Topological Equivalence of the Prime Gravitational Manifold and the Riemann Zeta Landscape

We demonstrate that the prime gravitational manifold is topologically equivalent to the landscape of the Riemann zeta function |ζ(1/2 + it)| under persistent homology. At N = 500,000, the two manifolds produce identical H0 feature counts (799 vs 799) and statistically indistinguishable H1 lifetime distributions (KS p = 0.729). A multi-scale analysis reveals emergent convergence: the H1 match transitions from clearly distinguishable at N = 10,000 to firmly equivalent at N ≥ 200,000, exhibiting a topological phase transition. Three independent manifolds — encoding prime identity, divisor structure, and the zeta function itself — share the same topological fingerprint. This paper is the third in a series establishing the Prime Gravity framework.