A real trained dot-product recommender's item embeddings carry a rank-one popularity-aligned direction that a prior-separation intervention removes

A real 128-dimensional dot-product matrix-factorization recommender is trained via Adam gradient descent on real Alibaba/Tianchi mobile-recommendation logs (459,105 training events, 1,413 users, 20,000 items) [cheng-2026-rank-one-popularity-component-recommender-scores] The trained item-embedding matrix's leading principal component correlates with log-item-popularity (PC1-log-popularity correlation 0.568), and a rank-one energy measure R1(E) and mean pairwise cosine similarity are elevated (0.0219 and 0.027 respectively), indicating a popularity-aligned anisotropic direction in the embedding geometry [cheng-2026-rank-one-popularity-component-recommender-scores] A causal prior-separation intervention -- moving the log-popularity term outside the dot product rather than absorbing it into the embeddings -- reduces the popularity-aligned score energy by 98.6% (0.01905 to 0.000265), with R1(E) dropping to 0.0157, mean pairwise cosine dropping to 0.006, and PC1-log-popularity correlation dropping to 0.078 (permutation test p=1/2001) [cheng-2026-rank-one-popularity-component-recommender-scores]