Over the past month, a conjecture that has stood for more than twenty years in numerical linear algebra suddenly gained two mutually reinforcing proof lines. Both target the same Crouzeix's conjecture, but they take markedly different paths.
The Problem Itself
The conjecture, proposed by Michel Crouzeix in 2004, can be compressed into a single inequality: for any complex matrix \(A\) and complex polynomial \(p\),
Here \(W(A)\) is the numerical range of the matrix, and the constant 2 cannot be reduced further: for a 2×2 Jordan block, \(\|A\|=1\), while the maximum of a linear function over the numerical range is only 1/2. The difficulty lies in the fact that non-normal matrices cannot be easily controlled via eigenvalues and spectral radius alone.
Line One: An AI-Assisted Public Research Record
Shanmu Jin's preprint, *The Numerical Range Is a 2-Spectral Set*, released v4 on August 7. Its abstract directly claims a proof of the conjecture, reducing the core to a mass-parameterized positive real completion construction. The proof contains a very specific sampling step: sampling at scaled conjugate eigenvalues of matrix-valued Herglotz kernels, adding a sample at the origin so that adjoint algebraic correction terms cancel; a weighted Gramian comparison and a limiting argument for general matrices then yield the constant 2.
Rarely, this preprint writes AI's contribution into its data availability statement: ChatGPT proposed the key ideas of scaled sampling and origin sampling, and also assisted with drafting, literature organization, and formatting; the author takes responsibility for the mathematical statements and citations. The preprint also explicitly notes: this version is not peer reviewed.
The corresponding public repository is more cautious. It calls the result a "candidate proof," preserving the LaTeX manuscript, a Lean 4 formalization directory, and axiom audits, and states that several independent computations and adversarial audits have not yet found specific errors. The repository's status still reads that formal peer review remains to be completed.
Line Two: A Shorter, Traditional Proof
On August 4, Emiel Lorist and Felix Schwenninger submitted *A solution to Crouzeix's conjecture* to arXiv, updating to v2 on August 17. The abstract's route is clean: combine tools previously used for weaker estimates with a simple perturbation lemma about 2-dilations, then apply them to iterates \(f^n\) in a double-layer potential representation to obtain the conjectured bound. The paper is 7 pages long, and its methodological emphasis differs from Jin's Herglotz kernel sampling.
The value of this line is not merely "one more proof." It gives reviewers two mutually checkable objects: one relying on matrix-valued Herglotz kernels, positive realness, and formalization tools; the other closing the estimate via potential representations and a perturbation lemma. Cross-validation between the two routes can more quickly expose shared hidden assumptions.
The Review Gate Is Still Lit
The most easily miswritten word right now is "solved." The preprint author's abstract uses explicit *complete proof* language, while the public repository retains the *candidate proof* status; the two are not contradictory—the former is the paper's claim, the latter is careful wording of the release process. There is no information about a completed independent peer review or formal journal acceptance, so the more accurate statement is: Crouzeix's conjecture has entered an auditable proof race, with an AI-assisted proof and an independent proof appearing simultaneously, but the verdict awaits the mathematical community completing its review.
The lesson for AI for Science here may outlast "who won the conjecture." AI can propose counterintuitive yet effective mathematical sampling points, and can also leave behind proof drafts, formalization checks, and audit records together. The difficulty accordingly shifts: who judges that a key lemma has no loopholes, who re-verifies the assumptions in Lean, and who confirms that "passing the audit" has not hidden unmodeled parts beyond the boundary.
Sources and Verifiable Links
1. Jin preprint v4: *The Numerical Range Is a 2-Spectral Set* https://www.preprints.org/manuscript/202607.1919 2. Lorist, Schwenninger: arXiv 2608.03841 v2 https://arxiv.org/abs/2608.03841 3. Jin's public research repository, Lean 4 and audit records https://github.com/jinshanmu/CrouzeixConjecture