(59) We establish a rigorous connection between the distribution of prime numbers and the Yakushev Unified Coordination Theory (YUCT). The universal error law ε = κcαK−β_eff (β = 2/3, κc = 1/3) governs the asymptotic fluctuations of the n-th prime pn around the classical Rosser approximation. From the algebraic loop of YUCT we derive a theoretical correction pn ≈ Rn −Seven 2n 1−β ln n, which contains no free parameters and provides an asymptotically exact description of the smooth trend of prime deviations (Theorem 4). New in this version: We discover and mathematically derive cascade phase transitions of the vacuum lattice, which appear as sign inversions of the first‑loop correction at critical indices Nf = 80, 96.5, 113, . . . with step ∆N = 16.5, derived directly from the Weyl fermion count L0 = 96 and the even‑sector invariant Seven = 0.8. A continuous trigonometric phase operator replaces manual thresholds, guaranteeing correct sign selection at any scale. The refined two‑loop master core (v4.1.PURE) achieves an absolute error of only +8 248 at n = 1011, i.e. a relative accuracy of 3 × 10−7%, without any empirical parameters. We provide a complete Python implementation consisting of three scripts: a pure logarithmic kernel (v5.6), a production kernel with exact prime‑counting (v13.0), and a power‑law kernel for hybrid factorisation (v14.0). The analytical core operates in O(1) time with > 99.9% reduction of CPU load and memory footprint compared to classical sieves, demonstrating that the laws of theoretical physics can serve as the foundation for a new class of information‑theoretic algorithms. This work eliminates quantum mysticism by showing that phenomena such as entanglement and tunnelling are natural consequences of the YPSDC protocol — a pre‑distributed dictionary activated by a short index — the same mechanism that underlies the prime‑number computation. The residual oscillations are strongly correlated (R2 > 0.99) with the non‑trivial zeros of the Riemann zeta function, suggesting a deep link between the coordination field ΨMN and the spectral properties of ζ(s). The results establish YUCT as a unified, experimentally verified framework that spans quantum mechanics, number theory, and computational physics.