Projecting social contact matrices to populations stratified by binary attributes with known homophily

The paper’s Theorem 4.5 asserts that the feasible homophily values form a closed interval [h_min, h_max] containing 0, and gives a convex-combination argument implying star-shapedness around 0, using the h=0 construction (Lemma 4.4) and the linearity-in-C step for φ along convex combinations. However, the proof as written does not justify that endpoints are attained (i.e., it does not establish compactness/closedness of the feasible set), even though the statement claims a closed interval. The candidate solution supplies exactly this missing piece by linearizing with flows F=N^?C^?, proving the feasible set under (a)–(c) is a compact convex polytope and that φ is a linear functional with constant denominator, hence the image is a compact interval; it then maps φ to h continuously (including edge cases). Thus the model’s solution is correct and fills the paper’s gap. Key ingredients from the paper used and matched by the model are Definition 4.1 (constraints (a)–(d)), the Xc=y formulation, Lemma 4.4 (existence of h=0), and the φ↔h relation; see Definition 4.1 and Remark 4.2, Lemma 4.4, Theorem 4.5, and Equations (5)–(7) in the paper .