A Universal Closed-Form Solution for the Exact Summation and Gradient of i.i.d. Heavy-Tailed Distributions

Version Update: Continuous Domain Generalization and Theoretical Refinement This version maintains absolute adherence to the original core premise established in Version 1—formalizing the aggregate response as a pointwise operator identity—while expanding the theoretical framework from discrete constraints to a continuous operational spectrum. Theoretical Premise:The core assertion remains invariant: the application of spatial-temporal convolution to co-existing profiles is a domain misallocation, and the aggregate response is governed strictly by coordinate-consistent linear superposition. This update clarifies the structural stability of the system without introducing any alterative modifications to the original thesis. Key Revisions: - Domain Generalization: The theoretical derivation has been elevated from discrete, homogeneous sequences to a continuous functional space. The unified operator formulation now natively accommodates non-homogeneous, arbitrarily scaled, and non-uniformly translated field configurations. - Conceptual Fortification: The text has been refined to emphasize the exactness and mathematical certainty of the deterministic summation directly within the functional domain, completing the transition of the exposition into a fully mature, continuous framework. - Terminological Alignment: Refinement of the Abstract, Discussion, and Conclusion to ensure absolute conceptual consistency with the continuous nature of the identity.