The Pantheon+ Analysis: SuperCal-Fragilistic Cross Calibration,
  Retrained SALT2 Light Curve Model, and Calibration Systematic Uncertainty
  We present here a re-calibration of the photometric systems used in the
Pantheon+ sample of Type Ia supernovae (SNe Ia) including those used for the
SH0ES distance-ladder measurement of H$_0$. We utilize the large and uniform
sky coverage of the public Pan-STARRS stellar photometry catalog to
cross-calibrate against tertiary standards released by individual SN Ia
surveys. The most significant updates over the `SuperCal' cross-calibration
used for the previous Pantheon and SH0ES analyses are: 1) expansion of the
number of photometric systems (now 25) and filters (now 105), 2) solving for
all filter offsets in all systems simultaneously in order to produce a
calibration uncertainty covariance matrix that can be used in
cosmological-model constraints, and 3) accounting for the change in the
fundamental flux calibration of the HST CALSPEC standards from previous
versions on the order of $1.5\%$ over a $\Delta \lambda$ of 4000~\AA. The
re-calibration of samples used for light-curve fitting has historically been
decoupled from the retraining of the light-curve model. Here, we are able to
retrain the SALT2 model using this new calibration and find the change in the
model coupled with the change to the calibration of the light-curves themselves
causes a net distance modulus change ($d\mu/dz$) of 0.04 mag over the redshift
range $0<z<1$. We introduce a new formalism to determine the systematic impact
on cosmological inference by propagating the covariance in fitted calibration
offsets through retraining simultaneously with light-curve fitting and find a
total calibration uncertainty impact of $\sigma_w=0.013$, which is roughly half
the size of the sample statistical uncertainty. Similarly, we find a systematic
SN calibration contribution to the SH0ES H$_0$ uncertainty is less than
0.2~km/s/Mpc, suggesting that SN Ia calibration cannot resolve the current
level of the `Hubble Tension'.
