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 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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