Lemma 10 ([Gaussian attenuation away from time zero]). For every δ>0 there is κ_δ∈(0,1) such that, for every finite connected subtree D , every v,u∈D , and every s≥δ , [C_vu^D(s)²] ≤θ^2r (1-κ_δ)^r, r=(v,u). The same inequality holds …

Choose ε_δ>0 with 2ε_δ/√2πδ≤1/2 . A Gaussian random variable Z of variance s≥δ , with any mean, has (|Z|≤ε_δ)≤1/2 by its density bound. Since a(z)^2≤a(z) ≤1-(1-θ^2)^2z, it follows that [a(Z)^2]≤1-κ_δ, κ_δ=(1-θ^2)/(2) ^2ε_δ∈(0,1). The estimate is uniform in the mean. Let v=v_0,…,v_r=u be the path in D . In the squared covariance formula, discard the initial factor (1-(m_v^D)^2)^2≤1 . At the fixed time s the remaining factors depend only on terminal observations. Conditional on all spins and all terminal Brownian coordinates other than B_ v_1 (s) , the cavity recursion has the form h_ v_1→v_0 ^D(s)=B_ v_1 (s)+z, where the shift z is fixed: all branch fields in this recursion exclude v_1 . The later fields h_ v_i→v_ i-1 ^D , i>1 , also exclude v_1 . Conditional on the spins, the terminal Brownian coordinates are independent Gaussians of variance s . Integrating B_ v_1 (s) first therefore removes the first squared factor at a cost at most 1-κ_δ , without changing the later ones. Integrate B_ v_2 (s),…,B_ v_r (s) in that order. Iteration proves, even conditional on the spins. At a leaf the shift is simply its planted spin drift; the argument still applies. No infinite side branch is used here. For fixed v,u,s , exhaust the tree by finite connected subtrees containing the path. Bounded conditional-expectation convergence from Lemma 1 ([Posterior identities]) gives C_ vu ^D(s)→C_ vu (s) in L^2 . Passing to the limit proves the whole-tree inequality.