Crouzeix's Conjecture

Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$ , $\lVert p(A)\rVert \leq 2 \max_{z \in W(A)} |p(z)|$ , where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Matrix analysis
Posed by: Michel Crouzeix
Year posed: 2004
Years open: 22y
Solved: 2026-07-27
Model: GPT-5.6 Sol; ChatGPT 5.6 Pro
Vendor: OpenAI
Collaborators: Shanmu Jin, Emiel Lorist, Felix L. Schwenninger
Verification: Independently expert-verified
Publication: Preprint
Significance: 35 / 100
Disclosed cost: —
Wikipedia: 2 languages

What was actually shown

Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown. The complete (matrix-coefficient) form, which contains this scalar theorem as the case m = 1, is claimed in the OpenAI math release (September 2026); see complete-crouzeix-conjecture.

What the AI did

For the first proof: Jin, a neurosurgery resident with no specialized mathematical training, reports that the key result (Theorem 2) emerged during an approximately sixteen-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode - a public prompt adapted from the Cycle Double Cover run, web access denied, a branching portfolio of subagent strategies under adversarial audit, and no human intervention once started. Jin then simplified and verified the output; the repository publishes the prompt, successive manuscripts, a Lean formalization and an axiom audit. For the independent second proof, Lorist and Schwenninger disclose that ChatGPT 5.6 Pro was used to review previous approaches to the weaker spectral constant $1 + \sqrt{2}$ and to identify a possible source of improvement in estimates involving iterates $f^n$ of extremal or approximately extremal functions.

Verification

Independently expert-verified, publicly on record: Townsend and Greenbaum's essay of 14 August 2026 states that both authors and Michel Crouzeix himself "have checked the proof thoroughly and believe that Dr. Jin's manuscript is correct" - the conjecture's own poser among the verifiers, and Greenbaum co-organized the 2017 AIM workshop on the problem. This site read that essay in full and audited Jin's repository (commit 9df0783): 82 Lean files with zero sorry, zero axiom declarations and zero native_decide with comments stripped, on toolchain v4.28.0, alongside an Annals-formatted manuscript and the complete autonomous-run prompt - though the Lean was not compiled here and its statement-to-conjecture correspondence not audited, so the tier rests on the expert endorsement, not the formalization. The independent second proof by Lorist and Schwenninger (arXiv:2608.03841) has no comparable public endorsement yet and the essay stops short of vouching for it. Neither manuscript is refereed.

Sources

Related entries

Built on byLorist-Schwenninger Remark 2

Submitted by NimbleRaven553 on 07 Aug 2026