A Rank-One Popularity Component in Dot-Product Recommender Scores:Population Theory and Prior-Separation Evidence

Representation anisotropy in recommender systems is often attributed to Transformer architectures. We identify a more general source in the conditional training distribution. For any encoder using a dot-product softmax decoder, the population-optimal score decomposes into pointwise mutual information, an item-marginal term log p(i), and a context-dependent offset. After centering, the item marginal produces a context-shared rank-one score component, while time-varying marginals induce a low-rank popularity subspace. This score-level result does not imply universal embedding collapse because its transfer to embeddings depends on factorization geometry. Experiments on synthetic data and public Alibaba and Tianchi interaction logs support the proposed mechanism. Separating log p(i) from the learned dot product reduces the measured popularity-aligned score energy by 98.6 percent in a matched intervention. Permutation tests confirm that this reduction is specific to the empirical popularity direction. These results explain a class of apparent representation degeneration as a decoder-level consequence of long-tailed item marginals rather than a property unique to Transformer encoders.