Cosmic Birefringence from Non-Orientable Spacetime Topology: Scenario, Prediction, and Observational Tests

Recent analyses of Cosmic Microwave Background (CMB) polarization from \textit{Planck}, \textit{WMAP}, and the Atacama Cosmology Telescope (ACT DR6) report evidence for isotropic cosmic birefringence $\beta \sim 0.2^\circ - 0.35^\circ$. We propose a scenario where cosmic birefringence arises from a $CPT$-symmetric cosmological bounce modeled as a non-orientable four-dimensional spacetime quotient $\mathcal{M}_4 = (\mathbb{R} \times S^3)/\mathbb{Z}_2$. Integrating out Standard Model fermions in this non-trivial $\text{Pin}^+(3,1)$ bundle induces a global spectral asymmetry governed by the Atiyah--Patodi--Singer $\eta$-invariant on the spatial section $\mathbb{RP}^3$. Using Peter--Weyl decomposition and Hurwitz zeta regularization, we evaluate $\eta(D) = 1/4$. Via the canonical Dai--Freed boundary phase $\Phi_{\rm DF} = (\pi/2)\eta(D) = \pi/8$ and exact 1-loop Chern--Simons matching, under the phase-matching ansatz $\Delta\theta_5 = \Phi_{\rm DF}$, the theory predicts a canonical birefringence angle $|\beta|_{\rm canon} = \frac{1}{2}|\theta_{\rm vac}| = \alpha_{\rm em}/2 \approx 0.2091^\circ$ (with an alternative direct phase map predicting $|\beta|_{\rm alt} = \alpha_{\rm em} \approx 0.4181^\circ$). The canonical prediction agrees with ACT DR6 ($\beta = 0.215^\circ \pm 0.074^\circ$, pull $-0.08\sigma$) and the global weighted mean ($\beta = 0.264^\circ \pm 0.058^\circ$, pull $-0.95\sigma$). In the expanding background, tight Thomson scattering before recombination randomizes polarization while Hubble friction locks the scalar condensate at $\chi \approx 0$; coherent birefringence accumulates during transparent free-streaming, while local Galactic dust is unrotated ($\beta_{\rm dust} \approx 0$). First-principles integration on $\mathbb{RP}^3$ proves strict scale invariance ($\beta_\ell = \text{const}$) for $R_0 \ge 15.8 \, c/H_0$. Realistic Fisher forecasts demonstrate that LiteBIRD and Simons Observatory will detect $\alpha_{\rm em}/2$ at $> 11.6\sigma$ and separate it from $\alpha_{\rm em}$ at $11.6\sigma$, providing a quantitative empirical test of spacetime topology.