Supplement A: Convergence of the Partial Fold Operator in Alpay Algebra

This technical note provides a concrete numerical and visual supplement to the symbolic framework of Alpay Algebra. It focuses on the convergence behavior of the Partial Fold Operator — a recursive identity evolution mechanism defined by Faruk Alpay. Using a two-dimensional state vector example, the paper derives the closed-form solution of state convergence under the partial fold rule and visualizes the trajectory toward the internal target state [1, 1]. The work demonstrates how symbolic identity compression, as proposed in Alpay Algebra, yields predictable and exponential convergence aligned with layer-wise recursion principles. This supplement includes a LaTeX-formatted explanation, mathematical proof, and animation visual. It is intended to accompany the main publication: Alpay Algebra: Foundations for Recursive Compression and Intelligent Evolution (DOI: 10.5281/zenodo.15306575). Author: Faruk Alpay ORCID: 0009-0009-2207-6528