We isolate and prove the central probabilistic estimate underlying a lower bound g_sf(k) ≫ φ^k (golden ratio) for the maximum number of coprime adjacent divisor pairs of a squarefree integer with k prime factors. Working in the tight-band (lexicographic) weight model, the problem decomposes over cardinality layers, and the contribution of layer j is the expected number of consecutive disjoint pairs among the C(k,j) size-j subset sums. The lemma below shows this is ≫ C(k-j, j) for j ≍ 0.276 k; summed over layers via sum_j C(k-j, j) = F_{k+1} this yields the base φ. The proof reduces the "no blocker in the gap" event to a truncated first-moment estimate whose inputs are two elementary one-dimensional density bounds for sums of uniform variables; the correlated (swap-type and internal) terms, which appear to threaten the estimate, are shown to be exponentially negligible by an elementary conditional-variance bound together with the hypergeometric concentration of the disjointness defect. (A companion to A Reduction of the Squarefree Coprime Adjacent Divisor Problem and a Conditional Golden-Ratio Lower Bound.)
Publication Date: 2026-06-19