{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:dean-s-conjecture-for-k-5",
  "title": "Dean's conjecture for $k=5$",
  "canonical_statement": "Every finite simple graph $G$ with minimum degree $\\delta(G)\\ge 5$ contains a simple cycle $C$ whose length satisfies $|C|\\equiv 0\\pmod 5$.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:dean-s-conjecture-for-k-5:event",
      "problem_id": "vibemathed:dean-s-conjecture-for-k-5",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-08-29T00:00:00.000Z",
      "title": "Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture",
      "summary": "Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for $k\\in\\{3,4\\}$ and they proved every $k\\ge6$, leaving $k=5$ open. If this proof stands, Dean's conjecture holds for all $k\\ge3$.\n\nNine finite proposition families in the bipartite and triangle-free branches are computer-assisted, with verifiers and certificates in a separately archived supplement. The reductions from arbitrary graphs to those finite state spaces are prose arguments in the paper and are not machine-checked.\n\nThe claim has not been refereed, and no independent mathematician has audited the graph-theoretic core. The author describes extensive model-assisted hostile auditing of his own argument, which is worth something but is not external review, and says so plainly.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:dean-s-conjecture-for-k-5:attempt"
      ],
      "method_family_ids": [
        "vibemathed:dean-s-conjecture-for-k-5:method"
      ],
      "source_assertion_ids": [
        "vibemathed:dean-s-conjecture-for-k-5:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:dean-s-conjecture-for-k-5:attempt",
      "problem_id": "vibemathed:dean-s-conjecture-for-k-5",
      "model": "GPT-5.6 Sol (primary); Claude Opus 5; GLM 5.3 Flash",
      "provider": "OpenAI (primary); Anthropic; Z.ai",
      "attempted_at": "2026-08-29T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "GPT-5.6 Sol was the primary proof-development and hostile-audit system. It contributed substantial portions of the graph-theoretic reductions and terminal arguments, audited smoothing-edge scope, path and cycle simplicity, branch exhaustiveness, and graph-to-certificate coverage, and assisted in constructing and checking the finite verifiers. Claude Opus 5 was used as an ancillary prover for certain problems. GLM 5.3 Flash agents were used with the Danus framework earlier on in the problem-solving process.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Computational supplement: verifiers, certificates, manifest",
          "url": "https://doi.org/10.5281/zenodo.22167084",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Luo, Ma and Zhao - Dean's conjecture for every k >= 6",
          "url": "https://arxiv.org/abs/2601.13552",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture",
          "url": "https://zenodo.org/records/22182448",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Dean's conjecture for $k=5$",
          "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:dean-s-conjecture-for-k-5:verification",
      "problem_id": "vibemathed:dean-s-conjecture-for-k-5",
      "solution_event_id": "vibemathed:dean-s-conjecture-for-k-5:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Site-confirmed: the computational supplement was replayed here in full on 30 August 2026, and it passes.\n\nWhat was run. Downloaded the supplement from its own Zenodo record (10.5281/zenodo.22167084), checked all 86 files against the shipped $\\texttt{MANIFEST\\_SHA256.txt}$ - 86 match, 0 mismatch, 0 missing - then executed its $\\texttt{run\\_all.ps1}$ driver end to end. Result: \"All certificate runs passed: 47\", exit 0, in 33 min 53 s. Every per-run line reported $\\texttt{exit=0}$, and the only occurrences of \"fail\" in the fresh verification record are four instances of \"failures: 0\". Both toolchains ran, Python and JavaScript, including the bipartite verifiers, whose output reports \"checked rows: 580, failures: 0\" and \"verified rich pairs: 78\".\n\nWhat this does NOT establish, and it is the larger half. In the supplement's own words: \"The programs verify only the finite propositions listed in PROPOSITION_MAP.md. The reductions from arbitrary graphs to those finite state spaces, and the proofs that a reported forbidden object expands to a simple cycle or path in the original graph, are mathematical arguments in the paper. The programs do not replace those graph-to-state theorems.\"\n\nSo the nine computer-assisted proposition families are exactly what the author says they are, and reproducibly so. The argument that carries them to every graph of minimum degree five is unrefereed prose that no independent mathematician has read. That is why the entry remains a candidate.",
      "sources": [
        {
          "label": "VibeMathed: Dean's conjecture for $k=5$",
          "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture",
          "url": "https://zenodo.org/records/22182448",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture",
      "url": "https://zenodo.org/records/22182448",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Dean's conjecture for $k=5$",
      "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:dean-s-conjecture-for-k-5:signal:verify_now",
      "problem_id": "vibemathed:dean-s-conjecture-for-k-5",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}