  We study the critical properties of the Ising-Higgs gauge theory in $(2+1)D$
along the self-dual line which have recently been a subject of debate. For the
first time, using machine learning techniques, we determine the low energy
operator content of the associated field theory. Our approach enables us to
largely refute the existence of an emergent current operator and with it the
standing conjecture that this transition is of the $XY^*$ universality class.
We contrast these results with the ones obtained for the $(2+1)D$ Ashkin-Teller
transverse field Ising model where we find the expected current operator. Our
numerical technique extends the recently proposed Real-Space Mutual Information
allowing us to extract sub-leading non-linear operators. This allows a
controlled and computationally scalable approach to target CFT spectrum and
discern universality classes beyond $(1+1)D$ from Monte Carlo data.
