The Conservation of Arithmetic as a Theorem of Lagrangian Quantum Field Theory

I prove that the rational prefactors of formulas in Lagrangian quantum field theory are constrained by a conservation law governing their prime factorization. The constraint decomposes every observable into a universal structure whose components are individually classifiable. The proof assembles twelve published results spanning number theory, spectral geometry, and quantum field theory. The composition is new. I call this the Conservation of Arithmetic.