A Formal Proof of Spectral Recovery in the Prime Gravity Pipeline: From Divisor Weights to Zeta Zeros via the Explicit Formula

We provide a rigorous proof that the two-stage Prime Gravity pipeline — divisor weight accumulation followed by Gaussian gravitational well construction and log-space Fourier analysis — necessarily recovers the nontrivial zeros of the Riemann zeta function as spectral peaks in the limit N → ∞. The proof proceeds by establishing that each computational step corresponds to a well-characterized mathematical operation: the divisor weight sieve implements the Fundamental Theorem of Arithmetic, the Gaussian wells construct a mollified Chebyshev function, and the log-space FFT approximates a Mellin transform whose poles are precisely the nontrivial zeros of ζ(s). We provide explicit error bounds showing convergence. We further state three open conjectures connecting the topological invariants of the prime gravitational manifold to the Riemann Hypothesis, including a proposed equivalence between persistence lifetime bounds and the critical line constraint. This paper is the formal companion to Gleason, 2026, DOI: 10.5281/zenodo.19421086, which presents the construction and experimental results.