Finite Boolean families from coupled target and guard profiles

For each integer $m\ge0$, there exist a dimension $n$ and a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ such that

where $C=32057676379612508934192581$. Here $s(f)$ is the maximum sensitivity over all inputs, and $bs(f,0)$ is block sensitivity at the all-zero input, witnessed by nonempty pairwise disjoint sensitive blocks. The dimension and function may depend on $m$. This finite-family theorem is the reusable combinatorial input for the separation exponent $\log(201838849)/\log(7922)$ and, in particular, for exponent $2.1301$.