# The complete Crouzeix conjecture > The complete Crouzeix conjecture holds: matrix-coefficient polynomials of an operator obey the sharp factor 2. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-325/ Last updated: 2026-10-14 **The complete Crouzeix conjecture holds: matrix-coefficient polynomials of an operator obey the sharp factor 2.** **Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 23, 2026 · **Field:** Functional analysis · **Entry:** 325 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. Plug a matrix into a polynomial and you get a new matrix; its size is hard to predict when the matrix is not normal. Crouzeix conjectured in 2004 that the size is at most twice the polynomial's largest value on the matrix's numerical range. The complete version lets the polynomial's coefficients be matrices too. Two papers prove this complete version with the sharp factor 2, for matrices and for all bounded operators, and Lean checks it. The second paper also proves a structural form with one change of inner product. ## At a glance - **First step:** 2004 (Michel Crouzeix, the constant-two numerical-range conjecture) - **Significance:** Major. It settles the matrix-coefficient form of Crouzeix's conjecture with the sharp constant, after earlier complete bounds of 11.08 and 1 + √2. - **Trust:** Formal proof listed - **Field:** Functional analysis (2 papers) ## The explanation Take the 2-by-2 matrix A with top row (0, 2) and bottom row (0, 0). Its only eigenvalue is 0. So looking at eigenvalues alone suggests A is tiny. But A stretches the vector (0, 1) to (2, 0), so its size, or norm, is 2. A matrix whose eigenvalues miss its true size like this is called nonnormal. Even a nonzero nilpotent matrix, one with a power equal to zero, has spectrum {0}[1]. A better guide is the numerical range W(A). It collects every value u*Au, over unit vectors u. For this A, the numerical range is the disc of radius 1 around 0. Now take the polynomial p(z) = z. Its largest size on W(A) is 1. But p(A) = A has norm 2. So the ratio here is exactly 2. Crouzeix conjectured in 2004 that 2 is always enough. For any polynomial p, the norm of p(A) should be at most twice the largest value of |p| on W(A). He proved a universal complete bound of 11.08 in 2007[2]. Crouzeix and Palencia later lowered it to 1 + √2, about 2.414[3]. Badea, Crouzeix and Delyon proved the complete constant two for all 2-by-2 matrices[4]. Recent preprints give scalar proofs with constant two[5]. Åhag, Czyż and Virtanen prove the complete form for matrices of size at most 3[6]. The word complete matters. Here the polynomial's coefficients are themselves m-by-m matrices. The question asks for the factor 2 for every m, every matrix size, and every degree. The obstacle is that a scalar proof need not survive this step. Coefficient matrices need not commute, so their order of multiplication matters[7]. The first paper gives a direct proof for matrices of every size and every coefficient size. This resolves the complete Crouzeix conjecture in its matrix form, including numerical ranges that are points or line segments[8]. The constant two cannot be lowered[9]. The example above shows why. The proof places the numerical range inside a smooth convex region. It then controls the polynomial through coefficient estimates along the region's boundary. The second paper proves the same sharp bound for every bounded operator on a complex Hilbert space[10]. A Hilbert space is a space with lengths and angles, possibly of infinite dimension. No separability assumption is needed[11]. It also proves a structural result for finite matrices. A change of inner product, distorting lengths by a factor of at most two, turns a related map into a contraction[12]. A contraction is a map that never increases lengths. One fixed boundary density then represents every matrix-valued test function. The same change and density work for every coefficient size and every function[13]. How sure is this? Lean, a computer proof checker, proves the complete inequality with constant 2 for every matrix and every coefficient size. It proves that no smaller constant works. It also covers every bounded operator on any complex Hilbert space, and the structural result for finite matrices. No specialist review has been published, but the main statements have Lean proofs. **What it doesn't show.** This does not improve the factor 2, which is the best possible, and the structural result is stated only for finite matrices. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major. Lean proves the sharp complete inequality for matrices and Hilbert-space operators; scalar constant-two proofs had already appeared in other recent preprints. ## Words used - **Norm of a matrix:** The largest factor by which the matrix can stretch a vector of length 1. - **Spectrum:** The set of eigenvalues of a matrix, or its analogue for an operator. - **Nonnormal matrix:** A matrix that does not commute with its conjugate transpose; its eigenvalues can badly underestimate its size. - **Nilpotent matrix:** A matrix some power of which is the zero matrix. - **Numerical range W(A):** The set of values u*Au over all unit vectors u; a compact convex set containing the eigenvalues. - **Complete inequality:** A version of the inequality where the polynomial's coefficients may be matrices of any size. - **Hilbert space:** A space with lengths and angles, like ordinary space, possibly of infinite dimension. - **Contraction:** A linear map that never increases the length of a vector. ## The original papers (2) - [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf) - September 26, 2026 · Formal proof listed - [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf) - September 23, 2026 · Formal proof listed - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/325.md) ## Quoted lines from the paper (13) [1] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > For a nonnormal matrix, the spectrum alone cannot provide the same control: even a nonzero nilpotent matrix has spectrum $\{0\}$. [2] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > Crouzeix formulated the scalar constant-two question in 2004 and proved a universal complete bound of $11.08$ in 2007. [3] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > Crouzeix and Palencia combined a positive double-layer potential with the Cauchy transform of conjugate boundary data to obtain the complete bound $1+\sqrt2$; Ransford and Schwenninger simplified its final norm argument. [4] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > For all base matrices of order two, Badea, Crouzeix, and Delyon obtained the complete constant two. [5] From [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf): > Recent preprints by Jin, Lorist and Schwenninger, and Luo present scalar constant-two proofs. [6] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > \AA hag, Czy{\.{z}}, and Virtanen prove the complete inequality for base matrices of order at most three, with arbitrary coefficient dimension. [7] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > Its complete character is the uniformity in $m$: a scalar proof need not survive matrix amplification, because coefficient matrices need not commute. [8] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments. [9] From [A direct proof of the complete Crouzeix inequality](https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf): > The constant two cannot be decreased uniformly over $n,m$, and $d$. [10] From [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf): > 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. [11] From [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf): > No separability assumption is needed. [12] From [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf): > 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. [13] From [The complete Crouzeix theorem: optimal similarity and a common positive boundary representation](https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf): > The same $S$, $\Lambda$, and $\kappa$ work for all $m$ and $v$. ## Other sources - [openai/math, Lean scope note for family 325](https://github.com/openai/math/blob/main/lean/docs/325.md): “The formalized result proves the bound with constant $2$ for every positive matrix and coefficient size, every polynomial degree, and arbitrary complex coefficients.” - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [The cotype–cotype conjecture under the approximation property](https://closedform.nishachay.tech/e/openai-math-2026-326/) - Functional analysis - [Donaldson's hypersymplectic deformation conjecture](https://closedform.nishachay.tech/e/openai-math-2026-341/) - Differential geometry - [Donaldson's tamed-to-compatible conjecture](https://closedform.nishachay.tech/e/openai-math-2026-342/) - Differential geometry - [The second Kahn–Kalai conjecture with an edge-count bound](https://closedform.nishachay.tech/e/openai-math-2026-176/) - Combinatorics - [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) - Theoretical computer science - [Functional analysis](https://closedform.nishachay.tech/science/mathematics/functional-analysis/) - Field · 11 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Open since the 2000s](https://closedform.nishachay.tech/open-since/2000s/) - Era · 48 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20325%5D%20)