Stassis Stashkevichyus
Version 1.0.0 (1 August 2026). Preprint; not peer reviewed. Abstract: Let Σ = {A, C, U, G} and C = Σ^3. A nucleotide bijection e: Σ → Z/4Z, followed by the binary-reflected Gray map γ: Z/4Z → F2^2, converts squared fourth-root chord distance into twice Hamming distance. For the class-compatible embedding e(A), e(C), e(U), e(G) = (0, 1, 2, 3), the proposed codon dissimilarity is therefore an ordinary weighted Hamming metric on six binary coordinates, not a new topological torus metric. We prove that the additive function C, previously treated only as a cost, is itself a metric for all α > 0, β ≥ 0, and positive positional weights; sqrt(C) is both its one-half snowflake and the Euclidean metric of an explicit embedding. We distinguish the group (Z/4Z)^3 from the metric representation F2^6, classify all 24 nucleotide embeddings into three postcomposition-D4 orbits, and identify 16 class-compatible and 8 class-incompatible embeddings. For the standard translation table, exhaustive integer enumeration gives 61 sense codons, 263 undirected one-step sense edges, and 67 synonymous edges, distributed by changed position as (4, 0, 63). With (w1, w2, w3) = (4, 2, 1) and α = β = 1, the score has AUROC = 0.888174 and tie-grouped AP = 0.771271, but the position-conditioned AUROC is only 0.703985, and position-only has the larger global AUROC 0.892971. A fixed rational sensitivity grid contains broad in-sample plateaus reaching AUROC = 0.940337 and AP = 0.820836, so the default parameters are not identifiable from this benchmark. The principal negative result concerns null models. Permuting amino-acid labels over fixed canonical fibers is exactly non-identifying because endpoint equality does not change. Exhaustive fiber-geometry-preserving nulls over all 13,824 position-preserving symbol automorphisms place the canonical result in only the upper 11.1% by AUROC and 18.5% by AP; over all 82,944 automorphisms of the Hamming graph, the corresponding upper-tail fractions are 9.26% and 8.64%. Thus the finite metric structure is exact, but this internal task does not establish unique biological optimization or external predictive value. The release also evaluates 24 single-valued NCBI translation tables and excludes three context-dependent tables for which translation is not a single-valued map of the required type. A companion external validation study finds no practically relevant incremental signal for missense functional effects on its frozen assay panel. Author: Stassis Stashkevichyus; Independent Researcher, Lithuania; ORCID 0009-0000-2294-705X; theobserver.of.multiverses@proton.me. Canonical GitHub release: https://github.com/Observer1117/weighted-gray-codon-geometry/releases/tag/v1.0.0 Frozen commit: 5523149f837b9a27be3fbf830231a261b9408090 Companion Article II: https://github.com/Observer1117/weighted-gray-codon-external-validation/releases/tag/v1.0.0 Licensing: manuscript and author-created figures/tables are CC BY 4.0; original code is MIT; third-party materials retain source-specific terms recorded in THIRD_PARTY_DATA.md.