= Dense model theorem for a multiplicative test family
Let $\mathcal F$ be a closed, convex, symmetric family of functions into $[-1,1]$ that contains the constant function $1$, and suppose its <dual test-function norm> is submultiplicative under pointwise products. If a nonnegative majorant $\nu$ has average at most one and $\|\nu-1\|_{\mathcal F}$ is exponentially small in $1/\varepsilon$, then every $0\leq g\leq\nu$ has a dense model $0\leq\widetilde g\leq1$ satisfying
$$
\|g-\widetilde g\|_{\mathcal F}\leq\varepsilon.
$$
The proof separates $g$ from the convex set of dense models, then approximates the positive part of the separating test by a polynomial. Submultiplicativity controls every power in that polynomial.
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