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

Abstract

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.