Metricity and Sharp Comparison for Rainio's Intrinsic Distance in Balls and the Disk

This consolidated preprint proves that Rainio's intrinsic function w is a metric on Euclidean balls in every dimension. In the disk it refutes the proposed constant in Conjecture 5.19 and proves the corrected optimal constant, approximately 1.076491339350692, with all equality cases. The proofs are analytic; exact algebra checks and a rational-coordinate counterexample accompany the sources. The definitions and original conjectures retain their attribution to Oona Rainio. Consolidated preprint edition 1.0, dated 5 November 2026. The main PDF and its source/certificate archive form one article. Conceptual origin and AI assistance. The initial idea and conceptual starting point of this investigation are based on the framework of the original geometric paper by Christos Panorias and Kostandina Theodhori, Canonical Parametric Representation of Plane Euclidean Geometry (DOI: 10.5281/zenodo.23837735). This attribution concerns the conceptual origin and methodological organization of the research; every mathematical premise required by the present proofs is stated explicitly here. OpenAI ChatGPT/Codex assisted verification of calculations, editorial polishing and selected, delimited arguments during proof development, including substantive mathematical reasoning in those arguments. Christos Panorias is the sole named author and retains responsibility for reviewing, adopting and submitting the mathematical claims and final text. Research preprint for scholarly scrutiny. Independent peer review, proof-assistant formalization and worldwide priority are not certified.