Asymptotic sensitivity separation exponent

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 any real number $a$, the proposition defined here holds exactly when, 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 with $S≤s(g)$ and $s(g)^{a-ε}≤b(g)$, where sensitivity values are regarded as real numbers in this inequality. No sign restriction is imposed on $a$ or $a-ε$. The function and dimension may depend on both $ε$ and $S$. The dimension is quantified over all natural numbers, including zero, but zero-dimensional functions have sensitivity zero and therefore cannot satisfy the required lower bound.