$C$-networks and the planar Ising inverse problem

We solve the inverse problem for Ising models on reduced planar graphs in a disk, i.e., recovering the edge coupling constants from the boundary spin correlations. A recursive solution to this problem was provided by Galashin–Pylyavskyy. Our solution is non-recursive. It is based on an Ising analog of the chamber ansatz which asserts that the inverse map should factor through variables on the graph that transform under the Ising Y-$\mathrm{\Delta }$ move according to the discrete CKP equation.