{
  "schemaVersion": "proofatlas.public-release-credit.v1",
  "id": "credit.release.bondy-longest-cycle-proof.v002",
  "slug": "bondy-longest-cycle-proof",
  "canonicalRoute": "/research/bondy-longest-cycle-proof/",
  "canonicalUrl": "https://proofatlas.ai/research/bondy-longest-cycle-proof/",
  "title": "A Formalized Proof of Bondy's Minimum-Degree Longest-Cycle Conjecture",
  "releasedAt": "2026-08-16T00:00:00Z",
  "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 proof in an unrelated implementation.",
    "Specialist mathematical review and ProofAtlas accepted-result review remain open.",
    "The release does not establish historical priority or verify every citation and expository statement in the companion paper.",
    "The retained audit is pinned to source commit 9e13b044d8821b587935fa3019d913699eadffa2 and does not cover the later local commit 9a6d94363495d2578fed87911e5a247af3ce3394."
  ],
  "identities": [
    {
      "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.bondy.lech-formalized-paper.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.bondy.gpt-formalization-development.v002",
      "relationship": "direct",
      "roles": [
        "computation",
        "exposition",
        "formalization",
        "gap_or_error_discovery",
        "proof_strategy"
      ],
      "summary": "OpenAI GPT-5.6 Pro has source-reported roles in mathematical exploration, proof development, computational testing, adversarial auditing, exposition, and AI-assisted formalization development."
    },
    {
      "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.bondy.lech-formalized-paper.v002",
      "relationship": "direct",
      "roles": [
        "exposition",
        "proof_strategy",
        "research_direction"
      ],
      "summary": "Lech Mazur is the paper author and accountable editor and directed the AI-assisted research, verification process, output selection, and reconciliation."
    }
  ],
  "reviews": [],
  "evidence": {
    "assets": [
      {
        "byteLength": 4540,
        "id": "asset.bondy.audit",
        "mediaType": "text/markdown",
        "sha256": "sha256:d9aad2727c140ede5b14c7882d0aedb725399b0d404c3455c7eeb18732b701b0"
      },
      {
        "byteLength": 5526,
        "id": "asset.bondy.body-markdown",
        "mediaType": "text/markdown",
        "sha256": "sha256:ce20c002891d4dc8720f763d234373a47b68ca8c5c2126764d0d1875cd01c174"
      },
      {
        "byteLength": 272966,
        "id": "asset.bondy.infographic",
        "mediaType": "image/png",
        "sha256": "sha256:65e1e895c1ab89de086a4a5573258805a4909c98d5d3fa7d52c67151ca725692"
      },
      {
        "byteLength": 225252,
        "id": "asset.bondy.paper-pdf",
        "mediaType": "application/pdf",
        "sha256": "sha256:f265525ea05387cf556df1f1516985e19c301bf65fc3d9a80f8755ee349d2846"
      },
      {
        "byteLength": 85350,
        "id": "asset.bondy.paper-tex",
        "mediaType": "application/x-tex",
        "sha256": "sha256:c06a5e25cc24ca6a37bc4e92fcd0427b423fe411f30047d53db6a0be3c6be77a"
      }
    ],
    "ledgerBinding": {
      "id": "event.credit.bondy.retained.v002",
      "sha256": "sha256:db023025559a76432d04d23b3cabbfed73d62d4f737155cf4891bf8f095514ea"
    },
    "sourceCommits": [
      "9134da370295ac4c1989070c341ba101ac3d9c17",
      "9e13b044d8821b587935fa3019d913699eadffa2"
    ],
    "subjectBindings": [
      {
        "id": "artifact.bondy.minimum-degree-longest-cycle-conjecture.lean-package.v001",
        "sha256": "sha256:4da056c076554550f1f1ddf9fca61b69b34859c39a4c5ccef91ca78066cb845e"
      },
      {
        "id": "problem.bondy.minimum-degree-longest-cycle-conjecture.v001",
        "sha256": "sha256:c2a1a659b3fb93049c2777a901d898559b7f750235315882087e511890ba7e85"
      },
      {
        "id": "research.bondy-longest-cycle-proof.v002",
        "sha256": "sha256:31c0437c9ac33f9281886f0e35c123eb660e918029437fd74701cc6831edfb9c"
      },
      {
        "id": "statement.bondy.minimum-degree-longest-cycle-conjecture.v001",
        "sha256": "sha256:72904fd7d9d449ebffbd13beebe084b65ad739095166c5dc2b0e5ca30db92f14"
      },
      {
        "id": "target.bondy.minimum-degree-longest-cycle-conjecture.v001",
        "sha256": "sha256:ef31aecae82320ac79b6d4400bc0d80fbffeb2b3924143a16d2335e3890f63bc"
      }
    ]
  },
  "rights": [
    {
      "assetId": "asset.bondy.audit",
      "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.bondy.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.bondy.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.bondy.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.bondy.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"
        }
      ]
    }
  ],
  "publicRecord": {
    "predecessorSnapshot": {
      "id": "credit.release.bondy-longest-cycle-proof.v001",
      "sha256": "sha256:12e003992e07b47557f05a2922e6aaf56cd1c605591eefc870b6fc1979e2b412"
    },
    "priority": {
      "status": "release_chronology"
    }
  },
  "citation": {
    "bibtex": "@misc{mazur-2026-bondy-formalized-proof,\n  author = {Mazur, Lech},\n  title = {A Formalized Proof of Bondy's Minimum-Degree Longest-Cycle Conjecture},\n  publisher = {ProofAtlas},\n  year = {2026},\n  date = {2026-08-16},\n  version = {revised public version of August 16, 2026},\n  url = {https://proofatlas.ai/research/bondy-longest-cycle-proof/}\n}\n",
    "jsonLd": {
      "@context": "https://schema.org",
      "@id": "https://proofatlas.ai/research/bondy-longest-cycle-proof/",
      "@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",
            "formalization",
            "gap_or_error_discovery",
            "proof_strategy"
          ]
        },
        {
          "@type": "Thing",
          "identifier": "actor.human.lech-mazur.v001",
          "name": "Lech Mazur",
          "roleName": [
            "exposition",
            "proof_strategy",
            "research_direction"
          ]
        }
      ],
      "datePublished": "2026-08-16",
      "headline": "A Formalized Proof of Bondy's Minimum-Degree Longest-Cycle Conjecture",
      "identifier": [
        {
          "propertyID": "url",
          "url": "https://proofatlas.ai/research/bondy-longest-cycle-proof/",
          "value": "https://proofatlas.ai/research/bondy-longest-cycle-proof/"
        }
      ],
      "publisher": {
        "@type": "Organization",
        "name": "ProofAtlas"
      },
      "url": "https://proofatlas.ai/research/bondy-longest-cycle-proof/",
      "version": "revised public version of August 16, 2026"
    },
    "plain": "Mazur, Lech. A Formalized Proof of Bondy's Minimum-Degree Longest-Cycle Conjecture. revised public version of August 16, 2026. ProofAtlas, 2026-08-16. https://proofatlas.ai/research/bondy-longest-cycle-proof/"
  }
}