Recovery of shared-cubic hidden-star networks from two differential boundary responses

Consider a network with labelled boundary terminals and unobserved interior vertices, each connected to every terminal, with no other edges in the network. On an edge incident to interior vertex $c$, the current law is $a_{ci}(v+\lambda_c v^3)$, where $a_{ci}>0$ and the positive ratio $\lambda_c$ is shared by all edges at that vertex. For two or more interior vertices, the differential boundary response at zero has a continuous ambiguity in the conductances. We prove that the response at zero together with the differential response at one nonzero, single-terminal bias determines all conductances and cubic ratios up to permutation of the interior vertices. The sufficient conditions are full column rank of two boundary-profile matrices and distinct component ratios in a matrix pencil. For a fixed nonzero bias, these conditions hold on an open dense set of full measure when each profile block has at least as many terminals as interior vertices. The proof combines differential-conductance reduction and established simultaneous diagonalization with an explicit recovery of physical scales. A two-center, five-terminal example illustrates every stage of the inverse. The result concerns exact differential data and the stated cubic network class; it makes no uniform stability or optimal terminal-count claim.