{
  "schemaVersion": "proofatlas.public-release-credit.v1",
  "id": "credit.release.berlekamp-domineering-temperature-counterexample.v002",
  "slug": "berlekamp-domineering-temperature-counterexample",
  "canonicalRoute": "/research/berlekamp-domineering-temperature-counterexample/",
  "canonicalUrl": "https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/",
  "title": "An Exact Counterexample to Berlekamp's Temperature-2 Conjecture for Domineering",
  "releasedAt": "2026-08-14T00:00:00-04:00",
  "releaseKind": "paper",
  "limitations": [
    "No independent external priority anchor or literature-completeness review is recorded.",
    "ProofAtlas has not rerun the substantial Lean build for this page update or reproduced the result in an unrelated implementation.",
    "The Lean theorem does not formalize the deletion-local audit or separately formalize the checkerboard unreachability of the raw 26-cell core.",
    "The Lean theorem uses the narrow HasValueTemperature witness interface and does not establish a general thermograph-invariance theorem for arbitrary game-equivalent presentations.",
    "The scoped Lean theorem is not an accepted ProofAtlas result and does not substitute for specialist peer review."
  ],
  "identities": [
    {
      "actorId": "actor.ai.openai-codex.v001",
      "actorRecord": {
        "id": "actor.ai.openai-codex.v001",
        "sha256": "sha256:d022a1afbaa619096cff658949045eb9cdafd20761a54bf4c4cf651c3371f9fd"
      },
      "citationLabel": "OpenAI Codex",
      "displayLabel": "OpenAI Codex",
      "identityMode": "pseudonym"
    },
    {
      "actorId": "actor.ai.openai-gpt-5-6-pro.v001",
      "actorRecord": {
        "id": "actor.ai.openai-gpt-5-6-pro.v001",
        "sha256": "sha256:223bc15cba0da58d1914fa4a036a27a5df6281316fb44207a9281748e1ee66b1"
      },
      "citationLabel": "OpenAI GPT-5.6 Pro",
      "displayLabel": "OpenAI GPT-5.6 Pro",
      "identityMode": "pseudonym"
    },
    {
      "actorId": "actor.human.lech-mazur.v001",
      "actorRecord": {
        "id": "actor.human.lech-mazur.v001",
        "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
      },
      "citationLabel": "Mazur, Lech",
      "displayLabel": "Lech Mazur",
      "identityMode": "real_name"
    }
  ],
  "authorship": [
    {
      "actor": {
        "actorId": "actor.human.lech-mazur.v001",
        "actorRecord": {
          "id": "actor.human.lech-mazur.v001",
          "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
        },
        "citationLabel": "Mazur, Lech",
        "displayLabel": "Lech Mazur",
        "identityMode": "real_name"
      },
      "contributionIds": [
        "contribution.berlekamp.lech-successor.v002"
      ],
      "order": 1,
      "role": "paper"
    }
  ],
  "credits": [
    {
      "actors": [
        {
          "actorId": "actor.ai.openai-gpt-5-6-pro.v001",
          "actorRecord": {
            "id": "actor.ai.openai-gpt-5-6-pro.v001",
            "sha256": "sha256:223bc15cba0da58d1914fa4a036a27a5df6281316fb44207a9281748e1ee66b1"
          },
          "citationLabel": "OpenAI GPT-5.6 Pro",
          "displayLabel": "OpenAI GPT-5.6 Pro",
          "identityMode": "pseudonym"
        }
      ],
      "contributionId": "contribution.berlekamp.gpt-development.v002",
      "relationship": "direct",
      "roles": [
        "computation",
        "exposition",
        "gap_or_error_discovery",
        "proof_strategy"
      ],
      "summary": "OpenAI GPT-5.6 Pro has source-reported roles in mathematical exploration, exact computation, adversarial auditing, and exposition."
    },
    {
      "actors": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ],
      "contributionId": "contribution.berlekamp.lech-successor.v002",
      "relationship": "direct",
      "roles": [
        "exposition",
        "proof_strategy",
        "research_direction"
      ],
      "summary": "Lech Mazur is the manuscript author and accountable editor and directed the rectangle-reachable witness, Lean formalization, audit, and exposition workflow."
    }
  ],
  "reviews": [
    {
      "independenceStatus": "not_asserted",
      "review": {
        "id": "review.draft.public_wording.problem.berlekamp-domineering-temperature-conjecture.v001.h71799412ace5.v005",
        "sha256": "sha256:be67bfe0340251cb2bc51e6f7a9c62e7c07c6b9a1e4090ea19abe493cc11dcef"
      },
      "reviewKind": "public_wording",
      "reviewer": {
        "actorId": "actor.ai.openai-codex.v001",
        "actorRecord": {
          "id": "actor.ai.openai-codex.v001",
          "sha256": "sha256:d022a1afbaa619096cff658949045eb9cdafd20761a54bf4c4cf651c3371f9fd"
        },
        "citationLabel": "OpenAI Codex",
        "displayLabel": "OpenAI Codex",
        "identityMode": "pseudonym"
      },
      "reviewerActorId": "actor.ai.openai-codex.v001",
      "scope": {
        "limitations": [],
        "statement": "Public-wording review",
        "subjectBindings": [
          {
            "id": "artifact.berlekamp-domineering-temperature-counterexample.reachable.v001",
            "sha256": "sha256:80b52f948261d63eeaeb4f9f4ff9c36b9e1c58c95c2c47e518ac52bb6f65948e"
          },
          {
            "id": "problem.berlekamp-domineering-temperature-conjecture.v001",
            "sha256": "sha256:1c406ab86cbe1dff355a003a140021c21e91fb1c6f3b2c08cc438815f077273d"
          },
          {
            "id": "research.berlekamp-domineering-temperature-counterexample.v002",
            "sha256": "sha256:66df14117cae013febc88645aa3dbbf364730b056e636ac477d72e91b6251108"
          },
          {
            "id": "statement.berlekamp-domineering-temperature-counterexample.reachable.v001",
            "sha256": "sha256:10becd52593e9694b53b3221c65c6c3c5ff323aa6be17836e2d5eec23c732f40"
          },
          {
            "id": "target.berlekamp-domineering-temperature-counterexample.v001",
            "sha256": "sha256:d069fa754d26bf641689c2862906977ca5ae54936d89bc06ca383486c98c73a8"
          }
        ]
      }
    }
  ],
  "evidence": {
    "assets": [
      {
        "byteLength": 6528,
        "id": "asset.berlekamp.body-markdown",
        "mediaType": "text/markdown",
        "sha256": "sha256:52989cb0a97f02e6bfa2e968f1200cd0701436eebda1ced839b2d112dbbf040a"
      },
      {
        "byteLength": 2214789,
        "id": "asset.berlekamp.infographic",
        "mediaType": "image/png",
        "sha256": "sha256:70357365c9b7ef5f5d8048d86cb912d905d346fb4ddcc5cf4d7ae466a68fd0c2"
      },
      {
        "byteLength": 133150,
        "id": "asset.berlekamp.paper-pdf",
        "mediaType": "application/pdf",
        "sha256": "sha256:f6d13859436b13dc1668ed3c1927197b284ce2c872a9a0ad8d81ca55ebbe80fc"
      },
      {
        "byteLength": 36510,
        "id": "asset.berlekamp.paper-tex",
        "mediaType": "application/x-tex",
        "sha256": "sha256:5a0331085be0f852e601347aa43c1c00197702a4b251ecc6edf3245df918eae7"
      },
      {
        "byteLength": 8997385,
        "id": "asset.berlekamp.public-packet",
        "mediaType": "application/zip",
        "sha256": "sha256:0e1c641075015acc27741605369d6c55719c837080a655a356db13adc5ebfb5d"
      }
    ],
    "ledgerBinding": {
      "id": "event.credit.berlekamp.retained.v002",
      "sha256": "sha256:6b824f18282ae39221e46cb041e370e6afdc9a00bf2daebd8f3b51e4f99f23eb"
    },
    "sourceCommits": [
      "9edd1d307a62bcd1b59c4450c8675fcdbb5a3ebe"
    ],
    "subjectBindings": [
      {
        "id": "artifact.berlekamp-domineering-temperature-counterexample.reachable.v001",
        "sha256": "sha256:80b52f948261d63eeaeb4f9f4ff9c36b9e1c58c95c2c47e518ac52bb6f65948e"
      },
      {
        "id": "problem.berlekamp-domineering-temperature-conjecture.v001",
        "sha256": "sha256:1c406ab86cbe1dff355a003a140021c21e91fb1c6f3b2c08cc438815f077273d"
      },
      {
        "id": "research.berlekamp-domineering-temperature-counterexample.v002",
        "sha256": "sha256:66df14117cae013febc88645aa3dbbf364730b056e636ac477d72e91b6251108"
      },
      {
        "id": "statement.berlekamp-domineering-temperature-counterexample.reachable.v001",
        "sha256": "sha256:10becd52593e9694b53b3221c65c6c3c5ff323aa6be17836e2d5eec23c732f40"
      },
      {
        "id": "target.berlekamp-domineering-temperature-counterexample.v001",
        "sha256": "sha256:d069fa754d26bf641689c2862906977ca5ae54936d89bc06ca383486c98c73a8"
      }
    ]
  },
  "rights": [
    {
      "assetId": "asset.berlekamp.body-markdown",
      "basisStatus": "rights_confirmed",
      "publicPermissions": [
        {
          "basis": "unresolved",
          "status": "unresolved",
          "use": "inclusion_in_paper"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "incorporation_into_public_result"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_derived_summary"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_raw_redistribution"
        }
      ],
      "rightsholders": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ]
    },
    {
      "assetId": "asset.berlekamp.infographic",
      "basisStatus": "rights_confirmed",
      "publicPermissions": [
        {
          "basis": "unresolved",
          "status": "unresolved",
          "use": "inclusion_in_paper"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "incorporation_into_public_result"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_derived_summary"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_raw_redistribution"
        }
      ],
      "rightsholders": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ]
    },
    {
      "assetId": "asset.berlekamp.paper-pdf",
      "basisStatus": "rights_confirmed",
      "publicPermissions": [
        {
          "basis": "unresolved",
          "status": "unresolved",
          "use": "inclusion_in_paper"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "incorporation_into_public_result"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_derived_summary"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_raw_redistribution"
        }
      ],
      "rightsholders": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ]
    },
    {
      "assetId": "asset.berlekamp.paper-tex",
      "basisStatus": "rights_confirmed",
      "publicPermissions": [
        {
          "basis": "unresolved",
          "status": "unresolved",
          "use": "inclusion_in_paper"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "incorporation_into_public_result"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_derived_summary"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_raw_redistribution"
        }
      ],
      "rightsholders": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ]
    },
    {
      "assetId": "asset.berlekamp.public-packet",
      "basisStatus": "rights_confirmed",
      "publicPermissions": [
        {
          "basis": "unresolved",
          "status": "unresolved",
          "use": "inclusion_in_paper"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "incorporation_into_public_result"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_derived_summary"
        },
        {
          "basis": "rightsholder_authorization",
          "status": "allowed",
          "use": "public_raw_redistribution"
        }
      ],
      "rightsholders": [
        {
          "actorId": "actor.human.lech-mazur.v001",
          "actorRecord": {
            "id": "actor.human.lech-mazur.v001",
            "sha256": "sha256:209175a11d71298f013b2258fa44fcc0a345550a5a14844a51f5cd11c531aef8"
          },
          "citationLabel": "Mazur, Lech",
          "displayLabel": "Lech Mazur",
          "identityMode": "real_name"
        }
      ]
    }
  ],
  "publicRecord": {
    "predecessorSnapshot": {
      "id": "credit.release.berlekamp-domineering-temperature-counterexample.v001",
      "sha256": "sha256:d8c9e6f1721914b44b236ecf063eb1f09a74c95819da2d6e1950671f8dc976e6"
    },
    "priority": {
      "status": "release_chronology"
    }
  },
  "citation": {
    "bibtex": "@misc{mazur-2026-berlekamp-domineering-counterexample,\n  author = {Mazur, Lech},\n  title = {An Exact Counterexample to Berlekamp's Temperature-2 Conjecture for Domineering},\n  publisher = {ProofAtlas},\n  year = {2026},\n  date = {2026-08-14},\n  version = {public version of 14 August 2026 with scoped Lean artifact},\n  url = {https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/}\n}\n",
    "jsonLd": {
      "@context": "https://schema.org",
      "@id": "https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/",
      "@type": "ScholarlyArticle",
      "additionalType": "paper",
      "author": [
        {
          "@type": "Thing",
          "identifier": "actor.human.lech-mazur.v001",
          "name": "Mazur, Lech"
        }
      ],
      "contributor": [
        {
          "@type": "Thing",
          "identifier": "actor.ai.openai-gpt-5-6-pro.v001",
          "name": "OpenAI GPT-5.6 Pro",
          "roleName": [
            "computation",
            "exposition",
            "gap_or_error_discovery",
            "proof_strategy"
          ]
        },
        {
          "@type": "Thing",
          "identifier": "actor.human.lech-mazur.v001",
          "name": "Lech Mazur",
          "roleName": [
            "exposition",
            "proof_strategy",
            "research_direction"
          ]
        }
      ],
      "datePublished": "2026-08-14",
      "headline": "An Exact Counterexample to Berlekamp's Temperature-2 Conjecture for Domineering",
      "identifier": [
        {
          "propertyID": "url",
          "url": "https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/",
          "value": "https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/"
        }
      ],
      "publisher": {
        "@type": "Organization",
        "name": "ProofAtlas"
      },
      "url": "https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/",
      "version": "public version of 14 August 2026 with scoped Lean artifact"
    },
    "plain": "Mazur, Lech. An Exact Counterexample to Berlekamp's Temperature-2 Conjecture for Domineering. public version of 14 August 2026 with scoped Lean artifact. ProofAtlas, 2026-08-14. https://proofatlas.ai/research/berlekamp-domineering-temperature-counterexample/"
  }
}