  Black hole images are theoretically predicted (under mild astrophysical
assumptions) to display a stack of lensed "photon rings" that carry information
about the underlying spacetime geometry. Despite vigorous efforts, no such ring
has been observationally resolved thus far. However, planning is now actively
under way for space missions targeting the first (and possibly the second)
photon rings of the supermassive black holes M87* and Sgr A*. In this work, we
study interferometric photon ring signatures in time-averaged images of Kerr
black holes surrounded by different astrophysical profiles. We focus on the
first, most easily accessible photon ring, which has a larger width-to-diameter
ratio than subsequent rings and whose image consequently lacks a sharply
defined diameter. Nonetheless, we show that it does admit a precise
angle-dependent diameter in visibility space, for which the Kerr metric
predicts a specific functional form that tracks the critical curve. We find
that a measurement of this interferometric ring diameter is possible for most
astrophysical profiles, paving the way for precision tests of strong-field
general relativity via near-future observations of the first photon ring.
