Lemma 15. For the reference experiment just defined, Q_n(E_ mid ) ≤δ_n:=1,Cn(F_+(n)+F_-(n)).

Fix (H,Z) in its retained support; in particular H_1,H_3≥n . Write N=N_* . By list-reversal invariance in Lemma 10 ([Exact-cut bridges]), the placed chronological chunks have their corresponding raw bridge laws. Retain the complete positive lower and middle chunks, concatenated into a path α from height 0 to its first hit of N-H_1 . Retain the negative upper and middle chunks only up to their first arrival at a departure site of α . A middle contact ensures that this arrival occurs before the negative middle endpoint at height H_3-1 . The contact site y must therefore satisfy H_3≤y_1≤N-H_1-1. Thus each path has made at least n progress in its own signed height before reaching the contact, and neither has crossed the global barriers -1,N . We explain the comparison with two paths in one environment before using any quenched exit probabilities. The unused positive upper chunk and negative lower chunk integrate to one. The remaining four bridge normalizations are bounded by p_+^ -2 p_-^ -2 . Consecutive bridges within a sign have disjoint departure layers, so their raw weights multiply to the raw weight of their concatenation. Summing over the negative completion after its retained prefix, and dropping its remaining bridge requirements, costs at most that prefix's raw weight: in each fixed environment the total conditional completion probability is at most one, and this inequality can then be averaged. This argument allows the completion to revisit its prefix's departure rows and does not factor their weights independently. The first-hit chunk boundaries determine the splits, so no extra multiplicity is introduced. The retained negative prefix has no departure site in Dep(α) : its terminal point is the first arrival there. Repeated sites within either path cause no difficulty. Lemma 1 ([Transfer through unused departure rows]) therefore identifies the product of their raw weights with the weight of the two paths sampled independently conditional on a common iid environment. For each α the first contact fixes the negative prefix uniquely, and α itself is stopped at a fixed first-hit height. We may thus sum these joint prefix weights without overcounting, and extend the paths as raw trajectories in the common environment. Only now, in that shared environment, define for interior sites Q_y^ω=P_ y,ω (T_ -1 <T_N), where the barriers are absolute first-coordinate heights. If Q_y^ω≥1/2 at the contact, the positive raw path has visited such a site after hitting n and before exiting the global slab. Conditional on the environment, stop it at its first such visit. The probability of a subsequent exit at -1 is at least 1/2 . Consequently the annealed probability of this visit event is at most 2P_0(T_n<T_ -1 <T_N)=2(u_n^+-u_N^+). If Q_y^ω<1/2 , the negative marginal gives the analogous bound 2(u_n^--u_N^-) . Indeed the almost-sure finite-height-interval exit property established in Section holds quenched for all these starting sites and barriers outside one null set. The probability of the other exit is therefore greater than 1/2 , with no common lower bound on the transition rows. Measured from the negative start N-1 , the global barriers N and -1 have signed relative heights -1 and N , respectively. Lateral translation by Z does not change its annealed law. Combining these alternatives with the bounded normalizations gives, uniformly in (H,Z) , Q_n(E_ mid H,Z) ≤C∑_ σ∈+,- (u_n^σ-u_N^σ) ≤CN(F_+(n)+F_-(n)), where the last step is Lemma 11 ([Renewal-tail comparison]). Finally E N≤12n E K . Average over the retained joint marginal of (H,Z) and take the minimum with 1 .