A Mathematical Curiosity: Stability and Convergence in the 0-Hecke Monoid of Affine Type $\tilde{A}_2$

We present a minimal algebraic structure — the 0-Hecke monoid $H_0(\tilde{A}_2)$ — defined by three generators and five relations. This rewriting system possesses remarkable properties: guaranteed convergence through termination and confluence, geometry controlled by discrete Ollivier-Ricci curvature, and a crystalline structure constrained by Kashiwara's theorems1. The simplicity of the axioms contrasts with the richness of the consequences2. This document is an invitation to explore this structure, with no claims beyond mathematics3.