We present a complete first-principles derivation of the Standard Model gauge groupSU(3)c × SU(2)L × U(1)Y with three chiral generations of fermions, hierarchical masses,CKM/PMNSmixing, theseesaw mechanism for neutrinos, and Einstein—Cartan gravity withtorsion from one arithmetic object. The underlying structure is the resonant spectral tripleof the Bost—Connes system deformed by shear terms sourced by the class-group torsion ofirregular primes. In the deep infrared, the Chamseddine—Connes spectral action is evaluatedunder the global constraint of weight-12 principalization at the mirror point s = 6 of theassociated L-function.The fundamental duality between analytic volume (hyperbolic geometry, spectral flow,Seeley—DeWitt expansion) and algebraic rigidity (stabilizer symmetries at the two complexmultiplication points, adelic closure, weight-12 condition) is resolved by the irregular-primeshear, generating the pressure gradient ∇P that drives all symmetry breaking and massgeneration. Gauge representations emerge from the 3-cycle at τ = ρ and 4-cycle at τ = i;the bifundamental field Φ acquires a dynamical VEV; the 12×12 mass matrix, includinga natural seesaw for neutrinos, follows from breathing corrections sourced by class-grouptorsion. Numerical diagonalization over the full table of 47 irregular primes yields hierarchical fermion masses, realistic mixing, and a non-zero Jarlskog invariant. Confinement, thepositive mass gap, emergent proper time as a lapse function, and the Born rule all followfrom the same infrared thermodynamics.This construction realizes a complete arithmetic unification in which every low-energystructure is forced by the irreducible duality between analytic volume and algebraic rigidityunder weight-12 principalization. No external parameters beyond the shear amplitudes ofthe irregular primes are required.