We present a self-contained technical development of the reduced variational structureunderlying the Relativistic Field Theory of Primes (RFTP) in the weight-12 sector. AfterDirac reduction by the global valence constraint associated with the modular discriminant∆(τ ), the effective dynamics on the reduced space are generated by a convex action functional whose long-time minimizers are atomic measures supported at irregular primes. Weconstruct three explicit objects: (i) a linear isomorphism from the odd Galois eigenspacesfurnished by the Herbrand—Ribet theorem to local sections of the global spinor bundle, intertwining Galois action with Gauss—Manin parallel transport; (ii) a quantitative exponential convergence result for the Hopf—Lax semigroup showing that every admissible initial measure converges to a unique atomic minimizer at a rate controlled by the spectral gap of theHessian; and (iii) a term-by-term identification of the Weierstrass product of the regularizeddeterminant of the limiting operator with that of the completed Riemann xi function Ξ(s).These constructions render rigorous a four-step variational obstruction that selects discretearithmetic configurations whenever a continuous configuration would require an effective local exponent exceeding two. The entire development is phrased in the language of deterministic Hamilton—Jacobi—Bellman dynamics on a reduced symplectic manifold, thereby supplying both the analytic core of the weight-12 theory and the notational and conceptual bridge to a universal formulation over all weights.