We present a complete analytic derivation of color confinement and a positive Yang—Millsmass gap within the Relativistic Field Theory of Primes (RFTP). The derivation is carriedout on a shear-deformed resonant spectral triple constructed from the Bost—Connes systemand its low-temperature KMS states. Irregular primes deform the underlying GNS quadraticform, generating a pressure gradient between the two complex-multiplication points τ = ρand τ = i on the modular curve. A continuous infrared renormalization-group flow thenproduces an infrared dichotomy: states whose phase lies outside the weight-12 lattice acquirean infinite entropy barrier (leading to confinement), while color-singlet states remain stable with a strictly positive mass gap controlled by the dominant shear amplitude.The proof relies on the asymptotic behavior of the heat kernel on the full GNS Hilbertspace, an effective Hamilton—Jacobi dynamics in which the arithmetic pressure gradient acts as the driving force, and the thermodynamic interpretation of the infrared divergenceas the failure of an arithmetic Legendre transform (principalization) for fractional-phase states. Both confinement (via the Wilson-loop area law) and the mass gap are shown toemerge directly from the infrared structure of a single spectral object without additionalassumptions or fine-tuning. The dynamical vacuum expectation value that will generatefermion masses in extensions of this framework is recovered as the equilibrium of the samedriven dynamics. This work establishes that color confinement and the mass gap are not accidental features of non-abelian gauge theories but inevitable consequences of the arithmetic geometry of the primes when viewed through the lens of a resonant spectral triple.