Constant Maps and Exponentially Small Sectors in ABJM Bethe Observables

The topologically twisted index of ABJM theory captures the microscopic entropy of magnetically charged supersymmetric $\mathrm{AdS}_4$ black holes. Building on the companion Letter arXiv:2608.04107, which reconstructed the finite-rank twisted superpotential at the universal twist, we determine the finite-rank index using high-precision Bethe-vacuum data, physics-informed symbolic regression, and integer relations. The Hessian, effective dilaton, and one-loop factors encode information beyond the on-shell twisted superpotential. The two observables share the same shifted rank. Within a declared finite class, we reconstruct the index constant map from the ABJM $S^3$ constant maps at levels $k$ and $k/2$, the constant map of the twisted superpotential, and elementary terms. We prove that it equals a convergent one-kernel integral for real $k>0$, resumming the large-$k$ expansion, fixing a previously numerical constant, and yielding type-IIA coefficients at arbitrary genus. We also prove that the Clausen expression in arXiv:2608.04107 equals an alternating Euler sum and an absolutely convergent integral, giving closed evaluations for two rational-argument Euler-sum families. At $k=1,2,4$, the exponentially suppressed sectors exhibit arithmetic structure. The reconstructed coefficients of the twisted superpotential obey a divisor-sum formula with a conjectural all-order Eichler-integral series associated with a weight-four Eisenstein form. The index coefficients reduce to two rational generators; one admits modular-product formulas reproducing every available coefficient and suggesting an all-order completion. Conditional on this continuation, modular transformations determine the nearest logarithmic singularity, radius of convergence, and leading coefficient growth. These results provide a finite-$N$ benchmark for quantum gravitational blocks and bulk black hole quantum entropy.