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A direct proof of the complete Crouzeix inequality
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Abstract
We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Nic Berkopec, Crouzeix: Folie à Deux (Washington University in St. Louis analysis seminar, 2026-10-19). The seminar abstract notes that the earlier scalar proof methods do not extend directly to the completely bounded case and explicitly associates that case with the OpenAI release. [family-level]
- trostaft on Hacker News, comment in the thread Sharing AI progress in mathematics (Hacker News, 2026-10-07). Brief comment naming family 325 among the release items of interest; the commenter says they have attended talks on the Crouzeix conjecture and suggests a connection to convergence rates for Krylov methods. [family-level]
Formalization
- OpenAI / mathcommunity mirror, Lean scope notes for family 325: The complete Crouzeix conjecture (lean/docs/325.md, release of 2026-10-06). The release's own scope document, listing five comparator statements; this paper's statement (complete inequality and sharpness) corresponds to the challenge
DirectCrouzeix.lean, with further challenges for the Hilbert-space and structural versions. [family-level] - Beltran12138, oai-math-recheck (GitHub repository, October 2026). Independent re-run of the release's Lean files for family 325: reports compilation, an axiom audit (only
propext,Classical.choice,Quot.sound), a statement-equivalence check againstDirectCrouzeix, and a negative control; the README stresses that faithfulness of the formal statement to the paper remains a human judgement. [family-level]
Related prior literature mentioned by others
- Emiel Lorist and Felix Schwenninger, A solution to Crouzeix’s conjecture (arXiv:2608.03841v2, 2026-08-17). Xie’s survey, §9.4.6, identifies this scalar predecessor and its explicit limitation that the commutativity argument does not directly survive matrix amplification. [family-level]
Possibly relevant literature (found by the agent, not checked)
- R. Clouâtre, M. Ostermann, T. Ransford, An abstract approach to the Crouzeix conjecture (arXiv:2011.10422, 2020/2022). Formulates the abstract (COR) conjecture and discusses a completely bounded version via dilation theory; relevant to the paper's uniformity-in-coefficient-dimension claim (Theorem 1.1) and not in its bibliography, although the cited Åhag–Czyż–Virtanen paper treats this conjecture.
- C. Badea, R. O'Loughlin, J. Virtanen, On the abstract approach to spectral constants: a proof of the Clouâtre–Ostermann–Ransford conjecture (arXiv:2609.03637, 2026-09-03). Proves the scalar-norm COR conjecture using positivity of real parts, an operator-valued Herglotz theorem and the Lorist–Schwenninger perturbation lemma; comparable to the positivity of $R+R^*$ used in the paper's block comparison (Section 3).
- K. R. Davidson, V. I. Paulsen, H. J. Woerdeman, Complete spectral sets and numerical range (arXiv:1612.05683, 2016-12-17). Shows that a complete $C$-spectral set is a complete $(C+C^{-1})/2$-numerical radius set; with constant 2 this gives a numerical-radius consequence of the paper's main theorem.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, 2026-10-12). Survey of 38 release families whose Section 9.4.6 compares both family-325 manuscripts, describing this paper's "ordered density comparison" (top-singular-pair compression of the Crouzeix–Palencia kernel with Hilbert–Schmidt/Faber-coefficient bounds) as "combination leaning new" and noting that its novelty comparison with Luo 2026 is incomplete. [AI-generated per source] [family-level]
- Latent Seminar, The complete Crouzeix inequality (2026-10-06). This exposition explains both the direct Faber-coefficient argument and the structural metric-and-density argument, including their different scopes, and explicitly disclaims independent verification of the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 325
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026,
author = {{OpenAI}},
title = {{A direct proof of the complete Crouzeix inequality}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf}{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/325.md. The Lean statements have not been checked by people against the paper's claims.
Family 325: The complete Crouzeix conjecture
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has $\lVert P[A]\rVert\le2\sup_{z\in W(A)}\lVert P(z)\rVert$, where $W(A)$ is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.
Other papers in this family: