Clifford Periodicity Unifies Tenfold Topological Classification — E8 Intelligence Research

Abstract

FINDING: Bott periodicity in real Clifford algebras generates the 10-fold way classification of topological insulators and superconductors, with an 8-fold periodicity underlying the 10 symmetry classes. | MATH: Real Clifford algebra periodicity: \( Cl_{p+8,q} \cong Cl_{p,q+8} \cong Cl_{p,q} \otimes M_{16}(\mathbb{R}) \) (Morita equivalence, 8-fold). Complex case: \( Cl_{n+2} \cong Cl_n \otimes M_2(\mathbb{C}) \) (2-fold). The 10-fold way arises from the 3 discrete symmetries (time-reversal \( T^2 = \pm 1 \), particle-hole \( C^2 = \pm 1 \), chiral \( S = TC \)) — yielding 10 Altland-Zirnbauer classes. Classification groups: \( \mathbb{Z} \) (unitary classes A, AIII), \( \mathbb{Z}_2 \) (symplectic/orthogonal classes AII, DIII, CII, D, C), \( 0 \) (trivial). Periodic table: \( K \)-theory \( K_\mathbb{R}^{-p}(\text{point}) \) and \( K_\mathbb{C}^{-p}(\text{point}) \) give the entries, with Bott periodicity \( K_\mathbb{R}^{-p} \cong K_\mathbb{R}^{-p-8} \), \( K_\mathbb{C}^{-p} \cong K_\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com