A Closed-Form Operator Identity for the Aggregation and Differentiation of Truncated Probability Distributions in L1(R)

Version 5 Update: Operational Topology and Operator-Theoretic Characterization This version maintains absolute adherence to the invariant core mathematical result and O(1) operational efficiency established since Version 1. While the exact analytical outcome and its performance behavior remain identical throughout the evolutionary timeline of this research, this update fundamentally elevates and enriches the theoretical apparatus surrounding the derivation. Theoretical Premise: The core assertion remains fully continuous and invariant: the application of standard spatial-temporal convolution to concurrent coordinate-indexed fields introduces an artificial domain misallocation, leading to unphysical geometric distortions. To formalize this transition, the framework has been successfully reframed from an applied stochastic signal summation problem into a rigorous Linear Operator Theory and Functional Analysis paradigm within the metrical closure of L1(R). Key Revisions: Axiomatic Refinement: Transitioned the conceptual language from stochastic estimation to a pure operator-theoretic taxonomy, employing smooth manifold constraints and local spatial support classifiers. Topological Proofs: Introduced formal deductive proofs validating the strict boundaries of product topology and confirming the mathematical necessity of the pointwise convex linear combination over standard convolution. Terminological & Structural Alignment: Overhauled the Abstract, Introduction, and Technical Sections to achieve absolute conceptual harmony with the foundational criteria of modern functional analysis journals.