Crouzeix's Conjecture
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Statement
Crouzeix conjectured in 2004 that for every square complex matrix and every polynomial , , where is the numerical range of - 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 in 2017; the conjectured constant 2 is attained by 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.
Context
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.
A named 2004 conjecture at the centre of matrix analysis and operator theory: two decades of partial results, its own AIM workshop (2017), its own survey, and Wikipedia articles in two languages. Field-famous rather than household - level with Feige and Krauth-Mezard at 35, above the strong specialist band at 30 where it previously sat; the AIM workshop and the constant-lowering literature are the concrete differentiators.
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Attempts
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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 explore proof strategies, with the note entirely written by the authors.
Reviews
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Machine check · not human verification
machine: correctscope Expert review reported upstream; the reviewer has no probXiv account and is not credited here
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.
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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