The replicated duality conjecture for the multicritical point of the square-lattice 7 Ising model

Let $n$ be the replica number, and let $K_c$ denote the coupling at the multicritical point of the $n$-replicated $7$ Ising model on the square lattice. Let $x_0$ and $x_0^{*}$ be the principal Boltzmann factors of the model and its dual, respectively. On the Nishimori line, $K=K_p$.

Multicritical-point conjecture. The exact location of the multicritical point is determined by $x_0=x_0^{*}$ with $K=K_p$, equivalently

$$e^{(n+1)K_c}+e^{-(n+1)K_c}=2^{-n/2}(e^{K_c}+e^{-K_c})^n.$$

This conjecture applies the duality fixed-point condition to locate the multicritical point. Its justification depends on the replica method, analyticity of the duality map, and the existence of the multicritical point on the Nishimori line; the source states that these issues lack formal proofs.