Polarization-dependent observational signatures of Weyl-coupled photons around a black hole
A photon coupled non-minimally to the Weyl tensor propagates differently in its two linear polarization states: the vacuum around a black hole becomes a birefringent medium. On a Schwarzschild background the coupling $α$ splits the photon sphere and the critical impact parameter of the two polarizations, producing a double shadow and a polarization-dependent photon ring; the two shadow edges separate by $62\%$ of the shadow radius at $α/M^2=0.75$. We compute a unified set of wave-optical and geometric-optical observables for both polarizations across a range of $α$: greybody factors, absorption cross sections, differential cross sections and their glories, a backward birefringence signal, shadows, photon rings, and null trajectories. The two-polarization scattering matrices are extracted with a Riccati--Hankel matching scheme that keeps the phase-sensitive observables reliable, and the geometric-optics chain is cross-validated with three independent ray tracers. The cleanest diagnostic is the backward signal: it is protected by parity and vanishes identically at $α=0$.Laboratory bounds on $α$ render these effects unobservable for supermassive black holes; order-unity couplings would require a black hole of primordial mass.