cs.ETSep 10, 2026

Beyond density snapshots: Dynamical memory in Chern-Simons reservoirs

Authors: Jyotiranjan Beuria, Venkatesh H. Chembrolu

Organizations: Center for Philosophical and Cognitive Sciences, ISS Delhi, India · Indian Institute of Technology Mandi, Mandi, India

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Abstract

How a reservoir encodes input history jointly in density and geometric degrees of freedom remains insufficiently understood. We address this mechanistic gap with a continuous density on an internal manifold coupled to an independently evolving Chern-Simons (CS) connection through a conserved current. On the same internal manifold, we compare reciprocal transport, instantaneous transverse reconstruction, local nonlinear feedback, and coupled CS evolution. These controls isolate how transport and independent connection evolution contribute to memory. We identify a current-generated exact connection component that the tested instantaneous reconstruction and scalar local closure miss. Resetting this component at fixed density and Gauss-law curl changes the future density response within the chosen gauge and drift law and the small effect persists under numerical refinement. We then distinguish dynamical memory from its readout accessibility. Although the four models have similar scalar-memory scores and density traces of pulse order, their circulating and longitudinal traces differ. With all measured features, coupled CS improves pulse-order decoding by 6.5±1.96.5\pm1.9 percentage points over the instantaneous-transverse control in paired seedwise comparisons, while remaining comparable to local feedback. Coupled CS shows less lag than the instantaneous and local-feedback controls on the first cycle, but more on subsequent cycles, revealing a dependence on preparation. We thus connect a field-theoretic mechanism of history retention to controlled tests of its dynamical role and computational use. The four-model framework provides a way to determine what an evolving transport field contributes when density snapshots and aggregate performance alone leave the memory mechanism unresolved.

Figures & tables

Figure 1: Study design and controlled model hierarchy. We use the same spatial domain, input protocol, numerical scheme, initial conditions, and seeds in all four models, and vary how density couples to the velocity or gauge connection. We test scalar memory, pulse-order dependence, and cyclic-drive lag with the reservoir-computing protocols. With the gauge-dynamics protocols, we test Chern-Simons consistency, snapshot reconstruction, connection reset, and feedback-dependent cross-response. We then separate properties shared by the field reservoir family from effects that require an independently evolving gauge connection.
Figure 2: Reconstruction and closure at tf≃4t_{f}\simeq 4 . (a) Mean Ea(t)E_{a}(t) with SEM band at α=0.70\alpha=0.70 . (b) Mean EvE_{v} for instantaneous transverse and fitted-local reconstruction, with SEM. (c) Their paired percentage difference (mean ±\pm SEM).
Figure 3: CS-equation residuals on the 16216^{2} grid with Δt=0.015\Delta t=0.015 and tf≃4t_{f}\simeq 4 . (a) Mean Gauss residual RGR_{G} . (b) Mean seedwise time-median spatial residual RSR_{S} .
DiagnosticResult
Maximum RGR_{G} , main run4.42×10−144.42\times 10^{-14}
Coupled spatial residual, Δt\Delta t1.07×10−21.07\times 10^{-2}
Coupled spatial residual, Δt/2\Delta t/25.30×10−35.30\times 10^{-3}
∥ρ∥2\left\lVert\rho\right\rVert_{2} , N=12/16/24N=12/16/240.261882/0.261881/0.2618810.261882/0.261881/0.261881
∥ρ∥2\left\lVert\rho\right\rVert_{2} , Δt/2\Delta t/20.2638870.263887
∥ρ∥2\left\lVert\rho\right\rVert_{2} , three noise seeds0.2618440.261844 to 0.2618880.261888
Table 1: Numerical validation of coupled CS. Unless indicated otherwise, values use the 16216^{2} grid and Δt=0.015\Delta t=0.015 . The two spatial-residual convergence runs have nominal duration 2.
ObservablemodelAp=0.50A_{\rm p}=0.50Ap=0.75A_{\rm p}=0.75Ap=1.00A_{\rm p}=1.00Ap=1.25A_{\rm p}=1.25
density contrast Δρ\Delta_{\rho} (%)reciprocal0.319±0.2100.319\pm 0.2100.369±0.2060.369\pm 0.2060.437±0.2010.437\pm 0.2010.523±0.1970.523\pm 0.197
 inst. transverse−0.010±0.001-0.010\pm 0.001−0.014±0.002-0.014\pm 0.002−0.018±0.002-0.018\pm 0.002−0.024±0.002-0.024\pm 0.002
 local feedback−0.011±0.002-0.011\pm 0.002−0.014±0.002-0.014\pm 0.002−0.018±0.002-0.018\pm 0.002−0.024±0.003-0.024\pm 0.003
 coupled CS00 (reference)00 (reference)00 (reference)00 (reference)
circulation contrast Δω\Delta_{\omega} (%)reciprocal−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )
 inst. transverse−0.009±0.002-0.009\pm 0.002−0.011±0.002-0.011\pm 0.002−0.014±0.002-0.014\pm 0.002−0.017±0.002-0.017\pm 0.002
Table 2: Endpoint pulse-order contrasts versus amplitude, after R=12R=12 zero-input steps. For each seed and ApA_{\rm p} , we average raw distances over W∈{8,12,16}W\in\{8,12,16\} before forming contrasts relative to coupled CS. Entries are ten-seed means ±\pm SEM; coupled CS is the exact reference.
Figure 4: Held-out readout diagnostics. (a) Scalar-memory coefficient MC(d)MC(d) for continuous delayed-input regression; curves and bands show the ten-seed mean and SEM, and legend values give C=∑dMC(d)C=\sum_{d}MC(d) . (b) Binary pulse-order classification accuracy for the five feature families. Cells give the mean ±\pm SEM over 50 outer folds, with ridge penalties selected by nested seed-block validation; chance accuracy is 0.50.5 .
Observablemodeltrel=0t_{\rm rel}=0trel=0.24t_{\rm rel}=0.24trel=0.48t_{\rm rel}=0.48trel=0.72t_{\rm rel}=0.72
density contrast Δρ\Delta_{\rho} (%)reciprocal0.552±0.2080.552\pm 0.2080.404±0.2000.404\pm 0.2000.299±0.1940.299\pm 0.1940.222±0.1890.222\pm 0.189
 inst. transverse−0.010±9.89×10−4-0.010\pm 9.89\times 10^{-4}−0.021±0.002-0.021\pm 0.002−0.030±0.003-0.030\pm 0.003−0.039±0.004-0.039\pm 0.004
 local feedback−0.010±0.002-0.010\pm 0.002−0.021±0.003-0.021\pm 0.003−0.030±0.003-0.030\pm 0.003−0.039±0.004-0.039\pm 0.004
 coupled CS00 (reference)00 (reference)00 (reference)00 (reference)
circulation contrast Δω\Delta_{\omega} (%)reciprocal−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )−100.000-100.000 ( Dω≃0D_{\omega}\simeq 0 )
 inst. transverse−0.008±0.001-0.008\pm 0.001−0.016±0.002-0.016\pm 0.002−0.023±0.003-0.023\pm 0.003−0.029±0.004-0.029\pm 0.004
Table 3: Post-drive pulse-order contrasts. Here trel=0t_{\rm rel}=0 is immediately after the second pulse; the endpoint in Table 2 is at trel=0.18t_{\rm rel}=0.18 . For each seed and time, we average raw distances over WW and ApA_{\rm p} before forming contrasts and ratios relative to coupled CS. Entries are ten-seed means ±\pm SEM; coupled CS is the exact reference.
Figure 5: First-cycle density lag without pre-equilibration. (a) Mean forward (solid) and backward (dashed) branches at rs=0.2083r_{s}=0.2083 ; bands show SEM. We show only the reciprocal and coupled-CS branches because the instantaneous transverse controls overlap visually. (b) Loop area AcycA_{\rm cyc} versus sweep rate. (c) Mean loop area with SEM error bars at the representative rate.
Figure 6: Incremental feedback response. (a) Background-subtracted excess ΔRχ\Delta R_{\chi} ; the dashed line gives the uncorrected α=0\alpha=0 scale. (b) Raw asymmetry RχR_{\chi} for coupled CS and the instantaneous baseline.
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