Abstract: Extracting the operator spectrum (conformal towers) of critical models with space-time dimensionality larger than 2 is a formidable numerical task, closely related to diagonalizing very large element-wise non-negative matrices. Here we demonstrate the ability of a new ML-based numerical tool (extended RSMI-NE) to tackle such problems. We focus on critical properties of the Ising-Higgs gauge theory in $(2+1)D$ along the self-dual line, which has recently been a subject of debate. We determine, for the first time, the low energy operator content of the associated field-theory. Our approach enables us to largely refute a standing conjecture about the universality class of this transition.