A Counterexample to the 1 + |a| Möbius Distortion Bound for Rainio's w-Function

This preprint disproves the inequality w_D(T_a(x),T_a(y)) ≤ (1+|a|) w_D(x,y) numerically suggested for Rainio’s established w-function next to inequality (4.11), printed page 15, in arXiv:2103.04397v2 (29 March 2021). The definition of w and the proposed estimate retain their attribution to Rainio. An explicit counterexample is a=1/8, x=(-3+4i)/100 and y=(-3-4i)/100. The distortion ratio is approximately 1.12781947064018, exceeding 1+|a|=1.125. The proof uses exact rational inequalities, including the strict lower bound (w_D(T_a(x),T_a(y))/w_D(x,y))² > 29178576/22970533 > 81/64. Every inverted denominator is proved positive. This is a counterexample to the Möbius distortion bound. It does not concern Rainio’s separate triangle-inequality conjecture 5.6, and it does not determine the exact optimal distortion function L_w(a). Relationship to the author’s Barrlund work: the counterexample is briefly stated and cited in Section 5 of the revised Sharp Möbius Distortion of the Second Barrlund Metric, version 1.2. A copy of this note and checker is also supplied in that work’s supplement for reproducibility. This separate deposit is the primary standalone presentation of the complete w-counterexample proof; it does not repeat the Barrlund p = 2 proof. The research package contains the three-page PDF, editable LaTeX, a Python standard-library exact-fractions checker and its certificate, discovery diagnostics, a literature-screening log and rights terms. Numerical exploration is documented solely as a discovery diagnostic; the proof is exact. A targeted literature search completed on 3 November 2026 did not identify a prior resolution of this specific w-case. The screening is not a certificate of worldwide priority, and the note has not undergone independent peer review. Methodological basis and AI assistance: The methodological basis of this work throughout is the author's canonical parametric framework for Euclidean geometry, presented in the foundational preprint Canonical Parametric Representation of Plane Euclidean Geometry (C. Panorias and K. Theodhori, 2026; https://doi.org/10.5281/zenodo.23837735). All stages of this study were developed under the author's direction and continuous mathematical oversight. Generative AI (OpenAI ChatGPT/Codex) assisted with calculations, symbolic and numerical checks, verification code, editing and typesetting, and contributed to the development of parts of the mathematical arguments during the research process. The author reviewed, checked and adopted all calculations, arguments and results, made all final mathematical decisions, and retains full responsibility for the content. Established definitions and prior results remain attributed to their original sources. Rights: Copyright © 2026 Christos Panorias. All rights reserved. Personal reading and private scholarly evaluation are permitted under the terms included in the files.