We develop a symplectic geometry of atomic shear measures supported on the irregular primes of a modular curve. Starting from a space of dis-crete measures subject to a global valence constraint, we perform a strict symplectic reduction and obtain a reduced phase space equipped with a canonical Darboux form. The local coordinate functions that extractthe individual shear amplitudes are shown to be in Liouville involution, generating a completely integrable system whose invariant level sets areLagrangian tori. By lifting the functional equation of the associated automorphic L-functions to an anti-symplectic involution on this phase space, and under the established essential self-adjointness of the clutched conical Dirac operator, we prove a Confinement Theorem: the only invariant Lagrangianleaves compatible with the reflection symmetry are those whose spectralimage lies on the critical line Re(s) = 1/2. We conclude with a brief, explicitly programmatic dictionary that explores possible links between the resulting arithmetic geometry and structures appearing in gauge theory and the Standard Model. This workextends earlier constructions developed under the working title Relativistic Field Theory of Primes (RFTP), in which the irregular primes, the weight-12 valence constraint, and the associated clutching data were first introduced as geometric ingredients of an arithmetic field theory.