A Within-Layer Adjacency Lemma: A small-ball first-moment estimate for disjoint adjacencies among subset sums

We prove a first-moment small-ball estimate for subset sums of i.i.d. uniform variables. The estimate is the within-layer input to a golden-ratio lower bound for Erdős problem #1100. Let x_1, ..., x_k be i.i.d. Uniform[0,1], let X(S) = sum_{i ∈ S} x_i, and for j = floor(βk) order the C(k,j) size-j sums on the line; let N(k,j) be the expected number of consecutive disjoint pairs — size-j sets S, T with S ∩ T = ∅ and no size-j sum strictly between X(S) and X(T). We show N(k,j) ≫ C(k-j, j) for β near 0.276. Summed over layers via sum_j C(k-j, j) = F_{k+1}, this is the core input to a golden-ratio lower bound g_sf(k) ≫ φ^k for coprime adjacent divisors of squarefree integers, developed in companion notes. The proof is a truncated first moment whose only analytic inputs are elementary: standard Irwin–Hall density facts and a Hoeffding tail bound, with no local limit theorem or Edgeworth expansion. Its one delicate point is the overlap-free "swap" blocker class, controlled by a codimension-two small-ball bound — blocking forces two independent signed forms into a window of width θ, costing θ² rather than θ, which makes that class exponentially negligible. (This is a companion to A Reduction of the Squarefree Coprime Adjacent Divisor Problem and a Conditional Golden-Ratio Lower Bound.)