An expanding-map counterexample and a bounded-orbit repair for an iterate-dependent fixed-point theorem

Independent technical comment on Fabiano, Bekri, Baklouti and Alkathiry, A Fixed-Point Theorem for Mappings Satisfying a Contractive Condition Involving T^rho, Axioms 15(5) (2026), 320, doi:10.3390/axioms15050320. The continuous map T(x)=3x on the complete space [1,infinity) satisfies the stated contractive inequality with rho=2 and coefficient sum 3/4 but has no fixed point. A family of expanding maps gives counterexamples for every fixed rho>=2 with arbitrarily small positive coefficient sums. The note identifies the uncontrolled future index in the published induction and independently shows that the proposed separating example fails the uniform coefficient condition. A complete orbital-diameter proof supplies a repair when at least one Picard orbit is bounded. Uniqueness remains valid; convergence is established for each bounded orbit, while a single bounded orbit does not imply convergence from every initial point. Version 1.1. Includes the PDF and a source ZIP containing LaTeX source, a README, and auxiliary exact-arithmetic checks. AI assistance and author responsibility are disclosed in the note. This is an independent comment, not an author-issued corrigendum or a journal-endorsed correction.