Z₂ Invariant via Fredholm Index in Interacting Topological Insulators — E8 Intelligence Research

FINDING: Z₂ parity in 3D topological insulators is a time-reversal-protected invariant, computable via Fredholm index theory and generalizable to many-body partition functions for interacting fermionic phases. MATH: - Z₂ invariant ν ∈ {0,1} = (Pθ⁺ − Pθ⁻)/2 mod 2, where Pθ± are parity eigenvalues of occupied Bloch states at time-reversal-invariant momenta (TRIMs). - Fredholm index: Ind(D) = dim ker D − dim coker D, with D a Dirac-type operator encoding time-reversal symmetry (T² = −1). - Many-body invariant: Z(partition function) on a non-orientable manifold (e.g., Möbius strip) — equals ±1, with −1 signifying nontrivial Z₂ phase. - For 3D: 8 TRIMs → ν₀; 3 weak invariants (ν₁ν₂ν₃) from 2D planes. CONNECTION: - The 8 TRIMs form a Z₂³ structure — isomorphic to the root lattice of D₄ (hypercubic symmetry), and the strong invariant ν₀ = sum of parities mod 2 mirrors the parity of a 4D hypercube's vertices. - Time-reversal T² = −1 corresponds to quaternion units i, j, k — the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com