Elliptic Integrals in Kerr Spacetime: Explicit Reduction, AGM Evaluation, and Photon-Sphere Universality

This work shows that the geodesic structure of Kerr spacetime admits a complete exact reduction to the three classical complete elliptic integrals K(k), E(k), and Π(n,k). The resulting expressions are evaluated with the Gauss-Legendre arithmetic-geometric mean (AGM), which converges quadratically and provides certified error bounds at every step. All fourteen Kerr observables considered here are unified by a single master generating function,Z(λ) = ∫ P(r,λ) / √R(r) dr.Eleven follow from direct elliptic evaluation, while the ISCO radius, horizon area, and Hawking temperature arise as a critical point and exact elliptic identities of the same structure. The result is a structural unification of Kerr geodesic physics: the apparent distinction between elliptic and algebraic observables disappears, and both dynamical and thermodynamical quantities are shown to arise from the same underlying algebraic curve. Three theorems establishing the universality of the photon sphere are also proved. Version 3 note:Unification of all 14 observables (including the three previously considered algebraic) under a single master generating function Z(λ) (Section 9), and a few other changes Version 2.1 note:Version 2 was inadvertently replaced by an earlier draft during upload. Version 2.1 restores the intended revised manuscript, including the updated abstract, conclusion, and companion-work conjecture. Version 2 note: Revised abstract and conclusion, clarified the distinction between the exact geodesic framework and the approximate spectral extension related to quasinormal modes and the neutron-star normal-mode transition, and added a working conjecture for companion work.