This work develops the second entry of the arithmetic geometry branch of the Conservative Motion Theory (CMT–F series). CMTF2 constructs a Fredholm–analytic formulation of the Taniyama–Shimura modularity theorem by establishing elliptic reflection positivity and analytic Fredholm continuity for the modular kernels associated with elliptic curves over ℚ and their corresponding weight-2 newforms. A key result is that the modular correspondence E \leftrightarrow f induces only trace-class perturbations between the associated CMT kernels, ensuring equality of their regularized determinants \Xi_{\kappa,E}(s)=\Xi_{\kappa,f}(s), and guaranteeing real-zero persistence across the arithmetic–analytic correspondence. This provides an analytic alternative to deformation-theoretic proofs of modularity (Wiles–Taylor), expressing modular equivalence as a conservation principle: \text{Modularity} = \text{Fredholm continuity} + \text{Reflection positivity}. The paper situates CMTF2 as the bridge between the Fermat (F I) and BSD (F III) programs, and as a core part of the unified arithmetic Fredholm framework developed across the CMT project.