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Lorist-Schwenninger Remark 2 positivity question

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.

Exact FrontierDelta

disproved→disproved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: site-confirmed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Lorist-Schwenninger Remark 2 positivity question · Lorist-Schwenninger Remark 2

Confidence: Not scored

Registry verification: site confirmed · announcement · resolved

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Claude Opus 5.0
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VibeMathed record: Lorist-Schwenninger Remark 2 positivity question

specialist registry · passed

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.

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Lorist-Schwenninger Remark 2 positivity question

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Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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GitHub

code · pending