Axiomatic Reconstruction of Functional Integration via Extended Endomorphic Closure: Resolving the Navier-Stokes Existence and Smoothness Problem

The construction of a mathematically rigorous foundation for the Three-Dimensional Navier-Stokes Equations on $\mathbb{R}^3$ is constrained by the failure of localized differential evolution over unbounded configuration spaces. Pointwise operations over these domains encounter local pointwise singularities and gradient explosions, necessitating exterior regularization 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 velocity field configuration space onto the constituent unit intervals of the Hilbert cube ($M^\infty$) 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 trajectory evolution directly within the bounded topological Radon measure space of the Hilbert cube. Consequently, universal viscous stabilization and a strictly bounded trajectory norm ($\mathcal{E}_m(t) \le \mathcal{E}_m(0) < \infty$) giving global $C^\infty$ smoothness emerge as necessary topological and geometric consequences of the endomorphic containment of functional trajectories within this compact topological domain. This non-perturbative, fully unified framework satisfies the structural requirements of the Clay specifications, shifting the foundational paradigm from external regularizations on unconstrained function spaces to the intrinsic topological stabilization of functional representation spaces.