Lemma 14 ([Communication costs and closed classes]). An endpoint edge from i to j satisfies (u_i-u_j)^2≤C_βei . For a contact edge witnessed by a fixed-position interval of length τ≤1 , (u_i-u_j)^2≤C_βτ-1 (e_i+e_j). The kernel K is …

The walk jump rate is uniformly bounded by a constant depending on ℓ . The probability of making no fresh jump during a period is at least e^ -Cβ ; the probability of any prescribed allowed seam image is at least 1/ℓ . Thus an endpoint entry of K is at least c_β . For a contact interval, require exactly one jump from line i to line j during that interval and no other jumps, and choose endpoint φ(j) . This gives K_ i,φ(j) ≥c_βτ . Compare u_i to u_j through u_ φ(j) , using also the endpoint bound for j ; this proves. A fresh walk changes skeleton line only through contact, and every possible seam endpoint has been included in Γ . Hence each component is closed under K . Its stochastic block has eigenvalue 1 . Every eigenvalue λ of that block satisfies |λ|≤1 , and 1 1-wλ =(1-wλ)/(|1-wλ|^2)>0. Its contribution to D^ -1 is therefore at least 1/q , from the eigenvalue 1 . This trace argument applies equally to non-diagonalizable blocks. Sum over the components and use .