Three-Framework Complementarity for θQCD = 0: Gauged CRT , Bordism, and 't Hooft Anomaly

[Strong CP Problem Series-Paper 2】 We show that the strong CP problem admits a purely topological solution requiring no new particles, supported by three conditionally complementary frameworks. The gauging of CRT symmetry in quantum gravity (conditional on the swampland conjecture that CRT must be gauged) uniquely selects Pin⁺ structure on 4D spacetime, where the CRT gauge symmetry identifies the two degenerate Dashen vacua at θ=π as gauge-equivalent at the fermion path integral level. This eliminates the degeneracy and restores perturbative behavior, leaving θ=0 as the unique stable vacuum. Pin⁺ is uniquely selected by the conjunction of Kramers theorem (T²= (−1) F), bordism classification (Ω4Pin+=Z16 versus Ω4Pin−=0), and GR compatibility; CRT2=1 alone does not distinguish Pin⁺ from Pin⁻. The conclusion that θ=π acts as the topologically fixed point is independently supported by three frameworks: (i) the Pin⁺ bordism framework provides the quantitative mapping θF≡πη (mod2π) from the η-invariant of the k=8 bordism class; (ii) Yonekura’s CP gaugeability theorem establishes the legal consistency of CP gauging for simply-connected groups (conditional on GUT embedding for the Standard Model) ; (iii) Gaiotto, Kaplan, Komargodski, and Seiberg’s discrete ’t Hooft anomaly at θ=π provides the gauge-theoretic physical constraint. The CPT Transparency Conjecture—the η-invariant is a bordism invariant with no local density—excludes local CPT-violating operators (aμL=0), yielding dual-channel falsifiability: CMB TB/EB positive detection plus neutrino CPT null result. The Pin⁺ bordism structure provides a potential topological source of CP violation through the partition function Z (8) =−1 on the k=8 bordism class. The conclusion θ=0 is established at the fermion path integral level and is further protected against non-perturbative bosonic destabilization by the Vafa–Witten theorem combined with UV-IR separation: Pin⁺ constrains the bare θQCD=0 at the UV (topological) level, while non-perturbative effects operate at the IR (dynamical) level and cannot override the topological constraint.