(57) We present a rigorous algebraic derivation that unifies the universal constants of the Yakushev Unified Coordination Theory (YUCT) — the coordination exponent β = 2/3 and the algebraic invariants S_odd = 1.2, S_even = 0.8 — with the universal constants of chaos theory: the Feigenbaum constants δ ≈ 4.6692 and α ≈ 2.5029. Using the exact analytic relation 2α² = πδ and the YUCT‑derived approximation π ≈ S_odd φ² (where φ is the golden ratio), we establish the fundamental equation 2α² ≈ (S_odd φ²) δ. This result demonstrates that the constants governing the emergence of complex order (β, S_odd) and those governing the universal route to chaos (α, δ) are not independent but are algebraically linked through the geometry of the circle and the golden ratio. We further strengthen the unification by analysing Wolfram’s Rule 30 cellular automaton — the epitome of computational irreducibility — within the YUCT framework. Treating Rule 30 as a system with minimal coordination efficiency (K_eff → 1), we derive an exact expression for the growth of coordination entropy and analytically determine the critical depth t_crit ≈ 157.7 at which the central column becomes perfectly mixing. The calculation uses only the universal constants κ_c = 1/3 and q = (3/2)^{1/3} from YUCT. A numerical verification script is provided, confirming entropy saturation at the predicted generation. The critical depth is shown to be algebraically related to the Feigenbaum constants via the relation ln t_crit ∼ δ/φ, establishing a direct bridge between deterministic chaos and the coordination error law. The small deviation Δπ = |π − S_odd φ²| ≈ 4.8×10⁻⁵ is interpreted as a falsifiable prediction of YUCT, pointing to a measurable granularity of physical geometry. Experimental tests using gravitational wave interferometry (LISA), cosmological surveys (Euclid), and quantum interferometry are proposed. This work provides the first mathematical bridge between YUCT, the renormalisation group theory of critical phenomena, and the theory of computational irreducibility, offering profound implications for the prediction of systemic collapse and the experimental verification of YUCT's core tenets. This record contains the complete multilingual version of the article (English, Russian, German, Chinese, Arabic, Japanese, Korean).