We present a unified arithmetic field theory in which a single topological defect at prime691 on the modular curve X(1) forces a singular Legendre point — the arithmetic breakdown of the classical map from canonical velocity to canonical momentum. Eta refraction disperses analytic volume from the flat Eisenstein background, which concentrates at the high-symmetry elliptic points of X(1). Conductor-9 triality clutching performs principalization, restoring an effective invertible Legendre map and allowing the standard variationalprinciples of Lagrangian and Hamiltonian mechanics to hold on a stable low-energy soliton.= The Ramanujan τ (n) coefficients supply the matrix elements of the quantized curvaturetorsion potential on the clutched bundle. The Leech lattice provides the global analytic volume capacity, while temperature scaling via the Boltzmann constant turns the arithmetic engine thermodynamic. Stationary action (Hamilton’s principle) selects the physical trajectories, and acceleration (time derivative of constraints) enforces global smoothness via the product formula. Explicit 1-to-1 mappings are given between RFTP arithmetic objects, classical mechanics, and quantum structures. Gravity and electromagnetism emerge as different projections of the same clutched density. The Higgs field is the radial principalization process, and the hydrogen soliton is a concrete low-temperature realization whose Balmer spectrum is quantized by the clutched modes with τ (n) transitions. The zeta zeros appear as the self-adjoint spectrum of the hamiltonized soliton, giving variational realizations of the Riemann Hypothesis and the Birch—Swinnerton-Dyer conjecture. Historical context forLagrangian/Hamiltonian mechanics, Jacobi, Liouville, and Dirac is provided throughout.