Sharp Möbius Distortion of the Second Barrlund Metric: A Proof of the Fujimura–Mocanu–Vuorinen Conjecture for p = 2

This preprint proves the p = 2 case of the disk-automorphism conjecture of Fujimura, Mocanu and Vuorinen in Barrlund's distance function and quasiconformal maps, arXiv:1903.12475v3, Conjecture 4.4 (https://arxiv.org/abs/1903.12475v3). The stated conjecture number refers to that specific inspected version. For every disk automorphism T_a and every pair x,y in the disk, the second Barrlund distance of the images is at most (1 + |a|) times the original distance. The proof applies to arbitrary pairs. It uses the known explicit midpoint formula, a norm-gap lemma with all positivity conditions, and a two-case estimate. It also proves strictness for distinct points when a is nonzero, the optimality of the constant by a limiting family, and the sharp inverse estimate. The explicit metric formula and the necessary sharpness lower bound are credited to the original authors, Theorems 3.27 and 4.5 of the cited arXiv version. Nasyrov's 2026 preprint (https://arxiv.org/abs/2605.25779v3) already proves the analogous p = 1 triangular-ratio theorem; that result is treated as prior work. The present additional upper inequality concerns p = 2. The supplement provides executable checks and the full article source. Methodological basis and AI assistance. The geometric methodology follows the canonical parametric framework of Canonical Parametric Representation of Plane Euclidean Geometry (https://doi.org/10.5281/zenodo.23837735) (Panorias and Theodhori, 2026), particularly normalization, algebraic formulation, verification and transfer. The arguments include both author-originated contributions and arguments developed with generative AI assistance. Initial calculations were carried out by hand and subsequently transcribed and checked with AI assistance, as stated by the author. AI also assisted verification code and text editing/typesetting. Responsibility for the final content rests with Christos Panorias. Access, rights and status. Publicly accessible preprint. Copyright (C) 2026 Christos Panorias. All rights reserved, with personal reading and private scholarly evaluation permitted as specified in the license. This work has not undergone independent peer review; its literature comparison does not certify worldwide priority.