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The complete Crouzeix theorem: optimal similarity and a common positive boundary representation
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Abstract
We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with condition number at most two. For the similar matrix, one continuous positive boundary density of mass the identity represents the evaluation of every matrix-valued function holomorphic near the closed domain.
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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; it lists comparator challenges for the complete polynomial inequality (
CompleteCrouzeix.lean), the optimal similarity and boundary representation (StructuralCrouzeix.lean), the Hilbert-space version (CrouzeixHilbert.lean) and supporting numerical-range geometry (HilbertCrouzeix.lean), together with holomorphic and rational extensions. [family-level] - Beltran12138, oai-math-recheck (GitHub repository, October 2026). Independent re-run of the release's Lean files for family 325, including rebuilds of the
CrouzeixHilbert,CompleteCrouzeixandStructuralCrouzeixstatements and a check that theCompleteCrouzeixproof closes theDirectCrouzeixchallenge; the README stresses that this is agreement between two OpenAI formalizations and that statement faithfulness 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). Discusses a completely bounded version of their abstract conjecture via dilation theory; relevant to the paper's claim that the closure of $W(A)$ is a complete 2-spectral set and its use of Paulsen/Arveson, and not in its bibliography.
- M. Hartz, J. E. McCarthy, From Clouatre-Ostermann-Ransford to Okubo-Ando (arXiv:2606.02922, 2026-06-01). Proves a completely bounded norm estimate for unital homomorphisms of operator algebras in the COR setting; related to the paper's identification of the optimal similarity constant with a completely bounded norm (Theorem 1.2 and its Paulsen-similarity discussion).
- 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 complete 2-spectral-set claim.
- 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 for homomorphisms into $B(H)$ via positivity of real parts and an operator-valued Herglotz theorem; comparable in method to the paper's positive boundary representation (Theorem 1.3).
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, discussing this paper's optimal-metric theorem (Stein LMI, $\kappa\in[1,2]$) and noting that the structural theorem holds for each enclosing domain separately, with no limiting similarity on the closure of $W(A)$ asserted. [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
- VibeMathed: Problem record
- Emergent Mind: family 325
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-complete-Crouzeix-theorem-September-23-2026,
author = {{OpenAI}},
title = {{The complete Crouzeix theorem: optimal similarity and a common positive boundary representation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf}{OAI:The-complete-Crouzeix-theorem-September-23-2026}},
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:
- A direct proof of the complete Crouzeix inequality · September 26, 2026