  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.
