Endomorphic Closure and Invariant Topological Pullbacks of Unbounded Integral Operators via Sequential Operator Compositions

This paper establishes a rigorous structural framework for invariant topological pullbacks that map continuous integral operators over unbounded supports onto the open unit interval M = (0, 1). Classical coordinate-dependent approaches to unbounded domains often suffer from tail instabilities or require artificial truncation parameters. To overcome these limitations, we configure the global geometry of the operator upfront at the functional level. The boundary horizons are intrinsically fixed by organizing the function space through the structural class H₀(R, M), which consists of strictly increasing and continuous homeomorphisms. By introducing the topological inverse mapping of the incoming function, the integration domains undergo a uniform partitioning directly over the invariant unit interval. This mechanism yields a stable functional composition that operates independently of derivative layers. Consequently, sequential map compositions achieve a strict endomorphic closure under recursive cascades within the invariant function space. Methodologically, the topological inverse mapping acts as an autonomous, coordinate-free tool. It establishes the exact spatial profile of the operator via homogeneous slice allocations without losing geometric information. Finally, this pullback configuration projects identical invariant representations across evolutionary equations, hereditary Volterra–Fredholm structures, fractional-order operations, and continuous functional convolutions. The resulting sequential composition cascade determines the exact finalized state at each step without generating dimensional degradation.MSC: Primary 47G10; Secondary 47A05, 47B38, 45G10, 28A25, 46E30