Lemma 42 ([Positive trace on a pure sign side]). In a local sector with only positive string boundary, the limiting bounded harmonic conditional mean has Brownian boundary trace at least c>0 , away from endpoints and incompatible sides. On …

The conditional law is log-concave and monotone in its boundary data and in its one-sided constraints. Lemma 41 ([Bounded local insertion]) gives a uniform positive chance that a tested positive string height exceeds some fixed b>0 , even after other positive sites have been restricted to (0,d] . A one-dimensional log-concave density on [0,1] with this tail bound assigns at most C d mass to [0,d] . One proof integrates the chord lower bounds for its logarithm between a point in [0,d] and a point carrying the prescribed tail mass. Thus, for small fixed d>0 , the tested height exceeds d with conditional probability at least 1/2 . Monotonicity makes the conditioning that all earlier indicators vanish the worst one. Successive uniform couplings show that the positive string heights dominate d times independent fair Bernoulli variables. Write A for the full original string-height vector. Couple its positive coordinates with independent fair pins B_x∈0,d so that B_x≤A_x on the positive portion, and extend this coupling to the remaining coordinates of A by disintegration. Conditional on (A,B) , sample the original Gibbs interior given A and a comparison Gibbs interior pinned to B on the positive portion and to d/2 on the remaining comparison boundary. Then replace this conditional product by its stationary synchronous joining. Lemma 13 ([Adapted freezing]) preserves the original field marginal and the comparison mixture. The latter has the required estimates uniformly in its pins, which lie in [0,d]⊂[-1/2,1/2] . Its law is invariant under h↦d-h , because this map sends the kernel with pins B to the kernel with pins d-B , and the fair product law is unchanged. Its mean is therefore d/2 . The filled lower tail of the comparison has zero fixed input on the positive portion. Theorem 4.1 ([Local reflected comparison]) and Proposition 33 ([Varying observed sets and filled values]) give zero trace for every limit of its mean there; no harmonic equation for the tail is used. Jensen's inequality bounds the negative part of the limiting mean difference by this nonnegative zero-trace limit. The two field means are bounded, harmonic away from the observed sets, and their remote contribution is bounded by a constant times the harmonic probability of reaching an endpoint or an incompatible side. This probability tends to zero along Brownian approaches to the pure portion, by Beurling. The lower trace is therefore at least d/2 . All bounds used are uniform for each deterministic admissible conditioning sequence, which gives the claimed sequential uniformity. Reflect the field for the negative statement.