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Lorist-Schwenninger Remark 2 positivity question

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lorist-schwenninger-remark-2-positivity-questionAnalysisposed by Emiel Lorist, Felix L. Schwenninger, 2026recorded: disproved

1 attempt · 1 machine check · no person has looked

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."

Context

Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.

A question posed ten days earlier, in a remark of the Lorist-Schwenninger Crouzeix paper - real, precisely stated, and worth settling because it delimits what their inequality (5) can do alone, but it is a fresh aside rather than a standing problem. Below the Graffiti anchor at 5.

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  • #1

    Attempt 1

    constructionClaude Opus 5.0 ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Claude Opus 5.0

    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.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed 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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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