{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:three-block-admm-identity-block-counterexample",
  "title": "Convergence of Three-Block ADMM with Identity Third Block",
  "canonical_statement": "After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of period 66, verified by exact checks along a piecewise-affine reduction path.",
  "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:three-block-admm-identity-block-counterexample:event",
      "problem_id": "vibemathed:three-block-admm-identity-block-counterexample",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM",
      "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_co_developed",
      "attempt_ids": [
        "vibemathed:three-block-admm-identity-block-counterexample:attempt"
      ],
      "method_family_ids": [
        "vibemathed:three-block-admm-identity-block-counterexample:method"
      ],
      "source_assertion_ids": [
        "vibemathed:three-block-admm-identity-block-counterexample:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:three-block-admm-identity-block-counterexample:attempt",
      "problem_id": "vibemathed:three-block-admm-identity-block-counterexample",
      "model": "GPT-5.6 Sol (Codex)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [
        "Kenan Xu",
        "Xiangfeng Wang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper is framed as AI-assisted discovery in its own title: using Codex with GPT-5.6 Sol, the authors construct the explicit rational counterexample candidate and verify it along a piecewise-affine reduction path, with exact checks establishing the period-66 nonconvergent orbit; the same Codex workflow guides a further study.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM",
          "url": "https://arxiv.org/abs/2608.14396",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Convergence of Three-Block ADMM with Identity Third Block",
          "url": "https://vibemathed.com/problem/three-block-admm-identity-block-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:three-block-admm-identity-block-counterexample:verification",
      "problem_id": "vibemathed:three-block-admm-identity-block-counterexample",
      "solution_event_id": "vibemathed:three-block-admm-identity-block-counterexample:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14396): the open status of the identity-third-block subclass is stated in the abstract and introduction, and the construction-and-verification account is as quoted. The exact checks were not re-run here. Days-old preprint, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: Convergence of Three-Block ADMM with Identity Third Block",
          "url": "https://vibemathed.com/problem/three-block-admm-identity-block-counterexample",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM",
          "url": "https://arxiv.org/abs/2608.14396",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM",
      "url": "https://arxiv.org/abs/2608.14396",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Convergence of Three-Block ADMM with Identity Third Block",
      "url": "https://vibemathed.com/problem/three-block-admm-identity-block-counterexample",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:three-block-admm-identity-block-counterexample:signal:watch_only",
      "problem_id": "vibemathed:three-block-admm-identity-block-counterexample",
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      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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"
}