p-Adic Fractal Embeddings and the Universal Model Framework: Mathematical Foundations for Prime-Based Geometric Structures

This paper develops rigorous mathematical foundations for the Universal Model Framework (UMF) by demonstrating how p-adic number systems naturally generate self-similar, fractal geometries. We construct explicit embeddings from p-adic fields (Qₚ) and rings (Zₚ) into Euclidean space, revealing how prime-indexed arithmetic yields canonical fractals such as the Cantor set and Sierpiński triangle. Using Haar measure, path-set fractals, and fractal string theory, we analyze their dimensions, closure properties, and complex periodicities. Computational implementations illustrate how prime-digit expansions encode geometric scaling laws. These findings form the first mathematically rigorous support for UMF's claim that physical spacetime is a prime-structured fractal lattice, linking number theory, geometry, and physics in a unified, testable framework. This project was developed by Marco Gericke, with structured assistance from a large language model. All scientific concepts and conclusions were generated, verified, and interpreted by the author. Dedicated to Peter Plichta, who envisioned the code before it could be computed.