Spectral Disjointness and Cumulative Boundedness in the Prime Gravity Manifold

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.