{"schema_version":"source-reader-v1","record":{"source_id":"petersen-coloring-paper","versions":[{"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}],"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: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: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}],"problems":[{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7}],"selected_version_id":"petersen-coloring:source:paper:v1","available_versions":[{"id":"petersen-coloring:source:paper:v1","revision":"v1","observed_on":"2026-10-14","title":"A 112-Vertex Counterexample to the Petersen Coloring Conjecture","publication":"preprint","content_sha256":"c960f9c1884b87a0df9fbd37b5e3d3fc583b57607a049e9b0b92fe12e86327bd","used_in_current_publications":true}],"usages":[{"id":"source-use-b69912be7700aff0f072","entry_id":"petersen-coloring:entry:alternative:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Second nonisomorphic counterexample reported","claim":{"id":"claim-petersen-112-symmetric-witness","version":1,"slug":"petersen-112-symmetric-witness","title":"A distinct symmetric 112-vertex counterexample"},"fragment":"claim-petersen-112-symmetric-witness","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-fd7b6ee3a1d89138dfe3","entry_id":"petersen-coloring:entry:alternative:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"construction","title":"Inspect the symmetric graph","claim":{"id":"claim-petersen-112-symmetric-witness","version":1,"slug":"petersen-112-symmetric-witness","title":"A distinct symmetric 112-vertex counterexample"},"fragment":"claim-petersen-112-symmetric-witness","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-82070224322d66c23904","entry_id":"petersen-coloring:entry:alternative:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"milestone","title":"Two explicit counterexamples and certificates","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-fccb696f698f33e02dbe","entry_id":"petersen-coloring:entry:alternative:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"question","title":"Petersen coloring conjecture","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-3efc2328d7ab897c6f66","entry_id":"petersen-coloring:entry:alternative:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"A distinct symmetric 112-vertex counterexample","claim":{"id":"claim-petersen-112-symmetric-witness","version":1,"slug":"petersen-112-symmetric-witness","title":"A distinct symmetric 112-vertex counterexample"},"fragment":"claim-petersen-112-symmetric-witness","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-0ef9e48c26b76e303246","entry_id":"petersen-coloring:entry:infinite:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Infinite family consequence reported","claim":{"id":"claim-petersen-infinitely-many-counterexamples","version":1,"slug":"petersen-infinitely-many-counterexamples","title":"Infinitely many Petersen-coloring counterexamples"},"fragment":"claim-petersen-infinitely-many-counterexamples","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-7807f371358cb896f066","entry_id":"petersen-coloring:entry:infinite:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"question","title":"Petersen coloring conjecture","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-394f2a9cd5c43bc8e608","entry_id":"petersen-coloring:entry:infinite:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Infinitely many Petersen-coloring counterexamples","claim":{"id":"claim-petersen-infinitely-many-counterexamples","version":1,"slug":"petersen-infinitely-many-counterexamples","title":"Infinitely many Petersen-coloring counterexamples"},"fragment":"claim-petersen-infinitely-many-counterexamples","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-9924cf79e4f89f490718","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Jaeger equivalence recalled and proved","claim":{"id":"claim-petersen-normal-five-equivalence","version":1,"slug":"petersen-normal-five-equivalence","title":"Petersen and normal-five colorings are equivalent"},"fragment":"claim-petersen-normal-five-equivalence","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-66c5c1c3af044f9e68ba","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"connection","title":"Jaeger’s theorem gives equivalent existence conditions for each loopless cubic graph.","claim":{"id":"claim-petersen-normal-five-equivalence","version":1,"slug":"petersen-normal-five-equivalence","title":"Petersen and normal-five colorings are equivalent"},"fragment":"claim-petersen-normal-five-equivalence","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-108476ddae808179993e","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"definition","title":"Incident edge-star","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-bb38a02a274fd09e579a","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"definition","title":"Bridgeless cubic graph","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-a0510cd64b76abc13beb","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"milestone","title":"Jaeger’s coloring question","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-a23db00a0412caf58107","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"milestone","title":"The normal-five formulation","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-4cbdfa816e546c93f904","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"question","title":"Petersen coloring conjecture","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-e6c8d82f27bb48a8c5d7","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"reading","title":"Read the coloring definition and Jaeger equivalence.","claim":null,"fragment":"provenance","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-6a52a1674c5b5af0e0fd","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Petersen and normal-five colorings are equivalent","claim":{"id":"claim-petersen-normal-five-equivalence","version":1,"slug":"petersen-normal-five-equivalence","title":"Petersen and normal-five colorings are equivalent"},"fragment":"claim-petersen-normal-five-equivalence","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-7380eb5d608a21e9930b","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Normal five-edge-colorings for bridgeless cubic graphs","claim":{"id":"claim-petersen-normal-five","version":1,"slug":"petersen-normal-five","title":"Normal five-edge-colorings for bridgeless cubic graphs"},"fragment":"claim-petersen-normal-five","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-a94ed18d8d3089417454","entry_id":"petersen-coloring:entry:question:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Every bridgeless cubic graph admits a Petersen coloring","claim":{"id":"claim-petersen-universal-coloring","version":1,"slug":"petersen-universal-coloring","title":"Every bridgeless cubic graph admits a Petersen coloring"},"fragment":"claim-petersen-universal-coloring","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-9bc4cd7ed1b6c4f15382","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Explicit main graph and noncolorability reported","claim":{"id":"claim-petersen-112-main-witness","version":1,"slug":"petersen-112-main-witness","title":"The main 112-vertex counterexample"},"fragment":"claim-petersen-112-main-witness","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-bad998d34624d9deb57e","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Counterexample reported with checked SAT certificates","claim":{"id":"claim-petersen-universal-coloring","version":1,"slug":"petersen-universal-coloring","title":"Every bridgeless cubic graph admits a Petersen coloring"},"fragment":"claim-petersen-universal-coloring","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-ffcd7919c07d6c43cca1","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"assessment","title":"Counterexample reported with checked SAT certificates","claim":{"id":"claim-petersen-normal-five","version":1,"slug":"petersen-normal-five","title":"Normal five-edge-colorings for bridgeless cubic graphs"},"fragment":"claim-petersen-normal-five","assessment":{"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},"superseded":false,"formalization":null,"update":null},{"id":"source-use-c8b6c5c656e885565935","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"construction","title":"Inspect the main graph","claim":{"id":"claim-petersen-112-main-witness","version":1,"slug":"petersen-112-main-witness","title":"The main 112-vertex counterexample"},"fragment":"claim-petersen-112-main-witness","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-1733ae8c74c4b47aa6fe","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"definition","title":"Bridgeless cubic graph","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-e6abacd138f3f7a9012d","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"milestone","title":"Two explicit counterexamples and certificates","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-ca08fd6a6d9caa1bd97d","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"proof","title":"Two independent coloring encodings","claim":{"id":"claim-petersen-112-main-witness","version":1,"slug":"petersen-112-main-witness","title":"The main 112-vertex counterexample"},"fragment":"claim-petersen-112-main-witness","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-01b7a9eb27e5edc81d3c","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"question","title":"Petersen coloring conjecture","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-472a2be5aae5591f7787","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"reading","title":"Read the main construction and exact labeled graph.","claim":null,"fragment":"provenance","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-23c386047b7e3c774bef","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Every bridgeless cubic graph admits a Petersen coloring","claim":{"id":"claim-petersen-universal-coloring","version":1,"slug":"petersen-universal-coloring","title":"Every bridgeless cubic graph admits a Petersen coloring"},"fragment":"claim-petersen-universal-coloring","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-a251ee8c795b982ef43f","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"Normal five-edge-colorings for bridgeless cubic graphs","claim":{"id":"claim-petersen-normal-five","version":1,"slug":"petersen-normal-five","title":"Normal five-edge-colorings for bridgeless cubic graphs"},"fragment":"claim-petersen-normal-five","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-ad0d45ddaec7741fd9a7","entry_id":"petersen-coloring:entry:witness:v1","problem":{"slug":"petersen-coloring","title":"Petersen coloring conjecture","publication_version":7},"kind":"statement","title":"The main 112-vertex counterexample","claim":{"id":"claim-petersen-112-main-witness","version":1,"slug":"petersen-112-main-witness","title":"The main 112-vertex counterexample"},"fragment":"claim-petersen-112-main-witness","assessment":null,"superseded":false,"formalization":null,"update":null}],"linked_entries":[],"linked_sources":[]}}