Fu-Kane-Mele Z₂ Invariant via Parity Eigenvalues at TRIMs — E8 Intelligence Research

FINDING: The Fu-Kane-Mele (FKM) invariant is a Z₂-valued topological index for time-reversal-invariant (TRI) insulators, computed via parity eigenvalues at four time-reversal invariant momenta (TRIMs) in 2D, with a cohomological generalization to quaternionic vector bundles. MATH: - Z₂ invariant: ν = ∏_{k∈TRIM} δ_k, where δ_k = ∏_{m} ξ_{2m}(k) (ξ = parity eigenvalue ±1). - TRIM points in 2D: Γ = (0,0), X = (π,0), Y = (0,π), M = (π,π) — a Z₂⁴ lattice of momenta. - Fu-Kane parity criterion: ν = ∏_{TRIM} δ_k mod 2. - Cohomological form: FKMM invariant in H¹(B; Z₂) for quaternionic bundles over involutive base spaces (arXiv:1603.09421). - Fredholm index formulation: ν = ind(F) mod 2, where F is a Fredholm operator encoding TRS (Shapiro). CONNECTION: - The four TRIMs form a **tetragonal lattice** (square Bravais lattice) — a 2D crystallographic point group with 4-fold rotational symmetry (C₄). - The Z₂ structure is a **binary parity** — analogous to the 0/1 dichotomy of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com