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Feasibility and Memory Mechanisms of Chern-Simons Context Reservoir Computation
Jyotiranjan Beuria, Venkatesh H. Chembrolu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
This paper investigates the feasibility and memory mechanisms of a Chern-Simons (CS) context reservoir for physical computing. The authors compare four models—reciprocal transport, instantaneous transverse reconstruction, local nonlinear feedback, and fully coupled conserved-current CS dynamics—to determine if evolving the gauge connection provides a computational advantage. Results show that while all models exhibit fading scalar memory, the fully coupled CS dynamics uniquely supports circulating and longitudinal history channels, yielding higher pulse-order accuracy (0.879 vs 0.679) and satisfying the spatial CS equation more accurately. The study concludes that the advantage is task-specific, arising from geometry- and order-sensitive processing rather than generic reservoir superiority.
Entities (9)
Relation Signals (8)
Fully Coupled CS Dynamics → outperforms → Instantaneous Transverse Reconstruction
confidence 95% · yields a combined-feature pulse-order accuracy of 0.879±0.035, compared with 0.679±0.065 for the instantaneous-transverse control.
Fully Coupled CS Dynamics → satisfies → Gauss Law
confidence 95% · the fully coupled CS dynamics propagates Gauss law to numerical precision
Gauge Connection → cannotbereconstructedfrom → Instantaneous Density Snapshot
confidence 92% · The evolved connection also cannot be reconstructed from an instantaneous density snapshot
Fully Coupled CS Dynamics → supports → Longitudinal History Channels
confidence 90% · coupled evolution supports circulating and longitudinal history channels simultaneously
Fully Coupled CS Dynamics → supports → Circulating History Channels
confidence 90% · coupled evolution supports circulating and longitudinal history channels simultaneously
Reciprocal Transport → exhibits → Fading Scalar Memory
confidence 85% · All four models exhibit fading scalar memory
Instantaneous Transverse Reconstruction → exhibits → Fading Scalar Memory
confidence 85% · All four models exhibit fading scalar memory
Local Nonlinear Feedback → exhibits →
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Abstract
Abstract:We investigate whether a Chern-Simons (CS) context reservoir is a viable computational substrate and whether evolving its gauge connection provides a benefit beyond simpler mechanisms. The reservoir state is a density fluctuation on a two-dimensional context manifold, whose drift is generated by a density-sourced connection. To separate generic reservoir behavior from gauge-specific effects, we compare four matched models: reciprocal transport, instantaneous transverse reconstruction, local nonlinear feedback, and fully coupled conserved-current CS dynamics. Across ten random seeds, the fully coupled CS dynamics propagates Gauss law to numerical precision, converges under spatial and temporal refinement, remains stable under constraint-compatible noise, and satisfies the spatial CS equation more accurately than the instantaneous controls. All four models exhibit fading scalar memory and distinguish matched pulse-order histories in density, with no resolved general advantage for coupled CS. The distinction appears in the flow geometry: coupled evolution supports circulating and longitudinal history channels simultaneously, retains them briefly after input removal, and yields a combined-feature pulse-order accuracy of \(0.879\pm0.035\), compared with \(0.679\pm0.065\) for the instantaneous-transverse control. The evolved connection also cannot be reconstructed from an instantaneous density snapshot or replaced by a fitted local multiplier. We therefore find a task-specific advantage for geometry- and order-sensitive processing, rather than generic reservoir superiority. Here ``topological'' refers to the gauge organization of the state; the reported memory and cyclic-lag measures are not topological invariants.
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- Source: https://arxiv.org/abs/2609.13315v1
- Canonical: https://arxiv.org/abs/2609.13315v1
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Feasibility and Memory Mechanisms of Chern–Simons Context Reservoir Computation Jyotiranjan Beuria Affiliation: Center for Philosophical and Cognitive Sciences, ISS Delhi, India Venkatesh H. Chembrolu Affiliation: Indian Institute of Technology Mandi, Mandi, India September 10, 2026 Abstract We investigate whether a Chern-Simons (CS) context reservoir is a viable computational substrate and whether evolving its gauge connection provides a benefit beyond simpler mechanisms. The reservoir state is a density fluctuation on a two-dimensional context manifold, whose drift is generated by a density-sourced connection. To separate generic reservoir behavior from gauge-specific effects, we compare four matched models: reciprocal transport, instantaneous transverse reconstruction, local nonlinear feedback, and fully coupled conserved-current CS dynamics. Across ten random seeds, the fully coupled CS dynamics propagates Gauss law to numerical precision, converges under spatial and temporal refinement, remains stable under constraint-compatible noise, and satisfies the spatial CS equation more accurately than the instantaneous controls. All four models exhibit fading scalar memory and distinguish matched pulse-order histories in density, with no resolved general advantage for coupled CS. The distinction appears in the flow geometry: coupled evolution supports circulating and longitudinal history channels simultaneously, retains them briefly after input removal, and yields a combined-feature pulse-order accuracy of 0.879±0.0350.879± 0.035, compared with 0.679±0.0650.679± 0.065 for the instantaneous-transverse control. The evolved connection also cannot be reconstructed from an instantaneous density snapshot or replaced by a fitted local multiplier. We therefore find a task-specific advantage for geometry- and order-sensitive processing, rather than generic reservoir superiority. Here “topological” refers to the gauge organization of the state; the reported memory and cyclic-lag measures are not topological invariants. I Introduction Physical reservoir computing is often presented as a pragmatic route to computation with materials: a fixed driven medium transforms input history into a measured state, and only a final linear readout is trained [21, 20, 35]. For physics, however, the central question is not only whether a substrate can perform a task. It is what kind of state variable carries the history. A scalar relaxation mode, a reciprocal density field, a circulating transverse flow, and a hysteretic current loop can all give nontrivial readout signals, but they represent different mechanisms. Without controlled comparisons these mechanisms are easily conflated: classification accuracy can increase even when the measured variable is ordinary density memory, while visually chiral motion need not imply a gauge-generated current. The gap addressed here is therefore mechanistic. Existing reservoir theory gives powerful abstract conditions for fading memory, information capacity, and universality [5, 7, 13]. Physical-reservoir reviews emphasize that the material substrate supplies the recurrent degrees of freedom [25, 12]. What is less developed is a continuum field model in which the memory-bearing observables are separated by construction, so that scalar memory, longitudinal density response, transverse circulation, chirality, hysteresis, and current-feedback consistency can be tested against one another. Such a separation is needed if a gauge or active-matter reservoir is to be interpreted as a physical mechanism rather than as a black-box nonlinear filter. Nonlinear dynamics already uses reservoir methods for state reconstruction and prediction, including chaotic-attractor reconstruction and Lyapunov-spectrum estimation [27], spatiotemporally chaotic prediction [26], long-time climate replication tests [14], chaotic-signal separation [19], convective-flow forecasting [17], and Hamiltonian dynamics [41]. Other studies examine how the chosen reservoir mechanism shapes the represented state space: random automata [33], explicit filtering [6], transport in reservoir coordinates [22], continuous-time synchronization and embedding [16], minimal models [30], and adaptive oscillator substrates [32]. These works motivate, but do not by themselves provide, a gauge-constrained field theory in which memory can be assigned to specific local observables. Active and driven matter make the same issue concrete because density, current, and broken time-reversal symmetry can be physical observables [28, 23, 3]. Flocking theories [38, 36], nonreciprocal active interactions [10, 40, 29, 18], and odd transport coefficients such as odd viscosity, odd elasticity, and odd diffusivity [2, 31, 15] show that transverse response and non-potential dynamics are natural in nonequilibrium media. They also show why nomenclature must be disciplined. Circulation, chirality, nonreciprocity, nonlinear map noncommutativity, and hysteresis are related phenomena, but none is a substitute for the others. In an earlier work [4], the authors introduced a density on an internal context manifold as a source for a U(1)U(1) Chern-Simons field and established transverse density-sourced response, current-feedback nonreciprocity, collective circulation, and hysteresis after a context-to-physical pushforward. That work did not ask which part of the driven response should be regarded as reservoir memory, nor did it separate an instantaneous Gauss-law reconstruction from a spatial gauge field evolved by the same conserved current that transports the density. Those are the two needs of the present paper. The main technical advance is the replacement of density-snapshot gauge reconstruction by a coupled density-gauge evolution in which one conserved current propagates both the density and the spatial Chern-Simons field. We organize the paper around two tests. The first is feasibility: can a driven topological context reservoir propagate its constraint, remain stable under refinement and compatible noise, and expose measurable local observables? The second is comparative value: after holding the drive, dissipation, discretization, and initial ensemble fixed, what does coupled CS evolution add beyond reciprocal flow, instantaneous transverse reconstruction, and a local nonlinear closure? The resulting four-model hierarchy separates density relaxation, longitudinal response, transverse circulation, dissipative lag, and an independently evolved gauge state. Its purpose is not to label every path-dependent signal as topological. Rather, it identifies which memory signatures require conserved-current gauge evolution and whether those signatures are large and accessible enough to justify the additional field dynamics. This provides a concrete criterion for deciding whether topological context reservoirs merit further theoretical and experimental development. The remainder of the paper is organized as follows. Section I defines the context reservoir and the local density and geometric observables used to diagnose memory. Section I formulates the gauge-constrained dynamics and the conserved-current treatment of the external drive, while Sec. IV constructs the four-model hierarchy used for controlled comparison. Section V describes the common numerical implementation. Sections VI and VII then specify, respectively, the reservoir-computing and gauge-dynamics diagnostics. Section VIII presents the results, Sec. IX discusses their mechanistic interpretation and limitations, and Sec. X summarizes the conclusions. I Context reservoir and local memory A physical reservoir has a state z(t)z(t), fixed internal dynamics, temporal input u(t)u(t), measured features R[z(t)]R[z(t)], and a trained linear readout only: z˙=F[z,u(t)],y^=b+R[z(t)]. z=F[z,u(t)], y=b+ w TR[z(t)]. (1) In the present system z=(ρ,a)z=(ρ,a), where ρ(c,t)ρ(c,t) is a zero-mean context-density fluctuation and a(c,t)a(c,t) is a spatial gauge one-form on context-manifold C. The field dynamics are fixed during training and testing. We distinguish two local notions of memory. • Density memory is a residual difference in ρ after different input histories. • Geometric local memory is a difference in circulation, divergence, or other flow observables reconstructed from a. These are measured state variables rather than trained internal coordinates. The linear readout tests their accessibility to a simple observation protocol. This separation between fixed dynamics, measured features, and a trained readout is essential to the reservoir-computing tests developed below. The input history must first be encoded by the autonomous response of the physical state; the feature map then determines which parts of that response are experimentally accessible, while the readout tests whether the encoded information can be recovered without modifying the reservoir itself. Keeping these roles distinct also makes the comparison between models mechanistic: a difference in performance can be traced to the available density or geometric state variables rather than to model-specific training of the internal dynamics. The delayed-input and pulse-order protocols of Sec. VI therefore use the same dynamical states both for direct memory diagnostics and for constructing the feature families given to the linear readout. I Gauge-constrained context dynamics I.1 The Chern–Simons formalism Let jμ=(ρ,j1,j2)j^μ=(ρ,j^1,j^2) be a conserved context current, ∂μjμ=∂tρ+∇iji=0. _μj^μ= _tρ+ _ij^i=0. (2) We introduce an Abelian gauge field aμa_μ governed by the Chern–Simons action S[a;j]=κ4π∫ℳ3d3xϵμνλaμ∂νaλ−∫ℳ3d3xaμjμ,S[a;j]= κ4π _M_3d^3x\,ε^μνλa_μ _νa_λ- _M_3d^3x\,a_μj^μ, (3) where ϵ012=+1ε^012=+1. Here, ℳ3=ℝt×M_3=R_t×C denotes the three-dimensional manifold formed by time and the two-dimensional context space C. Accordingly, d3x=dtd2xd^3x=dtd^2x with t∈ℝt _t and x∈x . Variation with respect to aμa_μ gives κ2πϵμνλ∂νaλ=jμ,g≡2πκ. κ2πε^μνλ _νa_λ=j^μ, g≡ 2πκ. (4) The use of a Chern–Simons term follows the standard role of this action as a first-order, metric-independent gauge response in two spatial dimensions [9, 8, 39, 42]. In terms of temporal gauge a0=0a_0=0 and coupling g, Equation 4 gives ∂1a2−∂2a1 _1a_2- _2a_1 =gρ, =gρ, (5) ∂ta1 _ta_1 =gj2,∂ta2=−gj1. =gj^2, _ta_2=-gj^1. (6) In the overdamped limit, inertial effects are neglected, and the context drift is assumed to respond instantaneously and linearly to the gauge one-form a. The overdamped context drift is =−γa,γ>0. v=-γ a, γ>0. (7) The coefficient γ>0γ>0 sets the mobility, while the minus sign indicates drift opposite to the effective gauge-induced force. The spatial Chern–Simons equations ∂ta1=gj2,∂ta2=−gj1 _ta_1=gj^2, _ta_2=-gj^1 (8) can be written compactly as ∂ta=gJ,J=(01−10). _ta=gJ j, J= pmatrix0&1\\ -1&0 pmatrix. (9) The above equations and their physical interpretation in context space dynamics are inherited from the authors’ previous work Ref. [4]. I.2 Modelling external drive In the reservoir-computing interpretation, the external drive represents an input signal that perturbs the internal context state rather than acting directly on the output. Each input channel (uℓ(t)u_ (t)) is therefore coupled to a prescribed context profile (ρin,ℓ(c) _ in, (c)), so that the signal injects a structured deformation into the context density. The subsequent transport, relaxation, and gauge-mediated dynamics transform this deformation into a history-dependent internal response, providing the memory and nonlinear state expansion required for reservoir computation. We begin with a driven density equation in which the phenomenological transport current is p=αρ−D∇ρ. j_ p=αρ v-D∇ρ. (10) Here, α controls the strength of advective transport generated by the context velocity v, while D>0D>0 is the diffusion coefficient. The parameter α is purely phenomenological and is unrelated to the Chern–Simons coupling. To model external driving, we supplement the transport dynamics with a phenomenological source term consisting of linear relaxation and prescribed input injection. We therefore choose s(c,t)=−τρ(c,t)+ε∑ℓ=12uℓ(t)ρin,ℓ(c),s(c,t)=-τρ(c,t)+ _ =1^2u_ (t) _ in, (c), (11) where τ>0τ>0 is the relaxation rate, ε sets the input strength, uℓ(t)u_ (t) are externally prescribed input signals, and ρin,ℓ(c) _ in, (c) are their spatial profiles on the context manifold C. The term −τρ-τρ relaxes the context density toward the reference state ρ=0ρ=0, while uℓ(t)ρin,ℓ(c)u_ (t) _ in, (c) injects the ℓ input signal with a prescribed spatial profile. This source is a modelling assumption and is not derived from the Chern-Simons action. The phenomenological transport current p j_ p and source s define the driven density equation ∂tρ+∇⋅p=s. _tρ+∇· j_ p=s. (12) This equation is not itself a continuity equation because the source s locally injects or removes context density. However, the Chern–Simons field equation requires the full spacetime current jμ=(ρ,)j^μ=(ρ, j) to satisfy ∂tρ+∇⋅=0. _tρ+∇· j=0. (13) To represent the driven dynamics using such a conserved current, we introduce a longitudinal current correction ∇ψ∇ψ, where ψ solves ∇2ψ=−s,∇^2ψ=-s, (14) and define =p+∇ψ. j= j_ p+∇ψ. (15) It then follows that ∇⋅=∇⋅p+∇2ψ=∇⋅p−s.∇· j=∇· j_ p+∇^2ψ=∇· j_ p-s. (16) Substitution into Eq. (12) gives ∂tρ+∇⋅=0, _tρ+∇· j=0, (17) so the completed current reproduces exactly the original driven density evolution while remaining compatible with the Chern–Simons constraint. On a closed context manifold C, the Poisson equation ∇2ψ=−s∇^2ψ=-s is solvable only when s has zero spatial mean. Indeed, ∫∇2ψV=∫∂∇ψ⋅S=0, _C∇^2ψ\,dV= _ ∇ψ· n\,dS=0, (18) because ∂=∅ = , i.e., a closed manifold has no boundary surface. Therefore, a necessary solvability condition is ∫sV=0. _Cs\,dV=0. (19) For the source specified in Eq. (11), this condition holds when the initial density and each input profile have zero spatial mean. Indeed, integrating Eq. (12) over C gives dt∫ρdV=−τ∫ρdV+ε∑ℓ=12uℓ(t)∫ρin,ℓdV. ddt _Cρ\,dV=-τ _Cρ\,dV+ _ =1^2u_ (t) _C _ in, \,dV. (20) Here we used the divergence theorem, ∫∇⋅pV=∫∂p⋅S=0, _C∇· j_ p\,dV= _ j_ p· n\,dS=0, since the closed manifold C has no boundary. Hence, if ∫ρ(c,0)V=0,∫ρin,ℓ(c)V=0, _Cρ(c,0)\,dV=0, _C _ in, (c)\,dV=0, (21) then ∫ρV=0 _Cρ\,dV=0 and consequently ∫sV=0 _Cs\,dV=0 for all times. Physically, the source may rearrange positive and negative density deviations within C, but it does not change their net spatial integral. The solution of Eq. (14) is determined only up to an additive constant, since ψ and ψ+Cψ+C generate the same current correction. We remove this ambiguity by imposing ∫ψV=0. _Cψ\,dV=0. (22) This normalization does not affect ∇ψ∇ψ; it merely selects a unique scalar potential. The correction ∇ψ∇ψ is therefore a mathematical completion of the externally driven model rather than an independently derived microscopic transport process. More precisely, among all corrections q satisfying ∇⋅=−s,∇· q=-s, (23) the longitudinal choice =∇ψ q=∇ψ has minimum L2L^2 norm. Since ∇2ψ=−s∇^2ψ=-s, it can be written as =∇ψ+,∇⋅=0. q=∇ψ+ w, ∇· w=0. (24) ∇ψ∇ψ is the longitudinal part, while w is the transverse, divergence-free part. The longitudinal component changes the local density through its divergence; the transverse component only redistributes or circulates current without changing ∇⋅q∇· q. On the closed manifold C, integration by parts gives ∫∇ψ⋅dV=−∫ψ∇⋅dV=0. _C∇ψ· w\,dV=- _Cψ\,∇· w\,dV=0. (25) Thus the longitudinal and divergence-free components are orthogonal, and ‖L22=‖∇ψ‖L22+‖L22≥‖∇ψ‖L22.\| q\|_L^2^2=\|∇ψ\|_L^2^2+\| w\|_L^2^2≥\|∇ψ\|_L^2^2. (26) Hence ∇ψ∇ψ is the minimum-L2L^2-norm correction compatible with the prescribed divergence. The minimum-L2L^2-norm prescription is important because Eq. (12) fixes only the divergence of the correction through Eq. (14), not the full correction itself. As shown by Eq. (15), any additional divergence-free component would leave Eq. (17) unchanged but would modify the current coupled to the Chern-Simons field. Choosing =∇ψ q=∇ψ therefore adds only the current required to restore conservation, without introducing independent transverse transport. IV Controlled field models All models use the same source term, diffusion, relaxation, input protocol, grid, initial condition, and phenomenological transport current p=αρ−D∇ρ. j_ p=αρ v-D∇ρ. (27) They also use the same conserved completion =p+∇ψ,∇2ψ=−s, j= j_ p+∇ψ, ∇^2ψ=-s, (28) so that ∂tρ+∇⋅=0. _tρ+∇· j=0. (29) The comparison therefore changes only the constitutive relation between the density, the gauge field, and the drift velocity. This provides a controlled hierarchy in which reciprocal response, transverse response, local nonlinear feedback, and independent gauge-field memory can be introduced one at a time. Reciprocal-gradient control. We first define a reciprocal control in which the context density generates an ordinary scalar potential, −∇2ϕ=gρ,-∇^2φ=gρ, (30) and the induced drift follows the potential gradient, rec=−γ∇ϕ=−γg∇(−∇2)−1ρ. v_ rec=-γ∇φ=-γ g\,∇(-∇^2)^-1ρ. (31) Because the inverse Laplacian is nonlocal, the velocity at a given point depends on the density distribution over the full context manifold. The flow is nevertheless purely gradient-directed, and hence ∇×rec=−γ∇×∇ϕ=0.∇× v_ rec=-γ\,∇×∇φ=0. (32) This model therefore retains nonlocal density-mediated transport while excluding transverse circulation. It serves as the baseline against which the effect of the Chern–Simons-induced rotation of the drift can be identified. Instantaneous transverse control. We use the scalar potential ϕφ determined by Eq. (30). Its gradient, ∇ϕ=(∂1ϕ,∂2ϕ),∇φ=( _1φ, _2φ), (33) points in the direction of the steepest increase of ϕφ. To obtain a transverse response, we rotate this gradient by 90∘90 using J=(01−10).J= pmatrix0&1\\ -1&0 pmatrix. (34) Thus, ac=J∇ϕ=(∂2ϕ,−∂1ϕ).a_ c=J∇φ=( _2φ,- _1φ). (35) The corresponding context drift is defined by 0=−γac. v_0=-γ a_ c. (36) The rotated field is perpendicular to the original gradient because ∇ϕ⋅ac=(∂1ϕ)(∂2ϕ)−(∂2ϕ)(∂1ϕ)=0.∇φ· a_ c=( _1φ)( _2φ)-( _2φ)( _1φ)=0. (37) Hence aca_ c points along the level curves of ϕφ, producing circulation around density-induced structures rather than motion directly along the potential gradient. The field is also divergence-free: ∇⋅ac=∂1∂2ϕ−∂2∂1ϕ=0,∇· a_ c= _1 _2φ- _2 _1φ=0, (38) where equality of mixed derivatives has been used. Its curl, however, is nonzero: ∂1ac,2−∂2ac,1 _1a_ c,2- _2a_ c,1 =∂1(−∂1ϕ)−∂2(∂2ϕ) = _1(- _1φ)- _2( _2φ) =−∇2ϕ=gρ. =-∇^2φ=gρ. (39) Therefore, aca_ c satisfies the Chern–Simons Gauss-law (Eq. 5) constraint at every instant. Since it is reconstructed directly from the current density ρ(t)ρ(t), this control retains the transverse spatial response but does not possess an independently evolving gauge-field memory. Solving Eq. (30) gives ϕ(t)=g(−∇2)−1ρ(t),φ(t)=g(-∇^2)^-1ρ(t), (40) and hence ac(t)=J∇ϕ(t)=gJ∇(−∇2)−1ρ(t)a_ c(t)=J∇φ(t)=gJ∇(-∇^2)^-1ρ(t) (41) and 0(t)=−γgJ∇(−∇2)−1ρ(t). v_0(t)=-γ gJ∇(-∇^2)^-1ρ(t). (42) Thus aca_ c is reconstructed entirely from the instantaneous density: first the Poisson equation is inverted, then the resulting gradient is rotated by 90∘90 . Local nonlinear-feedback control. To isolate the effect of state-dependent feedback from that of independent gauge-field memory, we introduce the phenomenological closure loc=(1+ηρ)0. v_ loc=(1+ηρ) v_0. (43) Here, η is the feedback-control parameter. It determines how strongly the instantaneous context density modifies the amplitude of the transverse drift: positive η enhances the local response where ρ>0ρ>0 and suppresses it where ρ<0ρ<0, while η=0η=0 recovers the instantaneous transverse control, loc|η=0=0. v_ loc |_η=0= v_0. (44) Equivalently, loc=0+ηρ0, v_ loc= v_0+ηρ\, v_0, (45) so the second term is a local nonlinear feedback correction whose strength can be varied continuously through η. This control is important because nonlinear feedback and dynamical memory can produce similar history-dependent responses. Equation (43) adds nonlinear state dependence while keeping the field instantaneous: loc(t) v_ loc(t) is determined entirely by ρ(t)ρ(t) and 0(t) v_0(t), with no independently evolving gauge variable. Comparing this model with the fully coupled Chern–Simons model therefore tests whether the observed effects arise from local nonlinear amplification alone or require memory stored in the gauge-field dynamics. Comparison with the fully coupled model. The three controls are compared with the fully coupled Chern–Simons model introduced in section I.1. Unlike the controls, the full model evolves the gauge field independently through the conserved current and therefore retains a history-dependent internal state that cannot, in general, be reconstructed from the instantaneous density alone. The comparison separates the effects of reciprocal transport, transverse circulation, local nonlinear feedback, and genuine gauge-field memory. The four models differ in one constitutive ingredient at a time: rec v_ rec =−γ∇(−∇2)−1gρ, =-γ∇(-∇^2)^-1gρ, (46) 0 v_0 =−γJ∇(−∇2)−1gρ, =-γ J∇(-∇^2)^-1gρ, (47) loc v_ loc =(1+ηρ)0, =(1+ηρ) v_0, (48) v =−γa,∂ta=−gJ. =-γ a, _ta=-gJ j. (49) The reciprocal control tests nonlocal density-mediated dynamics without circulation. The instantaneous transverse control adds circulation but removes independent gauge memory. The local closure adds nonlinear density feedback while remaining instantaneous. The coupled model alone retains transverse response, nonlinear current feedback, and a gauge field that integrates the history of the conserved current. V Numerical implementation All four models are evolved using the same spatial discretization, time-integration scheme, external drive, initial-condition ensemble, and random seeds, so that differences in the measured responses can be attributed to the constitutive dynamics rather than to the numerical protocol. This section defines that common implementation; the task-specific protocols are introduced in Secs. VI and VII. V.1 Domain and spectral discretization We take the context manifold to be the flat periodic torus =[0,L)2C=[0,L)^2, with side length L=2πL=2π and coordinates =(c1,c2) c=(c_1,c_2). The dynamical variables are the zero-mean density fluctuation ρ(c1,c2,t)ρ(c_1,c_2,t), the drift field v, the conserved current j, and, in the coupled model, the spatial gauge connection a=(a1,a2)a=(a_1,a_2). Spatial averages are denoted by ⟨⋅⟩ · , and the root-mean-square norm is defined as ‖f‖2=⟨|f|2⟩1/2. f _2= |f|^2 ^1/2. Unless stated otherwise, the simulations are performed on a 16×1616× 16 uniform grid. Spatial derivatives and Poisson inversions are evaluated spectrally using the integer Fourier wave vectors =(n1,n2) k=(n_1,n_2). For the instantaneous controls, the scalar potential is defined by the zero-mean Poisson problem −∇2ϕ=gρ,⟨ϕ⟩=0,ϕ^=gρ^||2(≠0).-∇^2φ=gρ, φ =0, φ_ k= g ρ_ k| k|^2 ( k≠ 0). (50) The reciprocal and instantaneous-transverse velocities are then rec=−γ∇ϕ v_ rec=-γ∇φ and 0=−γJ∇ϕ v_0=-γ J∇φ, respectively. The rotated field has curl gρgρ and hence satisfies the Gauss-law constraint at that instant. By contrast, the coupled-CS connection is evolved from the conserved current; Eq. (50) is used for that model only during initialization and diagnostic reconstruction. Nonlinear products, in particular the advective term ρρ v, are evaluated pseudospectrally. Such products generate wave numbers larger than those resolved by the discrete grid, which can be spuriously folded onto lower Fourier modes. We suppress this aliasing with the standard componentwise two-thirds truncation. Since the Nyquist wave number of an N-point grid is N/2N/2, retaining two thirds of the resolved bandwidth gives the cutoff (2/3)(N/2)=N/3(2/3)(N/2)=N/3. After each nonlinear multiplication we therefore set to zero all Fourier coefficients satisfying |n1|>N/3or|n2|>N/3.|n_1|>N/3 |n_2|>N/3. For the 16×1616× 16 grid, the integer modes retained by this criterion are |n1|,|n2|≤5|n_1|,|n_2|≤ 5. Applying the cutoff separately to both components is the tensor-product form of the two-thirds rule appropriate to the square Fourier grid. For scalar Poisson problems, the =0 k=0 Fourier coefficient of the inverse Laplacian is set to zero, consistently with the imposed zero-mean convention. In the coupled gauge-field evolution, the spatially uniform component of the vector connection is removed separately, since it is not fixed by the Gauss-law constraint on the torus. V.2 Time integration and parameters Time integration is carried out using a fourth-order Runge–Kutta scheme with time step Δt=0.015. t=0.015. The external inputs are sampled and held fixed during each integration step. The parameters used in the principal simulations are g g =0.45, =0.45, γ γ =0.65, =0.65, D D =0.045, =0.045, τ τ =0.12, =0.12, ε =0.20, =0.20, α α =0.70. =0.70. Here g is the density–gauge coupling, γ converts the connection into an overdamped drift, D is the diffusivity, τ is the linear density-relaxation rate, ε sets the input amplitude, and α controls the strength of advective current feedback. For the local nonlinear control we use η=0.35η=0.35, except in the closure-fitting analysis, where η is determined explicitly from the simulated data. V.3 Inputs, initial conditions, and ensembles We introduce two input channels through the fixed zero-mean profiles ρin,1 _ in,1 ∝cosc1+cos(c1+c2), c_1+ (c_1+c_2), ρin,2 _ in,2 ∝sinc2+sin(c1−c2), c_2+ (c_1-c_2), (51) each normalized to unit root-mean-square amplitude. Equal weights avoid a fitted shape parameter. These are the only source profiles; protocols vary only u1(t)u_1(t) and u2(t)u_2(t). Later Gaussian masks and packets are readouts and probes, not alternative inputs. When a nonzero random initial density is required, it is constructed as a sum of eight cosine modes with 1≤n1≤31≤ n_1≤ 3, −3≤n2≤3-3≤ n_2≤ 3, and independently sampled coefficients and phases. The field is made zero mean and normalized to the root-mean-square amplitude stated for the corresponding diagnostic. All four models use the same realization for a given seed. The coupled connection is initialized as a(0)=gJ∇(−∇2)−1ρ(0)a(0)=gJ∇(-∇^2)^-1ρ(0), using Eq. (50), while each control is initialized from the same density field. Unless stated otherwise, ensemble averages use ten predetermined seeds paired across the four models. Shaded bands show the standard error of the mean (SEM), σsample/10 _ sample/ 10, where σsample _ sample is the sample standard deviation across seeds. The controlled mechanism hierarchy itself is defined in Sec. IV; the shared realizations used here isolate those constitutive differences. VI Reservoir-computing protocols We test three aspects of reservoir behavior: recovery of past scalar inputs, discrimination of input order, and rate-dependent lag. The pulse-order analysis includes endpoint separation and held-out linear classification. Throughout, uℓ(t)u_ (t) in Eq. (11) is the scalar amplitude of the fixed spatial profile ρin,ℓ() _ in, ( c); it is not a component of the evolving density ρ(,t)ρ( c,t). VI.1 Scalar fading memory We first test whether the reservoir retains information about a past scalar input. Only the first input channel is used, so u2(t)=0u_2(t)=0 and u1(t)=u(t)u_1(t)=u(t). The source term becomes s(,t)=−τρ(,t)+εu(t)ρin,1().s( c,t)=-τρ( c,t)+ u(t) _ in,1( c). (52) The input consists of independent random amplitudes un∼[−1,1]u_n [-1,1]. Each amplitude is held constant for one symbol interval, u(t)=un,nTs≤t<(n+1)Ts,Ts=3Δt.u(t)=u_n, nT_s≤ t<(n+1)T_s, T_s=3 t. (53) The reservoir is therefore driven by a sequence …,un−2,un−1,un…,u_n-2,u_n-1,u_n. If its dynamics has fading memory, its state after receiving unu_n should still contain some information about the earlier input un−du_n-d, with this information generally becoming weaker as the delay d increases. At the end of symbol interval n, we summarize the density field by n x_n =((Reρ^,Imρ^)∈,⟨ρ2⟩)∈ℝ9, = ( (Re ρ_ k,Im ρ_ k )_ k , ρ^2 ) ^9, (54) =(1,0),(0,1),(1,1),(1,−1). =\(1,0),(0,1),(1,1),(1,-1)\. The set K is not unique: in principle, one could use any number of Fourier modes scattered across wave-vector space. We deliberately retain only the lowest nonzero axial and diagonal modes because they capture robust, large-scale density variations while keeping the readout low-dimensional. For a real density field, the corresponding negative modes contain no independent information. The moment ⟨ρ2⟩ ρ^2 adds the overall density variation to the feature set. This restricted feature set tests whether memory is accessible from simple, preselected observables rather than from an exhaustive representation of the field. We standardize the features using means and standard deviations calculated from the training data, producing n z_n; the same transformation is then applied to the test data. For each delay d=1,…,16d=1,…,16, we fit a separate linear readout, u^n,d u_n,d =bd+dn, =b_d+ β_d T z_n, (55) (bd,d) (b_d, β_d) =argminb,ℒd(b,), = _b, βL_d(b, β), ℒd(b,) _d(b, β) =∑n∈d[un−d−b−n]2 = _n _d [u_n-d-b- β T z_n ]^2 +10−4‖22, +10^-4 β _2^2, (56) where dT_d contains the valid training indices. Here un−du_n-d is the actual input presented d symbol intervals earlier, whereas u^n,d u_n,d is its estimate obtained only from the present reservoir features n z_n. The squared-error term makes this estimate close to the past input, and the ridge penalty stabilizes the fit when features are correlated. The intercept is not penalized. This is a continuous regression task, not a binary classification task. We evaluate the fitted readout on an independent test sequence using MC(d)=corr2[un−d,u^n,d],C=∑d=116MC(d),MC(d)=corr^2[u_n-d, u_n,d], C= _d=1^16MC(d), (57) where corrcorr is the Pearson correlation over valid test times n for a fixed d. If the present state retains the input from d symbols ago, then u^n,d u_n,d follows un−du_n-d across the test sequence and MC(d)MC(d) is close to one. If that input is no longer recoverable, MC(d)MC(d) is close to zero. Thus MC(d)MC(d) measures linearly accessible memory at delay d, while C summarizes it over the tested 16-symbol window. One delay unit equals three RK4 steps. Each run begins with 50 discarded washout symbols. The readout is then fitted over a 350-symbol sequence and evaluated on an independent 250-symbol sequence generated in a separate run. Their indices are local to the two streams, and each post-washout stream is reindexed from zero. For delay d, its first d states are omitted because their targets un−du_n-d would lie in the discarded washout portion. The resulting 350−d350-d training and 250−d250-d test pairs come from the two independent streams; the correlation in Eq. (57) uses only the latter. All four models start from zero density and receive the same streams for direct comparison. VI.2 Matched pulse-order histories We next test whether the final reservoir state depends on the order of two input events. Intuitively, the same two pulses are applied in opposite sequences. Because the initial state, total input, and final input are matched, a difference between the two final states means that the reservoir retains the path taken rather than only the accumulated drive. This provides a direct test of order-sensitive memory; the density, circulation, and longitudinal diagnostics below identify which dynamical channel carries that memory. Both source channels in Eq. (11) are active, and =(u1,u2) u=(u_1,u_2) denotes their amplitudes. A pulse on channel 1 means =(Ap,0) u=(A_ p,0), so that the injected part of the source is εApρin,1 A_ p _ in,1; a pulse on channel 2 similarly uses =(0,Ap) u=(0,A_ p) and injects εApρin,2 A_ p _ in,2. Let W, G, and R denote the pulse width, intervening gap, and final relaxation time, all measured in RK4 steps. For compactness, []m[ q]_m denotes a constant input vector q held for m steps, and ∥ \| denotes temporal concatenation: the block to its right begins when the block to its left ends. It is not a product. The two protocols are A u_A =[(Ap,0)]W∥[(0,0)]G∥[(0,Ap)]W∥[(0,0)]R, =[(A_ p,0)]_W \|[(0,0)]_G \|[(0,A_ p)]_W \|[(0,0)]_R, B u_B =[(0,Ap)]W∥[(0,0)]G∥[(Ap,0)]W∥[(0,0)]R. =[(0,A_ p)]_W \|[(0,0)]_G \|[(A_ p,0)]_W \|[(0,0)]_R. (58) Thus protocol A applies channel 1 before channel 2, whereas protocol B reverses their order. The reservoir is observed after the final zero-input relaxation interval, at tf=(2W+G+R)Δt_f=(2W+G+R) t. At this time, A(tf) u_A(t_f) =B(tf)=0, = u_B(t_f)=0, ∫0tfAt _0^t_f u_A\,dt =∫0tfBt=ApWΔt(1,1), = _0^t_f u_B\,dt=A_ pW t\,(1,1), (59) where the integrals are componentwise. The two histories therefore have the same duration, final input, and total input delivered to each channel; only the order differs. Any difference between their final states is therefore a history-dependent response rather than a consequence of unequal input exposure. We use W∈8,12,16W∈\8,12,16\, Ap∈0.50,0.75,1.00,1.25A_ p∈\0.50,0.75,1.00,1.25\, G=5G=5, and R=12R=12. For a time step Δt=0.015 t=0.015, the three widths correspond to pulse durations 0.120.12, 0.180.18, and 0.240.24. They probe short, intermediate, and longer exposure, while the four amplitudes probe the response across increasing input strength. This compact factorial grid tests whether an order effect persists across both duration and strength rather than at one selected drive; the values are protocol choices, not optimized model parameters. For a given seed, both protocols and all four models start from the same random density field of root-mean-square amplitude 0.040.04. VI.3 Endpoint separation of matched histories The two protocols in Eq. (58) have the same duration, final input, and time-integrated input in each channel; only the order of the two pulses differs. Comparing their final states therefore establishes whether opposite pulse orders leave distinguishable reservoir states. This test complements scalar memory: scalar recall measures how long an input amplitude remains recoverable, whereas path separation asks whether the state retains the sequence in which different inputs arrived. Resolving the separation into density, circulating, and compressive components also shows which physical channel carries that history dependence. Comparison with the three controls then determines whether the effect is specific to the coupled CS dynamics or can be reproduced by generic nonlinearity, imposed chirality, or reciprocal context coupling. To resolve local geometric responses, we define two Gaussian observation masks on the domain of side L. Both have standard deviation 0.15L0.15L and unit spatial mean, and they are centered at w,1=(L/4,L/3) c_w,1=(L/4,L/3) and w,2=(3L/4,3L/5) c_w,2=(3L/4,3L/5). These fixed, well-separated rational locations avoid simple square symmetries and are not fitted to the data. The path diagnostics use w=w1w=w_1, whereas the readout features below use both masks. At tft_f, we compute Dρ D_ρ =‖ρA−ρB‖2, = _A- _B _2, Dv D_v =‖A−B‖2, = v_A- v_B _2, (60) Dω D_ω =|⟨w(∇×A−∇×B)⟩|, = | w(∇× v_A-∇× v_B) |, D∥ D_ =|⟨w(∇⋅A−∇⋅B)⟩|. = | w(∇· v_A-∇· v_B) |. Subscripts A and B denote the two input orders. The first two quantities measure global density and velocity separation. The masked curl and divergence differences, DωD_ω and D∥D_ , measure local circulating and compressive traces, respectively. Because these observables have different physical units, we do not compare their absolute magnitudes. When raw curves overlap, Tables 1 and 2 use paired, dimensionless contrasts. For model m, ΔX(m)=100DX(m)−DX(CS)DX(CS)+10−20,X∈ρ,ω, _X^(m)=100\, D_X^(m)-D_X^( CS)D_X^( CS)+10^-20, X∈\ρ,ω\, (61) and the longitudinal ratio Q∥(m)=D∥(m)D∥(CS)+10−20.Q_ ^(m)= D_ ^(m)D_ ^( CS)+10^-20. (62) The superscript CS CS denotes the coupled-CS value for the same seed and pulse pair, and 10−2010^-20 regularizes a vanishing denominator. Thus ΔX(m)=0 _X^(m)=0 or Q∥(m)=1Q_ ^(m)=1 denotes equality with coupled CS. We form these paired quantities before computing their means and SEMs across seeds. To test persistence beyond the endpoint, we continue each matched-history pair after the second pulse with u1=u2=0u_1=u_2=0. The relaxation term −τρ-τρ in Eq. (11) remains active. Over the next 48 RK4 steps (0.720.72 time units), we record DρD_ρ, DωD_ω, and D∥D_ , together with the paired measures in Eqs. (61) and (62). A nonzero trace means that pulse order remains encoded for a finite time after the drive is removed; it does not imply permanent storage. VI.4 Linear decodability of pulse order The endpoint distances above establish whether the two input histories produce different states, but they do not show whether pulse order can be recovered by a simple reservoir readout. We therefore test whether the order label is linearly decodable from five feature families. Let =(1,0),(0,1),(1,1),(1,−1)K=\(1,0),(0,1),(1,1),(1,-1)\ and define the eight-component low-mode vector ℱ[ϕ]=(Reϕ^,Imϕ^)∈,F_K[φ]= (Re φ_ k,Im φ_ k )_ k , (63) with the modes ordered as listed in K. Writing ω=∇×ω=∇× v and θ=∇⋅θ=∇· v, the five feature vectors are ρ x_ρ =(ℱ[ρ],⟨ρ2⟩)∈ℝ9, = (F_K[ρ], ρ^2 ) ^9, ω x_ω =(ℱ[ω],⟨ω2⟩,⟨w1ω⟩,⟨w2ω⟩)∈ℝ11, = (F_K[ω], ω^2 , w_1ω , w_2ω ) ^11, ∥ x_ =(ℱ[θ],⟨θ2⟩,⟨w1θ⟩,⟨w2θ⟩)∈ℝ11, = (F_K[θ], θ^2 , w_1θ , w_2θ ) ^11, geom x_ geom =(ω,∥)∈ℝ22,all=(ρ,geom)∈ℝ31. =( x_ω, x_ ) ^22, x_ all=( x_ρ, x_ geom) ^31. (64) Thus the families separately probe density, circulating flow, and longitudinal flow; the last two test whether combining geometric channels or all measured channels improves order recovery. As in the scalar-memory test, the low modes provide a compact record of large-scale amplitude and spatial phase, while the quadratic moments summarize total variation. The two masked averages additionally retain localized signed information that a global moment would erase. Using separate families makes the readout diagnostic: it shows whether order is accessible from density, circulation, longitudinal compression, or only from their combination, rather than merely maximizing the number of predictors. For each run, the selected feature vector j x_j is sampled at tft_f and assigned the label yj=+1y_j=+1 for protocol A or −1-1 for protocol B. On the standardized training data, we fit f()=b+,min∑jb,[yj−f(j)]2+λ‖22,f( x)=b+ a T x, _b, a _j[y_j-f( x_j)]^2+λ a _2^2, (65) and predict the order from the sign of f. Although the classifier is fitted with a ridge squared-error loss, its target has only the two order labels, and its reported held-out accuracy is Acc=1Ntest∑j∈test[signf(j)=yj].Acc= 1N_ test _j 1\! [signf( x_j)=y_j ]. (66) It is therefore the fraction of unseen endpoints for which the pulse order is predicted correctly; chance accuracy is 1/21/2. This loss and metric are distinct from Eqs. (56) and (57). For each random initial-condition seed, we simulate every combination of the three pulse widths and four amplitudes, giving 12 parameter settings. At each setting we run both pulse orders, A and B. Each run provides one endpoint feature vector and its order label, so one seed contributes 12×2=2412× 2=24 labeled samples. We evaluate the classifier by nested leave-one-seed-out validation, keeping all observations from a seed in the same fold to prevent information leakage between its paired runs. In each outer fold, feature means and standard deviations are fitted only on the training data, the intercept b is unpenalized, and the ridge coefficient λ∈10−6,10−5,…,102λ∈\10^-6,10^-5,…,10^2\ is selected by an inner leave-one-seed-out validation over the nine training seeds. The resulting accuracy measures the linear accessibility of pulse order from a specified feature family; it is distinct from the scalar memory capacity C. Each outer fit therefore uses 216 labeled endpoints and at most 31 features, with 24 endpoints reserved for testing. However, the statistically independent ensemble consists of ten seed blocks, not 240 individual endpoints. We consequently report variation across held-out seeds and treat these accuracies as comparative diagnostics; differences comparable to their SEM are not interpreted as model advantages. VI.5 Cyclic-drive density lag To complement the pulse-order test, we measure how the density follows a cyclic input. A difference between the outward and return responses at the same input amplitude means that the present state still depends on the path taken; we call this finite-time dependence cyclic-drive lag. In Eq. (11), we sweep the existing input amplitude u1u_1 from −1-1 to 11 and back while holding u2=0.50u_2=0.50, which keeps the second channel active. The sweep is deterministic. Each branch uses NbN_b uniformly spaced input values, one per RK4 step of duration Δt=0.015 t=0.015; the return branch visits the same values in reverse. Thus a complete cycle has 2Nb2N_b steps. With the nominal rate rs=2/(NbΔt)r_s=2/(N_b t), the choices Nb=160,320,640,1280,2560N_b=160,320,640,1280,2560 give rs=0.8333,0.4167,0.2083,0.1042,0.0521r_s=0.8333,0.4167,0.2083,0.1042,0.0521. This factor-of-two scan tests whether slower driving reduces the lag or leads to accumulated lag or numerical instability. These are fixed protocol choices, not fitted parameters. We monitor Eρ=⟨ρ2⟩E_ρ= ρ^2 , which measures the overall density-pattern strength and is defined for every model. Vorticity is not used because it vanishes for the reciprocal control by construction. For outward and return branches Eρ+(u1)E_ρ^+(u_1) and Eρ−(u1)E_ρ^-(u_1), the integrated separation is Acyc=∫−11|Eρ+(u1)−Eρ−(u1)|du1.A_ cyc= _-1^1 |E_ρ^+(u_1)-E_ρ^-(u_1) |\,du_1. (67) Coincident branches give Acyc=0A_ cyc=0; larger values indicate stronger cyclic lag. Each run starts from a random density field of root-mean-square amplitude 0.080.08, with the same ten seeds used across models and rates. There is no pre-equilibration at u1=−1u_1=-1, so AcycA_ cyc includes initial relaxation. It is therefore a finite-cycle lag measure, not quasistatic hysteresis, multistability, or a topological invariant. Runs producing a nonfinite density or connection field are marked numerically unstable; means and SEMs use the finite runs. The representative branch rate is fixed as the middle point of the five-value factor-of-two scan, before comparing the outcomes. This gives rs=0.2083r_s=0.2083, the geometric midpoint of the tested rate range. Here Nb=640N_b=640, so one cycle contains 1280 steps. All four models complete this rate for all ten seeds. VII Gauge-dynamics protocols The reservoir protocols above establish observable memory and readout performance. Here we isolate what requires an independently evolved gauge connection. Three diagnostics answer successive questions. First, RGR_G and RSR_S test whether the numerical solution satisfies the two CS field equations. Second, EaE_a and EvE_v ask whether the evolved connection and velocity can be reconstructed from the instantaneous density. Third, ΔRχ R_χ measures the change in a local cross response caused by advective feedback after subtracting the background response. All tests use the numerical scheme of Sec. V; within a test, models share the same seed, initial density, and drive. All deterministic two-channel evolutions below use the common, nonoptimized drive u1(t)=sintu_1(t)= t, u2(t)=cos(2t)u_2(t)= (2t). The integer harmonics provide reproducible excitation without fitted frequency parameters. The scalar-memory test discussed earlier is different by design: independent random symbols ensure that delayed input reconstruction measures information retained by the reservoir rather than the predictability of a periodic drive. CS-equation residuals: RGR_G and RSR_S. We begin with ten random density fields of root-mean-square amplitude 0.120.12. Each is evolved for four time units under the common drive defined above. For these four-time-unit trajectories, the connection a=(a1,a2)a=(a_1,a_2), density ρ, and conserved current =(j1,j2) j=(j^1,j^2), we monitor the following normalized residuals because a discretized update need not preserve Gauss law or the spatial CS equation exactly: RG R_G =‖∂1a2−∂2a1−gρ‖2‖gρ‖2+10−14, = _1a_2- _2a_1-gρ _2 gρ _2+10^-14, (68) RS R_S =‖∂ta−g(j2,−j1)‖2‖g‖2+10−14. = _ta-g(j^2,-j^1) _2 g j _2+10^-14. (69) The residual RGR_G is nontrivial only for coupled CS: the instantaneous and local controls reconstruct ac[ρ]a_ c[ρ] and satisfy Gauss law by construction, while the reciprocal model has no CS connection. The residual RSR_S compares current-driven evolution with the two transverse reconstructions. The constant 10−1410^-14 only prevents division by zero. For coupled CS, ∂ta _ta is estimated by a trapezoidal finite difference. For the instantaneous control, ac=gJ∇(−∇2)−1ρa_ c=gJ∇(-∇^2)^-1ρ, so ∂tac=gJ∇(−∇2)−1∂tρ _ta_ c=gJ∇(-∇^2)^-1 _tρ; we evaluate it with Eq. (50) using ∂tρ _tρ in place of ρ. Thus its derivative is density-implied, not independently evolved. Identical initial ρ, a, and drive isolate this difference. We also repeat this test on 12212^2, 16216^2, and 24224^2 grids and with Δt t halved. These refinement runs use the same initial field and end at t=2t=2. A separate three-realization test adds weak Gaussian density noise, of amplitude 2×10−4Δt2× 10^-4 t, at every step. We remove its mean, apply the dealiasing filter, and construct the associated connection increment with Eq. (50). The perturbation therefore preserves Gauss law. Together, these tests check whether the residuals are robust to spatial and temporal resolution and to small constraint-compatible noise. Snapshot reconstruction: EaE_a and EvE_v. Constraint satisfaction alone does not show whether the evolved connection contains information absent from the instantaneous density. We quantify that difference below, writing CS≡=−γa v_ CS≡ v=-γ a for the full coupled velocity in Eq. (49): Ea E_a =‖a−ac[ρ]‖2‖a‖2+10−14, = a-a_ c[ρ] _2 a _2+10^-14, ac[ρ] a_ c[ρ] =gJ∇(−∇2)−1ρ, =gJ∇(-∇^2)^-1ρ, (70) Ev E_v =‖candidate−CS‖2‖CS‖2+10−14. = v_ candidate- v_ CS _2 v_ CS _2+10^-14. (71) Here EaE_a is nontrivial only for coupled CS: it vanishes for a connection reconstructed from the same density, while the reciprocal model has no CS connection. The velocity error EvE_v instead compares the coupled velocity with the instantaneous transverse field 0 v_0 and its fitted local closure in Eq. (43). We next ask whether the velocity difference can be absorbed into the local closure η=(1+ηρ)0 v_η=(1+ηρ) v_0 of Eq. (43). We vary the advective-current coefficient α in Eq. (10) and evolve ten coupled-CS seeds at α∈0,0.15,0.30,0.50,0.70,0.90α∈\0,0.15,0.30,0.50,0.70,0.90\. Each run uses 267 RK4 steps (tf=4.005≃4t_f=4.005 4) under the same drive, allowing the mismatch to develop before the endpoint fit. From the final coupled state we retain ρ and CS v_ CS, then reconstruct 0 v_0 from the same ρ. For each seed and α, one scalar η is chosen to minimize ‖(1+ηρ)0−CS‖2 (1+ηρ) v_0- v_ CS _2. Only η is fitted. The test therefore asks whether the correction CS−0 v_ CS- v_0 has the local form ηρ0ηρ v_0. We compare the resulting EvE_v with its instantaneous value at η=0η=0. Because η is optimized separately at each final state, this is a favorable snapshot test rather than a predictive fit. Failure to reduce EvE_v would therefore rule out even this fitted local replacement for the independent gauge state. Feedback-dependent cross-response: ΔRχ R_χ. We next test how a local density perturbation changes the velocity at a second site. The link is the gauge connection: Gauss law maps the density perturbation to a compatible change in a, which changes the velocity through =−γa v=-γ a. During the subsequent evolution, that velocity also changes adv=αρ j_ adv=αρ v, which feeds back into a. The measured cross-response is therefore a finite-time susceptibility, not a particle-transfer rate or propagation speed. For each of ten seeds, we drive a random density field of rms amplitude 0.120.12 for four time units, matching the field-residual protocol. We then apply a localized density probe at either 1=(L/4,L/3) c_1=(L/4,L/3) or 2=(3L/4,3L/5) c_2=(3L/4,3L/5); these well-separated locations avoid simple square symmetries. For a probe centered at i c_i, we define Gi() G_i( c) =exp[−dper(,i)22σprobe2], = \! [- d_ per( c, c_i)^22 _ probe^2 ], (72) δρi() δ _i( c) =Aprobe[Gi()−⟨Gi⟩]. =A_ probe [G_i( c)- G_i ]. Here dperd_ per is the shortest distance on the periodic square. The Gaussian GiG_i is a smooth bell-shaped profile that falls to e−1/2e^-1/2 of its peak at distance σprobe _ probe. We set σprobe=L/8 _ probe=L/8, spanning two grid spacings on the default 16216^2 grid. The subtraction of ⟨Gi⟩ G_i preserves the total mass, ⟨δρi⟩=0 δ _i =0. This is also required by periodic Gauss law, because the spatial integral of ∂1a2−∂2a1=gρ _1a_2- _2a_1=gρ vanishes, and by the Poisson inversion in Eq. (50), whose zero Fourier mode must vanish. The amplitude Aprobe=2×10−3A_ probe=2× 10^-3 is a finite-difference step, not a model parameter. Halving or doubling it changes the normalized response by less than 5×10−65× 10^-6 relative, confirming linear response. From the prepared background (ρ(0),a(0))(ρ^(0),a^(0)), probe i defines ρ(i)=ρ(0)+δρiρ^(i)=ρ^(0)+δ _i. Applying Eq. (50) to δρiδ _i supplies the matching connection increment δaiδ a_i, so a(i)=a(0)+δaia^(i)=a^(0)+δ a_i satisfies Gauss law and immediately gives δi=−γδaiδ v_i=-γδ a_i. We evolve the background and both perturbed states for 0.120.12 time units, holding the drive at its terminal value to avoid a discontinuity. We repeat this response test for α∈0,0.05,0.10,0.20,0.35,0.50,0.70α∈\0,0.05,0.10,0.20,0.35,0.50,0.70\. At α=0α=0, adv j_ adv is absent while the background and probes are unchanged, giving the no-feedback reference. The instantaneous control instead reconstructs 0 v_0 from each density snapshot without evolving a. After the response interval, (0) v^(0) denotes the background velocity and (i) v^(i) the velocity following probe i; superscripts label trajectories, not components. The opposite-site responses per unit probe amplitude are 12 χ_12 =(2)(1)−(0)(1)Aprobe, = v^(2)( c_1)- v^(0)( c_1)A_ probe, (73) 21 χ_21 =(1)(2)−(0)(2)Aprobe. = v^(1)( c_2)- v^(0)( c_2)A_ probe. Thus 12 χ_12 is the response at site 1 to a probe at site 2, and 21 χ_21 is the reverse response. An antisymmetric transverse response gives oppositely directed forward and reverse vectors, so their sum tests the failure of this cancellation. With χ=12+21 _χ= χ_12+ χ_21, we report Rχ R_χ =‖χ‖12‖+‖21‖+10−14, = _χ χ_12 + χ_21 +10^-14, ΔRχ(α) R_χ(α) =‖χ(α)−χ(0)‖12‖+‖21‖+10−14. = _χ(α)- _χ(0) χ_12 + χ_21 +10^-14. (74) The norm is Euclidean over velocity components, and each denominator uses the same α as its numerator. The raw RχR_χ includes background, probe-geometry, and discretization effects; ΔRχ R_χ subtracts the seedwise α=0α=0 cross-sum to isolate advective feedback. Figure 1: 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= _dMC(d). (b) Binary pulse-order classification accuracy, i.e., the held-out fraction classified as protocol A versus B correctly, for the five feature families. Cells give the mean ± SEM over ten outer folds, with ridge penalties selected by nested seed-block validation; chance accuracy is 0.50.5. Panel (b) is not computed from the scalar-memory loss in Eq. (56). VIII Results: reservoir feasibility and the value of dynamical CS coupling The simulations address the two questions posed in the Introduction. First, is a topological context reservoir numerically viable and capable of exposing recent input history to a linear readout? Second, does evolving the CS connection add anything that cannot be reproduced by reciprocal transport, instantaneous transverse reconstruction, or local nonlinear feedback? We answer the first question using scalar recall, pulse-order decoding, cyclic lag, and numerical robustness. We answer the second by resolving the connection equation and the density, circulating, and longitudinal memory channels separately. This distinction is important: generic reservoir feasibility does not by itself establish a benefit from CS dynamics. VIII.1 Fading memory and pulse-order accessibility We first ask whether the present reservoir state retains a linearly accessible trace of earlier scalar inputs. In Fig. 1(a), we find that MC(d)MC(d) is positive at short delays for all four models and decreases overall across the 16-symbol window. This is the expected signature of fading rather than permanent memory. The capacities C defined in Eq. (57) are 0.19±0.050.19± 0.05, 0.28±0.060.28± 0.06, 0.25±0.060.25± 0.06, and 0.27±0.060.27± 0.06 for reciprocal, instantaneous transverse, local-feedback, and coupled-CS dynamics, respectively (mean ± SEM over ten seeds). We find no resolved scalar-memory advantage for coupled CS except that C for the reciprocal dynamics is slightly weaker than the other three models. The result nevertheless establishes that all four evolving fields retain a linearly accessible record of recent input amplitudes. Table 1: Density and circulation separation for the matched pulse-order protocols. Values are ten-seed means ± SEM, averaged over three pulse widths. Contrasts are formed seedwise relative to coupled CS before averaging; its reference value is therefore exact. Observable model Ap=0.50A_ p=0.50 Ap=0.75A_ p=0.75 Ap=1.00A_ p=1.00 Ap=1.25A_ p=1.25 density contrast Δρ _ρ (%) reciprocal 0.319±0.2100.319± 0.210 0.369±0.2060.369± 0.206 0.437±0.2010.437± 0.201 0.523±0.1970.523± 0.197 inst. transverse −0.010±0.001-0.010± 0.001 −0.014±0.002-0.014± 0.002 −0.018±0.002-0.018± 0.002 −0.024±0.002-0.024± 0.002 local feedback −0.011±0.002-0.011± 0.002 −0.014±0.002-0.014± 0.002 −0.018±0.002-0.018± 0.002 −0.024±0.003-0.024± 0.003 coupled CS 00 (reference) 00 (reference) 00 (reference) 00 (reference) circulation contrast Δω _ω (%) reciprocal −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) inst. transverse −0.009±0.002-0.009± 0.002 −0.011±0.002-0.011± 0.002 −0.014±0.002-0.014± 0.002 −0.017±0.002-0.017± 0.002 local feedback 0.132±0.1140.132± 0.114 0.207±0.1150.207± 0.115 0.281±0.1150.281± 0.115 0.355±0.1160.355± 0.116 coupled CS 00 (reference) 00 (reference) 00 (reference) 00 (reference) Scalar recall does not test whether the reservoir retains the order in which different inputs arrived. The matched protocols in Eq. (58) isolate this question: the two histories have the same duration, final input, and integrated injection in Eq. (59), and differ only in pulse order. In Table 1, all four models produce distinct endpoint densities, but their contrasts relative to coupled CS are small. In Fig. 1(b), density-only decoding ranges from 0.6960.696 to 0.8040.804, with overlapping uncertainties. We therefore find that density alone records pulse order generically and does not identify the CS mechanism. In Fig. 1(b), a clearer distinction appears when the readout resolves the flow geometry. Circulation alone gives chance-level accuracy for reciprocal dynamics (0.4580.458), as expected from its gradient velocity, but reaches 0.6210.621 for both instantaneous-transverse and coupled-CS dynamics, which contain the rotated-gradient channel of Eq. (36). Longitudinal features provide complementary information and yield 0.690.69 for coupled CS. When all feature families are combined, we obtain 0.73±0.060.73± 0.06, 0.68±0.070.68± 0.07, 0.67±0.050.67± 0.05, and 0.88±0.030.88± 0.03 across the four-model ladder. In the paired seedwise comparison, coupled CS exceeds the instantaneous-transverse control by 0.20±0.070.20± 0.07. We therefore attribute the improvement to the joint accessibility of circulating and longitudinal order traces. This is a task-specific benefit for decoding input order, not a general enhancement of reservoir memory. VIII.2 Rate-dependent density lag Figure 2: Cyclic-drive density lag. (a) Mean forward (solid) and backward (dashed) branches at rs=0.2083r_s=0.2083; bands show SEM. Only the reciprocal and coupled-CS branches are shown because the instantaneous transverse controls overlap visually. (b) Loop area AcycA_ cyc versus sweep rate. (c) Mean loop area with SEM error bars at the representative rate. Missing reciprocal points denote unstable runs. We use the loop area AcycA_ cyc in Eq. (67) to quantify the failure of the density to retrace its outward path when the input is reversed. In Fig. 2(a), the separated forward and return branches show this lag directly. At the fastest sweep, we find nearly the same area for all four models. In Fig. 2(b), the coupled-CS loop becomes smaller than either instantaneous-transverse loop as the sweep is slowed. In Fig. 2(c), at the representative rate, the mean area is 0.2920.292 for coupled CS, compared with 0.4180.418 for both instantaneous controls and 2.0082.008 for the reciprocal model. The reciprocal evolution is unstable at the two slowest rates. We therefore find that the evolved connection improves tracking of a slowly varying input in this protocol. This loop remains a finite-time, dissipative response: because the initial ensemble is not pre-equilibrated, it should not be interpreted as quasistatic hysteresis or as a topological invariant. VIII.3 What the evolved connection adds The field equations distinguish the coupled model more sharply than the readout scores do. Gauss law fixes the density-sourced curl of the connection, whereas the spatial CS equation determines its current-driven evolution. An instantaneous reconstruction may therefore satisfy Gauss law without obeying the spatial equation. We test these two requirements separately using Eqs. (68) and (69). In Fig. 3(a), we find that the coupled update propagates Gauss law to numerical precision: the largest residual over ten seeds is 4.42×10−144.42× 10^-14. In Fig. 3(b), the ensemble-and-time median spatial residual is 1.39×10−21.39× 10^-2 for coupled CS, compared with 5.47×10−25.47× 10^-2 for the instantaneous- transverse reconstruction and 5.32×10−25.32× 10^-2 for local feedback. Thus reconstructing a connection that satisfies Gauss law at each instant is not sufficient to reproduce the current-driven CS dynamics. In Table 3, halving Δt t reduces the coupled residual from 1.07×10−21.07× 10^-2 to 5.30×10−35.30× 10^-3, indicating that the remaining error is dominated by temporal discretization of the diagnostic. Figure 3: CS-equation residuals at 16216^2, Δt=0.015 t=0.015, and tf≃4t_f 4. (a) Mean RG(t)R_G(t) with SEM band. (b) Mean ± SEM of the seedwise time-median RSR_S. Figure 4: Reconstruction and closure at tf≃4t_f 4. (a) Mean Ea(t)E_a(t) with SEM band at α=0.70α=0.70. (b) Mean EvE_v for instantaneous transverse and fitted-local reconstruction, with SEM. (c) Their paired percentage difference (mean ± SEM); α=0α=0 is omitted because both errors are at numerical zero. In Fig. 4(a), the mean connection mismatch grows to Ea=0.155E_a=0.155 at t=4t=4, even though the evolved and instantaneously reconstructed connections have the same curl to numerical precision. This is the geometric freedom left by Gauss law: it fixes the density-sourced curl, but not the nonuniform exact component of a. In temporal gauge, that component accumulates during current-driven evolution and therefore records history that is absent from a density snapshot. We next ask whether a fitted local multiplier can reproduce the resulting velocity. In Fig. 4(b), at α=0.90α=0.90, the instantaneous-transverse error is 0.19597±0.003010.19597± 0.00301, while the best local fit gives 0.19590±0.003010.19590± 0.00301 (mean ± SEM over ten seeds). In Fig. 4(c), the corresponding reduction is only 0.0325%±0.0058%0.0325\%± 0.0058\%, despite a fitted coefficient of (−2.50±0.25)×10−2(-2.50± 0.25)× 10^-2. We therefore find no meaningful improvement from the closure in Eq. (43). Local feedback remains a useful phenomenological control, but it is not a reduced representation of the history-dependent CS connection. VIII.4 Physical channels of post-input memory The endpoint observables in Eq. (60) show why an aggregate readout score does not identify the mechanism carrying pulse-order memory. In Table 1, the small density contrasts show that all four models distinguish the two orders by nearly the same amount. The underlying ensemble means are Dρ=2.097×10−3D_ρ=2.097× 10^-3 for reciprocal dynamics and approximately 2.088×10−32.088× 10^-3 for the three transverse models; their total-velocity distances are also nearly equal. Density and velocity separation are therefore generic signatures of nonlinear path dependence. The circulation and longitudinal observables separate the underlying kinematics. In Table 1, the reciprocal circulation is reported at numerical zero; its underlying value is 3.3×10−193.3× 10^-19, whereas the other three models give approximately 2.33×10−42.33× 10^-4. This vanishing value follows because the reciprocal velocity is a gradient. Conversely, the instantaneous-transverse longitudinal trace is numerically zero because the rotated-gradient velocity in Eq. (36) is divergence free. We find a longitudinal trace of 2.34×10−42.34× 10^-4 for reciprocal dynamics and smaller values of 2.02×10−62.02× 10^-6 and 2.90×10−62.90× 10^-6 for local feedback and coupled CS, respectively. The latter two mechanisms therefore retain both geometric channels, but only coupled CS does so while satisfying the spatial field equation in Eq. (69). Table 2: Post-input persistence of matched pulse-order traces at selected times. Values are ten-seed means ± SEM, averaged over three pulse widths and four amplitudes. Contrasts and ratios are formed seedwise relative to coupled CS before averaging; the coupled-CS reference is therefore exact. Observable model t=0t=0 t=0.24t=0.24 t=0.48t=0.48 t=0.72t=0.72 density contrast Δρ _ρ (%) reciprocal 0.552±0.2080.552± 0.208 0.404±0.2000.404± 0.200 0.299±0.1940.299± 0.194 0.222±0.1890.222± 0.189 inst. transverse −0.010±9.89×10−4-0.010± 9.89× 10^-4 −0.021±0.002-0.021± 0.002 −0.030±0.003-0.030± 0.003 −0.039±0.004-0.039± 0.004 local feedback −0.010±0.002-0.010± 0.002 −0.021±0.003-0.021± 0.003 −0.030±0.003-0.030± 0.003 −0.039±0.004-0.039± 0.004 coupled CS 00 (reference) 00 (reference) 00 (reference) 00 (reference) circulation contrast Δω _ω (%) reciprocal −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) −100.000-100.000 (num. zero) inst. transverse −0.008±0.001-0.008± 0.001 −0.016±0.002-0.016± 0.002 −0.023±0.003-0.023± 0.003 −0.029±0.004-0.029± 0.004 local feedback 0.285±0.1190.285± 0.119 0.266±0.1140.266± 0.114 0.249±0.1100.249± 0.110 0.233±0.1060.233± 0.106 coupled CS 00 (reference) 00 (reference) 00 (reference) 00 (reference) longitudinal ratio Q∥Q_ reciprocal 159.646±61.489159.646± 61.489 166.740±66.430166.740± 66.430 171.572±69.205171.572± 69.205 176.613±72.125176.613± 72.125 inst. transverse numerical zero numerical zero numerical zero numerical zero local feedback 0.755±0.3290.755± 0.329 0.755±0.3390.755± 0.339 0.742±0.3370.742± 0.337 0.730±0.3350.730± 0.335 coupled CS 1.0001.000 (reference) 1.0001.000 (reference) 1.0001.000 (reference) 1.0001.000 (reference) In Table 2, we follow these traces after the input is removed. The density separation decays smoothly for every model. Reciprocal dynamics retain a longitudinal trace but no circulation, the instantaneous-transverse model retains circulation but no longitudinal trace, and local-feedback and coupled-CS dynamics retain both. We therefore interpret the persistence as a short-lived record of pulse order in the flow components permitted by each closure. Because all traces decay while the relaxation term −τρ-τρ remains active, this is finite-time reservoir memory rather than asymptotic storage. VIII.5 Feedback-induced incremental response Figure 5: Incremental feedback response. (a) Background-subtracted excess ΔRχ R_χ; the dashed line gives the uncorrected α=0α=0 scale. (b) Raw asymmetry RχR_χ for coupled CS and the instantaneous baseline. Curves and bands show the ten-seed mean and SEM. In Fig. 5(b), the raw cross-response is already nonzero on the prepared inhomogeneous background. At α=0α=0, coupled CS gives Rχ=4.39×10−3R_χ=4.39× 10^-3, equal within rounding to the instantaneous control. We therefore cannot attribute this background value to advective feedback. After subtracting it seed by seed, we find in Fig. 5(a) that ΔRχ R_χ increases from 3.62×10−43.62× 10^-4 at α=0.05α=0.05 to 4.72×10−34.72× 10^-3 at α=0.70α=0.70. A fit to the four smallest nonzero strengths gives a slope of 0.983±0.0020.983± 0.002, consistent with the first-order contribution of αραρ v in Eq. (10). At weak coupling, the incremental response is small compared with the background; at the largest α, the two become comparable. We thus resolve a systematic feedback-induced correction, but the raw coupled and instantaneous responses remain of the same order. This diagnostic isolates advective feedback without demonstrating a general nonreciprocity advantage. VIII.6 Numerical robustness Table 3: Numerical validation of coupled CS. Unless indicated otherwise, values use the 16216^2 grid and Δt=0.015 t=0.015. The two spatial-residual convergence runs have duration 2. Diagnostic Result Maximum RGR_G, main run 4.42×10−144.42× 10^-14 Coupled spatial residual, Δt t 1.07×10−21.07× 10^-2 Coupled spatial residual, Δt/2 t/2 5.30×10−35.30× 10^-3 ‖ρ‖2 ρ _2, N=12/16/24N=12/16/24 0.261882/0.261881/0.2618810.261882/0.261881/0.261881 ‖ρ‖2 ρ _2, Δt/2 t/2 0.2638870.263887 ‖ρ‖2 ρ _2, three noise seeds 0.2618440.261844–0.2618880.261888 Finally, we test whether the coupled results survive reasonable changes in the numerical scheme. In Table 3, grid refinement changes the final density norm by only 5.5×10−65.5× 10^-6 between 12212^2 and 16216^2, and by 1.8×10−61.8× 10^-6 between 16216^2 and 24224^2, relative to the 16216^2 result. Halving the time step changes the norm by 0.77%0.77\%. With constraint-compatible noise, the largest relative change is 1.5×10−41.5× 10^-4, and every robustness run retains RG<2.0×10−14R_G<2.0× 10^-14. Together with the residual convergence reported above, these checks show that the propagated constraint and density response are not artifacts of the chosen grid, time step, or an exactly noiseless trajectory. Taken together, we find a numerically viable reservoir, but not one that is generically superior to all controls. Its specific contribution is an equation-consistent dynamical connection that stores geometric history in both circulating and longitudinal channels. IX Discussion IX.1 What the simulations establish Within the tested regime, the coupled CS equations provide a numerically viable reservoir substrate. The conserved-current update propagates Gauss law, converges under grid and time-step refinement, and remains stable under constraint-compatible noise. All four models exhibit fading memory, distinguish matched pulse-order histories, and retain finite-time post-input traces. These conclusions apply to the discretization and parameter range studied here. The controls delimit the computational claim. Scalar memory and density path separation show no resolved advantage for coupled CS, whereas the combined pulse-order readout favors it when both geometric channels are used. The cyclic-loop areas are finite-time, rate-dependent responses rather than topological invariants. The evidence therefore supports a task-specific, not generic, computational advantage. The distinctive CS result is instead equation-level and geometric. Reciprocal dynamics carry a longitudinal order trace without circulation, whereas instantaneous-transverse dynamics carry circulation without a longitudinal trace. Local feedback and coupled CS can support both channels, but only the coupled model evolves a connection that satisfies the spatial CS equation. Its history-dependent exact component is absent from an instantaneous density reconstruction and is not recovered by the fitted local multiplier. The incremental-response test further resolves a small correction linear in α, but not an independent nonreciprocity signature. IX.2 Potential computational role This interpretation fits the standard view of a reservoir as a nonlinear filter with fading memory and a trained readout [5, 21, 20, 13]. Reservoir methods have been used for chaotic attractors, signal separation, Hamiltonian learning, convection, and continuous-time embeddings [27, 19, 41, 17, 16]. Physical implementations include optical delays, soft bodies, spintronic oscillators, quantum ensembles, and adaptive oscillators [1, 24, 37, 11, 32]. The present addition is a gauge-constrained substrate in which the state decomposition, not only the readout score, is physically interpretable. Its most plausible use is therefore in tasks that address circulation, compression, orientation, or input order directly. The construction is also related to active and driven matter, where collective fields convert nonequilibrium drive into memory and transport [38, 36, 28, 23, 3]. Nonreciprocal and odd media show how transverse response can organize dynamics beyond potential relaxation [10, 40, 29, 18, 34, 2, 31, 15]. Here that role is isolated with reciprocal and instantaneous controls rather than inferred from a single model. IX.3 Limitations and next steps The present evidence for numerical feasibility does not establish experimental realizability. The context manifold is latent, the source completion models external drive, and the overdamped connection–drift relation requires calibration to a material substrate. An implementation must provide measurable proxies for density, current, circulation, and divergence. Relevant next steps are particle-based realizations with explicit context coordinates, response measurements about stationary backgrounds, and readout tasks matched to the symmetry of the geometric channels. X Conclusion This work asked two related questions: whether a density field coupled to a CS connection can operate as a reservoir, and whether evolving that connection adds anything beyond simpler nonlinear, reciprocal, or instantaneously chiral dynamics. The first question is answered positively within the regime studied. The conserved-current scheme propagates Gauss law to numerical precision, the spatial CS residual decreases under temporal refinement, and the principal observables remain stable under grid refinement and constraint-compatible noise. The coupled system also exhibits fading scalar memory, distinguishes matched input histories, and retains pulse-order information for a finite time after the input is removed. These results establish numerical feasibility, although not yet experimental realizability. The controls are essential to interpreting that feasibility. Scalar-memory capacity and density path separation are comparable across the four models, showing that neither fading memory nor order-dependent density alone is a specific consequence of CS dynamics. The cyclic density loops likewise measure finite-rate dissipative lag rather than a topological invariant. Consequently, the present results do not support the broad claim that a topological context reservoir is generically superior to conventional reservoir mechanisms. The more specific contribution of the coupled model is geometric and equation-level. An instantaneous density reconstruction fixes the density-sourced curl of the connection but misses the history-dependent exact component accumulated during current-driven evolution. A fitted local multiplier does not recover this missing state. Coupled CS can therefore retain circulating and longitudinal pulse-order traces simultaneously while satisfying the spatial CS equation; the reciprocal and instantaneous transverse controls each lack one of these properties. Consistent with this structure, the combined geometric readout reaches 0.879±0.0350.879± 0.035 pulse-order accuracy, compared with 0.679±0.0650.679± 0.065 for the instantaneous-transverse control. The two-site response test further isolates a feedback-induced correction that is nearly linear in α, as expected from the advective current αραρ v. These are targeted advantages of an evolved gauge state, not evidence of uniform improvement on every task. Topological context reservoirs are therefore worth pursuing when the task depends naturally on orientation, circulation, compression, input order, or other geometric aspects of a signal. The present model provides a controlled field-theoretic demonstration that such information can be stored in a dynamical connection and exposed to a simple linear readout. A stronger claim will require larger ensembles, tasks selected independently of the reported diagnostics, stationary-background response measurements, and a physical platform with measurable density and current fields. Until those tests are made, the appropriate conclusion is that coupled CS dynamics offer a promising specialized reservoir architecture rather than a universally better one. Declaration of generative AI and AI-assisted technologies Generative AI tools (ChatGPT and Grammarly) are used to assist with language editing, formatting, and clarity during manuscript preparation. The scientific content, conceptual framework, analyses, and conclusions are developed by the author. The author takes full responsibility for the integrity, originality, and accuracy of the content of this manuscript. References [1] L. 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