Achromatic, spin-odd Kerr EVPA as a null Frenet--Serret torsion integral on the photon ring

Abstract

We compute the achromatic gravitational imprint that Kerr spacetime leaves on linear polarization at the photon ring. Recasting parallel transport in a null Frenet--Serret frame yields a single scalar evolution law for the electric-vector position angle. On the observer's screen, the Kerr-minus-Schwarzschild pattern on the direct critical curve is nonzero, strictly odd under spin reversal after a half-turn azimuth relabelling, and tightly confined to a thin annulus. Using backward-shot, Carter-separated geodesics with midpoint RK2 transport, we achieve second-order convergence and degree-scale amplitudes that grow monotonically with spin and inclination (RMS $\simeq 0.5$--$2^\circ$ for $a/M\gtrsim 0.8$, $i\gtrsim 60^\circ$). Three independent constructions -- Frenet--Serret line integral, explicit Levi--Civita transport of the polarization vector, and the phase of the Walker--Penrose constant -- agree ray by ray. We then define a parity-odd ring estimator that is intrinsically achromatic after standard wavelength-squared regression, symmetry-protected against common even-parity systematics, and compressed into low azimuthal modes. This yields a minimal two-parameter template (spin and inclination) for mm/sub-mm polarimetry of horizon-scale rings in sources such as M87$^\ast$ and Sgr~A$^\ast$. The pipeline enables either a detection of the strong-field parallel-transport phase induced by frame dragging or informative upper limits.