We present the Relativistic Field Theory of Primes (RFTP) as a rst-principles derivationof low-energy physics from a single global object: the adelic spectral triple equipped with one rst-class constraint (the weight-12 central grading) in the min-plus semiring. Local sheardata at irregular primes evolve under the Hopf-Lax formula the explicit tropical linearevolution that realizes the dynamical Legendre transform. This single min-plus constraintgenerates a gauge symmetry (tropical equivalence), produces proper time as its evolution parameter, and yields an emergent EinsteinCartan geometry whose curvature 2-form is the tropical variety created by dierential relaxation across Galois eigenspace channels. Weprove that the functor from the category of such adelic spectral triples to the categoryof Lorentzian/EinsteinCartan geometries is an equivalence on the arithmetic subcategory.The Riemann Hypothesis appears as the statement that the tropical linear ow remains causal and dispersionless if and only if all non-trivial zeros lie on the critical line the unique stable tropical attractor of the theory. Higher SeeleyDeWitt coecients generate tropical polynomials in the shear amplitudes that control connement strength and the mass gap.