{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-1151",
  "title": "Erdős Problem #1151",
  "canonical_statement": "Let $\\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$ on the Chebyshev nodes. Prove that, for any closed $A\\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\\mathcal{L}^nf(x)$.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:erdos-1151:event",
      "problem_id": "vibemathed:erdos-1151",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-04-30T00:00:00.000Z",
      "title": "erdosproblems.com/1151",
      "summary": "An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-1151:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-1151:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-1151:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-1151:attempt",
      "problem_id": "vibemathed:erdos-1151",
      "model": "GPT-5.5 Pro, Codex",
      "provider": "OpenAI",
      "attempted_at": "2026-04-30T00:00:00.000Z",
      "human_collaborators": [
        "Przemysław Chojecki",
        "Allen Hart"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The solution was obtained with GPT-5.5 Pro using an explicit primitive-row decomposition of the Chebyshev-node measures; Theorem 1.1(a), the main contribution, was subsequently formalized largely autonomously by ChatGPT and Codex.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Write-up",
          "url": "https://www.ulam.ai/research/erdos1151.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Lean formalization of Theorem 1.1(a)",
          "url": "https://github.com/AllenGrahamHart/FormalConjectures-Bench/tree/main/formalizations/erdos1151",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/1151",
          "url": "https://www.erdosproblems.com/1151",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #1151",
          "url": "https://vibemathed.com/problem/erdos-1151",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-1151:verification",
      "problem_id": "vibemathed:erdos-1151",
      "solution_event_id": "vibemathed:erdos-1151:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Theorem 1.1(a), the main part of the contribution, is formalized in Lean and the formalization was confirmed correct on the forum; part (b) is unformalized because it depends on an Erdős result absent from mathlib. erdosproblems.com still lists the problem open.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #1151",
          "url": "https://vibemathed.com/problem/erdos-1151",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/1151",
          "url": "https://www.erdosproblems.com/1151",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
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  ],
  "sources": [
    {
      "label": "erdosproblems.com/1151",
      "url": "https://www.erdosproblems.com/1151",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #1151",
      "url": "https://vibemathed.com/problem/erdos-1151",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-1151:signal:verify_now",
      "problem_id": "vibemathed:erdos-1151",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}