Stable Comodule Deformations and the Synthetic Adams-Novikov Spectral Sequence

We study the Adams-Novikov spectral sequence in Fₚ-synthetic spectra, computing the synthetic analogs of BP and its cooperations to identify the synthetic Adams-Novikov E₂-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 Fₚ-synthetic spectra to deduce differentials in the synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem.