We present Hypatian Physics, a Lean4-verified framework that unifies quantum gravity, particle physics, and cosmology via a recursive algebraic-geometric formulation. Our approach formalizes fundamental structures such as recursive Jordan algebras, p-adic metric spaces, and recursive renormalization group flows. The theory yields empirically testable predictions, including gravitational wave echoes, non-zero fermion masses via Yukawa couplings, and fractal spacetime geometries consistent with cosmological observations.

Formal Proof of Recursive Topological Vertex, Recursive Mirror Symmetry, and Recursive Renormalization Group Flow. The sequence C(n)λµν converges to a unique limit C(∞)λµν. The convergence of the sequence is established through the geometric decay of the recursive terms and the application of the Banach Fixed-Point.