Iterative Expansion of Generalized Mapping Theory and Description of Taylor Series Expansion

Based on the quadruple framework (object set A, operation set F, result set B, and generation relation ⊢ ) proposed in the Generalized Mapping Theory [1], this paper makes a key extension to address the limitations of its recursive mechanism, constructing an enhanced generalized mapping framework for dynamic mathematical process modeling. Inspired by the ideas of "explicit operation set" in Chapter 2 and "multi-stage process modeling" in Chapter 4 of the preprint, this study innovatively introduces a state transfer mechanism to solve the core problem of original function information loss in Taylor series expansion. By modeling the Taylor expansion as a recursive sequence of operations, the new framework achieves three breakthroughs: first, it explicitly separates the derivative calculation rules from the state transfer path, extending the "behavior mathematization" concept of the preprint to the field of pure mathematics; second, it establishes a state-driven recursive paradigm that completely retains the function itself and expansion point parameters. Using Taylor expansion as a paradigmatic case, this study verifies the universal descriptive ability of the enhanced generalized mapping for continuous mathematical processes, providing a mathematical tool with both operation traceability and structural expressiveness for dynamic systems such as numerical solutions of differential equations and function approximation, further realizing the theoretical vision of "establishing a complete path of behavior-result generation relations" proposed in the preprint.