α = 1/k: A Geometric Framework for Standard Model Gauge Couplings and fermion mass ratios

This paper explores whether the Standard Model's coupling constants and fermion mass ratios can be derived from three-dimensional geometry rather than fitted to data. Starting from the dynamical symmetry of 3D Euclidean space (SU (3), via the Fradkin construction), a chain of mathematical theorems — through the centre of SU (3), complex multiplication on the elliptic curve y² = x³ − 1, and A4 modular symmetry at the level-3 congruence subgroup — produces Chern-Simons levels that correspond to the gauge coupling constants, and modular form values at the CM point that correspond to fermion mass ratios and mixing angles. The framework generates 21 zero-parameter predictions spanning gauge couplings, quark and lepton mass ratios, CKM and PMNS mixing angles, the CP-violating phase, neutrino mass splittings, and Newton's gravitational constant. Of these, 17 agree with measurement to better than 5% and all 21 to better than 10%. The headline formula is mc/mₜ = 1/360 — three integers (4, 3, 30) from the geometry, no fitting. The algebraic identity |Y₁ (τ₀) |² = 9/10, confirmed by the Chowla-Selberg formula at discriminant −3, underpins the mass predictions. The derivation chain has eleven links from 3D space to the Standard Model. The paper identifies which are theorems (nine), which are strongly motivated (one), and which remain explicit choices (one, affecting only the two weakest predictions). Falsifiable predictions include sin²θ₁₂ = 3/10, |Vᵤs| = 1/√20, the neutrino mass splitting ratio Δm²ₛol/Δm²ₐtm = 3/100, and δCP (PMNS) = 198. 4°. A standalone computation script reproducing all 21 predictions from a single input (dim = 3) is provided. This is a candidate geometric framework for the Standard Model, not a completed theory. The companion document details all working, verifications, and open questions.