{"ID":5675151,"CreatedAt":"2026-07-03T01:40:09.565152011Z","UpdatedAt":"2026-07-05T05:45:35.603470622Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.01746","arxiv_id":"2607.01746","title":"Finite-Lag Operator Geometry of Recurrent Representations","abstract":"Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\\mid x)$, estimated by a dense Gaussian source-smoothing operator. From this directed finite-lag law we derive a source-centered transport tensor $G_Δ$, which decomposes exactly into conditional spread and coherent displacement, and an antisymmetric coordinate circulation $W_Δ^ρ$, which summarizes directed lagged flow. We prove affine covariance with explicit metric dependence of scalar summaries, dense estimator stability on bounded trajectory clouds, and a finite-lag separation result showing that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry. A linear-Gaussian closed form calibrates the quantities in terms of the update $A_Δ$, source covariance, and innovation covariance. Controlled experiments validate the decomposition, circulation, covariance, and stability predictions. In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace, while coherent displacement fraction is metric and resolution dependent.","short_abstract":"Recurrent representations are trajectories, but representation geometry is often measured from static snapshots. We develop finite-lag operator geometry for recurrent hidden states from observed source-successor pairs $(X_t,X_{t+Δ})$. The primitive is the conditional transport law $Q_Δ(dy\\mid x)$, estimated by a dense...","url_abs":"https://arxiv.org/abs/2607.01746","url_pdf":"https://arxiv.org/pdf/2607.01746v1","authors":"[\"Kanishka Reddy\"]","published":"2026-07-02T06:00:32Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}