OpenAI’s general finite seed bounds

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 pair of natural numbers $d,M$ satisfying $1≤d$ and $9^{d+1}≤M$, there exist a natural number $n$ and a Boolean function $f$ on $n$ bits such that $f(0^n)$ is false, $s(f)≤2(d+2)M^{d+1}$, and $M^2(2M^2+1)^d≤b(f,0^n)$, where $0^n$ denotes the all-false input and the bounds are natural-number inequalities. The witness dimension is not specified; although its quantifier includes zero, the positive lower bound on $b(f,0^n)$ excludes zero-dimensional witnesses. The assumptions exclude $d=0$ and $M=0$.