Z₂ Parity as Time-Reversal Invariant via Fredholm Index and Many-Body Partition Functions — E8 Intelligence Research

FINDING: Z₂ parity in 3D topological insulators is a time-reversal-symmetry-protected invariant, computable via Fredholm index theory and generalizable to many-body fermionic systems via path-integral partition functions. | MATH: Z₂ invariant ν ∈ {0,1} = (−1)^ν = ∏_{i=1}^4 δ_i (parity of four time-reversal-invariant momenta); Fredholm index Ind(D) = dim ker D − dim coker D mod 2; many-body invariant = partition function Z[A] with orientation-reversing symmetry A, satisfying Z[A] = ±Z[−A] (sign = Z₂ class); SSH model belongs to BDI class (Altland-Zirnbauer: T²=+1, C²=+1, S=TC) | CONNECTION: The four TRIM points in 3D form a Z₂⁴ lattice — isomorphic to the D₄ root system (tetragonal symmetry, order 8). The parity product ∏δ_i = ±1 mirrors the determinant of a Weyl group element in D₄ (det = ±1). The Z₂ classification itself is the simplest non-trivial element of the cohomology group H¹(BG, Z₂) — a discrete symmetry breaking pattern analogous to the 0.618/0.382 golden ratio split in that Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com