  We prove a precise form of AdS bulk locality by deriving analytical two-sided
bounds on bulk Wilson coefficients. Our bounds are on the Wilson coefficients
themselves, rather than their ratios, as is typically found in the literature.
Inspired by the Celestial amplitudes program in flat space, we perform a
Celestial transform of the CFT Mellin amplitude of four identical scalars.
Using the crossing symmetric dispersion relation (CSDR), we express the
resulting amplitude in terms of crossing symmetric conformal partial waves. The
partial waves satisy remarkable positivity properties which along with the
unitarity of the CFT prove sufficient to derive the bounds. We then employ our
methods in the limit of large AdS radius and recover known bounds on flat space
Wilson coefficients and new bounds on their large radius corrections. We check
that the planar Mellin amplitude of four stress-tensor multiplets in
$\mathcal{N} = 4$ SYM satisfies our bounds. Finally, using null constraints, we
derive a form of low-spin dominance in AdS EFTs. We find that the low-spin
dominance is strongest in flat space and weakens as we move away from flat
space towards higher AdS curvature.
