  The compact multi-transiting systems discovered by Kepler challenge
traditional planet formation theories. These fall into two broad classes: (1)
formation further out followed by migration; (2) formation in situ from a disk
of gas and planetesimals. In the former, an abundance of resonant chains is
expected, which the Kepler data do not support. In the latter, required disk
mass surface densities may be too high. A recently proposed mechanism
hypothesizes that planets form in situ at the pressure trap associated with the
dead-zone inner boundary (DZIB) where radially drifting "pebbles" accumulate.
This scenario predicts planet masses ($M_p$) are set by the gap-opening process
that then leads to DZIB retreat, followed by sequential, inside-out planet
formation (IOPF). For typical disk accretion rates, IOPF predictions for $M_p$,
$M_p$ versus orbital radius $r$, and planet-planet separations are consistent
with observed systems. Here we investigate the IOPF prediction for how the
masses, $M_{p,1}$, of the innermost ("Vulcan") planets vary with $r$. We show
that for fiducial parameters, $M_{p,1}\simeq5.0(r/{\rm{0.1\:AU}})\:M_\oplus$,
independent of the disk's accretion rate at time of planet formation. Then,
using Monte Carlo sampling of a population of these innermost planets, we test
this predicted scaling against observed planet properties, allowing for
intrinsic dispersions in planetary densities and Kepler's observational biases.
These effects lead to a slightly shallower relation
$M_{p,1}\propto{r}^{0.9\pm0.2}$, which is consistent with
$M_{p,1}\propto{r}^{0.7\pm0.2}$ of the observed Vulcans. The normalization of
the relation constrains the gap-opening process, favoring relatively low
viscosities in the inner dead zone.
