{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:pure-o-sequences-log-concavity",
  "title": "Log-Concavity of Codimension-Three Pure O-Sequences",
  "canonical_statement": "For a pure O-sequence $h = (h_0, \\dots, h_e)$ of codimension three and type two, is $h_i^2 \\ge h_{i-1} h_{i+1}$ for every interior index $i$? The stated monomial case is proved; the broader level-Hilbert-function case remains open.",
  "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:pure-o-sequences-log-concavity:event",
      "problem_id": "vibemathed:pure-o-sequences-log-concavity",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-21T00:00:00.000Z",
      "title": "arXiv:2605.22763 - AlphaProof Nexus report",
      "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_discovered",
      "attempt_ids": [
        "vibemathed:pure-o-sequences-log-concavity:attempt"
      ],
      "method_family_ids": [
        "vibemathed:pure-o-sequences-log-concavity:method"
      ],
      "source_assertion_ids": [
        "vibemathed:pure-o-sequences-log-concavity:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:pure-o-sequences-log-concavity:attempt",
      "problem_id": "vibemathed:pure-o-sequences-log-concavity",
      "model": "AlphaProof Nexus",
      "provider": "Google DeepMind",
      "attempted_at": "2026-05-21T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Solved autonomously by AlphaProof Nexus, with the proof formally verified in Lean.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.22763 - AlphaProof Nexus report",
          "url": "https://arxiv.org/abs/2605.22763",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Log-Concavity of Codimension-Three Pure O-Sequences",
          "url": "https://vibemathed.com/problem/pure-o-sequences-log-concavity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:pure-o-sequences-log-concavity:verification",
      "problem_id": "vibemathed:pure-o-sequences-log-concavity",
      "solution_event_id": "vibemathed:pure-o-sequences-log-concavity:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763) and its accompanying repository.",
      "sources": [
        {
          "label": "VibeMathed: Log-Concavity of Codimension-Three Pure O-Sequences",
          "url": "https://vibemathed.com/problem/pure-o-sequences-log-concavity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.22763 - AlphaProof Nexus report",
          "url": "https://arxiv.org/abs/2605.22763",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2605.22763 - AlphaProof Nexus report",
      "url": "https://arxiv.org/abs/2605.22763",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Log-Concavity of Codimension-Three Pure O-Sequences",
      "url": "https://vibemathed.com/problem/pure-o-sequences-log-concavity",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:pure-o-sequences-log-concavity:signal:replay_ready",
      "problem_id": "vibemathed:pure-o-sequences-log-concavity",
      "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"
}