A Mathematical Curiosity — II — The Extended Monoid Mext = H₀(Ã₂) × H₀(Ã₂)ᵒᵖ

We extend the 0-Hecke monoid H₀(Ã₂) by adjoining a dual copy H₀(Ã₂)ᵒᵖ with commuting generators. The resulting structure Mext possesses 576 elements and exhibits unexpected algebraic properties: a natural grading by charge, spectral degeneracies following binomial coefficients, and correlations between discrete curvature and algebraic observables. The simplicity of the construction contrasts with the richness of the emergent structure. This document presents the mathematical facts without interpretation.