Haar integrals over the unitary group contain subleading terms that are
needed for unitarity. We study analogous effects in the time evolution
operators of JT gravity and Brownian SYK. In JT gravity with bulk matter
we find an explanation for the first subleading terms, and in Brownian
SYK we find configurations that can explain the full series. An
important role is played by slightly off-shell modes that are
exponentially amplified by chaos.