{
  "schemaVersion": "1.0",
  "recordType": "claim-register",
  "site": "Evidence Press",
  "claimCeiling": "These records expose authored claims, scoped evidence, replay tasks and producer or external receipts without upgrading correctness, novelty, independence, formal verification or review beyond the evidence explicitly recorded.",
  "itemSchema": "https://evidencepress.org/api/schemas/claim.schema.json",
  "count": 1,
  "claims": [
    {
      "schemaVersion": "1.0",
      "recordType": "claim",
      "claimId": "ep-claim:sha256:77529f500af949b9ee0b76694fc022d8dc06d9779628df896888b6f2c8470bb1",
      "claimKey": "z20-equals-6-main",
      "claimVersion": "1.0.0",
      "statementFingerprint": "sha256:117128e0c8c96aefb59c21d3f8d936c12ea5c5840fc4985873ac3d6cb7513faa",
      "statement": {
        "short": "z(20) = 6",
        "naturalLanguage": "For every finite simple graph G on 20 vertices, the cochromatic number z(G) is at most 6, and there exists a finite simple graph on 20 vertices whose cochromatic number is 6; equivalently, z(20)=6.",
        "latex": "z(20)=\\max_{|V(G)|=20} z(G)=6",
        "definitions": [
          "For a finite simple graph G, z(G) is the minimum number of parts in a partition of V(G) in which every part induces either a clique or an independent set.",
          "For n >= 1, z(n) is the maximum of z(G) over finite simple graphs G with n vertices."
        ]
      },
      "release": {
        "slug": "z20-equals-6",
        "version": "candidate-2026-07-26",
        "doi": "10.5281/zenodo.21647645",
        "status": "unrefereed-candidate",
        "repository": "https://github.com/ipitchford/z20-cochromatic",
        "sourceCommit": "3c7e520fdc0615f5c700761c2b1e5108dcc836e7",
        "sourceTag": "candidate-2026-07-26"
      },
      "scope": {
        "resolution": "finite-case",
        "domain": "finite graph theory; cochromatic numbers; fixed finite order n=20",
        "canonicalProblem": {
          "name": "Erdos problem 758 (the n=20 cochromatic-number case)",
          "url": "https://www.erdosproblems.com/758"
        },
        "assumptions": [
          "Graphs are finite, simple and undirected.",
          "The computer-assisted upper bound depends on the mathematical reduction, catalogue generation, graph-to-CNF encoding, exact CNFs, proof objects and checker behavior described separately below."
        ]
      },
      "nonClaims": [
        {
          "id": "no-external-reproduction",
          "statement": "This record does not claim independent external reproduction of the complete result."
        },
        {
          "id": "no-end-to-end-formalisation",
          "statement": "This record does not claim an end-to-end proof-assistant formalisation of the graph-theoretic theorem or its reduction to SAT."
        },
        {
          "id": "no-human-peer-review",
          "statement": "This record does not claim completed human mathematical peer review."
        },
        {
          "id": "no-formal-encoding-proof",
          "statement": "The formally verified LRAT checking core does not by itself verify catalogue generation, the graph-to-CNF semantic bridge or the theorem statement."
        },
        {
          "id": "no-independent-companion",
          "statement": "The companion VR2(K4)=20 release reuses these proof objects and is not an independent reproduction."
        }
      ],
      "obligations": [
        {
          "id": "lower-bound",
          "label": "Paley(17) lower bound",
          "state": "proof-text-present",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_proof.tex",
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/src/lower_bound_result.txt"
          ],
          "claimCeiling": "A hand-checkable lower-bound argument is present; this registry records its availability but is not a substitute for specialist review."
        },
        {
          "id": "reduction",
          "label": "Reduction to two order-16 Ramsey cores",
          "state": "partial",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_proof.tex",
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/src/gen44.c"
          ],
          "claimCeiling": "The manuscript gives the reduction and producer-side catalogue replay is reported; the catalogue program and reduction are not formally verified or independently reimplemented."
        },
        {
          "id": "cnf",
          "label": "Exact fixed-core CNF construction",
          "state": "producer-replayed",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/generate.py",
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/MANIFEST.sha256"
          ],
          "claimCeiling": "Exact CNF regeneration and hashes were reported on a producer-side machine; this does not establish the semantic bridge or independent reproduction."
        },
        {
          "id": "certificate",
          "label": "DRUP and LRAT refutation objects",
          "state": "producer-replayed",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/tree/candidate-2026-07-26/certificates"
          ],
          "claimCeiling": "The preserved proof objects were reported to refute the two exact preserved CNFs; this is narrower than the graph-theoretic theorem."
        },
        {
          "id": "checker",
          "label": "Certificate-checker behavior",
          "state": "producer-replayed",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/verification/verify_certificate.py",
            "https://github.com/tanyongkiam/cake_lpr"
          ],
          "claimCeiling": "Multiple checker verdicts, including a formally verified LRAT checking core, were reported on one machine; this is not end-to-end theorem formalisation or external reproduction."
        },
        {
          "id": "semantic-bridge",
          "label": "Graph statement to exact-CNF semantic correspondence",
          "state": "partial",
          "evidenceRefs": [
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/verification/verify_certificate.py",
            "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_replay_receipt_2026-07-26.md"
          ],
          "claimCeiling": "Deterministic semantic spot checks are reported, but no proof-assistant formalisation or independent clean-room implementation closes the semantic bridge."
        }
      ],
      "statuses": {
        "database": "registered-candidate",
        "authorClaimed": "candidate-proof",
        "formalVerification": "certificate-checking-core-only",
        "independentReproduction": "none-reported",
        "humanReview": "none-reported",
        "novelty": "repository-reported-not-independently-audited",
        "systemAssessment": "candidate-with-producer-replay-and-open-semantic-bridge",
        "assessedAt": "2026-08-29"
      },
      "dependencies": [
        {
          "id": "ramsey-r44",
          "kind": "cited-theorem",
          "statement": "R(4,4)=18.",
          "status": "published-classical-result",
          "sourceRef": "https://doi.org/10.4153/CJM-1955-001-4"
        },
        {
          "id": "order16-ramsey-cores",
          "kind": "external-classification-and-local-regeneration",
          "statement": "There are two non-isomorphic (4,4)-Ramsey graphs on 16 vertices used as the fixed cores.",
          "status": "producer-regenerated-not-independently-reimplemented",
          "sourceRef": "https://users.cecs.anu.edu.au/~bdm/data/ramsey.html"
        },
        {
          "id": "checker-assumptions",
          "kind": "computational-assumption",
          "statement": "The preserved programs, runtime, hardware and checker invocations behaved as recorded in the producer replay receipt.",
          "status": "producer-reported",
          "sourceRef": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_replay_receipt_2026-07-26.md"
        }
      ],
      "artifacts": [
        {
          "path": "paper/z20_equals_6_proof.pdf",
          "role": "candidate manuscript",
          "sha256": "sha256:045e826bd5d249456ae4759155256d7562c95b850fc83fa1de39283dd7691db3",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_proof.pdf"
        },
        {
          "path": "paper/z20_equals_6_proof.tex",
          "role": "manuscript source",
          "sha256": "sha256:fdbe212ffca1fb7d6b4724054f6fdb1b3ae4eff8ab03939e92208e81398ffdb6",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_proof.tex"
        },
        {
          "path": "paper/z20_equals_6_replay_receipt_2026-07-26.md",
          "role": "historical producer replay receipt",
          "sha256": "sha256:2ad26844ecdceb934d70b227753e256a65b201e7e6a09c448eb5fe54f18a00ee",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/paper/z20_equals_6_replay_receipt_2026-07-26.md"
        },
        {
          "path": "certificates/z20_core0.cnf",
          "role": "core 0 formula",
          "sha256": "sha256:cbfea7b0b2cb712ad8b4f8b129f74c7f010e879064414eaef76e74aa88be3f44",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/z20_core0.cnf"
        },
        {
          "path": "certificates/z20_core0.drup",
          "role": "core 0 DRUP refutation",
          "sha256": "sha256:3f7aa99011da35a3e2746348bab550eab3f918d10e8a6c619041c72e7033649a",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/z20_core0.drup"
        },
        {
          "path": "certificates/core0.lrat",
          "role": "core 0 LRAT refutation",
          "sha256": "sha256:df04c4615dd6e44fd81c870e0b82433b6116f0bea135550a393a6d41ad7a1d57",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/core0.lrat"
        },
        {
          "path": "certificates/z20_core1.cnf",
          "role": "core 1 formula",
          "sha256": "sha256:d3d96a3c2a719c753242673b81aabc3977c5e435bb5996b4c58c85a82d11dc5a",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/z20_core1.cnf"
        },
        {
          "path": "certificates/z20_core1.drup",
          "role": "core 1 DRUP refutation",
          "sha256": "sha256:ebe9821bb9939f1331f9908415cb8499f6c1d216c0143a0d048ff47533df446b",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/z20_core1.drup"
        },
        {
          "path": "certificates/core1.lrat",
          "role": "core 1 LRAT refutation",
          "sha256": "sha256:72b7cc4fcce0c4558b457bc14f6c795bc7401b1e7d990c5b679ebbda3708e06c",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/certificates/core1.lrat"
        },
        {
          "path": "verification/verify_certificate.py",
          "role": "separate implementation verifier supplied by the producers",
          "sha256": "sha256:821a5bb500f011befcf700c84823d4ec7fe9e3757491e8d8fcd768d4c683e913",
          "url": "https://github.com/ipitchford/z20-cochromatic/blob/candidate-2026-07-26/verification/verify_certificate.py"
        }
      ],
      "provenance": [
        {
          "name": "GPT 5.6 Sol",
          "role": "repository-reported mathematical and computational contribution",
          "type": "AI system"
        },
        {
          "name": "Claude Opus 4.8 / 5",
          "role": "repository-reported mathematical and computational contribution",
          "type": "AI system"
        },
        {
          "name": "Repository maintainer",
          "role": "problem selection, research mediation, repository maintenance and publication",
          "type": "human; individual identity not asserted in this Evidence Press record"
        }
      ],
      "license": "CC0-1.0",
      "supersedes": null,
      "sourceRefs": [
        "https://doi.org/10.5281/zenodo.21647645",
        "https://github.com/ipitchford/z20-cochromatic/releases/tag/candidate-2026-07-26",
        "https://evidencepress.org/releases/z20-equals-6/"
      ]
    }
  ]
}