Managing Entropy Through Recursive Depth: A Symbolic Optimization Model Using Alpay Algebra

This paper introduces a symbolic optimization problem solvable using Alpay Algebra, a recursive framework for managing state transitions, entropy, and stability. It focuses on a new class of symbolic decision-making problems: how to manage recursive depth over time in order to absorb disruptive entropy with minimal cost. Unlike traditional control theory or numerical optimization, this model operates entirely within symbolic recursion layers and formal identity operators. The system faces increasing entropy over three time steps and must decide when to apply a symbolic depth-increase operator to avoid penalties. The solution demonstrates how symbolic operators like R_compress and recursive depth (lambda layers) can be used to make optimal tradeoffs between cost and resilience. While this paper does not solve physical dynamical equations or probabilistic systems, it opens a new domain: symbolic entropy scheduling and recursive identity planning. This represents a novel contribution to symbolic computation, cognitive modeling, and theoretical AI behavior design. Author: Faruk Alpay ORCID: 0009-0009-2207-6528