Global uniqueness of Kerr black hole spin and inclination from a segment of the critical curve

Abstract: High-resolution observations are expected to probe the photon-ring structure of black hole images. The critical curve is the geometrically defined limiting locus on the observer's screen toward which successive higher-order photon subrings accumulate. For a Kerr black hole, the shape of this limiting curve depends on the dimensionless spin parameter $a$ and the inclination angle $i$. We investigate whether these parameters can be determined uniquely from a connected positive-length segment of the critical curve when the segment's position and orientation on the screen are unknown. For rotating non-extremal Kerr black holes with $0<a<1$ and $0&lt;i\leqπ/2$, we prove global uniqueness: if two such segments coincide as point sets after orientation-preserving rigid motions of the screen, then their Kerr parameter pairs coincide and the two rigid motions are identical. The proof uses an irreducible implicit polynomial to show that a segment determines the full algebraic curve containing it and then recovers the parameter pair $(a,i)$ from rigid-motion invariants of that polynomial.