Canonical Unitary Isomorphisms and the Topological Closure of $L^2(\mathbb{R}^n)$ Domains Under Isometric, Differential and Spectral Invariance

The construction of a mathematically rigorous foundation for continuous differential operations over unconstrained function spaces is constrained by the analytical failure of localized differential operators over unbounded configuration spaces. Pointwise operations over these domains encounter localized singularities and singular gradient growth, necessitating exterior regularization infrastructures. This paper demonstrates that these analytical failures are artifacts of the selected coordinate representation rather than inherent properties of the functional field itself. An axiomatic reconstruction of domain alignment is established by projecting the unconstrained, unbounded Euclidean space ($\mathbb{R}^n$) onto the interior of the bounded open unit ball coordinate domain ($\mathbb{B}^n$) via a static canonical unitary isomorphism operator ($\mathcal{U}_\Psi$) derived directly from first principles. Under this stabilized representation space, continuous functional operations remain governed exclusively by canonical unitary equivalence and the Friedrichs extension theorem, yielding a unique, closed, self-adjoint operator extension across the dense state domain. At the differential layer, the Jacobian-dependent divergence of fields is structurally eliminated via the concurrent, pointwise cancellation of the generalized coordinate Jacobian determinant against the inverse Radon-Nikodym density derivatives embedded within the operator structure. Concurrently, any asymptotic boundary divergence at the spatial limit is structurally absorbed into the geometric filtration of the canonical metric formulation, preserving the invariant properties of the space. Through this exact algebraic cancellation across the coordinate boundaries of the open unit ball ($\partial\mathbb{B}^n$), the underlying functional invariants remain strictly preserved, establishing the strict inner product integrity and absolute convergence of continuous functional operations within the $L^2$ Hilbert space class. Consequently, global domain stabilization and an absolute operator norm identity ($\Vert\mathcal{U}_\Psi\Vert \equiv 1$) emerge as necessary geometric consequences of the strict isometric isomorphism. This structural preservation permits the global regularity properties secured within the representation space to map directly onto the continuous function space of the original configuration without distortion, spectral leakage, or information loss, shifting the foundational paradigm from external regularizations on unconstrained function spaces to the intrinsic topological stabilization of aligned coordinate domains.MSC: Primary 47G10; Secondary 47B25, 46E30, 28A25