CPT Theorem on Non-Time-Orientable Spacetimes: A Fermion-Boson Dichotomy from Pin+ Anomaly

We establish the full CPT structure on non-time-orientable spacetimes, yielding a fermion-boson dichotomy: fermions acquire an additional phase −1 under the generalized CPT operation CPT' while bosons remain unaffected. The standard CPT theorem is recovered as the w₁ᵀ → 0 contraction limit of this complete theory—an irreversible contraction, as the Spin representation cannot reconstruct the Pin⁺ representation. Using the bordism classification Ω₄^{Pin⁺} = ℤ₁₆, the Pin⁺ representation theory (where the time-reversal generator e₁ satisfies e₁² = +1, implying unitary time reversal T' with (T')² = +1), and the bordism equivariance framework, we prove that: (1) on Pin⁺ manifolds with w₁ᵀ ≠ 0, the time reversal operator is unitary rather than antiunitary, and the CPT operation becomes CPT'; (2) the fermion Hilbert space splits into superselection sectors separated by the time-direction flip surface Σ, and by equivariance the anomaly phase θ = π (element 8 in ℤ₁₆) appears as a sector-mapping phase: T' = −U_Σ for fermions, T' = +U_Σ for bosons; (3) this yields CPT'|ψ_F⟩ = −(CPT)_std|ψ_F⟩ while bosons are unaffected—a parameter-independent, quantized dichotomy absent in previous CPT anomaly frameworks; (4) the anomaly phase and (T')² = +1 are fully consistent; (5) the standard CPT theorem is the irreversible contraction limit w₁ᵀ → 0 of this complete theory. Equivalently, our result establishes a no-go theorem: on non-time-orientable spacetimes, a CPT operator of the standard form and equal CPT behavior for fermions and bosons are mutually exclusive.