We demonstrate that the infrared dynamics of the resonant spectral triple in the Bost-Connes system, deformed by irregular-prime shear, implements a precise arithmetic analogueof the classical Legendre transform. Starting from a classical Poisson manifold whose coordinates are the shear amplitudes {Iu2032p} and the bifundamental mixing field Φ, we construct an explicit action functional whose Legendre transform yields a Hamiltonian on the cotangent bundle. The weight-12 winding condition is imposed as a classical first-class constraint, and Dirac reduction produces a reduced symplectic manifold Pred. Deformation quantization of this reduced space, driven by the modular derivation Dp acting asymmetrically through Φ on the two complex-multiplication stabilizers (Z/6Z at τ = ρ and Z/4Z at τ = i), generates a non-commutative star product whose leading commutator is non-vanishing and proportional to the stabilizer contrast transmitted by Φ. This framework naturally produces: (i) proper time τ as the integrated thermodynamic work along the flow, with lapse N ∝ det(∇2P) · Tr(Φu2020Φ); (ii) a non-removable relative phase between up- and down-type sectors, yielding the Jarlskog invariant as a direct dynamical consequence of the asymmetric action of the real endomorphism J (J2 = −Id) forced by the holonomy defect; and (iii) definite outcomes via deterministic Hamilton—Jacobi flow to a unique weight-12 attractor, with probabilities given by normalized infrared Boltzmann weights from the entropy functional S(ϕ). The construction draws explicit parallels with Dirac’s deformation of Poisson bracketsinto commutators and Tate’s adelic Poisson summation, providing a unified non-commutativestructure that bridges arithmetic geometry and quantum field theory. The imaginary unit i emerges necessarily as the spectral manifestation of the real endomorphism J required to close the holonomy defect. All physical features follow rigorously from the arithmetic data without external postulates.