The M-Operator System: A Novel Framework for the Exact Evaluation of Non-Elementary Integrals via Auxiliary Function Mapping

This paper formalizes the M-Operator System, a novel mathematical framework designed to systematize the selection and application of auxiliary functions g(x) within thecontext of Cauchy’s Mean Value Theorem. By defining the M-Operator as a functionalmapping M : A(t) → g(x), we establish a rigorous algebraic basis for the exact analytical evaluation of non-elementary integrals. We demonstrate that the invariant meanpoint c is an intrinsic eigenvalue-like property of the M-transformation. The paper provides a comprehensive classification of M-Operator kernels, ensuring a unified approachto transcendental and special function integration.