Endomorphic Closure and Invariant Topological Pullbacks of Recursive Functional Cascades

This paper establishes a homeomorphic pullback transformation law and functional closure for continuous spatial integral operators over unbounded domains within the internal topology of the function space. Let H(R,M) denote the class of strictly monotonic and absolutely continuous homeomorphisms mapping the unbounded real line R onto the open unit manifold M defined over the interval (0,1). This structural class induces an exact topological pullback, driven by the intrinsic geometry of the integrator function, which maps global Lebesgue–Stieltjes integrals identically onto M and constructs a coordinate-free representation within the invariant function space. Under this transformation, the associated Radon–Nikodym derivatives achieve a simultaneous and complete algebraic cancellation at the differential layer, yielding a stable functional composition independent of localized basis selections or artificial truncation variables. Within the open manifold M, the transformed representation establishes exact functional closure, and multi-fold convex combinations of operators proceed as stable recursive functional pairings while preserving the structural invariance of the functional class across arbitrary sequential mappings, thereby establishing an exact endomorphic closure of the invariant class under recursive operator compositions.(MSC2020): Primary 47G10; Secondary 47A05, 47B38, 28A25, 26A46.