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Refutation of unrestricted Hermitian Taylor majorization.

Abstract

Refutation of unrestricted Hermitian Taylor majorization.

Theorem 1.1 (The spectral comparison is false).

Lean statement: D5/S3/Quantum/Matrix/TaylorMajorization/Result.result

Proof. Machine-checked in Lean as D5/S3/Quantum/Matrix/TaylorMajorization/Result.result (✓ std3). ∎

Resolves. Problems/zhang-2026-taylor-coefficient-weak-majorization (refuted) by D5/S3/Quantum/Matrix/TaylorMajorization/Result.result.

Source. Repository-derived.

Acknowledgement. T. Zhang (2026). Weak majorization inequalities for the cubic and quartic coefficients of e^{(A+B)t} versus e^{At}e^{Bt}. DOI: 10.48550/arXiv.2601.07286. URL: https://arxiv.org/abs/2601.07286.

Commentary.

The claim for all Hermitian matrices and every order at least five is false. Use dimension two, A=Z-I, B=X and order five. Both matrices are Hermitian. Weak majorization would imply the real trace inequality by the full initial sum. The shifted Pauli family instead gives a strictly negative trace defect, contradicting that inequality. The example concerns Hermitian matrices without a positivity restriction.

References