Elliptic Integrals in Kerr Spacetime: Explicit Reduction, AGM Evaluation, and Master Generating Function

The geodesic structure of the Kerr spacetime, within the class of observables expressible as separated Hamilton-Jacobi integrals with rational prefactors, reduces exactly to three complete elliptic integrals: K(k), E(k), and Pi(n,k), evaluated via the arithmetic-geometric mean with certified error bounds at every step. All 14 geodesic observables are organised by a pair of separated generators Z_r and Z_theta built from the radial and polar Hamilton-Jacobi data. Eleven arise by direct evaluation, one (the ISCO radius) as the critical point k_r = 0 of the radial elliptic modulus, and two (horizon area A and Hawking temperature T_H) via exact elliptic identities in the horizon modulus k_E. Three universality theorems for the photon sphere are proved. Paper companion : Genus Transitions in Regular Rotating Black Holes: Algebraic Structure, Vanishing Cycles, and Frame-Dragging in Echo Observability Final version note: The explicit recovery map (Appendix A) is new: absent from previous versions, it collects the reduction path for all fourteen observables in one place. The master generating function, previously treated as a single object, has also been separated into radial and polar generators for structural clarity; the main text has been revised accordingly. Quasi-final version note: Fixed visual artefacts from LaTeX encoding. Added Cygnus X-1 and GRO J1655-40 as benchmark objects. Reformatted to RevTeX (PRD two-column). Pre-final version note: Version pre-final note: Title updated to "Elliptic Integrals in Kerr Spacetime: Explicit Reduction, AGM Evaluation, and Master Generating Function". Applications of the AGM engine to M87* and Sgr A* added (but the mathematical content itself remains unchanged).Minor typographical corrections and removal of informational artefacts from previous versions. Some visual artefacts persist, but as they proved resistant and are purely stylistic rather than informative, they were deemed not worth pursuing further (temporary). 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.