  In this paper we study both projective and non-projective constraints on
four-dimensional gravitational effective fields theories implied from
unitarity, causality and crossing, assuming perturbative UV completions in
$M_{\rm pl}$. We derive bounds on the Wilson coefficients of $R^3$ and
$D^{2n}R^4$ from its dispersive representation, utilizing both numerical
semi-definite programming and analytic geometry analysis. From the former, we
derive projective bounds on ratios of couplings and observe accumulation point
spectrum populating the boundary of the allowed region. For the latter we
consider the non-projective geometry of the EFThedron, which we relate to the
known $L$-moment problem in the literature. This allows us to move beyond
positivity and incorporate the upper bound from unitarity of the imaginary
parts of partial waves. This leads to sharp bounds on individual coefficients,
which are of order unity when normalized with respect to the UV scale. Finally,
the non-projective geometry also allows us to derive optimal bounds reflecting
assumptions of low-spin dominance, improving previous results. We complement
the analytic analysis with a simple linear programming approach that validates
the bounds.
