{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:cohn-elkies-bound-dimension-36",
  "title": "Tightness of the Cohn-Elkies Bound in Dimension 36",
  "canonical_statement": "Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension $36$ as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-$18$ modular forms for $\\Gamma_0(24)$, shows the two-point linear programming bound in dimension $36$ exceeds the density of the Kschischang-Pasupathy packing by a factor of at least $32.91$.",
  "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:cohn-elkies-bound-dimension-36:event",
      "problem_id": "vibemathed:cohn-elkies-bound-dimension-36",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36",
      "summary": "rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:cohn-elkies-bound-dimension-36:attempt"
      ],
      "method_family_ids": [
        "vibemathed:cohn-elkies-bound-dimension-36:method"
      ],
      "source_assertion_ids": [
        "vibemathed:cohn-elkies-bound-dimension-36:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:cohn-elkies-bound-dimension-36:attempt",
      "problem_id": "vibemathed:cohn-elkies-bound-dimension-36",
      "model": "Claude Fable 5, Claude Opus 4.8, Codex (GPT-5.6)",
      "provider": "Anthropic / OpenAI",
      "attempted_at": "2026-07-13T00:00:00.000Z",
      "human_collaborators": [
        "Rifat Jumagulov"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The disclosure reports substantial assistance: the Claude models were used for the construction of the certificate itself, for the verification tooling and for drafting, and Codex was used as an independent cross-check of the certificate computations.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36",
          "url": "https://arxiv.org/abs/2607.11319",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Tightness of the Cohn-Elkies Bound in Dimension 36",
          "url": "https://vibemathed.com/problem/cohn-elkies-bound-dimension-36",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:cohn-elkies-bound-dimension-36:verification",
      "problem_id": "vibemathed:cohn-elkies-bound-dimension-36",
      "solution_event_id": "vibemathed:cohn-elkies-bound-dimension-36:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The result is an explicit dual-feasible certificate, so it is checkable in principle by evaluating the constructed function; the paper reports an independent cross-check of the computations by a second model. We have not reproduced it. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Tightness of the Cohn-Elkies Bound in Dimension 36",
          "url": "https://vibemathed.com/problem/cohn-elkies-bound-dimension-36",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36",
          "url": "https://arxiv.org/abs/2607.11319",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36",
      "url": "https://arxiv.org/abs/2607.11319",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Tightness of the Cohn-Elkies Bound in Dimension 36",
      "url": "https://vibemathed.com/problem/cohn-elkies-bound-dimension-36",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:cohn-elkies-bound-dimension-36:signal:watch_only",
      "problem_id": "vibemathed:cohn-elkies-bound-dimension-36",
      "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"
}