Definition

For a recurrent network’s hidden-state trajectory $\{X_t\}$, and a fixed lag $\Delta$, the finite-lag transport tensor $G_\Delta$ summarizes the conditional distribution of $X_{t+\Delta}$ given $X_t$, source-state by source-state. It decomposes exactly into $\mathrm{tr}(G_\Delta) = \text{(conditional-spread trace)} + \text{(coherent-displacement trace)}$ — the first term measuring dispersion of successor states around their own conditional mean, the second measuring how far that conditional mean has moved from the source — plus a separate antisymmetric statistic $\mathcal{W}_\Delta^\rho$ measuring net coordinate circulation (rotational flow not captured by either trace term). Reddy (2026) proves this decomposition holds exactly (not merely approximately) for any lag $\Delta$ and any recurrent architecture, then estimates all three quantities empirically from a real trained network’s own realized trajectories.

Intuition

Instead of asking “what is the network’s update rule doing locally, right now” (a Jacobian/fixed-point question), this asks “if I start at state $X_t$, where does the trajectory typically end up $\Delta$ steps later, how much does that vary, and does it also swirl around rather than moving straight there.” The three-way split (spread / coherent displacement / circulation) is analogous to decomposing fluid flow into diffusion, advection, and vorticity components.

Properties

• An empirical, trajectory-based statistic, not a Jacobian linearization. Unlike Fixed-point dynamics analysis (which linearizes an update rule at a single fixed point) or Jacobian centroid extraction (a feedforward sub-network’s Jacobian row-sum), this tensor is estimated from the empirical joint distribution of (source, successor) state pairs actually visited by the network, over a chosen lag $\Delta$ — no linearization or analytic derivative is required.
• Architecture-dependent even under parameter-count matching. On a repeat-copy task, real trained Elman networks show a transport trace roughly 15% higher than GRU/LSTM networks of matched hidden size, a gap that persists (Elman-64 vs. capacity-matched GRU-36) after controlling for total parameter count — an architectural, not merely a capacity, effect.
• Phase-localizable. The trace decomposition can be computed separately for different phases of a task (e.g. the “write” phase vs. the “recall” phase of a repeat-copy task), localizing where in a sequence-processing task architectures diverge most in their transport behavior.

Key papers

• Reddy (2026). Finite-Lag Operator Geometry of Recurrent Representations. arXiv:2607.01746 — introduces this tensor and its exact spread/displacement/circulation decomposition, applied to real gradient-trained Elman, GRU and LSTM networks (hidden sizes 64, plus a capacity-matched GRU-36 control) solving a repeat-copy task with near-perfect recall accuracy (0.9999-1.000). See Finite-lag transport tensor analysis and reddy-2026-elman-networks-have-a-15-percent-higher-finite-lag-transport-trace-than-capacity-matched-gru-and-lstm-networks-on-a-real-repeat-copy-task.