Finite seed certificate implies an asymptotic separation

For a Boolean function $g$ on $m$ bits, let $s(g)$ be the maximum, over all inputs, of the number of individual coordinates whose flip changes its output, and let $b(g,y)$ be the maximum size of a family of pairwise disjoint nonempty coordinate sets whose separate flips at input $y$ each change its output; the empty family is permitted and has size zero. Write $b(g)$ for the maximum of $b(g,y)$ over all inputs $y$. For every natural number $n$, every Boolean function $f$ on $n$ bits, every input $x$, all natural numbers $A,B$ with $A>1$ and $B>1$, and every real $a≥0$, if $s(f)≤A$, $B≤b(f,x)$, and $A^a≤B$ after interpreting $A,B$ as real numbers, then the following holds: For every real $ε>0$ and every natural number $S≥2$, there exist a natural number $m$ and a Boolean function $g$ on $m$ bits such that $S≤s(g)$ and $s(g)^{a-ε}≤b(g)$, with the power taken in the reals. The witnesses may depend on both $ε$ and $S$, and $a-ε$ may be negative; zero-dimensional witnesses are ruled out by $S≤s(g)$. Although $n=0$ is allowed by its quantifier, its hypotheses cannot hold because then $b(f,x)=0$.