{"schema_version":"claim-reader-v1","record":{"available_versions":[1],"current_version":1,"assessment_summary":"consistent","claim":{"statement":"There is a simple connected bridgeless cubic graph H on 112 vertices and 168 edges, of girth five and both connectivities three, with no Petersen coloring or normal five-edge-coloring; H is nonisomorphic to G and has automorphism group D₃ of order six.","id":"claim-petersen-112-symmetric-witness","slug":"petersen-112-symmetric-witness","version":1,"kind":"theorem","title":"A distinct symmetric 112-vertex counterexample","assumptions":["The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph."],"scope":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","source_entry_ids":["petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"formal_declarations":[]},"sha256":"c7fdc21883fb9a383b73864954056882927d247d0375179e220d52c5c4a7a49c","contexts":[{"assessment_history":[{"report_id":"petersen-coloring:report:alternative:v1","source_id":"petersen-coloring-paper","source_version_id":"petersen-coloring:source:paper:v1","source_title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","source_label":"Second nonisomorphic counterexample reported","state":"resolved","scope":"Exact D₃-symmetric graph in Theorem 6.1.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","entry_id":"petersen-coloring:entry:alternative:v1","source_date":"2026-08-08"},{"report_id":"petersen-coloring:report:certificates:v1","source_id":"petersen-coloring-archive","source_version_id":"petersen-coloring:source:archive:v1","source_title":"Petersen counterexample: frozen construction and certificates","source_label":"External SAT and DRAT checks reported","state":"resolved","scope":"Four graph/formulation pairs; not an OpenMath execution.","observed_on":"2026-10-14","evidence_url":"https://zenodo.org/records/21845291","entry_id":"petersen-coloring:entry:certificates:v1","source_date":"2026-08-08"}],"reader_guide":{"reading":[],"definitions":[],"proofs":[],"bounds":[],"witnesses":[{"id":"petersen-coloring-symmetric-graph","claim":{"id":"claim-petersen-112-symmetric-witness","version":1},"kind":"graph","title":"Inspect the symmetric graph","fields":[{"label":"Order and size","value":"112 vertices; 168 edges"},{"label":"Coloring encodings","value":"Petersen: 3,640 variables and 68,324 clauses; normal five: 1,960 variables and 25,484 clauses"},{"label":"Certificate tools","value":"CaDiCaL 3.0.1 at c60730422e758ef1cebe7aeddf2dda31c996bf04; drat-trim at 2e3b2dc0ecf938addbd779d42877b6ed69d9a985"},{"label":"Normalized edge-list SHA-256","value":"0f2d8858110c6f012de7ddffa92fdbc709d7da630f199b0e3c81bb56eb6b35c7"}],"limitation":"The exact adjacency data identifies this witness. Inspecting neighbors does not verify noncolorability, and 112 is not asserted to be minimal.","source_entry_ids":["petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"graph":{"vertex_count":112,"edges":[[0,1],[0,5],[0,31],[1,2],[1,6],[2,7],[2,52],[3,5],[3,6],[3,11],[4,6],[4,7],[4,9],[5,7],[8,9],[8,10],[8,11],[9,14],[10,12],[10,20],[11,28],[12,13],[12,17],[13,14],[13,18],[14,19],[15,17],[15,18],[15,22],[16,18],[16,19],[16,74],[17,19],[20,21],[20,25],[21,22],[21,26],[22,27],[23,25],[23,26],[23,30],[24,26],[24,27],[24,109],[25,27],[28,29],[28,33],[29,30],[29,34],[30,35],[31,33],[31,34],[32,34],[32,35],[32,111],[33,35],[36,37],[36,41],[36,67],[37,38],[37,42],[38,43],[38,88],[39,41],[39,42],[39,47],[40,42],[40,43],[40,45],[41,43],[44,45],[44,46],[44,47],[45,50],[46,48],[46,56],[47,64],[48,49],[48,53],[49,50],[49,54],[50,55],[51,53],[51,54],[51,58],[52,54],[52,55],[53,55],[56,57],[56,61],[57,58],[57,62],[58,63],[59,61],[59,62],[59,66],[60,62],[60,63],[60,110],[61,63],[64,65],[64,69],[65,66],[65,70],[66,71],[67,69],[67,70],[68,70],[68,71],[68,109],[69,71],[72,73],[72,77],[72,103],[73,74],[73,78],[74,79],[75,77],[75,78],[75,83],[76,78],[76,79],[76,81],[77,79],[80,81],[80,82],[80,83],[81,86],[82,84],[82,92],[83,100],[84,85],[84,89],[85,86],[85,90],[86,91],[87,89],[87,90],[87,94],[88,90],[88,91],[89,91],[92,93],[92,97],[93,94],[93,98],[94,99],[95,97],[95,98],[95,102],[96,98],[96,99],[96,111],[97,99],[100,101],[100,105],[101,102],[101,106],[102,107],[103,105],[103,106],[104,106],[104,107],[104,110],[105,107],[108,109],[108,110],[108,111]],"edge_sha256":"0f2d8858110c6f012de7ddffa92fdbc709d7da630f199b0e3c81bb56eb6b35c7"}}],"contributors":null,"scopes":[]},"problem_id":"problem-petersen-coloring","slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7,"relationship":{"claim":{"id":"claim-petersen-112-symmetric-witness","version":1},"relation":"counterexample","relationship_note":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","result":"proved","report_ids":["petersen-coloring:report:alternative:v1","petersen-coloring:report:certificates:v1"],"independent_review":"pending"},"assessments":[{"report_id":"petersen-coloring:report:alternative:v1","source_id":"petersen-coloring-paper","source_version_id":"petersen-coloring:source:paper:v1","source_title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","source_label":"Second nonisomorphic counterexample reported","state":"resolved","scope":"Exact D₃-symmetric graph in Theorem 6.1.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","entry_id":"petersen-coloring:entry:alternative:v1","source_date":"2026-08-08"},{"report_id":"petersen-coloring:report:certificates:v1","source_id":"petersen-coloring-archive","source_version_id":"petersen-coloring:source:archive:v1","source_title":"Petersen counterexample: frozen construction and certificates","source_label":"External SAT and DRAT checks reported","state":"resolved","scope":"Four graph/formulation pairs; not an OpenMath execution.","observed_on":"2026-10-14","evidence_url":"https://zenodo.org/records/21845291","entry_id":"petersen-coloring:entry:certificates:v1","source_date":"2026-08-08"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"There is a simple connected bridgeless cubic graph H on 112 vertices and 168 edges, of girth five and both connectivities three, with no Petersen coloring or normal five-edge-coloring; H is nonisomorphic to G and has automorphism group D₃ of order six.","id":"claim-petersen-112-symmetric-witness","slug":"petersen-112-symmetric-witness","version":1,"kind":"theorem","title":"A distinct symmetric 112-vertex counterexample","assumptions":["The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph."],"scope":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","source_entry_ids":["petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"formal_declarations":[],"relation":"counterexample","relationship_note":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","result":"proved","report_ids":["petersen-coloring:report:alternative:v1","petersen-coloring:report:certificates:v1"],"independent_review":"pending"}],"entries":[{"id":"petersen-coloring:entry:alternative:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"alternative","url":"https://arxiv.org/html/2608.10012v1","wording":"A second, nonisomorphic 112-vertex graph has automorphism group D₃ of order six and separately checked coloring certificates.","wording_kind":"paraphrase","locator":"Theorem 6.1; Section 6","supersedes":null},{"id":"petersen-coloring:entry:certificates:v1","source_version_id":"petersen-coloring:source:archive:v1","item_id":"certificates","url":"https://zenodo.org/records/21845291","wording":"CaDiCaL 3.0.1 reports UNSAT; drat-trim reports VERIFIED for four DRAT proofs. Separate encoders, satisfiable controls and status-parsing negative controls support the reported checks.","wording_kind":"paraphrase","locator":"03_CERTIFICATION, 04_OPTIONAL_D3 and 05_DOCUMENTATION","supersedes":null}],"sources":[{"work":{"authors":["Bryce Putman"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"Putman reports extensive OpenAI-model assistance in discovery, search, verification and preparation; he reviewed the final claims and artifacts and accepts responsibility.","artifacts":[],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:paper:v1","source_id":"petersen-coloring-paper","title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","owner":"Bryce Putman","url":"https://arxiv.org/html/2608.10012v1","subject_scope":"Theorems 1.1 and 6.1, Proposition 2.1, Corollary 5.1, Appendices A–B and computational provenance.","revision":"v1","content_sha256":"c960f9c1884b87a0df9fbd37b5e3d3fc583b57607a049e9b0b92fe12e86327bd","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2608.10012v1","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90},{"work":{"authors":["Bryce Putman"],"kind":"research_project","publication":"project","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The author attributes discovery and verification assistance to OpenAI systems; source logs are external computational reports.","artifacts":[{"kind":"dataset","title":"Exact graphs, encodings, manifests and logs","url":"https://zenodo.org/records/21845291/files/PCC_counterexample_112_compact_ancillary_v1.1.0.zip","version":"1.1.0","assessment":"not_executed","note":"Small archive inspected; exact graph data imported. SAT solving and DRAT certificate checking have not been rerun by OpenMath.","sha256":"75e935442fcb5457a9cfd0fd07c6d0a5b5fca80e1aeedab6e7243e8ccde9287b"},{"kind":"dataset","title":"Four DRAT proof certificates","url":"https://zenodo.org/records/21845291/files/PCC_112_DRAT_PROOFS_v1.0.0.zip","version":"1.0.0 (unchanged certificate bytes in release 1.1.0)","assessment":"external_report","note":"The source reports successful drat-trim checks for both formulations of both graphs. These large certificate files were not executed by OpenMath.","sha256":"8af3eec414b652f05c56979dc148321535cdc51ff9cbd59dff278ef3d53d9832"}],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:archive:v1","source_id":"petersen-coloring-archive","title":"Petersen counterexample: frozen construction and certificates","owner":"Bryce Putman","url":"https://zenodo.org/records/21845291","subject_scope":"Version 1.1.0; exact graphs, encoders, manifests, logs and separate DRAT proof files.","revision":"1.1.0","content_sha256":null,"update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://zenodo.org/records/21845291","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}]}],"assessments":[{"report_id":"petersen-coloring:report:certificates:v1","source_id":"petersen-coloring-archive","source_version_id":"petersen-coloring:source:archive:v1","source_title":"Petersen counterexample: frozen construction and certificates","source_label":"External SAT and DRAT checks reported","state":"resolved","scope":"Four graph/formulation pairs; not an OpenMath execution.","observed_on":"2026-10-14","evidence_url":"https://zenodo.org/records/21845291","entry_id":"petersen-coloring:entry:certificates:v1","source_date":"2026-08-08"},{"report_id":"petersen-coloring:report:alternative:v1","source_id":"petersen-coloring-paper","source_version_id":"petersen-coloring:source:paper:v1","source_title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","source_label":"Second nonisomorphic counterexample reported","state":"resolved","scope":"Exact D₃-symmetric graph in Theorem 6.1.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","entry_id":"petersen-coloring:entry:alternative:v1","source_date":"2026-08-08"}],"sources":[{"work":{"authors":["Bryce Putman"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"Putman reports extensive OpenAI-model assistance in discovery, search, verification and preparation; he reviewed the final claims and artifacts and accepts responsibility.","artifacts":[],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:paper:v1","source_id":"petersen-coloring-paper","title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","owner":"Bryce Putman","url":"https://arxiv.org/html/2608.10012v1","subject_scope":"Theorems 1.1 and 6.1, Proposition 2.1, Corollary 5.1, Appendices A–B and computational provenance.","revision":"v1","content_sha256":"c960f9c1884b87a0df9fbd37b5e3d3fc583b57607a049e9b0b92fe12e86327bd","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2608.10012v1","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90},{"work":{"authors":["Bryce Putman"],"kind":"research_project","publication":"project","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The author attributes discovery and verification assistance to OpenAI systems; source logs are external computational reports.","artifacts":[{"kind":"dataset","title":"Exact graphs, encodings, manifests and logs","url":"https://zenodo.org/records/21845291/files/PCC_counterexample_112_compact_ancillary_v1.1.0.zip","version":"1.1.0","assessment":"not_executed","note":"Small archive inspected; exact graph data imported. SAT solving and DRAT certificate checking have not been rerun by OpenMath.","sha256":"75e935442fcb5457a9cfd0fd07c6d0a5b5fca80e1aeedab6e7243e8ccde9287b"},{"kind":"dataset","title":"Four DRAT proof certificates","url":"https://zenodo.org/records/21845291/files/PCC_112_DRAT_PROOFS_v1.0.0.zip","version":"1.0.0 (unchanged certificate bytes in release 1.1.0)","assessment":"external_report","note":"The source reports successful drat-trim checks for both formulations of both graphs. These large certificate files were not executed by OpenMath.","sha256":"8af3eec414b652f05c56979dc148321535cdc51ff9cbd59dff278ef3d53d9832"}],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:archive:v1","source_id":"petersen-coloring-archive","title":"Petersen counterexample: frozen construction and certificates","owner":"Bryce Putman","url":"https://zenodo.org/records/21845291","subject_scope":"Version 1.1.0; exact graphs, encoders, manifests, logs and separate DRAT proof files.","revision":"1.1.0","content_sha256":null,"update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://zenodo.org/records/21845291","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}],"entries":[{"id":"petersen-coloring:entry:alternative:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"alternative","url":"https://arxiv.org/html/2608.10012v1","wording":"A second, nonisomorphic 112-vertex graph has automorphism group D₃ of order six and separately checked coloring certificates.","wording_kind":"paraphrase","locator":"Theorem 6.1; Section 6","supersedes":null},{"id":"petersen-coloring:entry:certificates:v1","source_version_id":"petersen-coloring:source:archive:v1","item_id":"certificates","url":"https://zenodo.org/records/21845291","wording":"CaDiCaL 3.0.1 reports UNSAT; drat-trim reports VERIFIED for four DRAT proofs. Separate encoders, satisfiable controls and status-parsing negative controls support the reported checks.","wording_kind":"paraphrase","locator":"03_CERTIFICATION, 04_OPTIONAL_D3 and 05_DOCUMENTATION","supersedes":null}],"formalizations":[],"tasks":[],"assessment_state":"proved","publication":{"version":7,"sha256":"a3faba6224e32fa719f72a6f65116aa84fd67bf538fac506a94d4af0d41bcc18","published_at":"2026-10-21T12:38:41.845859+00:00","reason":"Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.","withdrawn":false,"draft":{"reader_guide":{"reading":[{"entry_id":"petersen-coloring:entry:question:v1","role":"original_question","note":"Read the coloring definition and Jaeger equivalence."},{"entry_id":"petersen-coloring:entry:witness:v1","role":"proof","note":"Read the main construction and exact labeled graph."},{"entry_id":"petersen-coloring:entry:certificates:v1","role":"proof","note":"Inspect the encodings, certificate manifests, logs and reproduction instructions."}],"definitions":[{"id":"petersen-coloring-star","term":"Incident edge-star","explanation":"The three edges meeting a cubic vertex. A Petersen coloring sends this whole set bijectively to the star at a vertex of the Petersen graph.","source_entry_ids":["petersen-coloring:entry:question:v1"]},{"id":"petersen-coloring-bridgeless","term":"Bridgeless cubic graph","explanation":"Every vertex has degree three, and removing any one edge does not disconnect its component. The reported witnesses are simple and connected.","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1"]}],"proofs":[{"id":"petersen-coloring-sat-proof","claim":{"id":"claim-petersen-112-main-witness","version":1},"title":"Two independent coloring encodings","method":"SAT encodings and externally checked DRAT certificates","summary":"Each possible coloring would satisfy its corresponding CNF. The reported UNSAT certificates exclude all satisfying assignments. Graph reconstruction, encoding semantics and certificate checking are distinct steps.","limitation":"A solver’s UNSAT message alone is insufficient. OpenMath inspected the small archive and graph data, but has not rerun the four DRAT checks.","dependencies":[{"id":"claim-petersen-normal-five-equivalence","version":1}],"source_entry_ids":["petersen-coloring:entry:witness:v1","petersen-coloring:entry:certificates:v1"],"report_ids":["petersen-coloring:report:main:v1","petersen-coloring:report:certificates:v1"]}],"bounds":[],"witnesses":[{"id":"petersen-coloring-main-graph","claim":{"id":"claim-petersen-112-main-witness","version":1},"kind":"graph","title":"Inspect the main graph","fields":[{"label":"Order and size","value":"112 vertices; 168 edges"},{"label":"Coloring encodings","value":"Petersen: 3,640 variables and 68,324 clauses; normal five: 1,960 variables and 25,484 clauses"},{"label":"Certificate tools","value":"CaDiCaL 3.0.1 at c60730422e758ef1cebe7aeddf2dda31c996bf04; drat-trim at 2e3b2dc0ecf938addbd779d42877b6ed69d9a985"},{"label":"Normalized edge-list SHA-256","value":"dc16cc18600cf77c8661b7baf89c7019f265299308541961ff884ea7187b4e8b"}],"limitation":"The exact adjacency data identifies this witness. Inspecting neighbors does not verify noncolorability, and 112 is not asserted to be minimal.","source_entry_ids":["petersen-coloring:entry:witness:v1","petersen-coloring:entry:certificates:v1"],"graph":{"vertex_count":112,"edges":[[0,1],[0,5],[0,31],[1,2],[1,6],[2,7],[2,100],[3,5],[3,6],[3,11],[4,6],[4,7],[4,9],[5,7],[8,9],[8,10],[8,11],[9,14],[10,12],[10,20],[11,28],[12,13],[12,17],[13,14],[13,18],[14,19],[15,17],[15,18],[15,22],[16,18],[16,19],[16,56],[17,19],[20,21],[20,25],[21,22],[21,26],[22,27],[23,25],[23,26],[23,30],[24,26],[24,27],[24,39],[25,27],[28,29],[28,33],[29,30],[29,34],[30,35],[31,33],[31,34],[32,34],[32,35],[32,37],[33,35],[36,37],[36,38],[36,39],[37,42],[38,64],[38,92],[39,78],[40,41],[40,45],[40,71],[41,42],[41,46],[42,47],[43,45],[43,46],[43,51],[44,46],[44,47],[44,49],[45,47],[48,49],[48,50],[48,51],[49,54],[50,52],[50,60],[51,68],[52,53],[52,57],[53,54],[53,58],[54,59],[55,57],[55,58],[55,62],[56,58],[56,59],[57,59],[60,61],[60,65],[61,62],[61,66],[62,67],[63,65],[63,66],[63,70],[64,66],[64,67],[65,67],[68,69],[68,73],[69,70],[69,74],[70,75],[71,73],[71,74],[72,74],[72,75],[72,108],[73,75],[76,77],[76,81],[76,107],[77,78],[77,82],[78,83],[79,81],[79,82],[79,87],[80,82],[80,83],[80,85],[81,83],[84,85],[84,86],[84,87],[85,90],[86,88],[86,96],[87,104],[88,89],[88,93],[89,90],[89,94],[90,95],[91,93],[91,94],[91,98],[92,94],[92,95],[93,95],[96,97],[96,101],[97,98],[97,102],[98,103],[99,101],[99,102],[99,106],[100,102],[100,103],[101,103],[104,105],[104,109],[105,106],[105,110],[106,111],[107,109],[107,110],[108,110],[108,111],[109,111]],"edge_sha256":"dc16cc18600cf77c8661b7baf89c7019f265299308541961ff884ea7187b4e8b"}},{"id":"petersen-coloring-symmetric-graph","claim":{"id":"claim-petersen-112-symmetric-witness","version":1},"kind":"graph","title":"Inspect the symmetric graph","fields":[{"label":"Order and size","value":"112 vertices; 168 edges"},{"label":"Coloring encodings","value":"Petersen: 3,640 variables and 68,324 clauses; normal five: 1,960 variables and 25,484 clauses"},{"label":"Certificate tools","value":"CaDiCaL 3.0.1 at c60730422e758ef1cebe7aeddf2dda31c996bf04; drat-trim at 2e3b2dc0ecf938addbd779d42877b6ed69d9a985"},{"label":"Normalized edge-list SHA-256","value":"0f2d8858110c6f012de7ddffa92fdbc709d7da630f199b0e3c81bb56eb6b35c7"}],"limitation":"The exact adjacency data identifies this witness. Inspecting neighbors does not verify noncolorability, and 112 is not asserted to be minimal.","source_entry_ids":["petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"graph":{"vertex_count":112,"edges":[[0,1],[0,5],[0,31],[1,2],[1,6],[2,7],[2,52],[3,5],[3,6],[3,11],[4,6],[4,7],[4,9],[5,7],[8,9],[8,10],[8,11],[9,14],[10,12],[10,20],[11,28],[12,13],[12,17],[13,14],[13,18],[14,19],[15,17],[15,18],[15,22],[16,18],[16,19],[16,74],[17,19],[20,21],[20,25],[21,22],[21,26],[22,27],[23,25],[23,26],[23,30],[24,26],[24,27],[24,109],[25,27],[28,29],[28,33],[29,30],[29,34],[30,35],[31,33],[31,34],[32,34],[32,35],[32,111],[33,35],[36,37],[36,41],[36,67],[37,38],[37,42],[38,43],[38,88],[39,41],[39,42],[39,47],[40,42],[40,43],[40,45],[41,43],[44,45],[44,46],[44,47],[45,50],[46,48],[46,56],[47,64],[48,49],[48,53],[49,50],[49,54],[50,55],[51,53],[51,54],[51,58],[52,54],[52,55],[53,55],[56,57],[56,61],[57,58],[57,62],[58,63],[59,61],[59,62],[59,66],[60,62],[60,63],[60,110],[61,63],[64,65],[64,69],[65,66],[65,70],[66,71],[67,69],[67,70],[68,70],[68,71],[68,109],[69,71],[72,73],[72,77],[72,103],[73,74],[73,78],[74,79],[75,77],[75,78],[75,83],[76,78],[76,79],[76,81],[77,79],[80,81],[80,82],[80,83],[81,86],[82,84],[82,92],[83,100],[84,85],[84,89],[85,86],[85,90],[86,91],[87,89],[87,90],[87,94],[88,90],[88,91],[89,91],[92,93],[92,97],[93,94],[93,98],[94,99],[95,97],[95,98],[95,102],[96,98],[96,99],[96,111],[97,99],[100,101],[100,105],[101,102],[101,106],[102,107],[103,105],[103,106],[104,106],[104,107],[104,110],[105,107],[108,109],[108,110],[108,111]],"edge_sha256":"0f2d8858110c6f012de7ddffa92fdbc709d7da630f199b0e3c81bb56eb6b35c7"}}],"contributors":{"summary":"François Jaeger posed the coloring question and established its normal-five formulation. Bryce Putman authored the 2026 construction and verification release, reporting extensive OpenAI-model assistance and accepting responsibility for the final work. CaDiCaL and drat-trim supply distinct solver and certificate-checking roles. Ma, Mattiolo, Steffen and Wolf supplied the theorem used for the infinitude consequence.","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1","petersen-coloring:entry:alternative:v1","petersen-coloring:entry:infinite:v1","petersen-coloring:entry:certificates:v1"]},"scopes":[],"question":{"scope_summary":"An edge-star mapping question, answered negatively by a finite source-reported obstruction.","detail_title":"Exact conventions and hypotheses","aspects":[],"source_entry_ids":["petersen-coloring:entry:question:v1"]},"remaining":{"mathematical":"Determine the smallest possible counterexample order; continue reviewing the certificate correspondence and weaker covering assertions separately.","assessment":"Independent mathematical review is pending. External certificates and manuscript proofs are attributed to their sources.","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1","petersen-coloring:entry:alternative:v1","petersen-coloring:entry:infinite:v1","petersen-coloring:entry:certificates:v1"]},"connections":{"statements":[{"claim":{"id":"claim-petersen-universal-coloring","version":1},"label":"Universal Petersen coloring"},{"claim":{"id":"claim-petersen-normal-five","version":1},"label":"Universal normal five-edge-coloring"}],"relationships":[]}},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"petersen-coloring-setup","version":6,"subject":{"kind":"problem","id":"problem-petersen-coloring","sha256":"7d7a7a67bd7aca3539fe6b6b3c9072d8f0c184321677841b8da113f3e23de5d8"},"purpose":"setup","title":"From the Petersen graph to the obstruction","caption":"Construction hierarchy for the main graph: F has 8 vertices, L has 36 and G has 112.","description":"Deleting two adjacent vertices leaves F. Four F copies and C give L; three L copies and another C give G. The source tables specify the semiedge joins.","assumptions":["Layout is schematic; coordinates and line lengths carry no mathematical scale."],"limitation":"This diagram shows copy counts, not every semiedge join. Use the exact adjacency list and source construction for the witness.","author":"OpenMath source curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-14","independent_review":"pending","source_entry_ids":["petersen-coloring:entry:witness:v1","petersen-coloring:entry:certificates:v1"],"payload":{"kind":"diagram_schematic","marks":[{"x":105,"y":45,"id":"p","kind":"label","end":null,"radius":null,"text":"Petersen · 10","anchor":"middle","dashed":false},{"x":75,"y":65,"id":"pf","kind":"segment","end":{"x":75,"y":110},"radius":null,"text":null,"anchor":"middle","dashed":false,"directed":true},{"x":75,"y":135,"id":"f","kind":"label","end":null,"radius":null,"text":"F · 8","anchor":"middle","dashed":false},{"x":250,"y":135,"id":"l","kind":"label","end":null,"radius":null,"text":"L = 4F + C · 36","anchor":"middle","dashed":false},{"x":125,"y":130,"id":"fl","kind":"segment","end":{"x":170,"y":130},"radius":null,"text":null,"anchor":"middle","dashed":false,"directed":true},{"x":250,"y":245,"id":"g","kind":"label","end":null,"radius":null,"text":"G = 3L + C · 112","anchor":"middle","dashed":false},{"x":250,"y":158,"id":"lg","kind":"segment","end":{"x":250,"y":215},"radius":null,"text":null,"anchor":"middle","dashed":false,"directed":true},{"x":385,"y":65,"id":"c","kind":"label","end":null,"radius":null,"text":"C · claw · 4","anchor":"middle","dashed":false},{"x":385,"y":85,"id":"cl","kind":"segment","end":{"x":310,"y":115},"radius":null,"text":null,"anchor":"middle","dashed":true},{"x":385,"y":85,"id":"cg","kind":"segment","end":{"x":310,"y":220},"radius":null,"text":null,"anchor":"middle","dashed":true}],"features":[{"id":"construction","label":"The 112-vertex main witness","claim":{"id":"claim-petersen-112-main-witness","version":1},"binding":"assertion","mark_ids":["p","pf","f","l","fl","g","lg","c","cl","cg"]}],"legend":"Numbers count vertices; each stage joins copies of the previous one through the claw C."}}],"visuals":[{"representation":{"id":"petersen-coloring-setup","version":6},"section":"question"}],"claim_relations":[{"claim":{"id":"claim-petersen-universal-coloring","version":1},"relation":"primary_formulation","relationship_note":"Parallel edges are allowed; the simple connected witness already lies in this class.","result":"refuted","report_ids":["petersen-coloring:report:refuted:v1"],"independent_review":"pending"},{"claim":{"id":"claim-petersen-normal-five","version":1},"relation":"equivalent_formulation","relationship_note":"A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.","result":"refuted","report_ids":["petersen-coloring:report:refuted:v1"],"independent_review":"pending","reading":{"placement":"formulation","note":"A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden."}},{"claim":{"id":"claim-petersen-normal-five-equivalence","version":1},"relation":"supporting_result","relationship_note":"The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.","result":"proved","report_ids":["petersen-coloring:report:equivalence:v1"],"independent_review":"pending"},{"claim":{"id":"claim-petersen-112-main-witness","version":1},"relation":"counterexample","relationship_note":"G is the exact labeled graph in Appendix A; its normalized sorted-edge-list digest identifies the SAT input. No minimum-order claim is made.","result":"proved","report_ids":["petersen-coloring:report:main:v1","petersen-coloring:report:certificates:v1"],"independent_review":"pending"},{"claim":{"id":"claim-petersen-112-symmetric-witness","version":1},"relation":"counterexample","relationship_note":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","result":"proved","report_ids":["petersen-coloring:report:alternative:v1","petersen-coloring:report:certificates:v1"],"independent_review":"pending"},{"claim":{"id":"claim-petersen-infinitely-many-counterexamples","version":1},"relation":"adjacent_result","relationship_note":"Corollary 5.1 combines the finite counterexample with the universal-target theorem of Ma, Mattiolo, Steffen and Wolf.","result":"proved","report_ids":["petersen-coloring:report:infinite:v1"],"independent_review":"pending"}],"connections":[{"id":"petersen-formulation-equivalence","source":{"id":"claim-petersen-universal-coloring","version":1},"target":{"id":"claim-petersen-normal-five","version":1},"relation":"equivalent_to","assertion":{"id":"claim-petersen-normal-five-equivalence","version":1},"explanation":"Jaeger’s theorem gives equivalent existence conditions for each loopless cubic graph.","limitation":"The positive universal assertions are both refuted by the reported witness; equivalence remains a valid theorem.","source_entry_ids":["petersen-coloring:entry:question:v1"],"independent_review":"pending"}],"milestones":[{"id":"petersen-coloring:milestone:jaeger","date":{"value":"1988"},"date_basis":"publication","category":"foundation","title":"Jaeger’s coloring question","summary":"The selected paper attributes the bridgeless cubic question to Jaeger’s 1988 chapter.","limitation":"This is a selected source-supported milestone, not an exhaustive history.","claims":[{"id":"claim-petersen-universal-coloring","version":1}],"source_entry_ids":["petersen-coloring:entry:question:v1"],"independent_review":"pending"},{"id":"petersen-coloring:milestone:equivalence","date":{"value":"1985"},"date_basis":"publication","category":"foundation","title":"The normal-five formulation","summary":"Jaeger’s earlier work establishes the coloring equivalence recalled in Proposition 2.1.","limitation":"This is a selected source-supported milestone, not an exhaustive history.","claims":[{"id":"claim-petersen-normal-five-equivalence","version":1}],"source_entry_ids":["petersen-coloring:entry:question:v1"],"independent_review":"pending"},{"id":"petersen-coloring:milestone:counterexample","date":{"value":"2026-08-08"},"date_basis":"preprint","category":"related_result","title":"Two explicit counterexamples and certificates","summary":"Putman publishes the graph constructions, encodings and external checks; the frozen release is version 1.1.0.","limitation":"This is a selected source-supported milestone, not an exhaustive history.","claims":[{"id":"claim-petersen-112-main-witness","version":1},{"id":"claim-petersen-112-symmetric-witness","version":1}],"source_entry_ids":["petersen-coloring:entry:witness:v1","petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"independent_review":"pending"}],"curation":{"checked_on":"2026-10-14","selection":"Canonical question, exact result scopes, proof evidence, contributors, history and explanatory examples.","search_scope":"Selected primary manuscript versions and their cited original literature, checked on 14 October 2026. This is not an exhaustive literature review.","formal_status":"SAT/DRAT certificates are available in the immutable archive. They are computational proof evidence, not an identified Lean or Isabelle theorem; OpenMath has not executed the certificates.","ai_status":"Model assistance is self-reported by Putman; specific model versions are not identified in the selected disclosure.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"Every finite loopless bridgeless cubic graph admits an edge map to the Petersen graph that maps each incident edge-star bijectively onto a target vertex’s edge-star.","id":"claim-petersen-universal-coloring","slug":"petersen-universal-coloring","version":1,"kind":"conjecture","title":"Every bridgeless cubic graph admits a Petersen coloring","assumptions":["Parallel edges are allowed; the simple connected witness already lies in this class."],"scope":"Parallel edges are allowed; the simple connected witness already lies in this class.","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1"],"formal_declarations":[]},{"statement":"Every finite loopless bridgeless cubic graph has a proper edge-coloring with at most five colors in which the union of the palettes at the endpoints of every edge has either three or five colors.","id":"claim-petersen-normal-five","slug":"petersen-normal-five","version":1,"kind":"conjecture","title":"Normal five-edge-colorings for bridgeless cubic graphs","assumptions":["A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden."],"scope":"A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1"],"formal_declarations":[]},{"statement":"For each finite loopless cubic multigraph, the existence of a Petersen coloring is equivalent to the existence of a normal five-edge-coloring.","id":"claim-petersen-normal-five-equivalence","slug":"petersen-normal-five-equivalence","version":1,"kind":"equivalence","title":"Petersen and normal-five colorings are equivalent","assumptions":["The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence."],"scope":"The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.","source_entry_ids":["petersen-coloring:entry:question:v1"],"formal_declarations":[]},{"statement":"There is a simple connected bridgeless cubic graph G on 112 vertices and 168 edges, of girth five and edge- and vertex-connectivity three, with no Petersen coloring and no normal five-edge-coloring.","id":"claim-petersen-112-main-witness","slug":"petersen-112-main-witness","version":1,"kind":"theorem","title":"The main 112-vertex counterexample","assumptions":["G is the exact labeled graph in Appendix A; its normalized sorted-edge-list digest identifies the SAT input. No minimum-order claim is made."],"scope":"G is the exact labeled graph in Appendix A; its normalized sorted-edge-list digest identifies the SAT input. No minimum-order claim is made.","source_entry_ids":["petersen-coloring:entry:witness:v1","petersen-coloring:entry:certificates:v1"],"formal_declarations":[]},{"statement":"There is a simple connected bridgeless cubic graph H on 112 vertices and 168 edges, of girth five and both connectivities three, with no Petersen coloring or normal five-edge-coloring; H is nonisomorphic to G and has automorphism group D₃ of order six.","id":"claim-petersen-112-symmetric-witness","slug":"petersen-112-symmetric-witness","version":1,"kind":"theorem","title":"A distinct symmetric 112-vertex counterexample","assumptions":["The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph."],"scope":"The Section 6 graph is separately encoded and certified; it is not a relabeling of the main graph.","source_entry_ids":["petersen-coloring:entry:alternative:v1","petersen-coloring:entry:certificates:v1"],"formal_declarations":[]},{"statement":"Infinitely many connected simple bridgeless cubic graphs admit no Petersen coloring.","id":"claim-petersen-infinitely-many-counterexamples","slug":"petersen-infinitely-many-counterexamples","version":1,"kind":"theorem","title":"Infinitely many Petersen-coloring counterexamples","assumptions":["Corollary 5.1 combines the finite counterexample with the universal-target theorem of Ma, Mattiolo, Steffen and Wolf."],"scope":"Corollary 5.1 combines the finite counterexample with the universal-target theorem of Ma, Mattiolo, Steffen and Wolf.","source_entry_ids":["petersen-coloring:entry:infinite:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"counterexample","extent":"full_statement","summary":"A source-reported 112-vertex counterexample refutes the universal coloring conjecture. A second nonisomorphic witness is separately certified; neither establishes the smallest possible order.","report_ids":["petersen-coloring:report:refuted:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-petersen-coloring","slug":"petersen-coloring","title":"Petersen coloring conjecture","question":"Can every bridgeless cubic graph be colored by the Petersen graph?","scope":"A Petersen coloring maps edges of a finite loopless cubic graph G to edges of the Petersen graph P. At each vertex of G, its three incident edges must map bijectively onto the three edges incident with some vertex of P. This is a local edge-star condition, not a proper three-edge-coloring. The reported witness is simple and connected; it therefore refutes the broader bridgeless cubic assertion.","aliases":["Jaeger conjecture","Normal five-edge-coloring","Petersen colourings"],"fields":["Graph theory"],"msc":["05C15","05C21"],"topics":[],"background":"undergraduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["petersen-coloring:entry:question:v1","petersen-coloring:entry:witness:v1","petersen-coloring:entry:alternative:v1","petersen-coloring:entry:infinite:v1","petersen-coloring:entry:certificates:v1"],"status_report_ids":["petersen-coloring:report:refuted:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"known","scope":"Source comparison and presentation; independent mathematical review remains pending.","rationale":"A source-reported 112-vertex counterexample refutes the universal coloring conjecture. A second nonisomorphic witness is separately certified; neither establishes the smallest possible order.","decided_on":"2026-10-14","review_due_on":"2027-01-14","report_ids":["petersen-coloring:report:refuted:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","url":"https://arxiv.org/html/2608.10012v1","note":"Theorems 1.1 and 6.1, Proposition 2.1, Corollary 5.1, Appendices A–B and computational provenance."},{"title":"Petersen counterexample: frozen construction and certificates","url":"https://zenodo.org/records/21845291","note":"Version 1.1.0; exact graphs, encoders, manifests, logs and separate DRAT proof files."}],"formalizations":[],"related":[{"slug":"cycle-double-cover","title":"Cycle double cover conjecture","relation":"related_result","note":"Petersen coloring is a stronger covering assertion. Refuting it does not refute cycle double cover; the covering question has its own existing Claims and evidence."}],"tasks":[],"insights":[{"id":"petersen-coloring-construction","title":"Build the obstruction from small pieces","summary":"Delete the endpoints of one Petersen edge to obtain the eight-vertex four-pole F. Join four copies of F and a four-vertex claw six-pole C into a 36-vertex four-pole L, then join three L copies and another C into G. The source specifies every semiedge join.","review":"pending","kind":"approach","source_entry_ids":["petersen-coloring:entry:witness:v1"],"claims":[{"id":"claim-petersen-112-main-witness","version":1}],"limitation":"Copy counts do not specify the graph without the precise joins and labeling map."},{"id":"petersen-coloring-impact","title":"A finite obstruction has an infinite-family consequence","summary":"The universal-target theorem cited in Corollary 5.1 turns the finite counterexample into an infinitude result for connected simple bridgeless cubic graphs.","review":"pending","kind":"impact","source_entry_ids":["petersen-coloring:entry:infinite:v1"],"claims":[{"id":"claim-petersen-infinitely-many-counterexamples","version":1}],"scenario":{"id":"claim-petersen-112-main-witness","version":1},"limitation":"This consequence does not assert that the witnesses refute weaker cycle-cover or flow conjectures."}],"coverage_gaps":["Independent OpenMath mathematical review and external proof execution remain pending."],"checked_on":"2026-10-14"},"sources":[{"work":{"authors":["Bryce Putman"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"Putman reports extensive OpenAI-model assistance in discovery, search, verification and preparation; he reviewed the final claims and artifacts and accepts responsibility.","artifacts":[],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:paper:v1","source_id":"petersen-coloring-paper","title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","owner":"Bryce Putman","url":"https://arxiv.org/html/2608.10012v1","subject_scope":"Theorems 1.1 and 6.1, Proposition 2.1, Corollary 5.1, Appendices A–B and computational provenance.","revision":"v1","content_sha256":"c960f9c1884b87a0df9fbd37b5e3d3fc583b57607a049e9b0b92fe12e86327bd","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2608.10012v1","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90},{"work":{"authors":["Bryce Putman"],"kind":"research_project","publication":"project","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The author attributes discovery and verification assistance to OpenAI systems; source logs are external computational reports.","artifacts":[{"kind":"dataset","title":"Exact graphs, encodings, manifests and logs","url":"https://zenodo.org/records/21845291/files/PCC_counterexample_112_compact_ancillary_v1.1.0.zip","version":"1.1.0","assessment":"not_executed","note":"Small archive inspected; exact graph data imported. SAT solving and DRAT certificate checking have not been rerun by OpenMath.","sha256":"75e935442fcb5457a9cfd0fd07c6d0a5b5fca80e1aeedab6e7243e8ccde9287b"},{"kind":"dataset","title":"Four DRAT proof certificates","url":"https://zenodo.org/records/21845291/files/PCC_112_DRAT_PROOFS_v1.0.0.zip","version":"1.0.0 (unchanged certificate bytes in release 1.1.0)","assessment":"external_report","note":"The source reports successful drat-trim checks for both formulations of both graphs. These large certificate files were not executed by OpenMath.","sha256":"8af3eec414b652f05c56979dc148321535cdc51ff9cbd59dff278ef3d53d9832"}],"published":{"value":"2026-08-08"}},"id":"petersen-coloring:source:archive:v1","source_id":"petersen-coloring-archive","title":"Petersen counterexample: frozen construction and certificates","owner":"Bryce Putman","url":"https://zenodo.org/records/21845291","subject_scope":"Version 1.1.0; exact graphs, encoders, manifests, logs and separate DRAT proof files.","revision":"1.1.0","content_sha256":null,"update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://zenodo.org/records/21845291","attribution":"Bryce Putman","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}],"entries":[{"id":"petersen-coloring:entry:question:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"question","url":"https://arxiv.org/html/2608.10012v1","wording":"The edge-star formulation is equivalent to the existence of a normal five-edge-coloring. The original conjecture concerns every bridgeless cubic graph.","wording_kind":"paraphrase","locator":"Introduction; Proposition 2.1; references to Jaeger (1985, 1988)","supersedes":null},{"id":"petersen-coloring:entry:witness:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"witness","url":"https://arxiv.org/html/2608.10012v1","wording":"The main graph has 112 vertices, 168 edges, girth five and edge- and vertex-connectivity three. Section 3 specifies its recursive construction.","wording_kind":"paraphrase","locator":"Theorem 1.1; Section 3; Appendix A","supersedes":null},{"id":"petersen-coloring:entry:alternative:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"alternative","url":"https://arxiv.org/html/2608.10012v1","wording":"A second, nonisomorphic 112-vertex graph has automorphism group D₃ of order six and separately checked coloring certificates.","wording_kind":"paraphrase","locator":"Theorem 6.1; Section 6","supersedes":null},{"id":"petersen-coloring:entry:infinite:v1","source_version_id":"petersen-coloring:source:paper:v1","item_id":"infinite","url":"https://arxiv.org/html/2608.10012v1","wording":"The finite counterexample together with the cited universal-target theorem implies infinitely many connected simple bridgeless cubic counterexamples.","wording_kind":"paraphrase","locator":"Corollary 5.1; Ma–Mattiolo–Steffen–Wolf (2025)","supersedes":null},{"id":"petersen-coloring:entry:certificates:v1","source_version_id":"petersen-coloring:source:archive:v1","item_id":"certificates","url":"https://zenodo.org/records/21845291","wording":"CaDiCaL 3.0.1 reports UNSAT; drat-trim reports VERIFIED for four DRAT proofs. Separate encoders, satisfiable controls and status-parsing negative controls support the reported checks.","wording_kind":"paraphrase","locator":"03_CERTIFICATION, 04_OPTIONAL_D3 and 05_DOCUMENTATION","supersedes":null}],"reports":[{"id":"petersen-coloring:report:refuted:v1","entry_id":"petersen-coloring:entry:witness:v1","state":"refuted","source_label":"Counterexample reported with checked SAT certificates","scope":"Universal Petersen coloring assertion; minimum counterexample order is not determined.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","supersedes":null},{"id":"petersen-coloring:report:main:v1","entry_id":"petersen-coloring:entry:witness:v1","state":"resolved","source_label":"Explicit main graph and noncolorability reported","scope":"Exact main labeled graph in Theorem 1.1.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","supersedes":null},{"id":"petersen-coloring:report:alternative:v1","entry_id":"petersen-coloring:entry:alternative:v1","state":"resolved","source_label":"Second nonisomorphic counterexample reported","scope":"Exact D₃-symmetric graph in Theorem 6.1.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","supersedes":null},{"id":"petersen-coloring:report:certificates:v1","entry_id":"petersen-coloring:entry:certificates:v1","state":"resolved","source_label":"External SAT and DRAT checks reported","scope":"Four graph/formulation pairs; not an OpenMath execution.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://zenodo.org/records/21845291","supersedes":null},{"id":"petersen-coloring:report:equivalence:v1","entry_id":"petersen-coloring:entry:question:v1","state":"resolved","source_label":"Jaeger equivalence recalled and proved","scope":"Finite loopless cubic multigraphs.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","supersedes":null},{"id":"petersen-coloring:report:infinite:v1","entry_id":"petersen-coloring:entry:infinite:v1","state":"resolved","source_label":"Infinite family consequence reported","scope":"Uses the separate Ma–Mattiolo–Steffen–Wolf theorem.","source_date":"2026-08-08","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2608.10012v1","supersedes":null}]},"status":{"state":"refuted","label":"Resolved · counterexample","extent":"full_statement","summary":"A source-reported 112-vertex counterexample refutes the universal coloring conjecture. A second nonisomorphic witness is separately certified; neither establishes the smallest possible order.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-14","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"]}},"selected_context":null}}