{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:sarkozy-sums-products-mod-p",
  "title": "Sárközy's Conjecture on Sums and Products Modulo a Prime",
  "canonical_statement": "For $A \\subseteq \\mathbb{F}_p$ let $A^* = (A+A) \\cup (AA)$. Sárközy conjectured that for all large primes, every set of size at least $c\\sqrt{p}$ has $A^* = \\mathbb{F}_p$-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.",
  "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:sarkozy-sums-products-mod-p:event",
      "problem_id": "vibemathed:sarkozy-sums-products-mod-p",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-03-31T00:00:00.000Z",
      "title": "arXiv",
      "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:sarkozy-sums-products-mod-p:attempt"
      ],
      "method_family_ids": [
        "vibemathed:sarkozy-sums-products-mod-p:method"
      ],
      "source_assertion_ids": [
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      ]
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  "attempts": [
    {
      "id": "vibemathed:sarkozy-sums-products-mod-p:attempt",
      "problem_id": "vibemathed:sarkozy-sums-products-mod-p",
      "model": "Harmonic Aristotle",
      "provider": "Harmonic",
      "attempted_at": "2026-03-31T00:00:00.000Z",
      "human_collaborators": [
        "Quanyu Tang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Aristotle produced formal Lean proofs of all four principal statements of the paper (the formalization is public), and \"was also used to assist in the preparation of this paper.\" The counterexample construction itself builds on a classical projective-geometric orbit.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2603.29992",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Sárközy's Conjecture on Sums and Products Modulo a Prime",
          "url": "https://vibemathed.com/problem/sarkozy-sums-products-mod-p",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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  "verifications": [
    {
      "id": "vibemathed:sarkozy-sums-products-mod-p:verification",
      "problem_id": "vibemathed:sarkozy-sums-products-mod-p",
      "solution_event_id": "vibemathed:sarkozy-sums-products-mod-p:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "All four principal statements formalized and checked in Lean; Wouter van Doorn assisted with the formalization. No independent expert review yet. Tier: the formalization was produced within the project (Aristotle, with van Doorn assisting); no independent statement audit.",
      "sources": [
        {
          "label": "VibeMathed: Sárközy's Conjecture on Sums and Products Modulo a Prime",
          "url": "https://vibemathed.com/problem/sarkozy-sums-products-mod-p",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2603.29992",
          "kind": "primary-mathematical-source",
          "license": null
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  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2603.29992",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Sárközy's Conjecture on Sums and Products Modulo a Prime",
      "url": "https://vibemathed.com/problem/sarkozy-sums-products-mod-p",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:sarkozy-sums-products-mod-p:signal:replay_ready",
      "problem_id": "vibemathed:sarkozy-sums-products-mod-p",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}