We demonstrate that non-orientable spacetime topology, characterized by a non-vanishing first Stiefel-Whitney class wT¹ ≠ 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 → θF = π → CPT² = (−1) F → non-orientable spacetime → wT¹ ≠ 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 Pinᶜ ℤ₈ ⊕ ℤ₂ classifications, with the forgetting map f (k) = (0, k mod 2) (mapping to the ℤ₂ factor of Ω⁴Pinᶜ, 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 = 4ηdw mod 8 equals the bulk invariant α = 8ηbulk mod 16 reduced modulo 8. Our work repositions wT¹ ≠ 0 from an arbitrary assumption to a motivated choice grounded in CPT symmetry, while maintaining honest boundaries about the distinction between “motivation” and “proof. ”
Fangyuan Hao (Tue,) studied this question.