Prime Gravity: A Riemannian Manifold Construction on the Integers with Emergent Prime Detection and Zeta Zero Recovery

Abstract

We construct a Riemannian manifold on the integers in which prime numbers emerge as geometric features - specifically, as the minima of a gravitational potential landscape induced by the divisibility structure of the natural numbers. The construction proceeds in two stages: (1) a divisor weight accumulation that identifies primes with 100% precision and recall as zero-weight positions, and (2) a Gaussian gravitational well model centered at detected primes with von Mangoldt weighting. We demonstrate that the spectral content of this manifold, extracted via Fourier analysis in logarithmic space (a discrete Mellin transform), recovers the first 20 nontrivial zeros of the Riemann zeta function with a mean alignment error of 0.121. Statistical significance is established at 5.4 sigma against density-matched null models. Two independent encodings of the same arithmetic prime identity and divisor structure are shown to share matching topological signatures under persistent homology despite zero pointwise correlation, establishing that the arithmetic truth resides at the topological level. No primes are specified as input. No free parameters are tuned. The integers enter; the zeta zeros emerge.