Topological Origin of CPT Realization on Non-Time-Orientable Spacetimes

Non-time-orientability (w₁ᵀ ≠ 0) selects the Pin⁺ framework, yielding two tiers of results. Tier 1 (theorem-level, signature-independent): the bordism group Ω₄ᴾⁱⁿ⁺ = ℤ₁₆ constrains fermion content, requiring right-handed neutrinos; the Arf invariant gives Arf(q) = 1, corresponding to θ_F = π (mod 2π) at the ℤ₂ level (the complete ℤ₁₆ invariant is θ_F = 2π νₖ/16); the anti-unitarity of T̂ with T² = -1 (McRae) implies Kramers degeneracy. Tier 2 (conjecture with strong evidence, Framework 2: CPT' acts on spinor components as a unitary Clifford-algebra operator): the Pin⁺-unique theorem T² = (-1)ᴲ---via the sector-mapping operator T' = T * γ⁵ and anomaly matching with e^(iθ_F) = -1---yields a fermion-boson dichotomy. The algebraic relation CPT'² = -I on fermionic spinor components (from {γ⁰, γ⁵} = 0) provides independent support but is not Pin⁺-unique. The bridge from spinor-component results to Fock-space operators remains an open problem. Four testable predictions follow: (1) CMB topology signatures, (2) Majorana neutrinos from θ_F = π, (3) right-handed neutrinos nν_R = 3 mod 16, and (4) Kramers degeneracy.