Abstract

We develop a non-Archimedean framework for the representation and analysis of genomic sequences based on the arithmetic and geometric structure of the ring of 5-adic integers. The proposed approach associates RNA sequences with points in a compact ultrametric space through an injective symbolic-to-arithmetic embedding that transforms genomic information into a hierarchical geometric object. We prove that the embedding is a global isometry between a natural symbolic prefix metric and the induced 5-adic metric, and we show that its image forms a compact Cantor-type subset of Z5. Building upon this representation, we formulate a continuous-time evolutionary model governed by a Vladimirov pseudo-differential operator. The resulting non-Archimedean diffusion equation provides a mathematically rigorous mechanism for describing evolutionary transitions across hierarchical genomic scales and admits an explicit fundamental solution obtained through 5-adic Fourier analysis. We further introduce a finite-resolution projection onto quotient rings of Z5 and develop an alignment-free phylogenetic inference framework based directly on the 5-adic valuation. The induced distance function is ultrametric and naturally encodes hierarchical relationships through shared symbolic prefixes. The proposed construction establishes a bridge between p-adic analysis, ultrametric geometry, pseudo-differential operators, and computational phylogenetics. As an illustration, we discuss its application to Hantavirus genomic sequences, demonstrating how hierarchical evolutionary organization can be represented within a unified non-Archimedean mathematical framework.

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