Never more than twice?
Two AI preprints claim the complete Crouzeix inequality, with matrix coefficients of any size
Covers result 325 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
3:51. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
The numerical range of a matrix A is the set of all values x*Ax over unit vectors x: a convex region of the complex plane that contains the eigenvalues. Crouzeix’s question is how well this region controls polynomials of A. Take the 2 × 2 matrix with a two in the corner. Its numerical range is the unit disk, the polynomial z is at most one there, yet A itself has norm two. Our computer confirms the ratio is exactly two.
In 2004 Crouzeix conjectured that two is the worst case for every matrix and every polynomial. Crouzeix proved a bound of 11.08 in 2007, and Crouzeix and Palencia improved it to 1 + √2 in 2017. In July and August 2026, three preprints by human authors gave proofs of the original version, with numbers as coefficients. At least two had help from OpenAI models. The complete version, with matrices as coefficients, was proved only for sizes up to three by three and a few special families, and was described as open in September.
Result 325 is two OpenAI papers, of 12 and 22 pages, dated September 26 and 23 and made public October 6, 2026. They claim the complete inequality: for every matrix A of any size and every polynomial whose coefficients are matrices of any size, the norm of P of A is at most two times the largest norm of P on the numerical range. The second, longer paper extends it to every bounded operator on any complex Hilbert space, and adds a structural theorem about the best change of metric.
What it doesn’t claim
Not claimed: the first proof of the original scalar conjecture, which the papers credit to the earlier preprints, or that two is the best constant as something new. That was known from the 2 × 2 example. Nothing is claimed for Banach spaces.
Where it stands
- Posted as preprints
- OpenAI, 2 papers, 12 and 22 pages, dated September 26 and 23, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Both proofs, the structural theorem and the Hilbert-space version (docs)
Last checked October 19, 2026. The papers are unrefereed and unchanged since release. Both proofs of the matrix inequality, the structural theorem, and the Hilbert space version are formalized in Lean, in five checks and about 39,000 lines, not run by us. Four outside projects re-ran them and report them passing. We found no public comment from specialists, and no reported error. OSVerify has not reviewed it.
How the proof goes
- A positive kernel from convexity. The direct proof puts the numerical range inside a smooth convex region and maps the outside of the unit circle onto the outside of that region. Convexity makes a certain kernel positive, and that traps a polynomial’s average size on the boundary between two sums of its coefficients. The resolvent of A, suitably weighted, has positive real part on the circle, because the numerical range lies inside the region.
- Both orders, then a 2-by-2 trick. The key step scales P to be at most one on the boundary, takes the top singular vectors of P of A, and tests both orders of multiplication with a test polynomial G, P times G and G times P, which is what matrix coefficients require. A 2 × 2 positivity trick then gives 2γ² ≤ 8, so γ, the norm, is at most two. The step to infinite dimensions, in the second paper, is a finite compression; our computer checks it on examples to about 15 digits. Our random small cases, 600 of them, peak at 1.53, and hill climbing reaches about 1.96, never above two.
The numbers in the video
- Ours, exact: for A = [[0,2],[0,0]] the numerical range is the unit disk and ‖A‖ = 2, so the ratio is exactly 2.
- Ours: 600 random 2 × 2 and 3 × 3 cases, with scalar and 2 × 2 matrix coefficients, peak at 1.53; hill climbing reaches 1.95, 1.91, 1.96 and 1.78; none above 2.
- Ours: the finite compression identity holds on examples with relative error about 5 × 10⁻¹⁶.
- Lean: five checks, 199 files and 39,133 lines in the import closures (counted by us; not run by us).
Read the source
- A direct proof of the complete Crouzeix inequality (PDF), 12 pages.
- The complete Crouzeix theorem: optimal similarity and a common positive boundary representation (PDF), 22 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 325.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
The numerical range of a matrix A is the set of all values x*Ax over unit vectors x: a convex region of the complex plane that contains the eigenvalues. Crouzeix’s question is how well this region controls polynomials of A. Take the 2 × 2 matrix with a two in the corner. Its numerical range is the unit disk, the polynomial z is at most one there, yet A itself has norm two. Our computer confirms the ratio is exactly two. In 2004 Crouzeix conjectured that two is the worst case for every matrix and every polynomial. Crouzeix proved a bound of 11.08 in 2007, and Crouzeix and Palencia improved it to 1 + √2 in 2017. In July and August 2026, three preprints by human authors gave proofs of the original version, with numbers as coefficients. At least two had help from OpenAI models. The complete version, with matrices as coefficients, was proved only for sizes up to three by three and a few special families, and was described as open in September.
1:10The claim
Result 325 is two OpenAI papers, of 12 and 22 pages, dated September 26 and 23 and made public October 6, 2026. They claim the complete inequality: for every matrix A of any size and every polynomial whose coefficients are matrices of any size, the norm of P of A is at most two times the largest norm of P on the numerical range. The second, longer paper extends it to every bounded operator on any complex Hilbert space, and adds a structural theorem about the best change of metric. Not claimed: the first proof of the original scalar conjecture, which the papers credit to the earlier preprints, or that two is the best constant as something new. That was known from the 2 × 2 example. Nothing is claimed for Banach spaces.
2:03How the proof goes
The direct proof puts the numerical range inside a smooth convex region and maps the outside of the unit circle onto the outside of that region. Convexity makes a certain kernel positive, and that traps a polynomial’s average size on the boundary between two sums of its coefficients. The resolvent of A, suitably weighted, has positive real part on the circle, because the numerical range lies inside the region. The key step scales P to be at most one on the boundary, takes the top singular vectors of P of A, and tests both orders of multiplication with a test polynomial G, P times G and G times P, which is what matrix coefficients require. A 2 × 2 positivity trick then gives 2γ² ≤ 8, so γ, the norm, is at most two. The step to infinite dimensions, in the second paper, is a finite compression; our computer checks it on examples to about 15 digits. Our random small cases, 600 of them, peak at 1.53, and hill climbing reaches about 1.96, never above two.
3:11Where it stands
The papers are unrefereed and unchanged since release. Both proofs of the matrix inequality, the structural theorem, and the Hilbert space version are formalized in Lean, in five checks and about 39,000 lines, not run by us. Four outside projects re-ran them and report them passing. We found no public comment from specialists, and no reported error. If it holds, the numerical range controls every matrix polynomial of every bounded operator on a Hilbert space within a factor of two, the best possible, whatever the size of its coefficient matrices.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.