Equivalency Mathematics: A Topological Computing Paradigm for Non-von Neumann Hardware Architectures

Modern computer science and physical simulations are deeply mired in a computational bottleneck. The traditional von Neumann architecture relies on the absolute precision of "Equality Mathematics." When dealing with highly complex non-linear systems such as fluid dynamics and protein folding, it faces insurmountable exponential complexity and extremely high energy consumption. This paper proposes a novel theoretical framework—"Equivalency Mathematics"—and its core model "H3QM." This model proves that complex physical phenomena can be dimensionally reduced to the convergence toward the Principle of Least Action within a topological space. We further demonstrate how to use Pick's Theorem to reduce complex calculus and tensor operations into discrete counting problems on a discrete topological lattice, and establishing an abstract topological computing model. By geometrizing mathematical operations, this paradigm drastically reduces system complexity, opening a path toward "Whole-System Science" for solving NP-Hard problems. This paper is available in three language versions: English, Simplified Chinese, and Traditional Chinese.