Non-Orientable Spacetime as the Topological Foundation of CPT Symmetry

We demonstrate that non-orientable spacetime topology, characterized by a non-vanishing first Stiefel-Whitney class w_T¹ ≠ 0, provides the natural geometric foundation for CPT-symmetric cosmology. Building on Boyle-Turok’s CPT-symmetric universe and the Pin⁺ cobordism classification, we establish a logical chain: CPT symmetry as spacetime structure → Pin⁺ structure → θ_F = π → CPT² = (−1)F → non-orientable spacetime → w_T¹ ≠ 0. The key step follows from the Kirby-Taylor theorem Ω⁴Pin⁺ = ℤ₁₆ and is independent of the unitarity/anti-unitarity of T̂ (McRae has shown T̂ is necessarily anti-unitary in Lorentzian signature [4]). The bordism-invariant nature of the η-invariant (via the APS index theorem) further implies the vanishing of CPT-odd SME coefficients. The Cross-Pin Structural Hierarchy connects the bulk ℤ₁₆ classification to domain wall ℤ₈ and Pin^c ℤ₈ ⊕ ℤ₂ classifications, with the forgetting map f(k) = (0, k mod 2) (mapping to the ℤ₂ factor of Ω⁴Pin^c, preserving only parity information). The bulk-to-domain-wall map S(k) = k mod 8 (the natural projection, with kernel {0, 8} ≅ ℤ₂) is determined by η-invariant consistency: the domain wall bordism invariant β = 4η_dw mod 8 equals the bulk invariant α = 8η_bulk mod 16 reduced modulo 8. Our work repositions w_T¹ ≠ 0 from an arbitrary assumption to a motivated choice grounded in CPT symmetry, while maintaining honest boundaries about the distinction between “motivation” and “proof.”