9.1 Geometric Theory of Superconductivity — From Mathematical Framework to Material Criteria

Using symplectic geometry, K-theory, noncommutative geometry, and the Atiyah-Singer index theorem as mathematical tools, we construct a complete geometric theory of the superconducting transition temperature. The core formula is rigorously derived from the entropy dilution condition for spectral gap direction instability. The formulas for thirteen classes of superconducting materials each have a clear geometric origin. The superconductivity three-question criterion provides a first-principles determination of whether any given material can superconduct or not, and the classification of superconducting types for all 230 space groups covers every crystal system. The topological superconducting material screening yields a commutation criterion between the lattice symmetry group and the Birkhoff mixing matrix.