Convergence of Three-Block ADMM with Identity Third Block

Precise 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.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6 Sol (Codex)

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.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

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.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-14

   AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• 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.