This paper establishes an empirical and theoretical framework connecting the spectral properties of the Prime Gravity fluctuation sequence Vₑrr (n) to the unconditional boundedness of its cumulative potential W (N). The central observation is domain-theoretic: Riemann zeta zeros generate log-periodic Ingham waves incommensurable with the linear Fourier basis, so projecting Vₑrr onto linear frequencies constitutes a disjointness test rather than a zero-detection apparatus. FFT peak amplitudes measured across eleven scales (N = 5, 000 to N = 10, 000, 000) grow as N⁰. 1326 near γ₁ = 14. 13 and N⁰. 1555 at the global maximum — both dramatically sub-random-walk (α ≪ 0. 5) — while the cumulative potential W (N) scales at α ≈ 0. 49, consistent with a Central Limit Theorem accumulation of √N incoherent channels. A companion Power Spectral Density experiment confirms a flat white-noise plateau across four decades of linear frequency, with no low-frequency energy accumulation. These results are interpreted as empirical evidence of hyperuniformity: the Sieve of Eratosthenes is maximally disjoint from linear oscillations, preventing any rogue-wave constructive interference in W (N). The paper identifies three algebraic bridges — zero topological entropy of the integer sieve, linear disjointness via Sarnak's Möbius Disjointness framework, and a CLT bound on W (N) — which together constitute a complete proof architecture. Combined with Paper 5's topological equivalence theorem (DOI: 10. 5281/zenodo. 19430053), closing all three bridges would unconditionally prove the Riemann Hypothesis. Appendix A presents a computational experiment using a novel Holographic Geodesic Transformer — a custom transformer architecture inheriting geodesic attention symmetrization from Project Obsidian — trained to minimize major arc spectral energy in Vₑrr. Five controlled versions (V1–V5) systematically varied the input feature set across arithmetic embeddings, exact Möbius-formula Ramanujan sums, and prime gap features. The Stripe Coefficient of Variation (a quantitative measure of attention phase-locking onto prime positions) converged monotonically from 1. 009 to 0. 185 as feature classes were added. The experiment empirically decomposes major arc cancellation into two finite components — modular structure (Ramanujan sums) and local density (Cramér-scale gap features) — and one irreducible infinite component identified as the Euler product tail of the Hardy-Littlewood singular series. This residual 0. 185 floor constitutes direct empirical evidence that Bridge 2 requires Halász's theorem or a Sarnak entropy classification operating over the full infinite prime product — confirming the paper's theoretical conclusion that no finite local or modular tool is sufficient.
Timothy Gleason (Sun,) studied this question.