First-Principles Derivation of the Boundary Chiral Phase on Non-Orientable Spacetime: Twisted Pin^+(3,1) Bundle, Anomaly Inflow, and Cosmic Birefringence

In non-orientable topological cosmology, cosmic birefringence arises from a $CPT$-symmetric cosmological bounce on the quotient spacetime M₄ = (R × S³)/Z₂. While matching the effective chiral rotation angle Δθ₅ to the Dai--Freed boundary phase ΦDF = π/8 predicts β = +αₑₘ/2 ≈ +0.20905^∘ in agreement with ACT DR6 and Planck polarimetry, this identification was previously introduced as a physical ansatz. In this paper, we derive this boundary phase matching within a consistent topological framework based on index theory, spectral geometry, and anomaly inflow. First, because the spatial cross-cap RP³ SO(3) is parallelizable (w(TRP³)=1 w₂ = 0), Kirby--Taylor obstruction theory and Chamblin's Lorentzian reduction guarantee that M₄ unconditionally admits Pin⁺(3,1) structures, whereas Pin⁻(3,1) is obstructed.Second, through Peter--Weyl analysis on Spin(4) SU(2)L × SU(2)R, central inversion acts by (-1)2jL, establishing eigenspace parity staggering from group representation theory. Hurwitz zeta regularization yields η(RP³) = 1/4, while the Lichnerowicz theorem proves that Dirac zero modes vanish identically ($h = 0$).Third, under the Atiyah--Patodi--Singer framework on the regulated cylinder (assuming extension to the Z₂ -twisted cross-cap), the vanishing Pontryagin class (p₁(TM) ≡ 0) and vanishing asymptotic phase (η(S³) = 0) localize the total phase Φₜₒₜₐₗ = π/8 onto the throat cross-cap.Fourth, matching ΦDF = π/8 in the cobordism group Ω₄Pin⁺ Z₁₆ fixes the generator ν = 1. Under the principal-branch convention, formulating fermions in the standard real-mass basis fixes the axial rotation to α = -π/16 (2α = -π/8). Combined with the Standard Model charge sum ∑f qf² = 8, anomaly inflow yields θvac = -αₑₘ.Finally, in Carroll--Field--Jackiw electrodynamics, an effective condensate rolling from χ ≈ 0 at recombination to θvac = -αₑₘ today yields βCMB = -1/2Δχ = +αₑₘ/2 ≈ +0.209054^∘, while local Galactic dust remains unrotated.