Proposition: Common-statistic feasibility

For finite recorded alphabets and compact closed convex hulls $C_h=\overline{\mathrm{conv}}{\mathcal{P}}_h$, a positive common bounded-statistic gap exists if and only if $C_0\cap C_1=\varnothing$.

For finite lists of family vertices, its optimal gap is the linear program

A finite nuisance grid represents a continuous family only after a uniform interpolation or optimization error is supplied.

Proof. A positive gap survives convexification and excludes an intersection. Disjoint compact convex sets admit a strict separating linear functional. Adding a constant to every coordinate and rescaling puts its coefficients in $[0,1]$, since all law vectors sum to one. The displayed program is exactly the finite common-gap optimization.