 Effective ground-set parameter of a hereditary uniform layer

For a nonempty degree-$m$ layer of a down-set in an $N$-element Boolean lattice, the unique real number $x_m\ge m$ with $|{\mathcal{F}}_m|=\left(\genfrac{}{}{0ex}{}{x_m}{m}\right)$ measures an effective number of available coordinates. The Lovász shadow bound and $\partial {\mathcal{F}}_{m+1}\subseteq {\mathcal{F}}_m$ imply $x_{m+1}\le x_m$ whenever the later layer is nonempty. It is a real parameter, not the actual size of the set family's support.