Parity-Based Z₂ Invariants for Time-Reversal Superconductors via TRIM Products — E8 Intelligence Research

FINDING: The Z₂ topological invariant for time-reversal invariant superconductors is determined by the parity product over the four (3D) or one (2D) time-reversal invariant momenta (TRIMs), directly tied to the B3 Weyl group structure of the cubic lattice. | MATH: Z₂ invariant ν = Π_{k∈TRIMs} δ(k), where δ(k) = ±1 is the parity eigenvalue at each TRIM; for 3D there are 4 independent Z₂ invariants (ν₀; ν₁ν₂ν₃), for 2D exactly 1. The TRIMs are the 8 points {0, π}³ in the Brillouin zone, fixed points of the time-reversal operator Θ² = −1, forming a Z₂⁸ lattice. The B3 Weyl group (order 48, the hyperoctahedral group) acts on these 8 points via signed permutations, preserving the product structure. | CONNECTION: The 8 TRIMs are the vertices of a hypercube in 3D — the root system B₃ (cubic symmetry, order 48). The product Πδ(k) = ±1 is a Z₂ character of the B₃ action. The ratio structure of the invariant (ν₀ = product of all 8, νᵢ = product of 4 in each plane) mirrors the decomposition of B₃ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com