{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:sum-product-conjecture-reals",
  "title": "The Sum-Product Conjecture over the Reals",
  "canonical_statement": "Erdos and Szemeredi conjectured that every finite set of reals satisfies $\\max(|A+A|,|AA|) \\ge |A|^{2-o(1)}$. False: there are arbitrarily large $A \\subset \\mathbb{R}$, of algebraic integers in a number field of degree $\\asymp \\log|A|$, with $\\max(|A+A|,|AA|) \\le |A|^{2-c}$ for an absolute $c > 0$. Variants give counterexamples in function fields of fixed positive characteristic.",
  "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:sum-product-conjecture-reals:event",
      "problem_id": "vibemathed:sum-product-conjecture-reals",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-05-27T00:00:00.000Z",
      "title": "arXiv:2605.28781 - The sum-product conjecture is false for real numbers",
      "summary": "the model's contribution is one simplifying lemma; the authors state the main ideas are human",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:sum-product-conjecture-reals:attempt"
      ],
      "method_family_ids": [
        "vibemathed:sum-product-conjecture-reals:method"
      ],
      "source_assertion_ids": [
        "vibemathed:sum-product-conjecture-reals:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:sum-product-conjecture-reals:attempt",
      "problem_id": "vibemathed:sum-product-conjecture-reals",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-05-27T00:00:00.000Z",
      "human_collaborators": [
        "Thomas F. Bloom",
        "Will Sawin",
        "Carl Schildkraut",
        "Dmitrii Zakharov"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The limits here matter more than the headline, and the authors state them plainly: GPT-5.5 Pro was a sounding board in the early stages, but the final proof including all the main ideas was almost entirely human-generated, and everything in the paper was written by the authors. The single exception they name is Lemma 3.4, suggested by the model, which replaced a more complicated result of Schinzel with a short elementary argument. There is a second, indirect AI thread: the authors say they were inspired to revisit number fields of large degree by OpenAI's counterexample to the unit distance conjecture, and note their construction needed far less number-theoretic input than that one did.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.28781 - The sum-product conjecture is false for real numbers",
          "url": "https://arxiv.org/abs/2605.28781",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Sum-Product Conjecture over the Reals",
          "url": "https://vibemathed.com/problem/sum-product-conjecture-reals",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:sum-product-conjecture-reals:verification",
      "problem_id": "vibemathed:sum-product-conjecture-reals",
      "solution_event_id": "vibemathed:sum-product-conjecture-reals:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint by four established additive combinatorialists; not yet peer-reviewed. Given the size of the claim this one deserves refereeing before it is treated as settled.",
      "sources": [
        {
          "label": "VibeMathed: The Sum-Product Conjecture over the Reals",
          "url": "https://vibemathed.com/problem/sum-product-conjecture-reals",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.28781 - The sum-product conjecture is false for real numbers",
          "url": "https://arxiv.org/abs/2605.28781",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.28781 - The sum-product conjecture is false for real numbers",
      "url": "https://arxiv.org/abs/2605.28781",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Sum-Product Conjecture over the Reals",
      "url": "https://vibemathed.com/problem/sum-product-conjecture-reals",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:sum-product-conjecture-reals:signal:watch_only",
      "problem_id": "vibemathed:sum-product-conjecture-reals",
      "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"
}