Mathematics as the Bridge Between Admissibility and Usability: A Structural Clarification Within the Paton System

This paper clarifies the structural role of mathematics within the Paton System. It establishes that mathematics operates on admissible systems rather than determining system validity. Admissibility defines whether a system may exist, while mathematics transforms admissible structure into usable, measurable, and operational forms. This positions mathematics as a bridge between structural validity and application across domains.