The microscopic geometry of Wasserstein separation: cone geodesics, the exact single-step barrier, and the failure of fractional transport

The Wasserstein separation of the extreme broken-phase KMS states of a localized Bost-Connes system is purified from an algorithmic allocation problem into a problem of metric geometry. Over the full Kantorovich polytope of single-step transport plans - fractional, randomized, or otherwise - the optimal cost equals the deterministic scan bound of the adaptive coupling, certified on the toy model by an exact linear program (5/12, with channel masses reproducing the scan allocation term by term): fractional transport gains nothing. The distance of a diagonal displacement is computed in closed form, d_L = w(1 + 1/sqrt(2)) < 2w at equal weights: the pointwise second-order-cone constraint forbids a test function from saturating two orthogonal directions at one point, so coordinated multi-step moves are strictly cheaper than sums of steps. Against the certified SOCP window, the entire residual factor above the single-step barrier is carried by such cone discounts. The gap is geometry, not strategy; the one remaining instrument is the exact solution of the discrete infinity-Laplacian geodesic problem for the cone metric, and with that identification the campaign file is closed.