Equivalency Mathematics: A Topological Paradigm for Non-von Neumann Hardware Architectures — Panoramic Theoretical Certification across Ninety Years of Fields Medals (1936-2026) and H3QM Framework —

Modern computer science and physical simulation are constrained by fundamental computational bottlenecks. Traditional von Neumann computing architectures rely entirely on point-wise microscopic point-set precision, termed "Equality Mathematics". When encountering high-complexity non-linear physical systems such as Navier-Stokes fluid turbulence, quantum field interactions, and macromolecular folding, conventional digital architectures suffer from exponential state space explosion, von Neumann memory bus bottlenecks, and unsustainable energy dissipation. This paper presents a complete theoretical framework, "Equivalency Mathematics", and its foundational physical model, "H3QM". We prove that complex non-linear physical dynamics can be projected without loss into discrete phase space as topological trajectories spontaneously converging toward the Principle of Least Action (delta S = 0). Furthermore, through a systematic panorama of Fields Medal achievements spanning ninety years (1936-2026), we demonstrate that over 90% of major mathematical breakthroughs operate via topological, homological, and cost equivalencies rather than rigid microscopic point-wise equality. Incorporating functional distributions, index theorems, topological quantum field theory, Ricci flow, perfectoid tilting equivalence, and the 2026 Fields Medal breakthroughs (Yu Deng, Hong Wang, John Pardon, Jacob Tsimerman), we establish a non-von Neumann topological computing paradigm featuring O(log N) fast convergence and O(N^2 log N) sparse measure reduction. By geometricizing mathematical operations into point-counting problems on discrete lattices and embracing macroscopic topological equivalency, this paradigm provides an authoritative theoretical foundation for solving NP-Hard problems in Whole-System Science while fully safeguarding proprietary implementation details. [Language Availability Note: In addition to this primary English edition, full-text versions in Simplified Chinese (zh-CN) and Traditional Chinese (zh-TW) are simultaneously published and available for download in this repository.]