Correctness of the proposed algorithm for solving the black-box problem

Let $\mathcal{E}$ be a well-connected network with an odd number of boundary nodes and response matrix $M_R(\mathcal{E})$. Let $A(\mathcal{E})$ be the matrix constructed from this response matrix, and let a Lam model be obtained from the decomposition in the paper. A minimal Lam model is one obtained from this model by Postnikov transformations. The weights of the minimal Lam model are reconstructed using the stated theorem, and the weights of the original Lam model are then recovered. Finally, let $\mathcal{E'}$ be the standard network constructed from these weights and the physical meaning of the decomposition parameters. Algorithm correctness conjecture. The following algorithm for solving the black-box problem is correct: construct $A(\mathcal{E})$ from $M_R(\mathcal{E})$; construct a Lam model and reduce it by Postnikov transformations to a minimal Lam model; reconstruct the weights of the minimal and original Lam models; construct a standard network $\mathcal{E'}$ equivalent to $\mathcal{E}$; and transform $\mathcal{E'}$ into $\mathcal{E}$ using star-triangle transformations. The proposed procedure is intended to connect the black-box inverse problem with the Berenstein–Fomin–Zelevinsky problem for totally non-negative symplectic matrices. Its correctness is presented as expected rather than established, and the reduction to a minimal Lam model is anticipated to admit multiple simple canonical choices.