Finite-Lag Operator Geometry of Recurrent Representations

• URL: http://arxiv.org/abs/2607.01746v1
• Date: Thu, 02 Jul 2026 06:00:32 GMT
• Title: Finite-Lag Operator Geometry of Recurrent Representations
• Abstract summary: We show that source-centered transport detects deterministic recurrent motion not recorded by infinitesimal carre-du-champ geometry.<n>In performance matched repeat-copy networks, the framework reveals architecture dependent differences in total transport scale and coherent displacement trace.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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