Phylogenetic distance deconvolution represents mixture-to-reference distances by weighted sources placed on a reference tree. We study the minimum number of strictly interior sources that explains a distance profile, and what incomplete references cost. On a positive-rational reduced tree we give an exact algorithm with 2^O(K) poly(B) bit complexity for a free positive mean pendant and any nonempty set of observed leaves. With arbitrary missing references and a prescribed pendant the problem is W[1]-hard in the source budget K, even on positive binary trees supplied with a strict explanation using at most K+1 sources. We then use the algorithm as an instrument. Removing references can only lower the minimum source count, never raise it. The direction is a theorem at every error tolerance and held without exception in 23,545 draws, so which references are missing matters more than how many. Two adversarially chosen references suffice, at the median, to destroy uniqueness of the minimum explanation, and that number does not grow with panel size (Spearman 0.203, p = 0.15, over 20 to 353 references) where random loss scales linearly. Discarding 7,931 of 10,575 references from a real 16S panel at random costs 1.8% of the tolerance at which one source suffices. Discarding comparably many as whole clades costs 90%. That tolerance needs neither ground truth nor an error model, which makes the model measurable on real data. Published distance estimates exceed it in 296 of 296 cases. The mixture inferred by the published heuristic reproduces real observations better than the configuration that generated them in 282 of 296. The phenomenon is not unobservable in practice. It is unobservable through a point estimate of the count. Methods for this problem should report the tolerance at which an explanation of a given size exists.

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