Free-Probability Kernels for Zero-Rollout Hyperparameter Selection in Reservoir Computing

# Free-Probability Kernels for Zero-Rollout Hyperparameter Selection in Reservoir Computing
Source: [https://arxiv.org/html/2608.20998](https://arxiv.org/html/2608.20998)
Sara Malacarne1, Andrea Ceni2, Claudio Gallicchio2Affiliation:1Telenor Research & Innovation, Oslo, Norway 2Department of Computer Science, University of Pisa, Pisa, Italy

###### Abstract

Reservoir computing \(RC\) couples a fixed recurrent dynamical system with a trained lightweight readout, but this efficiency is partly lost during hyperparameter selection: the recurrent gain, input scale, and leakage rate determine the reservoir’s stability and temporal processing regime and are usually tuned through many rollouts\. We introduce a deterministic, pilot\-informed selector for leaky linear reservoirs followed by coordinate\-wise nonlinear features\. Free probability yields cross\-lag propagation coefficients that summarize how the reservoir mixes past inputs\. In the large\-width limit, these coefficients define a deterministic temporal kernel that approximates the finite\-reservoir feature geometry\. Kernel ridge regression on a short labeled pilot sequence therefore ranks candidate operating regimes without instantiating or rolling out a reservoir, and the selected configuration transfers across widths\. Across ten synthetic temporal benchmarks, zero\-rollout selection obtains a mean deployment score of0\.7720\.772, compared with0\.7740\.774for exhaustive simulation\-based search, while avoiding156 600156\\,600selection rollouts\. With a small rollout budget, the proposed ranking provides the strongest mean performance at every tested budget and reaches the exhaustive reference using4\.8%4\.8\\%of its rollout cost\. On four public electricity\-transformer\-temperature \(ETT\) forecasting datasets, five retained candidates recover the exhaustive operating point on three datasets\. On multivariate cellular\-traffic forecasting, 15 rollouts per cell reach the 462\-rollout exhaustive reference and outperform random search and Bayesian optimization at low budgets\. These results position free\-probability kernels as deterministic surrogates for selecting reservoir operating regimes when validation rollouts are scarce\.

###### Index Terms:

Reservoir computing, Echo state networks, Free probability, Kernel methods, Hyperparameter selection

## IIntroduction

Reservoir computing \(RC\) provides an efficient framework for temporal sequence learning: a fixed recurrent system maps the input history into a high\-dimensional representation, while only a lightweight readout is trained\[[21](https://arxiv.org/html/2608.20998#bib.bib7),[32](https://arxiv.org/html/2608.20998#bib.bib8)\]\. Its canonical instance is the Echo State Network \(ESN\)\[[17](https://arxiv.org/html/2608.20998#bib.bib5)\]\. Although readout training is inexpensive, performance depends strongly on hyperparameters controlling the recurrent dynamics, particularly the recurrent gain, input scale, and leakage rate\. These parameters govern stability, memory, and the relative influence of recent and distant inputs, and their best values can vary substantially across tasks and temporal scales\.

In standard practice, these hyperparameters are selected by repeatedly instantiating finite\-width reservoirs, generating their state trajectories, fitting readouts, and evaluating validation error\. Consequently, the search cost grows with the candidate grid, the reservoir width, and the number of random realizations\. The selected configuration may also depend on the width used during search, so selection may need to be repeated when the deployment width changes\.

We address this bottleneck with a deterministic, task\-informed selector that requires no finite\-reservoir rollout during candidate ranking\. Given a short labelled input–output sequence from the downstream task, the method evaluates each candidate through a deterministic large\-width kernel\. The pilot sequence describes the prediction task; it is not obtained by running the candidate reservoirs\. The resulting ranking can be used directly as a zero\-rollout selector or as a pre\-screen that reduces the candidate grid before a small number of task\-direct finite\-reservoir evaluations\.

We study a leaky linear recurrence followed by coordinate\-wise nonlinear readout features\. For this architecture, the nonlinear feature Gram matrix converges at large reservoir width to a deterministic temporal kernel\. Its covariance is governed by mixed cross\-lag propagation moments of the recurrent matrix, which we compute using free probability \(FP\)\. For each operating pointθ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\), the resulting kernel approximates the feature geometry of the finite reservoir with those same parameters\. Kernel ridge regression on the pilot sequence can therefore rank the candidate operating points, after which the selected parameters are transferred to the finite reservoir used for deployment\.

Existing reservoir kernels are used mainly either to describe large\-width reservoir behaviour or as predictors in their own right\. Our use is different: for each candidateθ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\), the deterministic kernel approximates the feature Gram matrix of the finite reservoir with the same parameters\. We use this approximation to rank candidates on labelled pilot data, and then deploy the selectedθ⋆\\theta^\{\\star\}in the finite reservoir\.

The main analytical difficulty is that the same recurrent matrix is applied repeatedly over time, so the effects of inputs arriving at different lags are not independent\. Our derivation captures these cross\-time dependencies and shows that, at large width, the resulting nonlinear feature similarities converge to a deterministic kernel\. To the best of our knowledge, this is the first task\-informed, zero\-rollout method that uses such a kernel to select the hyperparameters of a finite reservoir\.

The main contributions are as follows:

1. 1\.We derive the deterministic large\-width post\-nonlinearity kernel of a leaky linear reservoir\. The analysis computes the mixed cross\-lag propagation moments generated by repeated use of the same recurrent matrix and establishes the coordinate\-level self\-averaging required by the nonlinear feature map\. We also obtain a controlled complete\-history extension and explicit kernels for several structured recurrent ensembles\.
2. 2\.We use this kernel as a task\-informed surrogate for finite\-reservoir hyperparameter selection\. Candidate values of\(σr,σin,α\)\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\)are ranked from labelled pilot data without instantiating any candidate reservoir\. If the deterministic validation score has a unique best candidate, then a sufficiently wide finite reservoir selects the same candidate with high probability\.
3. 3\.We evaluate zero\-rollout selection and FP\-based pre\-screening on synthetic temporal benchmarks, the public ETT forecasting datasets\[[37](https://arxiv.org/html/2608.20998#bib.bib32)\], and an operational cellular\-traffic forecasting task\. We compare against exhaustive task\-direct selection, a memory\-based proxy, nonlinear\-ESN selection, random search\[[3](https://arxiv.org/html/2608.20998#bib.bib15)\], and Bayesian optimization using the tree\-structured Parzen estimator \(TPE\)\[[2](https://arxiv.org/html/2608.20998#bib.bib19)\], under matched finite\-reservoir rollout budgets\.

Complete proofs, an explicit kernel catalogue for alternative recurrent random matrices, and additional experimental results are provided in Appendices[A](https://arxiv.org/html/2608.20998#A1)–[G](https://arxiv.org/html/2608.20998#A7)\.

## IIBackground and Problem Formulation

Existing approaches to reservoir hyperparameter selection can be divided broadly into task\-based search and dynamical proxies\. Task\-based methods evaluate candidate configurations according to their downstream validation performance\. The simplest examples are grid and random search\[[3](https://arxiv.org/html/2608.20998#bib.bib15)\], while more adaptive strategies include Bayesian optimization for reservoir parameters\[[24](https://arxiv.org/html/2608.20998#bib.bib28)\], gradient\-based optimization\[[33](https://arxiv.org/html/2608.20998#bib.bib25)\], and evolutionary methods\[[26](https://arxiv.org/html/2608.20998#bib.bib30)\]\. More efficient validation schemes reduce or reuse some of these evaluations\[[29](https://arxiv.org/html/2608.20998#bib.bib26),[22](https://arxiv.org/html/2608.20998#bib.bib27)\]\. Despite their different search strategies, these methods share the same basic cost: to assess a new candidate, they instantiate a finite reservoir, generate its state trajectory, fit a readout, and evaluate its predictions\.

A different strategy is to select or adapt reservoirs using properties of their internal dynamics\. Echo\-state\-property conditions and spectral\-radius guidelines identify stable operating regions\[[17](https://arxiv.org/html/2608.20998#bib.bib5),[36](https://arxiv.org/html/2608.20998#bib.bib22),[23](https://arxiv.org/html/2608.20998#bib.bib20)\], while memory capacity and Jacobian\-based indicators quantify aspects of temporal processing\[[35](https://arxiv.org/html/2608.20998#bib.bib31)\]\. Reservoir dynamics may also be adapted without labels, for example through intrinsic plasticity\[[30](https://arxiv.org/html/2608.20998#bib.bib21)\]\. These criteria provide useful information about stability, memory, or dynamical regime, but they are not designed to rank candidates directly according to the labelled downstream validation objective\. When they depend on realized states, they also still require a finite reservoir to be instantiated and run\.

Large\-width reservoir theory offers a third perspective: replacing the random finite system by a deterministic limiting description\. Hermans and Schrauwen formulated infinite\-width ESNs as recurrent kernel machines\[[14](https://arxiv.org/html/2608.20998#bib.bib23)\]\. Couillet et al\. used random\-matrix theory to characterize the training and testing behaviour of large linear ESNs\[[7](https://arxiv.org/html/2608.20998#bib.bib1)\], while Dong et al\. derived nonlinear recurrent kernels and used them directly for prediction\[[8](https://arxiv.org/html/2608.20998#bib.bib14)\]\. Related recurrent\-kernel constructions have since been developed for leaky, sparse, and deep reservoir topologies\[[9](https://arxiv.org/html/2608.20998#bib.bib24)\]\. Gonon, Grigoryeva, and Ortega\[[10](https://arxiv.org/html/2608.20998#bib.bib2)\]instead construct an infinite\-dimensional Volterra reservoir whose kernel defines a universal temporal predictive model\.

These works show that the behaviour of a large reservoir can often be represented by a deterministic kernel\. Their main purpose, however, is to characterize the reservoir or to use the limiting kernel itself as the predictive model\. We address a different question: whether such a kernel can replace finite\-reservoir rollouts during hyperparameter selection\. For each candidateθ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\), we construct the large\-width kernel corresponding to the finite reservoir with those same parameters\. Its performance on labelled pilot data is used to rank the candidates, after which the selectedθ⋆\\theta^\{\\star\}is transferred back to the finite reservoir used for deployment\.

We now introduce the reservoir family for which this correspondence is derived\. Consider the leaky linear recurrence

xt\+1=A​xt\+α​Win​ut,A=\(1−α\)​In\+α​Wr,x\_\{t\+1\}=Ax\_\{t\}\+\\alpha W\_\{\\rm in\}u\_\{t\},\\qquad A=\(1\-\\alpha\)I\_\{n\}\+\\alpha W\_\{r\},\(1\)equipped with coordinate\-wise nonlinear readout features

zt=ψ⁡\(xt\),y^t=Wout⊤​zt\.z\_\{t\}=\\psi\(x\_\{t\}\),\\qquad\\hat\{y\}\_\{t\}=W\_\{\\rm out\}^\{\\top\}z\_\{t\}\.\(2\)Here,ut∈ℝdinu\_\{t\}\\in\\mathbb\{R\}^\{d\_\{\\rm in\}\},xt,zt∈ℝnx\_\{t\},z\_\{t\}\\in\\mathbb\{R\}^\{n\}, andy^t,yt∈ℝdout\\hat\{y\}\_\{t\},y\_\{t\}\\in\\mathbb\{R\}^\{d\_\{\\rm out\}\}denote the input, linear state, nonlinear feature vector, prediction, and target, respectively\. The recurrent and input matricesWr∈ℝn×nW\_\{r\}\\in\\mathbb\{R\}^\{n\\times n\}andWin∈ℝn×dinW\_\{\\rm in\}\\in\\mathbb\{R\}^\{n\\times d\_\{\\rm in\}\}are randomly initialized and remain fixed, while onlyWout∈ℝn×doutW\_\{\\rm out\}\\in\\mathbb\{R\}^\{n\\times d\_\{\\rm out\}\}is fitted\. The leakage rate isα∈\(0,1\]\\alpha\\in\(0,1\], the reservoir is initialized atx0=0x\_\{0\}=0, andψ\\psiis applied coordinate\-wise\. The experiments useψ=tanh\\psi=\\tanh\. The scalings ofWrW\_\{r\}andWinW\_\{\\rm in\}introduce the recurrent gainσr\\sigma\_\{r\}and input scaleσin\\sigma\_\{\\rm in\}and are specified in Section[III](https://arxiv.org/html/2608.20998#S3)\.

Placing the nonlinearity after the recurrence separates temporal memory from nonlinear processing\. Related linear\-transition designs also appear in modern state\-space sequence models such as S4 and LRU\[[13](https://arxiv.org/html/2608.20998#bib.bib4),[28](https://arxiv.org/html/2608.20998#bib.bib9)\]\. Linear recurrent dynamics do not by themselves imply limited temporal modelling capacity: stable linear reservoirs equipped with sufficiently expressive polynomial or neural\-network readouts form universal approximation families for broad classes of fading\-memory filters\[[11](https://arxiv.org/html/2608.20998#bib.bib29),[12](https://arxiv.org/html/2608.20998#bib.bib3)\]\. In such architectures, the linear recurrence represents and propagates the input history, while the readout supplies the nonlinear approximation capacity\.

Our model uses the lightweight feature mapzt=tanh⁡\(xt\)z\_\{t\}=\\tanh\(x\_\{t\}\)followed by a trained linear output layer\. This coordinate\-wise readout is not itself covered verbatim by the broader universality theorem, but it has recently been shown to be competitive with fully nonlinear ESNs on a range of temporal tasks\[[18](https://arxiv.org/html/2608.20998#bib.bib6)\]\. More importantly for the present analysis, placing the nonlinearity outside the recurrent loop keeps the state linear in the propagated input history\.

Indeed, unrolling \([1](https://arxiv.org/html/2608.20998#S2.E1)\) fromx0=0x\_\{0\}=0gives

xt=α​∑k=0t−1Ak​Win​ut−1−k\.x\_\{t\}=\\alpha\\sum\_\{k=0\}^\{t\-1\}A^\{k\}W\_\{\\rm in\}u\_\{t\-1\-k\}\.\(3\)Because the same recurrent matrix is applied repeatedly over time, the covariance between states depends on the mixed normalized moments

1n​Tr⁡\(Ak​\(Aℓ\)⊤\)\.\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(A^\{k\}\(A^\{\\ell\}\)^\{\\top\}\\right\)\.\(4\)These moments form the reservoir’s cross\-lag propagation profile: diagonal terms measure the average squared gain of delayed echoes, whereas off\-diagonal terms measure their overlap\. Unlike the spectral radius, this profile also retains cross\-time coupling and non\-normal amplification\.

The next section derives the large\-width limits of these mixed moments, uses them to construct the nonlinear feature kernel, and applies the kernel to zero\-rollout candidate ranking\.

## IIIFree\-Probability Kernel for Temporal Model Selection

Sections[III\-A](https://arxiv.org/html/2608.20998#S3.SS1)and[III\-B](https://arxiv.org/html/2608.20998#S3.SS2)specify the reservoir family and introduce the empirical\-kernel notation\. The new theoretical results begin in Section[III\-C](https://arxiv.org/html/2608.20998#S3.SS3), where we derive the deterministic large\-width kernel at fixed context length; Sections[III\-D](https://arxiv.org/html/2608.20998#S3.SS4)and[III\-E](https://arxiv.org/html/2608.20998#S3.SS5)extend this construction to the complete washed\-out history and use it for hyperparameter selection\.

Throughout this section,θ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\)denotes a fixed candidate configuration\. To avoid overloading the notation, we suppress the dependence onθ\\thetainAA,WrW\_\{r\},WinW\_\{\\rm in\}, and the reservoir states, but retain it in the kernel and covariance quantities because the selector compares these objects across candidates\. The subscriptnnidentifies a finite\-width random kernel, while kernels withoutnnare deterministic large\-width limits\. The superscript\(L\)\(L\)denotes anLL\-context quantity, and\(∞\)\(\\infty\)denotes complete history\.

### III\-ALeaky linear reservoir and stability

We use the leaky linear reservoir model of Section[II](https://arxiv.org/html/2608.20998#S2), eq\. \([1](https://arxiv.org/html/2608.20998#S2.E1)\) with a scaled real Ginibre recurrent matrixWrW\_\{r\}and an independent Gaussian input matrixWinW\_\{\\rm in\}:

Wr\\displaystyle W\_\{r\}=σrn​Gr,\\displaystyle=\\frac\{\\sigma\_\{r\}\}\{\\sqrt\{n\}\}\\,G\_\{r\},Gr\\displaystyle G\_\{r\}∈ℝn×n,\\displaystyle\\in\\mathbb\{R\}^\{n\\times n\},\(5\)Win\\displaystyle W\_\{\\rm in\}=σindin​Gin,\\displaystyle=\\frac\{\\sigma\_\{\\rm in\}\}\{\\sqrt\{d\_\{\\rm in\}\}\}\\,G\_\{\\rm in\},Gin\\displaystyle G\_\{\\rm in\}∈ℝn×din,\\displaystyle\\in\\mathbb\{R\}^\{n\\times d\_\{\\rm in\}\},\(Gr\)i​j\\displaystyle\(G\_\{r\}\)\_\{ij\}∼i\.i\.d\.​𝒩​\(0,1\),\\displaystyle\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\(0,1\),\(Gin\)i​j\\displaystyle\(G\_\{\\rm in\}\)\_\{ij\}∼i\.i\.d\.​𝒩​\(0,1\),\\displaystyle\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\(0,1\),Gr\\displaystyle G\_\{r\}⟂Gin\.\\displaystyle\\perp G\_\{\\rm in\}\.The1/n1/\\sqrt\{n\}normalization keeps the spectrum and operator norm ofWrW\_\{r\}on anO⁡\(1\)O\(1\)scale as the reservoir width grows\. The normalization ofWinW\_\{\\rm in\}controls the scale of the injected input\. Indeed, for fixed deterministic vectorsu,v∈ℝdinu,v\\in\\mathbb\{R\}^\{d\_\{\\rm in\}\},

𝔼Win​\[Win​u​v⊤​Win⊤\]=σin2din​\(u⊤​v\)​In\.\\mathbb\{E\}\_\{W\_\{\\rm in\}\}\\\!\\left\[W\_\{\\rm in\}uv^\{\\top\}W\_\{\\rm in\}^\{\\top\}\\right\]=\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\(u^\{\\top\}v\)I\_\{n\}\.\(6\)In particular,

VarWin⁡\[\(Win​u\)i\]=σin2din​∥u∥2\.\\operatorname\{Var\}\_\{W\_\{\\rm in\}\}\\\!\\left\[\\left\(W\_\{\\rm in\}u\\right\)\_\{i\}\\right\]=\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\lVert u\\rVert^\{2\}\.\(7\)Our asymptotic analysis takesn→∞n\\to\\inftywithdind\_\{\\rm in\}fixed; no additional high\-dimensional assumption on the input vectors is required\. The independence ofGrG\_\{r\}andGinG\_\{\\rm in\}allows us to condition on the recurrent matrix while retaining an independent Gaussian input map\.

For the linear recurrence, the echo\-state property can be characterized exactly\. Letxtx\_\{t\}andxt′x\_\{t\}^\{\\prime\}be two trajectories driven by the same input sequence but initialized atx0x\_\{0\}andx0′x\_\{0\}^\{\\prime\}, respectively\. Their difference satisfies

xt−xt′=At​\(x0−x0′\)\.x\_\{t\}\-x\_\{t\}^\{\\prime\}=A^\{t\}\(x\_\{0\}\-x\_\{0\}^\{\\prime\}\)\.Hence, the influence of the initial condition vanishes for every pairx0,x0′x\_\{0\},x\_\{0\}^\{\\prime\}if and only ifAt→0A^\{t\}\\to 0ast→∞t\\to\\infty, which is equivalent to

ForWr=\(σr/n\)​GrW\_\{r\}=\(\\sigma\_\{r\}/\\sqrt\{n\}\)G\_\{r\}, the circular law places the large\-width eigenvalue cloud ofWrW\_\{r\}in a disk centered at the origin with radiusσr\\sigma\_\{r\}\[[5](https://arxiv.org/html/2608.20998#bib.bib11)\]\. Therefore, the eigenvalue cloud of

A=\(1−α\)​In\+α​WrA=\(1\-\\alpha\)I\_\{n\}\+\\alpha W\_\{r\}is asymptotically a disk centered at1−α1\-\\alphawith radiusα​σr\\alpha\\sigma\_\{r\}\. Its asymptotic spectral radius is consequently

ρFP​\(α,σr\):=\(1−α\)\+α​σr\.\\rho\_\{\\rm FP\}\(\\alpha,\\sigma\_\{r\}\):=\(1\-\\alpha\)\+\\alpha\\sigma\_\{r\}\.\(9\)Forα\>0\\alpha\>0, the asymptotic linear echo\-state conditionρFP<1\\rho\_\{\\rm FP\}<1is equivalent toσr<1\\sigma\_\{r\}<1\. A finite\-width robustness margin is introduced in Section[III\-E](https://arxiv.org/html/2608.20998#S3.SS5)\.

### III\-BReadout and finite\-reservoir kernel notation

The matricesWrW\_\{r\}andWinW\_\{\\rm in\}in eq\. \([5](https://arxiv.org/html/2608.20998#S3.E5)\) are never trained\. The readout features are

zt=ψ\(xt\),zt,i=ψ\(xt,i\),i=1,…,n,z\_\{t\}=\\psi\(x\_\{t\}\),\\qquad z\_\{t,i\}=\\psi\(x\_\{t,i\}\),\\quad i=1,\\ldots,n,\(10\)whereψ:ℝ→ℝ\\psi:\\mathbb\{R\}\\to\\mathbb\{R\}is applied coordinate\-wise\. The experiments useψ=tanh\\psi=\\tanh\. The theoretical kernel result assumesψ∈Cb1​\(ℝ\)\\psi\\in C\_\{b\}^\{1\}\(\\mathbb\{R\}\), meaning thatψ\\psiis bounded and continuously differentiable with bounded derivative\.

Let𝒯\\mathcal\{T\}and𝒱\\mathcal\{V\}denote the training and validation time indices of the pilot sequence\. Stacking the corresponding untruncated feature vectors gives

ZT=\(zt⊤\)t∈𝒯,ZV=\(zt⊤\)t∈𝒱\.Z\_\{T\}=\\begin\{pmatrix\}z\_\{t\}^\{\\top\}\\end\{pmatrix\}\_\{t\\in\\mathcal\{T\}\},\\qquad Z\_\{V\}=\\begin\{pmatrix\}z\_\{t\}^\{\\top\}\\end\{pmatrix\}\_\{t\\in\\mathcal\{V\}\}\.\(11\)Since the feature mapψ\\psiis fixed within each construction, we suppress it in the notation for the pilot kernel blocks\. The width\-normalized empirical kernel blocks for candidateθ\\thetaare

Kn,θ,T​T\\displaystyle K\_\{n,\\theta,TT\}=1n​ZT​ZT⊤,\\displaystyle=\\frac\{1\}\{n\}Z\_\{T\}Z\_\{T\}^\{\\top\},\(12\)Kn,θ,V​T\\displaystyle K\_\{n,\\theta,VT\}=1n​ZV​ZT⊤\.\\displaystyle=\\frac\{1\}\{n\}Z\_\{V\}Z\_\{T\}^\{\\top\}\.They are indexed by pilot time steps, not by reservoir coordinates\. WritingYTY\_\{T\}for the training targets, kernel ridge regression predicts on the validation block as

Y^V,n​\(θ,λ\)=Kn,θ,V​T​\(Kn,θ,T​T\+λ​I\)−1​YT\.\\widehat\{Y\}\_\{V,n\}\(\\theta,\\lambda\)=K\_\{n,\\theta,VT\}\\left\(K\_\{n,\\theta,TT\}\+\\lambda I\\right\)^\{\-1\}Y\_\{T\}\.\(13\)
The blocks in \([12](https://arxiv.org/html/2608.20998#S3.E12)\) are computed from the untruncated finite\-reservoir trajectory\. Section[III\-C](https://arxiv.org/html/2608.20998#S3.SS3)first derives deterministic large\-width limits for theirLL\-context counterparts\. Section[III\-D](https://arxiv.org/html/2608.20998#S3.SS4)then shows that, in the stable regime, the finite\-context kernels converge to the complete\-history kernel associated with the washed\-out recurrence\.

### III\-CFinite\-context free\-probabilistic kernel construction

Starting fromx0=0x\_\{0\}=0, the linear recurrence can be unrolled exactly:

xt=α​∑k=0t−1Ak​Win​ut−1−k\.x\_\{t\}=\\alpha\\sum\_\{k=0\}^\{t\-1\}A^\{k\}W\_\{\\rm in\}u\_\{t\-1\-k\}\.\(14\)For a fixed context lengthL<∞L<\\infty, the selector retains the most recentL\+1L\+1propagated inputs and uses the truncated state

xt\(L\):=α​∑k=0LAk​Win​ut−1−k\.x\_\{t\}^\{\(L\)\}:=\\alpha\\sum\_\{k=0\}^\{L\}A^\{k\}W\_\{\\rm in\}u\_\{t\-1\-k\}\.\(15\)The corresponding empirical nonlinear kernel entry at widthnnis

Kψ,n,θ\(L\)​\(t,s\):=1n​ψ​\(xt\(L\)\)⊤​ψ​\(xs\(L\)\)\.K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\):=\\frac\{1\}\{n\}\\,\\psi\(x\_\{t\}^\{\(L\)\}\)^\{\\top\}\\psi\(x\_\{s\}^\{\(L\)\}\)\.\(16\)
For the pilot train and validation sets, define the corresponding empiricalLL\-context blocks entrywise by

\[Kn,θ,T​T\(L\)\]t,s\\displaystyle\\bigl\[K\_\{n,\\theta,TT\}^\{\(L\)\}\\bigr\]\_\{t,s\}:=Kψ,n,θ\(L\)​\(t,s\),\\displaystyle:=K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\),t,s∈𝒯,\\displaystyle t,s\\in\\mathcal\{T\},\(17\)\[Kn,θ,V​T\(L\)\]t,s\\displaystyle\\bigl\[K\_\{n,\\theta,VT\}^\{\(L\)\}\\bigr\]\_\{t,s\}:=Kψ,n,θ\(L\)​\(t,s\),\\displaystyle:=K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\),t∈𝒱,s∈𝒯\.\\displaystyle t\\in\\mathcal\{V\},\\ s\\in\\mathcal\{T\}\.
Because the recurrence is linear, the state covariance is determined by mixed normalized traces of propagated recurrent matrices\. For fixedk,ℓ≥0k,\\ell\\geq 0, define

τk,ℓ:=∑j=0min⁡\(k,ℓ\)\(kj\)​\(ℓj\)​\(1−α\)k\+ℓ−2​j​\(α​σr\)2​j\.\\tau\_\{k,\\ell\}:=\\sum\_\{j=0\}^\{\\min\(k,\\ell\)\}\\binom\{k\}\{j\}\\binom\{\\ell\}\{j\}\(1\-\\alpha\)^\{k\+\\ell\-2j\}\(\\alpha\\sigma\_\{r\}\)^\{2j\}\.\(18\)These coefficients quantify how input components injected at two lags remain correlated after propagation through the reservoir\.

###### Proposition 1\(Ginibre cross\-lag propagation coefficients\)\.

Let

An=\(1−α\)​In\+α​Wr\(n\),Wr\(n\)=σrn​Gr\(n\)\.A\_\{n\}=\(1\-\\alpha\)I\_\{n\}\+\\alpha W\_\{r\}^\{\(n\)\},\\qquad W\_\{r\}^\{\(n\)\}=\\frac\{\\sigma\_\{r\}\}\{\\sqrt\{n\}\}G\_\{r\}^\{\(n\)\}\.For every fixedk,ℓ≥0k,\\ell\\geq 0,

1n​Tr⁡\(Ank​\(Anℓ\)⊤\)→L1τk,ℓ\.\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\)\\xrightarrow\{L^\{1\}\}\\tau\_\{k,\\ell\}\.\(19\)

The proof expands the mixed powers by the binomial theorem, uses the circular\-element limit for the normalized∗\*\-moments, and controls fluctuations by Gaussian concentration\. Complete details are given in Appendix[A](https://arxiv.org/html/2608.20998#A1)\.

The trace limit controls only an average over coordinates\. Because the activation is applied coordinate\-wise, we additionally require the diagonal propagation terms to self\-average\.

###### Lemma 1\(Averaged diagonal self\-averaging\)\.

LetAn=\(1−α\)​In\+α​Wr\(n\)A\_\{n\}=\(1\-\\alpha\)I\_\{n\}\+\\alpha W\_\{r\}^\{\(n\)\}\. For every fixedL<∞L<\\infty,

max0≤k,ℓ≤L⁡1n​∑i=1n\|\[Ank​\(Anℓ\)⊤\]i​i−τk,ℓ\|2→ℙ0\.\\displaystyle\\max\_\{0\\leq k,\\ell\\leq L\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\Bigl\|\\bigl\[A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\bigr\]\_\{ii\}\-\\tau\_\{k,\\ell\}\\Bigr\|^\{2\}\\xrightarrow\{\\mathbb\{P\}\}0\.\(20\)

Thus, although individual reservoir units remain random, their diagonal propagation profile becomes deterministic\. This coordinate\-level result is essential becauseψ\\psiacts neuronwise; dependence among different units is handled by the concentration step below\. The complete proof is given in Appendix[A](https://arxiv.org/html/2608.20998#A1)\.

Using the cross\-lag propagation coefficients, define the deterministic linear\-state covariance

Qθ\(L\)​\(t,s\):=α2​σin2din​∑k,ℓ=0Lτk,ℓ​ut−1−k⊤​us−1−ℓ,\\displaystyle Q\_\{\\theta\}^\{\(L\)\}\(t,s\):=\\alpha^\{2\}\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\sum\_\{k,\\ell=0\}^\{L\}\\tau\_\{k,\\ell\}\\,u\_\{t\-1\-k\}^\{\\top\}u\_\{s\-1\-\\ell\},\(21\)whereθ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\), and the associated2×22\\times 2covariance matrix

Σθ\(L\)​\(t,s\):=\(Qθ\(L\)​\(t,t\)Qθ\(L\)​\(t,s\)Qθ\(L\)​\(t,s\)Qθ\(L\)​\(s,s\)\)\.\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\):=\\begin\{pmatrix\}Q\_\{\\theta\}^\{\(L\)\}\(t,t\)&Q\_\{\\theta\}^\{\(L\)\}\(t,s\)\\\\ Q\_\{\\theta\}^\{\(L\)\}\(t,s\)&Q\_\{\\theta\}^\{\(L\)\}\(s,s\)\\end\{pmatrix\}\.\(22\)
The nonlinear feature kernel is obtained by passing the limiting Gaussian state covariance through the coordinate\-wise activation\.

###### Theorem 1\(Deterministic nonlinear reservoir kernel\)\.

Letψ∈Cb1​\(ℝ\)\\psi\\in C\_\{b\}^\{1\}\(\\mathbb\{R\}\)\. For fixedL,t,sL,t,sand a deterministic bounded input sequence\(ur\)r\(u\_\{r\}\)\_\{r\}, asn→∞n\\to\\infty,

Kψ,n,θ\(L\)​\(t,s\)→ℙKψ,θ\(L\)​\(t,s\),K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\xrightarrow\{\\mathbb\{P\}\}K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\),\(23\)where

Kψ,θ\(L\)​\(t,s\)\\displaystyle K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\):=𝔼⁡\[ψ⁡\(gt\)​ψ​\(gs\)\],\\displaystyle:=\\mathbb\{E\}\\\!\\left\[\\psi\(g\_\{t\}\)\\psi\(g\_\{s\}\)\\right\],\(24\)\(gt,gs\)\\displaystyle\(g\_\{t\},g\_\{s\}\)∼𝒩⁡\(0,Σθ\(L\)​\(t,s\)\)\.\\displaystyle\\sim\\mathcal\{N\}\\\!\\left\(0,\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\)\\right\)\.The convergence is in probability with respect to the joint randomness ofWrW\_\{r\}andWinW\_\{\\rm in\}\.

#### RC interpretation and proof idea

Conditional on the recurrent matrix, each reservoir unit receives a Gaussian projection of the same propagated input history\. Lemma[1](https://arxiv.org/html/2608.20998#Thmlemma1)shows that the unitwise covariance profiles self\-average, and concentration over the input weights removes the remaining finite\-width fluctuations\. Hence a wide random reservoir induces a deterministic temporal feature geometry at kernel level\. On any fixed pilot set, the empirical Gram blocks converge in Frobenius norm to their deterministic counterparts\. Complete details are given in Appendix[A](https://arxiv.org/html/2608.20998#A1)\.

The nonlinear expectation in \([24](https://arxiv.org/html/2608.20998#S3.E24)\) can be evaluated for the exact feature mapψ=tanh\\psi=\\tanhby deterministic low\-dimensional Gaussian quadrature\. For each pair\(t,s\)\(t,s\), the covariance matrixΣθ\(L\)​\(t,s\)\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\)defines a bivariate Gaussian law, andKtanh,θ\(L\)​\(t,s\)K\_\{\\tanh,\\theta\}^\{\(L\)\}\(t,s\)is obtained by Gauss–Hermite quadrature of𝔼⁡\[tanh⁡\(gt\)​tanh⁡\(gs\)\]\\mathbb\{E\}\[\\tanh\(g\_\{t\}\)\\tanh\(g\_\{s\}\)\]\. Since Theorem[1](https://arxiv.org/html/2608.20998#Thmtheorem1)holds for anyψ∈Cb1\\psi\\in C\_\{b\}^\{1\}, a closed\-form erf/arcsine surrogate is also available; its uniform entrywise approximation bound and empirical validation are given in Remark[1](https://arxiv.org/html/2608.20998#Thmremark1)and Appendix[F\-A](https://arxiv.org/html/2608.20998#A6.SS1)\. The experimental use of the exact kernel and of this surrogate is specified in Section[IV](https://arxiv.org/html/2608.20998#S4)\.

### III\-DComplete history and other recurrent ensembles

A finite context is the implementable selector; the complete\-history limit connects it to the fading\-memory interpretation of RC\. Whenσr<1\\sigma\_\{r\}<1, the reservoir forgets remote inputs geometrically, so increasingLLrecovers the washed\-out state with controlled error\. For a bounded input history andσr<1\\sigma\_\{r\}<1, define

Qθ\(∞\)​\(t,s\):=α2​σin2din​∑k,ℓ=0∞τk,ℓ​ut−1−k⊤​us−1−ℓ,Q\_\{\\theta\}^\{\(\\infty\)\}\(t,s\):=\\alpha^\{2\}\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\sum\_\{k,\\ell=0\}^\{\\infty\}\\tau\_\{k,\\ell\}\\,u\_\{t\-1\-k\}^\{\\top\}u\_\{s\-1\-\\ell\},\(25\)and letKψ,θ\(∞\)​\(t,s\)K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)be the Gaussian nonlinear kernel obtained from this covariance as in \([24](https://arxiv.org/html/2608.20998#S3.E24)\)\. The covariance series converges absolutely, and the finite\-context kernels converge geometrically to the complete\-history kernel:

Kψ,θ\(L\)​\(t,s\)⟶Kψ,θ\(∞\)​\(t,s\),L→∞\.K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\longrightarrow K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\),\\quad L\\to\\infty\.\(26\)Because the admissible candidate grid is finite, this convergence is uniform over the candidates used by the selector\.

For a complete bounded input history, let

xt\(∞\):=α​∑k=0∞Ak​Win​ut−1−kx\_\{t\}^\{\(\\infty\)\}:=\\alpha\\sum\_\{k=0\}^\{\\infty\}A^\{k\}W\_\{\\rm in\}u\_\{t\-1\-k\}\(27\)denote the complete washed\-out state\. For real Ginibre recurrence withσr<1\\sigma\_\{r\}<1, this series is well defined almost surely for all sufficiently largenn\. Strong convergence controls the powers of the non\-normal transition matrix and hence the finite\-width truncation tail\. Consequently, on every fixed finite pilot set,

Kψ,n,θ\(∞\)​\(t,s\):=1n​ψ​\(xt\(∞\)\)⊤​ψ​\(xs\(∞\)\)→n→∞ℙKψ,θ\(∞\)​\(t,s\)\.K\_\{\\psi,n,\\theta\}^\{\(\\infty\)\}\(t,s\):=\\frac\{1\}\{n\}\\psi\\\!\(x\_\{t\}^\{\(\\infty\)\}\)^\{\\top\}\\psi\\\!\(x\_\{s\}^\{\(\\infty\)\}\)\\xrightarrow\[n\\to\\infty\]\{\\mathbb\{P\}\}K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)\.\(28\)The quantitative truncation bounds and the full Ginibre argument are given in Appendix[B](https://arxiv.org/html/2608.20998#A2)\.

The same construction applies to recurrent ensembles with deterministic mixed propagation moments and diagonal self\-averaging\. The Haar\-orthogonal argument and explicit kernels for cyclic, circulant, skew\-symmetric, and complex\-valued recurrences are given in Appendices[C](https://arxiv.org/html/2608.20998#A3)–[D](https://arxiv.org/html/2608.20998#A4)\. The experiments in this paper use real Ginibre recurrence throughout\.

### III\-EZero\-rollout selection on temporal pilot data

The kernel now acts as a virtual reservoir during model selection\. For each operating point, it captures the large\-width temporal feature geometry of the corresponding finite reservoir without constructing recurrent weights or state trajectories\.

For each candidateθ=\(σr,σin,α\)\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\), the construction above replaces the empirical feature kernel of a large random reservoir by a deterministic kernel computed directly from a short labelled pilot sequence\. LetKsel,θ\(L\)K\_\{\\mathrm\{sel\},\\theta\}^\{\(L\)\}denote the deterministic selection kernel used in a given experiment\. In the experiments below, this is either the exacttanh\\tanhkernel evaluated by quadrature or the closed\-form erf surrogate described above; the precise choice is stated in Section[IV](https://arxiv.org/html/2608.20998#S4)\. For pilot train and validation index sets𝒯\\mathcal\{T\}and𝒱\\mathcal\{V\}, define the deterministic kernel blocks entrywise by

\[Kθ,T​T\(L\)\]t,s\\displaystyle\\bigl\[K\_\{\\theta,TT\}^\{\(L\)\}\\bigr\]\_\{t,s\}:=Ksel,θ\(L\)​\(t,s\),\\displaystyle:=K\_\{\{\\rm sel\},\\theta\}^\{\(L\)\}\(t,s\),t,s∈𝒯,\\displaystyle t,s\\in\\mathcal\{T\},\(29\)\[Kθ,V​T\(L\)\]t,s\\displaystyle\\bigl\[K\_\{\\theta,VT\}^\{\(L\)\}\\bigr\]\_\{t,s\}:=Ksel,θ\(L\)​\(t,s\),\\displaystyle:=K\_\{\{\\rm sel\},\\theta\}^\{\(L\)\}\(t,s\),t∈𝒱,s∈𝒯,\\displaystyle t\\in\\mathcal\{V\},\\ s\\in\\mathcal\{T\},where each entry is computed fromQθ\(L\)Q\_\{\\theta\}^\{\(L\)\}by either exact tanh quadrature or the closed\-form erf surrogate, as specified above\. For a ridge parameterλ\\lambda, the corresponding validation prediction is

Y^V​\(θ,λ\)=Kθ,V​T\(L\)​\(Kθ,T​T\(L\)\+λ​I\)−1​YT\.\\widehat\{Y\}\_\{V\}\(\\theta,\\lambda\)=K\_\{\\theta,VT\}^\{\(L\)\}\\left\(K\_\{\\theta,TT\}^\{\(L\)\}\+\\lambda I\\right\)^\{\-1\}Y\_\{T\}\.\(30\)
#### FP selector

Input:Pilot sequence\(ut,yt\)\(u\_\{t\},y\_\{t\}\), split into train/validation; context lengthLL; candidate gridΘ\\Theta; ridge gridΛ\\Lambda

Output:Selected hyperparameters

θ⋆\\theta^\{\\star\}and ranking regularizer

λsel⋆\\lambda^\{\\star\}\_\{\\rm sel\}
for*eachθ=\(σr,σin,α\)∈Θ\\theta=\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\)\\in\\Thetasatisfying the stability guard*do1

Compute

τk,ℓ​\(θ\)\\tau\_\{k,\\ell\}\(\\theta\)via \([18](https://arxiv.org/html/2608.20998#S3.E18)\) for

0≤k,ℓ≤L0\\leq k,\\ell\\leq L; Build

Qθ\(L\)Q\_\{\\theta\}^\{\(L\)\}via \([21](https://arxiv.org/html/2608.20998#S3.E21)\); compute

Ksel,θ\(L\)K\_\{\{\\rm sel\},\\theta\}^\{\(L\)\}; form

Kθ,T​T\(L\)K\_\{\\theta,TT\}^\{\(L\)\}and

Kθ,V​T\(L\)K\_\{\\theta,VT\}^\{\(L\)\}via \([29](https://arxiv.org/html/2608.20998#S3.E29)\);for*eachλ∈Λ\\lambda\\in\\Lambda*do2

Compute

Y^V​\(θ,λ\)\\widehat\{Y\}\_\{V\}\(\\theta,\\lambda\)via \([30](https://arxiv.org/html/2608.20998#S3.E30)\); compute

NMSEval\(L\)⁡\(θ,λ\)\\operatorname\{NMSE\}\_\{\\rm val\}^\{\(L\)\}\(\\theta,\\lambda\); end for3

end for4

return

\(θ⋆,λsel⋆\)=arg⁡minθ,λ​NMSEval\(L\)⁡\(θ,λ\)\(\\theta^\{\\star\},\\lambda^\{\\star\}\_\{\\rm sel\}\)=\\arg\\min\_\{\\theta,\\lambda\}\\operatorname\{NMSE\}\_\{\\rm val\}^\{\(L\)\}\(\\theta,\\lambda\);

Algorithm 1Zero\-rollout FP hyperparameter selectorAlgorithm[1](https://arxiv.org/html/2608.20998#algorithm1)presents the holdout\-ridge version used for the synthetic benchmarks\. For ETT and Telco, the inner selection ofλ\\lambdais instead performed by generalized cross\-validation on the training block, after which the candidate is scored on the chronological validation block\. The reservoir\-candidate ranking procedure is otherwise unchanged\.

All matrices in Algorithm[1](https://arxiv.org/html/2608.20998#algorithm1)are indexed by pilot time steps rather than reservoir coordinates: non×nn\\times nrecurrent matrix is formed and no finite reservoir is rolled out\. HereNMSEval\(L\)⁡\(θ,λ\)=∑t∈𝒱∥y^t−yt∥2/∑t∈𝒱∥yt−y¯𝒱∥2\\operatorname\{NMSE\}\_\{\\rm val\}^\{\(L\)\}\(\\theta,\\lambda\)=\\sum\_\{t\\in\\mathcal\{V\}\}\\lVert\\widehat\{y\}\_\{t\}\-y\_\{t\}\\rVert^\{2\}/\\sum\_\{t\\in\\mathcal\{V\}\}\\lVert y\_\{t\}\-\\bar\{y\}\_\{\\mathcal\{V\}\}\\rVert^\{2\}is the normalized mean\-squared error on the pilot validation block, withy¯𝒱\\bar\{y\}\_\{\\mathcal\{V\}\}the validation target mean\. After precomputing input inner products, each candidate kernel requiresO⁡\(Tpilot2​L2\)O\(T\_\{\\rm pilot\}^\{2\}L^\{2\}\)lag\-pair operations andO⁡\(Tpilot2\)O\(T\_\{\\rm pilot\}^\{2\}\)memory, followed by kernel ridge regression on theTtrain×TtrainT\_\{\\rm train\}\\times T\_\{\\rm train\}block\. Crucially, none of these costs depends on the deployment widthnn\. The regularizerλsel⋆\\lambda^\{\\star\}\_\{\\rm sel\}is used only for ranking; at deployment,λ\\lambdais re\-selected on the deployment training block over the same canonical grid, using the kernel\-scale feature convention of Section[IV](https://arxiv.org/html/2608.20998#S4), so that the selection\-stage and deployment\-stage regularizers are directly comparable\.

Theorem[1](https://arxiv.org/html/2608.20998#Thmtheorem1)concerns kernel entries; the selector, however, returns anarg⁡min\\arg\\minover a finite grid\. The following corollary records the corresponding decision\-level result\.

###### Corollary 1\(Consistency of FP selection\)\.

Assume that the deterministic kernel and the finite reservoir use the same feature map\. Fix finite candidate and ridge grids with strictly positive ridge parameters\. For any fixed context lengthLL, if the deterministic validation score has a unique minimizer, then finite\-width reservoir selection returns the same minimizer with probability tending to one asn→∞n\\to\\infty\.

In the stable regime, a unique complete\-history optimum is preserved by all sufficiently large finite contexts, and selection based on the complete finite\-reservoir trajectory converges to this optimum asn→∞n\\to\\infty\.

The result follows from kernel convergence, continuity of kernel ridge regression, and stability of a unique minimizer on a finite grid\. The complete statement and proof are given in Appendices[A](https://arxiv.org/html/2608.20998#A1)–[B](https://arxiv.org/html/2608.20998#A2)\.

The corollary concerns agreement on the pilot validation objective; it does not assert that the selected configuration minimizes deployment test error\. When several candidates are nearly tied, agreement may require a larger context length or reservoir width\. In the synthetic experiments, the selector uses the erf surrogate while the finite reservoir usestanh\\tanh, so the corollary does not apply verbatim\. The selection\-level impact of this approximation is evaluated separately in Remark[1](https://arxiv.org/html/2608.20998#Thmremark1)and Appendix[F\-A](https://arxiv.org/html/2608.20998#A6.SS1)\.

#### Stability guard

Equation \([9](https://arxiv.org/html/2608.20998#S3.E9)\) gives the asymptotic spectral radius of the Ginibre transition matrix\. Since the large\-width selector cannot detect realization\-specific instabilities of finite Ginibre matrices close to the stability boundary, we apply the fixed conservative guard

ρFP​\(α,σr\)≤1−δ,δ=0\.05\.\\rho\_\{\\rm FP\}\(\\alpha,\\sigma\_\{r\}\)\\leq 1\-\\delta,\\qquad\\delta=0\.05\.\(31\)The margin is a practical finite\-width safeguard rather than a claim of optimality, and it is kept fixed across deployment widths\. We apply the guard to every candidate grid before selection\. For recurrent matrices whose finite\-width spectral boundary is controlled exactly, such as scaled Haar\-orthogonal or cyclic\-shift recurrence, the additional margin is not required\.

## IVExperiments

### IV\-ASetup

We evaluate the selectors in three complementary regimes\. First, a suite of ten synthetic sequence benchmarks covers nonlinear temporal processing, explicit memory, delayed nonlinear transformations, and forecasting of chaotic dynamical systems\. Second, four public ETT\-small datasets \(ETTh1, ETTh2, ETTm1, and ETTm2\) test open\-data real\-world forecasting at hourly and 15\-minute sampling rates\. Third, a proprietary cellular\-traffic dataset tests operational multivariate forecasting on ten randomly selected network cells\. Thus, the evaluation spans controlled mechanistic tasks, reproducible public forecasting data, and a cell\-replicated operational application\.

Synthetic benchmarks:The synthetic suite contains NARMA\-kkfork∈\{10,20,30,50\}k\\in\\\{10,20,30,50\\\}\[[1](https://arxiv.org/html/2608.20998#bib.bib33)\], linear memory capacity \(MC\)\[[16](https://arxiv.org/html/2608.20998#bib.bib16)\], the nonlinear delayed\-input task of Inubushi and Yoshimura\[[15](https://arxiv.org/html/2608.20998#bib.bib34)\], and four forecasting tasks generated from chaotic dynamical systems: Lorenz63\[[19](https://arxiv.org/html/2608.20998#bib.bib35)\], Mackey–Glass with two delay/horizon settings\[[25](https://arxiv.org/html/2608.20998#bib.bib36)\], and 5D Lorenz96\[[20](https://arxiv.org/html/2608.20998#bib.bib37)\]\. Their deployment protocols and task parameters are summarised in Table[I](https://arxiv.org/html/2608.20998#S4.T1)\.

TABLE I:Synthetic task protocols\. Deployment train/test lengths are counted after washout\. Forecast horizons are given in task time steps\.#### Real\-data benchmarks

We use the public ETT\-small benchmark introduced in\[[37](https://arxiv.org/html/2608.20998#bib.bib32)\], comprising ETTh1, ETTh2, ETTm1, and ETTm2\. For ETT, the input consists of oil temperature \(OT\) and calendar covariates, and the target is future OT at physical horizons of 1, 6, and 12 hours\. ETTh1/ETTh2 are sampled hourly, whereas ETTm1/ETTm2 are sampled every 15 minutes\. The proprietary Telco benchmark uses hourly voice\-traffic key performance indicators \(KPIs\) together with calendar covariates, and jointly predicts both KPIs over horizons1:121\{:\}12for each of ten cells\. All real\-data splits are chronological\. Section[IV\-C](https://arxiv.org/html/2608.20998#S4.SS3)gives the context lengths, split sizes, anchor strides, and rollout details\.

#### Zero\-rollout and exhaustive selectors

FPis the proposed zero\-rollout selector \(Algorithm[1](https://arxiv.org/html/2608.20998#algorithm1)\)\. It ranks candidate operating points by deterministic kernel NMSE on labelled task\-specific pilot data, without instantiating a finite reservoir\. In the synthetic suite, three independently generated pilot sequences are used and each candidate is assigned its largest validation NMSE across the three pilots; this conservative aggregation penalizes pilot\-sensitive configurations\. The three pilot sequences play, for the deterministic FP selector, the role that reservoir seeds play for the simulation\-based selectors: FP is replicated over independently sampled task pilots, whereas the empirical selectors are replicated over finite\-reservoir realizations\.Memorynis a task\-agnostic empirical proxy\. At each grid point and selection widthnn, a ridge readout reconstructsD=2​nD=2ndelayed samples of a generic scalar inputut∼U⁡\(−1,1\)u\_\{t\}\\sim U\(\-1,1\)from the reservoir features\. Its score follows Jaeger’s memory\-capacity criterion\[[16](https://arxiv.org/html/2608.20998#bib.bib16)\]:

MemScore⁡\(θ\)=1D​∑d=1Dmax⁡\(0,1−∑t\(ut−d−u^t−d\)2∑t\(ut−d−u¯d\)2\),\\mathrm\{MemScore\}\(\\theta\)=\\frac\{1\}\{D\}\\sum\_\{d=1\}^\{D\}\\max\\\!\\left\(0,\\;1\-\\frac\{\\sum\_\{t\}\(u\_\{t\-d\}\-\\hat\{u\}\_\{t\-d\}\)^\{2\}\}\{\\sum\_\{t\}\(u\_\{t\-d\}\-\\bar\{u\}\_\{d\}\)^\{2\}\}\\right\),\(32\)whereu¯d\\bar\{u\}\_\{d\}is the sample mean of the lag\-ddtarget\. Scores are averaged over three reservoir seeds\.Directnevaluates the downstream validation task at widthnnand ranks candidates by their mean score over three reservoir seeds\.ESN Directnuses the same task\-direct protocol with the standard nonlinear ESN recurrencext\+1=\(1−α\)​xt\+α​tanh⁡\(Wr​xt\+Win​ut\)x\_\{t\+1\}=\(1\-\\alpha\)x\_\{t\}\+\\alpha\\tanh\(W\_\{r\}x\_\{t\}\+W\_\{\\rm in\}u\_\{t\}\)\.

#### Budgeted selectors

We additionally compare three selectors under a matched retained candidate budgetKK\.FP\-Kfirst ranks the admissible grid with FP at zero rollout cost and then applies Direct500only to the topKKcandidates; we write FP\-55, FP\-1010, etc\. for specific budgets\.Random\-KsamplesKKadmissible candidates uniformly without replacement and returns the candidate with the best Direct500validation score\.TPE\-K, a sequential Bayesian\-optimization baseline, uses the outcomes of previous trials to direct later evaluations toward promising regions of the search space\. Specifically, it applies the tree\-structured Parzen estimator \(TPE\)\[[2](https://arxiv.org/html/2608.20998#bib.bib19)\]to the discrete\(σr,σin,α\)\(\\sigma\_\{r\},\\sigma\_\{\\rm in\},\\alpha\)search axes forKKtask\-direct trials and returns the best observed candidate\.

Each task\-direct trial uses three selection seeds, so one run of any budgeted selector spends3​K3Kfinite\-reservoir rollouts; FP’s deterministic pre\-ranking adds none\. Random\-K and TPE\-K are repeated for 20 independent selection draws on the synthetic suite and 10 draws on ETT and Telco, reflecting the higher cost of the forecasting evaluations\. Each individual run still receives the same matched budget ofKKcandidates; repetitions are used only to estimate stochastic\-selector variability\.

#### Regularization protocol

All ridge regressions in this paper \(FP kernel selection, empirical selection, and deployment\) use a single canonical regularization gridλ∈\{10−12,10−11,…,102\}\\lambda\\in\\\{10^\{\-12\},10^\{\-11\},\\ldots,10^\{2\}\\\}expressed on the kernel scale\. Finite\-reservoir readouts are fitted on width\-normalized featureszt/nz\_\{t\}/\\sqrt\{n\}, so that the empirical Gram matrix1n​Z​Z⊤\\tfrac\{1\}\{n\}ZZ^\{\\top\}and the deterministic FP kernel live on the same scale and a givenλ\\lambdacorresponds to the same effective regularization at every stage and every reservoir width\. The selection rule forλ\\lambdadiffers by regime\. On the synthetic benchmarks, where the data\-generating processes are stationary and fresh i\.i\.d\.\-driven sequences can be generated,λ\\lambdais selected by the holdout\-ridge protocol on this grid\. On the real\-data ETT and Telco benchmarks,λ\\lambdais selected by generalized cross\-validation \(RidgeCV\) on the corresponding training block over the same grid, both inside finite\-reservoir selection rollouts and at deployment\. This uses the full training block for regularization selection, avoiding an additional dependence on a single chronological validation window in the nonstationary real\-data setting\.

#### FP kernel implementation

All finite\-reservoir deployments use coordinate\-wisetanh\\tanhfeatures\. For FP selection, we use the exacttanh\\tanhkernel, evaluated by Gauss–Hermite quadrature, on all real\-data benchmarks \(ETT and Telco\)\. For the synthetic benchmark suite and the synthetic timing sweeps, we use the closed\-form erf surrogate from Remark[1](https://arxiv.org/html/2608.20998#Thmremark1)\. This surrogate is used only as a computational accelerator in the controlled synthetic setting; Appendix[F\-A](https://arxiv.org/html/2608.20998#A6.SS1)verifies that matched erf\-surrogate and exact\-tanh\\tanhselections have negligible selection\-level impact on the checked synthetic settings\.

#### Synthetic benchmark protocol

The synthetic Cartesian grid contains6 4806\\,480raw points\. Applying the fixed stability guard \([31](https://arxiv.org/html/2608.20998#S3.E31)\) leaves5 2205\\,220admissible candidates\. Full\-grid empirical selection atnselect=500n\_\{\\rm select\}=500therefore uses5220×3=15 6605220\\times 3=15\\,660finite\-reservoir rollouts per task, or156 600156\\,600across the ten\-task suite\. FP ranks the same admissible grid with zero finite\-reservoir rollouts\. Its pilot sequences are generated after the task\-specific warmup and have lengthTpilot=500T\_\{\\rm pilot\}=500, split into 333 training and 167 validation time steps; the deterministic selection kernel uses contextL=50L=50\. The washout lengths in Table[I](https://arxiv.org/html/2608.20998#S4.T1)apply only to finite\-reservoir deployment, since FP selection does not instantiate or roll out a reservoir\.

After selection,θ⋆\\theta^\{\\star\}is fixed and evaluated with new reservoir seeds\. Intermediate\-width trends use 3 deployment seeds atn∈\{1 000,3 000,5 000,10 000\}n\\in\\\{1\\,000,3\\,000,5\\,000,10\\,000\\\}; then=20 000n=20\\,000synthetic comparison uses 10 deployment seeds\. In the posthoc synthetic budget sweep, selected configurations use 3 deployment seeds\.

#### Real\-data benchmark protocol

All forecasting experiments use physical horizons of 1, 6, and 12 hours\. A*forecast anchor*is a time index at which the preceding input history is used to predict the future target vector, and hence defines one forecasting example\. Because nearby anchors in the long ETT series produce strongly overlapping examples, we retain everyssth valid anchor, using the same anchor indices for all selectors\.

For each anchor, the finite\-reservoir state is obtained by a local rollout over the input history represented by the FP kernel\. In the FP construction,LLis the maximum retained lag, so lags0,…,L0,\\ldots,Lare included and each state usesL\+1L\+1input samples\. Thus,L=96L=96corresponds to 97 hourly samples spanning a maximum lag of 96 hours, whereasL=384L=384corresponds to 385 quarter\-hourly samples over the same maximum\-lag span\. The FP kernel is evaluated only between the selected forecast anchors\.

For ETT, the input contains oil temperature \(OT\) and calendar covariates, and the target is future OT\. ETTh is sampled hourly and usesL=96L=96, horizonsh∈\{1,6,12\}h\\in\\\{1,6,12\\\}, and anchor stride 43\. ETTm is sampled every 15 minutes and usesL=384L=384, horizonsh∈\{4,24,48\}h\\in\\\{4,24,48\\\}, and anchor stride 173\. Hence both variants use a 96\-hour maximum retained lag and the same physical forecast horizons\. ETTh uses8 640/2 880/2 8808\\,640/2\\,880/2\\,880train/validation/test rows, while ETTm uses34 560/11 520/11 52034\\,560/11\\,520/11\\,520rows\.

For the cellular\-traffic task, the input combines data and voice KPIs with sine–cosine calendar covariates, and the target is the joint forecast of both KPIs over horizons1:121\{:\}12\. Each cell uses 841 training, 360 validation, and 932 test rows, withL=96L=96\. Because the Telco series are shorter than ETT, every valid anchor is retained\.

In all forecasting experiments, one readout jointly predicts all horizons from a single reservoir state\. Accordingly, one configurationθ⋆\\theta^\{\\star\}is selected per ETT dataset or per Telco cell using the validation score aggregated over all horizons\. Every selector, including full\-grid Direct500, optimizes this same aggregate objective\. Selecting horizon\-specific configurations is also possible by restricting the targets to one horizon, but is not considered here\.

The ETT and Telco Cartesian grids each contain 245 raw candidates, of which 154 satisfy the stability guard\. All selectors operate on this same admissible set, so exhaustive empirical selection requires154×3=462154\\times 3=462finite\-reservoir rollouts per ETT dataset and per Telco cell\.

For FP selection, the chronological training and validation blocks define the*pilot split*: the deterministic kernel model is fitted on the training anchors and evaluated on the validation anchors\. Since each real\-data series is fixed, a single pilot split is used rather than averaging over independently generated pilot sequences\.

All real\-data models are deployed at widthn=20 000n=20\\,000with three fresh reservoir seeds\. For ETT, reported deviations are computed over deployment seeds for deterministic FP\-K and over independent selection draws for Random\-K and TPE\-K, after averaging deployment seeds within each draw\. For Telco, results are first averaged over selection draws within each cell and then summarized by the mean and standard deviation across the 10 cells\.

#### Metrics and statistical comparisons

Performance is reported as1−NRMSE1\-\\operatorname\{NRMSE\}; higher is better\. The source of averaging and variability \(tasks, deployment seeds, selection draws, or cells\) is specified in the corresponding experimental protocol\. Paired selector comparisons use Wilcoxon signed\-rank tests over tasks or cells, with Holm correction across the reported pairwise comparisons\. Stochastic budget repetitions are used to characterise selection variability, not as additional independent tasks\.

### IV\-BSynthetic benchmark results

The synthetic suite evaluates three complementary uses of the FP selector: zero\-rollout selection when no simulation budget is available, width\-portable deployment as the reservoir grows, and FP\-based pre\-screening when a small rollout budget can be spent\. The paragraphs below address these in turn\.

#### Zero\-rollout selection

Direct500attains the highest mean deployment score across the ten\-task suite, reaching0\.7740\.774atn=20 000n=20\\,000\(Table[II](https://arxiv.org/html/2608.20998#S4.T2)\)\. This is expected: it directly evaluates finite\-reservoir downstream validation performance during selection\. FP reaches0\.7720\.772while using zero finite\-reservoir search rollouts\. The task\-level comparison in Figure[1](https://arxiv.org/html/2608.20998#S4.F1)shows that FP is best or tied\-best on MC, NARMA30, NARMA50, and Lorenz96, and remains close to the best selector on several other tasks\. The largest gaps occur on Lorenz63 and Inubushi, where finite\-reservoir or ESN direct selection offers a clearer advantage\. FP also exceeds ESN Direct500on average atn=20 000n=20\\,000\(0\.7720\.772versus0\.7340\.734\), showing that the analytically tractable linear\-recurrence setting remains competitive in this suite\. Full task\-level scores are in Appendix[E](https://arxiv.org/html/2608.20998#A5)\.

TABLE II:Average deployment score across ten synthetic benchmarks under the holdout\-ridge protocol\. Scores are1−NRMSE1\-\\operatorname\{NRMSE\}; higher is better\. Mean±\\pmstandard deviation is across tasks\. Best and second\-best results are bold and underlined, respectively\.Fig\. 1:Deployment score atn=20 000n=20\\,000across the ten\-task synthetic suite under the holdout\-ridge protocol\. Scores are1−NRMSE1\-\\operatorname\{NRMSE\}; higher is better\. Error bars show the standard deviation over deployment seeds\.
#### Average\-rank comparison

Figure[2](https://arxiv.org/html/2608.20998#S4.F2)gives the rank\-based comparison atn=20 000n=20\\,000\. Direct500has the best mean rank \(1\.651\.65\), followed by FP \(2\.052\.05\), ESN Direct500\(2\.902\.90\), and Memory500\(3\.403\.40\)\. After Holm correction atα=0\.05\\alpha=0\.05, FP and Direct500are significantly better than Memory500, while the pairwise differences among FP, Direct500, and ESN Direct500are not significant\. The supported conclusion is that FP remains competitive with the task\-direct simulation\-based selectors while eliminating finite\-reservoir search rollouts\.

Fig\. 2:Average ranks across the ten synthetic tasks atn=20 000n=20\\,000\. Lower is better\. Horizontal cliques indicate no significant pairwise difference under Wilcoxon signed\-rank tests with Holm correction \(α=0\.05\\alpha=0\.05\)\.
#### Width portability and selection cost

The predictive comparison above is obtained under sharply different selection costs\. Figure[3](https://arxiv.org/html/2608.20998#S4.F3)reports measured full\-grid selection time at several selection widths\. FP does not takenselectn\_\{\\rm select\}as an algorithmic input: its cost depends on the pilot length, context length, candidate grid, and ridge grid, but not on reservoir width\. We therefore summarize repeated FP timings by a horizontal mean line; the variation among the individual measurements reflects run\-to\-run timing variability rather than width dependence\. The empirical selectors, by contrast, instantiate, roll out, and evaluate finite reservoirs at every candidate, so their measured cost increases withnselectn\_\{\\rm select\}; the connecting lines are only guides to the eye\.

At deployment, the same FP operating point is reused without rerunning the selector\. Its observed mean score increases from0\.6920\.692atn=1000n=1000to0\.7720\.772atn=20 000n=20\\,000, while the gap to Direct500narrows from0\.0280\.028to0\.0020\.002\(Table[II](https://arxiv.org/html/2608.20998#S4.T2)\)\. This narrowing is consistent with the large\-width construction: as deployment width increases, the empirical feature geometry approaches the deterministic kernel targeted by FP\. The result does not imply monotone improvement at every finite width\. The task\-level trends in Appendix[E](https://arxiv.org/html/2608.20998#A5)are consistent with this interpretation: FP is stable or improving over the reported widths on every task, whereas selectors tuned atnselect=500n\_\{\\rm select\}=500can exhibit non\-monotone trends on individual tasks\.

![Refer to caption](https://arxiv.org/html/2608.20998v1/plots/fig_compute_time_vs_width_3seed_new.png)Fig\. 3:Measured aggregate selection CPU time over the ten synthetic tasks as a function of selection widthnselectn\_\{\\rm select\}\. All finite\-reservoir selector points are measured; connecting lines are guides to the eye and do not represent fitted complexity laws\. FP does not takenselectn\_\{\\rm select\}as an input, so its repeated measured runtimes are summarized by a horizontal mean line, with individual markers and a shaded±1\\pm 1standard\-deviation band showing run\-to\-run timing variability\. FP uses the closed\-form erf surrogate employed for synthetic selection; see Sec\.[III\-C](https://arxiv.org/html/2608.20998#S3.SS3)\. Lower is better\.
#### Budgeted screening

The zero\-rollout selector can also serve as a pre\-screen when a small simulation budget is available\. FP first ranks the full grid without finite\-reservoir rollouts; Direct500is then evaluated only on the topKKcandidates\. Table[III](https://arxiv.org/html/2608.20998#S4.T3)compares this FP\-K strategy with Random\-K and TPE\-K, using the same retained candidate budget but reporting Random/TPE over 20 draws to estimate baseline variability\. FP\-K gives the strongest mean performance at every reported budget\. The advantage is largest at smallKKand shrinks asKKgrows, indicating convergence of the budgeted selectors\. After Holm correction across the eight FP\-vs\-baseline budget comparisons, the paired differences are directional rather than significant\. The main advantage of FP\-K is therefore budget efficiency and determinism: atK=250K=250, FP\-K matches the full\-grid Direct500reference within sampling variation \(0\.7770\.777versus0\.7740\.774\) using750750finite\-reservoir selection rollouts per task \(4\.8%4\.8\\%of the full grid\)\. Random\-K and TPE\-K use the same per\-run candidate budget, but their reported means are estimated over 20 stochastic draws and reach0\.7500\.750and0\.7690\.769, respectively\. The screened protocol therefore provides an intermediate operating mode between zero\-rollout selection and exhaustive task\-direct search, and motivates the low\-budget Telco experiments \(FP\-55, FP\-1010\) below\.

TABLE III:Budgeted synthetic comparison atn=20 000n=20\\,000\. Rows indicate the retained candidate budgetKK; scores are deployment1−NRMSE1\-\\operatorname\{NRMSE\}\(higher is better\), averaged across the ten tasks\. Percentages are relative to exhaustive Direct500 selection\.
#### Selection\-length sensitivity

An auxiliary coarse\-grid sweep over the selection\-sequence length \(Appendix[F](https://arxiv.org/html/2608.20998#A6)\) shows that FP is close to its highest peak already atTselect=250T\_\{\\rm select\}=250and reaches0\.7710\.771byTselect=500T\_\{\\rm select\}=500, whereas direct selection continues to benefit from longer sequences as additional task\-specific validation data provide a more reliable ranking of the candidate reservoirs\. Memory500remains constant because its task\-agnostic memory proxy selects the same candidate at every tested selection length\. FP is therefore efficient not only in finite\-reservoir rollouts but also in the length of the labelled pilot sequence required for selection\.

### IV\-CReal\-data benchmark results

We complement the synthetic suite with real\-data forecasting experiments derived from five datasets: the four public ETT\-small datasets \(ETTh1, ETTh2, ETTm1, and ETTm2\) and an operational cellular\-traffic dataset for ten randomly selected network cells\. The corresponding protocol details are given in the real\-data benchmark protocol of Section[IV](https://arxiv.org/html/2608.20998#S4)\. The ETT datasets provide reproducible public benchmarks, while the Telco task tests the selector in an operational multivariate setting where finite\-reservoir validation rollouts are costly\. We therefore focus on two questions: whether zero\-rollout FP selection remains competitive with exhaustive task\-direct search, and whether FP\-based screening allocates small rollout budgets more effectively than Random\-K and TPE\-K\.

#### Zero\-rollout selection

Table[IV](https://arxiv.org/html/2608.20998#S4.T4)reports dataset\-average scores for all selectors; horizon\-level results are in Appendix[G](https://arxiv.org/html/2608.20998#A7)\. On ETTm1, zero\-rollout FP is the strongest of the linear\-recurrence selectors \(0\.5050\.505versus0\.4990\.499for full\-grid Direct500\), while ESN Direct500is slightly higher \(0\.5080\.508\)\. At the longest ETTm horizons, the task\-agnostic Memory500proxy is the strongest selector \(Appendix[G](https://arxiv.org/html/2608.20998#A7)\)\. This reflects a short\-versus\-long\-horizon trade\-off under aggregate selection\. On ETTm2, for example, Memory500improves the 6\- and 12\-hour scores to0\.4740\.474and0\.5010\.501, compared with0\.4560\.456and0\.4420\.442for FP, but reduces the 1\-hour score from0\.8620\.862to0\.7620\.762\. On the ETT dataset averages, Memory500remains below the task\-informed selectors throughout \(Table[IV](https://arxiv.org/html/2608.20998#S4.T4)\)\. These results also indicate that the ETT candidate landscape is relatively flat: several operating points have nearly identical deployment scores, so stochastic search can catch up as the budget grows\.

Telco is different\. On the three representative horizons shown in Table[IV](https://arxiv.org/html/2608.20998#S4.T4), zero\-rollout FP trails full\-grid Direct500\(0\.5720\.572versus0\.6350\.635\), and the task\-agnostic Memory500proxy is also higher \(0\.6190\.619\)\. This shows that zero\-rollout ranking alone is not uniformly sufficient on real data\. However, the budgeted aggregate comparison in Table[V](https://arxiv.org/html/2608.20998#S4.T5)shows that only a very small task\-direct budget is needed to close the gap: FP\-55reaches0\.622±0\.1120\.622\\pm 0\.112against the full\-grid Direct500reference0\.621±0\.1130\.621\\pm 0\.113, using 15 rather than 462 selection rollouts per cell\.

TABLE IV:Three\-horizon real\-data forecasting summary atn=20 000n=20\\,000\. Each entry is the mean±\\pmstandard deviation over the three reported physical horizons \(1, 6, and 12 h\)\. Horizon\-level scores, whose deviations are instead over deployment seeds \(ETT\) or cells \(Telco\), are given in Sec\. S\-VII of the SM\. The Telco aggregate over horizons1:121\{:\}12is reported in Table[V](https://arxiv.org/html/2608.20998#S4.T5)\. Scores are1−NRMSE1\-\\operatorname\{NRMSE\}; higher is better\.
#### Budgeted screening

As on the synthetic suite, FP\-K ranks the grid deterministically at zero rollout cost and spends the task\-direct budget only on the retained candidates\. Table[V](https://arxiv.org/html/2608.20998#S4.T5)compares FP\-K with Random\-K and TPE\-K at matched candidate budgets: each individual selector evaluatesKKcandidates using three reservoir seeds, hence3​K3Kfinite\-reservoir rollouts\. Random\-K and TPE\-K are repeated for ten independent draws to estimate their variability; these repetitions use30​K30Kreported rollouts in total but do not increase the candidate budget available to any individual selector\. Their standard deviations therefore quantify selection instability: a single stochastic run has the same3​K3Kbudget but can be substantially below the reported mean\.

TABLE V:Budgeted real\-data forecasting comparison atn=20 000n=20\\,000\. Each selector evaluatesKKcandidates using3​K3Kfinite\-reservoir rollouts\.†\\daggerdenotes FP\-K significantly outperforming both Random\-K and TPE\-K after Holm correction\.On the public ETT benchmarks, FP\-K recovers the full\-grid Direct500operating point already atK=5K=5on ETTh1, ETTh2, and ETTm2, and selects the same operating point at every reported budget\. TPE\-K attains its strongest ETTh1 test score atK=10K=10, while the three screening methods converge at larger budgets\. On ETTh2, Random\-K and TPE\-K improve withKKbut remain below FP\-K even atK=50K=50; on ETTm2 they trail at small budgets and approach FP\-K only at largerKK\.

The operational Telco task is more discriminative and provides the strongest evidence for pilot\-informed screening\. Full\-grid Direct500reaches0\.621±0\.1130\.621\\pm 0\.113across cells, at154×3=462154\\times 3=462rollouts per cell\. FP\-55already matches or slightly exceeds this reference \(0\.622±0\.1120\.622\\pm 0\.112\) with 15 rollouts, compared with0\.5480\.548for Random\-K and0\.5720\.572for TPE\-K at the same budget, and FP\-K remains between0\.6210\.621and0\.6230\.623at every reported budget, compared with0\.5840\.584and0\.5930\.593for the baselines atK=10K=10\.

FP\-K beats both baselines on all ten cells atK=5K=5,K=10K=10, andK=25K=25; after Holm correction over the eight FP\-vs\-baseline budget comparisons, the adjusted values satisfypHolm≤0\.0078p\_\{\\rm Holm\}\\leq 0\.0078in all three cases\. FP\-50 recovers the full\-grid Direct500operating point with 150 rather than 462 rollouts, and byK=50K=50the methods have largely converged and the corrected differences are no longer significant\.

Taken together, the forecasting results show that FP ranking is most valuable when validation rollouts are scarce\. It either recovers the full\-grid operating point immediately, as on several ETT datasets, or allocates a small task\-direct budget to substantially better candidates than Random\-K and TPE\-K, as on the operational Telco task\.

## VDiscussion

#### Width\-independent tuning of reservoir dynamics

The large\-width kernel relocates selection cost from the reservoir to the pilot sequence\. Empirical selectors instantiate a reservoir for every grid point; FP instantiates none\. This has two concrete consequences\. First, the selection cost is independent of reservoir width: the sameθ⋆\\theta^\{\\star\}computed at pilot lengthTpilot=500T\_\{\\rm pilot\}=500can be deployed at anynnwithout rerunning selection\. In our experiments, its mean score improves over the reported widths, consistently with convergence toward the large\-width kernel\. Second, cost scales with pilot length rather than with reservoir width\. The selection\-length study \(Appendix[F](https://arxiv.org/html/2608.20998#A6)\) shows that FP plateaus atTselect=500T\_\{\\rm select\}=500, while task\-direct selection continues to improve; FP is therefore data\-efficient as well as simulation\-efficient\.

#### Deterministic screening versus stochastic search

Random\-K and TPE\-K are important baselines because they represent the finite\-rollout strategies a practitioner would normally try instead of full\-grid selection\. Their behavior also clarifies the role of FP\. FP is not a replacement for high\-budget sequential search; rather, it provides a deterministic ranking of the full candidate grid before any reservoir is instantiated\. The mechanism behind the low\-budget advantage is that the FP ranking is task\-informed from the outset: the pilot kernel already encodes the temporal structure of the downstream task, so the firstKKcandidates evaluated are concentrated in the relevant region of the grid\. Random\-K carries no task information, and TPE\-K must spend its earliest trials learning the response surface before it can exploit it\. The budgeted results show that this ranking is most valuable at smallKK\. On the synthetic suite, FP\-K gives the strongest mean score at every reported budget and matches the full\-grid Direct500reference within sampling variation using4\.8%4\.8\\%of the full\-grid rollout budget\. On Telco, exact\-tanh\\tanhFP\-K is already at the full\-grid Direct500level withK=5K=5and remains stable across the reported budgets, while Random\-K and TPE\-K require larger budgets to approach the same operating point\. The low\-budget differences against both stochastic baselines are significant forK≤25K\\leq 25after Holm correction \(pHolm≤0\.0078p\_\{\\rm Holm\}\\leq 0\.0078\)\. On flatter ETT landscapes, the stochastic baselines catch up asKKgrows, but their reported means are expectations over stochastic selector runs; a single run receives the same3​K3Kcandidate budget and can vary substantially, as reflected by the standard deviations\. Thus, the practical distinction is not only cost, but also determinism and stability of the selected operating point\.

#### Recurrent topology as a kernel design variable

The theory requires two properties of the recurrent matrix: asymptotic operator\-norm control and diagonal self\-averaging of the entries ofAnk​\(Anℓ\)⊤A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\(Lemma[1](https://arxiv.org/html/2608.20998#Thmlemma1)\)\. Ginibre matrices satisfy both with probability tending to one\. The same framework extends to cyclic\-shift and Haar\-orthogonal recurrences, and more generally to ensembles with deterministic cross\-lag propagation coefficients; the corresponding statements and coefficient catalogue are provided in Appendices[C](https://arxiv.org/html/2608.20998#A3)–[D](https://arxiv.org/html/2608.20998#A4)\. In this sense, the recurrent matrix is not merely an implementation detail: once its limiting memory coefficients are specified, it becomes part of the analytic kernel used for selection\.

#### Scope and extensions

The exact theory requires linear recurrent dynamics\. For a nonlinear ESN, the state is no longer a linear combination of propagated inputs, and the large\-width covariance involves time\-varying activation factors that depend on the input trajectory\. A frozen\-diagonal or mean\-field approximation would replace the exact trace moments with approximate ones, at the cost of the exactness guarantee\. Two further limitations delimit the intended use of the selector\. First, the pilot sequence must be drawn from the same task distribution as deployment: FP ranks operating points for the task the pilot describes, and its behavior under pilot–deployment distribution shift has not been studied here\. Second, FP ranks a fixed discrete candidate grid and does not refine candidates continuously; a natural hybrid uses FP to screen the grid and a sequential method to refine locally within the retained region\. Extending the framework to nonlinear recurrences, deep or stacked reservoir layers, and structured state\-space models are natural directions for future work\.

## VIConclusion

We introduced a deterministic zero\-rollout hyperparameter selector for reservoir\-based temporal sequence learning\. The selector uses a free\-probability large\-width kernel: the Gram matrix of nonlinear temporal features converges to a deterministic limit computable from a short labelled pilot sequence and the cross\-lag propagation coefficients of the reservoir ensemble\. This moves hyperparameter search offline: no reservoir is instantiated during selection, the cost is independent of deployment width, and the selected configuration transfers across widths without rerunning the search\.

Across ten temporal benchmarks covering nonlinear sequence processing, memory, and chaotic forecasting, Holm\-corrected pairwise tests find no significant difference between FP and the task\-direct simulation\-based selectors requiring156 600156\\,600finite\-reservoir selection rollouts, while FP is significantly better than the task\-agnostic Memory baseline\. In the budgeted synthetic sweep, FP\-K gives the strongest mean score at each reported budget and matches the full\-grid Direct500reference within sampling variation using only4\.8%4\.8\\%of the full\-grid rollout budget\. On public ETT data, FP\-K recovers the full\-grid Direct500operating point on three of four datasets already atK=5K=5\. On the ten\-cell Telco forecasting task, FP\-5 slightly exceeds full\-grid Direct500\(0\.6220\.622versus0\.6210\.621\) using 15 rather than 462 selection rollouts per cell, and FP\-K is significantly better than Random\-K/TPE\-K forK≤25K\\leq 25after Holm correction \(pHolm≤0\.0078p\_\{\\rm Holm\}\\leq 0\.0078\)\.

The results support free\-probability kernels as a practical tool for pilot\-informed reservoir model selection: they are most useful when finite\-reservoir validation rollouts are scarce, while remaining compatible with higher\-budget direct or sequential search\. Future work will extend the construction to nonlinear recurrences and broader classes of structured sequence models\.

## Acknowledgments

This work was supported by the Norwegian Research Council SFI NorwAI \(309834\); by EMERGE, a project funded by the European Innovation Council \(prj\. code 101070918\); and by NEURONE, a project funded by the European Union – Next Generation EU, M4C1 CUP I53D23003600006, under program PRIN 2022 \(prj\. code 20229JRTZA\)\.

## Appendix AFixed\-Context Kernel

### A\-AProof roadmap and notation

The proof follows four steps\. First, the mixed propagation traces converge to deterministic coefficientsτk,ℓ\\tau\_\{k,\\ell\}\. Second, the corresponding diagonal entries converge to the same coefficients on average over reservoir coordinates\. Third, conditional on the recurrent matrix, each coordinate pair is Gaussian; the diagonal result therefore identifies the limiting nonlinear feature kernel\. Finally, concentration with respect to the Gaussian input matrix removes the remaining finite\-width fluctuations\.

We use the same notation as in the main text\. For each widthnn, let

Wr\(n\)=σrn​Gr\(n\),An=\(1−α\)​In\+α​Wr\(n\),W\_\{r\}^\{\(n\)\}=\\frac\{\\sigma\_\{r\}\}\{\\sqrt\{n\}\}G\_\{r\}^\{\(n\)\},\\qquad A\_\{n\}=\(1\-\\alpha\)I\_\{n\}\+\\alpha W\_\{r\}^\{\(n\)\},where the entries ofGr\(n\)G\_\{r\}^\{\(n\)\}are independent standard Gaussian variables\. Independently,

Win\(n\)=σindin​Gin\(n\),W\_\{\\rm in\}^\{\(n\)\}=\\frac\{\\sigma\_\{\\rm in\}\}\{\\sqrt\{d\_\{\\rm in\}\}\}G\_\{\\rm in\}^\{\(n\)\},withGin\(n\)G\_\{\\rm in\}^\{\(n\)\}again standard Gaussian\. For deterministicu,v∈ℝdinu,v\\in\\mathbb\{R\}^\{d\_\{\\rm in\}\},

𝔼Win​\[Win\(n\)​u​v⊤​\(Win\(n\)\)⊤\]=σin2din​\(u⊤​v\)​In\.\\mathbb\{E\}\_\{W\_\{\\rm in\}\}\\\!\\left\[W\_\{\\rm in\}^\{\(n\)\}uv^\{\\top\}\(W\_\{\\rm in\}^\{\(n\)\}\)^\{\\top\}\\right\]=\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\(u^\{\\top\}v\)I\_\{n\}\.\(33\)
For a fixed maximum retained lagLL, define

xt\(L\):=α​∑k=0LAnk​Win\(n\)​ut−1−k\.x\_\{t\}^\{\(L\)\}:=\\alpha\\sum\_\{k=0\}^\{L\}A\_\{n\}^\{k\}W\_\{\\rm in\}^\{\(n\)\}u\_\{t\-1\-k\}\.\(34\)Thus, lags0,…,L0,\\ldots,Lare retained, for up toL\+1L\+1input time points\. For fixedk,ℓ≥0k,\\ell\\geq 0, set

τk,ℓ:=∑j=0min⁡\{k,ℓ\}\(kj\)​\(ℓj\)​\(1−α\)k\+ℓ−2​j​\(α​σr\)2​j\.\\tau\_\{k,\\ell\}:=\\sum\_\{j=0\}^\{\\min\\\{k,\\ell\\\}\}\\binom\{k\}\{j\}\\binom\{\\ell\}\{j\}\(1\-\\alpha\)^\{k\+\\ell\-2j\}\(\\alpha\\sigma\_\{r\}\)^\{2j\}\.\(35\)The deterministic linear\-state covariance and its associated2×22\\times 2matrix are

Qθ\(L\)​\(t,s\):=α2​σin2din​∑k,ℓ=0Lτk,ℓ​ut−1−k⊤​us−1−ℓ,Q\_\{\\theta\}^\{\(L\)\}\(t,s\):=\\alpha^\{2\}\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\sum\_\{k,\\ell=0\}^\{L\}\\tau\_\{k,\\ell\}\\,u\_\{t\-1\-k\}^\{\\top\}u\_\{s\-1\-\\ell\},\(36\)and

Σθ\(L\)​\(t,s\):=\(Qθ\(L\)​\(t,t\)Qθ\(L\)​\(t,s\)Qθ\(L\)​\(t,s\)Qθ\(L\)​\(s,s\)\)\.\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\):=\\begin\{pmatrix\}Q\_\{\\theta\}^\{\(L\)\}\(t,t\)&Q\_\{\\theta\}^\{\(L\)\}\(t,s\)\\\\ Q\_\{\\theta\}^\{\(L\)\}\(t,s\)&Q\_\{\\theta\}^\{\(L\)\}\(s,s\)\\end\{pmatrix\}\.\(37\)For a coordinate\-wise feature mapψ\\psi, define

Kψ,n,θ\(L\)​\(t,s\):=1n​ψ​\(xt\(L\)\)⊤​ψ​\(xs\(L\)\),K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\):=\\frac\{1\}\{n\}\\psi\(x\_\{t\}^\{\(L\)\}\)^\{\\top\}\\psi\(x\_\{s\}^\{\(L\)\}\),\(38\)and

Kψ,θ\(L\)​\(t,s\):=𝔼⁡\[ψ⁡\(gt\)​ψ​\(gs\)\]\.K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\):=\\mathbb\{E\}\\\!\\left\[\\psi\(g\_\{t\}\)\\psi\(g\_\{s\}\)\\right\]\.\(39\)Here\(gt,gs\)∼𝒩⁡\(0,Σθ\(L\)​\(t,s\)\)\(g\_\{t\},g\_\{s\}\)\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\)\)\.

Throughout, the input sequence is deterministic and bounded:U:=supt∥ut∥<∞U:=\\sup\_\{t\}\\lVert u\_\{t\}\\rVert<\\infty\. The symbolCCdenotes a finite constant that may change from line to line, may depend on fixed model parameters,LL,UU, andψ\\psi, but never onnn\.

We use repeatedly the standard normalized\-Ginibre bound

supn𝔼⁡\[‖Gr\(n\)n‖m\]<∞for every fixed​m\.\\sup\_\{n\}\\mathbb\{E\}\\\!\\left\[\\left\\\|\\frac\{G\_\{r\}^\{\(n\)\}\}\{\\sqrt\{n\}\}\\right\\\|^\{m\}\\right\]<\\infty\\qquad\\text\{for every fixed \}m\.\(40\)Indeed, the expected norm is at most2\+o⁡\(1\)2\+o\(1\)and Gaussian concentration gives a sub\-Gaussian upper tail\[[34](https://arxiv.org/html/2608.20998#bib.bib17)\]\. Consequently, every fixed power ofAnA\_\{n\}has uniformly bounded norm moments\.

### A\-BMixed propagation moments

###### Proposition 2\(Ginibre recurrent memory coefficients\)\.

For every fixedk,ℓ≥0k,\\ell\\geq 0,

1n​Tr⁡\(Ank​\(Anℓ\)⊤\)→L1τk,ℓ\.\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\)\\xrightarrow\{L^\{1\}\}\\tau\_\{k,\\ell\}\.\(41\)

###### Proof\.

For the proof, writeXn:=Gr\(n\)/nX\_\{n\}:=G\_\{r\}^\{\(n\)\}/\\sqrt\{n\}\. SinceInI\_\{n\}commutes withXnX\_\{n\}, with the binomial theorem we can writen−1​Tr⁡\(Ank​\(Anℓ\)⊤\)n^\{\-1\}\\operatorname\{Tr\}\\\!\\left\(A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\)as

∑i=0k∑j=0ℓ\(ki\)​\(ℓj\)​\(1−α\)k\+ℓ−i−j​\(α​σr\)i\+j​mn​\(i,j\),\\sum\_\{i=0\}^\{k\}\\sum\_\{j=0\}^\{\\ell\}\\binom\{k\}\{i\}\\binom\{\\ell\}\{j\}\(1\-\\alpha\)^\{k\+\\ell\-i\-j\}\(\\alpha\\sigma\_\{r\}\)^\{i\+j\}m\_\{n\}\(i,j\),\(42\)wheremn​\(i,j\):=n−1​Tr⁡\(Xni​\(Xn⊤\)j\)m\_\{n\}\(i,j\):=n^\{\-1\}\\operatorname\{Tr\}\(X\_\{n\}^\{i\}\(X\_\{n\}^\{\\top\}\)^\{j\}\)\.

Normalized real Ginibre matrices converge in expected∗\*\-moments to a standard circular elementcc\[[27](https://arxiv.org/html/2608.20998#bib.bib10), Sec\. 11\.6\.3\]\. The free Wick rule gives

φ⁡\(ci​\(c∗\)j\)=δi​j:\\varphi\\\!\\left\(c^\{i\}\(c^\{\*\}\)^\{j\}\\right\)=\\delta\_\{ij\}:a non\-crossing pairing of the ordered wordci​\(c∗\)jc^\{i\}\(c^\{\*\}\)^\{j\}exists only wheni=ji=j, and is then unique\. Hence𝔼⁡\[mn​\(i,j\)\]→δi​j\\mathbb\{E\}\\\!\\left\[m\_\{n\}\(i,j\)\\right\]\\to\\delta\_\{ij\}\.

It remains to control fluctuations\. Differentiatingmn​\(i,j\)m\_\{n\}\(i,j\)with respect to the Gaussian entries and summing the resulting position terms yields

∥∇mn​\(i,j\)∥2≤\(i\+j\)2n2​max⁡\{1,∥Xn∥2​\(i\+j−1\)\}\.\\lVert\\nabla m\_\{n\}\(i,j\)\\rVert^\{2\}\\leq\\frac\{\(i\+j\)^\{2\}\}\{n^\{2\}\}\\max\\bigl\\\{1,\\lVert X\_\{n\}\\rVert^\{2\(i\+j\-1\)\}\\bigr\\\}\.The Gaussian Poincaré inequality and \([40](https://arxiv.org/html/2608.20998#A1.E40)\) therefore giveVar⁡\(mn​\(i,j\)\)=O⁡\(n−2\)\\operatorname\{Var\}\(m\_\{n\}\(i,j\)\)=O\(n^\{\-2\}\)\. Thusmn​\(i,j\)→δi​jm\_\{n\}\(i,j\)\\to\\delta\_\{ij\}inL2L^\{2\}\. Substitution into the finite sum \([42](https://arxiv.org/html/2608.20998#A1.E42)\) gives \([41](https://arxiv.org/html/2608.20998#A1.E41)\) with the coefficients \([35](https://arxiv.org/html/2608.20998#A1.E35)\)\. ∎

### A\-CDiagonal self\-averaging

###### Lemma 2\(Averaged diagonal self\-averaging\)\.

For every fixedL<∞L<\\infty,

max0≤k,ℓ≤L⁡1n​∑i=1n\|\[Ank​\(Anℓ\)⊤\]i​i−τk,ℓ\|2→ℙ0\.\\max\_\{0\\leq k,\\ell\\leq L\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\|\\left\[A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\]\_\{ii\}\-\\tau\_\{k,\\ell\}\\right\|^\{2\}\\xrightarrow\{\\mathbb\{P\}\}0\.\(43\)

###### Proof\.

Fixk,ℓk,\\elland setBn:=Ank​\(Anℓ\)⊤B\_\{n\}:=A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\. Simultaneous permutation of the rows and columns ofGr\(n\)G\_\{r\}^\{\(n\)\}preserves its law\. Hence the diagonal entries ofBnB\_\{n\}are identically distributed and

𝔼⁡\[\(Bn\)i​i\]=𝔼⁡\[1n​Tr⁡Bn\]⟶τk,ℓ\\mathbb\{E\}\\\!\\left\[\(B\_\{n\}\)\_\{ii\}\\right\]=\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{n\}\\operatorname\{Tr\}B\_\{n\}\\right\]\\longrightarrow\\tau\_\{k,\\ell\}by Proposition[2](https://arxiv.org/html/2608.20998#Thmproposition2)\.

To control one diagonal entry, expandBnB\_\{n\}into the finitely many matrix words inXn:=Gr\(n\)/nX\_\{n\}:=G\_\{r\}^\{\(n\)\}/\\sqrt\{n\}andXn⊤X\_\{n\}^\{\\top\}\. For a wordM=F1⋯FrM=F\_\{1\}\\cdots F\_\{r\}, differentiation of\(M\)11\(M\)\_\{11\}at each occurrence ofXnX\_\{n\}orXn⊤X\_\{n\}^\{\\top\}gives

∑a,b\|∂\(M\)11∂\(Gr\(n\)\)a​b\|2≤r2n​max⁡\{1,∥Xn∥2​\(r−1\)\}\.\\sum\_\{a,b\}\\left\|\\frac\{\\partial\(M\)\_\{11\}\}\{\\partial\(G\_\{r\}^\{\(n\)\}\)\_\{ab\}\}\\right\|^\{2\}\\leq\\frac\{r^\{2\}\}\{n\}\\max\\bigl\\\{1,\\lVert X\_\{n\}\\rVert^\{2\(r\-1\)\}\\bigr\\\}\.After summing over the finitely many words, Gaussian Poincaré and \([40](https://arxiv.org/html/2608.20998#A1.E40)\) yieldVar⁡\(\(Bn\)11\)=O⁡\(n−1\)\\operatorname\{Var\}\(\(B\_\{n\}\)\_\{11\}\)=O\(n^\{\-1\}\)\. Therefore

𝔼⁡\[1n​∑i=1n\|\(Bn\)i​i−τk,ℓ\|2\]=𝔼⁡\[\|\(Bn\)11−τk,ℓ\|2\]⟶0\.\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\|\(B\_\{n\}\)\_\{ii\}\-\\tau\_\{k,\\ell\}\|^\{2\}\\right\]=\\mathbb\{E\}\\\!\\left\[\|\(B\_\{n\}\)\_\{11\}\-\\tau\_\{k,\\ell\}\|^\{2\}\\right\]\\longrightarrow 0\.Markov’s inequality and a finite union bound over0≤k,ℓ≤L0\\leq k,\\ell\\leq Lprove \([43](https://arxiv.org/html/2608.20998#A1.E43)\)\. ∎

Only the diagonal propagation terms are needed to identify the marginal Gaussian law of each coordinate pair entering the kernel\. Dependence between distinct reservoir coordinates is controlled separately by the concentration step below\.

### A\-DNonlinear feature\-kernel limit

###### Theorem 2\(Deterministic nonlinear reservoir kernel\)\.

Letψ∈Cb1​\(ℝ\)\\psi\\in C\_\{b\}^\{1\}\(\\mathbb\{R\}\)\. For fixedL,t,sL,t,sand a deterministic bounded input sequence,

Kψ,n,θ\(L\)​\(t,s\)→ℙKψ,θ\(L\)​\(t,s\)\(n→∞\)\.K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\xrightarrow\{\\mathbb\{P\}\}K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\qquad\(n\\to\\infty\)\.\(44\)The convergence is with respect to the joint randomness ofWr\(n\)W\_\{r\}^\{\(n\)\}andWin\(n\)W\_\{\\rm in\}^\{\(n\)\}\.

###### Proof\.

WriteMψ:=∥ψ∥∞M\_\{\\psi\}:=\\lVert\\psi\\rVert\_\{\\infty\}andLψ:=∥ψ′∥∞L\_\{\\psi\}:=\\lVert\\psi^\{\\prime\}\\rVert\_\{\\infty\}\. For a positive\-semidefinite2×22\\times 2matrixΣ\\Sigma, let

h⁡\(Σ\):=𝔼⁡\[ψ⁡\(g1\)​ψ​\(g2\)\],\(g1,g2\)∼𝒩⁡\(0,Σ\)\.h\(\\Sigma\):=\\mathbb\{E\}\\\!\\left\[\\psi\(g\_\{1\}\)\\psi\(g\_\{2\}\)\\right\],\\qquad\(g\_\{1\},g\_\{2\}\)\\sim\\mathcal\{N\}\(0,\\Sigma\)\.Coupling two Gaussian vectors through a common standard normal vector and using∥Σ1/2−\(Σ′\)1/2∥≤∥Σ−Σ′∥1/2\\lVert\\Sigma^\{1/2\}\-\(\\Sigma^\{\\prime\}\)^\{1/2\}\\rVert\\leq\\lVert\\Sigma\-\\Sigma^\{\\prime\}\\rVert^\{1/2\}\[[4](https://arxiv.org/html/2608.20998#bib.bib18), Thm\. X\.1\.1\]gives

\|h⁡\(Σ\)−h⁡\(Σ′\)\|≤2​2​Mψ​Lψ​∥Σ−Σ′∥1/2\.\|h\(\\Sigma\)\-h\(\\Sigma^\{\\prime\}\)\|\\leq 2\\sqrt\{2\}\\,M\_\{\\psi\}L\_\{\\psi\}\\lVert\\Sigma\-\\Sigma^\{\\prime\}\\rVert^\{1/2\}\.\(45\)
Conditional onAnA\_\{n\}, the pair\(xt,i\(L\),xs,i\(L\)\)\(x\_\{t,i\}^\{\(L\)\},x\_\{s,i\}^\{\(L\)\}\)is centered Gaussian\. Its covariance entries are

cn,i\(L\)​\(r,q\)=α2​σin2din​∑k,ℓ=0L\[Ank​\(Anℓ\)⊤\]i​i​ur−1−k⊤​uq−1−ℓ\.c\_\{n,i\}^\{\(L\)\}\(r,q\)=\\alpha^\{2\}\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\sum\_\{k,\\ell=0\}^\{L\}\\left\[A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\]\_\{ii\}u\_\{r\-1\-k\}^\{\\top\}u\_\{q\-1\-\\ell\}\.\(46\)LetCn,i\(L\)​\(t,s\)C\_\{n,i\}^\{\(L\)\}\(t,s\)be the corresponding2×22\\times 2covariance matrix\. By Lemma[2](https://arxiv.org/html/2608.20998#Thmlemma2),

1n​∑i=1n‖Cn,i\(L\)​\(t,s\)−Σθ\(L\)​\(t,s\)‖F2→ℙ0\.\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\\\|C\_\{n,i\}^\{\(L\)\}\(t,s\)\-\\Sigma\_\{\\theta\}^\{\(L\)\}\(t,s\)\\right\\\|\_\{F\}^\{2\}\\xrightarrow\{\\mathbb\{P\}\}0\.Together with \([45](https://arxiv.org/html/2608.20998#A1.E45)\), this implies

𝔼⁡\[Kψ,n,θ\(L\)​\(t,s\)∣An\]−Kψ,θ\(L\)​\(t,s\)→ℙ0\.\\mathbb\{E\}\\\!\\left\[K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\mid A\_\{n\}\\right\]\-K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\xrightarrow\{\\mathbb\{P\}\}0\.\(47\)
It remains to control fluctuations around the conditional mean\. Stack the entries ofGin\(n\)G\_\{\\rm in\}^\{\(n\)\}into a standard Gaussian vectorggand writext\(L\)=Mt​gx\_\{t\}^\{\(L\)\}=M\_\{t\}g,xs\(L\)=Ms​gx\_\{s\}^\{\(L\)\}=M\_\{s\}g\. Since

∥Mt∥≤α​σin​Udin​∑k=0L∥Ank∥,\\lVert M\_\{t\}\\rVert\\leq\\frac\{\\alpha\\sigma\_\{\\rm in\}U\}\{\\sqrt\{d\_\{\\rm in\}\}\}\\sum\_\{k=0\}^\{L\}\\lVert A\_\{n\}^\{k\}\\rVert,the conditional Gaussian Poincaré inequality gives

Var⁡\(Kψ,n,θ\(L\)​\(t,s\)∣An\)≤Cn​\(∑k=0L∥Ank∥\)2\.\\operatorname\{Var\}\\\!\\left\(K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\mid A\_\{n\}\\right\)\\leq\\frac\{C\}\{n\}\\left\(\\sum\_\{k=0\}^\{L\}\\lVert A\_\{n\}^\{k\}\\rVert\\right\)^\{2\}\.\(48\)The right\-hand side isOℙ​\(n−1\)O\_\{\\mathbb\{P\}\}\(n^\{\-1\}\)by \([40](https://arxiv.org/html/2608.20998#A1.E40)\)\. Hence the conditional fluctuation vanishes in probability, and \([47](https://arxiv.org/html/2608.20998#A1.E47)\) proves \([44](https://arxiv.org/html/2608.20998#A1.E44)\)\.

For reference, the same conditional\-mean calculation before applyingψ\\psi, together with the Gaussian quadratic\-form variance bound, also gives

1n​\(xt\(L\)\)⊤​xs\(L\)→ℙQθ\(L\)​\(t,s\)\.\\frac\{1\}\{n\}\(x\_\{t\}^\{\(L\)\}\)^\{\\top\}x\_\{s\}^\{\(L\)\}\\xrightarrow\{\\mathbb\{P\}\}Q\_\{\\theta\}^\{\(L\)\}\(t,s\)\.\(49\)On any fixed finite pilot set, a union bound gives entrywise and hence Frobenius\-norm convergence of the associated kernel blocks\. ∎

## Appendix BComplete History and Selection Consistency

The fixed\-context result is the object used directly by the selector\. This section shows that, in the stable regime, increasing the maximum retained lag recovers the complete washed\-out history\. The argument has three parts: a deterministic covariance\-tail bound, a uniform finite\-width power bound for Ginibre recurrence, and a final triangle\-inequality passage to the nonlinear kernel\.

### B\-ADeterministic complete\-history kernel

###### Proposition 3\(Complete\-history covariance and context limit\)\.

Assumeσr<1\\sigma\_\{r\}<1andU=supt∥ut∥<∞U=\\sup\_\{t\}\\lVert u\_\{t\}\\rVert<\\infty\. Define

Qθ\(∞\)​\(t,s\):=α2​σin2din​∑k,ℓ≥0τk,ℓ​ut−1−k⊤​us−1−ℓ\.Q\_\{\\theta\}^\{\(\\infty\)\}\(t,s\):=\\alpha^\{2\}\\frac\{\\sigma\_\{\\rm in\}^\{2\}\}\{d\_\{\\rm in\}\}\\sum\_\{k,\\ell\\geq 0\}\\tau\_\{k,\\ell\}u\_\{t\-1\-k\}^\{\\top\}u\_\{s\-1\-\\ell\}\.\(50\)The series converges absolutely\. With

rθ:=1−α\+α​σr2<1,r\_\{\\theta\}:=1\-\\alpha\+\\alpha\\sigma\_\{r\}^\{2\}<1,\|Qθ\(∞\)​\(t,s\)−Qθ\(L\)​\(t,s\)\|≤2​σin2​U2din​\(1−σr2\)​rθL\+1\.\\left\|Q\_\{\\theta\}^\{\(\\infty\)\}\(t,s\)\-Q\_\{\\theta\}^\{\(L\)\}\(t,s\)\\right\|\\leq\\frac\{2\\sigma\_\{\\rm in\}^\{2\}U^\{2\}\}\{d\_\{\\rm in\}\(1\-\\sigma\_\{r\}^\{2\}\)\}r\_\{\\theta\}^\{L\+1\}\.\(51\)IfKψ,θ\(∞\)K\_\{\\psi,\\theta\}^\{\(\\infty\)\}is obtained fromQθ\(∞\)Q\_\{\\theta\}^\{\(\\infty\)\}as in \([39](https://arxiv.org/html/2608.20998#A1.E39)\), then

\|Kψ,θ\(∞\)​\(t,s\)−Kψ,θ\(L\)​\(t,s\)\|≤Cθ​rθ\(L\+1\)/2\.\\left\|K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)\-K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\right\|\\leq C\_\{\\theta\}r\_\{\\theta\}^\{\(L\+1\)/2\}\.\(52\)In particular,

Kψ,θ\(L\)​\(t,s\)⟶Kψ,θ\(∞\)​\(t,s\)\(L→∞\)\.K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\longrightarrow K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)\\qquad\(L\\to\\infty\)\.\(53\)

###### Proof\.

The coefficientsτk,ℓ\\tau\_\{k,\\ell\}are nonnegative and have generating function

F⁡\(x,y\)\\displaystyle F\(x,y\):=∑k,ℓ≥0τk,ℓ​xk​yℓ\\displaystyle:=\\sum\_\{k,\\ell\\geq 0\}\\tau\_\{k,\\ell\}x^\{k\}y^\{\\ell\}=1\(1−\(1−α\)​x\)​\(1−\(1−α\)​y\)−α2​σr2​x​y\.\\displaystyle=\\frac\{1\}\{\(1\-\(1\-\\alpha\)x\)\(1\-\(1\-\\alpha\)y\)\-\\alpha^\{2\}\\sigma\_\{r\}^\{2\}xy\}\.\(54\)Atx=y=1x=y=1,

∑k,ℓ≥0τk,ℓ=1α2​\(1−σr2\),\\sum\_\{k,\\ell\\geq 0\}\\tau\_\{k,\\ell\}=\\frac\{1\}\{\\alpha^\{2\}\(1\-\\sigma\_\{r\}^\{2\}\)\},which proves absolute convergence for bounded inputs\. Moreover,

F⁡\(x,1\)=1α⁡\(1−rθ​x\),∑ℓ≥0τk,ℓ=1α​rθk\.F\(x,1\)=\\frac\{1\}\{\\alpha\(1\-r\_\{\\theta\}x\)\},\\qquad\\sum\_\{\\ell\\geq 0\}\\tau\_\{k,\\ell\}=\\frac\{1\}\{\\alpha\}r\_\{\\theta\}^\{k\}\.The complement of\{0,…,L\}2\\\{0,\\ldots,L\\\}^\{2\}is contained in\{k\>L\}∪\{ℓ\>L\}\\\{k\>L\\\}\\cup\\\{\\ell\>L\\\}\. By symmetry,

∑max⁡\{k,ℓ\}\>Lτk,ℓ≤2α​∑k\>Lrθk=2α2​\(1−σr2\)​rθL\+1\.\\sum\_\{\\max\\\{k,\\ell\\\}\>L\}\\tau\_\{k,\\ell\}\\leq\\frac\{2\}\{\\alpha\}\\sum\_\{k\>L\}r\_\{\\theta\}^\{k\}=\\frac\{2\}\{\\alpha^\{2\}\(1\-\\sigma\_\{r\}^\{2\}\)\}r\_\{\\theta\}^\{L\+1\}\.Multiplying byα2​σin2​U2/din\\alpha^\{2\}\\sigma\_\{\\rm in\}^\{2\}U^\{2\}/d\_\{\\rm in\}gives \([51](https://arxiv.org/html/2608.20998#A2.E51)\)\. The same bound applies to each entry ofΣθ\(∞\)−Σθ\(L\)\\Sigma\_\{\\theta\}^\{\(\\infty\)\}\-\\Sigma\_\{\\theta\}^\{\(L\)\}\. The Gaussian continuity estimate \([45](https://arxiv.org/html/2608.20998#A1.E45)\) then yields \([52](https://arxiv.org/html/2608.20998#A2.E52)\)\. ∎

Because the admissible candidate grid and pilot index sets are finite, the convergence in \([53](https://arxiv.org/html/2608.20998#A2.E53)\) is uniform over all candidates and pilot\-kernel entries used by the selector\.

### B\-BFinite\-width Ginibre tail

###### Proposition 4\(Uniform Ginibre power bound\)\.

Let

An=\(1−α\)​In\+α​σrn​Gr\(n\)A\_\{n\}=\(1\-\\alpha\)I\_\{n\}\+\\alpha\\frac\{\\sigma\_\{r\}\}\{\\sqrt\{n\}\}G\_\{r\}^\{\(n\)\}and assumeσr<1\\sigma\_\{r\}<1\. For any1<R<σr−11<R<\\sigma\_\{r\}^\{\-1\}, setqR:=1−α\+α​σr​R<1q\_\{R\}:=1\-\\alpha\+\\alpha\\sigma\_\{r\}R<1\. Almost surely, there are finite random constantsCRC\_\{R\}andNRN\_\{R\}such that, for alln≥NRn\\geq N\_\{R\}andm≥0m\\geq 0,

∥Anm∥≤CR​qRm\.\\lVert A\_\{n\}^\{m\}\\rVert\\leq C\_\{R\}q\_\{R\}^\{m\}\.\(55\)Consequently, for bounded input histories,

xt\(∞\):=α​∑k=0∞Ank​Win\(n\)​ut−1−kx\_\{t\}^\{\(\\infty\)\}:=\\alpha\\sum\_\{k=0\}^\{\\infty\}A\_\{n\}^\{k\}W\_\{\\rm in\}^\{\(n\)\}u\_\{t\-1\-k\}\(56\)is well defined for all sufficiently largenn, and

lim supn→∞1n​‖xt\(∞\)−xt\(L\)‖≤α​CR​σin​Udin​\(1−qR\)​qRL\+1\\limsup\_\{n\\to\\infty\}\\frac\{1\}\{\\sqrt\{n\}\}\\left\\\|x\_\{t\}^\{\(\\infty\)\}\-x\_\{t\}^\{\(L\)\}\\right\\\|\\leq\\frac\{\\alpha C\_\{R\}\\sigma\_\{\\rm in\}U\}\{\\sqrt\{d\_\{\\rm in\}\}\(1\-q\_\{R\}\)\}q\_\{R\}^\{L\+1\}\(57\)almost surely\.

###### Proof\.

WriteXn:=Gr\(n\)/nX\_\{n\}:=G\_\{r\}^\{\(n\)\}/\\sqrt\{n\}for this proof\. The normalized real Ginibre matrix converges strongly to a standard circular elementcc\[[31](https://arxiv.org/html/2608.20998#bib.bib13)\]\. A standard consequence of strong convergence is uniform resolvent control on compact subsets ofℂ∖spec⁡\(c\)\\mathbb\{C\}\\setminus\\operatorname\{spec\}\(c\): by hermitization, the smallest singular value ofz​In−XnzI\_\{n\}\-X\_\{n\}is eventually bounded away from zero uniformly on such compact sets\. Sincespec⁡\(c\)\\operatorname\{spec\}\(c\)is the closed unit disk\[[27](https://arxiv.org/html/2608.20998#bib.bib10)\], for everyR\>1R\>1there are almost surely finiteMR,NRM\_\{R\},N\_\{R\}such that

supn≥NRsup\|z\|=R‖\(z​In−Xn\)−1‖≤MR\.\\sup\_\{n\\geq N\_\{R\}\}\\sup\_\{\|z\|=R\}\\left\\\|\(zI\_\{n\}\-X\_\{n\}\)^\{\-1\}\\right\\\|\\leq M\_\{R\}\.\(58\)
Withf⁡\(z\)=1−α\+α​σr​zf\(z\)=1\-\\alpha\+\\alpha\\sigma\_\{r\}z, we haveAn=f⁡\(Xn\)A\_\{n\}=f\(X\_\{n\}\)\. The holomorphic functional calculus and \([58](https://arxiv.org/html/2608.20998#A2.E58)\) give

∥Anm∥\\displaystyle\\lVert A\_\{n\}^\{m\}\\rVert≤12​π​∫\|z\|=R\|f⁡\(z\)\|m​‖\(z​In−Xn\)−1‖​\|𝑑z\|\\displaystyle\\leq\\frac\{1\}\{2\\pi\}\\int\_\{\|z\|=R\}\|f\(z\)\|^\{m\}\\left\\\|\(zI\_\{n\}\-X\_\{n\}\)^\{\-1\}\\right\\\|\|dz\|≤R​MR​qRm\.\\displaystyle\\leq RM\_\{R\}q\_\{R\}^\{m\}\.Thus \([55](https://arxiv.org/html/2608.20998#A2.E55)\) holds withCR=R​MRC\_\{R\}=RM\_\{R\}\. Summing the geometric tail and using

∥Win\(n\)∥n⟶σindinalmost surely\\frac\{\\lVert W\_\{\\rm in\}^\{\(n\)\}\\rVert\}\{\\sqrt\{n\}\}\\longrightarrow\\frac\{\\sigma\_\{\\rm in\}\}\{\\sqrt\{d\_\{\\rm in\}\}\}\\qquad\\text\{almost surely\}proves \([57](https://arxiv.org/html/2608.20998#A2.E57)\)\. ∎

### B\-CComplete\-history kernel

###### Theorem 3\(Complete\-history Ginibre kernel\)\.

Under the assumptions of Theorem[2](https://arxiv.org/html/2608.20998#Thmtheorem2)and Proposition[4](https://arxiv.org/html/2608.20998#Thmproposition4), for every fixedt,st,s,

Kψ,n,θ\(∞\)​\(t,s\):=1n​ψ​\(xt\(∞\)\)⊤​ψ​\(xs\(∞\)\)→ℙKψ,θ\(∞\)​\(t,s\)\.K\_\{\\psi,n,\\theta\}^\{\(\\infty\)\}\(t,s\):=\\frac\{1\}\{n\}\\psi\(x\_\{t\}^\{\(\\infty\)\}\)^\{\\top\}\\psi\(x\_\{s\}^\{\(\\infty\)\}\)\\xrightarrow\{\\mathbb\{P\}\}K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)\.\(59\)

###### Proof\.

By boundedness and Lipschitz continuity ofψ\\psi,

\|Kψ,n,θ\(∞\)​\(t,s\)−Kψ,n,θ\(L\)​\(t,s\)\|\\displaystyle\\left\|K\_\{\\psi,n,\\theta\}^\{\(\\infty\)\}\(t,s\)\-K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\right\|≤Mψ​Lψn​\(∥xt\(∞\)−xt\(L\)∥\+∥xs\(∞\)−xs\(L\)∥\)\.\\displaystyle\\quad\\leq\\frac\{M\_\{\\psi\}L\_\{\\psi\}\}\{\\sqrt\{n\}\}\\left\(\\lVert x\_\{t\}^\{\(\\infty\)\}\-x\_\{t\}^\{\(L\)\}\\rVert\+\\lVert x\_\{s\}^\{\(\\infty\)\}\-x\_\{s\}^\{\(L\)\}\\rVert\\right\)\.\(60\)Proposition[4](https://arxiv.org/html/2608.20998#Thmproposition4)bounds this term geometrically inLL, uniformly for all sufficiently largenn\. For fixedLL, Theorem[2](https://arxiv.org/html/2608.20998#Thmtheorem2)gives

Kψ,n,θ\(L\)​\(t,s\)→ℙKψ,θ\(L\)​\(t,s\),K\_\{\\psi,n,\\theta\}^\{\(L\)\}\(t,s\)\\xrightarrow\{\\mathbb\{P\}\}K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\),while Proposition[3](https://arxiv.org/html/2608.20998#Thmproposition3)givesKψ,θ\(L\)​\(t,s\)→Kψ,θ\(∞\)​\(t,s\)K\_\{\\psi,\\theta\}^\{\(L\)\}\(t,s\)\\to K\_\{\\psi,\\theta\}^\{\(\\infty\)\}\(t,s\)\. The result follows from the triangle inequality by first choosingLLlarge and then lettingn→∞n\\to\\infty\. ∎

### B\-DSelection consistency

###### Corollary 2\(Consistency of FP selection\)\.

Assume that the deterministic kernel and the finite reservoir use the same feature map\. Fix finite candidate and ridge grids, with all ridge parameters strictly positive\. For any fixed maximum retained lagLL, if the deterministic validation score has a unique minimizer, then finite\-width reservoir selection returns the same minimizer with probability tending to one asn→∞n\\to\\infty\.

In the stable regime, a unique complete\-history optimum is preserved by all sufficiently large finite contexts, and selection based on the complete finite\-reservoir trajectory converges to this optimum asn→∞n\\to\\infty\.

###### Proof\.

LetTTandVVbe the fixed pilot training and validation indices\. Forλ≥λmin\>0\\lambda\\geq\\lambda\_\{\\min\}\>0, the prediction map

\(KT​T,KV​T\)⟼KV​T​\(KT​T\+λ​I\)−1​YT\(K\_\{TT\},K\_\{VT\}\)\\longmapsto K\_\{VT\}\(K\_\{TT\}\+\\lambda I\)^\{\-1\}Y\_\{T\}is continuous and uniformly Lipschitz on the fixed finite blocks\. This follows from∥\(KT​T\+λ​I\)−1∥≤λmin−1\\lVert\(K\_\{TT\}\+\\lambda I\)^\{\-1\}\\rVert\\leq\\lambda\_\{\\min\}^\{\-1\}and the resolvent identity\. The same is therefore true of the validation NMSE, assuming its denominator is nonzero\.

Theorem[2](https://arxiv.org/html/2608.20998#Thmtheorem2)gives convergence of every fixed\-context kernel entry\. A union bound over the finite pilot, candidate, and ridge grids yields

maxθ,λ⁡\|NMSEval,n\(L\)⁡\(θ,λ\)−NMSEval\(L\)⁡\(θ,λ\)\|→ℙ0\.\\max\_\{\\theta,\\lambda\}\\left\|\\operatorname\{NMSE\}\_\{\{\\rm val\},n\}^\{\(L\)\}\(\\theta,\\lambda\)\-\\operatorname\{NMSE\}\_\{\\rm val\}^\{\(L\)\}\(\\theta,\\lambda\)\\right\|\\xrightarrow\{\\mathbb\{P\}\}0\.A unique minimizer on a finite grid has a positive gap to the second\-best score\. Whenever the uniform error is less than half this gap, the empirical and deterministic minimizers agree\. This proves the fixed\-context statement\.

Proposition[3](https://arxiv.org/html/2608.20998#Thmproposition3)gives uniform convergence of the finite\-context deterministic scores to their complete\-history counterparts over the same finite grids\. Hence a unique complete\-history minimizer is also the finite\-context minimizer for all sufficiently largeLL\. Finally, Theorem[3](https://arxiv.org/html/2608.20998#Thmtheorem3)gives uniform convergence of the empirical complete\-history scores, and the same positive\-gap argument proves the final claim\. ∎

Near ties have a small score gap and may therefore require larger reservoir width or maximum lag before the ranking stabilizes\.

## Appendix CHaar\-Orthogonal Recurrent Matrix

The Ginibre proof uses Gaussian concentration to establish diagonal self\-averaging\. For a Haar\-orthogonal recurrent matrix, the same conclusion follows directly from orthogonal invariance\.

###### Proposition 5\(Haar\-orthogonal extension\)\.

LetOnO\_\{n\}be Haar distributed on the orthogonal group, independently ofWin\(n\)W\_\{\\rm in\}^\{\(n\)\}, and let

An=\(1−α\)​In\+α​σr​On\.A\_\{n\}=\(1\-\\alpha\)I\_\{n\}\+\\alpha\\sigma\_\{r\}O\_\{n\}\.For every fixedL<∞L<\\infty, Lemma[2](https://arxiv.org/html/2608.20998#Thmlemma2)holds with the coefficientsτk,ℓ\\tau\_\{k,\\ell\}in \([35](https://arxiv.org/html/2608.20998#A1.E35)\)\. Consequently, Theorem[2](https://arxiv.org/html/2608.20998#Thmtheorem2)holds with the same deterministic kernel\. Ifσr<1\\sigma\_\{r\}<1, the complete\-history conclusion also holds\.

###### Proof\.

Fixk,ℓk,\\elland setBn=Ank​\(Anℓ\)⊤B\_\{n\}=A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\. Orthogonal conjugation preserves the law ofBnB\_\{n\}\. Ifvvis independent and uniform on the unit sphere, then

\(Bn\)11​=𝑑​v⊤​Bn​v\.\(B\_\{n\}\)\_\{11\}\\overset\{d\}\{=\}v^\{\\top\}B\_\{n\}v\.The spherical fourth\-moment identity gives

𝔼⁡\[\|v⊤​Bn​v−1n​Tr⁡Bn\|2\|Bn\]≤2​∥Bn∥2n\+2\.\\mathbb\{E\}\\\!\\left\[\\left\|v^\{\\top\}B\_\{n\}v\-\\frac\{1\}\{n\}\\operatorname\{Tr\}B\_\{n\}\\right\|^\{2\}\\,\\middle\|\\,B\_\{n\}\\right\]\\leq\\frac\{2\\lVert B\_\{n\}\\rVert^\{2\}\}\{n\+2\}\.Moreover,∥Bn∥≤\(1−α\+α​σr\)k\+ℓ\\lVert B\_\{n\}\\rVert\\leq\(1\-\\alpha\+\\alpha\\sigma\_\{r\}\)^\{k\+\\ell\}\. The normalized traces of fixed Laurent polynomials inOnO\_\{n\}converge to the corresponding Haar\-unitary moments\[[6](https://arxiv.org/html/2608.20998#bib.bib12)\], which give the sameτk,ℓ\\tau\_\{k,\\ell\}as in \([35](https://arxiv.org/html/2608.20998#A1.E35)\)\. Thus

𝔼⁡\[\|\(Bn\)11−τk,ℓ\|2\]⟶0\.\\mathbb\{E\}\\\!\\left\[\\left\|\(B\_\{n\}\)\_\{11\}\-\\tau\_\{k,\\ell\}\\right\|^\{2\}\\right\]\\longrightarrow 0\.The diagonal entries are exchangeable, so averaging over them, followed by Markov’s inequality and a finite union bound, proves the diagonal self\-averaging statement\. The fixed\-context kernel follows from the same conditional\-Gaussian argument as Theorem[2](https://arxiv.org/html/2608.20998#Thmtheorem2)\. Forσr<1\\sigma\_\{r\}<1,

∥Anm∥≤\(1−α\+α​σr\)m,\\lVert A\_\{n\}^\{m\}\\rVert\\leq\(1\-\\alpha\+\\alpha\\sigma\_\{r\}\)^\{m\},which gives the complete\-history extension directly\. ∎

## Appendix DOther Recurrent Matrix Structures

The kernel derivation depends on the recurrent matrix only through the limits of

1n​Tr⁡\(Ank​\(Anℓ\)⊤\)\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}\\right\)and the corresponding diagonal self\-averaging property\. This section records these coefficients for several recurrent matrix structures\. Once they are known, the covariance and nonlinear kernel are obtained from \([36](https://arxiv.org/html/2608.20998#A1.E36)\)–\([39](https://arxiv.org/html/2608.20998#A1.E39)\) by replacingτk,ℓ\\tau\_\{k,\\ell\}with the appropriate coefficientηk,ℓ\\eta\_\{k,\\ell\}\. No additional kernel notation is needed\.

### D\-AGinibre, Haar\-orthogonal, and cyclic\-shift matrices

For the real Ginibre matrix used in the experiments,

Wr\(n\)=σrn​Gr\(n\),W\_\{r\}^\{\(n\)\}=\\frac\{\\sigma\_\{r\}\}\{\\sqrt\{n\}\}G\_\{r\}^\{\(n\)\},Proposition[2](https://arxiv.org/html/2608.20998#Thmproposition2)givesηk,ℓ=τk,ℓ\\eta\_\{k,\\ell\}=\\tau\_\{k,\\ell\}\. Proposition[5](https://arxiv.org/html/2608.20998#Thmproposition5)shows that the same coefficients hold forWr\(n\)=σr​OnW\_\{r\}^\{\(n\)\}=\\sigma\_\{r\}O\_\{n\}withOnO\_\{n\}Haar orthogonal\.

They also hold for the deterministic cyclic\-shift matrixCnC\_\{n\}, defined by

Cn​ej=ej\+1​\(mod​n\),Wr\(n\)=σr​Cn\.C\_\{n\}e\_\{j\}=e\_\{j\+1\\;\(\{\\rm mod\}\\;n\)\},\\qquad W\_\{r\}^\{\(n\)\}=\\sigma\_\{r\}C\_\{n\}\.Indeed,Cn⊤=Cn−1C\_\{n\}^\{\\top\}=C\_\{n\}^\{\-1\}and, for fixedi,ji,jand all sufficiently largenn,

1n​Tr⁡\(Cni​\(Cn⊤\)j\)=δi​j\.\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(C\_\{n\}^\{i\}\(C\_\{n\}^\{\\top\}\)^\{j\}\\right\)=\\delta\_\{ij\}\.The three structures therefore produce the same fixed\-context coefficients\. Their complete\-history control differs: Ginibre uses Proposition[4](https://arxiv.org/html/2608.20998#Thmproposition4), whereas the orthogonal matrices satisfy the direct geometric bound

∥Anm∥≤\(1−α\+α​σr\)m\.\\lVert A\_\{n\}^\{m\}\\rVert\\leq\(1\-\\alpha\+\\alpha\\sigma\_\{r\}\)^\{m\}\.

### D\-BCirculant matrices

Let

Cn=Fn∗​diag⁡\(γn,0,…,γn,n−1\)​Fn,Wr\(n\)=σr​Cn,C\_\{n\}=F\_\{n\}^\{\*\}\\operatorname\{diag\}\(\\gamma\_\{n,0\},\\ldots,\\gamma\_\{n,n\-1\}\)F\_\{n\},\\qquad W\_\{r\}^\{\(n\)\}=\\sigma\_\{r\}C\_\{n\},whereFnF\_\{n\}is the unitary discrete Fourier matrix\. Assume that the empirical distribution of the uniformly bounded Fourier eigenvalues converges weakly toν\\nu\. Then

ηk,ℓcirc=∫ℂ\(1−α\+α​σr​z\)k​\(1−α\+α​σr​z¯\)ℓ​𝑑ν​\(z\)\.\\eta\_\{k,\\ell\}^\{\\rm circ\}=\\int\_\{\\mathbb\{C\}\}\(1\-\\alpha\+\\alpha\\sigma\_\{r\}z\)^\{k\}\(1\-\\alpha\+\\alpha\\sigma\_\{r\}\\overline\{z\}\)^\{\\ell\}\\,d\\nu\(z\)\.\(62\)If the matrices are generated by a limiting Fourier symbolg:\[0,2​π\]→ℂg:\[0,2\\pi\]\\to\\mathbb\{C\}, define

aθ​\(ω\)=1−α\+α​σr​g​\(ω\)\.a\_\{\\theta\}\(\\omega\)=1\-\\alpha\+\\alpha\\sigma\_\{r\}g\(\\omega\)\.Then

ηk,ℓcirc=12​π​∫02​πaθ​\(ω\)k​aθ​\(ω\)¯ℓ​𝑑ω\.\\eta\_\{k,\\ell\}^\{\\rm circ\}=\\frac\{1\}\{2\\pi\}\\int\_\{0\}^\{2\\pi\}a\_\{\\theta\}\(\\omega\)^\{k\}\\overline\{a\_\{\\theta\}\(\\omega\)\}^\{\\,\\ell\}\\,d\\omega\.BecauseAnk​\(Anℓ\)⊤A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\\top\}is circulant, every diagonal entry equals its normalized trace; diagonal self\-averaging is therefore automatic\. Complete history follows whenever

supω∈\[0,2​π\]\|aθ​\(ω\)\|<1\.\\sup\_\{\\omega\\in\[0,2\\pi\]\}\|a\_\{\\theta\}\(\\omega\)\|<1\.The cyclic shift is the special caseg⁡\(ω\)=ei​ωg\(\\omega\)=e^\{i\\omega\}\.

### D\-CGaussian skew\-symmetric matrices

Let

Sn=Gn−Gn⊤2​n,Wr\(n\)=σr​Sn\.S\_\{n\}=\\frac\{G\_\{n\}\-G\_\{n\}^\{\\top\}\}\{\\sqrt\{2n\}\},\\qquad W\_\{r\}^\{\(n\)\}=\\sigma\_\{r\}S\_\{n\}\.SinceSn⊤=−SnS\_\{n\}^\{\\top\}=\-S\_\{n\}, the Hermitian matrixi​SniS\_\{n\}has the standard semicircular limit\[[27](https://arxiv.org/html/2608.20998#bib.bib10)\]\. Writing

d​μsc​\(x\)=4−x22​π​𝟏\[−2,2\]​\(x\)​d​x,d\\mu\_\{\\rm sc\}\(x\)=\\frac\{\\sqrt\{4\-x^\{2\}\}\}\{2\\pi\}\\mathbf\{1\}\_\{\[\-2,2\]\}\(x\)\\,dx,the coefficients are

ηk,ℓskew=∫−22\(1−α\+i​α​σr​x\)k​\(1−α−i​α​σr​x\)ℓ​d​μsc​\(x\)\.\\eta\_\{k,\\ell\}^\{\\rm skew\}=\\int\_\{\-2\}^\{2\}\(1\-\\alpha\+i\\alpha\\sigma\_\{r\}x\)^\{k\}\(1\-\\alpha\-i\\alpha\\sigma\_\{r\}x\)^\{\\ell\}\\,d\\mu\_\{\\rm sc\}\(x\)\.\(63\)Permutation symmetry and the Gaussian\-Poincaré argument of Lemma[2](https://arxiv.org/html/2608.20998#Thmlemma2)give diagonal self\-averaging\. SinceAnA\_\{n\}is normal, complete history follows under the asymptotic condition

\(1−α\)2\+4​α2​σr2<1\.\\sqrt\{\(1\-\\alpha\)^\{2\}\+4\\alpha^\{2\}\\sigma\_\{r\}^\{2\}\}<1\.

### D\-DComplex\-valued matrices

For complex recurrent matrices, transpose is replaced by the Hermitian adjoint\. Normalized complex Ginibre and scaled Haar\-unitary matrices again satisfy

ηk,ℓℂ:=limn→∞1n​Tr⁡\(Ank​\(Anℓ\)∗\)=τk,ℓ\\eta\_\{k,\\ell\}^\{\\mathbb\{C\}\}:=\\lim\_\{n\\to\\infty\}\\frac\{1\}\{n\}\\operatorname\{Tr\}\\\!\\left\(A\_\{n\}^\{k\}\(A\_\{n\}^\{\\ell\}\)^\{\*\}\\right\)=\\tau\_\{k,\\ell\}\[[27](https://arxiv.org/html/2608.20998#bib.bib10),[6](https://arxiv.org/html/2608.20998#bib.bib12)\]\. The corresponding Hermitian linear\-state covariance is obtained from \([36](https://arxiv.org/html/2608.20998#A1.E36)\) by replacing the input inner product withut−1−k∗​us−1−ℓu\_\{t\-1\-k\}^\{\*\}u\_\{s\-1\-\\ell\}\. A nonlinear feature kernel additionally requires a specified complex activation or a realification convention\. The experiments in this paper use real\-valued recurrent matrices\.

TABLE VI:Recurrent\-matrix catalogue\. The coefficient column gives the quantities substituted forτk,ℓ\\tau\_\{k,\\ell\}in \([36](https://arxiv.org/html/2608.20998#A1.E36)\); the nonlinear kernel is then formed as in \([39](https://arxiv.org/html/2608.20998#A1.E39)\)\.

## Appendix EFull Task\-Level Deployment Scores

Table[VII](https://arxiv.org/html/2608.20998#A5.T7)gives per\-task scores for the four primary selectors, including ESN Direct500, at all five deployment widths \(n∈\{1 000,3 000,5 000,10 000,20 000\}n\\in\\\{1\\,000,3\\,000,5\\,000,10\\,000,20\\,000\\\}\), while Figure[4](https://arxiv.org/html/2608.20998#A5.F4)shows the corresponding width trends\.

TABLE VII:Full task\-level deployment scores under the holdout\-ridge protocol\. Values are1−NRMSE1\-\\operatorname\{NRMSE\}mean±\\pmstd over deployment seeds\. Intermediate widths use three deployment seeds;n=20 000n=20\\,000uses ten deployment seeds\. Best score per task and width is bolded; second\-best is underlined\. Identical FP and Direct500rows \(e\.g\., MC\) indicate that both selectors chose the same configuration\.![Refer to caption](https://arxiv.org/html/2608.20998v1/plots/score_vs_n_all_tasks.png)Fig\. 4:Deployment score vs\. reservoir widthnnthroughn=20 000n=20\\,000for the ten\-task, four\-selector comparison\. Shaded bands show±1\\pm 1s\.d\. over deployment seeds; then=20 000n=20\\,000column uses ten deployment seeds and the intermediate widths use three\. Higher is better\.
## Appendix FSelection\-Length Sensitivity and Selection Time

Table[VIII](https://arxiv.org/html/2608.20998#A6.T8)and Figures[6](https://arxiv.org/html/2608.20998#A6.F6)–[6](https://arxiv.org/html/2608.20998#A6.F6)report the auxiliary coarse\-grid sweep over the selection\-sequence lengthTselectT\_\{\\rm select\}summarized in Section[IV\-B](https://arxiv.org/html/2608.20998#S4.SS2): deployment performance atn=20 000n=20\\,000and per\-task selection CPU time as a function ofTselectT\_\{\\rm select\}, atnselect=500n\_\{\\rm select\}=500\.

TABLE VIII:Selection\-length sensitivity in an auxiliary coarse\-grid synthetic sweep\. Scores are deployment1−NRMSE1\-\\operatorname\{NRMSE\}atn=20 000n=20\\,000, averaged over the ten synthetic tasks;±\\pmdenotes task\-level standard deviation\. The selection lengthTselectT\_\{\\rm select\}is used only during hyperparameter selection\. FP uses the closed\-form erf\-surrogate selection kernel from Remark[1](https://arxiv.org/html/2608.20998#Thmremark1); matched exact\-tanh\\tanhchecks are reported in Sec\.[F\-A](https://arxiv.org/html/2608.20998#A6.SS1)\. Memory500usesD=min⁡\(Tselect,1000\)D=\\min\(T\_\{\\rm select\},1000\)delayed targets and selects the same candidate throughout this sweep\. Best entries are bolded and second\-best entries are underlined\.![Refer to caption](https://arxiv.org/html/2608.20998v1/plots/selection_length_sensitivity.png)Fig\. 5:Mean deployment score atn=20 000n=20\\,000as a function ofTselectT\_\{\\rm select\}\. Higher is better\. FP reaches its best observed mean atTselect=500T\_\{\\rm select\}=500; Direct500and ESN Direct500improve with longer sequences\. Memory500selects the same candidate across this sweep and is therefore flat; its proxy usesD=min⁡\{Tselect,1000\}D=\\min\\\{T\_\{\\rm select\},1000\\\}\.
![Refer to caption](https://arxiv.org/html/2608.20998v1/plots/fig_compute_time_vs_T_n500_avg.png)Fig\. 6:Mean selection CPU time per task averaged over the ten\-task synthetic suite as a function ofTselectT\_\{\\rm select\}\(nselect=500n\_\{\\rm select\}=500\)\. All points are measured runs\. FP uses the closed\-form erf\-surrogate selection kernel\. Lower is better\.

### F\-ASurrogate\-kernel validation

The main synthetic selection and timing experiments use the closed\-form erf surrogate of Remark[1](https://arxiv.org/html/2608.20998#Thmremark1)\. All real\-data forecasting experiments, both ETT and Telco, use the exacttanh\\tanhkernel evaluated by Gauss–Hermite quadrature\. The surrogate is therefore used only as a computational accelerator in the controlled synthetic setting\.

We validated this choice by comparing matched erf\-surrogate and exact\-tanh\\tanhFP selections on the synthetic suite\. Across the checked selection lengths, the largest task\-level deployment\-score difference was6\.3×10−36\.3\\times 10^\{\-3\}, and the mean difference wasO⁡\(10−3\)O\(10^\{\-3\}\), well below the task\-level variability in the deployment tables\. This supports using the surrogate for the synthetic selection and timing sweeps\. By contrast, the real\-data experiments are reported with the exacttanh\\tanhkernel, so no real\-data conclusion relies on the surrogate approximation\.

## Appendix GForecasting Horizon\-Level Results

Table[IX](https://arxiv.org/html/2608.20998#A7.T9)gives the underlying horizon\-level scores\.

TABLE IX:Horizon\-level forecasting results atn=20 000n=20\\,000\. Scores are1−NRMSE1\-\\operatorname\{NRMSE\}; higher is better\. ETTh is sampled hourly, so horizon steps coincide with physical hours; ETTm is sampled every 15 minutes, and horizons are given in steps with the physical horizon in parentheses\. ETT deviations are over three deployment seeds for deterministic selectors; Telco deviations are across ten cells\. Telco selection optimizes the aggregate over horizons 1–12; the table shows three representative horizons, so their mean differs from the aggregate scores of Table IV of the main paper\.
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