{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-106",
  "title": "Erdős Problem #106",
  "canonical_statement": "If $f(n)$ is the maximum total side length of $n$ interior-disjoint squares packed in the unit square, is $f(k^2 + 1) = k$? An exact rational configuration packs $17$ squares with total side length greater than $4$, refuting the identity at $k = 4$.",
  "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:erdos-106:event",
      "problem_id": "vibemathed:erdos-106",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "erdosproblems.com/106",
      "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:erdos-106:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-106:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-106:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-106:attempt",
      "problem_id": "vibemathed:erdos-106",
      "model": "Claude Opus 5",
      "provider": "Anthropic",
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The public Lean source credits Codex as formal author; The packing was discovered by Claude Opus 5 (Anthropic) against search infrastructure built and run by the submitter, and verified independently in exact rational arithmetic via separating-axis certificates.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Construction and verification code (GitHub)",
          "url": "https://github.com/Sprite143/erdos-106-counterexample",
          "kind": "code",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/106",
          "url": "https://www.erdosproblems.com/106",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #106",
          "url": "https://vibemathed.com/problem/erdos-106",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-106:verification",
      "problem_id": "vibemathed:erdos-106",
      "solution_event_id": "vibemathed:erdos-106:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A 613-line, placeholder-free Lean proof of the counterexample; the erdosproblems.com page has now the result and is showing the conjecture as disproved.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #106",
          "url": "https://vibemathed.com/problem/erdos-106",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/106",
          "url": "https://www.erdosproblems.com/106",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/106",
      "url": "https://www.erdosproblems.com/106",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #106",
      "url": "https://vibemathed.com/problem/erdos-106",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-106:signal:replay_ready",
      "problem_id": "vibemathed:erdos-106",
      "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"
}