{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63",
  "title": "Borsuk Conjecture lowest-ever counterexample (N=63)",
  "canonical_statement": "Borsuk's conjecture asked whether every bounded set in $\\mathbb{R}^n$ can be partitioned into $n+1$ subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in $\\mathbb{R}^{63}$ whose smaller-diameter subsets have at most 5 points, so at least $\\lceil 321/5\\rceil = 65 > 64$ parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for $4 \\le n \\le 62$. The construction modifies Bondarenko's $G_2(4)$ two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:event",
      "problem_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63",
      "type": "disproved",
      "status": "partial",
      "occurred_at": "2026-05-26T00:00:00.000Z",
      "title": "Max Grinsztajn's proof note and certificates",
      "summary": "Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:attempt"
      ],
      "method_family_ids": [
        "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:method"
      ],
      "source_assertion_ids": [
        "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:attempt",
      "problem_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-05-26T00:00:00.000Z",
      "human_collaborators": [
        "Max Grinsztajn"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "For the first solve, Grinsztajn's README states: \"The construction and proof were obtained with assistance from GPT-5.5 Pro\", with a dedicated \"Disclose GPT assistance\" commit; no finer division of labour is given, so the tier is the floor for an unspecific disclosure. The independent August 2026 rediscovery by Nicholas Konz with Claude (Fable 5 and Opus 5) carries a much fuller disclosure - Claude produced the counterexample and an exact certificate over $\\mathbb{Q}(\\sqrt{222})$ - and would rate ai-discovered on its own, but the entry's tier follows the solve it records, which is the first one.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Tao's optimization-problems ledger, constant 28a - credits the 63 bound to Grinsztajn",
          "url": "https://teorth.github.io/optimizationproblems/constants/28a.html",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Independent rediscovery by Konz + Claude, August 2026: write-up, coordinates and verifier",
          "url": "https://nickk124.github.io/borsuk/",
          "kind": "independent",
          "license": null
        },
        {
          "label": "Wikipedia: Borsuk's conjecture",
          "url": "https://en.wikipedia.org/wiki/Borsuk%27s_conjecture",
          "kind": "wikipedia",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Max Grinsztajn's proof note and certificates",
          "url": "https://github.com/maaxgrin/borsuk-63-counterexample",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Borsuk Conjecture lowest-ever counterexample (N=63)",
          "url": "https://vibemathed.com/problem/borsuk-conjecture-lowest-ever-counterexample-n-63",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:verification",
      "problem_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63",
      "solution_event_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Both derivations reproduced by this site on 12 August 2026, independently of each other. For the first solve (Grinsztajn, May 2026): the repository's exact verifier - pure Python integer arithmetic over F16, read before running - was executed locally and passes all checks: it rebuilds the G2(4) strongly regular graph with parameters (416,100,36,20), the B1/B2/B3/C partition and degree data behind the dimension drop, and the clique obstructions forcing every smaller-diameter subset to size at most 5. The repo's GitHub creation date of 2026-05-26 is not forgeable after the fact, and Terence Tao's optimization-problems ledger (constant 28a) independently credits the 63 bound to Grinsztajn, citing this repository. For the August rediscovery (Konz + Claude): we ran the author's stand-alone verifier against the published 321x63 coordinate file and confirmed affine dimension exactly 63, the squared-distance spectrum (53-sqrt(222))/156, 1/4 and 1/3, and independence number 5 for the diameter graph by Bron-Kerbosch, forcing ceil(321/5) = 65 parts where Borsuk allows 64; the distance-class gap is far wider than any float tolerance. Neither write-up is peer-reviewed; neither is on arXiv.",
      "sources": [
        {
          "label": "VibeMathed: Borsuk Conjecture lowest-ever counterexample (N=63)",
          "url": "https://vibemathed.com/problem/borsuk-conjecture-lowest-ever-counterexample-n-63",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Max Grinsztajn's proof note and certificates",
          "url": "https://github.com/maaxgrin/borsuk-63-counterexample",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Max Grinsztajn's proof note and certificates",
      "url": "https://github.com/maaxgrin/borsuk-63-counterexample",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Borsuk Conjecture lowest-ever counterexample (N=63)",
      "url": "https://vibemathed.com/problem/borsuk-conjecture-lowest-ever-counterexample-n-63",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63:signal:recent_partial_result",
      "problem_id": "vibemathed:borsuk-conjecture-lowest-ever-counterexample-n-63",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}