The 600-Cell as Universal Algebraic Substrate: Physical Constants, Biomolecular Structure, and Linguistic Universals from E8 Invariant Theory

The regular 4-polytope known as the 600-cell ${3, 3, 5}$ — with 120 vertices realizing the binary icosahedral group 2I as quaternions in $S^3$ — exhibits structural correspondences with empirical patterns across several unrelated domains. We catalog and analyze: (i) decomposition of five empirical formulas for dimensionless Standard Model parameters (proton/electron mass ratio, lepton mass ratios, fine structure constant, Weinberg angle) into algebraic primitives of 2I/E#sub[8]; the integer exponents appearing in these formulas (7, 11, 14 = 2·7) are drawn from the eight $E_8$ invariant exponents ${1, 7, 11, 13, 17, 19, 23, 29}$. A 200,000-trial Monte Carlo null test on the m_p/m_e formula template finds *zero* random parameter choices reaching 12-digit agreement, against our formula's 13.1-digit precision; analogous null tests for the other four constants show weaker but positive separation from the random best. (ii) An algebraic completion argument for the 22 proteinogenic amino acids predicts hydroxyproline and hydroxylysine as candidates for direct genetic encoding, completing the inscribed 24-cell. (iii) Statistical association between seven crystal systems and seven non-polar 2I conjugacy classes ($chi^2 = 44.58$ on 98 elements with internally-assigned classes; external assignment test pending). (iv) Cross-linguistic agreement of synonymy density with the algebraic value $30^2 / 120 = 7.5$ (8/8 corpora within ±2.5). (v) The 600-cell as the 4-dimensional substrate of icosahedral quasicrystals via the cut-and-project construction. We propose that the nine irreducible representations $V_i$ of 2I (dimensions 1, 2, 2, 3, 3, 4, 4, 5, 6) serve as universal categories operating across atoms, biomolecules, morphemes, and rotation frequencies. The framework is supported by 56 computational experiments and 30 formalized hypothesis projects, with falsifiable predictions and clearly demarcated speculative components