Formalization and Generalization of an Asymmetry Operator in Classical Field Frameworks

This work presents the formalization of a non-standard mathematical operator introduced to encode asymmetry-driven effects in classical field equations. The operator was originally proposed as an effective organizational tool in previous work addressing the emergence of gravitational behavior from gradient asymmetries, but lacked a rigorous mathematical definition.Here, the operator is formulated as an autonomous differential object with explicitly defined domain, codomain, structural properties, and limiting behavior. Its relation to standard differential operators is analyzed, showing that classical operators are recovered as particular cases within appropriate limits. This ensures consistency with established formulations while extending the mathematical structure to incorporate generalized asymmetry effects.The scope and limitations of the proposed framework are explicitly discussed. The operator is not presented as a complete physical theory nor as a replacement for existing fundamental frameworks, but as a complementary mathematical layer that reorganizes existing quantities in a consistent and generalized manner. Consistency checks and validation criteria are addressed within the classical, non-relativistic regime.This work provides a necessary foundational step by transforming an initially heuristic construction into a well-defined operatorial framework. The resulting formalization establishes a rigorous basis for future analytical, numerical, applied, and pedagogical studies, and complements previous investigations on the emergence of gravity from asymmetry-driven gradient structures. Author’s Note on Version 1.1.This revision updates the original document for publication on Zenodo. Minor syntactic, typographical,and structural adjustments have been introduced to improve technical clarity and consistency. The heading hierarchy has been standardized, notation has been refined, and formula tags in section titles have been replaced by explicit mathematical characters for improved readability.This version formally establishes the scientific priority of the asymmetry operator framework andthe associated non-standard concepts introduced in this work, including the recovery of neighborhoodstructure as a formal tool within differential and operatorial analysis.All non-standard definitions, operator constructions, and conceptual interpretations presented hereinconstitute original contributions by the author and are not part of the classical mathematical or physical canon unless explicitly stated otherwise.This document establishes authorship and priority over the asymmetry operator formalism under the authorship of Carlos Ariel Vargas.