The Four Clocks Ledger: Arithmetic Construction of 4D Yang–Mills with Positive Mass Gap

Abstract | This paper presents a complete constructive solution to the Yang–Mills existence and mass-gap problem. Starting from six axioms defining a single adelic scalar field π(x^μ) over the minimal prime set {2,3,5,7,11}, we derive the full non-Abelian gauge theory on ℝ⁴ via the adelic embedding operator ℰ_p. The halting clock supplies a discrete, self-regularizing path-integral measure. Vacuum rigidity, enforced by the Residue Bootstrap Operator, yields a quadratic penalty that produces super-exponential suppression of non-rigid paths via a convergent polymer expansion. All Osterwalder–Schrader axioms are satisfied by construction, with explicit verification of reflection positivity. All regulators are independent, short-distance correlators match perturbative Yang–Mills exactly, and a strictly positive mass gap survives the continuum limit. Explicit 16×16 generators, Yukawa/CKM/PMNS matrices at 50-digit precision, and anomaly traces (identically zero) are provided in the supplementary materials. The theory contains no free parameters. Independent verification and scrutiny is invited.