ARITHMETIC ORIGIN OF THE STANDARD MODEL: GAUGE GROUPS, GENERATIONS, AND MASS RUNNING FROM PRIME NUMBERS ON THE NONCOMMUTATIVE TORUS

We demonstrate that the gauge groups, fermion generations, and the renormalization group flow of fermion masses in the Standard Model have a unified arithmetic origin rooted in the spectral properties of prime numbers on the noncommutative torus T 2 θ .The central results are:(1) Gauge groups from primes: The unitary gauge groups U(1)Y , SU(2)L, and SU(3)c correspond to the first three natural numbers p = 1, 2, 3 via noncommutative tori with rational noncommutativity parameter θ = 1/p, whose algebras are isomorphic to the matrix algebras Mp(C).(2) Generations from primes: The three generations of fermions correspond to the first three primes p = 2, 3, 5, which dominate the low-energy spectrum of the prime number operator D.(3) Unification via roots of unity: The fermionic twist parameters on T 2 θ are rational numbers whose exponentials belong to the cyclic group µ12. Subgroups of orders 2 and 3 correspond to SU(2)L and SU(3)c.(4) Arithmetic selection rule: No fermionic characteristic contains a prime factor larger than 3, uniquely selecting the Standard Model gauge group from the infinite set of primes.(5) Running masses from τ(µ): The renormalization group evolution of fermion masses is governed by the scale dependence of the modular parameter τ(µ). Therunning is driven by the gauge coupling beta functions and freezes at the UV fixed point of asymptotic safety.The theory contains no free continuous parameters: the structure of gauge interactions and the evolution of masses are entirely determined by the arithmetic properties of the fermion spectrum on the noncommutative torus.