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 with M = 800 sample points, the two manifolds produce statistically indistinguishable H1 (loop) lifetime distributions (KS p = 0.729) and closely matching H0 (connected component) lifetime distributions (mean lifetimes within 1%). 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 — produce converging persistence lifetime distributions, suggesting a universal topological fingerprint of arithmetic. v2.0: Corrected H0 count discussion (799 = M-1 is an algebraic property of Vietoris-Rips, not a property of the landscape); emphasis shifted to lifetime distributions as the meaningful geometric comparison.