# The precise statement being claimed

This file pins exactly what the entry asserts, so a reviewer can check the
formal claim against the informal question before checking any arithmetic.
It is the analogue of the Lean statement anchor used in earlier submissions:
no proof, just the statement, stated once and unambiguously.

## The question, as posed

Lorist and Schwenninger, "A solution to Crouzeix's conjecture", arXiv:2608.03841v1
(4 August 2026), prove their Lemma 1 — a perturbation result for 2-dilations —
and then remark:

> **Remark 2.** Inequality (5) alone implies that kappa <= 1 + sqrt(1 - Re<E_1 T x, x>).
> Thus, if it were known that Re<E_1 T x, x> >= 0, the assertion of Lemma 1 would
> already follow. However, it is unclear whether Re<E_1 T x, x> >= 0 in general and
> the lower bound in (4) already suffices in our proof.

The objects are those of Lemma 1 and its proof:

- H a finite-dimensional complex Hilbert space, T in L(H), kappa := ||T||;
- K a Hilbert space, Q in L(K) a contraction, V in L(H,K) an isometry;
- E_n := 2 V* Q*^n V - T*^n, assumed uniformly bounded and to commute with T;
- x in H with ||x|| = 1 and T*T x = kappa^2 x, chosen in the branch kappa > 1
  (the proof opens "If kappa <= 1, there is nothing to prove, so suppose kappa > 1").

Inner products are linear in the first variable (their Section 1). The quantity
Re<E_1 T x, x> is invariant under that choice: swapping to the second variable
replaces the inner product by its complex conjugate, and only its real part is used.

## The claim

**The answer is no.** There is a configuration satisfying every hypothesis of
Lemma 1, with kappa > 1, and Re<E_1 T x, x> < 0.

Concretely, taking Omega to be the open unit disk and building V, Q and E_1 exactly
as the proof of their Theorem 3 builds them —
K = L^2(dOmega, |dsigma|; H), (Vx)(sigma) = 2^{-1/2} P_Omega(sigma)^{1/2} x,
Q = multiplication by f, T = f(A) — with

    A = [[1/2, 9/10], [0, 1/2]],        f(z) = (8 - 19 z)/(19 - 8 z),

we have

    kappa = sqrt(594 sqrt(90709) + 178918)/500 = 1.196358697123234...  > 1,
    Re<E_1 T x, x> = -4/95 - 1188 sqrt(90709)/8617355 = -0.083626247971819...  < 0.

Both are algebraic numbers; the second has minimal polynomial
163729745 X^2 + 13787768 X + 8000. The largest singular value of T is simple, so x
is determined up to a phase and no choice is being made.

**Minimality.** No 1x1 configuration can be a counterexample: there T = f(a) is a
scalar with a in Omega, so kappa = |f(a)|/||f||_inf <= 1 by the maximum principle
and the branch kappa > 1 is never entered. So 2x2 is the smallest possible, and it
is attained.

## What is NOT claimed

- Nothing about the correctness of the paper. Inequalities (4) and (5) both hold at
  this witness, and so does the theorem (kappa <= 2). Only the shortcut proposed in
  Remark 2 fails.
- No claim about non-disk domains Omega. Every configuration here uses a disk.
- No claim of priority over the Crouzeix conjecture itself, which is settled by two
  independent 2026 proofs (Jin; Lorist-Schwenninger) and is not at issue.