The Universal Action Threshold: A Unified Topological Solution to the Yang-Mills Mass Gap and the Infinite Mass Electron

The most profound unresolved problem in the Standard Model of particle physics is the Yang-Mills existence and mass gap hypothesis. Standard continuous Quantum Field Theory (ℝ⁴) cannot resolve this mathematically due to infinite-energy ultraviolet divergences at zero spatial separation. This paper provides a cross-disciplinary solution by applying the Cartesian Relativity Framework (CRF), modeling spacetime as a discrete, integer-based coordinate topology. By introducing the Universal Action Threshold (A_T), we demonstrate that the transition from unrendered energy to localized mass is governed by a strict, invariant energy-to-mass threshold. We prove the universality of this threshold by algebraically extracting the QED Schwinger Limit, the Heisenberg Uncertainty Principle, and the Bekenstein Bound from the exact same fundamental formula, unifying their embodied structural floors. We further demonstrate how this discrete topology organically eradicates the infinite self-energy of the electron, eliminating the need for renormalization. Furthermore, we demonstrate that "virtual particles" failing this rendering threshold are pruned by the lattice algorithm, exerting zero macroscopic gravitational tension, thereby organically resolving the Cosmological Constant Problem (the Vacuum Catastrophe). Applying this framework to non-Abelian gauge fields, we mathematically derive the Yang-Mills mass gap (Δ) as the mandatory, strictly positive mass required to balance the topological force interaction of gluons against the universe's fundamental structural limit (ħ). Finally, we resolve the Wightman axiomatic trap by proposing the ℝ⁵ postulate, demonstrating that continuous ℝ⁴ geometry incorrectly permits energy to scale infinitely, and that the mass gap is the definitive projection eigenvalue of unrendered fields mapping onto physical reality.