  Recently it was proposed that the theory space of effective field theories
with consistent UV completions can be described as a positive geometry, termed
the EFThedron. In this paper we demonstrate that at the core, the geometry is
given by the convex hull of the product of two moment curves. This makes
contact with the well studied bi-variate moment problem, which in various
instances has known solutions, generalizing the Hankel matrices of couplings
into moment matrices. We are thus able to obtain analytic expressions for
bounds, which perfectly match numerical results from semi-definite programing
methods. Furthermore, we demonstrate that crossing symmetry in the IR imposes
non-trivial constraints on the UV spectrum. In particular, permutation
invariance for identical scalar scattering requires that any UV completion
beyond the scalar sector must contain arbitrarily high spins, including at
least all even spins $\ell\le28$, with the ratio of spinning spectral functions
bounded from above, exhibiting large spin suppression. The spinning spectrum
must also include at least one state satisfying a bound $m^2_{J}<M^2 \frac{(
J^2-12) ( J^4 - 32 J^2 +204)}{8 (150 - 43 J^2 + 2 J^4)}$, where
$J^2=\ell(\ell+1)$, and $M^2$ is the mass of the heaviest spin 2 state in the
spectrum.
