W≡0 Global Topological Polarity Conservation Axiom: Strict Academic Definition, Mathematical Paradigm and Universal Physical Judgment Scale

Abstract The core dilemma behind the century-long exploration of unified field theories lies in the lack of a unique, global and self-consistent underlying axiom system. All mainstream candidate theories rely on multiple independent hypotheses and artificially fitted parameters, failing to fundamentally resolve the ontological division between general relativity and quantum mechanics. Based on the philosophical foundational achievements of the zero stage, this paper presents the rigorous academic definition and mathematical paradigm of the W≡0 Global Topological Polarity Conservation Axiom for the first time. This paper clarifies that the entire cosmic universe is defined as a topological space , on which the global topological potential functional is identically equal to zero, meaning the net global polarity balance always remains zero; any local polarity breaking must be accompanied by opposite polarity compensation across the whole domain. This paper demonstrates the rationality of adopting differential topology and algebraic topology as the fundamental mathematical language of the theory, constructs a complete mathematical expression system for the axiom, and strictly distinguishes constraint identities as meta-rules from dynamic equations governing physical laws. On this basis, the four major TOE (Theory of Everything) access criteria proposed in the zero stage are upgraded into a semi-quantitative system accessible for academic verification. This paper also releases the unified symbol table and terminology dictionary (V1.0) applicable to all nine research stages, and clarifies the applicable scope of the axiom as well as logical interfaces for subsequent research stages. The W≡0 axiom and its criterion system established in this paper constitute the sole immutable underlying foundation and ultimate judgment criterion of the entire W≡0 unified field theory school. All theoretical derivations and experimental designs in subsequent stages must trace their origins back to this foundation. Keywords: W≡0 axiom; global topological conservation; polarity conservation; unified field theory; mathematical paradigm; TOE access criteria