Clifford Algebra Periodicity Unifies the 10-Fold Way Classification — E8 Intelligence Research

Abstract

FINDING: Bott periodicity in real Clifford algebras generates the 10-fold way classification of topological insulators/superconductors, unifying symmetry classes with periodic table structure. | MATH: Real Clifford algebra periodicity: Cl(p+8,q) ≅ Cl(p,q+8) ≅ Cl(p,q) ⊗ Mat(16,ℝ); 10-fold way = 2 complex (A, AIII) + 8 real (AI, BDI, D, DIII, AII, CII, C, CI) symmetry classes; classification groups: ℤ, ℤ₂, ℤ₂, 0, ℤ, 0, 0, ℤ (in d=0); Bott periodicity: π_k(O) = π_{k+8}(O), π_k(U) = π_{k+2}(U); Clifford algebra chain: Cl₀→Cl₁→...→Cl₇→Cl₈ with Cl₈ ≅ Mat(16,ℝ). | CONNECTION: The 8-fold real periodicity mirrors the 8-dimensional lattice structure of the octonions and the E₈ root system; the 10-fold way = 8 real + 2 complex, where 8 is the dimension of the octonionic division algebra and 2 is the complex dimension; the period-8 structure corresponds to the 8 real Clifford algebras Cl₀...Cl₇, whose dimensions follow 1,2,4,8,8,4,2,1 — the same doubling sequence as division algebras (ℝ, ℂ, ℍ, 𝕆); Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com