Flytrons within the Law of the Unity of Being of the Triune Substance

We prove the existence of a mass gap in quantum Yang–Mills theory within the ZEBTS axiomatic framework, addressing the Clay Millennium Prize problem. Our approach relies on the flytron model, introducing a discrete‑arithmetic structure that encodes gauge bundle topology and controls the spectrum. Each state is assigned an integer weight derived from holonomy traces and arithmetic invariants, determining energy via $E_\pi = \Lambda \cdot \log \Psi(\pi)$ as a discrete topological quantum number. Discreteness ensures no continuous spectrum near zero, while topological dependence guarantees persistence across all sectors. The Euclidean theory satisfies the Osterwalder–Schrader axioms, yielding a valid Hilbert space, self‑adjoint Hamiltonian, and local observables. We prove $\Psi(\pi) \ge 2$ for nontrivial loops, establishing an explicit lower bound $E_\pi \ge \Lambda \log 2$ so $(0, \Lambda \log 2)$ is empty. This is formalized in Theorem MG for $\mathrm{SU}(N)$ Yang–Mills theory, confirming a gauge‑invariant, stable mass gap. Our method unites topology, spectral theory, operator algebras, and arithmetic into a coherent structure for nonperturbative QFT, validating the foundational principle that key physical properties emerge rigorously from a unified mathematical framework.