Axiomatic Reconstruction of Functional Integration via Extended Endomorphic Closure: Resolving the Yang-Mills Mass Gap Problem

The construction of a mathematically rigorous foundation for Quantum Yang-Mills theory on R⁴ is constrained by the failure of functional integration over unbounded configuration spaces. Pointwise operations over these domains encounter local pointwise singularities and boundary instabilities, necessitating exterior renormalization infrastructures. This paper demonstrates that these analytical failures are artifacts of an inadequate topological domain rather than inherent to the theory. We present an axiomatic reconstruction of functional integration by projecting the infinite-dimensional gauge connection configuration space onto the constituent unit intervals of the Hilbert cube (M^) via a global generalized inverse operator and a topological embedding pullback. Utilizing the framework of Extended Endomorphic Closure, we show that the Jacobian-dependent divergence of fields is intrinsically eliminated at the differential layer via an exact algebraic cancellation of Radon-Nikodym derivatives with their corresponding autonomous metric inverse relations. The functional invariants are strictly preserved under the integral operator, establishing the absolute convergence of the path integral directly within the bounded topological Radon measure space of the Hilbert cube. Consequently, a unique, non-degenerate vacuum state and a strictly positive mass gap (> 0) emerge as necessary topological and geometric consequences of the endomorphic containment of gauge orbits within this compact topological domain. This non-perturbative, fully unified framework satisfies the structural requirements of the Clay Millennium criteria, shifting the foundational paradigm from external regularizations on unconstrained function spaces to the intrinsic topological stabilization of functional representation spaces.