A Mathematical Curiosity — III — Thermodynamic Properties of the Extended Monoid Mext(Ã₂)

We study the extended 0-Hecke monoid Mext = H₀(Ã₂) × H₀(Ã₂)ᵒᵖ under the involutive transformation π ↦ 2π − 1, which maps idempotent generators to spin operators. The resulting six-spin system inherits the braid relations as geometric constraints on two coupled triangular graphs. Monte Carlo simulations reveal: (1) a phase transition at critical temperature Tc ≈ 1.1; (2) near-constant discrete curvature κ across all temperatures (variation < 2%); (3) nonzero absolute magnetization ⟨|M|⟩ → 2 as T → 0, indicating spontaneous symmetry breaking. The ground state cannot be neutral: geometric frustration forbids symmetric configurations. These properties emerge from the algebraic structure without additional assumptions.