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\lefttitle{Y. Wang, F. Ren, S. Chen and H. Tang}
\righttitle{Symbolic feedback laws for hydrodynamic cloaking and illusion}

\title{Interpretable feedback laws for active hydrodynamic cloaking and illusion of the fluidic pinball}

\author{Yanqi Wang\aff{1,3}, Feng Ren\aff{2}, Shiyi Chen\aff{3} \and Hui Tang\aff{1}}

\affiliation{\aff{1}Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, PR China
\aff{2}School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an 710072, PR China
\aff{3}Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo 315200, PR China}

\corresau{Hui Tang, \email{h.tang@polyu.edu.hk}}

\begin{document}
\maketitle

\begin{abstract}
Hydrodynamic cloaking and illusion require a controller to manipulate not only an integrated force but also the information transported by a nonlinear wake. We study these observer-centred control objectives using the fluidic pinball, a compact array of three independently rotating cylinders. Proximal-policy-optimization controllers are first trained from the forces on the cylinders and three downstream velocity probes. We then use symbolic regression to determine whether the successful neural policies contain a compact and transferable feedback structure. Candidate expressions are constrained by the reflection symmetry of the pinball and are accepted only after deployment in closed-loop lattice-Boltzmann simulations; one-step regression accuracy alone is shown to be an unreliable selection criterion. For K\'arm\'an-street cloaking, a joint expression extracted across diameter-based Reynolds numbers $\Rey_D=25$--$200$ separates into an approximately steady counter-rotation of the rear cylinders and dynamic front-cylinder rate--lift feedback. The symbolic controller obtains downstream similarities of $0.847$, $0.888$, $0.845$ and $0.806$, respectively, and transfers without refitting to a Lamb vortex dipole and a Taylor vortex monopole with similarities of $0.949$ and $0.905$. Illusion of target-cylinder wakes admits a common target-error law near target diameters $0.75D$--$1.0D$, but transfer deteriorates as the target scale departs from the naturally controllable wake. At $1.5D$, the neural action contains a high-frequency modulation that is not represented by the present instantaneous feature library, exposing a clear symbolic-model regime boundary. A compact correction-field analysis provides independent spatial support for the actuator interpretation. The results show that deep reinforcement learning can serve as a policy-discovery stage from which a physically interpretable cloak backbone is distilled, while also identifying the conditions under which target-dependent temporal structure must be retained.
\end{abstract}

\begin{keywords}
hydrodynamic cloaking, flow control, symbolic regression, deep reinforcement learning, fluidic pinball
\end{keywords}

\section{Introduction}
\label{sec:introduction}

A bluff body leaves a spatially and temporally extended signature in the surrounding fluid. The signature includes a mean velocity deficit, separated shear layers, coherent vortices, pressure fluctuations and characteristic frequencies. These quantities are not merely consequences of the force on the body: they are also information from which a downstream observer may infer the presence, size or state of an upstream object. Active flow control may therefore be formulated not only as drag reduction, lift control or suppression of vortex-induced vibration, but also as control of the information transmitted by the wake.

This viewpoint gives two related objectives. \emph{Hydrodynamic cloaking} requires the downstream flow to approach the background that would have reached the observer in the absence of the controlled body. For a steady background this reference is an undisturbed channel flow; for an incoming vortex street or isolated vortex it is the corresponding disturbance propagated without the body. \emph{Hydrodynamic illusion} instead requires the downstream flow to reproduce the non-zero signature of a different target object. The body remains present and exchanges momentum with the fluid in both problems. Cloaking is consequently not defined as disappearance of the local near-body disturbance, and illusion is not defined by force matching alone. The relevant criterion is the field or signal perceived outside the immediate actuation region.

Nonlinear separated wakes make these objectives difficult. Instability, mean-flow deformation, vortex formation and convection are coupled, and the observer responds only after a finite propagation delay. A controller may reduce drag while leaving a readily detectable wake, or suppress a dominant frequency while producing an incorrect mean deficit. Illusion is more restrictive still because frequency, phase, amplitude and spatial organisation must be retuned towards a non-zero target.

Deep reinforcement learning (DRL) is attractive for this setting because it can discover feedback strategies directly through interaction with the flow, without requiring a tractable model of the controlled Navier--Stokes dynamics. Its principal limitation is interpretability. A successful neural policy does not by itself reveal which measurements are essential, whether the actions contain a reusable physical mechanism, or why a policy transfers between incident disturbances. This limitation is especially important for hydrodynamic perception control: a high scalar reward establishes performance only in the chosen observer space and does not identify the mechanism that produces the downstream correction.

The present study uses the fluidic pinball as a controlled laboratory for this question. Three equal circular cylinders form an equilateral triangle with one cylinder facing the incident flow. Independent rotation provides a compact three-input actuator capable of changing stagnation points, gap flow, base bleed, separation and shedding phase. The uncontrolled pinball also undergoes Hopf, pitchfork and higher-frequency transitions as the Reynolds number increases. It is therefore simple enough for repeated high-fidelity simulation but rich enough to expose nonlinear, multi-attractor control dynamics.

The central hypothesis is that successful cloaking is organised by a reusable actuator-side compensation mechanism. The pinball creates a blockage-induced velocity deficit and an additional force signature. The learned policy compensates the mean deficit through rear-cylinder counter-rotation and regulates the residual phase and antisymmetric force through dynamic front-cylinder actuation. We test this hypothesis by distilling the observation-to-action map with symbolic regression (SR). Crucially, symbolic expressions are judged in closed-loop computational fluid dynamics (CFD), not only by their ability to fit actions recorded from the DRL trajectory. A compact correction-field decomposition is then used as an independent check of the spatial interpretation.

The principal contributions are as follows. First, cloaking and illusion are formulated as observer-centred wake-matching problems for a nonlinear bluff-body flow. Secondly, a symmetry-constrained SR procedure is developed for post hoc distillation of a trained PPO policy, with closed-loop CFD as the decisive model-selection test. Thirdly, a common K\'arm\'an-cloak law is identified across $\Rey_D=25$--$200$ and transferred without refitting to periodic and transient incident disturbances. Fourthly, illusion is shown to reuse this compensation backbone only over a finite target-scale range; high-frequency neural actuation at larger mismatch reveals the information missing from the present symbolic state.

The paper is organised as follows. Section~\ref{sec:problem} defines the physical configuration and the Reynolds-number convention. Section~\ref{sec:methods} describes the numerical environment, DRL policy and SR protocol. Section~\ref{sec:srresults} presents the symbolic controllers and their closed-loop transfer. Sections~\ref{sec:oid} and~\ref{sec:ccd} provide independent spatial and spatio-temporal tests of the mechanism. Conclusions are given in \S~\ref{sec:conclusion}.

\subsection{Relation to interpretable flow control and system identification}

The aim of extracting an analytical policy has precedents in machine-learning control, sparse system identification and reduced-order modelling, but the present task differs from each of these in a consequential way. Genetic-programming control searches directly over mathematical expressions and has exposed physically meaningful feedback in separated flows \citep{Gautier2015,Li2017}. Such an approach makes interpretability native to the optimisation, but repeated evaluation of candidate laws in a large CFD environment is costly. Here PPO first explores a flexible policy class, after which symbolic regression asks whether the successful behaviour occupies a much smaller algebraic class. This ordering treats the neural controller as a discovery instrument rather than as the final scientific result.

Sparse identification of nonlinear dynamics (SINDy) provides a second point of reference \citep{Brunton2016,Loiseau2018Galerkin}. SINDy and constrained sparse Galerkin regression identify parsimonious evolution equations from a prescribed library. They have established two lessons that are directly relevant here: structural constraints can matter more than small changes in residual error, and a model must be assessed by its long-time dynamics rather than its derivative fit alone. In the present work, however, the principal pipeline is PySR applied to the policy map, not SINDy applied to the Navier--Stokes state. SINDy is used only as a historical sparse-regression reference and a cross-diagnostic in the difficult $1.5D$ illusion case. There is no DANTE pipeline in this study, and no claim is made that the discovered expressions are governing equations.

The distinction between policy imitation and closed-loop equivalence is central. Supervised policy distillation usually minimises an action discrepancy over states visited by the teacher. In a convectively delayed flow, replacing the teacher changes those states. A formula can accurately interpolate the PPO trajectory yet introduce a small phase bias that compounds until the downstream signal no longer matches the reference. Conversely, a low-$R^2$ formula can preserve the stabilising or tracking geometry that matters in closed loop while ignoring high-variance action details. We therefore place closed-loop CFD inside the selection loop conceptually, even though the computational pipeline performs fitting and validation as separate reproducible stages.

The fluidic pinball is especially suitable for this test. Its unforced dynamics pass through successive Hopf and symmetry-breaking bifurcations, and its force observables obey symmetry-constrained low-order structure \citep{Deng2020,Loiseau2018Galerkin}. Rotation of the downstream pair can create base bleed, while front-cylinder rotation can supply phase-sensitive forcing; related roles emerged in gradient-enriched machine-learning control \citep{CornejoMaceda2021}. These prior results provide physical hypotheses, not labels supplied to PySR. A credible symbolic law should recover actuator roles compatible with this dynamical organisation and should remain useful beyond one recorded waveform.

Finally, the term \emph{cloaking} is used here in an observer-centred and explicitly limited sense. Active hydrodynamic metamaterial studies show that cancellation of a body-induced disturbance can be designed over restricted regimes, while also exposing stability limitations at finite Reynolds number \citep{Urzhumov2012}. Our sparse-probe similarity measures whether selected downstream signals are recovered; it is not evidence of pointwise invisibility throughout the domain. Full-field correction analyses reduce, but do not eliminate, this observer dependence. Likewise, illusion denotes reproduction of selected force and wake signatures of a target cylinder, not a claim that every external measurement would classify the pinball as that cylinder.

\section{Flow configuration and control objectives}
\label{sec:problem}

\subsection{Fluidic pinball}

The controlled body consists of three circular cylinders of diameter $D$. Their centres form an equilateral triangle of side $1.5D$, leaving a gap of $0.5D$ between adjacent surfaces. The upstream cylinder is denoted by $F$ and the downstream upper and lower cylinders by $T$ and $B$. The three angular velocities form the actuation vector
\begin{equation}
 \vect{\omega}(t)=(\omega_F,\omega_T,\omega_B)^{\mathsf T},
\end{equation}
The Legacy solver stores the applied rotational command in velocity units as $\omega_i$; the SR pipeline therefore defines $\alpha_i=\omega_i/U_0$. This project-specific variable should not be confused with a dimensional angular frequency. The corresponding no-slip wall velocity is prescribed by the solver convention below.
On cylinder $i$, the no-slip surface velocity is
\begin{equation}
 \vect{u}_{w,i}=\omega_i R
 \begin{pmatrix}-\sin\theta_i\\ \cos\theta_i\end{pmatrix},
 \qquad R=D/2.
\end{equation}
Three downstream probes measure both velocity components. The force on each pinball cylinder is also available to the controller.

\draftfigure{Insert the fluidic-pinball geometry, disturbance generator and three downstream probes.}{Computational arrangement and observer-centred control objectives. The three pinball cylinders rotate independently; the reference trajectory is recorded without the pinball for cloaking and with a separate target cylinder for illusion.}{fig:configuration}

The incompressible flow satisfies
\begin{align}
 \nabla\boldsymbol{\cdot}\vect{u}&=0,\\
 \rho_f\left(\frac{\partial\vect{u}}{\partial t}
 +\vect{u}\boldsymbol{\cdot}\nabla\vect{u}\right)
 &=-\nabla p+\mu\nabla^2\vect{u}+\vect{f}_e,
\end{align}
where $\vect{f}_e$ represents the force density used to impose the moving curved boundaries.

\subsection{Reynolds-number convention}
\label{sec:reconv}

Throughout this paper, the physical Reynolds number follows the fluidic-pinball convention and is based on the diameter of one pinball cylinder,
\begin{equation}
 \Rey_D=\frac{U_0D}{\nu}.
 \label{eq:red}
\end{equation}
The project-level solver and several historical file names use a different reference length, $2D$, and therefore store
\begin{equation}
 \Rey_{\mathrm{code}}=\frac{U_0(2D)}{\nu}=2\Rey_D.
 \label{eq:recode}
\end{equation}
These quantities are not interchangeable. All Reynolds numbers reported in the text, tables and conclusions below are $\Rey_D$ unless the code-level quantity is explicitly identified.

The mechanism dataset used for the present SR study was generated with the Legacy configuration on a $1280\times512$ lattice, with $D=20$ lattice units, $U_0=0.01$, a parabolic inlet and no-slip upper and lower walls. A newer V5 training environment exists on a $2000\times600$ lattice with a uniform inlet and free-slip walls, but it is not mixed with the numerical coefficients reported here. The distinction is necessary because the two environments have different plant dynamics, action decoders and observation normalisation. The present expressions and performance values refer to the Legacy analysis dataset.

\subsection{Cloaking and illusion}

Let $\vect{q}_{\mathrm{in}}$ denote the incident reference flow, $\vect{q}_{\mathrm{blk}}$ the flow with a fixed pinball and $\vect{q}_{\mathrm{ctl}}$ the controlled flow. Cloaking seeks
\begin{equation}
 \mathcal{H}\vect{q}_{\mathrm{ctl}}(t)
 \simeq \mathcal{H}\vect{q}_{\mathrm{in}}(t),
 \label{eq:cloakobj}
\end{equation}
where $\mathcal{H}$ is the downstream observation operator. For K\'arm\'an cloaking, $\vect{q}_{\mathrm{in}}$ contains the street generated by an upstream cylinder but no pinball. For the Lamb and Taylor cases it contains an isolated dipole or monopole-like event.

For illusion, a separate target cylinder generates $\vect{q}_{\mathrm{tar}}$, and the objective becomes
\begin{equation}
 \mathcal{H}\vect{q}_{\mathrm{ctl}}(t)
 \simeq \mathcal{H}\vect{q}_{\mathrm{tar}}(t).
 \label{eq:illusionobj}
\end{equation}
The target is non-zero and changes with target diameter. Sensor agreement, force response and phase-aligned fields are evaluated separately so that no single reward component is equated with complete invisibility.

\section{Numerical and data-driven methods}
\label{sec:methods}

\subsection{Lattice-Boltzmann environment}

The flow is advanced with a GPU-accelerated multiple-relaxation-time lattice-Boltzmann method on a D2Q9 lattice. The populations obey
\begin{equation}
 f_i(\vect{x}+\vect{c}_i\Delta t,t+\Delta t)-f_i(\vect{x},t)
 =-\left[\vect{M}^{-1}\vect{S}
 (\vect{m}-\vect{m}^{\mathrm{eq}})\right]_i.
\end{equation}
Curved ghost-node interpolation imposes the rotating no-slip cylinder boundaries. The low lattice velocity $U_0=0.01$ limits compressibility effects. Cylinder rotation, force integration and sensor sampling are coupled directly to the Python control environment, allowing each policy decision to be followed by a prescribed number of CFD steps without restarting the solver.

\subsection{Observation, action and reward}

For cloaking, the observation contains six force components and six downstream velocity components,
\begin{equation}
 \vect{o}_t=
 (F_{F,x},F_{F,y},F_{T,x},F_{T,y},F_{B,x},F_{B,y},
 u_1,v_1,u_2,v_2,u_3,v_3)^{\mathsf T},
 \label{eq:obs}
\end{equation}
with scene-specific physical scaling. Illusion augments this state with the target drag and lift reconstructed at the current target phase. The PPO output $\vect{a}_t\in[-1,1]^3$ is converted to physical rotation using the action scale and bias of the corresponding Legacy scene. SR is fitted only after undoing this decoder.

The reward combines force and downstream-signal objectives. For cloak, Gaussian force scores encourage small additional mean drag and lift,
\begin{equation}
 r_D=\exp(-K_D C_D^2),\qquad
 r_L=\exp(-K_L C_L^2),
\end{equation}
while a signal score compares the six probe channels with the reference. For illusion, $C_D$ and $C_L$ are replaced by the differences between the pinball assembly and the target-cylinder forces.

Temporal similarity is based on dynamic time warping (DTW). For vector sequences $X=\{\vect{x}_i\}$ and $Y=\{\vect{y}_j\}$, the accumulated cost is
\begin{equation}
 D(i,j)=\lVert\vect{x}_i-\vect{y}_j\rVert_2+
 \min\{D(i-1,j-1),D(i-1,j),D(i,j-1)\}.
\end{equation}
The distance is normalised by sequence length and a scene-specific fluctuation scale, and then mapped to a bounded similarity. DTW accommodates modest phase and convection-time offsets, but the full-field comparisons are additionally phase aligned.

\subsection{PPO policy discovery}

The actor and critic contain two hidden layers of width 64 with sinusoidal activation,
\begin{equation}
 \vect{h}_1=\sin(\vect{W}_1\vect{o}+\vect{b}_1),\qquad
 \vect{h}_2=\sin(\vect{W}_2\vect{h}_1+\vect{b}_2).
\end{equation}
The policy is trained with the clipped PPO objective
\begin{equation}
 \mathcal{L}^{\mathrm{clip}}=
 \mathbb{E}_t\left[
 \min\left(\rho_t\widehat A_t,
 \operatorname{clip}(\rho_t,1-\epsilon,1+\epsilon)\widehat A_t\right)
 \right].
\end{equation}
The neural policy is used as a discovery mechanism. Its deterministic trajectories supply the force, sensor and physical-action records used for SR; the policy weights are not inspected to infer the mechanism.

\subsection{Symbolic-regression target and feature libraries}
\label{sec:srmethod}

The SR target is the dimensionless applied actuation
\begin{equation}
 \vect{\alpha}_t=\frac{\vect{\omega}_t}{U_0}
 = (\alpha_F,\alpha_T,\alpha_B)^{\mathsf T}.
 \label{eq:srtarget}
\end{equation}
Converting the stored PPO action to $\vect{\alpha}$ before regression is essential: otherwise the fitted coefficients combine policy physics with scene-dependent decoder scale and bias.

Candidate variables include total drag and lift,
\begin{equation}
 C_{d,\mathrm{tot}}=\sum_i C_{d,i},\qquad
 C_{l,\mathrm{tot}}=\sum_i C_{l,i},\qquad
 C_{d,\mathrm{rear}}=C_{d,T}+C_{d,B},
\end{equation}
selected velocity probes, target-relative force errors for illusion, and finite-time rates. For a generic signal $g$,
\begin{equation}
 \dot g(t)\simeq\frac{g(t)-g(t-\Delta t_c)}{\Delta t_c},
 \label{eq:rate}
\end{equation}
where $\Delta t_c$ is the physical control interval rather than one raw LBM step. The joint K\'arm\'an library also includes viscosity and selected action-rate variables to represent changes in phase dynamics across $\Rey_D$.

The search uses PySR, whose evolutionary search alternates mutation, crossover, simplification and numerical coefficient optimisation over populations of expression trees \citep{Cranmer2023}. The binary operator set is restricted to $\{+,-,\times,\div\}$ and the unary set to a square operation. Protected numerical evaluation rejects non-finite candidates. This deliberately conservative grammar avoids introducing trigonometric or transcendental functions merely because they can interpolate a finite trajectory. Expression complexity is the weighted node count; constants and variables have unit cost, while compound operations incur their tree cost. The Pareto front is ranked by loss improvement per added unit of complexity, but its nominal winner is not accepted automatically.

Four nested feature classes are considered: instantaneous force and sensor variables; phase-state variables with temporal derivatives; a cross-$\Rey_D$ physics-rate library including viscosity and action rates; and an illusion library including target errors. The purpose is not to maximise library size but to determine the smallest closed-loop state that preserves the policy behaviour. Table~\ref{tab:features} records the physical meaning and reflection parity of the features used in the reported searches.

\begin{table}
 \centering
 \caption{Symbolic-regression features. ``Even'' and ``odd'' denote parity under centreline reflection; paired quantities are evaluated after exchanging upper and lower channels. Rates use the physical control interval $\Delta t_c$.}
 \label{tab:features}
 \begin{tabular}{p{0.19\textwidth}p{0.43\textwidth}p{0.12\textwidth}p{0.16\textwidth}}
  \toprule
  symbol & definition or role & parity & library\\
  \midrule
  $u_m,u_a,u_c$ & streamwise velocities at the three downstream probes & paired/even & all\\
  $v_a$ & selected transverse probe velocity & odd & cloak\\
  $C_{d,\mathrm{tot}}$ & total pinball drag coefficient & even & all\\
  $C_{d,\mathrm{rear}}$ & sum of rear-cylinder drag coefficients & even & all\\
  $C_{l,\mathrm{tot}}$ & total pinball lift coefficient & odd & all\\
  $C_{l,\mathrm{diff}}$ & symmetry-adapted rear lift difference & parity-adapted & cloak\\
  $\dot a_F,\dot a_T,\dot a_B$ & decoded physical-action rates & transformed & cross-$\Rey_D$\\
  $\mu g$ & viscosity-weighted feature $g$ & parity of $g$ & cross-$\Rey_D$\\
  $C_{d,\mathrm{err}},C_{l,\mathrm{err}}$ & pinball-minus-target force errors & even/odd & illusion\\
  $\dot u_a,\dot C_l,\dot C_{d,\mathrm{err}},\dot C_{l,\mathrm{err}}$ & finite-time phase and error rates & inherited & phase/illusion\\
  \bottomrule
 \end{tabular}
\end{table}

\subsection{Trajectory collection and decoder audit}

The fitting records are deterministic PPO rollouts collected after training. At every control decision the archive stores the six force components, six probe velocities and three normalised policy outputs. Illusion records additionally contain the target force phase used by the environment. The four K\'arm\'an datasets are stored under the code-level labels $\Rey_{\mathrm{code}}=50$, 100, 200 and 400 and therefore cover $\Rey_D=25$, 50, 100 and 200; the joint illusion fit uses the $0.75D$ and $1.0D$ targets. Lamb and Taylor records are held out from the K\'arm\'an fitting and are used only for transfer evaluation.

Undoing the action decoder is a necessary audit step rather than a change of units made for presentation. In the Legacy environment the normalised action $a_i\in[-1,1]$ is mapped according to
\begin{equation}
 \alpha_i=s a_i+b_i,
 \label{eq:decoder}
\end{equation}
where $s=8$ with biases $(0,-4,4)$ for K\'arm\'an, $s=8$ with $(0,-2,2)$ for illusion, and $s=4$ with $(0,-4,4)$ for the transient-vortex scenes. The precise sign and ordering follow the Legacy cylinder convention. Regression on $a_i$ would make a constant rear rotation appear smaller or disappear into the decoder bias; equation~\eqref{eq:decoder} is therefore inverted before constructing the targets and re-applied consistently during deployment.

The control interval is also part of the identified model. K\'arm\'an and transient-vortex scenes normally use 800 LBM steps per decision, while the $0.75D$, $1.0D$ and $1.5D$ illusion scenes use 400, 600 and 800 steps, respectively. A derivative in the symbolic library is a finite difference over this decision interval, not over one lattice update. Thus changing the sampling interval changes both the information available to feedback and the numerical value of a rate feature. The dedicated $\Rey_D=200$ ($\Rey_{\mathrm{code}}=400$) sensitivity test halves the interval to 400 steps and doubles the number of validation decisions so that the physical horizon remains comparable.

\subsection{PySR fitting and model-selection protocol}

For each channel, samples from the relevant scenes are concatenated after scene-consistent feature construction. Front and shared-rear heads are searched separately. Candidate expressions are first screened for finite values and action bounds, then evaluated on withheld trajectory segments. Search depth and population evolution produce a family of loss--complexity compromises rather than one privileged expression. We retain candidates that improve materially over a constant baseline without adding structurally redundant terms.

Model selection then proceeds through four filters. The first is offline action prediction, reported with $R^2$ only as a diagnostic. The second is a symbolic audit of dimensions, decoder convention and $\mathbb{Z}_2$ equivariance. The third is a deployment audit: every term must remain active when the expression generates its own actions. The fourth and decisive filter is closed-loop CFD similarity over a horizon long enough for actuation information to convect from the pinball to the observer. Force histories and boundedness are inspected alongside similarity. This hierarchy is designed to reject formulas that exploit correlations peculiar to the teacher trajectory.

The pipeline is reproducible in four stages: deterministic PPO inference, PySR fitting, CFD validation and figure generation. It should not be confused with an end-to-end symbolic reinforcement-learning algorithm. PPO and PySR are trained sequentially, and validation is an independent CFD rollout. This separation permits the neural and symbolic controllers to be compared from the same flow state with the same normalisation, actuator smoothing and observation schedule.

\subsection{Reflection symmetry}

Reflection about the centreline exchanges $T$ and $B$ and changes the sign of transverse quantities and rotation. The action transforms as
\begin{equation}
 \mathcal{G}_a
 \begin{bmatrix}\alpha_F\\\alpha_T\\\alpha_B\end{bmatrix}
 =
 \begin{bmatrix}-\alpha_F\\-\alpha_B\\-\alpha_T\end{bmatrix}.
 \label{eq:gsym}
\end{equation}
One front relation and one shared rear relation are fitted; the lower-cylinder law is generated by evaluating the shared relation on the reflected state. This removes degeneracy caused by the strong correlation of the rear actions and prevents symmetry-related actuators from being interpreted as independent mechanisms.

\subsection{Closed-loop model selection}
\label{sec:clselection}

Each candidate expression is subjected to three tests. First, its one-step action fit is measured on held-out PPO data. Secondly, its dimensions, decoder convention, deployment state and reflection symmetry are audited. Thirdly, the PPO decoder is replaced by the symbolic expression and the controller is deployed in CFD from the same initial condition and with the same sampling and actuation filtering.

The third test is decisive because a policy surrogate changes the state distribution on which all later actions are evaluated. In the present trajectories, a full-lag expression reached a one-step $R^2\simeq0.94$ but only $0.62$ closed-loop similarity. A lower-dimensional expression with $R^2\simeq0.32$ reached approximately $0.75$ similarity. Regression accuracy alone would therefore have selected the inferior controller. Terms that are correlated on the PPO trajectory but inactive after deployment are also removed. In particular, an early rear-action-rate term vanished because the deployed rear action was approximately constant.

\section{Symbolic feedback laws and closed-loop performance}
\label{sec:srresults}

\subsection{A joint law for K\'arm\'an cloaking}

Scene-specific fits at four Reynolds numbers used different but correlated force and phase variables. Literal equality of the expressions was therefore not used as evidence of a shared mechanism. A stronger test was performed by fitting all four datasets jointly and deploying one algebraic structure in every scene.

The raw joint PySR candidate retained in the formula archive is
\begin{equation}
 \alpha_F^{\mathrm{raw}}=0.38505\,\dot a_B+\dot a_F-14.95165\,\mu C_{l,\mathrm{tot}}.
 \label{eq:rawfrontlaw}
\end{equation}
It has a nominal offline $R^2$ of unity on the highly correlated joint trajectory. That number is not interpreted as perfect identification. When equation~\eqref{eq:rawfrontlaw} is deployed with the shared rear head, the rear action is constant, hence $\dot a_B=0$ after initialisation. The first term is therefore a teacher-trajectory correlate, not an independently exercised feedback route. Retaining it in the paper's mechanism statement would falsely suggest dynamic rear feedback.

Because the shared rear head is constant, $\dot a_B$ becomes inactive after initialisation. The reduced relation used only for mechanism interpretation is
\begin{equation}
 \boxed{\alpha_F=\dot a_F-14.952\,\mu C_{l,\mathrm{tot}}},
 \label{eq:frontlaw}
\end{equation}
and the rear relation reduces to
\begin{equation}
 \boxed{\alpha_T\simeq3.414,\qquad \alpha_B\simeq-3.414.}
 \label{eq:rearlaw}
\end{equation}
Here $\dot a_F$ is the backward-looking front action-rate feature formed with the scene control interval, and $\mu$ is the viscosity variable used in the joint library. The numerical coefficients depend on the Legacy normalisation, control interval and action convention and are not proposed as universal constants. The closed-loop values in table~\ref{tab:crossre} validate the archived raw expression~\eqref{eq:rawfrontlaw}; the reduced equation~\eqref{eq:frontlaw} has not yet been assigned a separate validation record and must not be read as an independently tested replacement.

Equations~\eqref{eq:frontlaw}--\eqref{eq:rearlaw} separate the controller into a dynamic front contribution and a persistent antisymmetric rear contribution. The rear pair modifies the gap flow and supplies mean streamwise momentum to compensate the pinball velocity deficit. The front cylinder responds to the phase-rate and antisymmetric force state, regulating the residual lift and wake phase.

\subsection{Cross-$\Rey_D$ validation}

Table~\ref{tab:crossre} reports closed-loop performance. One symbolic expression is deployed without changing its algebraic structure.

\begin{table}
 \centering
 \caption{Closed-loop K\'arm\'an-cloak similarity. All Reynolds numbers are based on one pinball-cylinder diameter.}
 \label{tab:crossre}
 \begin{tabular}{cccc}
  \toprule
  $\Rey_D$ & PPO & joint SR & comment\\
  \midrule
  25  & 0.961 & 0.847 & successful transfer, below PPO\\
  50  & 0.954 & 0.888 & closest agreement with PPO\\
  100 & 0.884 & 0.845 & robust intermediate-$\Rey_D$ transfer\\
  200 & 0.795 & 0.806 & 0.819 with shorter control interval\\
  \bottomrule
 \end{tabular}
\end{table}

The symbolic controller remains effective over the full range. At $\Rey_D=200$ ($\Rey_{\mathrm{code}}=400$), reducing the control interval improves similarity from $0.806$ to $0.819$, indicating that temporal resolution contributes to the high-$\Rey_D$ degradation. This observation does not establish sampling as the only cause: the PPO baseline is also weaker, and the wake contains shorter coupled time scales.

The important result is structural rather than numerical equality with PPO. A joint expression fitted to combined data retains most of the closed-loop performance across a factor of eight in $\Rey_D$. Reynolds-number dependence enters through the rate construction and viscosity-weighted lift feedback instead of through four unrelated policies.

\draftfigure{Insert the SR cross-$\Rey_D$ performance figure and representative PPO/SR action traces.}{Closed-loop validation of the joint cloak law. The figure should compare PPO, symbolic and uncontrolled similarity at each $\Rey_D$ and show that the rear actions are approximately constant while the front action carries the dynamic response.}{fig:sr-crossre}

\subsection{Transfer from periodic to transient disturbances}

The same law was applied without refitting to two transient-vortex cases. The Lamb dipole and Taylor monopole differ qualitatively from a periodic street and are not labelled in the symbolic expression. Table~\ref{tab:vortextransfer} shows that the SR law remains competitive with PPO.

\begin{table}
 \centering
 \caption{Transfer of the joint K\'arm\'an SR law to transient disturbances.}
 \label{tab:vortextransfer}
 \begin{tabular}{lcc}
  \toprule
  incident disturbance & PPO & transferred SR\\
  \midrule
  Lamb vortex dipole & 0.942 & 0.949\\
  Taylor vortex monopole & 0.916 & 0.905\\
  \bottomrule
 \end{tabular}
\end{table}

The transfer is difficult to explain by waveform memorisation because the law contains neither a disturbance identifier nor an explicit target waveform. It uses local force and action-phase information that remains available when the incident field changes. The Taylor result is slightly below PPO and therefore also defines the scope of the claim: the relation is a reusable cloak backbone, not a universally optimal transient controller.

\subsection{Physical interpretation}

The joint relation may be written schematically as
\begin{equation}
 \vect{\alpha}(t)\simeq
 \underbrace{(0,\alpha_0,-\alpha_0)^{\mathsf T}}_{\text{mean rear compensation}}
 +\underbrace{(\alpha_F(t),0,0)^{\mathsf T}}_{\text{front phase--force regulation}}.
 \label{eq:mechanism}
\end{equation}
Opposite rear rotation accelerates the gap flow and changes the base-bleed jet, providing a mean correction to the blocked wake. The front cylinder carries most of the time-dependent action and changes the instantaneous antisymmetric state. This decomposition is consistent with known actuator roles in the fluidic pinball, but the present objective differs from conventional drag reduction or stabilisation: the corrected wake must retain an imposed incident event at a downstream observer.

The physical conclusion is deliberately narrower than the algebraic expression. The transferable finding is the division into persistent rear compensation and dynamic front rate--lift feedback. The coefficient in \eqref{eq:frontlaw} is tied to the dataset and is not a new nondimensional fluid constant.

\subsection{Illusion as target-dependent retuning}

Illusion adds a non-zero target state. Compact scene-specific laws are obtained near the natural controllable scale. For target diameter $0.75D$,
\begin{equation}
 \alpha_F=-0.169(C_{l,\mathrm{tot}}+\dot C_{l,\mathrm{tot}})-1.240,
 \label{eq:ill075}
\end{equation}
whereas for $1.0D$,
\begin{equation}
 \alpha_F=0.0123(\dot u_a+u_a+26.5).
 \label{eq:ill100}
\end{equation}
These formulas use different local coordinates around different target manifolds but both retain low-dimensional force or phase feedback.

A joint law fitted to the $0.75D$ and $1.0D$ targets is
\begin{align}
 \alpha_F={}&C_{d,\mathrm{tot}}-(C_{d,\mathrm{err}}+5.428)
 +0.00978(\dot u_a+u_a),
 \label{eq:illfront}\\
 \alpha_T={}&0.535\left[C_{d,\mathrm{err}}
 -(C_{d,\mathrm{rear}}-C_{l,\mathrm{err}})\right]+2.782,
 \label{eq:illrear}
\end{align}
with the lower action generated by reflection. It reaches full-horizon similarities of $0.982$ and $0.958$ on the two fitted target diameters in the current validation JSON files; the corresponding tail-window values are $0.807$ and $0.886$. The distinction shows that a high aggregate score need not imply uniform long-time retention. Thus moderate illusion retains mean-flow correction, phase information and force regulation, but expresses them relative to target error.

\begin{table}
 \centering
 \caption{Cross-diameter deployment of the joint illusion law.}
 \label{tab:illusion}
 \begin{tabular}{ccl}
  \toprule
  target diameter & joint-SR similarity & interpretation\\
  \midrule
  $0.5D$  & 0.854 & partial transfer\\
  $0.6D$  & 0.939 & strong transfer\\
  $0.75D$ & $\approx0.98$ & fitted near-native regime\\
  $1.0D$  & 0.958 & fitted near-native regime\\
  $1.2D$  & 0.849 & beginning of degradation\\
  $1.5D$  & -- & not represented by present library\\
  $2.0D$  & 0.675 & weak transfer\\
  \bottomrule
 \end{tabular}
\end{table}

Table~\ref{tab:illusion} defines a finite generalisation range. The shared law is effective while the target wake remains near the natural controllable state, but it degrades as the target scale changes.

\draftfigure{Insert the illusion diameter-generalisation curve. Add the $1.5D$ PPO spectrum and autocorrelation only after the original trajectory has been restored and the diagnostics regenerated.}{Target-scale dependence of the symbolic illusion controller. Near-native targets admit a compact target-error law, whereas no useful symbolic closed-loop controller has yet been validated for $1.5D$ with the present feature library.}{fig:illusion-regimes}

The result supports
\begin{equation}
 \text{illusion}=\text{shared compensation}+\text{target-specific retuning},
\end{equation}
not a universal formula for all target cylinders. The individual formulas in equations~\eqref{eq:ill075} and~\eqref{eq:ill100} are useful local descriptions, whereas equations~\eqref{eq:illfront}--\eqref{eq:illrear} test whether a common target-error coordinate survives across scenes. The latter is the stronger generalisation test even when an individual fit has a better offline score.

The coefficient signs in the joint front law deserve explicit attention. The archived PySR expression is
\begin{equation}
 C_{d,\mathrm{tot}}-(C_{d,\mathrm{err}}+5.4277)
 -(-0.0097839)(\dot u_a+u_a),
\end{equation}
so its final phase term is positive after resolving the double negative. Writing the unresolved archive syntax alongside the simplified mathematical law prevents a transcription error between the machine-readable formula and the manuscript.

Cross-diameter deployment is performed without introducing target diameter as an input. Consequently, the degradation curve measures extrapolation of a law whose only target information is supplied through the instantaneous target-error features. A separate diameter-marker regression can fit a diameter-dependent expression, but it answers an easier question and is not the principal result reported here. The $0.8D$ validation, omitted from the compact table for readability, gives a similarity of $0.908$ and follows the same near-training-range trend.

\subsection{The $1.5D$ regime boundary}

The $1.5D$ case is a failure of the present symbolic representation rather than a failure of neural control. The stored PPO baseline remains effective, whereas PySR returns a degenerate lag-copy relation with nominal one-step $R^2=1$ and no useful canonical symbolic closed-loop validation. Exploratory analysis indicates rapid periodic modulation, but the previously quoted frequency ratio, lag-two autocorrelation and feature correlations cannot presently be recomputed because the underlying trajectory NPZ is absent from the archived analysis tree. Those numerical diagnostics are therefore withheld pending restoration of the raw trajectory.

Both PySR and sparse linear identification then favour lagged-action proxies. Such expressions predict one step but do not identify a mechanism: the lagged action stands in for a rapidly switching internal phase that is absent from the feature library. The appropriate response is not to report the proxy as a physical law. The case instead shows that an explicit oscillator, a sufficient delay embedding or a target-scale variable is required. SR is therefore useful both when it succeeds and when it exposes a missing state coordinate.

\section{Observable-inferred decomposition of the active correction}
\label{sec:oid}

Symbolic regression answers which measured quantities are mapped to which rotations, but an action-space law does not identify the spatial structures responsible for force and downstream matching. We therefore analyse the active correction field independently. Proper orthogonal decomposition (POD) provides an energy-ranked basis, while observable-inferred decomposition (OID) rotates a retained POD subspace towards covariance with a selected output, following the broader observable-correlation viewpoint of extended POD \citep{Boree2003}. This distinction is important because a low-energy near-body structure can dominate integrated force, whereas an energetic convected structure can dominate a downstream probe.

\subsection{Correction fields and OID construction}

For aligned vectorised correction snapshots $\vect{x}_t$, let
\begin{equation}
 \vect{X}=\vect{U}\vect{\Sigma}\vect{V}^{\mathsf T},\qquad
 \vect{z}_t=\vect{U}_r^{\mathsf T}\vect{x}_t.
\end{equation}
For an observable $\vect{y}_t$, the reduced cross-covariance is
\begin{equation}
 \vect{C}_{zy}=N^{-1}\sum_t\vect{z}_t\vect{y}_t^{\mathsf T}
 =\vect{U}_y\vect{\Sigma}_y\vect{V}_y^{\mathsf T},
\end{equation}
and the OID field directions are $\vect{\psi}^{(y)}_k=\vect{U}_r\vect{u}_{y,k}$. Force, downstream-signature, suppression and action observables are treated separately while sharing the underlying correction subspace. OID is therefore not a latent-state reconstruction of PPO and is not used to modify the SR law.

Five POD modes capture $99.97\%$ of steady-cloak correction energy, $99.90\%$ for K\'arm\'an cloak, $99.93\%$ and $99.91\%$ for the $0.75D$ and $1.0D$ illusions, and $97.90\%$ for $1.5D$. This compactness concerns the active change, not the full multi-scene flow. The lower value at $1.5D$ is consistent with a correction distributed over additional scales.

\begin{table}
 \centering
 \caption{Two-coordinate held-out prediction from observable-informed and energy-ranked coordinates. Negative POD $R^2$ means that the two-coordinate predictor is worse than the held-out mean.}
 \label{tab:oidpred}
 \begin{tabular}{llrr}
  \toprule
  scene & observable & OID $R^2$ & POD $R^2$\\
  \midrule
  K\'arm\'an cloak & force & 0.750 & 0.418\\
  illusion $0.75D$ & force & 0.435 & -2.426\\
  illusion $0.75D$ & signature & 0.661 & -0.034\\
  illusion $1.0D$ & force & 0.671 & -0.237\\
  illusion $1.0D$ & signature & 0.586 & -0.160\\
  illusion $1.5D$ & force & 0.640 & 0.264\\
  illusion $1.5D$ & signature & 0.315 & 0.060\\
  \bottomrule
 \end{tabular}
\end{table}

Table~\ref{tab:oidpred} shows that observable ranking, rather than extra state dimension, produces the predictive advantage. The K\'arm\'an force result is particularly clear. For the corresponding delayed signature error, successful cloaking leaves very little target variance; an $R^2$ close to zero is consequently not evidence that the correction is absent. Metrics that normalise by target variance become ill-conditioned precisely when the error is nearly eliminated.

\subsection{Task-dependent force--signature geometry}

The leading force and signature directions vary systematically with objective. In steady cloak, the signed force--suppression overlap is $+0.763$: reducing natural fluctuations and regulating force rely on related near-wake changes. The full-field fluctuation RMS decreases by $99.43\%$, lift RMS by $83.3\%$, and recirculation area by $38.5\%$, while recirculation length changes by only $3.2\%$. The correction narrows and quietens the wake rather than simply deleting its streamwise recirculation extent.

In K\'arm\'an cloak, the leading force--signature overlap is approximately $-0.034$ and remains close to zero over the tested convective delays. This near-orthogonality is physically plausible: the controller must regulate additional pinball force while preserving, not suppressing, the incoming street. For illusion the signed overlaps are $-0.082$, $-0.495$ and $-0.932$ for $0.75D$, $1.0D$ and $1.5D$. The $0.75D$ value is rank-sensitive and supports only partial separation; the latter two are more stable. A negative overlap is an orientation within the retained correction subspace, not proof of dynamically independent channels.

Raw observations predict K\'arm\'an PPO action with $R^2=0.956$, whereas two force-OID coordinates explain only about $22.5\%$ of action variance. This contrast reinforces the division of labour: SR approximates the observation-to-action map, while OID identifies output-relevant structures in the action-induced field. Neither should be substituted for the other.

\draftfigure{Insert the selected OID field-analysis panels after final figure assembly.}{Observable-informed correction modes and held-out prediction. Force relevance is concentrated in the body-connected near wake, whereas signature relevance shifts towards the convected correction near the observer. Error bars should show retained-rank or leave-one-cycle-out sensitivity where available.}{fig:oid}

\section{Lagged canonical-correlation decomposition and propagation}
\label{sec:ccd}

OID is instantaneous. Canonical-correlation decomposition (CCD) augments the observable with delays to distinguish the source correction near the cylinders from its downstream descendant. For delays $\{\tau_q\}_{q=1}^Q$, define
\begin{equation}
 \vect{P}_t=[\vect{p}(t+\tau_1),\ldots,\vect{p}(t+\tau_Q)]
\end{equation}
and
\begin{equation}
 \vect{C}_{PZ}=N^{-1}\sum_t\vect{P}_t^{\mathsf T}\vect{z}_t
 =\vect{R}\vect{\Sigma}\vect{W}^{\mathsf T}.
\end{equation}
The field directions are $\vect{\psi}^{\mathrm{CCD}}_k=\vect{U}_r\vect{w}_k$. Periodic cases use leave-one-cycle-out tests; the steady case is evaluated with suppression and recirculation diagnostics. Delay embedding is not a claim of causality by itself, but the known ordering of cylinder actuation, near-wake response and sensor arrival gives the correlation a physically constrained interpretation.

\subsection{Common cloak correction and spatial zones}

Across steady, K\'arm\'an, Lamb and Taylor cloak, the dominant correction is a positive streamwise-velocity increment behind the pinball, with dipole-like velocity and concentrated vorticity changes around the rotating cylinders. In the common analysis normalisation, correction-field RMS values are $0.196$, $0.397$, $0.146$ and $0.188$, respectively. Similarity of structure does not mean equality of amplitude: the periodic K\'arm\'an task requires the largest correction because it must remove pinball distortion while transmitting an incoming street.

The field is examined in overlapping near-body, body-wake and sensor zones. Force-correlated directions are strongest around the surfaces, separated shear layers and body-connected rolled-up vorticity, consistent with impulse-based interpretations of instantaneous force. Delayed signature directions extend farther downstream as the active correction is convected and deformed. The zones are diagnostic masks, not independent subsystems.

Action-CCD associates the leading correction with the persistent rear rotation; higher directions carry the dynamic front contribution. Force-CCD emphasises the body-connected source, whereas signature-CCD emphasises the transported consequence. This yields the physically testable ordering
\begin{equation}
 \text{rotation}\rightarrow\text{near-body correction}
 \rightarrow\text{wake transport}\rightarrow\text{observer signal},
\end{equation}
although the present evidence is entirely numerical and no experimental measurements are included.

\subsection{Illusion target-correction comparison}

For illusion, the active correction can be compared with the correction required to transform the blocked pinball field into the target-cylinder field. At retained rank six, leading-mode absolute overlaps are $0.383$, $0.926$ and $0.922$ for target diameters $0.75D$, $1.0D$ and $1.5D$; at rank ten they become $0.320$, $0.684$ and $0.661$. The $1.0D$ target provides the clearest robust evidence of a shared leading correction. The $0.75D$ match is partial and rank-sensitive. At $1.5D$, a target-like leading spatial direction coexists with higher-order and temporal disagreement, consistent with the PPO high-frequency action and the failure of the instantaneous SR library.

These results refine, rather than simply repeat, the SR interpretation. Mean deficit compensation can remain spatially recognisable even when the temporal policy needed to realise it cannot be represented by the current symbolic coordinates. A strong leading-mode overlap is therefore not sufficient evidence for a transferable feedback law.

\draftfigure{Insert phase-aligned active corrections and lagged CCD modes after assembling the final SR, OID and CCD panels.}{Propagation of the active correction. Columns should distinguish action-, force- and delayed-signature-informed structures, with near-body, body-wake and sensor zones marked. Lamb and Taylor panels test whether the positive streamwise cloak correction survives a change from periodic to transient incidence.}{fig:ccd}

\section{Integrated mechanism and limitations}
\label{sec:field}

SR maps measurements to actions but does not show where those actions modify the flow. We therefore use the active correction
\begin{equation}
 \Delta\vect{q}_{\mathrm{ctl}}
 =\vect{q}_{\mathrm{ctl}}-\vect{q}_{\mathrm{blk}}
 \label{eq:dqctl}
\end{equation}
rather than decomposing the raw controlled field. For illusion, the correction required to transform the blocked pinball wake into the target is
\begin{equation}
 \Delta\vect{q}_{\mathrm{tar}}
 =\vect{q}_{\mathrm{tar}}-\vect{q}_{\mathrm{blk}}.
\end{equation}
This subtraction separates the active change from the incident disturbance and natural pinball wake.

Five POD modes capture $99.97\%$ of the steady-cloak correction energy, $99.90\%$ for K\'arm\'an cloak, $99.93\%$ and $99.91\%$ for the $0.75D$ and $1.0D$ illusions, and $97.90\%$ for the $1.5D$ illusion. The actuator therefore modifies the flow within a compact subspace even though the complete flow is not low-dimensional over all scenes.

The leading spatial correction shared by steady, K\'arm\'an, Lamb and Taylor cloak is a positive streamwise-velocity increment behind the pinball, accompanied by near-body dipole-like velocity and vorticity changes. This is the field counterpart of the rear counter-rotation in \eqref{eq:rearlaw}: it compensates the pinball-induced velocity deficit. Time-dependent near-body changes are consistent with the front phase--force term in \eqref{eq:frontlaw}.

Observable-informed projections further show why force matching and downstream matching must not be conflated. Two observable-informed coordinates predict K\'arm\'an force with $R^2=0.750$, compared with $0.418$ for two energy-ranked POD coordinates. For illusion, the same two-coordinate comparison gives force/signature $R^2$ values of $0.435/0.661$, $0.671/0.586$ and $0.640/0.315$ for $0.75D$, $1.0D$ and $1.5D$, respectively; the corresponding POD values are substantially lower. The force-relevant structure is concentrated in the body-connected near wake, whereas the signature-relevant structure is its convected descendant near the observer.

\draftfigure{Insert the phase-aligned active correction fields for steady, K\'arm\'an, Lamb and Taylor cloak, together with the leading force- and signature-informed modes.}{Spatial consistency of the symbolic mechanism. Rear-cylinder compensation produces a common downstream streamwise increment, while force- and signature-relevant projections emphasise the near-body source and convected descendant, respectively.}{fig:correction-fields}

These field results are not used to fit the symbolic law. They provide an independent consistency check for the mechanism chain
\begin{equation}
 \text{measurements}
 \rightarrow\text{compact actuation}
 \rightarrow\text{near-body correction}
 \rightarrow\text{body-wake transport}
 \rightarrow\text{downstream signature}.
 \label{eq:chain}
\end{equation}
They also explain why a common actuator mechanism can have different force and signature projections in steady suppression, incident-street preservation and target-wake synthesis.

\section{Discussion}
\label{sec:discussion}

\subsection{Why closed-loop validation changes the SR problem}

Policy distillation differs fundamentally from ordinary supervised regression. The target data are generated by a controller that shapes its own state distribution. Once the symbolic surrogate replaces PPO, each action changes all future observations. High one-step accuracy can preserve correlations that are incidental to the PPO trajectory while accumulating a systematic phase error in deployment. Closed-loop CFD must therefore be treated as part of symbolic model selection rather than as a final illustration.

This criterion also limits algebraic over-interpretation. Correlated forces, sensor values and action lags can yield multiple formulas with similar offline error. Reflection symmetry, dimensional construction, deployment activity and transfer performance are used to select a canonical representative. The identified mechanism is the robust actuator-role decomposition, not the uniqueness of every algebraic term.

\subsection{Scope of the common cloak backbone}

The strongest evidence for a reusable mechanism is the combination of cross-$\Rey_D$ and cross-disturbance transfer. The joint law retains substantial performance from $\Rey_D=25$ to 200 and transfers from a periodic street to isolated vortices. This suggests that the controller responds primarily to the additional distortion introduced by the pinball. The incident field changes the timing and amplitude of the response, but rear momentum compensation and front phase--force regulation remain useful.

The result does not imply complete invariance. Sampling sensitivity increases at high $\Rey_D$, the Taylor transient remains imperfect, and the formula has been evaluated on the state manifolds reached by the trained policies. Arbitrary initial conditions, different sensor locations or different actuator limits may require a new rate coordinate or refitting.

\subsection{Why illusion has a regime boundary}

Cloaking cancels the pinball contribution relative to an incident background; illusion must additionally create the difference between the corrected pinball wake and a target wake. When target and natural controllable scales are close, target-error feedback retunes the cloak backbone. As the target diameter grows, the controller must reorganise frequency, amplitude and spatial scale more strongly. The $1.5D$ policy appears to use a rapid internal phase state, and the $2.0D$ transfer is weak. The symbolic failure is consequently physically informative: the control state requires a temporal coordinate that cannot be reconstructed from the present instantaneous force and probe library.

\subsection{What the symbolic law establishes}

The evidence supports three levels of claim, which should not be collapsed. At the empirical level, one algebraic structure provides useful closed-loop K\'arm\'an cloaking over a factor of eight in $\Rey_D$ and transfers to two transient disturbances. At the mechanistic level, the deployment audit and field correction support a division between mean rear compensation and dynamic front regulation. At the universal level, however, the available evidence is insufficient: coefficients depend on the Legacy plant, and neither three-dimensional flow nor alternative actuator geometry has been tested. The paper's main contribution is therefore a transferable mechanism within a documented numerical family, not a universal constitutive law for hydrodynamic cloaks.

The raw-versus-canonical distinction in equations~\eqref{eq:rawfrontlaw} and~\eqref{eq:frontlaw} illustrates why symbolic interpretability requires intervention. An expression tree does not become a mechanism merely by being short. The raw $\dot a_B$ term is mathematically legitimate on the PPO trajectory and improves neither deployment activity nor physical explanation once the rear head becomes constant. Removing it is analogous to testing a putative causal input by changing the operating distribution: the symbolic controller itself supplies that intervention. This is also why the nominal $R^2=1$ attached to the archived joint formula should not dominate interpretation.

The implicit appearance of $\dot a_F$ warrants similar caution. Written literally, equation~\eqref{eq:frontlaw} contains the derivative of the output being computed. In implementation it is an available finite difference from the previous applied action, so the law is a discrete dynamic controller rather than an algebraic loop. If $k$ indexes control decisions,
\begin{equation}
 \dot a_F^k=\frac{a_F^{k-1}-a_F^{k-2}}{\Delta t_c}
\end{equation}
is formed from stored applied actions before $\alpha_F^k$ is evaluated. This convention must accompany any reproduction because using the newly requested action in the difference would define a different controller. The rate term acts as a compact phase coordinate, but it is not asserted to equal a material derivative or a state derivative of the fluid.

\subsection{Implications for sensor and controller design}

The extracted feature set suggests a hierarchy for future controller design. Mean rear compensation requires little dynamic information and could plausibly be supplied as a calibrated operating-point schedule. The front cylinder requires an antisymmetric force measurement and a temporal coordinate. The downstream probes remain important to PPO training and reward evaluation, but their limited appearance in the canonical K\'arm\'an law suggests that the distilled controller can use local force feedback once the compensation operating point has been discovered. This is a hypothesis about the tested closed-loop manifold, not a demonstration that probes can be removed under noise, drift or unseen incidence.

The illusion law differs because target errors enter directly. A target wake is not specified by one scalar diameter in the principal joint formula; it is represented by phase-dependent force errors and a downstream phase coordinate. This design allows the same expression to interpolate moderate target changes, but it also explains why extrapolation fails. As target scale grows, the available error coordinates may no longer resolve whether a mismatch should be corrected by changing mean momentum, shedding phase or oscillation frequency. The degenerate lag-copy fit and absence of a useful symbolic closed-loop result reveal this ambiguity; the detailed spectrum should be restored from the original trajectory before publication.

A practical successor to the present library would introduce a minimal oscillator state rather than indiscriminately adding lags. Two quadrature variables estimated from a narrow-band force or probe signal could encode phase without using the previous action as a proxy. Alternatively, a delay-coordinate state could be selected by observability and closed-loop tests. Both choices would increase complexity and should be justified by improvement at $1.5D$ without degrading the compact laws at $0.75D$ and $1.0D$. The present negative result supplies a concrete benchmark for that extension.

\subsection{Numerical provenance and reproducibility}

All mechanism coefficients and validation values in this article originate from the Legacy $1280\times512$ parabolic-inlet, no-slip-wall dataset. This provenance is repeated because a superficially similar V5 environment uses $2000\times600$ nodes, uniform inflow and free-slip walls. V5 may be appropriate for new training studies, but mixing its trajectories with the Legacy regression would conflate changes in blockage, wall boundary layers, action maps and normalisation. A future cross-plant study should treat V5 as an external validation domain and should report whether coefficients are refitted, not silently pool the two datasets.

The computational record contains machine-readable scene definitions, formula files, deterministic PPO trajectories and closed-loop validation summaries. Formula JSON files retain feature names and archive syntax; validation JSON files retain scene, controller mode, number of decisions and similarity. The scene registry is the source of truth when summary documents disagree. One such example is the $1.0D$ illusion similarity: the registry records $0.958$ for the current canonical validation, while an earlier results summary reports approximately $0.970$. We have retained approximate wording in the narrative where archival summaries differ and recommend regenerating the final table from a frozen registry immediately before submission.

Figure placeholders intentionally compile without copied graphics. The suggested paths point to generated SR plots in the analysis directory, but the files have not been assumed to exist beside the Overleaf source. Before submission, each selected PDF should be copied into a manuscript figure directory, its plotted Reynolds-number labels should be audited against the $\Rey_D$ convention, and any caption language such as ``universal'' should be weakened to ``shared over the tested cases''. OID and CCD panels require separate assembly from their field outputs and uncertainty checks.

\subsection{Limitations}

The study is two-dimensional and uses numerical trajectories. No experimental results are included, and the simulations should not be described as experimental validation. Three-dimensional instability, measurement noise, actuator bandwidth and laboratory delay may change the extracted coefficients and possibly the preferred features. The reported SR coefficients belong to the Legacy solver configuration and should not be transplanted directly to the newer V5 plant. The correction-field records contain a limited number of independent cycles, and the larger-target cases require longer trajectories and explicit frequency-state features.

Similarity is defined in a sparse downstream observer space and is not equivalent to pointwise cancellation everywhere. Full-field correction and force results reduce this ambiguity but do not establish invisibility to every possible observer. Finally, symbolic expressions are empirical closed-loop surrogates, not governing equations for the Navier--Stokes dynamics.

\subsection{Comparison of cloak and illusion identification problems}

Cloak and illusion produce superficially similar regression tables but pose different identification problems. For cloak, the desired observer signal is inherited from the incident flow. The controller should remove the incremental distortion created by the pinball, so force and phase variables can be interpreted relative to a recurring compensation task. This is why a formula trained on a periodic street can remain useful for isolated vortices: the incident waveform changes, but the additional blockage and body-connected distortion retain common actuator-side features.

For illusion, the reference itself changes. The error is generated jointly by the pinball dynamics and a target cylinder that is absent from the controlled domain. A target-force harmonic reconstruction supplies phase information during training, but the controlled flow can depart from that phase manifold in deployment. The joint formula succeeds near $0.75D$--$1.0D$ because target errors remain informative local coordinates there. Beyond this range, equal instantaneous errors can demand different actions depending on target phase and frequency history. The cross-diameter decline is thus not merely ordinary extrapolation in a scalar parameter; it is loss of state identifiability in the chosen feature space.

This difference also clarifies why a diameter marker is not a complete remedy. A marker can tell PySR which target family is requested, but it does not supply the missing oscillator phase at $1.5D$. Conversely, a phase coordinate without target scale may distinguish switching direction while failing to choose the required mean correction. A robust large-range illusion controller will probably require both a target descriptor and a dynamic phase state. Establishing the minimal pair is a natural next symbolic-identification problem.

\subsection{Why the field analyses remain secondary to SR}

OID and CCD strengthen the mechanism claim but do not replace the central policy result. A spatial mode can recur across cases because geometrically similar actuators create similar local disturbances, even if the feedback rules that time those disturbances are unrelated. The cross-$\Rey_D$ and cross-disturbance closed-loop deployments directly test the feedback relation; field decompositions then ask whether its inferred actuator roles have a consistent consequence. This ordering avoids using visually similar vorticity panels as proof of controller equivalence.

Conversely, SR alone could mistake a compact correlation for physics. The positive downstream streamwise correction is the expected spatial consequence of rear counter-rotation compensating a blocked mean wake. The body-connected force modes and convected signature modes explain why the same action can influence two reward channels at different locations and times. The agreement is therefore triangulation: policy transfer, deployment activity and field organisation support one another while retaining distinct uncertainties.

The field analyses also constrain the language of invisibility. Sparse-sensor similarity can be high even when correction energy remains near the cylinders, and OID deliberately selects structures relevant to one observable. CCD shows transport towards the sensor zone, not cancellation for every possible observer. A full observability claim would require dense multi-component measurements over a prescribed exterior region and robustness to alternative observer placement. Those tests are beyond the current dataset.

\subsection{Prospects for closed-loop symbolic discovery}

The present workflow uses CFD validation after PySR search because evaluating every evolutionary candidate in CFD would be prohibitive. A future method could incorporate closed-loop information more efficiently through staged screening. Offline Pareto fronts could first be filtered by symmetry, boundedness and local sensitivity. A differentiable or data-driven short-horizon surrogate could then reject candidates with obvious phase drift, after which only a small set would enter full CFD. Importantly, the final acceptance criterion should remain the original solver, since surrogate agreement on the teacher distribution recreates the problem identified here.

Multiple independent PPO seeds would provide another valuable axis. If different neural policies converge to the same symbolic actuator roles but different coefficients or correlated coordinates, the shared structure would be stronger evidence of mechanism. If they instead produce distinct successful laws, the result would reveal policy multiplicity: several feedback strategies may generate the same observer signature. Such multiplicity is scientifically relevant and should not be hidden by selecting one best seed.

Noise and delay studies should likewise be performed at the symbolic level. Compact formulas can be more transparent but less forgiving than neural policies with implicit smoothing. Rate features amplify measurement noise, and the $\dot a_F$ coordinate depends on actuator telemetry and timing. Low-pass differentiation, state observers or explicit oscillator coordinates could improve robustness, but each changes the identified law. Reporting the filtering operator and its delay would be as important as reporting the coefficients.

Finally, the symbolic law offers a tractable object for stability and sensitivity analysis. Around a periodic controlled orbit one could linearise the discrete controller together with a reduced flow response and examine Floquet multipliers as the control interval changes. Such an analysis could determine whether the SI400 improvement at $\Rey_D=200$ ($\Rey_{\mathrm{code}}=400$) results from increased phase margin or simply finer waveform tracking. The current data establish sensitivity but do not resolve that mechanism, so this remains a proposed extension rather than a conclusion.

\section{Conclusions}
\label{sec:conclusion}

DRL policies for hydrodynamic cloaking and illusion of the fluidic pinball have been distilled into explicit feedback laws and tested in closed-loop CFD. The main conclusions are:

\begin{enumerate}
 \item One-step action fit is insufficient for policy distillation. A model with $R^2\simeq0.94$ can perform substantially worse in closed loop than a lower-dimensional model with $R^2\simeq0.32$. Closed-loop CFD, symmetry and deployment relevance must be included in model selection.
 \item K\'arm\'an cloaking across $\Rey_D=25$--200 admits a common actuator-side structure: approximately steady and opposite rear-cylinder rotation supplies mean wake compensation, while front-cylinder rate--lift feedback regulates phase and residual force.
 \item The same symbolic law transfers without refitting to Lamb and Taylor transient vortices, supporting a reusable cloak backbone rather than memorisation of a periodic waveform.
 \item Illusion near target diameters $0.75D$--$1.0D$ reuses the compensation mechanism with target-error feedback. Generalisation deteriorates as target scale departs from this regime.
 \item The $1.5D$ target reveals a high-frequency policy state absent from the current feature library. This regime requires an oscillator, delay embedding or explicit target-frequency coordinate rather than a lagged-action proxy.
 \item The active correction field is strongly low-dimensional and contains the downstream streamwise increment predicted by the symbolic rear-cylinder mechanism. Force and downstream signature are different observable projections of this common correction.
\end{enumerate}

The broader result is that DRL and symbolic regression play complementary roles. DRL discovers high-performing strategies in a nonlinear delayed environment; symmetry-constrained SR identifies the smallest transferable feedback relation; and field analysis tests whether that relation has a consistent spatial consequence. This combination turns hydrodynamic cloaking from a black-box control demonstration into a testable mechanism for active wake-information management.

\section*{Supplementary data}
Supplementary material will include the scene definitions, symbolic expressions, closed-loop validation records and correction-field datasets.

\section*{Acknowledgements}
The authors acknowledge the computational resources and discussions that supported this work.

\section*{Funding}
Funding information will be added in the final manuscript.

\section*{Declaration of interests}
The authors report no conflict of interest.

\section*{Author ORCIDs}
Author ORCID information will be added before submission.

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