{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample",
  "title": "Kusner's Conjecture on Equilateral Sets in $\\ell_p^n$",
  "canonical_statement": "Kusner conjectured in 1983 that the maximum number of points in $\\mathbb{R}^n$ that are pairwise at $\\ell_p$-distance one is exactly $n+1$ for every $2 < p < \\infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample:event",
      "problem_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "A counterexample to Kusner's conjecture on equilateral sets",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:kusner-conjecture-equilateral-sets-counterexample:attempt"
      ],
      "method_family_ids": [
        "vibemathed:kusner-conjecture-equilateral-sets-counterexample:method"
      ],
      "source_assertion_ids": [
        "vibemathed:kusner-conjecture-equilateral-sets-counterexample:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample:attempt",
      "problem_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample",
      "model": "GPT-5.6 Sol, Claude Fable 5",
      "provider": "OpenAI, Anthropic",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Logan R. Chalmers"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's disclosure: GPT-5.6 Sol assisted in implementing the computational search strategy in code, and Claude Fable 5 was used as a tool in drafting. No mathematical step is attributed to a model by name, but the search that produced the configuration is the load-bearing computation.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Zenodo data and verification package",
          "url": "https://doi.org/10.5281/zenodo.21911503",
          "kind": "code",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "A counterexample to Kusner's conjecture on equilateral sets",
          "url": "https://arxiv.org/abs/2608.14013",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Kusner's Conjecture on Equilateral Sets in $\\ell_p^n$",
          "url": "https://vibemathed.com/problem/kusner-conjecture-equilateral-sets-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample:verification",
      "problem_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample",
      "solution_event_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14013): the disclosure is verbatim as quoted, Kusner's 1983 attribution is in the introduction, and the data is deposited on Zenodo (10.5281/zenodo.21911503). The exact-arithmetic certificate was not re-run here. Days-old preprint, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Kusner's Conjecture on Equilateral Sets in $\\ell_p^n$",
          "url": "https://vibemathed.com/problem/kusner-conjecture-equilateral-sets-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "A counterexample to Kusner's conjecture on equilateral sets",
          "url": "https://arxiv.org/abs/2608.14013",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "A counterexample to Kusner's conjecture on equilateral sets",
      "url": "https://arxiv.org/abs/2608.14013",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Kusner's Conjecture on Equilateral Sets in $\\ell_p^n$",
      "url": "https://vibemathed.com/problem/kusner-conjecture-equilateral-sets-counterexample",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample:signal:watch_only",
      "problem_id": "vibemathed:kusner-conjecture-equilateral-sets-counterexample",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "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"
}