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 $\mathcal{M}_4 = (\mathbb{R} \times S^3)/\mathbb{Z}_2$. While matching the effective chiral rotation angle $\Delta\theta_5$ to the Dai--Freed boundary phase $\Phi_{\rm DF} = \pi/8$ predicts $\beta = +\alpha_{\rm em}/2 \approx +0.20905^\circ$ in agreement with ACT DR6 and \textit{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 $\mathbb{RP}^3 \cong SO(3)$ is parallelizable ($w(T\mathbb{RP}^3)=1 \implies w_2 = 0$), Kirby--Taylor obstruction theory and Chamblin's Lorentzian reduction guarantee that $\mathcal{M}_4$ unconditionally admits $\text{Pin}^+(3,1)$ structures, whereas $\text{Pin}^-(3,1)$ is obstructed.Second, through Peter--Weyl analysis on $\text{Spin}(4) \cong SU(2)_L \times SU(2)_R$, central inversion acts by $(-1)^{2j_L}$, establishing eigenspace parity staggering from group representation theory. Hurwitz zeta regularization yields $\eta(\mathbb{RP}^3) = 1/4$, 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 $\mathbb{Z}_2$ -twisted cross-cap), the vanishing Pontryagin class ($p_1(TM) \equiv 0$) and vanishing asymptotic phase ($\eta(S^3) = 0$) localize the total phase $\Phi_{\rm total} = \pi/8$ onto the throat cross-cap.Fourth, matching $\Phi_{\rm DF} = \pi/8$ in the cobordism group $\Omega_4^{\text{Pin}^+} \cong \mathbb{Z}_{16}$ fixes the generator $\nu = 1$. Under the principal-branch convention, formulating fermions in the standard real-mass basis fixes the axial rotation to $\alpha = -\pi/16$ ($2\alpha = -\pi/8$). Combined with the Standard Model charge sum $\sum_f q_f^2 = 8$, anomaly inflow yields $\theta_{\rm vac} = -\alpha_{\rm em}$.Finally, in Carroll--Field--Jackiw electrodynamics, an effective condensate rolling from $\chi \approx 0$ at recombination to $\theta_{\rm vac} = -\alpha_{\rm em}$ today yields $\beta_{\rm CMB} = -\frac{1}{2}\Delta\chi = +\alpha_{\rm em}/2 \approx +0.209054^\circ$, while local Galactic dust remains unrotated.