afb687222882d04966aeedbbcf258baa160739","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v0a\n- v0a --refutes--> v28\n- v28 --refutes--> v1f\n- v1f --refutes--> v0d\n- v0d --refutes--> v0e\n- v0e --supports--> v14\n- v14 --supports--> v1d\n- v1d --supports--> v2c\n- v2c --refutes--> v07\n- v07 --supports--> v0c\n- v0c --refutes--> v06\n- v06 --refutes--> v15\n- v15 --supports--> v26\n- v26 --refutes--> v2e\n- v2e --refutes--> v13\n- v13 --refutes--> v25\n- v25 --supports--> v03\n- v03 --supports--> v20\n- v20 --refutes--> v19\n- v19 --supports--> v05\n- v05 --supports--> v18\n- v18 --supports--> v21\n- v21 --supports--> v1b\n- v1b --refutes--> v27\n- v27 --refutes--> v30\n- v30 --supports--> v1e\n- v1e --supports--> v17\n- v17 --refutes--> v0b\n- v0b --refutes--> v2b\n- v2b --refutes--> v08\n- v08 --refutes--> v02\n- v02 --refutes--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v05 --refutes--> v29\n- v06 --supports--> v24\n- v0a --refutes--> v11\n- v0b --refutes--> v01\n- v0c --supports--> v12\n- v0e --supports--> v1c\n- v0e --supports--> v2d\n- v17 --supports--> v1a\n- v17 --supports--> v22\n- v20 --supports--> v00\n- v21 --refutes--> v2a\n- v23 --refutes--> v16\n- v23 --supports--> v10\n- v25 --supports--> v2f\n- v2c --refutes--> v04\n- v2c --supports--> v0f\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":"f70131a7ea506cc86b725934186edc43e07703f5913f70c021d097ad2566d3ea","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2f --supports--> v1b\n- v1b --supports--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v01\n- v01 --supports--> v0d\n- v0d --refutes--> v1c\n- v1c --supports--> v18\n- v18 --supports--> v04\n- v04 --refutes--> v1a\n- v1a --supports--> v1f\n- v1f --refutes--> v2d\n- v2d --refutes--> v22\n- v22 --refutes--> v12\n- v12 --supports--> v17\n- v17 --refutes--> v19\n- v19 --refutes--> v2a\n- v2a --refutes--> v25\n- v25 --supports--> v0e\n- v0e --supports--> v30\n- v30 --supports--> v2e\n- v2e --supports--> v00\n- v00 --refutes--> v06\n- v06 --refutes--> v1e\n- v1e --supports--> v16\n- v16 --supports--> v2b\n- v2b --supports--> v03\n- v03 --supports--> v15\n- v15 --refutes--> v28\n- v28 --refutes--> v1d\n- v1d --refutes--> v07\n- v07 --refutes--> v02\n- v02 --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v20\n- v02 --refutes--> v05\n- v02 --supports--> v24\n- v04 --supports--> v27\n- v06 --refutes--> v0b\n- v0d --supports--> v13\n- v15 --refutes--> v0f\n- v15 --supports--> v2c\n- v17 --refutes--> v26\n- v18 --refutes--> v08\n- v1d --supports--> v14\n- v1e --refutes--> v0a\n- v22 --refutes--> v29\n- v25 --refutes--> v11\n- v2b --refutes--> v0c\n- v2d --supports--> v10\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":"d15ccf8a7d00cd28d65fcd6b0265fbe14febce83826194ed2d919274b3376429","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v20\n- v20 --refutes--> v11\n- v11 --supports--> v1b\n- v1b --supports--> v2a\n- v2a --supports--> v0d\n- v0d --refutes--> v25\n- v25 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v2f\n- v2f --refutes--> v03\n- v03 --supports--> v1f\n- v1f --refutes--> v14\n- v14 --supports--> v29\n- v29 --supports--> v2e\n- v2e --refutes--> v1e\n- v1e --refutes--> v16\n- v16 --supports--> v07\n- v07 --supports--> v1c\n- v1c --supports--> v0a\n- v0a --refutes--> v0c\n- v0c --refutes--> v01\n- v01 --refutes--> v21\n- v21 --refutes--> v1a\n- v1a --supports--> v2c\n- v2c --refutes--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v28\n- v28 --refutes--> v05\n- v05 --supports--> v26\n- v26 --supports--> v0b\n- v0b --supports--> v13\n- v13 --refutes--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --refutes--> v24\n- v10 --supports--> v04\n- v10 --supports--> v27\n- v11 --supports--> v22\n- v16 --refutes--> v15\n- v16 --supports--> v19\n- v1a --refutes--> v09\n- v1c --supports--> v17\n- v1e --refutes--> v0f\n- v1e --refutes--> v23\n- v21 --supports--> v0e\n- v25 --refutes--> v00\n- v28 --supports--> v30\n- v2a --refutes--> v02\n- v2a --refutes--> v1d\n- v2f --supports--> v2d\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":"67014bd90307d3eff15eac7490ce159dfa26fb803c6c7aad9c8eae3a6a64ab95","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --supports--> v2c\n- v2c --refutes--> v23\n- v23 --refutes--> v17\n- v17 --refutes--> v1f\n- v1f --supports--> v20\n- v20 --supports--> v27\n- v27 --supports--> v19\n- v19 --refutes--> v02\n- v02 --supports--> v28\n- v28 --refutes--> v0d\n- v0d --supports--> v2b\n- v2b --refutes--> v03\n- v03 --supports--> v15\n- v15 --refutes--> v1d\n- v1d --refutes--> v30\n- v30 --supports--> v16\n- v16 --supports--> v29\n- v29 --supports--> v1b\n- v1b --refutes--> v0f\n- v0f --supports--> v2f\n- v2f --supports--> v13\n- v13 --refutes--> v05\n- v05 --supports--> v11\n- v11 --supports--> v25\n- v25 --refutes--> v14\n- v14 --supports--> v10\n- v10 --refutes--> v18\n- v18 --refutes--> v04\n- v04 --refutes--> v00\n- v00 --refutes--> v07\n- v07 --supports--> v1a\n- v1a --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v2d\n- v04 --refutes--> v0a\n- v07 --refutes--> v2a\n- v10 --supports--> v06\n- v11 --refutes--> v1e\n- v14 --supports--> v0e\n- v16 --supports--> v0c\n- v18 --refutes--> v2e\n- v19 --refutes--> v08\n- v19 --refutes--> v1c\n- v20 --supports--> v22\n- v23 --supports--> v0b\n- v2b --supports--> v24\n- v2c --supports--> v09\n- v30 --refutes--> v26\n- v30 --supports--> v21\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":"6529461c23d495946ff19844fed206d4392b467057cf3a9fb69070801833a9ea","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2c --refutes--> v2b\n- v2b --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v2f\n- v2f --supports--> v04\n- v04 --supports--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v30\n- v30 --supports--> v1d\n- v1d --supports--> v20\n- v20 --supports--> v0f\n- v0f --refutes--> v0c\n- v0c --supports--> v17\n- v17 --refutes--> v24\n- v24 --refutes--> v23\n- v23 --supports--> v00\n- v00 --refutes--> v15\n- v15 --supports--> v01\n- v01 --refutes--> v27\n- v27 --refutes--> v1b\n- v1b --supports--> v11\n- v11 --supports--> v1a\n- v1a --refutes--> v12\n- v12 --supports--> v1c\n- v1c --refutes--> v21\n- v21 --refutes--> v28\n- v28 --supports--> v13\n- v13 --supports--> v29\n- v29 --refutes--> v02\n- v02 --refutes--> v14\n- v14 --supports--> v22\n- v22 --refutes--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v2d\n- v01 --refutes--> v16\n- v01 --refutes--> v18\n- v05 --refutes--> v1f\n- v0b --refutes--> v07\n- v0b --refutes--> v25\n- v0d --refutes--> v09\n- v0f --supports--> v1e\n- v13 --supports--> v03\n- v15 --refutes--> v0e\n- v17 --supports--> v06\n- v1b --refutes--> v0a\n- v1b --supports--> v2a\n- v1c --refutes--> v26\n- v30 --refutes--> v19\n- v30 --supports--> v2e\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":"1d4ae2a7ede0d127de28a8346f3cbe8b50fd23f87209290ff3840399ce083fb1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v2e\n- v2e --refutes--> v02\n- v02 --refutes--> v27\n- v27 --supports--> v30\n- v30 --refutes--> v2d\n- v2d --refutes--> v1c\n- v1c --supports--> v26\n- v26 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v06\n- v06 --supports--> v17\n- v17 --supports--> v03\n- v03 --supports--> v21\n- v21 --refutes--> v24\n- v24 --refutes--> v22\n- v22 --refutes--> v28\n- v28 --refutes--> v11\n- v11 --refutes--> v2c\n- v2c --supports--> v2b\n- v2b --refutes--> v18\n- v18 --supports--> v0b\n- v0b --refutes--> v29\n- v29 --supports--> v1f\n- v1f --supports--> v12\n- v12 --refutes--> v14\n- v14 --refutes--> v04\n- v04 --supports--> v0a\n- v0a --refutes--> v19\n- v19 --supports--> v00\n- v00 --supports--> v1d\n- v1d --supports--> v13\n- v13 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v0d\n- v04 --supports--> v25\n- v09 --refutes--> v10\n- v0a --refutes--> v08\n- v0b --supports--> v23\n- v0e --supports--> v15\n- v17 --refutes--> v01\n- v19 --refutes--> v20\n- v1c --supports--> v05\n- v21 --refutes--> v07\n- v22 --refutes--> v1a\n- v24 --refutes--> v1b\n- v24 --refutes--> v2a\n- v26 --supports--> v1e\n- v2b --refutes--> v0c\n- v2d --supports--> v2f\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":"f6a74e838f4c370856397de3692df748c208f6be82273182c7879f4992188671","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --supports--> v2f\n- v2f --supports--> v11\n- v11 --supports--> v2e\n- v2e --refutes--> v17\n- v17 --refutes--> v27\n- v27 --refutes--> v2a\n- v2a --supports--> v22\n- v22 --refutes--> v02\n- v02 --supports--> v26\n- v26 --refutes--> v04\n- v04 --refutes--> v1f\n- v1f --supports--> v24\n- v24 --refutes--> v15\n- v15 --refutes--> v28\n- v28 --refutes--> v16\n- v16 --refutes--> v21\n- v21 --supports--> v08\n- v08 --supports--> v0b\n- v0b --refutes--> v05\n- v05 --refutes--> v29\n- v29 --refutes--> v20\n- v20 --supports--> v12\n- v12 --supports--> v2c\n- v2c --supports--> v06\n- v06 --supports--> v2d\n- v2d --refutes--> v14\n- v14 --supports--> v09\n- v09 --supports--> v00\n- v00 --supports--> v03\n- v03 --supports--> v10\n- v10 --supports--> v0f\n- v0f --supports--> v0d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --refutes--> v0e\n- v09 --supports--> v25\n- v0b --supports--> v0a\n- v0f --refutes--> v23\n- v11 --refutes--> v1b\n- v12 --refutes--> v19\n- v12 --supports--> v1d\n- v16 --refutes--> v18\n- v16 --supports--> v01\n- v16 --supports--> v07\n- v20 --refutes--> v13\n- v20 --supports--> v1a\n- v26 --refutes--> v2b\n- v26 --supports--> v0c\n- v28 --refutes--> v30\n- v2c --refutes--> v1c\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":"92bfe7a181c8538a1b0b334395db5bd4ff47d7dd87b2fe9947f59bcbc737ddea","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v09 --refutes--> v0a\n- v0a --supports--> v24\n- v24 --refutes--> v05\n- v05 --supports--> v10\n- v10 --supports--> v20\n- v20 --supports--> v18\n- v18 --supports--> v2a\n- v2a --supports--> v01\n- v01 --supports--> v28\n- v28 --supports--> v0f\n- v0f --refutes--> v2d\n- v2d --refutes--> v23\n- v23 --supports--> v29\n- v29 --refutes--> v1c\n- v1c --supports--> v0d\n- v0d --supports--> v22\n- v22 --supports--> v17\n- v17 --supports--> v0c\n- v0c --refutes--> v12\n- v12 --supports--> v19\n- v19 --refutes--> v2e\n- v2e --supports--> v27\n- v27 --supports--> v00\n- v00 --refutes--> v30\n- v30 --refutes--> v14\n- v14 --refutes--> v1a\n- v1a --refutes--> v07\n- v07 --refutes--> v2b\n- v2b --refutes--> v08\n- v08 --supports--> v16\n- v16 --supports--> v26\n- v26 --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v00 --refutes--> v25\n- v01 --supports--> v21\n- v07 --supports--> v02\n- v09 --supports--> v2f\n- v0f --supports--> v2c\n- v10 --refutes--> v13\n- v17 --supports--> v0b\n- v18 --refutes--> v03\n- v18 --refutes--> v06\n- v18 --refutes--> v1f\n- v18 --supports--> v1b\n- v22 --supports--> v15\n- v26 --supports--> v1e\n- v28 --refutes--> v11\n- v2b --supports--> v1d\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":"75edfb87d02bfb0a980b8d787405eab227c4481c35f9094869cd87fb8e229a92","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v12\n- v12 --refutes--> v24\n- v24 --refutes--> v26\n- v26 --supports--> v25\n- v25 --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v02\n- v02 --refutes--> v16\n- v16 --refutes--> v13\n- v13 --supports--> v15\n- v15 --supports--> v28\n- v28 --refutes--> v2f\n- v2f --supports--> v17\n- v17 --refutes--> v0c\n- v0c --supports--> v08\n- v08 --refutes--> v23\n- v23 --refutes--> v03\n- v03 --supports--> v05\n- v05 --refutes--> v11\n- v11 --refutes--> v1d\n- v1d --refutes--> v0d\n- v0d --supports--> v19\n- v19 --supports--> v30\n- v30 --refutes--> v27\n- v27 --refutes--> v1f\n- v1f --refutes--> v29\n- v29 --refutes--> v22\n- v22 --refutes--> v2b\n- v2b --supports--> v0a\n- v0a --supports--> v1b\n- v1b --refutes--> v1a\n- v1a --supports--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v10\n- v05 --supports--> v0e\n- v0d --supports--> v20\n- v0f --supports--> v21\n- v0f --supports--> v2d\n- v13 --refutes--> v01\n- v16 --supports--> v1c\n- v19 --refutes--> v1e\n- v1a --supports--> v07\n- v1b --refutes--> v14\n- v1f --supports--> v2c\n- v26 --supports--> v04\n- v28 --supports--> v0b\n- v29 --refutes--> v2e\n- v2f --refutes--> v00\n- v2f --refutes--> v18\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":"7c0ee124601a8b7202b9dc71c6f08d65983fc20d7b0ee41c29cc841cb8e999d6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v06 --refutes--> v2a\n- v2a --supports--> v0f\n- v0f --supports--> v12\n- v12 --supports--> v0a\n- v0a --refutes--> v2d\n- v2d --supports--> v17\n- v17 --supports--> v03\n- v03 --refutes--> v13\n- v13 --supports--> v2e\n- v2e --supports--> v23\n- v23 --supports--> v0e\n- v0e --supports--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v04\n- v04 --refutes--> v2b\n- v2b --refutes--> v2f\n- v2f --refutes--> v2c\n- v2c --refutes--> v24\n- v24 --refutes--> v0c\n- v0c --refutes--> v05\n- v05 --supports--> v01\n- v01 --refutes--> v16\n- v16 --supports--> v27\n- v27 --supports--> v11\n- v11 --refutes--> v1e\n- v1e --supports--> v21\n- v21 --supports--> v00\n- v00 --supports--> v07\n- v07 --supports--> v18\n- v18 --refutes--> v19\n- v19 --refutes--> v20\n- v20 --supports--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0a --supports--> v30\n- v10 --supports--> v1c\n- v12 --refutes--> v22\n- v15 --refutes--> v1d\n- v15 --supports--> v09\n- v17 --supports--> v1f\n- v18 --refutes--> v1a\n- v18 --supports--> v02\n- v20 --refutes--> v28\n- v20 --supports--> v14\n- v21 --refutes--> v0d\n- v21 --refutes--> v1b\n- v27 --supports--> v0b\n- v2a --supports--> v25\n- v2c --refutes--> v29\n- v2f --refutes--> v08\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":"219277c5de20991122fa88b9c0da4631b8e5a1d65c0b407407d7b7716e41272d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v07\n- v07 --refutes--> v16\n- v16 --supports--> v27\n- v27 --supports--> v17\n- v17 --refutes--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v0d\n- v0d --refutes--> v1b\n- v1b --refutes--> v00\n- v00 --refutes--> v2c\n- v2c --refutes--> v18\n- v18 --supports--> v2f\n- v2f --supports--> v02\n- v02 --supports--> v2a\n- v2a --supports--> v1f\n- v1f --supports--> v20\n- v20 --supports--> v1c\n- v1c --refutes--> v2e\n- v2e --refutes--> v2d\n- v2d --refutes--> v03\n- v03 --refutes--> v04\n- v04 --refutes--> v26\n- v26 --supports--> v25\n- v25 --supports--> v14\n- v14 --supports--> v05\n- v05 --supports--> v01\n- v01 --supports--> v0f\n- v0f --refutes--> v23\n- v23 --supports--> v1a\n- v1a --supports--> v09\n- v09 --refutes--> v08\n- v08 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v11\n- v02 --refutes--> v29\n- v03 --refutes--> v12\n- v05 --supports--> v06\n- v08 --refutes--> v0e\n- v0a --supports--> v28\n- v0b --supports--> v1d\n- v0f --supports--> v2b\n- v14 --refutes--> v22\n- v17 --refutes--> v10\n- v1a --supports--> v13\n- v20 --refutes--> v24\n- v23 --supports--> v15\n- v25 --supports--> v30\n- v26 --supports--> v21\n- v2f --refutes--> v0c\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":"ddd714a09c31ade5c4758079481d4ff14758963315081dd645079cddf01684c2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --refutes--> v01\n- v01 --refutes--> v06\n- v06 --refutes--> v0c\n- v0c --refutes--> v2e\n- v2e --supports--> v21\n- v21 --supports--> v2a\n- v2a --supports--> v13\n- v13 --supports--> v2d\n- v2d --refutes--> v26\n- v26 --refutes--> v05\n- v05 --supports--> v10\n- v10 --refutes--> v1e\n- v1e --refutes--> v28\n- v28 --supports--> v2f\n- v2f --refutes--> v18\n- v18 --supports--> v0a\n- v0a --supports--> v15\n- v15 --supports--> v1d\n- v1d --refutes--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v09\n- v09 --refutes--> v0e\n- v0e --supports--> v00\n- v00 --supports--> v27\n- v27 --refutes--> v02\n- v02 --supports--> v07\n- v07 --supports--> v30\n- v30 --refutes--> v0b\n- v0b --refutes--> v17\n- v17 --supports--> v0f\n- v0f --refutes--> v2b\n- v2b --supports--> v2c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v0d\n- v05 --refutes--> v03\n- v05 --supports--> v23\n- v06 --refutes--> v08\n- v0a --refutes--> v29\n- v1c --refutes--> v25\n- v1e --supports--> v22\n- v20 --refutes--> v1b\n- v26 --supports--> v16\n- v28 --refutes--> v14\n- v28 --refutes--> v1a\n- v2b --refutes--> v1f\n- v2b --refutes--> v24\n- v2d --refutes--> v12\n- v2e --supports--> v19\n- v30 --refutes--> v04\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":"e2b394edc0f0a0f58ee1a5e1eab45f16521aea30c42de1f910bef8affaee38ad","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --refutes--> v1a\n- v1a --supports--> v13\n- v13 --refutes--> v2b\n- v2b --refutes--> v1d\n- v1d --refutes--> v2d\n- v2d --refutes--> v07\n- v07 --supports--> v23\n- v23 --supports--> v00\n- v00 --refutes--> v24\n- v24 --refutes--> v1b\n- v1b --refutes--> v01\n- v01 --refutes--> v20\n- v20 --refutes--> v2e\n- v2e --supports--> v12\n- v12 --supports--> v2f\n- v2f --supports--> v25\n- v25 --supports--> v11\n- v11 --refutes--> v28\n- v28 --refutes--> v2a\n- v2a --refutes--> v2c\n- v2c --refutes--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v1e\n- v1e --supports--> v08\n- v08 --supports--> v14\n- v14 --supports--> v02\n- v02 --refutes--> v27\n- v27 --supports--> v22\n- v22 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v06\n- v06 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v18\n- v03 --refutes--> v04\n- v03 --refutes--> v26\n- v03 --supports--> v21\n- v07 --supports--> v1f\n- v10 --refutes--> v29\n- v11 --supports--> v16\n- v15 --supports--> v0b\n- v1b --supports--> v17\n- v1e --supports--> v0a\n- v23 --refutes--> v0f\n- v25 --refutes--> v30\n- v27 --refutes--> v0d\n- v27 --supports--> v1c\n- v28 --supports--> v19\n- v2c --supports--> v0c\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":"3718b06b92a0ba818cd8e6689af468f9557744463fbcf4e24c2c92115e7877df","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v14\n- v14 --refutes--> v17\n- v17 --supports--> v26\n- v26 --refutes--> v03\n- v03 --supports--> v07\n- v07 --refutes--> v11\n- v11 --supports--> v06\n- v06 --supports--> v21\n- v21 --supports--> v25\n- v25 --supports--> v16\n- v16 --supports--> v2b\n- v2b --refutes--> v19\n- v19 --supports--> v18\n- v18 --supports--> v1f\n- v1f --refutes--> v01\n- v01 --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v2f\n- v2f --supports--> v0b\n- v0b --refutes--> v1a\n- v1a --refutes--> v2a\n- v2a --refutes--> v30\n- v30 --refutes--> v2e\n- v2e --supports--> v0e\n- v0e --supports--> v0d\n- v0d --supports--> v15\n- v15 --refutes--> v10\n- v10 --refutes--> v0f\n- v0f --supports--> v1c\n- v1c --supports--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v2c\n- v0d --supports--> v09\n- v0e --supports--> v2d\n- v0f --supports--> v1d\n- v10 --supports--> v22\n- v11 --supports--> v04\n- v15 --refutes--> v13\n- v16 --supports--> v23\n- v18 --refutes--> v08\n- v18 --supports--> v1e\n- v19 --supports--> v29\n- v1f --refutes--> v02\n- v25 --refutes--> v28\n- v26 --refutes--> v20\n- v2a --refutes--> v24\n- v2b --supports--> v12\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":"19740431de8baef0d3c3ef99d1dccb0730017a4c3599a537640bf71abb5332c6","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v29 --supports--> v14\n- v14 --refutes--> v0a\n- v0a --refutes--> v09\n- v09 --refutes--> v21\n- v21 --supports--> v2b\n- v2b --supports--> v0e\n- v0e --supports--> v01\n- v01 --supports--> v03\n- v03 --refutes--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v04\n- v04 --supports--> v2c\n- v2c --refutes--> v30\n- v30 --supports--> v0d\n- v0d --refutes--> v05\n- v05 --refutes--> v11\n- v11 --refutes--> v27\n- v27 --supports--> v07\n- v07 --refutes--> v17\n- v17 --supports--> v1f\n- v1f --supports--> v20\n- v20 --refutes--> v28\n- v28 --refutes--> v15\n- v15 --supports--> v2a\n- v2a --refutes--> v0b\n- v0b --refutes--> v1a\n- v1a --refutes--> v25\n- v25 --supports--> v08\n- v08 --refutes--> v1c\n- v1c --refutes--> v13\n- v13 --refutes--> v18\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --refutes--> v26\n- v09 --supports--> v1b\n- v0d --supports--> v12\n- v11 --supports--> v2e\n- v14 --refutes--> v19\n- v14 --refutes--> v2f\n- v15 --supports--> v1d\n- v17 --supports--> v2d\n- v1a --supports--> v1e\n- v21 --refutes--> v24\n- v21 --supports--> v23\n- v25 --supports--> v16\n- v2b --refutes--> v02\n- v2b --refutes--> v0c\n- v2b --refutes--> v10\n- v30 --supports--> v22\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":"e5e6730475669473d120a73d00ab536627a207e172f6b43c6fc1881476f998d2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1e --refutes--> v29\n- v29 --supports--> v11\n- v11 --refutes--> v19\n- v19 --refutes--> v18\n- v18 --supports--> v25\n- v25 --refutes--> v1d\n- v1d --refutes--> v07\n- v07 --supports--> v16\n- v16 --supports--> v28\n- v28 --supports--> v09\n- v09 --supports--> v20\n- v20 --refutes--> v1b\n- v1b --refutes--> v10\n- v10 --supports--> v26\n- v26 --supports--> v24\n- v24 --supports--> v2e\n- v2e --supports--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v0c\n- v0c --refutes--> v1c\n- v1c --refutes--> v2f\n- v2f --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v05\n- v05 --refutes--> v12\n- v12 --refutes--> v08\n- v08 --refutes--> v0e\n- v0e --supports--> v2d\n- v2d --refutes--> v1f\n- v1f --refutes--> v06\n- v06 --refutes--> v30\n- v30 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v0a\n- v05 --supports--> v15\n- v06 --refutes--> v2c\n- v09 --refutes--> v2b\n- v0c --supports--> v22\n- v0c --supports--> v2a\n- v0e --refutes--> v01\n- v0e --refutes--> v17\n- v1b --refutes--> v03\n- v1d --supports--> v0d\n- v1f --refutes--> v27\n- v1f --supports--> v21\n- v25 --supports--> v02\n- v26 --refutes--> v23\n- v28 --refutes--> v04\n- v28 --supports--> v14\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":"c769778a4fe148c24685c3ead8fcd53502ee5935aa21dd81715f382d3ab8f3b1","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v12\n- v12 --refutes--> v28\n- v28 --supports--> v0e\n- v0e --refutes--> v0d\n- v0d --supports--> v20\n- v20 --refutes--> v00\n- v00 --supports--> v11\n- v11 --refutes--> v16\n- v16 --supports--> v17\n- v17 --supports--> v27\n- v27 --refutes--> v01\n- v01 --supports--> v1f\n- v1f --supports--> v06\n- v06 --refutes--> v14\n- v14 --refutes--> v1d\n- v1d --supports--> v30\n- v30 --refutes--> v2d\n- v2d --refutes--> v0c\n- v0c --supports--> v25\n- v25 --supports--> v2e\n- v2e --supports--> v29\n- v29 --supports--> v0f\n- v0f --supports--> v26\n- v26 --supports--> v15\n- v15 --supports--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v08\n- v08 --supports--> v1c\n- v1c --refutes--> v05\n- v05 --refutes--> v1b\n- v1b --refutes--> v18\n- v18 --refutes--> v1a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v2c\n- v06 --supports--> v2f\n- v0c --refutes--> v24\n- v0c --supports--> v2a\n- v10 --refutes--> v2b\n- v13 --supports--> v09\n- v15 --refutes--> v22\n- v15 --supports--> v0b\n- v17 --supports--> v23\n- v1b --supports--> v02\n- v1c --refutes--> v1e\n- v26 --refutes--> v03\n- v29 --refutes--> v0a\n- v29 --supports--> v04\n- v2d --refutes--> v21\n- v30 --refutes--> v07\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":"c5d25cb14ff291154f689d26ca26ddefb0b336743ca680f025469005c8a5574d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --supports--> v00\n- v00 --refutes--> v15\n- v15 --supports--> v14\n- v14 --refutes--> v1c\n- v1c --refutes--> v0e\n- v0e --refutes--> v22\n- v22 --refutes--> v16\n- v16 --refutes--> v09\n- v09 --refutes--> v27\n- v27 --supports--> v1d\n- v1d --supports--> v10\n- v10 --refutes--> v24\n- v24 --supports--> v0a\n- v0a --refutes--> v30\n- v30 --supports--> v1e\n- v1e --supports--> v1a\n- v1a --supports--> v07\n- v07 --supports--> v25\n- v25 --refutes--> v29\n- v29 --supports--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v02\n- v02 --supports--> v2d\n- v2d --supports--> v23\n- v23 --refutes--> v1f\n- v1f --refutes--> v26\n- v26 --refutes--> v19\n- v19 --refutes--> v13\n- v13 --supports--> v04\n- v04 --refutes--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v2c\n- v00 --supports--> v01\n- v0a --refutes--> v0b\n- v0c --refutes--> v12\n- v0f --refutes--> v18\n- v0f --refutes--> v2a\n- v13 --supports--> v1b\n- v16 --supports--> v0d\n- v1a --refutes--> v05\n- v1a --refutes--> v28\n- v1a --supports--> v21\n- v23 --refutes--> v08\n- v23 --refutes--> v17\n- v23 --supports--> v06\n- v24 --refutes--> v2f\n- v30 --refutes--> v2e\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":"4075ac86790625f683c939eec74a6c86aaf4de64863471d0ea7c2fb8f2ccaa28","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v25\n- v25 --supports--> v11\n- v11 --refutes--> v28\n- v28 --refutes--> v0e\n- v0e --supports--> v05\n- v05 --supports--> v1b\n- v1b --supports--> v0a\n- v0a --supports--> v00\n- v00 --supports--> v30\n- v30 --supports--> v23\n- v23 --supports--> v0b\n- v0b --supports--> v1c\n- v1c --refutes--> v21\n- v21 --refutes--> v0d\n- v0d --supports--> v1d\n- v1d --refutes--> v10\n- v10 --supports--> v2b\n- v2b --supports--> v0f\n- v0f --supports--> v2f\n- v2f --refutes--> v01\n- v01 --supports--> v27\n- v27 --refutes--> v17\n- v17 --supports--> v2c\n- v2c --supports--> v12\n- v12 --supports--> v08\n- v08 --supports--> v02\n- v02 --refutes--> v06\n- v06 --supports--> v09\n- v09 --supports--> v1e\n- v1e --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v2e\n- v07 --refutes--> v19\n- v0d --refutes--> v24\n- v0d --supports--> v1f\n- v0e --refutes--> v15\n- v10 --supports--> v29\n- v12 --supports--> v18\n- v13 --refutes--> v2a\n- v1b --supports--> v2d\n- v1d --refutes--> v0c\n- v23 --supports--> v14\n- v23 --supports--> v1a\n- v25 --supports--> v16\n- v27 --refutes--> v26\n- v2f --supports--> v20\n- v30 --refutes--> v22\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":"fbbd15e84112d3deac798662bc28bc7593f5ab1b21ea9c4aa4487eb0d3be5f75","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v12\n- v12 --refutes--> v2f\n- v2f --refutes--> v2b\n- v2b --refutes--> v0b\n- v0b --supports--> v06\n- v06 --refutes--> v1b\n- v1b --supports--> v29\n- v29 --refutes--> v23\n- v23 --refutes--> v1a\n- v1a --refutes--> v08\n- v08 --refutes--> v2c\n- v2c --refutes--> v0c\n- v0c --supports--> v2e\n- v2e --refutes--> v17\n- v17 --refutes--> v1e\n- v1e --supports--> v05\n- v05 --refutes--> v13\n- v13 --refutes--> v24\n- v24 --supports--> v21\n- v21 --supports--> v16\n- v16 --supports--> v22\n- v22 --refutes--> v25\n- v25 --refutes--> v0e\n- v0e --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v18\n- v18 --refutes--> v00\n- v00 --supports--> v03\n- v03 --refutes--> v15\n- v15 --supports--> v20\n- v20 --supports--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v27\n- v05 --supports--> v02\n- v06 --supports--> v09\n- v0e --supports--> v0a\n- v12 --supports--> v11\n- v17 --refutes--> v04\n- v18 --supports--> v26\n- v1b --refutes--> v0f\n- v1e --refutes--> v01\n- v21 --refutes--> v2a\n- v23 --refutes--> v1c\n- v23 --supports--> v2d\n- v24 --refutes--> v07\n- v25 --supports--> v28\n- v25 --supports--> v30\n- v2f --refutes--> v1d\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":"466578591c9f6eb93301da56f1e516cb3bea5e4207fa805bfb2c1612bf6620be","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --supports--> v0d\n- v0d --supports--> v01\n- v01 --supports--> v26\n- v26 --refutes--> v09\n- v09 --refutes--> v10\n- v10 --refutes--> v16\n- v16 --refutes--> v11\n- v11 --supports--> v0b\n- v0b --supports--> v29\n- v29 --refutes--> v2e\n- v2e --refutes--> v08\n- v08 --refutes--> v17\n- v17 --supports--> v20\n- v20 --refutes--> v0f\n- v0f --refutes--> v18\n- v18 --refutes--> v2a\n- v2a --supports--> v23\n- v23 --refutes--> v2f\n- v2f --supports--> v19\n- v19 --refutes--> v21\n- v21 --refutes--> v04\n- v04 --supports--> v1d\n- v1d --supports--> v2d\n- v2d --supports--> v24\n- v24 --refutes--> v22\n- v22 --refutes--> v15\n- v15 --supports--> v1b\n- v1b --supports--> v05\n- v05 --refutes--> v0c\n- v0c --refutes--> v07\n- v07 --supports--> v2c\n- v2c --supports--> v0a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v13\n- v01 --supports--> v1e\n- v04 --refutes--> v00\n- v08 --supports--> v06\n- v09 --refutes--> v1a\n- v09 --supports--> v28\n- v15 --refutes--> v30\n- v18 --supports--> v1f\n- v1b --refutes--> v03\n- v22 --supports--> v25\n- v23 --refutes--> v2b\n- v24 --refutes--> v02\n- v24 --refutes--> v14\n- v2a --refutes--> v12\n- v2d --supports--> v1c\n- v2f --refutes--> v0e\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":"b83a73e616ea3268a307a0d3499d56b7ae03971aaa3abff2c9024a86d26762e0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v21\n- v21 --refutes--> v0a\n- v0a --supports--> v09\n- v09 --supports--> v2a\n- v2a --supports--> v20\n- v20 --refutes--> v01\n- v01 --supports--> v05\n- v05 --refutes--> v24\n- v24 --supports--> v04\n- v04 --refutes--> v0c\n- v0c --supports--> v1a\n- v1a --refutes--> v11\n- v11 --refutes--> v10\n- v10 --supports--> v25\n- v25 --supports--> v03\n- v03 --refutes--> v29\n- v29 --refutes--> v08\n- v08 --refutes--> v17\n- v17 --refutes--> v2c\n- v2c --refutes--> v1d\n- v1d --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v16\n- v16 --supports--> v06\n- v06 --refutes--> v19\n- v19 --supports--> v1f\n- v1f --refutes--> v02\n- v02 --supports--> v30\n- v30 --supports--> v26\n- v26 --supports--> v2e\n- v2e --refutes--> v1e\n- v1e --supports--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v28\n- v05 --refutes--> v13\n- v0c --refutes--> v1b\n- v0c --supports--> v23\n- v10 --refutes--> v2f\n- v17 --supports--> v15\n- v19 --refutes--> v12\n- v19 --refutes--> v27\n- v1e --supports--> v1c\n- v20 --supports--> v22\n- v21 --supports--> v0f\n- v24 --refutes--> v0b\n- v2e --supports--> v0d\n- v30 --refutes--> v07\n- v30 --supports--> v14\n- v30 --supports--> v2d\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":"d09da466ddc7714ed7b5301a0e084f082e85198f7c41a46c2726872f53d3606c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v11\n- v11 --refutes--> v23\n- v23 --refutes--> v09\n- v09 --refutes--> v30\n- v30 --refutes--> v2a\n- v2a --refutes--> v0b\n- v0b --refutes--> v2d\n- v2d --supports--> v2f\n- v2f --refutes--> v10\n- v10 --refutes--> v15\n- v15 --supports--> v0a\n- v0a --refutes--> v29\n- v29 --supports--> v2b\n- v2b --supports--> v22\n- v22 --supports--> v1e\n- v1e --refutes--> v26\n- v26 --supports--> v0d\n- v0d --supports--> v21\n- v21 --supports--> v1c\n- v1c --refutes--> v03\n- v03 --refutes--> v06\n- v06 --supports--> v25\n- v25 --refutes--> v0f\n- v0f --refutes--> v1b\n- v1b --refutes--> v2e\n- v2e --refutes--> v17\n- v17 --supports--> v05\n- v05 --supports--> v14\n- v14 --supports--> v27\n- v27 --refutes--> v0c\n- v0c --refutes--> v24\n- v24 --supports--> v16\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v12\n- v05 --refutes--> v01\n- v06 --supports--> v04\n- v09 --refutes--> v08\n- v09 --refutes--> v20\n- v0a --supports--> v1f\n- v0d --refutes--> v07\n- v0d --supports--> v18\n- v0f --supports--> v1a\n- v11 --refutes--> v13\n- v15 --supports--> v2c\n- v1e --refutes--> v28\n- v22 --refutes--> v19\n- v23 --refutes--> v0e\n- v29 --supports--> v1d\n- v2f --supports--> v00\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":"af971d16faf9cd29754153daef48f7fd4982e73448e51e9897754bdfae987f42","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v00\n- v00 --supports--> v25\n- v25 --supports--> v0d\n- v0d --supports--> v1d\n- v1d --refutes--> v1e\n- v1e --refutes--> v21\n- v21 --supports--> v03\n- v03 --refutes--> v13\n- v13 --refutes--> v07\n- v07 --supports--> v16\n- v16 --supports--> v22\n- v22 --refutes--> v12\n- v12 --supports--> v2f\n- v2f --refutes--> v2d\n- v2d --refutes--> v2c\n- v2c --refutes--> v05\n- v05 --supports--> v08\n- v08 --refutes--> v06\n- v06 --supports--> v20\n- v20 --refutes--> v30\n- v30 --supports--> v10\n- v10 --refutes--> v2e\n- v2e --refutes--> v0f\n- v0f --supports--> v15\n- v15 --supports--> v0a\n- v0a --refutes--> v11\n- v11 --supports--> v27\n- v27 --refutes--> v2a\n- v2a --refutes--> v0e\n- v0e --refutes--> v23\n- v23 --refutes--> v0b\n- v0b --supports--> v24\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v09\n- v0a --refutes--> v04\n- v0b --supports--> v02\n- v0e --refutes--> v18\n- v0f --supports--> v26\n- v10 --refutes--> v01\n- v11 --refutes--> v1c\n- v12 --supports--> v28\n- v15 --refutes--> v29\n- v15 --supports--> v0c\n- v23 --supports--> v2b\n- v25 --supports--> v14\n- v2a --refutes--> v1a\n- v2a --supports--> v1f\n- v2f --refutes--> v19\n- v30 --refutes--> v1b\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":"d6860a2e92375aa58c68861abdfc3d0865afe3f7193af682802e675de4a1e0c8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v07 --refutes--> v0a\n- v0a --supports--> v02\n- v02 --supports--> v23\n- v23 --refutes--> v1f\n- v1f --supports--> v16\n- v16 --supports--> v0b\n- v0b --refutes--> v06\n- v06 --supports--> v25\n- v25 --refutes--> v20\n- v20 --refutes--> v21\n- v21 --refutes--> v28\n- v28 --refutes--> v1c\n- v1c --refutes--> v0c\n- v0c --supports--> v2e\n- v2e --supports--> v1a\n- v1a --supports--> v2d\n- v2d --refutes--> v13\n- v13 --refutes--> v2c\n- v2c --supports--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v0f\n- v0f --refutes--> v24\n- v24 --supports--> v14\n- v14 --supports--> v19\n- v19 --refutes--> v05\n- v05 --supports--> v03\n- v03 --refutes--> v0e\n- v0e --supports--> v1e\n- v1e --supports--> v2a\n- v2a --supports--> v08\n- v08 --refutes--> v15\n- v15 --supports--> v09\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v1b\n- v0a --refutes--> v18\n- v0c --refutes--> v26\n- v10 --refutes--> v17\n- v14 --refutes--> v11\n- v14 --refutes--> v22\n- v19 --supports--> v27\n- v1a --refutes--> v2f\n- v1a --supports--> v1d\n- v1e --refutes--> v01\n- v23 --supports--> v00\n- v24 --refutes--> v04\n- v24 --supports--> v30\n- v28 --refutes--> v2b\n- v28 --supports--> v0d\n- v28 --supports--> v29\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":"345727c1c4d688fe1a708010ddff8aa9fcd298bcbb55bf86b58a62b99c81aee8","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v20 --supports--> v24\n- v24 --refutes--> v1a\n- v1a --refutes--> v2e\n- v2e --refutes--> v23\n- v23 --supports--> v05\n- v05 --refutes--> v15\n- v15 --supports--> v12\n- v12 --refutes--> v08\n- v08 --supports--> v04\n- v04 --supports--> v00\n- v00 --refutes--> v17\n- v17 --refutes--> v1b\n- v1b --supports--> v0a\n- v0a --refutes--> v0e\n- v0e --refutes--> v19\n- v19 --supports--> v16\n- v16 --refutes--> v13\n- v13 --supports--> v1c\n- v1c --supports--> v0d\n- v0d --supports--> v29\n- v29 --refutes--> v21\n- v21 --supports--> v27\n- v27 --refutes--> v26\n- v26 --supports--> v18\n- v18 --supports--> v22\n- v22 --refutes--> v2a\n- v2a --refutes--> v1f\n- v1f --supports--> v2d\n- v2d --refutes--> v0c\n- v0c --refutes--> v0f\n- v0f --supports--> v09\n- v09 --supports--> v2c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v30\n- v08 --refutes--> v28\n- v09 --supports--> v06\n- v0c --supports--> v14\n- v12 --supports--> v1e\n- v17 --supports--> v03\n- v19 --refutes--> v2f\n- v1f --supports--> v10\n- v1f --supports--> v2b\n- v20 --refutes--> v07\n- v21 --refutes--> v0b\n- v21 --supports--> v01\n- v21 --supports--> v02\n- v23 --supports--> v25\n- v24 --refutes--> v11\n- v27 --refutes--> v1d\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":"87b27be522d5d051b58d2a44a7503328d8470a2d2af5b0e9a09d2651a0c4131f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --supports--> v07\n- v07 --refutes--> v20\n- v20 --supports--> v28\n- v28 --refutes--> v2f\n- v2f --refutes--> v11\n- v11 --refutes--> v24\n- v24 --refutes--> v1d\n- v1d --refutes--> v22\n- v22 --refutes--> v05\n- v05 --supports--> v0d\n- v0d --supports--> v21\n- v21 --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v27\n- v27 --refutes--> v00\n- v00 --supports--> v15\n- v15 --refutes--> v04\n- v04 --refutes--> v30\n- v30 --refutes--> v09\n- v09 --supports--> v0f\n- v0f --supports--> v29\n- v29 --supports--> v18\n- v18 --supports--> v1a\n- v1a --refutes--> v26\n- v26 --refutes--> v1e\n- v1e --refutes--> v2c\n- v2c --refutes--> v01\n- v01 --refutes--> v1c\n- v1c --refutes--> v16\n- v16 --supports--> v2e\n- v2e --supports--> v2a\n- v2a --refutes--> v2d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1f\n- v04 --supports--> v0e\n- v06 --refutes--> v0c\n- v09 --supports--> v13\n- v11 --refutes--> v03\n- v15 --supports--> v23\n- v21 --refutes--> v2b\n- v21 --supports--> v02\n- v22 --refutes--> v19\n- v22 --supports--> v14\n- v26 --refutes--> v08\n- v29 --supports--> v12\n- v2a --refutes--> v0b\n- v2c --refutes--> v25\n- v2f --refutes--> v10\n- v2f --refutes--> v17\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":"d528a4a28398423374b9af26f1e0bbea54428e3abd07f797ef19c2e6ecb49b69","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v12 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v02\n- v02 --refutes--> v16\n- v16 --supports--> v27\n- v27 --refutes--> v25\n- v25 --refutes--> v09\n- v09 --supports--> v2f\n- v2f --supports--> v22\n- v22 --supports--> v19\n- v19 --supports--> v0b\n- v0b --refutes--> v14\n- v14 --supports--> v11\n- v11 --supports--> v18\n- v18 --refutes--> v2e\n- v2e --supports--> v00\n- v00 --supports--> v28\n- v28 --refutes--> v1c\n- v1c --refutes--> v1f\n- v1f --refutes--> v26\n- v26 --supports--> v01\n- v01 --refutes--> v20\n- v20 --refutes--> v06\n- v06 --supports--> v05\n- v05 --refutes--> v10\n- v10 --supports--> v1d\n- v1d --supports--> v1a\n- v1a --supports--> v08\n- v08 --supports--> v30\n- v30 --refutes--> v21\n- v21 --refutes--> v2b\n- v2b --supports--> v0e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v04\n- v01 --refutes--> v2a\n- v06 --supports--> v15\n- v10 --refutes--> v07\n- v10 --supports--> v13\n- v11 --supports--> v1b\n- v1d --refutes--> v0a\n- v1d --supports--> v2c\n- v1f --supports--> v17\n- v1f --supports--> v23\n- v20 --refutes--> v2d\n- v25 --refutes--> v29\n- v25 --supports--> v1e\n- v28 --refutes--> v24\n- v2f --refutes--> v0f\n- v2f --supports--> v03\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":"3e82ac5909dbb84198943866e38a42a7b3cec040dec01db44d4973a1a393869b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v27 --refutes--> v22\n- v22 --supports--> v15\n- v15 --refutes--> v1e\n- v1e --refutes--> v09\n- v09 --supports--> v25\n- v25 --supports--> v01\n- v01 --supports--> v2b\n- v2b --refutes--> v0d\n- v0d --refutes--> v0f\n- v0f --refutes--> v16\n- v16 --supports--> v2c\n- v2c --refutes--> v19\n- v19 --refutes--> v1b\n- v1b --supports--> v02\n- v02 --supports--> v2f\n- v2f --refutes--> v23\n- v23 --supports--> v18\n- v18 --refutes--> v0c\n- v0c --supports--> v30\n- v30 --refutes--> v14\n- v14 --supports--> v07\n- v07 --refutes--> v0b\n- v0b --supports--> v2a\n- v2a --supports--> v21\n- v21 --refutes--> v05\n- v05 --supports--> v03\n- v03 --supports--> v1f\n- v1f --refutes--> v1d\n- v1d --refutes--> v1a\n- v1a --supports--> v10\n- v10 --supports--> v28\n- v28 --refutes--> v12\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0e\n- v03 --refutes--> v06\n- v03 --refutes--> v13\n- v05 --supports--> v04\n- v10 --supports--> v08\n- v16 --refutes--> v0a\n- v18 --refutes--> v29\n- v19 --supports--> v2d\n- v1a --refutes--> v11\n- v1a --refutes--> v2e\n- v1a --supports--> v1c\n- v21 --refutes--> v00\n- v22 --supports--> v24\n- v25 --refutes--> v17\n- v2b --refutes--> v26\n- v2f --supports--> v20\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":"d9a6f70910889986d831d617d967ede7b585a8b6d2640ac490f02ca6d4679651","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v17 --refutes--> v1a\n- v1a --refutes--> v1c\n- v1c --refutes--> v13\n- v13 --supports--> v22\n- v22 --refutes--> v2d\n- v2d --refutes--> v02\n- v02 --refutes--> v03\n- v03 --supports--> v30\n- v30 --refutes--> v2a\n- v2a --refutes--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --supports--> v2b\n- v2b --refutes--> v21\n- v21 --supports--> v04\n- v04 --refutes--> v05\n- v05 --supports--> v11\n- v11 --supports--> v23\n- v23 --supports--> v27\n- v27 --supports--> v1f\n- v1f --refutes--> v29\n- v29 --refutes--> v1d\n- v1d --refutes--> v15\n- v15 --supports--> v2c\n- v2c --supports--> v1b\n- v1b --supports--> v0b\n- v0b --refutes--> v00\n- v00 --refutes--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v10\n- v10 --refutes--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v01 --supports--> v26\n- v05 --refutes--> v2e\n- v0b --refutes--> v25\n- v0e --supports--> v0c\n- v10 --refutes--> v24\n- v11 --supports--> v28\n- v11 --supports--> v2f\n- v12 --refutes--> v09\n- v12 --supports--> v0d\n- v14 --supports--> v18\n- v15 --supports--> v20\n- v17 --supports--> v1e\n- v1a --supports--> v19\n- v1f --refutes--> v0a\n- v2c --refutes--> v16\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":"8095c753d10dbd5d63c117b943d11b66e3a97c91c1148eb3190655e352f3f60d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v02 --supports--> v16\n- v16 --refutes--> v1a\n- v1a --supports--> v22\n- v22 --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --refutes--> v1d\n- v1d --supports--> v05\n- v05 --supports--> v0d\n- v0d --refutes--> v14\n- v14 --supports--> v2b\n- v2b --supports--> v17\n- v17 --supports--> v28\n- v28 --refutes--> v2d\n- v2d --refutes--> v08\n- v08 --refutes--> v13\n- v13 --refutes--> v00\n- v00 --supports--> v09\n- v09 --refutes--> v25\n- v25 --refutes--> v2a\n- v2a --supports--> v0f\n- v0f --refutes--> v07\n- v07 --supports--> v12\n- v12 --refutes--> v2e\n- v2e --refutes--> v1e\n- v1e --supports--> v21\n- v21 --refutes--> v23\n- v23 --supports--> v1c\n- v1c --refutes--> v0b\n- v0b --supports--> v01\n- v01 --refutes--> v15\n- v15 --supports--> v18\n- v18 --refutes--> v19\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v0a\n- v05 --supports--> v24\n- v07 --supports--> v1b\n- v08 --refutes--> v26\n- v08 --refutes--> v27\n- v09 --refutes--> v20\n- v0c --refutes--> v10\n- v0f --supports--> v2c\n- v13 --refutes--> v03\n- v13 --supports--> v11\n- v16 --supports--> v2f\n- v1d --supports--> v30\n- v1e --refutes--> v1f\n- v22 --refutes--> v06\n- v23 --refutes--> v29\n- v25 --supports--> v0e\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":"93fc9476453759343ec18e2172ba4dcb7b29e9a2d5f3d87f25976e72d8d76d08","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1d --supports--> v26\n- v26 --refutes--> v22\n- v22 --supports--> v11\n- v11 --refutes--> v21\n- v21 --refutes--> v0a\n- v0a --refutes--> v02\n- v02 --supports--> v15\n- v15 --refutes--> v2c\n- v2c --refutes--> v28\n- v28 --supports--> v07\n- v07 --refutes--> v30\n- v30 --supports--> v1e\n- v1e --refutes--> v1b\n- v1b --supports--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v2d\n- v2d --supports--> v04\n- v04 --supports--> v12\n- v12 --refutes--> v27\n- v27 --supports--> v2a\n- v2a --refutes--> v08\n- v08 --refutes--> v16\n- v16 --supports--> v25\n- v25 --supports--> v1a\n- v1a --supports--> v19\n- v19 --refutes--> v0c\n- v0c --supports--> v03\n- v03 --supports--> v24\n- v24 --refutes--> v18\n- v18 --supports--> v09\n- v09 --supports--> v13\n- v13 --refutes--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0e\n- v04 --supports--> v1c\n- v04 --supports--> v1f\n- v09 --supports--> v17\n- v09 --supports--> v2b\n- v0f --supports--> v06\n- v13 --refutes--> v14\n- v15 --supports--> v0b\n- v16 --refutes--> v20\n- v19 --supports--> v10\n- v1a --supports--> v29\n- v1d --refutes--> v00\n- v26 --supports--> v2e\n- v28 --supports--> v23\n- v2d --supports--> v0d\n- v30 --supports--> v2f\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":"44ffc37c8dfbd0e8625d1499c4e0cf21ce0417a7bb391b4edf120e80116674f0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --refutes--> v22\n- v22 --supports--> v13\n- v13 --refutes--> v2c\n- v2c --refutes--> v0d\n- v0d --refutes--> v19\n- v19 --supports--> v1e\n- v1e --supports--> v1d\n- v1d --supports--> v2f\n- v2f --supports--> v11\n- v11 --supports--> v2d\n- v2d --refutes--> v07\n- v07 --supports--> v0c\n- v0c --supports--> v02\n- v02 --supports--> v2a\n- v2a --refutes--> v29\n- v29 --refutes--> v05\n- v05 --refutes--> v1c\n- v1c --refutes--> v12\n- v12 --supports--> v00\n- v00 --refutes--> v18\n- v18 --refutes--> v25\n- v25 --refutes--> v14\n- v14 --refutes--> v09\n- v09 --supports--> v1a\n- v1a --refutes--> v20\n- v20 --refutes--> v17\n- v17 --refutes--> v1f\n- v1f --supports--> v2b\n- v2b --refutes--> v06\n- v06 --refutes--> v15\n- v15 --refutes--> v0f\n- v0f --supports--> v08\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1b\n- v06 --refutes--> v2e\n- v06 --supports--> v01\n- v0c --refutes--> v10\n- v0c --supports--> v30\n- v15 --supports--> v0a\n- v15 --supports--> v21\n- v15 --supports--> v27\n- v1a --refutes--> v28\n- v1c --supports--> v03\n- v1d --refutes--> v0e\n- v1e --refutes--> v26\n- v20 --refutes--> v16\n- v29 --supports--> v0b\n- v2b --refutes--> v04\n- v2b --supports--> v24\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":"7448c940d30e4cd3061083a6d759b90e9ef218876a803e3fb6b672c5bebfc718","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1c --refutes--> v2d\n- v2d --refutes--> v1b\n- v1b --refutes--> v09\n- v09 --refutes--> v14\n- v14 --refutes--> v21\n- v21 --refutes--> v18\n- v18 --supports--> v07\n- v07 --supports--> v1f\n- v1f --refutes--> v1d\n- v1d --supports--> v2b\n- v2b --refutes--> v30\n- v30 --supports--> v26\n- v26 --refutes--> v22\n- v22 --supports--> v25\n- v25 --supports--> v0e\n- v0e --supports--> v11\n- v11 --supports--> v12\n- v12 --supports--> v08\n- v08 --supports--> v06\n- v06 --supports--> v20\n- v20 --refutes--> v15\n- v15 --supports--> v0a\n- v0a --supports--> v29\n- v29 --supports--> v03\n- v03 --refutes--> v1a\n- v1a --refutes--> v23\n- v23 --refutes--> v24\n- v24 --refutes--> v19\n- v19 --refutes--> v16\n- v16 --supports--> v2e\n- v2e --supports--> v0d\n- v0d --supports--> v0b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v10\n- v07 --supports--> v2a\n- v08 --refutes--> v02\n- v09 --supports--> v0c\n- v09 --supports--> v28\n- v0a --refutes--> v04\n- v0e --refutes--> v05\n- v0e --refutes--> v0f\n- v0e --refutes--> v2c\n- v0e --supports--> v13\n- v1a --refutes--> v17\n- v1a --supports--> v00\n- v1c --refutes--> v27\n- v23 --refutes--> v2f\n- v23 --supports--> v1e\n- v25 --supports--> v01\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":"a1d9364fc8a86063d9e7aec02cd5b583b98e84f6f1bef2b48b0db97ef07ab4ad","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v1d\n- v1d --refutes--> v2e\n- v2e --supports--> v0f\n- v0f --supports--> v08\n- v08 --supports--> v16\n- v16 --supports--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v15\n- v15 --refutes--> v27\n- v27 --refutes--> v01\n- v01 --supports--> v11\n- v11 --refutes--> v21\n- v21 --supports--> v10\n- v10 --supports--> v19\n- v19 --refutes--> v0c\n- v0c --refutes--> v2f\n- v2f --supports--> v07\n- v07 --supports--> v05\n- v05 --refutes--> v22\n- v22 --refutes--> v1b\n- v1b --refutes--> v2d\n- v2d --refutes--> v0b\n- v0b --supports--> v12\n- v12 --refutes--> v04\n- v04 --supports--> v09\n- v09 --refutes--> v06\n- v06 --refutes--> v29\n- v29 --supports--> v2b\n- v2b --supports--> v30\n- v30 --refutes--> v17\n- v17 --refutes--> v25\n- v25 --supports--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0b --refutes--> v28\n- v0f --supports--> v0e\n- v10 --supports--> v13\n- v11 --supports--> v23\n- v15 --refutes--> v1e\n- v16 --refutes--> v03\n- v16 --refutes--> v24\n- v19 --supports--> v1f\n- v1b --supports--> v0d\n- v22 --refutes--> v2c\n- v22 --supports--> v02\n- v29 --refutes--> v26\n- v2b --refutes--> v1c\n- v2b --refutes--> v2a\n- v2d --supports--> v20\n- v30 --refutes--> v00\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":"1d7e9f174c71958290bfe50f15d0607f6c21bfd210bc707d60d100a0519155ca","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v16 --refutes--> v1a\n- v1a --supports--> v17\n- v17 --supports--> v22\n- v22 --refutes--> v04\n- v04 --refutes--> v2f\n- v2f --refutes--> v0e\n- v0e --refutes--> v1d\n- v1d --supports--> v01\n- v01 --supports--> v09\n- v09 --refutes--> v30\n- v30 --refutes--> v00\n- v00 --refutes--> v26\n- v26 --refutes--> v03\n- v03 --refutes--> v1f\n- v1f --supports--> v1b\n- v1b --supports--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v12\n- v12 --supports--> v14\n- v14 --supports--> v27\n- v27 --refutes--> v07\n- v07 --supports--> v1c\n- v1c --refutes--> v2c\n- v2c --refutes--> v06\n- v06 --supports--> v19\n- v19 --supports--> v2e\n- v2e --refutes--> v25\n- v25 --refutes--> v20\n- v20 --refutes--> v2d\n- v2d --supports--> v0f\n- v0f --refutes--> v0b\n- v0b --supports--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v07 --refutes--> v02\n- v09 --supports--> v11\n- v0b --supports--> v29\n- v14 --refutes--> v18\n- v14 --refutes--> v1e\n- v16 --supports--> v24\n- v22 --refutes--> v0a\n- v22 --supports--> v05\n- v27 --refutes--> v08\n- v27 --refutes--> v21\n- v2c --refutes--> v23\n- v2d --refutes--> v10\n- v2d --supports--> v2a\n- v2e --supports--> v28\n- v2f --supports--> v13\n- v30 --supports--> v15\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":"b54e32568eb9aa3deecd667588bf35e38679f7be84508f15bfb4862d32fe2616","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --supports--> v1a\n- v1a --refutes--> v20\n- v20 --supports--> v06\n- v06 --refutes--> v00\n- v00 --supports--> v2c\n- v2c --supports--> v18\n- v18 --supports--> v0b\n- v0b --refutes--> v28\n- v28 --supports--> v04\n- v04 --supports--> v10\n- v10 --supports--> v17\n- v17 --refutes--> v0d\n- v0d --refutes--> v21\n- v21 --refutes--> v19\n- v19 --supports--> v1c\n- v1c --supports--> v30\n- v30 --refutes--> v1e\n- v1e --refutes--> v24\n- v24 --refutes--> v0c\n- v0c --refutes--> v23\n- v23 --supports--> v05\n- v05 --supports--> v11\n- v11 --refutes--> v14\n- v14 --supports--> v02\n- v02 --supports--> v08\n- v08 --supports--> v2f\n- v2f --supports--> v2b\n- v2b --refutes--> v1b\n- v1b --refutes--> v22\n- v22 --refutes--> v29\n- v29 --refutes--> v1f\n- v1f --supports--> v0f\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v2a\n- v10 --refutes--> v13\n- v19 --supports--> v12\n- v1c --refutes--> v07\n- v1f --refutes--> v09\n- v1f --refutes--> v15\n- v1f --refutes--> v27\n- v1f --supports--> v2d\n- v20 --refutes--> v0a\n- v20 --supports--> v2e\n- v21 --refutes--> v1d\n- v22 --supports--> v03\n- v22 --supports--> v0e\n- v25 --supports--> v16\n- v28 --refutes--> v01\n- v2b --refutes--> v26\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":"ee6bbd22036f312cecf0fb55e644ecfd8f605d73b3d9a4994841cd0e76bc0997","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2f --refutes--> v09\n- v09 --refutes--> v1a\n- v1a --refutes--> v16\n- v16 --refutes--> v1c\n- v1c --supports--> v29\n- v29 --refutes--> v1d\n- v1d --refutes--> v01\n- v01 --supports--> v18\n- v18 --refutes--> v03\n- v03 --supports--> v12\n- v12 --supports--> v2e\n- v2e --refutes--> v0b\n- v0b --refutes--> v2a\n- v2a --supports--> v07\n- v07 --supports--> v14\n- v14 --supports--> v0e\n- v0e --supports--> v06\n- v06 --supports--> v22\n- v22 --refutes--> v25\n- v25 --supports--> v0f\n- v0f --supports--> v11\n- v11 --supports--> v2b\n- v2b --supports--> v1b\n- v1b --refutes--> v1f\n- v1f --refutes--> v17\n- v17 --refutes--> v13\n- v13 --refutes--> v20\n- v20 --supports--> v24\n- v24 --refutes--> v19\n- v19 --refutes--> v2d\n- v2d --refutes--> v0d\n- v0d --refutes--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v1e\n- v06 --supports--> v15\n- v11 --refutes--> v02\n- v11 --supports--> v27\n- v16 --refutes--> v00\n- v16 --refutes--> v2c\n- v18 --supports--> v08\n- v1b --supports--> v0a\n- v1b --supports--> v28\n- v1f --refutes--> v10\n- v22 --refutes--> v04\n- v24 --supports--> v23\n- v25 --supports--> v0c\n- v25 --supports--> v26\n- v29 --refutes--> v21\n- v2f --supports--> v30\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":"f609d01d124c6108580dde264832567f8779d880b1218b7eb0d80e4f1e464137","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v0a\n- v0a --refutes--> v2c\n- v2c --supports--> v1c\n- v1c --supports--> v05\n- v05 --supports--> v1a\n- v1a --supports--> v01\n- v01 --refutes--> v19\n- v19 --refutes--> v2a\n- v2a --refutes--> v04\n- v04 --supports--> v22\n- v22 --refutes--> v03\n- v03 --supports--> v12\n- v12 --supports--> v07\n- v07 --refutes--> v08\n- v08 --supports--> v0f\n- v0f --refutes--> v18\n- v18 --supports--> v20\n- v20 --supports--> v0d\n- v0d --supports--> v00\n- v00 --supports--> v14\n- v14 --refutes--> v21\n- v21 --refutes--> v09\n- v09 --supports--> v13\n- v13 --supports--> v0e\n- v0e --refutes--> v2e\n- v2e --supports--> v17\n- v17 --supports--> v2f\n- v2f --supports--> v23\n- v23 --supports--> v27\n- v27 --refutes--> v10\n- v10 --refutes--> v16\n- v16 --supports--> v02\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v28\n- v04 --supports--> v26\n- v09 --refutes--> v1f\n- v0f --refutes--> v15\n- v10 --supports--> v0b\n- v12 --supports--> v24\n- v16 --refutes--> v1b\n- v16 --supports--> v11\n- v19 --supports--> v06\n- v1c --refutes--> v1d\n- v22 --refutes--> v30\n- v22 --supports--> v1e\n- v22 --supports--> v29\n- v2a --refutes--> v2d\n- v2e --refutes--> v2b\n- v2f --supports--> v25\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":"04461f3a4339e7788cc21573bc4622352af3c6b73bba24bad93729abcc258823","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --supports--> v2b\n- v2b --supports--> v29\n- v29 --refutes--> v11\n- v11 --refutes--> v09\n- v09 --supports--> v2a\n- v2a --supports--> v2f\n- v2f --refutes--> v07\n- v07 --refutes--> v0a\n- v0a --supports--> v21\n- v21 --refutes--> v0b\n- v0b --refutes--> v27\n- v27 --refutes--> v06\n- v06 --supports--> v1e\n- v1e --refutes--> v14\n- v14 --supports--> v19\n- v19 --supports--> v04\n- v04 --supports--> v2d\n- v2d --supports--> v1c\n- v1c --refutes--> v25\n- v25 --supports--> v26\n- v26 --refutes--> v2c\n- v2c --supports--> v17\n- v17 --supports--> v12\n- v12 --refutes--> v23\n- v23 --refutes--> v18\n- v18 --supports--> v1f\n- v1f --refutes--> v20\n- v20 --supports--> v00\n- v00 --refutes--> v13\n- v13 --supports--> v01\n- v01 --refutes--> v1a\n- v1a --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v0a --refutes--> v02\n- v11 --refutes--> v0d\n- v15 --supports--> v1d\n- v17 --refutes--> v2e\n- v17 --refutes--> v30\n- v19 --supports--> v1b\n- v19 --supports--> v22\n- v20 --supports--> v24\n- v23 --supports--> v0c\n- v25 --refutes--> v10\n- v27 --supports--> v16\n- v2a --refutes--> v28\n- v2d --supports--> v0e\n- v2f --refutes--> v03\n- v2f --supports--> v08\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":"8375d8ebb2dcbce042de93c3d2e422e924a80c5efddb5432c165f0773da73f78","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --refutes--> v2e\n- v2e --supports--> v22\n- v22 --refutes--> v1b\n- v1b --supports--> v2d\n- v2d --supports--> v04\n- v04 --refutes--> v26\n- v26 --refutes--> v2f\n- v2f --refutes--> v1f\n- v1f --refutes--> v1c\n- v1c --supports--> v01\n- v01 --supports--> v06\n- v06 --refutes--> v29\n- v29 --refutes--> v07\n- v07 --supports--> v12\n- v12 --supports--> v11\n- v11 --refutes--> v2c\n- v2c --supports--> v16\n- v16 --supports--> v20\n- v20 --refutes--> v1e\n- v1e --supports--> v08\n- v08 --supports--> v23\n- v23 --refutes--> v27\n- v27 --supports--> v19\n- v19 --supports--> v24\n- v24 --refutes--> v25\n- v25 --refutes--> v0f\n- v0f --refutes--> v15\n- v15 --refutes--> v14\n- v14 --refutes--> v13\n- v13 --refutes--> v03\n- v03 --refutes--> v10\n- v10 --refutes--> v2a\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v02\n- v07 --supports--> v1a\n- v08 --refutes--> v05\n- v11 --refutes--> v30\n- v13 --supports--> v0a\n- v14 --supports--> v21\n- v19 --supports--> v0c\n- v1b --supports--> v1d\n- v23 --refutes--> v2b\n- v23 --supports--> v00\n- v23 --supports--> v28\n- v26 --refutes--> v0e\n- v29 --supports--> v17\n- v2c --refutes--> v0d\n- v2d --supports--> v18\n- v2f --refutes--> v09\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":"35ed66f1ccf65851fccdafe383941bc5fd48f51bca8536a5aea0ec188f639f4b","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v24 --supports--> v08\n- v08 --supports--> v26\n- v26 --supports--> v09\n- v09 --supports--> v0f\n- v0f --refutes--> v17\n- v17 --supports--> v0e\n- v0e --refutes--> v00\n- v00 --supports--> v28\n- v28 --supports--> v04\n- v04 --refutes--> v2b\n- v2b --refutes--> v1e\n- v1e --refutes--> v01\n- v01 --supports--> v27\n- v27 --refutes--> v2f\n- v2f --refutes--> v1c\n- v1c --refutes--> v2a\n- v2a --supports--> v0a\n- v0a --refutes--> v10\n- v10 --refutes--> v14\n- v14 --supports--> v1b\n- v1b --refutes--> v05\n- v05 --refutes--> v06\n- v06 --supports--> v07\n- v07 --supports--> v25\n- v25 --supports--> v22\n- v22 --refutes--> v1d\n- v1d --refutes--> v2d\n- v2d --refutes--> v03\n- v03 --supports--> v21\n- v21 --refutes--> v2c\n- v2c --supports--> v1a\n- v1a --supports--> v15\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v0d\n- v17 --refutes--> v13\n- v17 --supports--> v0c\n- v17 --supports--> v19\n- v1a --supports--> v02\n- v1b --refutes--> v0b\n- v1c --supports--> v20\n- v1d --refutes--> v11\n- v1e --refutes--> v1f\n- v22 --supports--> v2e\n- v24 --refutes--> v16\n- v24 --supports--> v23\n- v27 --refutes--> v29\n- v2c --refutes--> v30\n- v2f --refutes--> v12\n- v2f --supports--> v18\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":"5c7a98a8d3067d53e52ecf01c2b672054a78979717d063a8392f891e6fc14aef","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --refutes--> v13\n- v13 --supports--> v2b\n- v2b --supports--> v0f\n- v0f --supports--> v10\n- v10 --supports--> v26\n- v26 --supports--> v25\n- v25 --supports--> v0a\n- v0a --supports--> v01\n- v01 --supports--> v22\n- v22 --refutes--> v0e\n- v0e --supports--> v08\n- v08 --supports--> v19\n- v19 --refutes--> v14\n- v14 --supports--> v0d\n- v0d --refutes--> v2a\n- v2a --refutes--> v05\n- v05 --supports--> v27\n- v27 --refutes--> v11\n- v11 --refutes--> v00\n- v00 --refutes--> v1a\n- v1a --refutes--> v16\n- v16 --refutes--> v09\n- v09 --refutes--> v1f\n- v1f --supports--> v0c\n- v0c --supports--> v1e\n- v1e --refutes--> v2e\n- v2e --supports--> v02\n- v02 --refutes--> v2d\n- v2d --refutes--> v29\n- v29 --refutes--> v07\n- v07 --supports--> v04\n- v04 --supports--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v30\n- v09 --refutes--> v28\n- v0a --supports--> v2c\n- v0e --refutes--> v12\n- v0e --refutes--> v20\n- v10 --supports--> v0b\n- v13 --refutes--> v06\n- v13 --refutes--> v21\n- v18 --refutes--> v24\n- v19 --supports--> v2f\n- v26 --refutes--> v15\n- v27 --supports--> v1c\n- v29 --refutes--> v23\n- v2a --refutes--> v17\n- v2a --supports--> v1d\n- v2d --refutes--> v03\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":"68205c6d9508e8afba8568bddcb696e96f1612f49f86366da54239df86c08fa7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v26\n- v26 --supports--> v10\n- v10 --supports--> v21\n- v21 --refutes--> v1c\n- v1c --supports--> v13\n- v13 --supports--> v25\n- v25 --supports--> v0e\n- v0e --refutes--> v1a\n- v1a --refutes--> v17\n- v17 --supports--> v2d\n- v2d --refutes--> v09\n- v09 --refutes--> v1f\n- v1f --refutes--> v2c\n- v2c --refutes--> v08\n- v08 --refutes--> v0c\n- v0c --refutes--> v14\n- v14 --supports--> v00\n- v00 --refutes--> v05\n- v05 --supports--> v0f\n- v0f --refutes--> v2e\n- v2e --refutes--> v2a\n- v2a --refutes--> v18\n- v18 --supports--> v1b\n- v1b --supports--> v12\n- v12 --supports--> v30\n- v30 --supports--> v11\n- v11 --refutes--> v06\n- v06 --supports--> v27\n- v27 --supports--> v03\n- v03 --refutes--> v16\n- v16 --refutes--> v22\n- v22 --supports--> v1e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v06 --supports--> v0d\n- v09 --supports--> v04\n- v0e --supports--> v07\n- v0f --refutes--> v20\n- v11 --refutes--> v28\n- v13 --refutes--> v02\n- v14 --refutes--> v2b\n- v17 --supports--> v29\n- v17 --supports--> v2f\n- v1a --supports--> v0b\n- v1c --refutes--> v0a\n- v21 --supports--> v24\n- v23 --supports--> v01\n- v23 --supports--> v15\n- v2e --refutes--> v1d\n- v30 --supports--> v19\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":"31d49626214f6555a782346cdee6566379078b7314ce3e3375fbefab8f796c0f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0a --refutes--> v2f\n- v2f --refutes--> v0e\n- v0e --refutes--> v24\n- v24 --supports--> v03\n- v03 --refutes--> v0d\n- v0d --supports--> v23\n- v23 --refutes--> v02\n- v02 --supports--> v30\n- v30 --supports--> v2a\n- v2a --refutes--> v2d\n- v2d --refutes--> v1d\n- v1d --refutes--> v1b\n- v1b --refutes--> v29\n- v29 --supports--> v1c\n- v1c --supports--> v28\n- v28 --refutes--> v17\n- v17 --supports--> v26\n- v26 --supports--> v15\n- v15 --supports--> v27\n- v27 --refutes--> v1e\n- v1e --refutes--> v2e\n- v2e --refutes--> v06\n- v06 --supports--> v0c\n- v0c --refutes--> v1a\n- v1a --refutes--> v00\n- v00 --supports--> v08\n- v08 --refutes--> v21\n- v21 --refutes--> v05\n- v05 --refutes--> v20\n- v20 --supports--> v22\n- v22 --supports--> v1f\n- v1f --supports--> v07\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v04\n- v02 --refutes--> v18\n- v05 --refutes--> v0b\n- v05 --refutes--> v12\n- v08 --refutes--> v01\n- v0d --supports--> v2c\n- v15 --refutes--> v10\n- v15 --supports--> v09\n- v1a --supports--> v0f\n- v1c --supports--> v19\n- v1e --refutes--> v25\n- v1f --supports--> v11\n- v20 --refutes--> v16\n- v26 --supports--> v13\n- v2d --supports--> v2b\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":"221031192fd7dbd0c27b49a058082bf21f213cd42bfe5ce7b984182564e59c61","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v14 --supports--> v05\n- v05 --refutes--> v2a\n- v2a --refutes--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v1a\n- v1a --refutes--> v06\n- v06 --refutes--> v1b\n- v1b --refutes--> v0d\n- v0d --supports--> v0f\n- v0f --refutes--> v01\n- v01 --refutes--> v24\n- v24 --supports--> v16\n- v16 --refutes--> v1e\n- v1e --refutes--> v0a\n- v0a --supports--> v15\n- v15 --refutes--> v19\n- v19 --supports--> v13\n- v13 --refutes--> v02\n- v02 --supports--> v2c\n- v2c --refutes--> v30\n- v30 --refutes--> v07\n- v07 --supports--> v29\n- v29 --supports--> v23\n- v23 --refutes--> v0b\n- v0b --supports--> v20\n- v20 --refutes--> v2d\n- v2d --supports--> v25\n- v25 --refutes--> v12\n- v12 --refutes--> v10\n- v10 --refutes--> v28\n- v28 --supports--> v2f\n- v2f --refutes--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v08\n- v05 --supports--> v1f\n- v07 --refutes--> v0e\n- v0b --supports--> v0c\n- v0d --supports--> v26\n- v13 --supports--> v18\n- v15 --refutes--> v21\n- v16 --refutes--> v2e\n- v19 --refutes--> v09\n- v23 --refutes--> v1d\n- v23 --supports--> v2b\n- v24 --supports--> v00\n- v24 --supports--> v17\n- v25 --refutes--> v04\n- v29 --refutes--> v22\n- v2c --refutes--> v27\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":"660175973fa894e9062debb5016f5b3048e26b6bc36dbe641818d8ab97331238","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v29 --refutes--> v27\n- v27 --refutes--> v09\n- v09 --supports--> v03\n- v03 --supports--> v25\n- v25 --refutes--> v04\n- v04 --supports--> v2a\n- v2a --supports--> v2b\n- v2b --supports--> v18\n- v18 --supports--> v20\n- v20 --supports--> v1c\n- v1c --supports--> v11\n- v11 --supports--> v24\n- v24 --refutes--> v00\n- v00 --supports--> v26\n- v26 --supports--> v14\n- v14 --refutes--> v0f\n- v0f --supports--> v10\n- v10 --refutes--> v13\n- v13 --refutes--> v16\n- v16 --supports--> v23\n- v23 --refutes--> v15\n- v15 --refutes--> v0d\n- v0d --refutes--> v0c\n- v0c --supports--> v1f\n- v1f --supports--> v17\n- v17 --refutes--> v28\n- v28 --supports--> v1b\n- v1b --refutes--> v08\n- v08 --supports--> v1a\n- v1a --supports--> v2c\n- v2c --refutes--> v2f\n- v2f --refutes--> v2e\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v06\n- v03 --refutes--> v1d\n- v0c --refutes--> v05\n- v0d --supports--> v30\n- v0f --refutes--> v12\n- v0f --refutes--> v19\n- v11 --refutes--> v01\n- v16 --supports--> v1e\n- v1a --supports--> v07\n- v23 --refutes--> v21\n- v23 --refutes--> v2d\n- v25 --supports--> v0a\n- v26 --refutes--> v02\n- v27 --supports--> v0b\n- v28 --refutes--> v0e\n- v2f --supports--> v22\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":"c4f956ead32d18d83a5895863928b488d19ac36d9d25f29ee888345a73c85971","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0b --supports--> v18\n- v18 --refutes--> v17\n- v17 --supports--> v30\n- v30 --supports--> v1e\n- v1e --refutes--> v2f\n- v2f --refutes--> v1d\n- v1d --refutes--> v11\n- v11 --supports--> v24\n- v24 --refutes--> v2a\n- v2a --refutes--> v1b\n- v1b --supports--> v09\n- v09 --refutes--> v16\n- v16 --supports--> v08\n- v08 --refutes--> v12\n- v12 --refutes--> v19\n- v19 --refutes--> v00\n- v00 --refutes--> v1f\n- v1f --refutes--> v10\n- v10 --supports--> v23\n- v23 --supports--> v22\n- v22 --supports--> v29\n- v29 --supports--> v14\n- v14 --supports--> v05\n- v05 --supports--> v28\n- v28 --supports--> v15\n- v15 --refutes--> v01\n- v01 --supports--> v0a\n- v0a --refutes--> v1c\n- v1c --refutes--> v25\n- v25 --refutes--> v13\n- v13 --supports--> v02\n- v02 --supports--> v04\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v0d\n- v09 --refutes--> v26\n- v0b --supports--> v0f\n- v0b --supports--> v20\n- v12 --refutes--> v2c\n- v16 --supports--> v0e\n- v18 --supports--> v0c\n- v19 --refutes--> v06\n- v1b --refutes--> v27\n- v22 --supports--> v21\n- v24 --refutes--> v07\n- v25 --supports--> v03\n- v25 --supports--> v2b\n- v25 --supports--> v2d\n- v28 --refutes--> v2e\n- v2a --supports--> v1a\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":"e8718b8b466c9c88a69f7395d52ec99c663b82a4394db33c10df02a3580a586f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0e --supports--> v22\n- v22 --refutes--> v25\n- v25 --supports--> v10\n- v10 --refutes--> v1c\n- v1c --refutes--> v0d\n- v0d --supports--> v11\n- v11 --supports--> v20\n- v20 --refutes--> v03\n- v03 --supports--> v2e\n- v2e --refutes--> v17\n- v17 --supports--> v2c\n- v2c --supports--> v0c\n- v0c --refutes--> v1d\n- v1d --supports--> v24\n- v24 --supports--> v21\n- v21 --supports--> v2f\n- v2f --supports--> v00\n- v00 --refutes--> v26\n- v26 --refutes--> v1a\n- v1a --supports--> v18\n- v18 --supports--> v2b\n- v2b --refutes--> v1b\n- v1b --refutes--> v09\n- v09 --supports--> v0b\n- v0b --refutes--> v0f\n- v0f --refutes--> v12\n- v12 --refutes--> v14\n- v14 --supports--> v1f\n- v1f --refutes--> v28\n- v28 --supports--> v05\n- v05 --refutes--> v27\n- v27 --refutes--> v23\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v0c --refutes--> v19\n- v0d --refutes--> v29\n- v11 --refutes--> v15\n- v11 --supports--> v01\n- v14 --supports--> v30\n- v17 --refutes--> v07\n- v18 --refutes--> v04\n- v1c --supports--> v06\n- v1c --supports--> v13\n- v1f --refutes--> v02\n- v1f --refutes--> v1e\n- v20 --refutes--> v16\n- v21 --supports--> v2d\n- v22 --refutes--> v2a\n- v25 --supports--> v0a\n- v2e --refutes--> v08\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":"a3f369770ba1dc6d38a244c42fed8d985db8bee772de27b4ea963e5d0854c742","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v25\n- v25 --refutes--> v24\n- v24 --refutes--> v17\n- v17 --supports--> v13\n- v13 --refutes--> v22\n- v22 --supports--> v29\n- v29 --supports--> v1d\n- v1d --supports--> v2d\n- v2d --supports--> v1a\n- v1a --supports--> v03\n- v03 --supports--> v04\n- v04 --supports--> v10\n- v10 --refutes--> v20\n- v20 --supports--> v0d\n- v0d --supports--> v0f\n- v0f --supports--> v1f\n- v1f --refutes--> v0c\n- v0c --supports--> v0a\n- v0a --supports--> v01\n- v01 --refutes--> v23\n- v23 --supports--> v11\n- v11 --supports--> v14\n- v14 --refutes--> v07\n- v07 --refutes--> v27\n- v27 --supports--> v09\n- v09 --refutes--> v06\n- v06 --supports--> v21\n- v21 --refutes--> v19\n- v19 --supports--> v00\n- v00 --refutes--> v28\n- v28 --supports--> v2e\n- v2e --refutes--> v1b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v2c\n- v03 --supports--> v18\n- v06 --supports--> v2a\n- v0a --refutes--> v15\n- v0d --supports--> v2b\n- v13 --supports--> v2f\n- v17 --refutes--> v08\n- v19 --refutes--> v1e\n- v1d --refutes--> v02\n- v1d --supports--> v26\n- v21 --supports--> v1c\n- v22 --supports--> v12\n- v22 --supports--> v16\n- v28 --refutes--> v30\n- v29 --supports--> v0b\n- v2d --supports--> v0e\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":"28aa9811801bc4c8723bb4676712ac490c3bb84c1ce10500396261874f8db33f","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v13 --supports--> v1e\n- v1e --supports--> v26\n- v26 --supports--> v2e\n- v2e --supports--> v29\n- v29 --supports--> v22\n- v22 --refutes--> v1b\n- v1b --refutes--> v09\n- v09 --supports--> v0e\n- v0e --supports--> v23\n- v23 --refutes--> v2f\n- v2f --supports--> v21\n- v21 --refutes--> v12\n- v12 --supports--> v11\n- v11 --refutes--> v1f\n- v1f --supports--> v07\n- v07 --refutes--> v2b\n- v2b --refutes--> v27\n- v27 --supports--> v2d\n- v2d --refutes--> v1c\n- v1c --supports--> v04\n- v04 --supports--> v06\n- v06 --supports--> v30\n- v30 --refutes--> v01\n- v01 --supports--> v05\n- v05 --supports--> v0f\n- v0f --supports--> v28\n- v28 --refutes--> v0c\n- v0c --refutes--> v0a\n- v0a --supports--> v1d\n- v1d --supports--> v02\n- v02 --refutes--> v18\n- v18 --supports--> v00\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v15\n- v09 --refutes--> v0d\n- v0a --refutes--> v17\n- v0c --refutes--> v2c\n- v0c --supports--> v03\n- v12 --refutes--> v19\n- v13 --refutes--> v14\n- v18 --supports--> v24\n- v1e --refutes--> v08\n- v22 --supports--> v2a\n- v23 --supports--> v10\n- v26 --refutes--> v25\n- v27 --refutes--> v0b\n- v28 --refutes--> v1a\n- v28 --supports--> v16\n- v2e --supports--> v20\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":"e7a52f41fb34f9c66fb8ffed7ea177079def3bdf14cd5a3d5d509714c5e16770","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2b --supports--> v12\n- v12 --refutes--> v17\n- v17 --refutes--> v25\n- v25 --supports--> v02\n- v02 --supports--> v03\n- v03 --supports--> v29\n- v29 --supports--> v04\n- v04 --supports--> v1d\n- v1d --refutes--> v06\n- v06 --supports--> v09\n- v09 --supports--> v1e\n- v1e --supports--> v23\n- v23 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --refutes--> v21\n- v21 --supports--> v26\n- v26 --refutes--> v05\n- v05 --refutes--> v2e\n- v2e --supports--> v0a\n- v0a --refutes--> v0b\n- v0b --refutes--> v08\n- v08 --refutes--> v27\n- v27 --refutes--> v1c\n- v1c --supports--> v16\n- v16 --supports--> v1f\n- v1f --refutes--> v2f\n- v2f --supports--> v0f\n- v0f --supports--> v24\n- v24 --refutes--> v30\n- v30 --refutes--> v2a\n- v2a --refutes--> v18\n- v18 --supports--> v01\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --supports--> v20\n- v04 --supports--> v22\n- v05 --refutes--> v1b\n- v05 --supports--> v0e\n- v10 --supports--> v28\n- v12 --supports--> v15\n- v16 --refutes--> v14\n- v1d --supports--> v2c\n- v1e --refutes--> v2d\n- v1e --supports--> v07\n- v23 --refutes--> v0c\n- v23 --supports--> v1a\n- v24 --supports--> v13\n- v2b --refutes--> v11\n- v2e --supports--> v19\n- v2f --supports--> v00\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":"f9c6f08953627b50e7ee85fa36c05f9dbe92fd104378bb0f763da0c43122d301","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v1f --refutes--> v25\n- v25 --supports--> v12\n- v12 --refutes--> v2b\n- v2b --refutes--> v1a\n- v1a --supports--> v05\n- v05 --refutes--> v17\n- v17 --refutes--> v1b\n- v1b --refutes--> v22\n- v22 --supports--> v2a\n- v2a --refutes--> v0a\n- v0a --supports--> v0c\n- v0c --supports--> v21\n- v21 --supports--> v09\n- v09 --refutes--> v30\n- v30 --supports--> v1e\n- v1e --supports--> v29\n- v29 --supports--> v20\n- v20 --refutes--> v2e\n- v2e --refutes--> v08\n- v08 --refutes--> v02\n- v02 --supports--> v2d\n- v2d --supports--> v06\n- v06 --supports--> v03\n- v03 --supports--> v23\n- v23 --refutes--> v0b\n- v0b --supports--> v07\n- v07 --refutes--> v0f\n- v0f --supports--> v00\n- v00 --refutes--> v15\n- v15 --refutes--> v01\n- v01 --supports--> v19\n- v19 --supports--> v10\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v18\n- v03 --refutes--> v1c\n- v05 --refutes--> v2f\n- v08 --refutes--> v27\n- v15 --supports--> v0e\n- v1a --refutes--> v2c\n- v1b --refutes--> v16\n- v1b --supports--> v11\n- v1f --supports--> v28\n- v20 --refutes--> v13\n- v23 --supports--> v24\n- v29 --refutes--> v14\n- v2e --supports--> v1d\n- v30 --refutes--> v26\n- v30 --supports--> v04\n- v30 --supports--> v0d\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":"98a9b0576b9646bc29f19eaca6d2456142935ba50a71591bb0bb2d314ef541ac","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v03 --refutes--> v20\n- v20 --refutes--> v09\n- v09 --supports--> v17\n- v17 --supports--> v04\n- v04 --supports--> v24\n- v24 --supports--> v1f\n- v1f --refutes--> v07\n- v07 --supports--> v23\n- v23 --supports--> v26\n- v26 --supports--> v06\n- v06 --supports--> v2d\n- v2d --supports--> v1c\n- v1c --refutes--> v2e\n- v2e --refutes--> v11\n- v11 --supports--> v19\n- v19 --supports--> v25\n- v25 --refutes--> v01\n- v01 --supports--> v2a\n- v2a --refutes--> v2b\n- v2b --refutes--> v0a\n- v0a --supports--> v12\n- v12 --refutes--> v10\n- v10 --supports--> v08\n- v08 --refutes--> v0d\n- v0d --refutes--> v14\n- v14 --supports--> v0e\n- v0e --refutes--> v18\n- v18 --refutes--> v21\n- v21 --refutes--> v1d\n- v1d --refutes--> v00\n- v00 --refutes--> v2c\n- v2c --supports--> v27\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v15\n- v09 --refutes--> v02\n- v09 --refutes--> v16\n- v12 --supports--> v0c\n- v14 --refutes--> v0b\n- v14 --refutes--> v1b\n- v18 --refutes--> v2f\n- v1d --refutes--> v28\n- v1d --supports--> v1e\n- v1f --refutes--> v29\n- v1f --supports--> v13\n- v21 --refutes--> v05\n- v26 --refutes--> v1a\n- v2a --refutes--> v0f\n- v2b --refutes--> v22\n- v2b --refutes--> v30\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":"2c112b9502f09db3ffeb60e1b2ff3e2689b4b9ad286fcf95a7344dbd74cb719c","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2d --refutes--> v02\n- v02 --refutes--> v10\n- v10 --refutes--> v0d\n- v0d --supports--> v29\n- v29 --refutes--> v03\n- v03 --refutes--> v17\n- v17 --refutes--> v28\n- v28 --supports--> v1a\n- v1a --refutes--> v0f\n- v0f --refutes--> v13\n- v13 --supports--> v2f\n- v2f --refutes--> v05\n- v05 --supports--> v0e\n- v0e --supports--> v30\n- v30 --refutes--> v2c\n- v2c --supports--> v1e\n- v1e --supports--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v08\n- v08 --supports--> v2b\n- v2b --refutes--> v11\n- v11 --supports--> v1f\n- v1f --refutes--> v07\n- v07 --supports--> v25\n- v25 --refutes--> v23\n- v23 --refutes--> v1b\n- v1b --refutes--> v2a\n- v2a --refutes--> v12\n- v12 --supports--> v00\n- v00 --supports--> v27\n- v27 --supports--> v0c\n- v0c --refutes--> v22\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v04 --refutes--> v15\n- v04 --supports--> v26\n- v07 --refutes--> v1d\n- v08 --refutes--> v2e\n- v08 --supports--> v21\n- v0d --refutes--> v24\n- v0e --supports--> v18\n- v10 --refutes--> v0b\n- v1a --refutes--> v20\n- v1b --supports--> v09\n- v1b --supports--> v16\n- v23 --refutes--> v01\n- v2a --refutes--> v14\n- v2b --supports--> v06\n- v2b --supports--> v0a\n- v2c --supports--> v1c\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":"4b2470e55ddcff246d442b0d2d88d0a695cb717efc678766dfdb8242fff62db7","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2d --supports--> v0f\n- v0f --refutes--> v1b\n- v1b --supports--> v12\n- v12 --refutes--> v1d\n- v1d --supports--> v13\n- v13 --supports--> v1a\n- v1a --refutes--> v30\n- v30 --refutes--> v09\n- v09 --supports--> v01\n- v01 --supports--> v20\n- v20 --refutes--> v22\n- v22 --refutes--> v21\n- v21 --supports--> v14\n- v14 --supports--> v28\n- v28 --supports--> v02\n- v02 --supports--> v19\n- v19 --supports--> v00\n- v00 --supports--> v15\n- v15 --supports--> v18\n- v18 --supports--> v29\n- v29 --refutes--> v2a\n- v2a --supports--> v08\n- v08 --supports--> v2b\n- v2b --supports--> v11\n- v11 --supports--> v27\n- v27 --refutes--> v16\n- v16 --refutes--> v0e\n- v0e --refutes--> v07\n- v07 --supports--> v25\n- v25 --supports--> v23\n- v23 --refutes--> v10\n- v10 --supports--> v1c\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v08 --supports--> v24\n- v10 --supports--> v17\n- v11 --refutes--> v2c\n- v13 --refutes--> v1f\n- v15 --refutes--> v06\n- v15 --refutes--> v0d\n- v15 --supports--> v26\n- v16 --refutes--> v1e\n- v16 --supports--> v0b\n- v19 --supports--> v0a\n- v1a --supports--> v04\n- v1d --supports--> v2f\n- v23 --supports--> v2e\n- v27 --supports--> v03\n- v2a --refutes--> v05\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":"f53676d5bc84a7509d858bbccf1da5e3bf587ba3c8f1efbd1c3e915051e677dc","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v05 --refutes--> v25\n- v25 --supports--> v1b\n- v1b --refutes--> v1c\n- v1c --refutes--> v2a\n- v2a --refutes--> v2c\n- v2c --supports--> v06\n- v06 --supports--> v1d\n- v1d --refutes--> v01\n- v01 --refutes--> v27\n- v27 --refutes--> v18\n- v18 --supports--> v08\n- v08 --supports--> v2e\n- v2e --refutes--> v0c\n- v0c --supports--> v24\n- v24 --refutes--> v12\n- v12 --supports--> v1f\n- v1f --supports--> v2d\n- v2d --refutes--> v1a\n- v1a --supports--> v15\n- v15 --supports--> v0b\n- v0b --refutes--> v07\n- v07 --refutes--> v26\n- v26 --refutes--> v28\n- v28 --refutes--> v30\n- v30 --refutes--> v2f\n- v2f --refutes--> v14\n- v14 --refutes--> v23\n- v23 --refutes--> v2b\n- v2b --refutes--> v0e\n- v0e --refutes--> v02\n- v02 --supports--> v22\n- v22 --supports--> v03\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v05 --refutes--> v10\n- v07 --supports--> v0a\n- v07 --supports--> v0f\n- v0b --refutes--> v21\n- v0b --supports--> v19\n- v0c --refutes--> v11\n- v18 --refutes--> v16\n- v18 --refutes--> v20\n- v1b --refutes--> v04\n- v27 --supports--> v1e\n- v27 --supports--> v29\n- v2a --refutes--> v0d\n- v2b --refutes--> v00\n- v2b --refutes--> v13\n- v2f --supports--> v17\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":"8f3e0dca6fce4a19864d9425394d43c8120f456ae3d557f6e954999f59f87e36","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v19 --refutes--> v00\n- v00 --supports--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v26\n- v26 --refutes--> v2e\n- v2e --refutes--> v0e\n- v0e --refutes--> v1c\n- v1c --refutes--> v0d\n- v0d --refutes--> v2f\n- v2f --refutes--> v16\n- v16 --refutes--> v13\n- v13 --refutes--> v2b\n- v2b --refutes--> v25\n- v25 --refutes--> v2a\n- v2a --refutes--> v12\n- v12 --refutes--> v04\n- v04 --refutes--> v30\n- v30 --refutes--> v23\n- v23 --refutes--> v0c\n- v0c --supports--> v03\n- v03 --refutes--> v15\n- v15 --refutes--> v18\n- v18 --supports--> v1f\n- v1f --supports--> v1e\n- v1e --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v0f\n- v0f --supports--> v05\n- v05 --refutes--> v14\n- v14 --supports--> v02\n- v02 --refutes--> v2d\n- v2d --supports--> v21\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v08\n- v03 --refutes--> v1d\n- v04 --supports--> v2c\n- v0b --refutes--> v28\n- v0c --refutes--> v07\n- v0e --refutes--> v01\n- v10 --refutes--> v24\n- v12 --supports--> v29\n- v17 --refutes--> v1b\n- v17 --supports--> v22\n- v1f --supports--> v11\n- v2a --refutes--> v0a\n- v2a --supports--> v1a\n- v2b --refutes--> v27\n- v2b --supports--> v06\n- v30 --supports--> v20\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":"f9d7bdccbf7714fe676bd8939e5061ccd8bb01c34e8801ab1366fa591e65bae2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2e --supports--> v12\n- v12 --refutes--> v0c\n- v0c --refutes--> v2a\n- v2a --refutes--> v06\n- v06 --supports--> v2d\n- v2d --refutes--> v1b\n- v1b --refutes--> v28\n- v28 --refutes--> v13\n- v13 --supports--> v02\n- v02 --supports--> v1c\n- v1c --supports--> v20\n- v20 --refutes--> v0e\n- v0e --refutes--> v0f\n- v0f --supports--> v08\n- v08 --refutes--> v1e\n- v1e --supports--> v1d\n- v1d --refutes--> v07\n- v07 --supports--> v25\n- v25 --supports--> v2c\n- v2c --supports--> v09\n- v09 --refutes--> v16\n- v16 --refutes--> v10\n- v10 --refutes--> v24\n- v24 --supports--> v30\n- v30 --supports--> v2b\n- v2b --refutes--> v0b\n- v0b --supports--> v26\n- v26 --refutes--> v27\n- v27 --refutes--> v15\n- v15 --refutes--> v04\n- v04 --refutes--> v0a\n- v0a --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v2f\n- v07 --refutes--> v29\n- v07 --supports--> v22\n- v0b --supports--> v03\n- v10 --refutes--> v14\n- v15 --supports--> v23\n- v16 --refutes--> v18\n- v1c --refutes--> v00\n- v1d --supports--> v19\n- v27 --refutes--> v1a\n- v28 --refutes--> v0d\n- v28 --supports--> v1f\n- v2c --refutes--> v21\n- v2d --supports--> v17\n- v2e --supports--> v05\n- v30 --refutes--> v01\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":"4d6032f17309c36048155d4d7a2ee59e8e0efb99d49d5dd6f3b8776a03e60d5d","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v0c --refutes--> v14\n- v14 --supports--> v1b\n- v1b --supports--> v15\n- v15 --supports--> v30\n- v30 --supports--> v24\n- v24 --supports--> v05\n- v05 --refutes--> v02\n- v02 --supports--> v20\n- v20 --refutes--> v13\n- v13 --refutes--> v0a\n- v0a --refutes--> v1a\n- v1a --refutes--> v0d\n- v0d --refutes--> v19\n- v19 --refutes--> v00\n- v00 --supports--> v1c\n- v1c --refutes--> v25\n- v25 --refutes--> v23\n- v23 --refutes--> v10\n- v10 --refutes--> v03\n- v03 --supports--> v16\n- v16 --supports--> v2c\n- v2c --refutes--> v2e\n- v2e --refutes--> v21\n- v21 --supports--> v08\n- v08 --refutes--> v01\n- v01 --refutes--> v0f\n- v0f --supports--> v06\n- v06 --refutes--> v11\n- v11 --supports--> v1d\n- v1d --refutes--> v04\n- v04 --refutes--> v12\n- v12 --supports--> v17\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v2f\n- v04 --supports--> v28\n- v06 --refutes--> v18\n- v0c --refutes--> v26\n- v10 --refutes--> v07\n- v10 --supports--> v2d\n- v13 --supports--> v2b\n- v16 --refutes--> v29\n- v1a --supports--> v0b\n- v1b --refutes--> v1e\n- v1d --refutes--> v22\n- v1d --refutes--> v2a\n- v23 --supports--> v1f\n- v2e --refutes--> v09\n- v2e --refutes--> v0e\n- v2e --refutes--> v27\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":"3e42d7eb5e4ac8b748a214bc205dda4e9db61871709552484c07ce08780d5485","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v23 --supports--> v28\n- v28 --supports--> v24\n- v24 --refutes--> v19\n- v19 --refutes--> v18\n- v18 --refutes--> v14\n- v14 --supports--> v02\n- v02 --refutes--> v30\n- v30 --refutes--> v01\n- v01 --supports--> v1a\n- v1a --refutes--> v13\n- v13 --supports--> v2d\n- v2d --refutes--> v0c\n- v0c --refutes--> v00\n- v00 --supports--> v16\n- v16 --refutes--> v1b\n- v1b --supports--> v0f\n- v0f --refutes--> v1c\n- v1c --supports--> v26\n- v26 --supports--> v1d\n- v1d --refutes--> v06\n- v06 --supports--> v0a\n- v0a --refutes--> v0d\n- v0d --refutes--> v12\n- v12 --supports--> v17\n- v17 --refutes--> v10\n- v10 --supports--> v09\n- v09 --supports--> v0b\n- v0b --supports--> v0e\n- v0e --refutes--> v07\n- v07 --refutes--> v2c\n- v2c --supports--> v1f\n- v1f --supports--> v2b\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --supports--> v04\n- v06 --refutes--> v2a\n- v06 --supports--> v21\n- v0c --refutes--> v2e\n- v0c --supports--> v11\n- v0c --supports--> v25\n- v0d --supports--> v20\n- v10 --supports--> v2f\n- v16 --refutes--> v03\n- v18 --supports--> v27\n- v1b --supports--> v29\n- v1c --supports--> v05\n- v24 --refutes--> v08\n- v26 --refutes--> v22\n- v26 --supports--> v15\n- v2d --supports--> v1e\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":"500a2fb8d2f390e9a3d14d57e76746efd63ba4b700857b33ce52f995fb2a8767","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v15 --refutes--> v2d\n- v2d --supports--> v03\n- v03 --refutes--> v09\n- v09 --supports--> v25\n- v25 --refutes--> v0c\n- v0c --refutes--> v04\n- v04 --supports--> v14\n- v14 --refutes--> v0f\n- v0f --refutes--> v06\n- v06 --refutes--> v2e\n- v2e --supports--> v0d\n- v0d --supports--> v0b\n- v0b --supports--> v23\n- v23 --supports--> v20\n- v20 --refutes--> v1b\n- v1b --refutes--> v13\n- v13 --refutes--> v18\n- v18 --refutes--> v1d\n- v1d --refutes--> v07\n- v07 --refutes--> v30\n- v30 --refutes--> v21\n- v21 --refutes--> v2b\n- v2b --refutes--> v2f\n- v2f --supports--> v05\n- v05 --refutes--> v1c\n- v1c --supports--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v24\n- v24 --supports--> v1a\n- v1a --supports--> v00\n- v00 --refutes--> v02\n- v02 --supports--> v26\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --refutes--> v22\n- v03 --refutes--> v08\n- v03 --refutes--> v11\n- v03 --supports--> v19\n- v06 --supports--> v29\n- v0c --refutes--> v01\n- v0f --refutes--> v16\n- v10 --supports--> v27\n- v15 --supports--> v1e\n- v18 --supports--> v1f\n- v20 --supports--> v2c\n- v21 --supports--> v28\n- v24 --supports--> v0e\n- v2b --supports--> v2a\n- v2f --refutes--> v0a\n- v30 --refutes--> v17\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":"bc8e00346497b9323a71bc01093b2bef13f94a0582d9fba5c902b0674fd549e0","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v2e --supports--> v30\n- v30 --refutes--> v23\n- v23 --supports--> v1c\n- v1c --refutes--> v14\n- v14 --refutes--> v12\n- v12 --refutes--> v24\n- v24 --refutes--> v1b\n- v1b --supports--> v05\n- v05 --refutes--> v09\n- v09 --supports--> v29\n- v29 --refutes--> v07\n- v07 --refutes--> v13\n- v13 --refutes--> v18\n- v18 --refutes--> v26\n- v26 --refutes--> v06\n- v06 --supports--> v0d\n- v0d --refutes--> v04\n- v04 --refutes--> v03\n- v03 --refutes--> v27\n- v27 --supports--> v17\n- v17 --supports--> v00\n- v00 --refutes--> v0f\n- v0f --supports--> v2b\n- v2b --supports--> v0b\n- v0b --refutes--> v20\n- v20 --refutes--> v01\n- v01 --supports--> v2c\n- v2c --refutes--> v16\n- v16 --supports--> v0a\n- v0a --supports--> v25\n- v25 --refutes--> v11\n- v11 --refutes--> v1d\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --refutes--> v1e\n- v01 --supports--> v08\n- v03 --refutes--> v0e\n- v06 --refutes--> v15\n- v07 --refutes--> v21\n- v09 --refutes--> v2f\n- v0b --refutes--> v10\n- v17 --refutes--> v19\n- v18 --supports--> v2d\n- v20 --supports--> v0c\n- v23 --supports--> v1f\n- v26 --refutes--> v22\n- v29 --refutes--> v02\n- v29 --supports--> v1a\n- v2e --supports--> v28\n- v30 --refutes--> v2a\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":"47b13f540b67fd27d4c9c6723db8e586bdcbd21e99565cc3bbaceb208b2a97d2","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v25 --supports--> v2d\n- v2d --refutes--> v28\n- v28 --supports--> v02\n- v02 --refutes--> v01\n- v01 --refutes--> v1d\n- v1d --supports--> v11\n- v11 --refutes--> v18\n- v18 --supports--> v15\n- v15 --refutes--> v03\n- v03 --refutes--> v12\n- v12 --refutes--> v1e\n- v1e --supports--> v07\n- v07 --supports--> v1b\n- v1b --supports--> v00\n- v00 --refutes--> v08\n- v08 --supports--> v27\n- v27 --supports--> v06\n- v06 --supports--> v22\n- v22 --refutes--> v16\n- v16 --refutes--> v10\n- v10 --supports--> v0d\n- v0d --supports--> v2e\n- v2e --supports--> v04\n- v04 --refutes--> v19\n- v19 --supports--> v1a\n- v1a --supports--> v29\n- v29 --supports--> v2c\n- v2c --refutes--> v23\n- v23 --supports--> v30\n- v30 --refutes--> v2f\n- v2f --refutes--> v0b\n- v0b --refutes--> v14\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v00 --supports--> v0f\n- v03 --refutes--> v2b\n- v04 --refutes--> v0a\n- v04 --refutes--> v13\n- v19 --refutes--> v09\n- v19 --refutes--> v21\n- v1d --supports--> v05\n- v22 --refutes--> v26\n- v22 --supports--> v0c\n- v25 --refutes--> v24\n- v27 --refutes--> v17\n- v27 --refutes--> v20\n- v28 --refutes--> v1c\n- v2c --refutes--> v0e\n- v2d --supports--> v1f\n- v2f --refutes--> v2a\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":"d5d0dd0ff6d61047a33503a35ba80be9806394c967ceb5c3cf8d09bbf873853e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v22 --supports--> v21\n- v21 --supports--> v14\n- v14 --refutes--> v03\n- v03 --supports--> v1f\n- v1f --supports--> v0e\n- v0e --supports--> v10\n- v10 --supports--> v25\n- v25 --refutes--> v00\n- v00 --refutes--> v01\n- v01 --refutes--> v2c\n- v2c --refutes--> v1c\n- v1c --supports--> v12\n- v12 --supports--> v04\n- v04 --refutes--> v15\n- v15 --refutes--> v2b\n- v2b --refutes--> v0d\n- v0d --refutes--> v18\n- v18 --refutes--> v2f\n- v2f --refutes--> v11\n- v11 --supports--> v0c\n- v0c --refutes--> v26\n- v26 --supports--> v0a\n- v0a --refutes--> v23\n- v23 --supports--> v02\n- v02 --refutes--> v2d\n- v2d --refutes--> v2e\n- v2e --supports--> v30\n- v30 --supports--> v09\n- v09 --supports--> v1b\n- v1b --refutes--> v27\n- v27 --refutes--> v20\n- v20 --supports--> v05\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v01 --refutes--> v07\n- v03 --refutes--> v06\n- v09 --refutes--> v17\n- v09 --refutes--> v1e\n- v0a --refutes--> v16\n- v0e --refutes--> v24\n- v15 --refutes--> v0b\n- v15 --refutes--> v2a\n- v1f --supports--> v19\n- v1f --supports--> v1d\n- v23 --refutes--> v13\n- v26 --supports--> v0f\n- v27 --supports--> v08\n- v27 --supports--> v28\n- v27 --supports--> v29\n- v2d --refutes--> v1a\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":"5061e5e5af1fe6dd933c4c99d78a52d7dccb4f0fb6ec04eae4b08d4021eb7c2e","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v18 --supports--> v2a\n- v2a --refutes--> v2e\n- v2e --supports--> v0c\n- v0c --refutes--> v19\n- v19 --supports--> v25\n- v25 --refutes--> v1e\n- v1e --supports--> v16\n- v16 --refutes--> v03\n- v03 --supports--> v24\n- v24 --refutes--> v23\n- v23 --refutes--> v22\n- v22 --refutes--> v10\n- v10 --supports--> v06\n- v06 --supports--> v13\n- v13 --refutes--> v14\n- v14 --supports--> v0d\n- v0d --supports--> v1f\n- v1f --supports--> v0e\n- v0e --supports--> v21\n- v21 --supports--> v12\n- v12 --refutes--> v26\n- v26 --refutes--> v28\n- v28 --supports--> v1d\n- v1d --supports--> v2c\n- v2c --supports--> v2b\n- v2b --supports--> v1b\n- v1b --refutes--> v02\n- v02 --refutes--> v07\n- v07 --refutes--> v15\n- v15 --refutes--> v1a\n- v1a --refutes--> v00\n- v00 --refutes--> v11\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v02 --supports--> v09\n- v07 --refutes--> v27\n- v0d --refutes--> v0b\n- v0e --supports--> v05\n- v13 --supports--> v29\n- v14 --supports--> v2f\n- v16 --supports--> v1c\n- v19 --supports--> v30\n- v1e --refutes--> v0f\n- v21 --refutes--> v20\n- v21 --refutes--> v2d\n- v22 --supports--> v0a\n- v24 --refutes--> v01\n- v25 --refutes--> v17\n- v26 --refutes--> v04\n- v28 --refutes--> v08\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":"5184edf96f2aca21d01a5843dfbfdf1ddee51705e1bc0901cc82d90dd81fe637","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":"ordered"}
{"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":"[VERIFIED PATH EDGES; ROOT-TO-TARGET ORDER]\n- v11 --supports--> v2d\n- v2d --refutes--> v10\n- v10 --refutes--> v12\n- v12 --supports--> v22\n- v22 --refutes--> v27\n- v27 --refutes--> v1c\n- v1c --refutes--> v2c\n- v2c --refutes--> v03\n- v03 --supports--> v29\n- v29 --supports--> v09\n- v09 --refutes--> v08\n- v08 --refutes--> v14\n- v14 --refutes--> v0f\n- v0f --refutes--> v2b\n- v2b --refutes--> v21\n- v21 --supports--> v26\n- v26 --supports--> v15\n- v15 --supports--> v13\n- v13 --supports--> v04\n- v04 --refutes--> v1b\n- v1b --supports--> v2a\n- v2a --supports--> v06\n- v06 --refutes--> v1a\n- v1a --supports--> v0c\n- v0c --supports--> v1e\n- v1e --refutes--> v0d\n- v0d --supports--> v2f\n- v2f --refutes--> v0e\n- v0e --refutes--> v25\n- v25 --supports--> v07\n- v07 --supports--> v23\n- v23 --refutes--> v28\n\n[OTHER GRAPH EDGES; NOT ON THE TARGET PATH]\n- v03 --supports--> v20\n- v06 --supports--> v30\n- v07 --supports--> v24\n- v11 --refutes--> v01\n- v12 --refutes--> v0b\n- v12 --refutes--> v16\n- v15 --supports--> v00\n- v23 --supports--> v1f\n- v25 --supports--> v1d\n- v2b --supports--> v17\n- v2c --supports--> v0a\n- v2c --supports--> v2e\n- v2d --refutes--> v05\n- v2d --supports--> v02\n- v2f --supports--> v18\n- v2f --supports--> v19\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":"28041c386f38f125badc16eafd39a1af5f57277a39b338f24eae630f98245aca","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":"ordered"}