Boundary-Distance Profiles and Sharp Extrema for Barrlund Metrics: A Proof of Rainio's Conjecture 5.5

This preprint proves the complete ball-inclusion conjecture of Oona Rainio, Inclusion Properties of the Triangular Ratio Metric Balls (https://doi.org/10.1007/s41980-023-00837-w), Bulletin of the Iranian Mathematical Society 49 (2023), article 91, Conjecture 5.5. Corollary 11.2 states all four sharp thresholds in the original form and verifies their exact equivalence to the formulas proved in the paper. Corollary 5.1 of Rainio's paper is an earlier result about the j-star metric and is not identified as an open conjecture. The paper derives a one-parameter boundary-distance profile for Barrlund metrics on arbitrary proper domains for 1 ≤ p ≤ 2, and a supporting-half-space representation on convex domains. In the real unit ball, it determines the complete fixed-center minimum and maximum on a prescribed hyperbolic sphere for 1 ≤ p ≤ 2, with all equality cases. The sharp upper spectrum extends to every finite p > 2. A separate analytic argument gives the exact lower spectrum for p = 4; no complete lower spectrum for the remaining exponents above two is asserted. The planar supporting-half-plane representation is explicitly attributed to Nasyrov, Theorem 3 (https://arxiv.org/abs/2605.25779v3). Its consequence that artanh of the triangular-ratio distance is a metric (FMV Conjecture 2.3, arXiv v3) is included as a known-theorem consequence, without a priority claim. Written proofs are complete independently of the numerical diagnostics. 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.