Euclidean Construction (OS) of Non-Abelian Yang–Mills in R 4 via Continuous Gauge-Covariant Regularization Induced by the TFMR Mother Kernel and Demonstration of Mass Gap by Exponential Decay of Gauge-Invariant Observables

This work presents a rigorous and fully traceable mathematical construction of pure Yang–Mills theory in four-dimensional Euclidean space, specifically aimed at addressing the Clay Mathematics Institute problem: the non-perturbative existence of the theory and the proof of a strictly positive mass gap in the gauge-invariant sector. The novelty of this approach lies in the introduction of a continuous gauge-covariant regulator, generated directly from the Theory of Multidimensional Feedback Flow (TFMR) through the so-called Universal Mother Kernel. This kernel is not a heuristic or secondary formal construction; it is a deterministic object that acts as a functional substrate from which both physical correlations and the spectral structure associated with the mass gap emerge. The architecture of the manuscript is designed to explicitly map each requirement of the Osterwalder–Schrader (OS) axioms, from regularity and symmetry to reflection positivity (OS3), one of the most delicate aspects of the problem. Instead of imposing conventional gauge-fixing procedures or introducing lattice-type discretizations with continuum limits that obscure essential details, a continuous functional regularization integrated into the modal flow of the TFMR kernel is proposed. This approach not only preserves exact gauge invariance at every level of the regularization but also enables the construction of well-defined Schwinger functions, the fulfillment of the OS axioms, and the derivation of a physical Hilbert space reconstruction with a Hamiltonian bounded from below. One of the main achievements of this work is the constructive and quantitative demonstration of the mass gap through the analysis of the functional transfer operator and the use of Doeblin-type bounds on the functional semigroup derived from the Mother Kernel. A uniform exponential decay of gauge-invariant correlators is obtained, allowing the establishment of the existence of a value Δ > 0 separated from the vacuum in the thermodynamic limit. This result satisfies the spectral formulation of the mass gap in a mathematically rigorous manner, not merely physical or numerical. In addition, the manuscript includes two key technical appendices: One dedicated to the formalization of the kernel insertion as a continuous regulator without compromising the OS axioms,Another focused on the TFMR ↔ Yang–Mills dimensional dictionary, ensuring consistency in units, constants, and fundamental scales such as the quantum chromodynamics scale parameter and the strong coupling constant g. The overall result is a complete, verifiable, modular, and Clay-grade framework for pure Yang–Mills theory in R⁴, within which existence, reconstruction, and the mass gap are not merely postulated but established step by step through explicit functional construction.