Formalizing Habit Evolution: A Recursive Symbolic Model Using Alpay Algebra

This paper presents a formal application of Alpay Algebra, a symbolic-recursive mathematical system introduced in 2025, to the modeling of habit formation. It demonstrates how recursive state transformations—via operators like the partial fold can model the shift from externally-driven resistance to internally-stabilized behavior. The specific problem solved is a symbolic convergence case: modeling the formation of a daily exercise routine over recursive steps until symbolic alignment is reached. While this work does not address traditional continuous-time differential systems or physical dynamical models, it introduces a novel formalism for recursive identity transitions and symbolic behavioral convergence. The model works entirely in discrete symbolic space and is best interpreted as a framework for cognitive dynamics, habit engineering, and symbolic AI behavior modeling. This is not a control-theoretic Lyapunov solution, nor a general optimization algorithm, but a first step toward a new class of problems: recursive symbolic cognition and formal identity evolution. Author: Faruk Alpay ORCID: 0009-0009-2207-6528