Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities

Let $S$ be a finite sum of independent Bernoulli random variables. We ask how large the mean must be before the atom at $k+r$ can strictly exceed the atom at $k$. For the threshold $T_{k,r}:=\inf \{\mathbb{E}S:\mathbb{P}(S=k+r)>\mathbb{P}(S=k)\}$, the adjacent case recovers the classical Darroch boundary $k+1\mathrm{/}(k+2)$, while every gap $r\ge 3$ has threshold $k+1$. Distance two is the unique nonadjacent exception: $T_{k,2}=k+{\mathrm{\Delta }}_k$, where ${\mathrm{\Delta }}_k$ and its staircase optimizer are explicit and ${\mathrm{\Delta }}_k=1\mathrm{/}2+1\mathrm{/}\sqrt{2k}-1\mathrm{/}(4k)+O(k^{-3\mathrm{/}2})$. The proof combines the classical fixed-mean reduction to shifted binomial laws with an exact discrete optimization. The result also yields sharp restrictions on nonadjacent coefficient reversals in real-rooted polynomials with nonnegative coefficients.