The Geometrization Mechanism of Mathematical Structures: A Theory of Spatialization, Cognitive Utility, and Structural Unification

This paper proposes a unified, mechanism-based theory explaining why and how diverse mathematical structures admit geometric interpretations and why geometric perspectives repeatedly drive mathematical discovery. We call this the Geometrization Mechanism. The mechanism is presented at three interacting levels: (1) a formal categorical and topological spatialization mapping that associates algebraic/analytic structures to spaces (or space-like objects) subject to explicit axioms; (2) a collection of compatibility and utility principles explaining why spatializations preserve critical structure and compress complexity; and (3) a cognitive-epistemic account — the Cognitive Utility Hypothesis — explaining why geometric representations are especially productive for human mathematical reasoning. This theory subsumes many classical instances (spectrum of a ring, Lie groups, moduli spaces, sheaf/stack constructions), while providing novel criteria for when a structure admits productive geometrization, how to measure its effectiveness, and how to systematically seek new geometric incarnations. The paper formulates precise definitional machinery (spatialization functors, invariants, locality measures), proves general propositions about information compression and invariance transfer, and outlines empirical predictions for mathematical cognition. This mechanismized account aims to be both descriptive (explaining historical patterns) and prescriptive (guiding future theory-building).