{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes",
  "title": "Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes",
  "canonical_statement": "Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the spectral-to-tiling direction for those whose prime factors grow rapidly. Until now no cyclic group with an arbitrary number of distinct divisors was known to satisfy it.",
  "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:fuglede-square-free-cyclic-rapid-primes:event",
      "problem_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "arXiv",
      "summary": "A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:fuglede-square-free-cyclic-rapid-primes:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
        "vibemathed:fuglede-square-free-cyclic-rapid-primes:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:fuglede-square-free-cyclic-rapid-primes:attempt",
      "problem_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes",
      "model": "ChatGPT + Gemini",
      "provider": null,
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [
        "Gábor Somlai"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author writes that the work at this pace would not have been possible without ChatGPT and Gemini, which assisted in verifying the proof, collecting the relevant literature and earlier results, and formatting the text. Verification and search are the bottom tier by our own definition; the mathematics is the author's.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26534",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes",
          "url": "https://vibemathed.com/problem/fuglede-square-free-cyclic-rapid-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:fuglede-square-free-cyclic-rapid-primes:verification",
      "problem_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes",
      "solution_event_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent check. The models are credited with verifying the author's proof rather than producing it. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes",
          "url": "https://vibemathed.com/problem/fuglede-square-free-cyclic-rapid-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26534",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.26534",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes",
      "url": "https://vibemathed.com/problem/fuglede-square-free-cyclic-rapid-primes",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:fuglede-square-free-cyclic-rapid-primes:signal:recent_partial_result",
      "problem_id": "vibemathed:fuglede-square-free-cyclic-rapid-primes",
      "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"
}