Constant capture counts and witnessed realization on the Boolean cube

I study weighted families of labelled points in the Boolean cube. Each point carries a choice of half the coordinates. An order of the signed coordinate symbols captures some of these labelled points, and a mask is a family in which no order captures two members. The paper characterizes all real weighted arrays whose capture count is independent of the order. For nonnegative arrays whose label weights sum to one at each cube point, it also characterizes when the minimum possible capture bound can be attained by a nonnegative weighted sum of masks, with the weight prescribed at every labelled point. An integer consequence gives a sharp uniform threshold. For d ≥ 2, a nonnegative integer array with exactly K times the binomial coefficient (2d choose d) occurrences and capture count at most K can be partitioned into K maximum masks, without changing any point or label, whenever 1 ≤ K ≤ 7. The threshold cannot be increased uniformly in d. An eight-dimensional example shows that identical capture counts need not determine the cost of realization by weighted masks. This 19-page author preprint contains the proofs and explicit constructions, together with the funding, competing-interest, data-availability and AI-assistance statements. This deposit is a preprint, not a journal version of record. The separate quantitative companion is available at https://doi.org/10.5281/zenodo.22758028. None of its estimates is used in a proof in this article.