Abstract
<jats:p>We present the complete mathematical derivation of the Yakushev Law of Coordination — the fundamental principle that governs all stable structures in the Universe. The law is formulated in a single statement and is derived from one axiom (fractal scale invariance) and the definition of the coordination protocol YPSDC. We prove three structural theorems: the minimal dictionary entropy is exactly two bits; the spatial dimension is necessarily three; and time is an emergent measure of coordination imperfection. The universal error law is derived in its correct steady power-law form ε = κcαK−β eff with β = 2/3 ≈ 0.67 and κc = 1/3. The algebraic loop constants are Sodd = 1.2, Seven = 0.8. The scaling quantum q = (3/2)1/3 generates the universal mass ladder. The fine-structure constant α −1 ≈ 137.036 follows from the topological invariant of the compact fibre F18 and the universal shift σ = (Sodd − Seven)/2 = 0.20 without any empirical fit. The cosmological constant ΩΛ = 2/3 emerges from the same topology. The electroweak scale is obtained from the Weyl fermion count L = 96 with a geometric regulator (κcπYUCT) −1/2 that eliminates any phenomenological fit. As a further consequence, the same algebraic loop yields a parameter-free asymptotic correction to the distribution of prime numbers (Theorem 4). Three formalised postulates – the universal error law, quantised depth, and phase periodicity – provide a closed analytic model for prime fluctuations, bounded by the Planck scale. The law makes eight concrete, falsifiable predictions across physics, biology, number theory and cosmology, and contains no adjustable continuous parameters.</jats:p>