{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-662",
  "title": "Erdős Problem #662",
  "canonical_statement": "Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.",
  "plain_summary": "VibeMathed reports this item as open. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "open",
  "solution_events": [
    {
      "id": "vibemathed:erdos-662:event",
      "problem_id": "vibemathed:erdos-662",
      "type": "disproved",
      "status": "open",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "erdosproblems.com/662",
      "summary": "natural readings of the ambiguous historical statement are disproved",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-662:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-662:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-662:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-662:attempt",
      "problem_id": "vibemathed:erdos-662",
      "model": "GPT-5.6 starships (Claude Fable 5 reviewer)",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-13T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "erdosproblems.com/662",
          "url": "https://www.erdosproblems.com/662",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #662",
          "url": "https://vibemathed.com/problem/erdos-662",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-662:verification",
      "problem_id": "vibemathed:erdos-662",
      "solution_event_id": "vibemathed:erdos-662:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Finite certificates and variant statements Lean-checked; community status pending.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #662",
          "url": "https://vibemathed.com/problem/erdos-662",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/662",
          "url": "https://www.erdosproblems.com/662",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/662",
      "url": "https://www.erdosproblems.com/662",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #662",
      "url": "https://vibemathed.com/problem/erdos-662",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-662:signal:watch_only",
      "problem_id": "vibemathed:erdos-662",
      "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"
}