Topic profiles represent publications, authors, and other scholarly entities as probability distributions over a fixed set of topics, but flat total variation treats every pair of distinct pure-topic profiles as maximally separated and therefore ignores taxonomic proximity. From a rooted weighted taxonomy, we derive a cardinality-normalized linear operator that maps the leaf topics to points in the original probability simplex and exactly realizes a normalized lowest-common-ancestor ultrametric under total variation. The operator is doubly stochastic and positive definite; within the class of nonnegative edge-cluster Gram operators, its normalization is uniquely determined on the reduced branching tree. Applying the same invertible operator to arbitrary topic mixtures yields a nondegenerate hierarchy-aware metric that contracts flat total variation, differs from the tree-Wasserstein distance on mixtures, and can be evaluated in O(|V|+L) time and memory without forming the dense matrix. In a frozen OpenAlex taxonomy with 4,516 terminal Topics, raw dissimilarities between Topic texts showed consistent ordinal alignment with taxonomic proximity, while only 3 of 253 calibrated internal nodes required monotonic correction. Encoder choice nevertheless affected individual height estimates. The framework exactly realizes a supplied weighted hierarchy; text is used only to initialize its node heights, and distances between scholarly topic profiles are then computed in the induced geometry.

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