{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture",
  "title": "The Foregger–Sinkhorn Tie-Point Conjecture",
  "canonical_statement": "The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \\times 8$ counterexample exists, built on the unique root $\\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.",
  "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:foregger-sinkhorn-tie-point-conjecture:event",
      "problem_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture",
      "type": "disproved",
      "status": "candidate",
      "occurred_at": "2026-08-13T00:00:00.000Z",
      "title": "A counterexample to the Foregger-Sinkhorn tie-point conjecture",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:foregger-sinkhorn-tie-point-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:foregger-sinkhorn-tie-point-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:foregger-sinkhorn-tie-point-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:foregger-sinkhorn-tie-point-conjecture:attempt",
      "problem_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture",
      "model": "GPT-5.6-sol, Claude Fable 5",
      "provider": "OpenAI, Anthropic",
      "attempted_at": "2026-08-13T00:00:00.000Z",
      "human_collaborators": [
        "Yair Lavi"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's acknowledgement, in full: \"The proof of Theorem 1 was carried out by GPT-5.6-sol and Claude Fable 5, under the guidance of the author. The author has reviewed the resulting proof arguments. Responsibility for the final text rests with the author.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "A counterexample to the Foregger-Sinkhorn tie-point conjecture",
          "url": "https://arxiv.org/abs/2608.13025",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Foregger–Sinkhorn Tie-Point Conjecture",
          "url": "https://vibemathed.com/problem/foregger-sinkhorn-tie-point-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:foregger-sinkhorn-tie-point-conjecture:verification",
      "problem_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture",
      "solution_event_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Independently recomputed by this site on 21 August 2026 from the paper's own data (arXiv:2608.13025v1), at 60-digit precision: $\\beta$ is the unique root of the stated cubic in the stated bracket, on which the cubic is monotone; the matrix is doubly stochastic and nonnegative; its 19-entry support is nearly decomposable, being fully indecomposable while the removal of any single support entry destroys that; the permanent agrees with the paper's closed form to 1e-61; the cofactor gap agrees with its closed form to 4e-61 and exceeds the claimed 2047/240100; and the prescribed zero is not a tie point, with exactly one witnessing support entry. The conjecture's hypothesis also holds: the face is four-dimensional, and a 6561-point grid plus 61 local descents found nothing on it with a smaller permanent. Still a days-old preprint with no independent review, hence a candidate.",
      "sources": [
        {
          "label": "VibeMathed: The Foregger–Sinkhorn Tie-Point Conjecture",
          "url": "https://vibemathed.com/problem/foregger-sinkhorn-tie-point-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "A counterexample to the Foregger-Sinkhorn tie-point conjecture",
          "url": "https://arxiv.org/abs/2608.13025",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "A counterexample to the Foregger-Sinkhorn tie-point conjecture",
      "url": "https://arxiv.org/abs/2608.13025",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Foregger–Sinkhorn Tie-Point Conjecture",
      "url": "https://vibemathed.com/problem/foregger-sinkhorn-tie-point-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:foregger-sinkhorn-tie-point-conjecture:signal:verify_now",
      "problem_id": "vibemathed:foregger-sinkhorn-tie-point-conjecture",
      "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"
}