∞-Groupoid Cosmology H; On the Fundamental Role of the Generalized Gauss–Bonnet Theorem in the ∞-Groupoid Cosmology Framework

This paper aims to elucidate the central role of the generalized Gauss–Bonnettheorem and its modern extensions in constructing the theoretical framework of∞-Groupoid Cosmology. ∞-groupoids provide a mathematical language beyondtraditional manifolds for describing the microscopic structure of spacetime, but theproblem of “topology–geometry correspondence” poses a fundamental challenge.We argue that the extensions of the classical Gauss–Bonnet theorem to higher categories, operator theory, and arithmetic geometry precisely provide the rigorousmathematical tools to address this challenge. These extended forms collectivelyconstitute a clearly layered, functionally complementary “meta-theory tower”, realizing the quantification of “topology locking geometry”, and provide an intrinsicmechanism for explaining non-perturbative cosmological predictions such as thefine evolution exponent of the cosmological constant, topological imprints in theprimordial perturbation spectrum, and the arithmetic origin of black hole entropy.This paper systematically demonstrates how this framework endows ∞-GroupoidCosmology with concrete computational power and physical falsifiability.