The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

For $A\in M_n(\mathbb C)$ with numerical range $W(A)=\{x^*Ax:\|x\|=1\}$ and a matrix polynomial $P(z)=\sum_{j=0}^d B_jz^j$ with $B_j\in M_m(\mathbb C)$ , put $P[A]=\sum_j A^j\otimes B_j$ . Crouzeix conjectured in 2004 that $\|p(A)\|\le2\max_{W(A)}|p|$ for scalar polynomials (the case $m=1$ ), and in 2007 proved the complete bound with constant 11.08; Crouzeix and Palencia lowered the complete constant to $1+\sqrt2$ . The complete conjecture asks for the constant 2 uniformly in $m$ , $n$ and $d$ , that is, that $W(A)$ is a complete 2-spectral set. Does $\|P[A]\|\le2\sup_{z\in W(A)}\|P(z)\|$ hold for all such $A$ and $P$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Matrix analysis; numerical range and spectral sets
Posed by: Michel Crouzeix (scalar form 2004; complete form as named in the later literature)
Year posed: —
Years open: —
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Shown: for every bounded operator $A$ on any complex Hilbert space (no separability assumed) and every matrix polynomial $P$ , $\|P[A]\|\le2\sup_{W(A)}\|P\|$ , with 2 sharp; the same bound holds for finite matrix-valued functions holomorphic near $\overline{W(A)}$ and for rational functions with poles outside it. For a matrix and an admissible convex domain containing $W(A)$ , an optimal similarity with condition number at most 2 makes the conformal image contractive, with one positive boundary density representing all matrix-valued analytic evaluations. It contains the scalar Crouzeix theorem as the case $m=1$ .

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 23, 2026) gives a structural proof via an optimal similarity and a common positive boundary density; a second manuscript (September 26, 2026) gives a separate direct proof of the finite-matrix inequality. The manuscripts cite the scalar proofs of Jin, of Lorist and Schwenninger, and of Luo.

Verification

No independent mathematician has checked this yet. Checked here: the complete inequality in Corollaries 6.1 and 6.3 of the principal manuscript and Theorem 1.1 of the direct companion were read against the complete conjecture. lean/formalization.yaml lists main results for both manuscripts (CompleteCrouzeix, DirectCrouzeix, StructuralCrouzeix). The CompleteCrouzeix comparator was read here: for all $n,m\ge1$ , $d$ , $A\in M_n$ and $B_j\in M_m$ , $\|\sum_k A^k\otimes B_k\|\le2\sup_{W(A)}\|\sum_k z^kB_k\|$ , and 2 is the least such universal constant. This states the headline claim for matrices. The operator version, challenge CrouzeixHilbert, is not in the formalization catalogue; it is linked from lean/docs/325.md, its solution module exists at the pinned commit, and its statement (arbitrary complex Hilbert space, plus holomorphic and rational extensions and sharpness) was read here. Not rebuilt here.

The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

What was actually shown

What the AI did

Verification

Sources

Discussion