Lemma 13 ([Cancellation after exposure]). For every cell B , |∑_ i∈ H; x_i∈B f_i| ≤C√|B|/q.

Return to the original axes, with field direction Y , and put T_B=∑_ x∈B f_xY_x . The ordinary Laplace transform F_B(z)=⟨e^ zT_B ⟩_b is zero-free on |z|<cb : all slot fields in direction Y have positive real part there, so Lemma 3 ([Zero-free layered traces]) applies. Take its analytic logarithm equal to zero at 0 . Spectral calculus for T_B gives F_B(z)≤|z|S|B|≤Cb|B| (|z|<cb). The Borel--Carath\'eodory inequality on this disk, followed by Cauchy's second-derivative estimate on a smaller concentric disk, therefore yields _b(T_B)=( F_B)''(0)≤C|B|/b≤C|B|/q. Translation invariance and ∑_ x∈B f_x=0 give ⟨T_B⟩_b=0 . In the color representation its conditional expectation given a is (T_B a) =-12∑_ i∈A_0; x_i∈B f_i =12∑_ i∈ H; x_i∈B f_i. Here exposed cycles are fair and every hole cycle is up; the total sum of the slot coefficients on B is zero. Conditional Jensen and Cauchy--Schwarz prove.