{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-326",
  "title": "Erdős Problem #326",
  "canonical_statement": "Does there exist $A=\\{a_1<a_2<\\cdots\\}\\subset \\mathbb{N}$ which is a minimal basis of order $2$ (every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property) such that $\\lim_{k\\to \\infty}a_k/k^2=c$ for some $c\\neq 0$? A claimed construction gives a minimal basis with $A(x)=C\\sqrt{x}+O(1)$, answering the question affirmatively; Erdős and Graham had conjectured a negative answer.",
  "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-326:event",
      "problem_id": "vibemathed:erdos-326",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-06-14T00:00:00.000Z",
      "title": "erdosproblems.com/326",
      "summary": "Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:erdos-326:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-326:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-326:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-326:attempt",
      "problem_id": "vibemathed:erdos-326",
      "model": "GPT-5.5, Aristotle, Codex",
      "provider": "OpenAI, Harmonic",
      "attempted_at": "2026-06-14T00:00:00.000Z",
      "human_collaborators": [
        "Aron Bhalla"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Per the author's disclosure, most of the mathematics is his own, with GPT-5.5 used to stress-test ideas, suggest revisions, identify gaps and write up some proofs; the solution was then formalized over several weeks with Aristotle, Codex and GPT-5.5 into a roughly 15,000-line Lean proof confirming all claims in the manuscript.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Manuscript (latest version)",
          "url": "https://drive.google.com/file/d/1VKaFmiMWWMW7NME-L47HVGSWOoBt9sug/view?usp=sharing",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/326",
          "url": "https://www.erdosproblems.com/326",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #326",
          "url": "https://vibemathed.com/problem/erdos-326",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-326:verification",
      "problem_id": "vibemathed:erdos-326",
      "solution_event_id": "vibemathed:erdos-326:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The author reports a ~15,000-line Lean formalization, type-checkable online, confirming all claims of the manuscript. It has not been independently audited for statement fidelity, and erdosproblems.com has not accepted the claim: the site's owner found the AI-written exposition hard to digest while stressing that this was not a correctness objection.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #326",
          "url": "https://vibemathed.com/problem/erdos-326",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/326",
          "url": "https://www.erdosproblems.com/326",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/326",
      "url": "https://www.erdosproblems.com/326",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #326",
      "url": "https://vibemathed.com/problem/erdos-326",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-326:signal:verify_now",
      "problem_id": "vibemathed:erdos-326",
      "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"
}