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. Our result requires no new particles or symmetries beyond w₁ᵀ ≠ 0 and w₂(M) = 0, which have independent testable consequences including CMB TB/EB = 0 and infinite fermion-boson decoherence ratio.