Border Observations and the Number of Rank-Two Matrix Completions

Let H be a connected bipartite graph specifying interior observations of a generic rank-two matrix. Suppose every entry in a border row and a border column is also observed, except their intersection. We determine the number of completions from the cycle structure of H. A tree gives a one-dimensional generic fiber; a unicyclic graph whose cycle has length 2k gives exactly k-1 complex completions; and a graph with at least two independent cycles gives a unique completion. For a cycle with the border intersection restored, we give an irreducible circuit polynomial of degree 2k-1, of degree k-1 in that intersection and degree one in each other variable. The proof uses a Schur complement and a greatest common divisor calculation for two cycle polynomials. The rank-one ingredients and small circuit examples are known and credited. This restricted theorem does not classify all rank-two circuits or settle random-regular matrix completion. This is an unrefereed preprint prepared with AI assistance; no independent review, formal verification, or absolute priority certification is claimed.