{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:the-erdos-borwein-constant-is-2-dense",
  "title": "The Erdos-Borwein Constant is 2-Dense",
  "canonical_statement": "Is the Erdos-Borwein Constant 2-Dense?",
  "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:the-erdos-borwein-constant-is-2-dense:event",
      "problem_id": "vibemathed:the-erdos-borwein-constant-is-2-dense",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-09-07T00:00:00.000Z",
      "title": "GitHub repository with Lean proof",
      "summary": "This work proves that the Erdős–Borwein constant's binary expansion contains every finite binary sequence as a consecutive block, each occurring infinitely often. This implies that the Erdős–Borwein constant is 2-dense.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:the-erdos-borwein-constant-is-2-dense:attempt"
      ],
      "method_family_ids": [
        "vibemathed:the-erdos-borwein-constant-is-2-dense:method"
      ],
      "source_assertion_ids": [
        "vibemathed:the-erdos-borwein-constant-is-2-dense:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:the-erdos-borwein-constant-is-2-dense:attempt",
      "problem_id": "vibemathed:the-erdos-borwein-constant-is-2-dense",
      "model": "ChatGPT-6 Astra",
      "provider": "OpenAI",
      "attempted_at": "2026-09-07T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": null,
      "failed_routes": [],
      "artifacts": [
        {
          "label": "PrimeInputs.lean: the two external estimates, stated exactly",
          "url": "https://github.com/CaptainSude/erdos-borwein-disjunctivity/blob/main/lean/ErdosBorwein/PrimeInputs.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Lean verification record and axiom audit",
          "url": "https://github.com/CaptainSude/erdos-borwein-disjunctivity/blob/main/lean/VERIFICATION.md",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "GitHub repository with Lean proof",
          "url": "https://github.com/CaptainSude/erdos-borwein-disjunctivity/releases/tag/v1.0.0",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Erdos-Borwein Constant is 2-Dense",
          "url": "https://vibemathed.com/problem/the-erdos-borwein-constant-is-2-dense",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:the-erdos-borwein-constant-is-2-dense:verification",
      "problem_id": "vibemathed:the-erdos-borwein-constant-is-2-dense",
      "solution_event_id": "vibemathed:the-erdos-borwein-constant-is-2-dense:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The Lean development proves disjunctivity of the binary expansion conditional on two hypotheses supplied as theorem arguments, not as axioms: AGP, the Alford-Granville-Pomerance estimate in the form of Vandehey's Proposition 2.1, and PrimeIntervalSupply, a standard prime number theorem consequence bounding the primes in (L, 2L) below by L/(3 log L). Both are published theorems rather than conjectures, so the mathematics is conditional only on known results; neither is formalized here. Read directly from lean/ErdosBorwein/PrimeInputs.lean on 8 September 2026. The audited endpoints depend on propext, Classical.choice and Quot.sound only. Because the statement is the author's own rather than anchored to a canonical tracker, and because what the kernel certifies is the implication, this takes the statement-unaudited tier. No specialist in analytic number theory has read the argument.",
      "sources": [
        {
          "label": "VibeMathed: The Erdos-Borwein Constant is 2-Dense",
          "url": "https://vibemathed.com/problem/the-erdos-borwein-constant-is-2-dense",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "GitHub repository with Lean proof",
          "url": "https://github.com/CaptainSude/erdos-borwein-disjunctivity/releases/tag/v1.0.0",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "GitHub repository with Lean proof",
      "url": "https://github.com/CaptainSude/erdos-borwein-disjunctivity/releases/tag/v1.0.0",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Erdos-Borwein Constant is 2-Dense",
      "url": "https://vibemathed.com/problem/the-erdos-borwein-constant-is-2-dense",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:the-erdos-borwein-constant-is-2-dense:signal:verify_now",
      "problem_id": "vibemathed:the-erdos-borwein-constant-is-2-dense",
      "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"
}