Why 137: Masses, Couplings, and Mixing Angles from Z₉

This paper derives 32 fundamental quantities — particle masses, the electroweak vacuum expectation value, gauge coupling constants, mixing angles, and two cosmological parameters — from a single algebraic structure: the integers modulo 9 under multiplication. The framework requires one axiom (the choice of algebraic structure), one physical rule (the Born rule for transition probabilities), and one energy scale (the electron mass). Everything else follows from number theory. The results span 12 orders of magnitude and include: absolute masses for all three charged leptons, the proton (to 0.05 parts per billion), neutron, all six quarks, the W and Z bosons, the Higgs boson, and selected mesons; all three gauge coupling constants (electromagnetic, weak mixing, and strong); all four PMNS neutrino mixing parameters; the Cabibbo angle and CKM hierarchy; the neutrino mass-squared hierarchy; the dark energy density of the universe; and the number of particle generations. An algebraic uniqueness theorem proves that Z9 is the only modular ring whose structural arithmetic simultaneously produces 1/alpha = 137 and m_p/m_e = 1836. A conservative statistical analysis places the probability of these results arising by coincidence at less than 10^-65 on the full set. The Standard Model requires 19 free parameters. This framework requires 1.