Why θ = π is Physically Unacceptable on Pin+ Manifolds: Topological Constraint Meets Cosmological Exclusion

[Strong CP Problem Series-Paper 3] We establish that the strong CP parameter θ_QCD is constrained to zero on Pin⁺ manifolds through a two-stage argument: (i) a topological constraint from the Pin⁺ structure that restricts θ to {0, π}, and (ii) a physical constraint that excludes θ = π due to its cosmological consequences. Specifically, the time-reversal symmetry inherent in Pin⁺ manifolds forces θ ∈ {0, π}, while θ = π triggers spontaneous CP violation and domain wall formation that is cosmologically unacceptable. This resolves the tension identified by Witten (2015), who showed that θ = π is topologically well-defined on Pin⁺ manifolds as a topological insulator state—our analysis confirms this topological legitimacy while demonstrating that θ = π is excluded not by topology alone but by physical cosmological constraints. We note that θ_QCD = 0 constrains only the bare angle; the physical parameter θ̄ = θ_QCD + arg det M_q additionally requires arg det M_q = 0, which is a separate question beyond the scope of this work. We further establish that this two-stage argument is the unique consistent resolution within the Pin⁺ bordism framework, where the Pin⁺ bordism group Ω₄Pin⁺ = Z₁₆ provides the quantitative topological foundation. The result θ = 0 is thus not a direct consequence of Pin⁺ structure alone, but of Pin⁺ structure combined with the physical requirement of cosmological viability.