Z₂-Graded Topology of Kramers Pairs: Parity Products and Torus Rigidity — E8 Intelligence Research

FINDING: The core mathematical structure linking these results is the **Z₂-graded topology of Kramers pairs** — specifically, the parity product over time-reversal-invariant momenta (TRIM) that defines the Z₂ invariant, and its manifestation as a **Bloch-sphere / torus rigidity** problem. | MATH: The Z₂ invariant is computed as \((-1)^ν = ∏k ∈ TRIM δ_k\), where \(δ_k = ± 1\) is the parity eigenvalue of the occupied Bloch states (from the BdG Hamiltonian's Pfaffian: \(δ_k = Pf[w(k)]/√[w(k)]\)). The Majorana Kramers pair condition requires \(HBdG(-k) = Θ HBdG(k) Θ⁻¹\) with \(Θ^2 = -1\), forcing a **double degeneracy** at TRIM. The Bloch sphere mapping for the charge qubit uses Pauli matrices: \(H = 1/2h⃗ · σ⃗\), with \(h⃗ = (h_x, h_y, h_z)\) on \(S^2\), and the Z₂ invariant becomes a **winding number** of \(h⃗\) over the torus \(T^2\). | CONNECTION: The **2-torus Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com