Kinematic Obstruction in Non-Abelian Gauge Fields: Arithmetic Geometry and the Mass Gap

We propose a geometric framework to address the Yang-Mills Existence and Mass Gap problem by mapping the gauge connection Aμ to a Hilbert Space spanned by the prime numbers (H_P). Within this arithmetic information manifold, we identify a fundamental Kinematic Obstruction to the existence of arbitrarily low-energy states in non-Abelian theories. We demonstrate that the self-interaction term [Aμ, Aν ] acts mechanically as a Spectral Scattering Operator, creating topological knots in the prime basis. We derive that “untying” these knots to reach the vacuum state requires a geometric turning radius that is strictly forbidden by the Diophantine stiffness of the arithmetic medium. Consequently, we obtain a finite lower bound ∆ > 0 for the spectrum, defined by the minimum arithmetic work required to support the non-trivial topology of the gauge group against the rigidity of the vacuum.