Real K-Theory Bott Periodicity and the 8-Fold Topological Classification — E8 Intelligence Research

FINDING: Real K-theory Bott periodicity (8-fold) underpins the Altland-Zirnbauer classification of topological insulators/superconductors, with Z2 invariants and mod-2 Chern numbers as the operative topological charges. | MATH: Bott periodicity for real K-theory: KO^n(X) ≅ KOⁿ⁺⁸(X). The 8-fold clock: KO⁻ⁿ = KOⁿ⁺⁸. For the 2-sphere S², KO(S²) ≅ Z ⊕ Z/2 (the Z/2 is the mod-2 Chern number / Fu-Kane invariant). The Altland-Zirnbauer 10-fold way: 2 complex classes (A, AIII) with Z and Z invariants; 8 real classes (AI, BDI, D, DIII, AII, CII, C, CI) with invariants Z, Z, Z2, Z2, Z, Z, Z2, Z2 respectively. The mod-2 index ν ∈ Z2 = π_0 of the classifying space, computed via ν = (1/2π)∮_S² Tr(F) mod 2 for time-reversal invariant systems. | CONNECTION: The 8-fold periodicity is the real Clifford algebra periodicity: Clp+8,q ≅ Clp,q ⊗ M_16(R). This maps to the 8-fold symmetry of the root system of E8 (the exceptional Lie group), whose Coxeter number is 30, and whose Weyl group orde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com