Lorist-Schwenninger Remark 2 positivity question
orist 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 , so positivity of would prove the lemma outright. They write: "it is unclear whether in general."
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Numerical range / Crouzeix's conjecture
- Posed by
- Emiel Lorist, Felix L. Schwenninger
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-08-14
- Model
- Claude Opus 5.0
- Vendor
- Anthropic
- Collaborators
- —
- Verification
- Site-confirmed
- Publication
- Announced
- Significance
- 4 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
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.
What the AI did
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.
Verification
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.
Sources
Related entries
- Builds onCrouzeix's conjecture
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