A counterexample to a Berezin-number bound for a Moore–Penrose pseudo-mean

This technical correction note examines Proposition 2.24 and related statements in M. R. Jabbarzadeh and M. Sohrabi, On a pseudo-mean for Hilbert space operators, Linear and Multilinear Algebra 74(12) (2026), 1629–1644, DOI: 10.1080/03081087.2026.2679717. Exact counterexamples use a positive projection and a positive invertible operator on a two-dimensional analytic reproducing-kernel Hilbert space. A Hardy-space projection family excludes any universal multiplicative constant repair of the squared Berezin-number term. Valid replacement estimates are proved. The note also distinguishes false auxiliary statements, support and domain clarifications, and a Furuta-inequality repair that preserves the conclusion of Proposition 2.23. It does not claim that all results of the reviewed article are false. Version 1.0 includes the six-page PDF, editable LaTeX source, README and a standard-library Python script for exact arithmetic verification. The reviewed publisher PDF is not redistributed. AI assistance and responsibility: OpenAI ChatGPT assisted with English-language improvement, computational checks and auxiliary mathematical analysis, including derivations and counterexample verification. The author is responsible for the construction and validity of the counterexamples, the statements of the results, the proofs, and the final content.