The Wasserstein separation of the broken phases: a finite-dimensional reduction and the quadratic chaos bound

For a localized Bost-Connes system in its symmetry-broken phase, the Connes-Rieffel distance W_1 between the two extreme KMS states was previously squeezed between w_0 Pi_infinity(beta) and w_0. This paper sharpens both sides. A finite-dimensional reduction sandwiches W_1 between Pi_(N,infinity) W_1^(N) and Pi_(N,infinity) W_1^(N) + 2 rho_N, where the truncated distance W_1^(N) is an exactly solvable second-order cone program, the lower bound is exact on truncations, and the tail variance rho_N^2 = 2 sumj>N w_j^2 t_j / (1-t_j)^2 is explicit. Testing against the degree-one odd Walsh chaos yields the closed-form lower bound Pi_infinity (w_0^2 + 4 sum_j w_j^2 t_j^2/(1-t_j)^2)^(1/2), strictly sharper than the squeeze at every finite beta and collapsing onto it in the deep-freeze limit. Near-optimal test functions are shown to be odd and asymptotically finite-level. Two questions remain and are recorded: a matching upper bound, and whether the optimal profile is parity-measurable.