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Lorist-Schwenninger Remark 2 positivity question

Analysis · posed by Emiel Lorist, Felix L. Schwenninger, 2026 · disproved

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Statement

Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives κ≤1+1−ℜ⟨E1Tx,x⟩\kappa \le 1 + \sqrt{1 - \Re\langle E_1 Tx,x\rangle}, so positivity of ℜ⟨E1Tx,x⟩\Re\langle E_1 Tx,x\rangle would prove the lemma outright. They write: "it is unclear whether ℜ⟨E1Tx,x⟩≥0\Re\langle E_1 Tx,x\rangle \ge 0 in general."

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Claude Opus 5.0

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    — the result was found by a model.

    The counterexample was found and verified computationally by Claude Opus 5.0 in a single session. The search identified the mechanism (the centres of the domain and the numerical range must be separated to make the quantity negative), found a 2x2 witness, reduced it to a closed-form algebraic expression, and verified the sign in exact arithmetic. The sharp consequence — that inequality (5) alone recovers exactly the Crouzeix-Palencia constant 1+sqrt(2) — was derived in the same session. The human operator directed the investigation to Remark 2 and approved the final write-up; all mathematical content was produced by the model.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    Reproduced by this site on 17 August 2026. Remark 2 was confirmed verbatim in the Lorist-Schwenninger LaTeX source, including the exact sentence "it is unclear whether Re<E_1 Tx,x> >= 0 in general". The counterexample repository pins the claim in a statement file and was run here: the exact sympy certificate gives m = -4/95 - 1188*sqrt(90709)/8617355 = -0.0836..., with the sign certified through its minimal polynomial rather than floating point, and the independent mpmath implementation (no shared code) agrees; both also confirm the paper's own bounds still hold at the witness, so the counterexample refutes the remark's hope without touching Lemma 1. The mechanism note - the domain centre and numerical-range centre must separate - matches what the certificate shows. No human peer review; the tier records this site's own reproduction.

    Repeated from the source; nothing was checked here.

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