Möbius–Lorentz Quotients and Null Pin Holonomy: Affine Normal Forms, Null-Displacement Loci, Mapping-Torus Topology, Pin Lifts, and Mirror-Residue Pairing

We develop a rigorous mathematical framework for Lorentzian Möbius-type quotients M_g = R1,3 / where g = (P_k, a) is an orientation-reversing affine Lorentz isometry. Seven theorems are organized in a six-part structure: 1. Theorem 1 (Affine-Square Law): g^2 = (I, 2a_⊥) is a pure translation.2. Theorem 2 (Affine Normal Form): g is conjugate to g_0 = (P_k, a_⊥) via T_b with b = (a^k / 2)e_k, yielding g_0(s, u, y) = (-s, u + l, y) and M_g ≅ M° × R^2 where M° is the open Möbius strip.3. Theorem 3 (Null-Displacement Locus): N_g = {x : q_g(x) = 0} is non-empty iff a_⊥^2 ≥ 0. When non-empty, it consists of parallel timelike affine hyperplanes at x^k = (a^k ± √(a_⊥^2)) / 2.4. Theorem 4 (Topology and Pin Structures): M_g ≅ M° × R^2 implies π_1(M_g) ≅ Z, w_1(M_g) ≠ 0, w_1^2 = 0, and w_2 = 0. Consequently, M_g admits both Pin+ and Pin- structures.5. Theorem 5 (Affine Pin Lift Square): g~^2 = (εI, 2a_⊥) in G~ = Pin(1,3) ⋉ R1,3, where ε = ±1 for Pin+/Pin-.6. Theorem 6 (Mirror-Residue Pairing): I^σ_+ = -I_bar.7. Theorem 7 (Phase-Controlled Mirror Interference): |A_α|^2 = 4|I|^2 sin^2((α - 2φ)/2), where α is an independent U(1) character phase. Three conceptually distinct layers of data (Pin central sign ε, U(1) character χ, and spinor holonomy ρPin(g~)) are strictly decoupled. QFT applications are presented as a conditional research programme with open problems.