The Crouzeix constant

Upper bound, last-listed row

| $2$ | [LS2026] | Lorist–Schwenninger: independent proof by a different route, combining previous tools for weaker estimates with a "simple perturbation lemma for 2-dilations" applied to the iterates $f^n$ in the double-layer potential representation. See the note on priority below. |text edited 2026-08-27 — bound unchanged

Lower bound, last-listed row

| $2$ | [C2007] | Achieved by $p(z)=z$ and $A=\begin{pmatrix}0 & 2\\\\ 0 & 0\end{pmatrix}$, for which $W(A)$ is the unit disk. |tracked since 2026-07-24 — no movement seen yet

Outside the source table: a claim in OpenAI's math release

• Family 325, “The complete Crouzeix conjecture”, claims the complete Crouzeix inequality with constant 2, and that no smaller constant works.

  Its Lean statement, CompleteCrouzeix.lean:

  theorem main : UniversalBound 2 ∧ ∀ C : ℝ, UniversalBound C → 2 ≤ C

  With one-by-one coefficient matrices (m = 1) the file's inequality is this constant's, with C = 2: the value this constant's table already lists last as an upper bound. The second half, that no constant below 2 works, is about the version with matrix coefficients.

  Status: a Lean statement of it is published with a proof by its authors. We read the statement and it matches this constant's definition; we have not compiled the proof, and we have not read the paper.

A claim in a preprint release, read at openai/math@fd4aeeb on 2026-10-12. It is not in the source table, and this ledger has not changed the rows above because of it.