Topological Constraint on the QCD Vacuum Angle from Pin+ Structure on Non-Time-Orientable Spacetimes

We prove that the QCD topological angle θ is constrained to zero on non-time-orientable spacetimes equipped with Pin⁺ structure, providing a topological resolution of the strong CP problem without introducing axions or new symmetries. The argument proceeds in five steps. (1) A non-vanishing first Stiefel-Whitney class w₁ᵀ ≠ 0 of the time-direction bundle, together with w₂ (M) = 0, implies that M is non-time-orientable and admits a Pin⁺ structure. (2) The QCD θ-term changes sign under orientation reversal: θ → −θ. (3) On a Pin⁺ manifold, orientation reversal is a topological necessity — orientation-reversing hypersurfaces Σ must exist. (4) Since orientation reversal is intrinsic to the Pin⁺ structure, θ and −θ must describe identical physics on M, forcing θ = 0 or θ = π; equivalently, the parameter space collapses from S¹ to S¹/ℤ₂ with fixed points 0, π. (5) The value θ = π is excluded because it admits no self-consistent vacuum: the Pin⁺ compatibility operator T' connects the two would-be degenerate vacua, and since T' is a representation-theoretic compatibility condition (not a dynamical symmetry), it cannot be spontaneously broken. This selects θ = 0 as the unique stable vacuum, yielding θ̄ = 0 provided arg det Mq = 0. The conclusion θ = 0 is independently reinforced by three convergent channels: CPT topological locking — the bordism-invariant definition of CPT forces (aL) = 0 in the Standard Model Extension via a separate topological channel — the three-fold invisibility of the anomaly element 8 ∈ ℤ₁₆, and CRT gauging — which derives Pin⁺ as a theorem rather than assumption. Our result requires no new particles or symmetries beyond w₁ᵀ ≠ 0 and w₂ (M) = 0, which have independently testable consequences including CMB TB/EB polarization from global topological CPT modification and infinite fermion-boson decoherence ratio, and is falsifiable through dual-channel experiments: positive detection of CMB TB/EB (global topological observable) and negative exclusion of neutrino CPT violation (local SME observable).