Background: The Crouzeix Conjecture
Proposed by Michel Crouzeix in 2004, the conjecture can be compressed into a single inequality: for any complex matrix A and complex polynomial p,
W(A) is the numerical range of A. The constant 2 is sharp: for a 2-by-2 Jordan block with \|A\|=1, the supremum of any linear function over W(A) is only 1/2. The difficulty is that non-normal matrices cannot be controlled cleanly by eigenvalues or spectral radius.
Line 1: AI-Assisted Preprint with Public Research Artifacts
Shanmu Jin's preprint "The Numerical Range Is a 2-Spectral Set" was posted as v4 on August 7 on Preprints.org. The abstract asserts a complete proof and reduces the argument to a positive-real completion construction with a mass parameterization. A specialized sampling step is central: drawing samples at the scaled conjugate eigenvalues of a matrix-valued Herglotz kernel together with an origin sample, so that the adjoint algebraic correction terms cancel; a weighted Gramian comparison and a limiting argument over general matrices then yield the constant 2.
The preprint is unusual in documenting AI contributions inside its data-availability statement: ChatGPT is credited with proposing the scaled and origin sampling ideas and assisting with wording, reference整理, and typesetting, while the author takes responsibility for mathematical statements and citations. The preprint also explicitly flags that the current version has not been peer-reviewed.
The associated GitHub repository is more cautious. It labels the work a "candidate proof," preserving the LaTeX manuscript, a Lean 4 formalization index, and an axiom audit, and reports that several independent computations and adversarial audits have not yet surfaced concrete errors. The repository status still states that formal peer review is pending.
Line 2: A Shorter Traditional Proof
On August 4, Emiel Lorist and Felix Schwenninger posted "A solution to Crouzeix's conjecture" on arXiv; v2 followed on August 17. The abstract sketches a clean route: combine tools previously used for weaker estimates with a simple perturbation lemma for 2-dilations, then apply them to iterated f^n in a two-layer potential representation to obtain the conjectured bound. The paper is 7 pages, and the methodological emphasis differs from Jin's Herglotz-kernel sampling.
The value of this line goes beyond having a second proof. It gives reviewers two cross-checkable objects: one leaning on matrix-valued Herglotz kernels, positive realness, and formalization, the other collapsing estimates via potential representations and a perturbation lemma. Cross-validation between the two routes should more quickly expose any shared hidden assumptions.
The Peer-Review Gate
The word most easily miswritten here is "solved." The preprint author uses the unambiguous phrase "complete proof" in the abstract, while the public repository retains the candidate-proof label; the difference is not contradictory, since one is a paper claim and the other is a process-status description. No completed independent peer review or formal journal acceptance has been found, so the more accurate framing is: the Crouzeix conjecture has entered an auditable proof race, with an AI-assisted proof and an independent proof appearing in parallel, but the verdict still awaits the mathematics community's peer review.
The takeaway for AI for Science may outlast the conjecture itself. AI can propose unconventional but effective mathematical sampling points, and it can leave behind proof drafts, formalization checks, and audit logs together. The hard part shifts accordingly: who verifies that the key lemmas have no gaps, who re-checks the hypotheses inside Lean, and who confirms that "passes audit" has not hidden unmodeled pieces near 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 and Schwenninger: arXiv 2608.03841 v2 — https://arxiv.org/abs/2608.03841 3. Jin's public research repository, Lean 4, and audit log — https://github.com/jinshanmu/CrouzeixConjecture