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 $\mathrm{BP}$ 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.