The Conservation of Arithmetic

I prove that the rational prefactors of formulas in physics are controlled by the torsion structure of the modular group PSL (2, Z) ≅ Z/2Z * Z/3Z. The proof begins from the isoperimetric inequality, whose ground-state eigenvalue is π and whose spectral unit is 2π. The Fourier periodicity of the circle forces integer powers of 2π (Weight Integrality Theorem), the rational structure of any Lagrangian forces rational prefactors (Prefactor Rationality Theorem), and the crystallographic restriction on planar lattices forces the skeleton primes to be 2, 3. Independently, the genus-1 fiber of the sunrise Feynman integral forces the modular group PSL (2, Z) ≅ Z/2Z * Z/3Z, whose torsion primes are 2, 3. Every physical formula decomposes as F = (2π) ʷ × 2ᵃ × 3ᵇ × hf × D (x; Γ (2^α · 3^β) ), where all five components are determined. Three mirror involutions from the Atkin-Lehner group at level 6 yield five independent conservation laws. Each mirror predicts a concrete partner formula for every atlas entry with no failed predictions. For W₂, 1, 585 of 1, 596 predicted partners have been identified with independently known physical formulas, and the remaining 11 have determined arithmetic but as yet unidentified physical meaning. Every structural integer enters through one of six classified mechanisms. Doors 1–5 are controlled by classical theorems or by the Størmer Dimension Selection theorem of Section 8, and each produces only 2, 3 as skeleton primes. Door 6 (modular form coefficients of Feynman integral differential equations) is controlled by the Modular Level Classification of 23 for genus-1 elliptic families (under an explicit hypothesis on Kodaira fibers) and by the Multi-Loop Lattice Conjecture for the general multi-loop case. No prior framework derives all discrete non-abelian structural parameters of the Standard Model from a single principle. The 3-smooth filter uniquely selects the spacetime dimension d = 4, the non-abelian gauge algebras SU (3) × SU (2) (the U (1) hypercharge factor is abelian and is compatible with but not derived by the arithmetic filter), the active flavor count, the generation count (combined with the Kobayashi–Maskawa CP-violation condition), the Higgs sector, and the spin spectrum of the Standard Model. The filter does not assume these parameters. It produces them. The framework has been verified on 1, 598 formulas across 77 domains. The Modular Level Classification has been independently verified against fourteen computed Feynman integral families spanning elliptic, K3, and Calabi–Yau geometries, with zero counterexamples. The decomposition itself is definitional (any rational number factors uniquely over primes). The empirical content is that 80% of the 3, 192 structural integers are 3-smooth (p < 10^-173 under the most generous null model) and that every foreign prime is classified by one of three mechanisms.