  Recently, the conditional SAGE certificate has been proposed as a sufficient
condition for signomial positivity over a convex set. In this article, we show
that the conditional SAGE certificate is $\textit{complete}$. That is, for any
signomial $f(\mathbf{x}) = \sum_{j=1}^{\ell}c_j \exp(\mathbf{A}_j\mathbf{x})$
defined by rational exponents that is positive over a compact convex set
$\mathcal{X}$, there is $p \in \mathbb{Z}_+$ and a specific positive definite
function $w(\mathbf{x})$ such that $w(\mathbf{x})^p f(\mathbf{x})$ may be
verified by the conditional SAGE certificate. The completeness result is
analogous to Positivstellensatz results from algebraic geometry, which
guarantees representation of positive polynomials with sum of squares
polynomials. The result gives rise to a convergent hierarchy of lower bounds
for constrained signomial optimization over an $\textit{arbitrary}$ compact
convex set that is computable via the conditional SAGE certificate.
