Luoshu Chiral Potts Model: Exact Ground State and First-Order Phase Transition

The Luoshu Chiral Potts Model (LCPM) is defined on the Luoshu lattice—a 3×3 magic square—with directed generation (step +1) and restriction (step +2) bonds on the Z₅ state space. We prove that the only configurations satisfying all 31 bonds are the five Uniform Shift Configurations (USC), establishing a ground-state degeneracy exactly equal to |Z₅|. In the temporally extended Yunqi Potts Model (YPM), three all-or-nothing theorems (A, B, C) guarantee that each of the nine palaces either matches entirely or fails entirely, with no intermediate states possible. This algebraic structure forces a first-order phase transition at J_t^c = 288/295 ≈ 0.9763, corresponding to a block-level binary model with non-standard bond energy E_bond = -(9J_t)/4(1-s)(1-s'). We further show: (i) the minimum cooperative unit equals five steps, determined by the Z₅ internal symmetry rather than external parameters; (ii) noise robustness up to ρ_c = 0.80, with the exact identity Δ_B = Δ_A²; (iii) transfer matrix analysis at finite temperature showing that the sharp first-order transition becomes a smooth crossover consistent with 1D Landau-Peierls expectations, while remaining effectively sharp for β ≳ 5.