  We consider testing a composite null hypothesis $\mathcal{P}$ against a point
alternative $\mathsf{Q}$ using e-variables, which are nonnegative random
variables $X$ such that $\mathbb{E}_\mathsf{P}[X] \leq 1$ for every $\mathsf{P}
\in \mathcal{P}$. This paper establishes a fundamental result: under no
conditions whatsoever on $\mathcal{P}$ or $\mathsf{Q}$, there exists a special
e-variable $X^*$ that we call the numeraire, which is strictly positive and
satisfies $\mathbb{E}_\mathsf{Q}[X/X^*] \leq 1$ for every other e-variable $X$.
In particular, $X^*$ is log-optimal in the sense that
$\mathbb{E}_\mathsf{Q}[\log(X/X^*)] \leq 0$. Moreover, $X^*$ identifies a
particular sub-probability measure $\mathsf{P}^*$ via the density $d
\mathsf{P}^*/d \mathsf{Q} = 1/X^*$. As a result, $X^*$ can be seen as a
generalized likelihood ratio of $\mathsf{Q}$ against $\mathcal{P}$. We show
that $\mathsf{P}^*$ coincides with the reverse information projection (RIPr)
when additional assumptions are made that are required for the latter to exist.
Thus $\mathsf{P}^*$ is a natural definition of the RIPr in the absence of any
assumptions on $\mathcal{P}$ or $\mathsf{Q}$. In addition to the abstract
theory, we provide several tools for finding the numeraire and RIPr in concrete
cases. We discuss several nonparametric examples where we can indeed identify
the numeraire and RIPr, despite not having a reference measure. Our results
have interpretations outside of testing in that they yield the optimal Kelly
bet against $\mathcal{P}$ if we believe reality follows $\mathsf{Q}$. We end
with a more general optimality theory that goes beyond the ubiquitous
logarithmic utility. We focus on certain power utilities, leading to reverse
R\'enyi projections in place of the RIPr, which also always exist.
