Stable comodule deformations and the synthetic Adams-Novikov spectral sequence

We study the Adams-Novikov spectral sequence in ${\mathbb{F}}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{B}\mathrm{P}$ and its cooperations to identify the synthetic Adams-Novikov ${\mathrm{E}}_2$-page, computed in a range with a synthetic algebraic Novikov spectral sequence. We then identify deformations associated to the Cartan-Eilenberg and algebraic Novikov spectral sequences in terms of stable comodule categories, categorifying an algebraic Novikov spectral sequence result of Gheorghe-Wang-Xu. We then apply Isaksen-Wang-Xu methods in ${\mathbb{F}}_2$-synthetic spectra to deduce differentials in the $p=2$ synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem.