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We present here a re-calibration of the photometric systems used in thePantheon+ sample of Type Ia supernovae (SNe Ia) including those used for theSH0ES distance-ladder measurement of H$_0$. We utilize the large and uniformsky coverage of the public Pan-STARRS stellar photometry catalog tocross-calibrate against tertiary standards released by individual SN Iasurveys. The most significant updates over the `SuperCal' cross-calibrationused for the previous Pantheon and SH0ES analyses are: 1) expansion of thenumber of photometric systems (now 25) and filters (now 105), 2) solving forall filter offsets in all systems simultaneously in order to produce acalibration uncertainty covariance matrix that can be used incosmological-model constraints, and 3) accounting for the change in thefundamental flux calibration of the HST CALSPEC standards from previousversions on the order of $1.5\%$ over a $\Delta \lambda$ of 4000~\AA. There-calibration of samples used for light-curve fitting has historically beendecoupled from the retraining of the light-curve model. Here, we are able toretrain the SALT2 model using this new calibration and find the change in themodel coupled with the change to the calibration of the light-curves themselvescauses a net distance modulus change ($d\mu/dz$) of 0.04 mag over the redshiftrange $0<z<1$. We introduce a new formalism to determine the systematic impacton cosmological inference by propagating the covariance in fitted calibrationoffsets through retraining simultaneously with light-curve fitting and find atotal calibration uncertainty impact of $\sigma_w=0.013$, which is roughly halfthe size of the sample statistical uncertainty. Similarly, we find a systematicSN calibration contribution to the SH0ES H$_0$ uncertainty is less than0.2~km/s/Mpc, suggesting that SN Ia calibration cannot resolve the currentlevel of the `Hubble Tension'.

The Pantheon+ Analysis: SuperCal-Fragilistic Cross Calibration, Retrained SALT2 Light Curve Model, and Calibration Systematic Uncertainty