Exact Modular Completion of the ABJM Effective Twisted Superpotential

The effective twisted superpotential governs the supersymmetric partition functions of three-dimensional $\mathcal{N}=2$ theories in the Cardy limit and, at large $N$, the gravitational blocks that are glued into the entropy function of supersymmetric $\mathrm{AdS}_4$ black holes. We determine its on-shell structure for $\mathrm{U}(N)_k\times\mathrm{U}(N)_{-k}$ ABJM theory at the universal twist, using a machine-learning discovery pipeline---physics-informed symbolic regression with integer-relation detection---applied to Bethe-vacuum data of up to $800$ digits. Its perturbative part terminates after two terms, a shifted-rank $3/2$-power and an $N$-independent constant map, which we obtain in closed form for every integer level. Its type-IIA genus expansion admits a closed coefficient formula at arbitrary genus, involving binomial and Bernoulli-number terms, and is asymptotic. For $k=1,2,4$ the entire finite-$N$ remainder is a single absolutely convergent divisor-sum $q$-series, obtained by applying a first-order differential operator to the Eichler integral of a weight-four Eisenstein series on $\Gamma_0(2)$ or $\Gamma_0(4)$. This closed form reproduces the data to $\sim\!10^{-797}$.