长程记忆的代价：序列模型中的状态、上下文与稳定性复杂度

# The Cost of Long Memory: State, Context, and Stability Complexity in Sequence Models
Source: [https://arxiv.org/html/2610.08816](https://arxiv.org/html/2610.08816)
Yuheng SongAffiliation:School of Management, Huazhong University of Science and TechnologyEmail:[slark4560@gmail\.com](mailto:)

###### Abstract

Long\-range temporal dependence poses a resource question for sequence models: for a specified predictive\-memory law, how much state, context, or dynamical criticality is required in order to forecast accurately? We study this question directly in forecasting risk\. For algebraically decaying predictive memory, we prove matching upper and lower approximation bounds for exponential and finite\-state modes\. The bestrr\-mode forecast error decays ase−Θ⁡\(r\)e^\{\-\\Theta\(\\sqrt\{r\}\)\}, so reaching forecast errorτ\\tauneedsr=Θ⁡\(log2⁡\(1/τ\)\)r=\\Theta\(\\log^\{2\}\(1/\\tau\)\)states or modes\. Earlier curse\-of\-memory results establish broad limitations of stable recurrent models under different approximation notions; here both sides match for one canonical predictive target in forecast risk, which fixes the optimal resource exponent for that target\. We then show that genuine fractional long memory changes the geometry itself\. In particular, forecast error is measured after fractional integration, prediction from a finite context of lengthLLhas an exact1/L1/Lleading order, and a fixed fractional strengthddkeeps the square\-log state\-complexity law\. Near the short\-memory boundary, we identify the relevantd2d^\{2\}andd4d^\{4\}scales and give a uniform constructive law in the intermediate regime\. For nonlinear contextual recurrences with uniformly contractive state dynamics, we derive an exponential first\-chaos envelope and an explicit necessary condition that relates forecast accuracy to the contraction margin\. Vanishing forecasting error on an algebraic target forces the recurrence quantitatively toward criticality, a condition that is necessary and not by itself sufficient\. Finite\-sample Kullback–Leibler calculations further connect the predictive geometry to statistical information\. Theorem\-matched experiments with contractive state\-space, gated recurrent, and attention models reproduce the state and stability predictions\. Matched exponential\-memory controls locate the floor in the compatibility between the target tail and the contraction margin rather than in algebraic decay as a functional form, since an exponential target matched to the contraction is fit almost exactly\.

## 1Introduction

Long\-context sequence modeling is often discussed in qualitative terms\. Some architectures are said to remember longer, some recurrent parameterizations are more stable, and attention can expose an arbitrarily large finite context\. For forecasting, long memory is not a binary property: a target process specifies a concrete predictive\-memory law, and an architecture should be judged by the forecasting error it achieves with a finite amount of representational resource\.

This paper studies three such resources\.*State complexity*asks how many exponential modes or recurrent states are required to approximate a target memory to errorτ\\tau\.*Context complexity*asks how much observed history is needed before finite\-past prediction approaches the infinite\-past Bayes risk\.*Stability complexity*asks, for a nonlinear recurrent model constrained to stay uniformly contractive, how close to the stability boundary it must move before an algebraic predictive memory can be approximated accurately\.

A growing literature already studies the curse of memory in recurrent and state\-space models, robust long\-memory parameterizations, and latent\-state resources\. Given the well\-known broad fact that uniformly stable recurrence tends to forget exponentially, we ask a sharper target\-specific question\.

*Given a predictive\-memory law, what is the optimal forecasting error achievable with a fixed state dimension, context length, or stability margin?*

This target\-specific view yields matching resource laws, so the exponent itself is fully determined\.

##### Our results\.

We obtain four results\. Throughout,\(Xt\)t∈ℤ\(X\_\{t\}\)\_\{t\\in\\mathbb\{Z\}\}is a centered stationary process and the task is one\-step forecasting from its own past: a memory kernelb=\(bj\)j≥1b=\(b\_\{j\}\)\_\{j\\geq 1\}gives the predictormt​\(b\)=∑j≥1bj​Xt\+1−jm\_\{t\}\(b\)=\\sum\_\{j\\geq 1\}b\_\{j\}X\_\{t\+1\-j\}, and the best such predictor is the Bayes forecastmt⋆=𝔼\[Xt\+1∣Xs,s≤t\]m\_\{t\}^\{\\star\}=\\mathbb\{E\}\[X\_\{t\+1\}\\mid X\_\{s\},\\,s\\leq t\], whose kernela⋆=\(aj⋆\)j≥1a^\{\\star\}=\(a^\{\\star\}\_\{j\}\)\_\{j\\geq 1\}we call the predictive memory\. The excess forecast risk of a candidatebbis the extra one\-step mean squared error it incurs,

ℰ⁡\(b\)=𝔼​\(Xt\+1−mt​\(b\)\)2−𝔼​\(Xt\+1−mt⋆\)2=𝔼​\(∑j≥1\(bj−aj⋆\)​Xt\+1−j\)2,\\mathcal\{E\}\(b\)=\\mathbb\{E\}\\bigl\(X\_\{t\+1\}\-m\_\{t\}\(b\)\\bigr\)^\{2\}\-\\mathbb\{E\}\\bigl\(X\_\{t\+1\}\-m\_\{t\}^\{\\star\}\\bigr\)^\{2\}=\\mathbb\{E\}\\Bigl\(\\sum\_\{j\\geq 1\}\(b\_\{j\}\-a^\{\\star\}\_\{j\}\)X\_\{t\+1\-j\}\\Bigr\)^\{2\},\(1\)where the second equality holds because the Bayes error is orthogonal to the past\. This is a quadratic form in the coefficient error, and its geometry is fixed by the spectrum ofXX\. The architecture class𝒜r\\mathcal\{A\}\_\{r\}consists of the memory kernels generated by anrr\-dimensional linear state and truncated to a finite context, with therr\-mode exponential kernels as its diagonal subclass\.

For stable algebraic predictive memoryaj⋆=c​j−αa^\{\\star\}\_\{j\}=cj^\{\-\\alpha\}with decay exponentα\>1\\alpha\>1, we prove matching upper and lower bounds

c1​e−C1​r≤infb∈𝒜rℰ⁡\(b\)≤C2​e−c2​r,c\_\{1\}e^\{\-C\_\{1\}\\sqrt\{r\}\}\\leq\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)\\leq C\_\{2\}e^\{\-c\_\{2\}\\sqrt\{r\}\},and therefore, within this architecture class and for this target, the state or head complexityr⋆​\(τ\)r^\{\\star\}\(\\tau\), the smallestrrthat reaches forecast errorτ\\tau, isΘ⁡\(log2⁡\(1/τ\)\)\\Theta\(\\log^\{2\}\(1/\\tau\)\)\(Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)\)\. The upper bound comes from exponential\-sum compression and the lower bound from the Hankel spectrum of the target, so the resource question for this canonical long\-memory target is settled on both sides\. For fractional long memory with a fixed fractional strengthdd, the coefficient error is no longer measured in the ordinaryℓ2\\ell\_\{2\}\-norm, since the forecast\-risk norm applies a fractional\-integration operator \(Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)\)\. In this geometry, the gap between prediction from the most recentLLobservations and prediction from the infinite past has the sharp leading orderVL−V∞∼V∞​d2/LV\_\{L\}\-V\_\{\\infty\}\\sim V\_\{\\infty\}d^\{2\}/L, whereVLV\_\{L\}andV∞V\_\{\\infty\}are the corresponding Bayes prediction variances\. A fixedddkeeps the square\-log state\-complexity law, and the short\-memory limitd→0d\\to 0exposes thed2d^\{2\}andd4d^\{4\}boundary scales and admits a uniform constructive law between them \(Corollaries[3](https://arxiv.org/html/2610.08816#Thmtheorem3)and[4](https://arxiv.org/html/2610.08816#Thmtheorem4)and Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)\)\. For an innovation\-driven genuinely nonlinear contextual recurrence whose state Jacobian contracts uniformly with factorρ<1\\rho<1and whose input and readout derivatives are bounded byKKandMM, the lag\-jjfirst\-chaos memory coefficientβj\\beta\_\{j\}, the projection of the predictor onto thejjth past innovation, satisfies\|βj\|≤M​K​ρj−1\|\\beta\_\{j\}\|\\leq MK\\rho^\{j\-1\}forj≥2j\\geq 2, which yields an explicit forecasting floor on algebraic targets and the necessary condition−log⁡ρ≲τ1/\(2​α−1\)​log⁡\(e/τ\)\-\\log\\rho\\lesssim\\tau^\{1/\(2\\alpha\-1\)\}\\log\(e/\\tau\)\(Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)and Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\)\. This is a forecasting\-specific quantitative refinement of the established curse of memory\. Finally, in Gaussian causal models, the trajectory\-level conditional Kullback–Leibler divergence is exactly the accumulated forecast risk, and for fractional stationary blocks we control the finite\-section Fisher–Hartwig boundary and extend the logarithmic correction to any fixed number of reflection\-symmetric moving or merging fractional poles under a cluster\-integrability margin \(Prop\.[8](https://arxiv.org/html/2610.08816#Thmtheorem8)and discussions\)\.

### 1\.1Related works

##### The curse of memory in recurrent and state\-space models\.

A substantial literature shows that stable recurrent and state\-space models face long\-memory limitations\. These results formalize related obstructions under different notions rather than one common metric\.[Li et al\. \(2021\)](https://arxiv.org/html/2610.08816#bib.bib32)work with worst\-case functional approximation and the optimization dynamics that accompany it,[Wang et al\. \(2024\)](https://arxiv.org/html/2610.08816#bib.bib45)establish inverse\-approximation results in that same worst\-case sense for nonlinear recurrent networks, and[Wang & Xue \(2023\)](https://arxiv.org/html/2610.08816#bib.bib44)characterize exponential decay for stacked state\-space models through a derivative\-based memory function\. All three are distinct from the stationary forecast\-risk geometry used here\. StableSSM relates stable approximation to memory decay and proposes reparameterizations that ease optimization\([Wang & Li, 2024](https://arxiv.org/html/2610.08816#bib.bib43)\)\. HOPE uses Hankel coordinates to obtain a robust long\-memory state\-space parameterization\([Yu et al\., 2025b](https://arxiv.org/html/2610.08816#bib.bib52)\), initialization and timescale choices are shown to control the effective memory of a diagonal state\([Liu & Li, 2025](https://arxiv.org/html/2610.08816#bib.bib34)\), and the transfer function of a linear time\-invariant layer carries a frequency bias that can be tuned\([Yu et al\., 2025a](https://arxiv.org/html/2610.08816#bib.bib51)\)\. Selective architectures are known to have a receptive field shorter than their nominal context\. DeciMamba and LongMamba enlarge that receptive field in pretrained models without additional training\([Ben\-Kish et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib5);[Ye et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib49)\), whereas Block\-Biased Mamba instead modifies the S6 unit itself, combining block\-wise selective dynamics with a channel\-specific bias to improve expressiveness, inductive bias, and training stability on long\-range tasks\([Yu & Erichson, 2025](https://arxiv.org/html/2610.08816#bib.bib50)\)\. Our contribution, however, is a forecast\-risk envelope whose constants carry both the target tail and the contraction margin, so that the obstruction can be inverted into an explicit accuracy\-to\-criticality requirement\.

##### State resources and long\-context modeling\.

Recent theory asks how the latent\-state dimension should scale with long\-context information\. Information\-theoretic arguments lower bound the latent state a model needs in order to carry the mutual information present in long contexts\([Chen et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib10)\), and the architectures for which that question is asked are the modern state\-space and gated recurrent families\([Gu et al\., 2022](https://arxiv.org/html/2610.08816#bib.bib24);[Smith et al\., 2023](https://arxiv.org/html/2610.08816#bib.bib38);[Gu & Dao, 2023](https://arxiv.org/html/2610.08816#bib.bib23);[Dao & Gu, 2024](https://arxiv.org/html/2610.08816#bib.bib12);[Yang et al\., 2024](https://arxiv.org/html/2610.08816#bib.bib47);[Yang et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib48)\)\. Our formulation fixes the target predictive\-memory law instead and asks for the optimal approximation error as a function of state dimension, which complements those results\. This produces the explicit root\-exponential curve and its square\-log inverse\.

##### Power\-law and fractional architectures\.

Power\-law memories can be implemented through sums of exponentials and have recently appeared in long\-context architectures\. For example, VORT uses adaptive power\-law memory together with recurrent exponential compression\([Mlaiki, 2026](https://arxiv.org/html/2610.08816#bib.bib35)\)\. We study a different metric and a different asymptotic question, in which the horizon is not fixed externally, and approximation is evaluated in forecast risk over the infinite predictive law\. Classical exponential\-sum approximation results\([Beylkin & Monzón, 2005](https://arxiv.org/html/2610.08816#bib.bib7);[Beylkin & Monzón, 2010](https://arxiv.org/html/2610.08816#bib.bib8);[Braess & Hackbusch, 2005](https://arxiv.org/html/2610.08816#bib.bib9)\)supply our constructive tools, together with quadrature designed for fractional kernels\([Baffet, 2019](https://arxiv.org/html/2610.08816#bib.bib3)\)\. The matching lower bound combines the Eckart–Young principle for unconstrained low\-rank approximation in Frobenius norm\([Eckart & Young, 1936](https://arxiv.org/html/2610.08816#bib.bib15)\)with the singular\-value asymptotics of Hankel operators\([Widom, 1966](https://arxiv.org/html/2610.08816#bib.bib46)\), for which[Adamjan et al\. \(1971\)](https://arxiv.org/html/2610.08816#bib.bib1);[Peller \(2003\)](https://arxiv.org/html/2610.08816#bib.bib36)provide the structured approximation background\.

##### Fractional prediction and Fisher–Hartwig theory\.

Fractionally differenced processes are classical\([Granger & Joyeux, 1980](https://arxiv.org/html/2610.08816#bib.bib21);[Hosking, 1981](https://arxiv.org/html/2610.08816#bib.bib26);[Beran, 1994](https://arxiv.org/html/2610.08816#bib.bib6)\), and the finite\-past prediction and partial\-autocorrelation theory we use is developed by[Inoue \(2002\)](https://arxiv.org/html/2610.08816#bib.bib29);[Inoue & Kasahara \(2006\)](https://arxiv.org/html/2610.08816#bib.bib30)\. Asymptotic estimation of the memory parameter is a separate line\([Fox & Taqqu, 1986](https://arxiv.org/html/2610.08816#bib.bib18);[Giraitis et al\., 1997](https://arxiv.org/html/2610.08816#bib.bib19);[Lieberman & Phillips, 2004](https://arxiv.org/html/2610.08816#bib.bib33);[Hualde & Robinson, 2011](https://arxiv.org/html/2610.08816#bib.bib27)\)\. When the location of the pole is itself unknown, the problem changes character\([Giraitis et al\., 2001](https://arxiv.org/html/2610.08816#bib.bib20);[Hidalgo, 2005](https://arxiv.org/html/2610.08816#bib.bib25);[Arteche, 2020](https://arxiv.org/html/2610.08816#bib.bib2)\)\. Classical Toeplitz theory supplies the operator and determinant background\([Grenander & Szegő, 1958](https://arxiv.org/html/2610.08816#bib.bib22)\), while Fisher–Hartwig asymptotics treat symbols with singularities\([Ehrhardt & Silbermann, 1997](https://arxiv.org/html/2610.08816#bib.bib16);[Deift et al\., 2011](https://arxiv.org/html/2610.08816#bib.bib14)\)\. The uniform and merging\-singularity regimes are what our fixed\-pole\-count statement consumes\([Claeys & Krasovsky, 2015](https://arxiv.org/html/2610.08816#bib.bib11);[Fahs, 2021](https://arxiv.org/html/2610.08816#bib.bib17)\)\. Our contribution is the sequence\-model resource interpretation, in which the singular spectrum changes the predictive norm, the context law, and the state or head complexity relevant to forecasting\.

## 2Predictive Memory as a Resource Problem

### 2\.1Forecast\-risk geometry

Let the Bayes one\-step predictor have memorya⋆=\(aj⋆\)j≥1a^\{\\star\}=\(a\_\{j\}^\{\\star\}\)\_\{j\\geq 1\}and letℰ⁡\(a\)\\mathcal\{E\}\(a\)denote the excess forecast risk \([1](https://arxiv.org/html/2610.08816#S1.E1)\)\. WriteA⁡\(z\)=∑j≥1aj​zjA\(z\)=\\sum\_\{j\\geq 1\}a\_\{j\}z^\{j\}for the transfer function of a coefficient sequenceaa, andA⋆A^\{\\star\}for that ofa⋆a^\{\\star\}\. For a stationary process with spectral densityfXf\_\{X\}, Parseval turns \([1](https://arxiv.org/html/2610.08816#S1.E1)\) into the spectral form

ℰ⁡\(a\)=12​π​∫−ππ\|A⁡\(e−i​ω\)−A⋆​\(e−i​ω\)\|2​fX​\(ω\)​𝑑ω\.\\mathcal\{E\}\(a\)=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|A\(e^\{\-i\\omega\}\)\-A^\{\\star\}\(e^\{\-i\\omega\}\)\|^\{2\}f\_\{X\}\(\\omega\)\\,\\mathrm\{d\}\\omega\.\(2\)We write‖u‖pred,fX2\\\|u\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}for the same quadratic form applied to a coefficient perturbationuu, so thatℰ⁡\(a\)=‖a−a⋆‖pred,fX2\\mathcal\{E\}\(a\)=\\\|a\-a^\{\\star\}\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}, and call it the predictive norm\. Under a regular spectrum, meaning0<m≤fX≤M<∞0<m\\leq f\_\{X\}\\leq M<\\infty, this equals constant coefficientℓ22\\ell\_\{2\}^\{2\}error\. Under genuine fractional long memory, the upper equivalence fails, and Sec\.[4](https://arxiv.org/html/2610.08816#S4)identifies the correct singular geometry\.

### 2\.2Resources, targets, and regimes

Fix the architecture class𝒜r\\mathcal\{A\}\_\{r\}ofrr\-state kernels truncated to a finite context, taken as a union over contexts, and write its best forecast error asinfb∈𝒜rℰ⁡\(b\)\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)\. Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)states the class precisely, and App\.[A](https://arxiv.org/html/2610.08816#A1)verifies its Hankel\-rank property\. The state\-complexity function is

r⋆​\(τ\)=min⁡\{r:infb∈𝒜rℰ⁡\(b\)≤τ\},r^\{\\star\}\(\\tau\)=\\min\\\{r:\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)\\leq\\tau\\\},andrd⋆​\(τ\)r\_\{d\}^\{\\star\}\(\\tau\)denotes the same quantity withℰ\\mathcal\{E\}replaced by the fractional riskℰd\\mathcal\{E\}\_\{d\}of Sec\.[4](https://arxiv.org/html/2610.08816#S4)\. For finite context, letVLV\_\{L\}be the Bayes prediction variance from the most recentLLobservations andV∞V\_\{\\infty\}its infinite\-past limit\. For a nonlinear recurrence, the stability resource is the contraction marginδ=−log⁡ρ\\delta=\-\\log\\rho, where‖Ds​F‖op≤ρ<1\\\|D\_\{s\}F\\\|\_\{\\rm op\}\\leq\\rho<1\. Two target families carry these questions: algebraic memoryaj⋆=c​j−αa^\{\\star\}\_\{j\}=cj^\{\-\\alpha\}with amplitudeccand exponentα\>1\\alpha\>1, and fractional memory of strengthd∈\(0,1/2\)d\\in\(0,1/2\)\. Sec\.[3](https://arxiv.org/html/2610.08816#S3),[4](https://arxiv.org/html/2610.08816#S4), and[5](https://arxiv.org/html/2610.08816#S5)analyze the three resources in turn, and Table[1](https://arxiv.org/html/2610.08816#S2.T1)states the resulting laws\.

The three questions are asked in different regimes\. Sec\.[3](https://arxiv.org/html/2610.08816#S3)and[5](https://arxiv.org/html/2610.08816#S5)assume a stationary purely nondeterministic process with a regular spectrum, which is what keeps forecast risk comparable to coefficient error\. Sec\.[4](https://arxiv.org/html/2610.08816#S4)deliberately drops that assumption for a single spectral pole with a bounded regular factor\. Sec\.[5](https://arxiv.org/html/2610.08816#S5)assumes the derivative bounds on a forward\-invariant domain containing the stationary solution\. Full statements are in App\.[A](https://arxiv.org/html/2610.08816#A1),[C](https://arxiv.org/html/2610.08816#A3), and[D](https://arxiv.org/html/2610.08816#A4)\.

Table 1:The three resource laws\. The state law is two\-sided, the context law is exact at leading order, and the stability law is necessary and not sufficient\. The related\-work discussion of the introduction places each against the nearest existing line\.

## 3Sharp State Complexity of Algebraic Memory

A recurrent layer carrying anrr\-dimensional linear state, with transition matrixA∈ℝr×rA\\in\\mathbb\{R\}^\{r\\times r\}, input mapv∈ℝrv\\in\\mathbb\{R\}^\{r\}, and readoutu∈ℝru\\in\\mathbb\{R\}^\{r\}, generates the memory coefficientsbj=u⊤​Aj−1​vb\_\{j\}=u^\{\\top\}A^\{j\-1\}v, and therefore finite Hankel matrices of rank at mostrr; the resourcerrcan be read as the number of exponential temporal heads, diagonal recurrent modes, or linear state dimensions inside that class\. A positive algebraic memory, on the other side, admits a Laplace representation and can be compressed by quadrature into a sum of exponentials\. These two facts are what make the two directions meet, the rank constraint bounding every candidate from below and the quadrature construction supplying a candidate from above\.

###### Theorem 1\(Sharp state complexity of algebraic predictive memory\)\.

Consider a stable regular\-spectrum forecasting model with target memoryaj⋆=c​j−αa\_\{j\}^\{\\star\}=cj^\{\-\\alpha\},α\>1\\alpha\>1\. Let𝒜r\\mathcal\{A\}\_\{r\}denote the union, over finite contexts, ofrr\-dimensional linear\-state kernels restricted to that context, therr\-mode exponential kernels forming the diagonal subclass\. There exist constantsc1,c2,C1,C2\>0c\_\{1\},c\_\{2\},C\_\{1\},C\_\{2\}\>0andr0r\_\{0\}, depending only on the fixed model, such that forr≥r0r\\geq r\_\{0\},

c1​e−C1​r≤infb∈𝒜rℰ⁡\(b\)≤C2​e−c2​r\.c\_\{1\}e^\{\-C\_\{1\}\\sqrt\{r\}\}\\leq\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)\\leq C\_\{2\}e^\{\-c\_\{2\}\\sqrt\{r\}\}\.\(3\)Equivalently, we have

r⋆​\(τ\)=Θα​\(log2⁡1τ\)\(τ→0\)\.r^\{\\star\}\(\\tau\)=\\Theta\_\{\\alpha\}\\\!\\left\(\\log^\{2\}\\frac\{1\}\{\\tau\}\\right\)\\qquad\(\\tau\\to 0\)\.\(4\)The same exponent holds for the analytic algebraic Laplace\-mixture class stated in App\.[B](https://arxiv.org/html/2610.08816#A2)\.

Curse\-of\-memory and inverse\-approximation results establish limitations of stable recurrence on slowly decaying targets\([Li et al\., 2021](https://arxiv.org/html/2610.08816#bib.bib32);[Wang et al\., 2024](https://arxiv.org/html/2610.08816#bib.bib45)\), and modern state\-space work develops parameterizations aimed at improving long\-memory representation and optimization\([Wang & Li, 2024](https://arxiv.org/html/2610.08816#bib.bib43);[Yu et al\., 2025b](https://arxiv.org/html/2610.08816#bib.bib52)\)\. Those statements are made under different approximation notions and over different target classes, and App\.[B](https://arxiv.org/html/2610.08816#A2)separates the three notions before proving anything\. Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)settles a narrower question completely: for one canonical algebraic predictive target measured in forecast risk, both sides of the approximation curve match, which fixes the resource exponent rather than establishing only that the resource must grow\. The class is the linear\-state one just described, and Sec\.[5](https://arxiv.org/html/2610.08816#S5)treats a broader nonlinear class under an explicit contraction hypothesis\.

For the upper bound, the Laplace representation ofj−αj^\{\-\\alpha\}and positive exponential quadrature give a finite\-context relative approximation withr=O⁡\(log2⁡\(1/τ\)\)r=O\(\\log^\{2\}\(1/\\tau\)\)modes\. For the lower bound, everyrr\-state candidate has Hankel rank at mostrr, while finite sections of the algebraic Hankel operator have singular values on a root\-exponential scale\. A corrected Ky Fan comparison transfers that spectral obstruction into forecast risk\. App\.[B](https://arxiv.org/html/2610.08816#A2)contains the full proof and constants\.

## 4Singular Long Memory and Context Complexity

### 4\.1Fractional benchmark and predictive geometry

Consider FARIMA\(0,d,0\)\(0,d,0\), whereBBis the backshift operator\(B​X\)t=Xt−1\(BX\)\_\{t\}=X\_\{t\-1\}and\(1−B\)d\(1\-B\)^\{d\}is its binomial fractional power,

\(1−B\)d​Xt=εt,εt∼iidN⁡\(0,σ2\),0<d<12,\(1\-B\)^\{d\}X\_\{t\}=\\varepsilon\_\{t\},\\qquad\\varepsilon\_\{t\}\\stackrel\{\{\\scriptstyle\\rm iid\}\}\{\{\\sim\}\}N\(0,\\sigma^\{2\}\),\\qquad 0<d<\\tfrac\{1\}\{2\},\(5\)with spectral densityfd​\(ω\)=σ2​\|1−e−i​ω\|−2​df\_\{d\}\(\\omega\)=\\sigma^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\. Its autoregressive predictive\-memory coefficients satisfy

aj​\(d\)=d​Γ​\(j−d\)Γ⁡\(1−d\)​Γ​\(j\+1\)∼dΓ⁡\(1−d\)​j−1−d\.a\_\{j\}\(d\)=\\frac\{d\\,\\Gamma\(j\-d\)\}\{\\Gamma\(1\-d\)\\Gamma\(j\+1\)\}\\sim\\frac\{d\}\{\\Gamma\(1\-d\)\}j^\{\-1\-d\}\.LetIdI\_\{d\}be the discrete fractional\-integration map with transfer function\(1−z\)−d\(1\-z\)^\{\-d\}, and writeℰd\\mathcal\{E\}\_\{d\}for the excess forecast risk \([2](https://arxiv.org/html/2610.08816#S2.E2)\) taken under the fractional model \([5](https://arxiv.org/html/2610.08816#S4.E5)\)\.

###### Theorem 2\(Fractional predictive geometry\)\.

Suppose

fX​\(ω\)=\|1−e−i​ω\|−2​d​g​\(ω\),0<mg≤g⁡\(ω\)≤Mg<∞\.f\_\{X\}\(\\omega\)=\|1\-e^\{\-i\\omega\}\|^\{\-2d\}g\(\\omega\),\\qquad 0<m\_\{g\}\\leq g\(\\omega\)\\leq M\_\{g\}<\\infty\.For every finitely supported coefficient perturbationuu,

mg​‖Id​u‖22≤‖u‖pred,fX2≤Mg​‖Id​u‖22\.m\_\{g\}\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}\\leq\\\|u\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}\\leq M\_\{g\}\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}\.\(6\)For pure fractional noise the identity is exact up to the factorσ2\\sigma^\{2\},

ℰd​\(a⁡\(d\)\+u\)=σ2​‖Id​u‖22\.\\mathcal\{E\}\_\{d\}\(a\(d\)\+u\)=\\sigma^\{2\}\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}\.

Under a spectral pole the relevant geometry departs from ordinary coefficient error, because low\-frequency perturbations are amplified by fractional integration\. The two\-sided bound transports the boundedness ofggthrough the fractional\-integration isometry, and the pure\-noise identity is exact because the fractional density carries no additional regular factor\. The full argument is in App\.[C](https://arxiv.org/html/2610.08816#A3)\.

### 4\.2Consequences for the two resources

The change of geometry moves one resource law and leaves the other in place\.

###### Corollary 3\(Sharp finite\-context prediction law\)\.

For the fractional model \([5](https://arxiv.org/html/2610.08816#S4.E5)\),

VL−V∞=V∞​d2L\+o⁡\(L−1\)\.V\_\{L\}\-V\_\{\\infty\}=V\_\{\\infty\}\\frac\{d^\{2\}\}\{L\}\+o\(L^\{\-1\}\)\.\(7\)For a fixed finite set of separated fractional poles with a sufficiently smooth positive short\-memory envelope, the leading coefficient becomes the corresponding sum of squared pole strengths\.

The1/L1/Llaw differs from the naive unweighted coefficient\-tail calculation, because the predictive metric itself is singular\. The leading order comes from the partial autocorrelations of the fractional process evaluated in the predictive norm, and App\.[C](https://arxiv.org/html/2610.08816#A3)gives the proof\.

###### Corollary 4\(Fractional state complexity\)\.

For every fixedd∈\(0,1/2\)d\\in\(0,1/2\), the bestrr\-dimensional linear\-state approximation in fractional forecast risk satisfies

cd​e−Cd​r≤infb∈𝒜rℰd​\(b\)≤Cd​e−cd​r,rd⋆​\(τ\)=Θd​\(log2⁡1τ\)\.c\_\{d\}e^\{\-C\_\{d\}\\sqrt\{r\}\}\\leq\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\_\{d\}\(b\)\\leq C\_\{d\}e^\{\-c\_\{d\}\\sqrt\{r\}\},\\qquad r\_\{d\}^\{\\star\}\(\\tau\)=\\Theta\_\{d\}\\\!\\left\(\\log^\{2\}\\frac\{1\}\{\\tau\}\\right\)\.\(8\)

The upper bound is attained inside the diagonal subclass, by a positiverr\-mode exponential construction obtained from exponential quadrature applied to the fractionally reweighted kernel, and the lower bound reads off the fractional Hankel spectrum; both are carried out in App\.[C](https://arxiv.org/html/2610.08816#A3)\.

### 4\.3The short\-memory boundary

The fixed\-ddconstants in \([8](https://arxiv.org/html/2610.08816#S4.E8)\) cannot be held fixed asd→0d\\to 0\. In absolute forecast risk the memory itself disappears,

ℰd​\(0\)=σ2​\{Γ⁡\(1−2​d\)Γ​\(1−d\)2−1\}=π2​σ26​d2\+O⁡\(d3\)\.\\mathcal\{E\}\_\{d\}\(0\)=\\sigma^\{2\}\\left\\\{\\frac\{\\Gamma\(1\-2d\)\}\{\\Gamma\(1\-d\)^\{2\}\}\-1\\right\\\}=\\frac\{\\pi^\{2\}\\sigma^\{2\}\}\{6\}d^\{2\}\+O\(d^\{3\}\)\.\(9\)HenceO⁡\(1\)O\(1\)memory suffices onceτ≳σ2​d2\\tau\\gtrsim\\sigma^\{2\}d^\{2\}\. In the intermediate window we have a uniform constructive refinement\.

###### Theorem 5\(Uniform intermediate small\-ddupper bound\)\.

Fixd0<1/2d\_\{0\}<1/2\. There are constantsc0,C0,Cc\_\{0\},C\_\{0\},Csuch that, whenever

0<d≤d0,C0​σ2​d4≤τ≤c0​σ2​d2,0<d\\leq d\_\{0\},\\qquad C\_\{0\}\\sigma^\{2\}d^\{4\}\\leq\\tau\\leq c\_\{0\}\\sigma^\{2\}d^\{2\},there is a truncated positive exponential kernel with forecast risk at mostτ\\tauand

rd⋆​\(τ\)≤C​\[1\+log⁡\(σ2​d2τ\)\]2\.r\_\{d\}^\{\\star\}\(\\tau\)\\leq C\\left\[1\+\\log\\left\(\\frac\{\\sigma^\{2\}d^\{2\}\}\{\\tau\}\\right\)\\right\]^\{2\}\.\(10\)

The construction linearizes the fractional coefficients asaj​\(d\)=d/j\+O⁡\(d2\)a\_\{j\}\(d\)=d/j\+O\(d^\{2\}\)and truncates a positive harmonic exponential kernel, with the details in App\.[C](https://arxiv.org/html/2610.08816#A3)\. A matching uniform lower law in this boundary layer, and the deeper regimeτ≪σ2​d4\\tau\\ll\\sigma^\{2\}d^\{4\}, remain open\.

## 5Quantitative Stability Barriers for Recurrent Memory

### 5\.1Relation to existing curse\-of\-memory theory

Several formal results constrain stable recurrent approximation of slowly decaying targets, under different notions and for different model classes\.[Li et al\. \(2021\)](https://arxiv.org/html/2610.08816#bib.bib32)study worst\-case approximation and the optimization limitations that accompany it for linear recurrent models,[Wang et al\. \(2024\)](https://arxiv.org/html/2610.08816#bib.bib45)give inverse\-approximation results in the same worst\-case sense for nonlinear recurrent networks, and[Wang & Xue \(2023\)](https://arxiv.org/html/2610.08816#bib.bib44)establish exponential memory decay for stacked state\-space models through a derivative\-based memory function\. The notions are not interchangeable, and none of them is the stationary forecast risk used here\. StableSSM and HOPE are representative modern responses to this difficulty\([Wang & Li, 2024](https://arxiv.org/html/2610.08816#bib.bib43);[Yu et al\., 2025b](https://arxiv.org/html/2610.08816#bib.bib52)\)\. We derive a forecast\-risk lower bound whose constants expose the target tail and the contraction margin, which lets the qualitative obstruction be inverted into an explicit requirement linking accuracy and criticality\.

### 5\.2Contextual nonlinear recurrence

The predictors in this section are no longer given by a coefficient kernel, so for a general predictorm^t\\widehat\{m\}\_\{t\}we keep the same symbol and writeℰ⁡\(m^\):=𝔼⁡\[\(mt⋆−m^t\)2\]\\mathcal\{E\}\(\\widehat\{m\}\):=\\mathbb\{E\}\[\(m\_\{t\}^\{\\star\}\-\\widehat\{m\}\_\{t\}\)^\{2\}\]for its excess forecasting risk in this innovation\-driven noiseless Gaussian benchmark, wheremt⋆m\_\{t\}^\{\\star\}is a fixed target functional of the innovation sequence, specialized to the algebraic form in Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)\. LetZt∼iidN⁡\(0,1\)Z\_\{t\}\\stackrel\{\{\\scriptstyle\\rm iid\}\}\{\{\\sim\}\}N\(0,1\)and consider

St=F⁡\(St−1,Zt\),m^t=G⁡\(St,Zt\),S\_\{t\}=F\(S\_\{t\-1\},Z\_\{t\}\),\\qquad\\widehat\{m\}\_\{t\}=G\(S\_\{t\},Z\_\{t\}\),with a stationary solution and derivative budgets on its forward\-invariant state domain

‖Ds​F​\(s,z\)‖op≤ρ<1,‖Dz​F​\(s,z\)‖≤K,‖Ds​G​\(s,z\)‖≤M\.\\\|D\_\{s\}F\(s,z\)\\\|\_\{\\rm op\}\\leq\\rho<1,\\qquad\\\|D\_\{z\}F\(s,z\)\\\|\\leq K,\\qquad\\\|D\_\{s\}G\(s,z\)\\\|\\leq M\.
###### Theorem 6\(Uniform contraction gives an exponential first\-chaos envelope\)\.

Define the lag\-jjfirst\-chaos coefficient

βj=𝔼⁡\[m^t​Zt\+1−j\]\.\\beta\_\{j\}=\\mathbb\{E\}\[\\widehat\{m\}\_\{t\}Z\_\{t\+1\-j\}\]\.Under the regularity conditions in App\.[D](https://arxiv.org/html/2610.08816#A4),

\|βj\|≤M​K​ρj−1,j≥2\.\|\\beta\_\{j\}\|\\leq MK\\rho^\{j\-1\},\\qquad j\\geq 2\.\(11\)For the algebraic Gaussian targetmt⋆=∑j≥1c​j−α​Zt\+1−jm\_\{t\}^\{\\star\}=\\sum\_\{j\\geq 1\}cj^\{\-\\alpha\}Z\_\{t\+1\-j\}withc\>0c\>0andα\>1\\alpha\>1, orthogonal projection onto the first Wiener chaos gives

ℰ⁡\(m^\)≥∑j≥1\(c​j−α−βj\)2\.\\mathcal\{E\}\(\\widehat\{m\}\)\\geq\\sum\_\{j\\geq 1\}\(cj^\{\-\\alpha\}\-\\beta\_\{j\}\)^\{2\}\.\(12\)

The envelope follows from a Gaussian Stein identity together with contraction of the state Jacobian across lags, and the projection bound is orthogonality in the first Wiener chaos\. App\.[D](https://arxiv.org/html/2610.08816#A4)gives the proof\.

### 5\.3Accuracy forces near\-critical dynamics

###### Corollary 7\(Necessary quantitative near\-criticality\)\.

Under the assumptions of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6), in particularα\>1\\alpha\>1, with the derivative budgetsKKandMMand the target amplitudeccheld fixed: For sufficiently smallδ=−log⁡ρ\\delta=\-\\log\\rho,

ℰ⁡\(m^\)≥C​\[δlog⁡\(e/δ\)\]2​α−1\.\\mathcal\{E\}\(\\widehat\{m\}\)\\geq C\\left\[\\frac\{\\delta\}\{\\log\(e/\\delta\)\}\\right\]^\{2\\alpha\-1\}\.\(13\)Consequently, for all sufficiently smallτ\\tau, ifℰ⁡\(m^\)≤τ\\mathcal\{E\}\(\\widehat\{m\}\)\\leq\\tau, then necessarily

−log⁡ρ≤C′​τ1/\(2​α−1\)​log⁡eτ\.\-\\log\\rho\\leq C^\{\\prime\}\\tau^\{1/\(2\\alpha\-1\)\}\\log\\frac\{e\}\{\\tau\}\.\(14\)

The condition is necessary and not by itself sufficient: approaching criticality removes the fixed\-contraction obstruction, and it does not on its own guarantee an accurate representation of algebraic memory\. The lower envelope inverts the exponential first\-chaos tail against the algebraic target, and the sharp form is proved in App\.[D](https://arxiv.org/html/2610.08816#A4)\.

## 6Finite\-Sample Information Consequences

### 6\.1Conditional trajectory information equals forecast risk

Letθ↦aθ\\theta\\mapsto a\_\{\\theta\}index causal Gaussian models as in Sec\.[2](https://arxiv.org/html/2610.08816#S2)with a common innovation varianceσ2\\sigma^\{2\}, letPθ,TP\_\{\\theta,T\}be the law ofX1,…,XTX\_\{1\},\\dots,X\_\{T\}under memoryaθa\_\{\\theta\}, and writeℰθ​\(b\)\\mathcal\{E\}\_\{\\theta\}\(b\)for the excess risk \([2](https://arxiv.org/html/2610.08816#S2.E2)\) of a candidatebbwhen the truth isaθa\_\{\\theta\}\.

###### Proposition 8\(Conditional Gaussian information and forecast identity\)\.

For two Gaussian causal predictors with common innovation varianceσ2\\sigma^\{2\}, conditioning on the infinite past gives

𝔼θDKL\(Pθ,T\(⋅∣ℱ0\)∥Pϑ,T\(⋅∣ℱ0\)\)=T2​σ2ℰθ\(aϑ\)\.\\mathbb\{E\}\_\{\\theta\}\\,D\_\{\\mathrm\{KL\}\}\\\!\\left\(P\_\{\\theta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)\\,\\\|\\,P\_\{\\vartheta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)\\right\)=\\frac\{T\}\{2\\sigma^\{2\}\}\\mathcal\{E\}\_\{\\theta\}\(a\_\{\\vartheta\}\)\.\(15\)

The same predictive geometry therefore controls both approximation and statistical distinguishability\. The identity is the conditional Gaussian relative\-entropy computation given in App\.[E](https://arxiv.org/html/2610.08816#A5)\.

### 6\.2Fractional finite\-section corrections

WritePd\(T\)P\_\{d\}^\{\(T\)\}for the law of a length\-TTblock of the fractional process \([5](https://arxiv.org/html/2610.08816#S4.E5)\),P0\(T\)P\_\{0\}^\{\(T\)\}for the white\-noise law, andD¯\(d∥0\)\\overline\{D\}\(d\\\|0\)for the Kullback–Leibler divergence rate per observation between the two stationary laws\. For fractional noise against white noise,

0≤TD¯\(d∥0\)−DKL\(Pd\(T\)∥P0\(T\)\)≤Cd0d2log\(eT\),0\\leq T\\overline\{D\}\(d\\\|0\)\-D\_\{\\mathrm\{KL\}\}\(P\_\{d\}^\{\(T\)\}\\\|P\_\{0\}^\{\(T\)\}\)\\leq C\_\{d\_\{0\}\}d^\{2\}\\log\(eT\),\(16\)so the finite stationary block has the same local information constant as the entropy\-rate experiment on thed=h/Td=h/\\sqrt\{T\}scale\. For any fixed number of pure fractional poles with a*fixed*strength vector satisfying a uniform cluster\-integrability margin, meaning that the strengths summed over every merging cluster stay bounded away from1/21/2so that each cluster limit remains inside the stationary fractional class, uniform Fisher–Hartwig asymptotics give anO⁡\(log⁡T\)O\(\\log T\)finite\-section correction that is uniform over reflection\-symmetric pole locations even when those locations move withTTand merge\. The uniformity is over locations only; we do not claim the corresponding strength\-uniform refinement for triangular arrays of vanishing pole strengths\. App\.[E](https://arxiv.org/html/2610.08816#A5)gives the exact determinant formulas and states precisely what fails to be uniform when the number or strengths of the singularities vary withTT\.

## 7Experiments

The experiments are matched to the theorems\. App\.[G](https://arxiv.org/html/2610.08816#A7)gives the architecture definitions, the width\-independent budget enforcement, the training and seed protocol, and the complete numerical tables\. Fig\.[1](https://arxiv.org/html/2610.08816#S7.F1)numerically evaluates the deterministic resource quantities from Sec\.[3](https://arxiv.org/html/2610.08816#S3)and[4](https://arxiv.org/html/2610.08816#S4), and Fig\.[2](https://arxiv.org/html/2610.08816#S7.F2)tests the width and stability consequences in controlled sequence models\.

Figure 1:Computed resource laws\. Every point is computed directly from deterministic theoretical quantities, through closed\-form formulas, spectral quadrature, or controlled deterministic optimization\. Panel \(a\) brackets the bestrr\-mode algebraic\-memory error between a constructive exponential\-quadrature upper bound and the Hankel rank\-rrlower bound, on an axis spaced inr\\sqrt\{r\}so thate−Θ⁡\(r\)e^\{\-\\Theta\(\\sqrt\{r\}\)\}appears as a straight line\. The lower curve is evaluated on finite Hankel sections and is therefore a numerical surrogate for the operator singular values, not the operator spectrum itself\. The two exponent constants differ, which is what Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)asserts\. Panel \(b\) shows the exact fractional finite\-past prediction gap normalized byd2d^\{2\}, whose collapse onto one confirms the sharp1/L1/Llaw of Corollary[3](https://arxiv.org/html/2610.08816#Thmtheorem3)\. Panel \(c\) shows the exact zero\-memory fractional risk againstdd, compared with\(π2​σ2/6\)​d2\(\\pi^\{2\}\\sigma^\{2\}/6\)d^\{2\}, with the intermediate resource window of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)shaded\. The shaded region indicates the schematicd4d^\{4\}\-to\-d2d^\{2\}scaling window only; the theorem’s window isC0​σ2​d4≤τ≤c0​σ2​d2C\_\{0\}\\sigma^\{2\}d^\{4\}\\leq\\tau\\leq c\_\{0\}\\sigma^\{2\}d^\{2\}and its constants are suppressed in the plot\.Figure 2:Theorem\-matched recurrent and attention experiment\. Panel \(a\) shows the first\-chaos memory shape for the algebraic target, a contractive diagonal SSM, a bounded nonlinear gated recurrence, and attention\. The numbers in the legend are log\-log slopes fitted over the plotted lag range; they summarize the shape on that range and are not estimates of a theoretical exponent, since Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)predicts an exponential envelope rather than a power law for the contractive models\. Panel \(b\) shows a width sweep at fixed contraction and width\-independent derivative budget, together with two exponential\-target controls\. Because the three targets carry different variances, the vertical axis is the unexplained fraction of target variance, meaning held\-out mean squared error divided by the variance of the target\. One control is matched to the contraction scale and is fit almost exactly\. The other is still exponential and decays more slowly than the contraction, and it leaves a floor above the algebraic one\. For the diagonal state\-space model, that floor is essentially width\-independent, while the remaining recurrent curves improve at small widths and then plateau well above zero\. An exponential target is therefore both the easiest and the hardest case here, which shows that functional form alone does not explain the floor in this benchmark and points instead to compatibility with the contraction envelope\. Panel \(c\) shows a stability sweep asρ→1\\rho\\to 1, with the upper axis reparameterized byδ=−log⁡ρ\\delta=\-\\log\\rho, in which the diagonal model keeps improving while the gated model retains a residual floor\.##### Resource\-law calculations\.

Panels \(b\) and \(c\) of Fig\.[1](https://arxiv.org/html/2610.08816#S7.F1)are closed\-form Gamma\-function identities, and panel \(a\) combines a numerical evaluation of the spectral\-risk identity for each candidate kernel with a controlled deterministic search over the quadrature scalings at a hard budget ofrrmodes\. The fractional context panel uses the exact finite\-past prediction variance, and the short\-memory panel uses the Gamma\-function expression \([9](https://arxiv.org/html/2610.08816#S4.E9)\)\.

##### Architecture\-facing experiment\.

At fixedρ\\rhothe models satisfy the stated budgets; for the gated model these hold on its invariant domain, while theρ\\rho\-sweep is qualitative\. At fixedρ\\rhowe vary the state width, and at fixed width we varyρ\\rho\. Two exponential targets are included as specificity controls, holding the architecture, the budgets, the optimizer, the data splits, and the matched initialization seeds fixed and changing only the target, the algebraic sweep additionally including one further seed\. The first is matched to the contraction scale, with decay equal toρ\\rho\. The diagonal eigenvalues areρ\\rhotimes a sigmoid and therefore lie strictly belowρ\\rho, which places this target on the closure of the model class rather than in its interior, approachable to high accuracy without being realized exactly at finite width\. The second decays more slowly than the contraction, so its tail leaves the envelope of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)even though it is still exponential\. The three targets carry different variances, so we report the unexplained fraction of target variance, meaning held\-out mean squared error divided by the variance of the target\. For the diagonal state\-space model at the largest width, the boundary\-matched control is fit to about7×10−67\\times 10^\{\-6\}, the algebraic target to about1\.1×10−21\.1\\times 10^\{\-2\}, and the slower exponential to about4\.5×10−24\.5\\times 10^\{\-2\}, with the last of these insensitive to width\. The gated recurrence shows the same ordering with a much smaller gap on the boundary\-matched control\. The controls therefore isolate tail compatibility rather than functional form, since an exponential target is fit essentially exactly when it is matched to the contraction and leaves a width\-insensitive floor when it decays more slowly, with the algebraic target lying between the two\. This argues against generic optimization or finite\-width capacity as the sole explanation for the observed algebraic\-memory floor\. The targets here are noiseless linear functionals of the input, so the target functional is directly identified, in contrast with the stochastic forecasting setting of[Song \(2026\)](https://arxiv.org/html/2610.08816#bib.bib39), where trained Transformers attain forecasting error close to the matched autoregressive benchmark while recovering the generating memory law only loosely\. In the stability sweep the diagonal state\-space model improves steadily asρ\\rhoapproaches one, while the gated model stops improving beyond a moderate contraction and retains a residual floor\. AtL=64L=64, the largest\-ρ\\rhopoints are finite\-context illustrations rather than asymptotic tests; the gated floor may also contain higher\-chaos variance\.

## 8Discussion and Limitations

The sharp state\-complexity theorem concerns the specified finite\-state and exponential architecture class, and it is not an unrestricted bound for arbitrary neural networks\. The nonlinear stability theorem requires uniform contraction and derivative budgets, and selective or noncontractive architectures can fall outside this class\. The near\-criticality law is necessary only\. All three resource questions are posed for a scalar target\. App\.[F](https://arxiv.org/html/2610.08816#A6)gives a multivariate shared\-factor extension, related to tensor and factor vector autoregression\([Wang et al\., 2022](https://arxiv.org/html/2610.08816#bib.bib42);[Huang et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib28)\), together with the additional singularities that head collisions create, which are specific to that parameterization\. The small\-ddarchitecture boundary is not fully resolved below thed4d^\{4\}scale, and a growing number of moving Fisher–Hartwig singularities remains outside the finite\-section information theory developed here\. These boundaries matter because modern selective state\-space models can change the relevant representation and stability classes, and they need not stay inside a fixed one\. Two changes of setting would alter the conclusions rather than weaken them\. A contraction budget that varies across the state would replace the single margin of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)by a spectrum of margins, and a readout outside the derivative budgets would remove the uniform sensitivity control needed to derive the geometric envelope on the first\-chaos coefficients\. Both are realistic descriptions of selective architectures\([Gu & Dao, 2023](https://arxiv.org/html/2610.08816#bib.bib23);[Ben\-Kish et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib5)\), and both sit outside the class treated here\.

## Acknowledgment

We thank Yu Ji for helpful discussion during the preparation of this work\. The code, which includes the synthetic generators, the model\-selection procedures, the train, validation, and test split logic, the initialization controls, the checkpoint metadata, and scripts that reproduce every main\-text and appendix figure, is available from the corresponding author on reasonable request\. This manuscript was prepared with AI\-assisted drafting and editing\. The authors are responsible for verifying every mathematical claim, citation, experiment, and final wording\.

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## Appendix APredictive\-Risk and Hankel Preliminaries

### A\.1Stationary forecast\-risk identity

Let\(Xt\)t∈ℤ\(X\_\{t\}\)\_\{t\\in\\mathbb\{Z\}\}be centered, stationary, and purely nondeterministic, withℱt=σ\(Xs:s≤t\)\\mathcal\{F\}\_\{t\}=\\sigma\(X\_\{s\}:s\\leq t\)and Bayes forecastmt⋆=𝔼⁡\[Xt\+1∣ℱt\]=∑j≥1aj⋆​Xt\+1−jm\_\{t\}^\{\\star\}=\\mathbb\{E\}\[X\_\{t\+1\}\\mid\\mathcal\{F\}\_\{t\}\]=\\sum\_\{j\\geq 1\}a\_\{j\}^\{\\star\}X\_\{t\+1\-j\}\. The innovationεt\+1=Xt\+1−mt⋆\\varepsilon\_\{t\+1\}=X\_\{t\+1\}\-m\_\{t\}^\{\\star\}satisfies𝔼⁡\[εt\+1∣ℱt\]=0\\mathbb\{E\}\[\\varepsilon\_\{t\+1\}\\mid\\mathcal\{F\}\_\{t\}\]=0and is therefore orthogonal to every square\-integrableℱt\\mathcal\{F\}\_\{t\}\-measurable variable\. For a candidate kernelaawritemt​\(a\)=∑j≥1aj​Xt\+1−jm\_\{t\}\(a\)=\\sum\_\{j\\geq 1\}a\_\{j\}X\_\{t\+1\-j\}, and letℰ⁡\(a\)\\mathcal\{E\}\(a\)be the excess of𝔼⁡\[\(Xt\+1−mt​\(a\)\)2\]\\mathbb\{E\}\[\(X\_\{t\+1\}\-m\_\{t\}\(a\)\)^\{2\}\]over its value ata⋆a^\{\\star\}\.

With the conventionγ⁡\(h\)=\(2​π\)−1​∫−ππei​h​ω​fX​\(ω\)​𝑑ω\\gamma\(h\)=\(2\\pi\)^\{\-1\}\\int\_\{\-\\pi\}^\{\\pi\}e^\{ih\\omega\}f\_\{X\}\(\\omega\)\\,\\mathrm\{d\}\\omegaandΔj=aj−aj⋆\\Delta\_\{j\}=a\_\{j\}\-a\_\{j\}^\{\\star\}, the orthogonality of the innovation and Parseval’s identity give \([2](https://arxiv.org/html/2610.08816#S2.E2)\), namely

ℰ⁡\(a\)=12​π​∫−ππ\|Δ⁡\(e−i​ω\)\|2​fX​\(ω\)​𝑑ω\.\\mathcal\{E\}\(a\)=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|\\Delta\(e^\{\-i\\omega\}\)\|^\{2\}f\_\{X\}\(\\omega\)\\,\\mathrm\{d\}\\omega\.\(17\)The argument is the standard one\. Partial sums∑j≤JΔj​Xt\+1−j\\sum\_\{j\\leq J\}\\Delta\_\{j\}X\_\{t\+1\-j\}are Cauchy inL2L\_\{2\}wheneverfXf\_\{X\}is bounded above andΔ∈ℓ2\\Delta\\in\\ell\_\{2\}, the limit isℱt\\mathcal\{F\}\_\{t\}\-measurable and therefore orthogonal toεt\+1\\varepsilon\_\{t\+1\}, and the spectral representation converts the second moment of the limit into the displayed integral\. For a perturbationu∈ℓ2u\\in\\ell\_\{2\}we write

‖u‖pred,fX2:=12​π​∫−ππ\|U⁡\(e−i​ω\)\|2​fX​\(ω\)​𝑑ω,\\\|u\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}:=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|U\(e^\{\-i\\omega\}\)\|^\{2\}f\_\{X\}\(\\omega\)\\,\\mathrm\{d\}\\omega,\(18\)which is the seminorm appearing in Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)\.

### A\.2Regular and singular spectra

Under the regular condition0<m≤fX≤M<∞0<m\\leq f\_\{X\}\\leq M<\\infty, inserting the bounds into \([17](https://arxiv.org/html/2610.08816#A1.E17)\) and applying Parseval gives

m​‖a−a⋆‖22≤ℰ⁡\(a\)≤M​‖a−a⋆‖22,m\\\|a\-a^\{\\star\}\\\|\_\{2\}^\{2\}\\leq\\mathcal\{E\}\(a\)\\leq M\\\|a\-a^\{\\star\}\\\|\_\{2\}^\{2\},\(19\)so forecast risk and coefficient error are equivalent up to constants, and the truncation of the truth at contextLLhas risk betweenmmandMMtimes∑j\>L\(aj⋆\)2\\sum\_\{j\>L\}\(a\_\{j\}^\{\\star\}\)^\{2\}\.

The two halves of \([19](https://arxiv.org/html/2610.08816#A1.E19)\) are used by different results and should be separated\. The lower half needs onlyfX≥m\>0f\_\{X\}\\geq m\>0and is what converts an approximation obstruction into a risk lower bound, so Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)survives on spectra that are unbounded above\. The upper half needsfX≤Mf\_\{X\}\\leq Mand is what converts a construction into a risk upper bound\. Sec\.[4](https://arxiv.org/html/2610.08816#S4)deliberately violates the upper half, since a spectral pole at the origin makesMMinfinite, and App\.[C](https://arxiv.org/html/2610.08816#A3)shows that the correct replacement is not a weaker constant but a different norm\.

### A\.3Finite\-state kernels and Hankel rank

For a sequenceaaand an integernndefine the Hankel blockHn​\(a\)=\[ai\+j−1\]i,j=1nH\_\{n\}\(a\)=\[a\_\{i\+j\-1\}\]\_\{i,j=1\}^\{n\}, which involves only the first2​n−12n\-1coefficients, and writes1≥s2≥⋯s\_\{1\}\\geq s\_\{2\}\\geq\\cdotsfor singular values\. Anrr\-dimensional linear state with transitionA∈ℝr×rA\\in\\mathbb\{R\}^\{r\\times r\}and mapsu,v∈ℝru,v\\in\\mathbb\{R\}^\{r\}producesbj=u⊤​Aj−1​vb\_\{j\}=u^\{\\top\}A^\{j\-1\}v, and the factorization\[Hn​\(b\)\]i​j=\(u⊤​Ai−1\)​\(Aj−1​v\)\[H\_\{n\}\(b\)\]\_\{ij\}=\(u^\{\\top\}A^\{i\-1\}\)\(A^\{j\-1\}v\)exhibitsHn​\(b\)H\_\{n\}\(b\)as a product of ann×rn\\times rand anr×nr\\times nmatrix, so

rank⁡Hn​\(b\)≤rfor every​n\.\\operatorname\{rank\}H\_\{n\}\(b\)\\leq r\\qquad\\text\{for every \}n\.\(20\)The same bound holds for anyrr\-mode exponential kernelbj=∑q≤rwq​ρqj−1b\_\{j\}=\\sum\_\{q\\leq r\}w\_\{q\}\\rho\_\{q\}^\{\\,j\-1\}, sinceHn​\(b\)=∑qwq​zq​zq⊤H\_\{n\}\(b\)=\\sum\_\{q\}w\_\{q\}z\_\{q\}z\_\{q\}^\{\\top\}withzq=\(1,ρq,…,ρqn−1\)⊤z\_\{q\}=\(1,\\rho\_\{q\},\\dots,\\rho\_\{q\}^\{n\-1\}\)^\{\\top\}\. The architecture class must record the finite context as well as the state dimension, because a hard truncation of a state\-space sequence is in general no longer a state\-space sequence and no longer has globally bounded Hankel rank\. We therefore let𝒜r\\mathcal\{A\}\_\{r\}denote the union over finite contexts of the truncations ofrr\-state kernels,

𝒜r:=⋃L≥1\{b:bj=u⊤Aj−1v1\{j≤L\},A∈ℝr′×r′,u,v∈ℝr′,r′≤r\},\\mathcal\{A\}\_\{r\}:=\\bigcup\_\{L\\geq 1\}\\Bigl\\\{b:\\ b\_\{j\}=u^\{\\top\}A^\{j\-1\}v\\,\\mathbf\{1\}\\\{j\\leq L\\\},\\ A\\in\\mathbb\{R\}^\{r^\{\\prime\}\\times r^\{\\prime\}\},\\ u,v\\in\\mathbb\{R\}^\{r^\{\\prime\}\},\\ r^\{\\prime\}\\leq r\\Bigr\\\},\(21\)and we callLLthe context of that member\. Therr\-mode exponential kernels form the diagonal\-state subclass, obtained by takingAAdiagonal with entriesρq\\rho\_\{q\}and absorbing the weights intouu\.

The only property of \([21](https://arxiv.org/html/2610.08816#A1.E21)\) used below is the following restricted form of \([20](https://arxiv.org/html/2610.08816#A1.E20)\)\. For a member with contextLLand anynnwith2​n−1≤L2n\-1\\leq L, the blockHn​\(b\)H\_\{n\}\(b\)involves only coefficients with index at mostLL, so it coincides with the corresponding block of the underlying untruncated state\-space sequence and therefore satisfiesrank⁡Hn​\(b\)≤r\\operatorname\{rank\}H\_\{n\}\(b\)\\leq r\. No rank claim is made for blocks that reach past the context, and none is needed\. A lower bound over𝒜r\\mathcal\{A\}\_\{r\}applies simultaneously torrexponential modes,rrdiagonal recurrent states, andrrgeneral linear states, in each case at every finite context\.

### A\.4Ky Fan and low\-rank approximation inequalities

###### Lemma 9\(Low\-rank Frobenius approximation\)\.

Ifrank⁡\(B\)≤r<n\\operatorname\{rank\}\(B\)\\leq r<nthen‖M−B‖F2≥∑q\>rsq​\(M\)2\\\|M\-B\\\|\_\{F\}^\{2\}\\geq\\sum\_\{q\>r\}s\_\{q\}\(M\)^\{2\}for everyM∈ℝn×nM\\in\\mathbb\{R\}^\{n\\times n\}\.

###### Proof\.

LetPPproject onto the column space ofBB, so\(I−P\)​B=0\(I\-P\)B=0andrank⁡P≤r\\operatorname\{rank\}P\\leq r\. Projection does not increase the Frobenius norm, so‖M−B‖F2≥‖\(I−P\)​M‖F2=tr⁡\(M​M⊤​\(I−P\)\)\\\|M\-B\\\|\_\{F\}^\{2\}\\geq\\\|\(I\-P\)M\\\|\_\{F\}^\{2\}=\\operatorname\{tr\}\(MM^\{\\top\}\(I\-P\)\)\. DiagonalizingM​M⊤MM^\{\\top\}and writingpqp\_\{q\}for the diagonal ofPPin that basis gives∑qsq2​\(1−pq\)\\sum\_\{q\}s\_\{q\}^\{2\}\(1\-p\_\{q\}\)with0≤pq≤10\\leq p\_\{q\}\\leq 1and∑qpq≤r\\sum\_\{q\}p\_\{q\}\\leq r, and since thesqs\_\{q\}decrease the sum is at least∑q\>rsq2\\sum\_\{q\>r\}s\_\{q\}^\{2\}\. ∎

The second ingredient is the multiplicity count that converts a Hankel discrepancy back to coefficient error\. Writingwn,kw\_\{n,k\}for the number of pairs\(i,j\)\(i,j\)in\{1,…,n\}2\\\{1,\\dots,n\\\}^\{2\}withi\+j−1=ki\+j\-1=k,

‖Hn​\(e\)‖F2=∑k=12​n−1wn,k​ek2,1≤wn,k≤n,\\\|H\_\{n\}\(e\)\\\|\_\{F\}^\{2\}=\\sum\_\{k=1\}^\{2n\-1\}w\_\{n,k\}e\_\{k\}^\{2\},\\qquad 1\\leq w\_\{n,k\}\\leq n,\(22\)so∑k≤2​n−1ek2≥n−1​‖Hn​\(e\)‖F2\\sum\_\{k\\leq 2n\-1\}e\_\{k\}^\{2\}\\geq n^\{\-1\}\\\|H\_\{n\}\(e\)\\\|\_\{F\}^\{2\}\. A coefficient appears on up tonnantidiagonals of the block, so a comparison that ignoreswn,kw\_\{n,k\}counts that mass more than once, and the factorn−1n^\{\-1\}is the correction\. This is the step referred to in App\.[B](https://arxiv.org/html/2610.08816#A2)as the corrected Ky Fan indexing\.

### A\.5Gaussian first\-chaos projection

LetZ=\(Zt\)Z=\(Z\_\{t\}\)be independent standard Gaussians and letm^t\\widehat\{m\}\_\{t\}be a square\-integrable function ofZZ\. The Wiener chaos decomposition writesL2​\(σ​\(Z\)\)L\_\{2\}\(\\sigma\(Z\)\)as an orthogonal sum⨁p≥0ℋp\\bigoplus\_\{p\\geq 0\}\\mathcal\{H\}\_\{p\}, and the projection onto the first chaos is

P1​m^t=∑j≥1βj​Zt\+1−j,βj=𝔼⁡\[m^t​Zt\+1−j\],P\_\{1\}\\widehat\{m\}\_\{t\}=\\sum\_\{j\\geq 1\}\\beta\_\{j\}Z\_\{t\+1\-j\},\\qquad\\beta\_\{j\}=\\mathbb\{E\}\[\\widehat\{m\}\_\{t\}Z\_\{t\+1\-j\}\],\(23\)the coefficients being well defined because theZt\+1−jZ\_\{t\+1\-j\}are orthonormal\.

If the target is itself in the first chaos,mt⋆=∑j≥1aj⋆​Zt\+1−jm\_\{t\}^\{\\star\}=\\sum\_\{j\\geq 1\}a\_\{j\}^\{\\star\}Z\_\{t\+1\-j\}, then orthogonality of the chaoses gives

𝔼⁡\[\(mt⋆−m^t\)2\]≥‖mt⋆−P1​m^t‖22=∑j≥1\(aj⋆−βj\)2,\\mathbb\{E\}\[\(m\_\{t\}^\{\\star\}\-\\widehat\{m\}\_\{t\}\)^\{2\}\]\\ \\geq\\ \\\|m\_\{t\}^\{\\star\}\-P\_\{1\}\\widehat\{m\}\_\{t\}\\\|\_\{2\}^\{2\}\\ =\\ \\sum\_\{j\\geq 1\}\(a\_\{j\}^\{\\star\}\-\\beta\_\{j\}\)^\{2\},\(24\)since the higher chaoses ofm^t\\widehat\{m\}\_\{t\}are orthogonal tomt⋆m\_\{t\}^\{\\star\}and only add to the error\. Inequality \([24](https://arxiv.org/html/2610.08816#A1.E24)\) is what makes a statement about the linear part of a nonlinear predictor into a lower bound on its full risk, and it is the reason Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)can control an arbitrary nonlinear recurrence through a single sequenceβ\\beta\. Nothing in \([24](https://arxiv.org/html/2610.08816#A1.E24)\) asserts that the higher chaoses are small, only that they cannot help\.

## Appendix BSharp State Complexity of Algebraic Memory

Throughout this appendix the target isaj⋆=c​j−αa\_\{j\}^\{\\star\}=cj^\{\-\\alpha\}withα\>1\\alpha\>1inside a stable regular\-spectrum forecasting model, so that \([19](https://arxiv.org/html/2610.08816#A1.E19)\) holds with constantsmα,Mαm\_\{\\alpha\},M\_\{\\alpha\}determined by\(α,c,σ\)\(\\alpha,c,\\sigma\)\.

### B\.1Comparison with existing approximation metrics

The literature contains several distinct notions of memory approximation, and the constants and even the exponents depend on which one is used\. Since Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)is a statement about one of them, we separate the three before proving anything\.

#### B\.1\.1Functional\-memory approximation in curse\-of\-memory theory

Related curse\-of\-memory results formalize different notions of memory obstruction rather than one common metric\.[Li et al\. \(2021\)](https://arxiv.org/html/2610.08816#bib.bib32)and the inverse\-approximation results of[Wang et al\. \(2024\)](https://arxiv.org/html/2610.08816#bib.bib45)use worst\-case functional approximation over a class of inputs, whereas[Wang & Xue \(2023\)](https://arxiv.org/html/2610.08816#bib.bib44)characterize exponential decay through a derivative\-based memory function for stacked state\-space models\. None of these notions is the stationary forecast risk used here, and the separation is the one already drawn in Sec\.[5](https://arxiv.org/html/2610.08816#S5)\. Those results give explicit rates in their own metrics, and what they do not provide is a two\-sided resource curve for one fixed predictive target in forecast risk, which is the question asked here\. The constructions used for their upper directions are likewise not optimized for a single fixed target\. We do not claim to sharpen those statements on their own terms, since neither the metric nor the target class coincides with ours\.

#### B\.1\.2Finite\-horizon uniform approximation in power\-law recurrent constructions

A second notion fixes a horizon\[1,T\]\[1,T\]and asks for uniform approximation of a power\-law kernel on that window by a sum of exponentials, which is the setting of recent architectural constructions\([Mlaiki, 2026](https://arxiv.org/html/2610.08816#bib.bib35)\)\. The horizon is externally imposed and the error is measured on it, so the resulting complexity depends onTTand does not by itself say anything about the infinite predictive law\.

#### B\.1\.3Infinite predictive\-law approximation in forecast risk

Our metric is \([2](https://arxiv.org/html/2610.08816#S2.E2)\), the horizon is not externally fixed, and the infimum ininfb∈𝒜rℰ⁡\(b\)\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)is taken over the context as well as over the state\. This is the notion under which both sides match, and it is the one relevant to forecasting because it is the risk actually incurred\. A construction that is good on a finite window is not automatically good here, since the unseen tail contributes to \([2](https://arxiv.org/html/2610.08816#S2.E2)\) whether or not it lies inside the window\.

### B\.2Laplace representation of algebraic kernels

###### Lemma 10\(Power laws as continuous exponential mixtures\)\.

For everyα\>0\\alpha\>0andx\>0x\>0,

x−α=1Γ⁡\(α\)​∫0∞tα−1​e−x​t​𝑑t\.x^\{\-\\alpha\}=\\frac\{1\}\{\\Gamma\(\\alpha\)\}\\int\_\{0\}^\{\\infty\}t^\{\\alpha\-1\}e^\{\-xt\}\\,\\mathrm\{d\}t\.\(25\)In particularj−αj^\{\-\\alpha\}is a positive continuous mixture of the exponential atomse−t​je^\{\-tj\}\.

###### Proof\.

Substitutes=x​ts=xtin the definitionΓ⁡\(α\)=∫0∞sα−1​e−s​𝑑s\\Gamma\(\\alpha\)=\\int\_\{0\}^\{\\infty\}s^\{\\alpha\-1\}e^\{\-s\}\\,\\mathrm\{d\}s\. ∎

Lem\.[10](https://arxiv.org/html/2610.08816#Thmtheorem10)is the whole reason a state can represent algebraic memory at all\. The mixture is positive, so the discretization below produces positive weights and stable modes rather than a signed cancellation, and this is what keeps the construction inside𝒜r\\mathcal\{A\}\_\{r\}with a controlled norm\.

### B\.3Positive exponential quadrature

Substitutingt=eut=e^\{u\}in \([25](https://arxiv.org/html/2610.08816#A2.E25)\) gives

j−α=1Γ⁡\(α\)​∫−∞∞eα​u​e−j​eu​𝑑u,j^\{\-\\alpha\}=\\frac\{1\}\{\\Gamma\(\\alpha\)\}\\int\_\{\-\\infty\}^\{\\infty\}e^\{\\alpha u\}e^\{\-je^\{u\}\}\\,\\mathrm\{d\}u,\(26\)whose integrand decays likeeα​ue^\{\\alpha u\}asu→−∞u\\to\-\\inftyand doubly exponentially asu→\+∞u\\to\+\\infty, and is analytic in the strip\|Im⁡u\|<π/2\|\\operatorname\{Im\}u\|<\\pi/2whereRe⁡eu\>0\\operatorname\{Re\}e^\{u\}\>0\. Applying the trapezoidal rule with stephhon a truncated range therefore incurs a discretization error of ordere−π2/he^\{\-\\pi^\{2\}/h\}and a left\-truncation error of ordereα​umine^\{\\alpha u\_\{\\min\}\}\. Each node contributes one exponential mode with positive weight, sinceλ=eu\>0\\lambda=e^\{u\}\>0and the weighth​eα​u/Γ⁡\(α\)h\\,e^\{\\alpha u\}/\\Gamma\(\\alpha\)is positive\.

The upper bound uses one external approximation result, which we state exactly as it is used\.

###### Fact 11\(Positive exponential approximation of a power law\)\.

Fixα\>0\\alpha\>0\. For every0<ϱ<10<\\varrho<1and every relative tolerance0<η≤e−10<\\eta\\leq e^\{\-1\}there are an integerRR, positive weightswqw\_\{q\}, and positive exponentspqp\_\{q\}with

\|x−α−∑q=1Rwq​e−pq​x\|≤η​x−α\(ϱ≤x≤1\),\\Bigl\|x^\{\-\\alpha\}\-\\sum\_\{q=1\}^\{R\}w\_\{q\}e^\{\-p\_\{q\}x\}\\Bigr\|\\leq\\eta\\,x^\{\-\\alpha\}\\qquad\(\\varrho\\leq x\\leq 1\),\(27\)and, for fixedα\\alpha, the construction may be chosen with

R≤Cα​\(1\+log⁡η−1\)​\(1\+log⁡ϱ−1\+log⁡η−1\+log⁡log⁡η−1\),R\\leq C\_\{\\alpha\}\\bigl\(1\+\\log\\eta^\{\-1\}\\bigr\)\\bigl\(1\+\\log\\varrho^\{\-1\}\+\\log\\eta^\{\-1\}\+\\log\\log\\eta^\{\-1\}\\bigr\),\(28\)whereCαC\_\{\\alpha\}depends only onα\\alpha\. This is the exponential\-sum approximation of[Beylkin & Monzón \(2005\)](https://arxiv.org/html/2610.08816#bib.bib7);[Beylkin & Monzón \(2010\)](https://arxiv.org/html/2610.08816#bib.bib8), in the relative\-error form on a bounded interval bounded away from the origin\.

###### Theorem 12\(Finite\-context exponential compression\)\.

Fixα\>0\\alpha\>0, an integerL≥2L\\geq 2, and0<η≤e−10<\\eta\\leq e^\{\-1\}\. There are positivebqb\_\{q\}andλq\\lambda\_\{q\},1≤q≤R1\\leq q\\leq R, with

\|j−α−∑q=1Rbq​e−λq​\(j−1\)\|≤η​j−α\(1≤j≤L\),\\Bigl\|j^\{\-\\alpha\}\-\\sum\_\{q=1\}^\{R\}b\_\{q\}e^\{\-\\lambda\_\{q\}\(j\-1\)\}\\Bigr\|\\leq\\eta\\,j^\{\-\\alpha\}\\qquad\(1\\leq j\\leq L\),\(29\)and

R≤Cα​\(1\+log⁡η−1\)​\(1\+log⁡L\+log⁡η−1\+log⁡log⁡η−1\)\.R\\leq C\_\{\\alpha\}\\bigl\(1\+\\log\\eta^\{\-1\}\\bigr\)\\bigl\(1\+\\log L\+\\log\\eta^\{\-1\}\+\\log\\log\\eta^\{\-1\}\\bigr\)\.\(30\)

###### Proof\.

Apply Fact[11](https://arxiv.org/html/2610.08816#Thmtheorem11)withϱ=1/L\\varrho=1/L, so that \([27](https://arxiv.org/html/2610.08816#A2.E27)\) holds on\[1/L,1\]\[1/L,1\]and \([28](https://arxiv.org/html/2610.08816#A2.E28)\) readsR≤Cα​\(1\+log⁡η−1\)​\(1\+log⁡L\+log⁡η−1\+log⁡log⁡η−1\)R\\leq C\_\{\\alpha\}\(1\+\\log\\eta^\{\-1\}\)\(1\+\\log L\+\\log\\eta^\{\-1\}\+\\log\\log\\eta^\{\-1\}\)\. Setx=j/Lx=j/L, which lies in\[1/L,1\]\[1/L,1\]for1≤j≤L1\\leq j\\leq L\. Thenj−α=L−α​x−αj^\{\-\\alpha\}=L^\{\-\\alpha\}x^\{\-\\alpha\}, and substituting the approximant gives

\|j−α−L−α∑qwqe−pqj/L\|≤ηj−α\(1≤j≤L\)\.\\Bigl\|j^\{\-\\alpha\}\-L^\{\-\\alpha\}\\sum\_\{q\}w\_\{q\}e^\{\-p\_\{q\}j/L\}\\Bigr\|\\leq\\eta\\,j^\{\-\\alpha\}\\qquad\(1\\leq j\\leq L\)\.Puttingλq=pq/L\>0\\lambda\_\{q\}=p\_\{q\}/L\>0andbq=L−α​wq​e−λq\>0b\_\{q\}=L^\{\-\\alpha\}w\_\{q\}e^\{\-\\lambda\_\{q\}\}\>0turns the right\-hand sum into∑qbq​e−λq​\(j−1\)\\sum\_\{q\}b\_\{q\}e^\{\-\\lambda\_\{q\}\(j\-1\)\}, which is \([29](https://arxiv.org/html/2610.08816#A2.E29)\) with positive weights and positive rates\. The count is unchanged, giving \([30](https://arxiv.org/html/2610.08816#A2.E30)\)\. ∎

The error is relative rather than absolute, which matters because the risk weights every lag and an absolute guarantee would be vacuous at largejj\. Balancing the two error sources in \([26](https://arxiv.org/html/2610.08816#A2.E26)\) atRRnodes giveslog⁡η−1≍π​α​R\\log\\eta^\{\-1\}\\asymp\\pi\\sqrt\{\\alpha R\}, which is the origin of the square root and is consistent with \([30](https://arxiv.org/html/2610.08816#A2.E30)\)\.

### B\.4Finite\-context upper bound

###### Corollary 13\(Risk ofRRexponential modes at contextLL\)\.

WithRRas in \([30](https://arxiv.org/html/2610.08816#A2.E30)\) there isb∈𝒜Rb\\in\\mathcal\{A\}\_\{R\}supported on\{1,…,L\}\\\{1,\\dots,L\\\}with

ℰ⁡\(b\)≤Mα​c2​\{ζ⁡\(2​α\)​η2\+L1−2​α2​α−1\}\.\\mathcal\{E\}\(b\)\\leq M\_\{\\alpha\}c^\{2\}\\Bigl\\\{\\zeta\(2\\alpha\)\\eta^\{2\}\+\\frac\{L^\{1\-2\\alpha\}\}\{2\\alpha\-1\}\\Bigr\\\}\.\(31\)

###### Proof\.

By \([19](https://arxiv.org/html/2610.08816#A1.E19)\) it suffices to bound‖b−a⋆‖22\\\|b\-a^\{\\star\}\\\|\_\{2\}^\{2\}\. On\{1,…,L\}\\\{1,\\dots,L\\\}the relative bound \([29](https://arxiv.org/html/2610.08816#A2.E29)\) gives a squared error at mostη2​c2​∑j≤Lj−2​α≤η2​c2​ζ​\(2​α\)\\eta^\{2\}c^\{2\}\\sum\_\{j\\leq L\}j^\{\-2\\alpha\}\\leq\\eta^\{2\}c^\{2\}\\zeta\(2\\alpha\), and beyondLLthe candidate vanishes so the error is the tailc2​∑j\>Lj−2​α≤c2​L1−2​α/\(2​α−1\)c^\{2\}\\sum\_\{j\>L\}j^\{\-2\\alpha\}\\leq c^\{2\}L^\{1\-2\\alpha\}/\(2\\alpha\-1\)\. ∎

###### Corollary 14\(Constructive root\-exponential curve\)\.

There areCL,Cup,cup\>0C\_\{L\},C\_\{\\rm up\},c\_\{\\rm up\}\>0depending only on the fixed model such that for every sufficiently largerrthere isLr≤exp⁡\(CL​r\)L\_\{r\}\\leq\\exp\(C\_\{L\}\\sqrt\{r\}\)andb∈𝒜rb\\in\\mathcal\{A\}\_\{r\}supported on\{1,…,Lr\}\\\{1,\\dots,L\_\{r\}\\\}withℰ⁡\(b\)≤Cup​e−cup​r\\mathcal\{E\}\(b\)\\leq C\_\{\\rm up\}e^\{\-c\_\{\\rm up\}\\sqrt\{r\}\}\. Equivalently, reaching riskτ\\tauneeds at mostCmode​log2⁡\(Cup/τ\)C\_\{\\rm mode\}\\log^\{2\}\(C\_\{\\rm up\}/\\tau\)modes for another fixed constantCmode\>0C\_\{\\rm mode\}\>0\.

###### Proof\.

Chooseη\\etaandLLso that the two terms of Corollary[13](https://arxiv.org/html/2610.08816#Thmtheorem13)are comparable, that isL1−2​α≍η2L^\{1\-2\\alpha\}\\asymp\\eta^\{2\}, solog⁡L≍log⁡η−1/\(2​α−1\)\\log L\\asymp\\log\\eta^\{\-1\}/\(2\\alpha\-1\)\. Then \([30](https://arxiv.org/html/2610.08816#A2.E30)\) givesR≤C​\(1\+log⁡η−1\)2R\\leq C\(1\+\\log\\eta^\{\-1\}\)^\{2\}, and inverting yieldslog⁡η−1≍R\\log\\eta^\{\-1\}\\asymp\\sqrt\{R\}and henceℰ≤Cup​e−cup​R\\mathcal\{E\}\\leq C\_\{\\rm up\}e^\{\-c\_\{\\rm up\}\\sqrt\{R\}\}\. ∎

### B\.5Hankel operator formulation

Let𝐇α\\mathbf\{H\}\_\{\\alpha\}be the Hankel operator onℓ2​\(ℕ0\)\\ell\_\{2\}\(\\mathbb\{N\}\_\{0\}\)with symbol\(i\+j\+1\)−α\(i\+j\+1\)^\{\-\\alpha\}, and letGN\(α\)G^\{\(\\alpha\)\}\_\{N\}be its leadingN×NN\\times Nsection\. Forα\>1\\alpha\>1the entries are square summable, so𝐇α\\mathbf\{H\}\_\{\\alpha\}is Hilbert–Schmidt, hence compact and self\-adjoint, and being a moment matrix of a positive measure it is positive semidefinite\. By the Eckart–Young principle the best unconstrained rank\-rrapproximation in Frobenius norm is governed by the tail of the singular spectrum\([Eckart & Young, 1936](https://arxiv.org/html/2610.08816#bib.bib15)\), and classical Hankel\-operator theory supplies the structured spectral and approximation background used here\([Widom, 1966](https://arxiv.org/html/2610.08816#bib.bib46);[Adamjan et al\., 1971](https://arxiv.org/html/2610.08816#bib.bib1);[Peller, 2003](https://arxiv.org/html/2610.08816#bib.bib36)\)\. By \([20](https://arxiv.org/html/2610.08816#A1.E20)\) every member of𝒜r\\mathcal\{A\}\_\{r\}has Hankel rank at mostrr\.

### B\.6Finite\-section singular\-value lower scale

###### Lemma 15\(Finite sections retain the relevant singular value\)\.

LetPNP\_\{N\}project onto the firstNNcoordinates and put𝐇α,N=PN​𝐇α​PN\\mathbf\{H\}\_\{\\alpha,N\}=P\_\{N\}\\mathbf\{H\}\_\{\\alpha\}P\_\{N\}\. Then‖𝐇α−𝐇α,N‖op≤Cα​N1−α\\\|\\mathbf\{H\}\_\{\\alpha\}\-\\mathbf\{H\}\_\{\\alpha,N\}\\\|\_\{\\rm op\}\\leq C\_\{\\alpha\}N^\{1\-\\alpha\}\. Consequently there areκα,cα,Cα\>0\\kappa\_\{\\alpha\},c\_\{\\alpha\},C\_\{\\alpha\}\>0andr0r\_\{0\}such that, withNr=⌈eκα​r⌉N\_\{r\}=\\lceil e^\{\\kappa\_\{\\alpha\}\\sqrt\{r\}\}\\rceil, one hasNr\>rN\_\{r\}\>rand

sr\+1​\(GNr\(α\)\)≥cα​e−Cα​r\(r≥r0\)\.s\_\{r\+1\}\\bigl\(G^\{\(\\alpha\)\}\_\{N\_\{r\}\}\\bigr\)\\ \\geq\\ c\_\{\\alpha\}e^\{\-C\_\{\\alpha\}\\sqrt\{r\}\}\\qquad\(r\\geq r\_\{0\}\)\.\(32\)

###### Proof\.

For the operator bound,𝐇α−𝐇α,N\\mathbf\{H\}\_\{\\alpha\}\-\\mathbf\{H\}\_\{\\alpha,N\}is supported on the indices withmax⁡\(i,j\)≥N\\max\(i,j\)\\geq N, whose entries are at most\(i\+j\+1\)−α\(i\+j\+1\)^\{\-\\alpha\}\. Bounding the operator norm by the Hilbert–Schmidt norm and comparing the resulting sum with∫N∞x1−2​α​𝑑x\\int\_\{N\}^\{\\infty\}x^\{1\-2\\alpha\}\\,\\mathrm\{d\}x, which converges becauseα\>1\\alpha\>1, gives‖𝐇α−𝐇α,N‖op≤Cα​N1−α\\\|\\mathbf\{H\}\_\{\\alpha\}\-\\mathbf\{H\}\_\{\\alpha,N\}\\\|\_\{\\rm op\}\\leq C\_\{\\alpha\}N^\{1\-\\alpha\}\.

For the singular value we use one external input, namely the classical root\-exponential scale for the singular values of the Hankel operator with power\-law symbol, in the form

sr\+1​\(𝐇α\)≥cα​e−Cα​rfor all large​r,s\_\{r\+1\}\(\\mathbf\{H\}\_\{\\alpha\}\)\\ \\geq\\ c\_\{\\alpha\}e^\{\-C\_\{\\alpha\}\\sqrt\{r\}\}\\qquad\\text\{for all large \}r,\(33\)which follows from the spectral theory of Hankel matrices developed by[Widom \(1966\)](https://arxiv.org/html/2610.08816#bib.bib46), specifically the singular\-value asymptotics for Hankel operators with power\-law symbols established there;[Peller \(2003\)](https://arxiv.org/html/2610.08816#bib.bib36)restates that theory systematically and is the more convenient modern entry point\. Only the stretched\-exponential scale in \([33](https://arxiv.org/html/2610.08816#A2.E33)\) is used, and no claim is made about the exact constantCCor about the leading asymptotic coefficient\. Weyl’s inequality for the perturbation𝐇α,N=𝐇α−\(𝐇α−𝐇α,N\)\\mathbf\{H\}\_\{\\alpha,N\}=\\mathbf\{H\}\_\{\\alpha\}\-\(\\mathbf\{H\}\_\{\\alpha\}\-\\mathbf\{H\}\_\{\\alpha,N\}\)gives

sr\+1​\(GN\(α\)\)≥sr\+1​\(𝐇α\)−‖𝐇α−𝐇α,N‖op≥cα​e−Cα​r−Cα′​N1−α\.s\_\{r\+1\}\\bigl\(G^\{\(\\alpha\)\}\_\{N\}\\bigr\)\\ \\geq\\ s\_\{r\+1\}\(\\mathbf\{H\}\_\{\\alpha\}\)\-\\\|\\mathbf\{H\}\_\{\\alpha\}\-\\mathbf\{H\}\_\{\\alpha,N\}\\\|\_\{\\rm op\}\\ \\geq\\ c\_\{\\alpha\}e^\{\-C\_\{\\alpha\}\\sqrt\{r\}\}\-C\_\{\\alpha\}^\{\\prime\}N^\{1\-\\alpha\}\.Chooseκα:=2​Cα/\(α−1\)\\kappa\_\{\\alpha\}:=2C\_\{\\alpha\}/\(\\alpha\-1\)andNr=⌈eκα​r⌉N\_\{r\}=\\lceil e^\{\\kappa\_\{\\alpha\}\\sqrt\{r\}\}\\rceil, so thatCα′​Nr1−α≤Cα′​e−\(α−1\)​κα​r=Cα′​e−2​Cα​rC\_\{\\alpha\}^\{\\prime\}N\_\{r\}^\{1\-\\alpha\}\\leq C\_\{\\alpha\}^\{\\prime\}e^\{\-\(\\alpha\-1\)\\kappa\_\{\\alpha\}\\sqrt\{r\}\}=C\_\{\\alpha\}^\{\\prime\}e^\{\-2C\_\{\\alpha\}\\sqrt\{r\}\}, which is below12​cα​e−Cα​r\\tfrac\{1\}\{2\}c\_\{\\alpha\}e^\{\-C\_\{\\alpha\}\\sqrt\{r\}\}for all largerr\. The displayed difference is then at least12​cα​e−Cα​r\\tfrac\{1\}\{2\}c\_\{\\alpha\}e^\{\-C\_\{\\alpha\}\\sqrt\{r\}\}; decreasing the constantcαc\_\{\\alpha\}in the statement by a factor of two gives \([32](https://arxiv.org/html/2610.08816#A2.E32)\), andNr\>rN\_\{r\}\>rholds for largerrsinceeκα​re^\{\\kappa\_\{\\alpha\}\\sqrt\{r\}\}grows faster thanrr\. ∎

The two\-step structure is what the argument needs\. The singular values of the infinite operator decay root\-exponentially, which is the classical Widom scale, but a lower bound onℰ\\mathcal\{E\}must be expressed through a finite section, because only finitely many coefficients enter a Hankel block\. Lem\.[15](https://arxiv.org/html/2610.08816#Thmtheorem15)says the section can be taken exponentially large inr\\sqrt\{r\}while still retaining the\(r\+1\)\(r\+1\)st singular value up to a constant, and this is exactly the size at which the truncation errorN1−αN^\{1\-\\alpha\}is below the singular value being retained\.

### B\.7Corrected Ky Fan indexing in the singular\-value split

Combining Lem\.[9](https://arxiv.org/html/2610.08816#Thmtheorem9)with a Hankel block of orderNNbounds‖HN​\(b−a⋆\)‖F2\\\|H\_\{N\}\(b\-a^\{\\star\}\)\\\|\_\{F\}^\{2\}below by∑q\>rsq​\(GN\(α\)\)2\\sum\_\{q\>r\}s\_\{q\}\(G^\{\(\\alpha\)\}\_\{N\}\)^\{2\}, but the passage back to coefficients must use \([22](https://arxiv.org/html/2610.08816#A1.E22)\) and therefore loses a factorNN\. The block of orderNNinvolves the first2​N−12N\-1coefficients, so a context of at least2​N−12N\-1is required for the block to be a genuine restriction of the candidate, and the admissibility condition is2​N−1≤L2N\-1\\leq L\. Writing the two steps in the wrong order, that is applying Ky Fan to a block larger than the context allows, would compare against coefficients the candidate has not been asked to produce\.

The loss of the factorN=Nr=⌈eκα​r⌉N=N\_\{r\}=\\lceil e^\{\\kappa\_\{\\alpha\}\\sqrt\{r\}\}\\rceilis what makes the constant in the exponent of the lower bound differ from the constant in the upper bound\. Both are of the forme−Θ⁡\(r\)e^\{\-\\Theta\(\\sqrt\{r\}\)\}becauselog⁡Nr≍r\\log N\_\{r\}\\asymp\\sqrt\{r\}, so dividing byNrN\_\{r\}changes the constant in front ofr\\sqrt\{r\}and not the exponent\. Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)asserts matching rates and not matching constants, and this is where the discrepancy enters\.

### B\.8Proof of Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)

###### Proof of Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)\.

The upper bound is Corollary[14](https://arxiv.org/html/2610.08816#Thmtheorem14)\.

For the lower bound, fixr≥r0r\\geq r\_\{0\}, letN=NrN=N\_\{r\}from Lem\.[15](https://arxiv.org/html/2610.08816#Thmtheorem15), and letb∈𝒜rb\\in\\mathcal\{A\}\_\{r\}be arbitrary, with contextLLas in \([21](https://arxiv.org/html/2610.08816#A1.E21)\)\. IfL<2​N−1L<2N\-1then the coefficients ofbbbeyondLLvanish while those ofa⋆a^\{\\star\}are of orderj−αj^\{\-\\alpha\}, so the tail term alone givesℰ⁡\(b\)≥mα​c2​∑j\>Lj−2​α≥ctail​e−Ctail​r\\mathcal\{E\}\(b\)\\geq m\_\{\\alpha\}c^\{2\}\\sum\_\{j\>L\}j^\{\-2\\alpha\}\\geq c\_\{\\rm tail\}e^\{\-C\_\{\\rm tail\}\\sqrt\{r\}\}withCtail=\(2​α−1\)​καC\_\{\\rm tail\}=\(2\\alpha\-1\)\\kappa\_\{\\alpha\}after absorbing the ceiling and the factor2​N−12N\-1intoctailc\_\{\\rm tail\}\. Otherwise2​N−1≤L2N\-1\\leq L, so the blockHN​\(b\)H\_\{N\}\(b\)lies inside the context and, by the restricted rank statement following \([21](https://arxiv.org/html/2610.08816#A1.E21)\), has rank at mostrr\. Lem\.[9](https://arxiv.org/html/2610.08816#Thmtheorem9)applied withM=c​GN\(α\)M=cG^\{\(\\alpha\)\}\_\{N\}gives

‖HN​\(b−a⋆\)‖F2≥c2​∑q\>rsq​\(GN\(α\)\)2≥c2​sr\+1​\(GN\(α\)\)2,\\\|H\_\{N\}\(b\-a^\{\\star\}\)\\\|\_\{F\}^\{2\}\\ \\geq\\ c^\{2\}\\sum\_\{q\>r\}s\_\{q\}\\bigl\(G^\{\(\\alpha\)\}\_\{N\}\\bigr\)^\{2\}\\ \\geq\\ c^\{2\}s\_\{r\+1\}\\bigl\(G^\{\(\\alpha\)\}\_\{N\}\\bigr\)^\{2\},and \([22](https://arxiv.org/html/2610.08816#A1.E22)\) converts this to‖b−a⋆‖22≥c2​N−1​sr\+1​\(GN\(α\)\)2\\\|b\-a^\{\\star\}\\\|\_\{2\}^\{2\}\\geq c^\{2\}N^\{\-1\}s\_\{r\+1\}\(G^\{\(\\alpha\)\}\_\{N\}\)^\{2\}\. Inserting \([32](https://arxiv.org/html/2610.08816#A2.E32)\) andN=⌈eκα​r⌉N=\\lceil e^\{\\kappa\_\{\\alpha\}\\sqrt\{r\}\}\\rceil, and using the lower half of \([19](https://arxiv.org/html/2610.08816#A1.E19)\), gives

ℰ⁡\(b\)≥mα​c2​cα2​e−κα​r​e−2​Cα​r\.\\mathcal\{E\}\(b\)\\ \\geq\\ m\_\{\\alpha\}c^\{2\}c\_\{\\alpha\}^\{2\}e^\{\-\\kappa\_\{\\alpha\}\\sqrt\{r\}\}e^\{\-2C\_\{\\alpha\}\\sqrt\{r\}\}\.TakingC1=max⁡\{Ctail,κα\+2​Cα\}C\_\{1\}=\\max\\\{C\_\{\\rm tail\},\\kappa\_\{\\alpha\}\+2C\_\{\\alpha\}\\\}and decreasingc1\>0c\_\{1\}\>0to cover both context branches givesℰ⁡\(b\)≥c1​e−C1​r\\mathcal\{E\}\(b\)\\geq c\_\{1\}e^\{\-C\_\{1\}\\sqrt\{r\}\}\. Taking the infimum overb∈𝒜rb\\in\\mathcal\{A\}\_\{r\}and overLLgives \([3](https://arxiv.org/html/2610.08816#S3.E3)\)\.

For \([4](https://arxiv.org/html/2610.08816#S3.E4)\), apply the elementary inversion that ifc1​e−C1​rγ≤infb∈𝒜rℰ⁡\(b\)≤C2​e−c2​rγc\_\{1\}e^\{\-C\_\{1\}r^\{\\gamma\}\}\\leq\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)\\leq C\_\{2\}e^\{\-c\_\{2\}r^\{\\gamma\}\}for largerr, thenr⋆​\(τ\)=Θ⁡\(log1/γ⁡\(1/τ\)\)r^\{\\star\}\(\\tau\)=\\Theta\(\\log^\{1/\\gamma\}\(1/\\tau\)\)\. Hereγ=1/2\\gamma=1/2, which givesΘ⁡\(log2⁡\(1/τ\)\)\\Theta\(\\log^\{2\}\(1/\\tau\)\)\. ∎

### B\.9Analytic algebraic Laplace\-mixture extension

The same exponent holds for a broader class\. Fixα\>1\\alpha\>1and let the analytic algebraic Laplace\-mixture class consist of the positive sequences

aj=∫0∞e−\(j−1\)​t​tα−1​h​\(t\)​𝑑t,j≥1,a\_\{j\}=\\int\_\{0\}^\{\\infty\}e^\{\-\(j\-1\)t\}t^\{\\alpha\-1\}h\(t\)\\,\\mathrm\{d\}t,\\qquad j\\geq 1,\(34\)wherehhis positive on\[0,∞\)\[0,\\infty\)withh⁡\(0\)\>0h\(0\)\>0, extends holomorphically to an open set containing both a neighbourhood\{\|z\|<r0\}\\\{\|z\|<r\_\{0\}\\\}of the origin for somer0\>0r\_\{0\}\>0and a sectorΣϑ=\{z≠0:\|arg⁡z\|<ϑ\}\\Sigma\_\{\\vartheta\}=\\\{z\\neq 0:\|\\arg z\|<\\vartheta\\\}with0<ϑ<π/20<\\vartheta<\\pi/2, and satisfies

\|h⁡\(z\)\|≤Ch​e−μ​ℜ⁡z\(z∈Σϑ\)\|h\(z\)\|\\leq C\_\{h\}e^\{\-\\mu\\Re z\}\\qquad\(z\\in\\Sigma\_\{\\vartheta\}\)\(35\)for constantsCh,μ\>0C\_\{h\},\\mu\>0\. The three requirements are used separately below, so none is decorative\. Positivity ofhhmakes the mixture a moment representation and keeps the constructed modes positive, the valueh⁡\(0\)\>0h\(0\)\>0is what the local Loewner step consumes, and the exponential decay \([35](https://arxiv.org/html/2610.08816#A2.E35)\) together with sectorial analyticity is what the dyadic Bernstein\-ellipse quadrature consumes\. Every suchaasatisfiesaj∼h⁡\(0\)​Γ​\(α\)​j−αa\_\{j\}\\sim h\(0\)\\Gamma\(\\alpha\)j^\{\-\\alpha\}and is squeezed between two positive multiples ofj−αj^\{\-\\alpha\}, so its tail obeys∑j\>Laj2=Θ⁡\(L1−2​α\)\\sum\_\{j\>L\}a\_\{j\}^\{2\}=\\Theta\(L^\{1\-2\\alpha\}\)\. Ordinary algebraic memory and the fractional kernel of App\.[C](https://arxiv.org/html/2610.08816#A3)both belong to this class\.

Two statements carry over\. On the upper side, the geometry that the composite quadrature needs is uniform across dyadic Laplace intervals, and it is worth stating exactly, since it is the only place where sectorial analyticity is consumed\.

###### Lemma 16\(Dyadic Bernstein\-ellipse geometry\)\.

Let\[a,2​a\]⊂\(0,∞\)\[a,2a\]\\subset\(0,\\infty\)and lett=a⁡\(3\+x\)/2t=a\(3\+x\)/2map\[−1,1\]\[\-1,1\]onto it\. For every Bernstein ellipseEρE\_\{\\rho\}with1<ρ<3\+2​21<\\rho<3\+2\\sqrt\{2\}, the image satisfies

minEρ⁡ℜ⁡t=a4​\(6−ρ−ρ−1\)\>0,\\min\_\{E\_\{\\rho\}\}\\Re t=\\frac\{a\}\{4\}\\bigl\(6\-\\rho\-\\rho^\{\-1\}\\bigr\)\>0,so the image lies in the open right half\-plane, and forρ\>1\\rho\>1small enough it lies inside any prescribed sectorΣϑ\\Sigma\_\{\\vartheta\}\. Consequently\|e−\(j−1\)​t\|≤e−cρ​\(j−1\)​a\|e^\{\-\(j\-1\)t\}\|\\leq e^\{\-c\_\{\\rho\}\(j\-1\)a\}uniformly onEρE\_\{\\rho\}\. On the endpoint interval\[0,L−1\]\[0,L^\{\-1\}\]the affine image of a fixed ellipse obeys\|t\|≤Cρ/L\|t\|\\leq C\_\{\\rho\}/L, hence for all sufficiently largeLLit lies inside the fixed holomorphic neighbourhood of the origin; the finitely many remainingLLare absorbed into the constants\. Moreover\|\(j−1\)​t\|≤Cρ\|\(j\-1\)t\|\\leq C\_\{\\rho\}and\|e−\(j−1\)​t\|≤eCρ\|e^\{\-\(j\-1\)t\}\|\\leq e^\{C\_\{\\rho\}\}for everyj≤Lj\\leq L\.

###### Proof\.

A point ofEρE\_\{\\rho\}isx=12​\(ζ\+ζ−1\)x=\\tfrac\{1\}\{2\}\(\\zeta\+\\zeta^\{\-1\}\)with\|ζ\|∈\[ρ−1,ρ\]\|\\zeta\|\\in\[\\rho^\{\-1\},\\rho\], soℜ⁡x≥−12​\(ρ\+ρ−1\)\\Re x\\geq\-\\tfrac\{1\}\{2\}\(\\rho\+\\rho^\{\-1\}\)andℜ⁡t≥a2​\{3−12​\(ρ\+ρ−1\)\}\\Re t\\geq\\tfrac\{a\}\{2\}\\\{3\-\\tfrac\{1\}\{2\}\(\\rho\+\\rho^\{\-1\}\)\\\}, which is the stated quantity and is positive exactly whenρ\+ρ−1<6\\rho\+\\rho^\{\-1\}<6, that isρ<3\+2​2\\rho<3\+2\\sqrt\{2\}\. The sector statement follows becauseEρE\_\{\\rho\}shrinks to\[−1,1\]\[\-1,1\]asρ→1\\rho\\to 1and the image of\[−1,1\]\[\-1,1\]is the compact interval\[a,2​a\]\[a,2a\]on the positive axis\. The exponential bound is\|e−\(j−1\)​t\|=e−\(j−1\)​ℜ⁡t\|e^\{\-\(j\-1\)t\}\|=e^\{\-\(j\-1\)\\Re t\}with the displayed lower bound onℜ⁡t\\Re t, and the endpoint statement is the triangle inequality on the image of a fixed ellipse under a map of sizeO⁡\(1/L\)O\(1/L\)\. ∎

The compression then follows in three steps, each contributing one factor to the head count\. First, the decay \([35](https://arxiv.org/html/2610.08816#A2.E35)\) lets the Laplace integral be truncated atTη=C⁡\(1\+log⁡η−1\)T\_\{\\eta\}=C\(1\+\\log\\eta^\{\-1\}\)with relative error at mostη/3\\eta/3uniformly injj, with no dependence onLLat all\. The point is that the discarded mass carries the same power ofjjasaja\_\{j\}itself, so the comparison is scale free\. Writingqj:=j−1\+μq\_\{j\}:=j\-1\+\\mu, the bound\|h⁡\(t\)\|≤Ch​e−μ​t\|h\(t\)\|\\leq C\_\{h\}e^\{\-\\mu t\}gives

Rj​\(T\):=∫T∞e−\(j−1\)​t​tα−1​\|h⁡\(t\)\|​𝑑t≤Ch​∫T∞e−qj​t​tα−1​𝑑t=Ch​qj−α​Γ​\(α,qj​T\),R\_\{j\}\(T\):=\\int\_\{T\}^\{\\infty\}e^\{\-\(j\-1\)t\}t^\{\\alpha\-1\}\|h\(t\)\|\\,\\mathrm\{d\}t\\ \\leq\\ C\_\{h\}\\int\_\{T\}^\{\\infty\}e^\{\-q\_\{j\}t\}t^\{\\alpha\-1\}\\,\\mathrm\{d\}t\\ =\\ C\_\{h\}q\_\{j\}^\{\-\\alpha\}\\Gamma\(\\alpha,q\_\{j\}T\),after the substitutionu=qj​tu=q\_\{j\}t\. The upper incomplete Gamma function obeysΓ\(α,x\)≤Cαe−x/2\\Gamma\(\\alpha,x\)\\leq C\_\{\\alpha\}e^\{\-x/2\}forx≥1x\\geq 1, the polynomial prefactor being absorbed into the weaker exponential, andqj≥μ\>0q\_\{j\}\\geq\\mu\>0withqj≍μjq\_\{j\}\\asymp\_\{\\mu\}j, soRj\(T\)≤Cα,μe−μT/2j−αR\_\{j\}\(T\)\\leq C\_\{\\alpha,\\mu\}e^\{\-\\mu T/2\}j^\{\-\\alpha\}\. Sinceh⁡\(0\)\>0h\(0\)\>0givesaj≥c​j−αa\_\{j\}\\geq cj^\{\-\\alpha\}on the same class, the ratio obeysRj​\(T\)/aj≤C​e−c​TR\_\{j\}\(T\)/a\_\{j\}\\leq Ce^\{\-cT\}uniformly injj, andTη=C⁡\(1\+log⁡η−1\)T\_\{\\eta\}=C\(1\+\\log\\eta^\{\-1\}\)suffices\. Second, partition\[L−1,Tη\]\[L^\{\-1\},T\_\{\\eta\}\]dyadically and treat\[0,L−1\]\[0,L^\{\-1\}\]separately; the number of intervals isM=O⁡\(1\+log⁡L\+log⁡log⁡η−1\)M=O\(1\+\\log L\+\\log\\log\\eta^\{\-1\}\)\. Third, apply the analytic Gauss–Jacobi estimate of[Baffet \(2019\)](https://arxiv.org/html/2610.08816#bib.bib3)on each interval withJJnodes\. Lem\.[16](https://arxiv.org/html/2610.08816#Thmtheorem16)supplies the uniform ellipse geometry that estimate requires\. On a dyadic interval\[a,2​a\]\[a,2a\], the ellipse bound gives an absolute quadrature error at mostC​ρ−2​J​e−c⁡\(j−1\)​a​aα≤C​ρ−2​J​j−αC\\rho^\{\-2J\}e^\{\-c\(j\-1\)a\}a^\{\\alpha\}\\leq C\\rho^\{\-2J\}j^\{\-\\alpha\}, and the endpoint interval obeys the sameC​ρ−2​J​j−αC\\rho^\{\-2J\}j^\{\-\\alpha\}bound after the preceding neighbourhood argument\. Summing theMMinterval errors therefore gives at mostC​M​ρ−2​J​j−α≤Cquad​M​ρ−2​J​ajCM\\rho^\{\-2J\}j^\{\-\\alpha\}\\leq C\_\{\\rm quad\}M\\rho^\{\-2J\}a\_\{j\}\. ChoosingJ=O⁡\(1\+log⁡η−1\+log⁡M\)J=O\(1\+\\log\\eta^\{\-1\}\+\\log M\)makes that at mostη/3\\eta/3, and sinceM=O⁡\(1\+log⁡L\+log⁡log⁡η−1\)M=O\(1\+\\log L\+\\log\\log\\eta^\{\-1\}\)the total node countR=M​JR=MJremainsO⁡\{\(1\+log⁡L\+log⁡η−1\)2\}O\\\{\(1\+\\log L\+\\log\\eta^\{\-1\}\)^\{2\}\\\}, which is the analogue of Thm\.[12](https://arxiv.org/html/2610.08816#Thmtheorem12)\.

The lower side needs care, and a global Loewner comparison is not available under these hypotheses\. A positive analytic boundedhhmay decay to zero ast→∞t\\to\\infty, so the two mixing measures need not be comparable on the whole half\-line, and we do not assumehhbounded away from zero\. The correct argument is local\. The hypothesish⁡\(0\)\>0h\(0\)\>0from \([34](https://arxiv.org/html/2610.08816#A2.E34)\), together with continuity, givest0\>0t\_\{0\}\>0andc0\>0c\_\{0\}\>0withh⁡\(t\)≥c0h\(t\)\\geq c\_\{0\}on\[0,t0\]\[0,t\_\{0\}\]\. Writingv⁡\(t\)=\(1,e−t,e−2​t,…\)⊤v\(t\)=\(1,e^\{\-t\},e^\{\-2t\},\\dots\)^\{\\top\}and splitting the Laplace representation of𝐇α\\mathbf\{H\}\_\{\\alpha\}att0t\_\{0\}as𝐇α=𝐇α\(0\)\+𝐓α\\mathbf\{H\}\_\{\\alpha\}=\\mathbf\{H\}\_\{\\alpha\}^\{\(0\)\}\+\\mathbf\{T\}\_\{\\alpha\}, the bounde−t≤1e^\{\-t\}\\leq 1on\[0,t0\]\[0,t\_\{0\}\]gives the Loewner inequality𝐇⁡\(a\)⪰c0​Γ​\(α\)​𝐇α\(0\)\\mathbf\{H\}\(a\)\\succeq c\_\{0\}\\Gamma\(\\alpha\)\\mathbf\{H\}\_\{\\alpha\}^\{\(0\)\}on the low\-ttpiece alone\. The tail operator is exponentially compressible, since‖\(I−PN\)​v​\(t\)‖22≤Ct0​e−2​N​t0\\\|\(I\-P\_\{N\}\)v\(t\)\\\|\_\{2\}^\{2\}\\leq C\_\{t\_\{0\}\}e^\{\-2Nt\_\{0\}\}fort≥t0t\\geq t\_\{0\}gives‖𝐓α−PN​𝐓α​PN‖op≤C​e−c​N\\\|\\mathbf\{T\}\_\{\\alpha\}\-P\_\{N\}\\mathbf\{T\}\_\{\\alpha\}P\_\{N\}\\\|\_\{\\rm op\}\\leq Ce^\{\-cN\}and hencesN\+1​\(𝐓α\)≤C​e−c​Ns\_\{N\+1\}\(\\mathbf\{T\}\_\{\\alpha\}\)\\leq Ce^\{\-cN\}\. Applying the Ky Fan inequalitys2​N\+1​\(A\+B\)≤sN\+1​\(A\)\+sN\+1​\(B\)s\_\{2N\+1\}\(A\+B\)\\leq s\_\{N\+1\}\(A\)\+s\_\{N\+1\}\(B\)to𝐇α=𝐇α\(0\)\+𝐓α\\mathbf\{H\}\_\{\\alpha\}=\\mathbf\{H\}\_\{\\alpha\}^\{\(0\)\}\+\\mathbf\{T\}\_\{\\alpha\}, together with Widom’s scale for𝐇α\\mathbf\{H\}\_\{\\alpha\}and the exponentially small tail term, transfers the root\-exponential lower scale to𝐇α\(0\)\\mathbf\{H\}\_\{\\alpha\}^\{\(0\)\}and therefore, through the Loewner inequality, to𝐇⁡\(a\)\\mathbf\{H\}\(a\), givingsn​\(𝐇⁡\(a\)\)≥ca​e−Ca​ns\_\{n\}\(\\mathbf\{H\}\(a\)\)\\geq c\_\{a\}e^\{\-C\_\{a\}\\sqrt\{n\}\}\. The doubled index in the Ky Fan step is what changes the constant in the exponent and not the exponent itself\.

Combining the two sides givesinfb∈𝒜rℰ⁡\(b\)=e−Θ⁡\(r\)\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\(b\)=e^\{\-\\Theta\(\\sqrt\{r\}\)\}andr⋆​\(τ\)=Θ⁡\(log2⁡\(1/τ\)\)r^\{\\star\}\(\\tau\)=\\Theta\(\\log^\{2\}\(1/\\tau\)\)over the whole class, so the law is not an artifact of the exact power\.

### B\.10Current\-token affine\-gating benchmark

The theorem restricts the architecture class, so it is worth recording one enlargement that does not change the exponent\. Consider a recurrence whose transition is affine in the previous state with coefficients depending on the current input,St=A⁡\(Zt\)​St−1\+b⁡\(Zt\)S\_\{t\}=A\(Z\_\{t\}\)S\_\{t\-1\}\+b\(Z\_\{t\}\)andm^t=c​\(Zt\)⊤​St−1\+d⁡\(Zt\)\\widehat\{m\}\_\{t\}=c\(Z\_\{t\}\)^\{\\top\}S\_\{t\-1\}\+d\(Z\_\{t\}\), on the noiseless iid Gaussian benchmark with target in the first chaos\. Assume a stationary causal solution with

𝔼​‖St‖2<∞,𝔼​m^t2<∞,\\mathbb\{E\}\\\|S\_\{t\}\\\|^\{2\}<\\infty,\\qquad\\mathbb\{E\}\\widehat\{m\}\_\{t\}^\{2\}<\\infty,taken as hypotheses rather than derived: a random affine recursion can have a stationary law under𝔼​log⁡‖A⁡\(Z\)‖op<0\\mathbb\{E\}\\log\\\|A\(Z\)\\\|\_\{\\rm op\}<0while that law has an infinite second moment, so a moment condition on the maps alone does not deliver them\. Assume in addition

𝔼⁡\[\(1\+Z2\)​\{‖A⁡\(Z\)‖op2\+‖b⁡\(Z\)‖2\+‖c⁡\(Z\)‖2\+d​\(Z\)2\}\]<∞\.\\mathbb\{E\}\\bigl\[\(1\+Z^\{2\}\)\\bigl\\\{\\\|A\(Z\)\\\|\_\{\\rm op\}^\{2\}\+\\\|b\(Z\)\\\|^\{2\}\+\\\|c\(Z\)\\\|^\{2\}\+d\(Z\)^\{2\}\\bigr\\\}\\bigr\]<\\infty\.This last condition is stronger than necessary, but it ensures that the Bochner expectationsA¯\\bar\{A\},c¯\\bar\{c\}andvvbelow and the first\-chaos covariances are well defined\. The readout is taken before the current update\. A gate that reads after the update,m^t=c~​\(Zt\)⊤​St\+d~​\(Zt\)\\widehat\{m\}\_\{t\}=\\widetilde\{c\}\(Z\_\{t\}\)^\{\\top\}S\_\{t\}\+\\widetilde\{d\}\(Z\_\{t\}\), is the same predictor withc⁡\(z\)=A​\(z\)⊤​c~​\(z\)c\(z\)=A\(z\)^\{\\top\}\\widetilde\{c\}\(z\)andd⁡\(z\)=c~​\(z\)⊤​b​\(z\)\+d~​\(z\)d\(z\)=\\widetilde\{c\}\(z\)^\{\\top\}b\(z\)\+\\widetilde\{d\}\(z\), and is therefore covered whenever those induced maps satisfy the moment condition above; that is a mild restriction rather than an automatic consequence, since the inducedccinvolves a product of two input\-dependent factors\. LetμS=𝔼​St\\mu\_\{S\}=\\mathbb\{E\}S\_\{t\},S~t=St−μS\\widetilde\{S\}\_\{t\}=S\_\{t\}\-\\mu\_\{S\},A¯=𝔼​A​\(Zt\)\\bar\{A\}=\\mathbb\{E\}A\(Z\_\{t\}\),c¯=𝔼​c​\(Zt\)\\bar\{c\}=\\mathbb\{E\}c\(Z\_\{t\}\), and define the centered injectionξ⁡\(z\)=A⁡\(z\)​μS\+b⁡\(z\)−μS\\xi\(z\)=A\(z\)\\mu\_\{S\}\+b\(z\)\-\\mu\_\{S\}, for which𝔼​ξ​\(Zt\)=0\\mathbb\{E\}\\xi\(Z\_\{t\}\)=0\. ThenS~t=A¯​S~t−1\+\{A⁡\(Zt\)−A¯\}​S~t−1\+ξ⁡\(Zt\)\\widetilde\{S\}\_\{t\}=\\bar\{A\}\\widetilde\{S\}\_\{t\-1\}\+\\\{A\(Z\_\{t\}\)\-\\bar\{A\}\\\}\\widetilde\{S\}\_\{t\-1\}\+\\xi\(Z\_\{t\}\)\. Writingv=𝔼⁡\[Zt​ξ​\(Zt\)\]v=\\mathbb\{E\}\[Z\_\{t\}\\xi\(Z\_\{t\}\)\], independence gives𝔼⁡\[S~t−1​Zt−k\]=A¯k−1​v\\mathbb\{E\}\[\\widetilde\{S\}\_\{t\-1\}Z\_\{t\-k\}\]=\\bar\{A\}^\{k\-1\}v, and hence the first\-chaos coefficients satisfy

βj=c¯⊤​A¯j−2​v\(j≥2\)\.\\beta\_\{j\}=\\bar\{c\}^\{\\top\}\\bar\{A\}^\{\\,j\-2\}v\\qquad\(j\\geq 2\)\.\(36\)The shifted sequence therefore has Hankel rank at most the state dimension, by \([20](https://arxiv.org/html/2610.08816#A1.E20)\) applied to \([36](https://arxiv.org/html/2610.08816#A2.E36)\)\.

Consequently the entire lower\-bound argument applies to the averaged transition, and the bestmm\-state gated risk on this benchmark is again bracketed byc1​e−C1​mc\_\{1\}e^\{\-C\_\{1\}\\sqrt\{m\}\}andC2​e−c2​mC\_\{2\}e^\{\-c\_\{2\}\\sqrt\{m\}\}, som⋆​\(τ\)=Θ⁡\(log2⁡\(1/τ\)\)m^\{\\star\}\(\\tau\)=\\Theta\(\\log^\{2\}\(1/\\tau\)\)\. A gated linear attention head with key widthdkd\_\{k\}contributes at mostdkd\_\{k\}such modes throughρℓ=𝔼​αℓ​\(Zt\)\\rho\_\{\\ell\}=\\mathbb\{E\}\\alpha\_\{\\ell\}\(Z\_\{t\}\), soHHheads give at most∑hdk,h\\sum\_\{h\}d\_\{k,h\}linear predictive modes and the same bound withrrreplaced by that budget\. Data\-dependent gating of this affine kind therefore buys no better exponent, and Sec\.[5](https://arxiv.org/html/2610.08816#S5)is needed precisely because it treats recurrences that are not affine in the state\.

## Appendix CFractional and Singular Predictive Geometry

### C\.1Predictive Hilbert\-space completion

App\.[A](https://arxiv.org/html/2610.08816#A1)assumedfXf\_\{X\}bounded above, which fails at a spectral pole\. The identity \([17](https://arxiv.org/html/2610.08816#A1.E17)\) survives, but its right\-hand side is no longer comparable to a coefficient norm and the space of admissible perturbations must be constructed rather than assumed\.

Letc00c\_\{00\}be the finitely supported sequences and letℋfX\\mathcal\{H\}\_\{f\_\{X\}\}be the completion ofc00c\_\{00\}under the seminorm \([18](https://arxiv.org/html/2610.08816#A1.E18)\), with elements written as equivalence classes𝐮=\[u\(k\)\]\\mathbf\{u\}=\[u^\{\(k\)\}\]of Cauchy sequences\.

###### Proposition 17\(Forecast risk on the singular predictive space\)\.

Do not assumefXf\_\{X\}bounded above\. For𝐮∈ℋfX\\mathbf\{u\}\\in\\mathcal\{H\}\_\{f\_\{X\}\}the filtersUt\(k\)=∑j≥1uj\(k\)​Xt\+1−jU\_\{t\}^\{\(k\)\}=\\sum\_\{j\\geq 1\}u\_\{j\}^\{\(k\)\}X\_\{t\+1\-j\}are Cauchy inL2L\_\{2\}with a limitUt​\(𝐮\)U\_\{t\}\(\\mathbf\{u\}\)depending only on the class, and the perturbed predictormt𝐮=mt⋆\+Ut​\(𝐮\)m\_\{t\}^\{\\mathbf\{u\}\}=m\_\{t\}^\{\\star\}\+U\_\{t\}\(\\mathbf\{u\}\)satisfies

ℰ⁡\(𝐮\)=𝔼​\{Xt\+1−mt𝐮\}2−𝔼​\{Xt\+1−mt⋆\}2=‖𝐮‖pred,fX2\.\\mathcal\{E\}\(\\mathbf\{u\}\)=\\mathbb\{E\}\\\{X\_\{t\+1\}\-m\_\{t\}^\{\\mathbf\{u\}\}\\\}^\{2\}\-\\mathbb\{E\}\\\{X\_\{t\+1\}\-m\_\{t\}^\{\\star\}\\\}^\{2\}=\\\|\\mathbf\{u\}\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}\.\(37\)

###### Proof\.

For finitely supporteduuthe spectral computation of App\.[A](https://arxiv.org/html/2610.08816#A1)gives𝔼⁡\[\(Ut\(k\)−Ut\(l\)\)2\]=‖u\(k\)−u\(l\)‖pred,fX2\\mathbb\{E\}\[\(U\_\{t\}^\{\(k\)\}\-U\_\{t\}^\{\(l\)\}\)^\{2\}\]=\\\|u^\{\(k\)\}\-u^\{\(l\)\}\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}, so a Cauchy sequence in the predictive seminorm maps to a Cauchy sequence inL2L\_\{2\}, and equivalent sequences have the same limit\. The limit isℱt\\mathcal\{F\}\_\{t\}\-measurable and therefore orthogonal toεt\+1\\varepsilon\_\{t\+1\}, so the excess risk equals𝔼⁡\[Ut​\(𝐮\)2\]\\mathbb\{E\}\[U\_\{t\}\(\\mathbf\{u\}\)^\{2\}\], which is the limit of‖u\(k\)‖pred,fX2\\\|u^\{\(k\)\}\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}\. ∎

The perturbed predictor in \([37](https://arxiv.org/html/2610.08816#A3.E37)\) is defined as an affine object and not as a coordinatewise coefficient sum, because an abstract element ofℋfX\\mathcal\{H\}\_\{f\_\{X\}\}need not have a convergent coefficient expansion in general\. For the fractional class below it does, and Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)is what shows this, so Prop\.[17](https://arxiv.org/html/2610.08816#Thmtheorem17)is the general statement and Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)is the case in which coordinates are recovered\.

### C\.2Fractional integration operator

LetIdI\_\{d\}be the discrete fractional\-integration operator with transfer function\(1−z\)−d\(1\-z\)^\{\-d\}, so that\(Id​u\)n=∑j≤nπn−j​\(d\)​uj\(I\_\{d\}u\)\_\{n\}=\\sum\_\{j\\leq n\}\\pi\_\{n\-j\}\(d\)u\_\{j\}withπk​\(d\)=Γ⁡\(k\+d\)/\(Γ⁡\(d\)​Γ​\(k\+1\)\)\\pi\_\{k\}\(d\)=\\Gamma\(k\+d\)/\(\\Gamma\(d\)\\Gamma\(k\+1\)\)\. Multiplication by\(1−e−i​ω\)−d\(1\-e^\{\-i\\omega\}\)^\{\-d\}on the Fourier side is exactly division by the singular factor of the spectrum, soIdI\_\{d\}is the change of variables that turns a singular predictive norm into an ordinaryℓ2\\ell\_\{2\}norm\.

For0<d<1/20<d<1/2the operator is unbounded onℓ2\\ell\_\{2\}but is defined onc00c\_\{00\}, and the compositionId​I−dI\_\{d\}I\_\{\-d\}is the identity there\. Its role below is not as a bounded map but as a coordinate change under which the predictive norm becomes Euclidean\.

### C\.3FARIMA coefficients and exact singular geometry

For FARIMA\(0,d,0\)\(0,d,0\)as in \([5](https://arxiv.org/html/2610.08816#S4.E5)\) the spectral density isfd​\(ω\)=σ2​\|1−e−i​ω\|−2​df\_\{d\}\(\\omega\)=\\sigma^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}, and the autoregressive predictive\-memory coefficients are

aj​\(d\)=d​Γ​\(j−d\)Γ⁡\(1−d\)​Γ​\(j\+1\)∼dΓ⁡\(1−d\)​j−1−d\.a\_\{j\}\(d\)=\\frac\{d\\,\\Gamma\(j\-d\)\}\{\\Gamma\(1\-d\)\\Gamma\(j\+1\)\}\\sim\\frac\{d\}\{\\Gamma\(1\-d\)\}j^\{\-1\-d\}\.\(38\)Two facts are worth separating\. The coefficients decay likej−1−dj^\{\-1\-d\}, so their unweighted tail is∑j\>Laj​\(d\)2=Θ⁡\(L−1−2​d\)\\sum\_\{j\>L\}a\_\{j\}\(d\)^\{2\}=\\Theta\(L^\{\-1\-2d\}\)\. The predictive tail is different, and truncating the exact kernel at contextLLgives

ℰd​\(a\(L\)​\(d\)\)=Θd​\(L−1\),\\mathcal\{E\}\_\{d\}\\bigl\(a^\{\(L\)\}\(d\)\\bigr\)=\\Theta\_\{d\}\(L^\{\-1\}\),\(39\)which is strictly larger than the coefficient tail\. The discrepancy betweenL−1L^\{\-1\}andL−1−2​dL^\{\-1\-2d\}is the whole content of the singular geometry, and it is why a calculation that measures coefficient error would get the context law wrong\.

##### Proof of \([39](https://arxiv.org/html/2610.08816#A3.E39)\)\.

For pure fractional noise, Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)gives the exact identityℰd​\(a\(L\)​\(d\)\)=σ2​‖Id​a\>L​\(d\)‖22\\mathcal\{E\}\_\{d\}\(a^\{\(L\)\}\(d\)\)=\\sigma^\{2\}\\\|I\_\{d\}a^\{\>L\}\(d\)\\\|\_\{2\}^\{2\}\. WriteA\>L​\(ω\)=∑j\>Laj​\(d\)​e−i​j​ωA\_\{\>L\}\(\\omega\)=\\sum\_\{j\>L\}a\_\{j\}\(d\)e^\{\-ij\\omega\}for the omitted tail\. Sinceaj\(d\)≍dj−1−da\_\{j\}\(d\)\\asymp\_\{d\}j^\{\-1\-d\}and is eventually decreasing, for\|ω\|≤L−1\|\\omega\|\\leq L^\{\-1\},

\|A\>L​\(ω\)\|≤∑j\>Laj​\(d\)≤Cd​L−d\.\|A\_\{\>L\}\(\\omega\)\|\\leq\\sum\_\{j\>L\}a\_\{j\}\(d\)\\leq C\_\{d\}L^\{\-d\}\.ForL−1<\|ω\|≤πL^\{\-1\}<\|\\omega\|\\leq\\pi, Abel summation and the standard bound on geometric partial sums give

\|A\>L​\(ω\)\|≤Cd​L−1−d\|ω\|\.\|A\_\{\>L\}\(\\omega\)\|\\leq C\_\{d\}\\frac\{L^\{\-1\-d\}\}\{\|\\omega\|\}\.Using\|1−e−i​ω\|≍\|ω\|\|1\-e^\{\-i\\omega\}\|\\asymp\|\\omega\|on\[−π,π\]\[\-\\pi,\\pi\]and splitting the exact fractional risk at\|ω\|=L−1\|\\omega\|=L^\{\-1\}yields

‖Id​a\>L​\(d\)‖22≤Cd​\{L−2​d​∫0L−1ω−2​d​𝑑ω\+L−2−2​d​∫L−1πω−2−2​d​𝑑ω\}≤Cd​L−1\.\\\|I\_\{d\}a^\{\>L\}\(d\)\\\|\_\{2\}^\{2\}\\leq C\_\{d\}\\left\\\{L^\{\-2d\}\\int\_\{0\}^\{L^\{\-1\}\}\\omega^\{\-2d\}\\,\\mathrm\{d\}\\omega\+L^\{\-2\-2d\}\\int\_\{L^\{\-1\}\}^\{\\pi\}\\omega^\{\-2\-2d\}\\,\\mathrm\{d\}\\omega\\right\\\}\\leq C\_\{d\}L^\{\-1\}\.This proves the upper bound in \([39](https://arxiv.org/html/2610.08816#A3.E39)\)\. For the lower bound, hard truncation is one admissibleLL\-context predictor, so its excess risk is at least the optimal finite\-context gapVL−V∞≍dL−1V\_\{L\}\-V\_\{\\infty\}\\asymp\_\{d\}L^\{\-1\}from Corollary[3](https://arxiv.org/html/2610.08816#Thmtheorem3)\.

### C\.4Proof of Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)

###### Proof of Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)\.

WritefX​\(ω\)=\|1−e−i​ω\|−2​d​g​\(ω\)f\_\{X\}\(\\omega\)=\|1\-e^\{\-i\\omega\}\|^\{\-2d\}g\(\\omega\)with0<mg≤g≤Mg0<m\_\{g\}\\leq g\\leq M\_\{g\}\. For a finitely supporteduuwith transfer functionUU,

‖u‖pred,fX2=12​π​∫−ππ\|U⁡\(e−i​ω\)\|2​\|1−e−i​ω\|−2​d​g​\(ω\)​𝑑ω\.\\\|u\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|U\(e^\{\-i\\omega\}\)\|^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}g\(\\omega\)\\,\\mathrm\{d\}\\omega\.The transfer function ofId​uI\_\{d\}uisU⁡\(e−i​ω\)​\(1−e−i​ω\)−dU\(e^\{\-i\\omega\}\)\(1\-e^\{\-i\\omega\}\)^\{\-d\}, whose squared modulus is exactly\|U\|2​\|1−e−i​ω\|−2​d\|U\|^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\. Inserting the two\-sided bound onggand applying Parseval toId​uI\_\{d\}ugives \([6](https://arxiv.org/html/2610.08816#S4.E6)\)\. For pure fractional noiseg≡σ2g\\equiv\\sigma^\{2\}, so both inequalities are equalities andℰd​\(a⁡\(d\)\+u\)=σ2​‖Id​u‖22\\mathcal\{E\}\_\{d\}\(a\(d\)\+u\)=\\sigma^\{2\}\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}\.

For the last claim, since\|1−e−i​ω\|≤2\|1\-e^\{\-i\\omega\}\|\\leq 2we havefX≥2−2​d​mgf\_\{X\}\\geq 2^\{\-2d\}m\_\{g\}, so convergence in the predictive norm implies convergence inℓ2\\ell\_\{2\}\. Every abstract element ofℋfX\\mathcal\{H\}\_\{f\_\{X\}\}therefore has canonicalℓ2\\ell\_\{2\}coefficient coordinates in this class, andIdI\_\{d\}extends uniquely and continuously toℋfX\\mathcal\{H\}\_\{f\_\{X\}\}with closed image\. ∎

The theorem should be read as identifying the correct inner product rather than as a perturbation of the regular case\. Low\-frequency coefficient error is amplified by\|1−e−i​ω\|−2​d\|1\-e^\{\-i\\omega\}\|^\{\-2d\}, so two candidates with the sameℓ2\\ell\_\{2\}error can have very different forecast risk depending on where in the spectrum their error sits\.

### C\.5Finite\-context prediction and partial autocorrelations

LetVLV\_\{L\}be the variance of the best linear predictor from the lastLLobservations andV∞V\_\{\\infty\}its limit\. Two classical facts drive Corollary[3](https://arxiv.org/html/2610.08816#Thmtheorem3)\. First, the Durbin–Levinson recursion givesVL=V0​∏k≤L\(1−ϕk​k2\)V\_\{L\}=V\_\{0\}\\prod\_\{k\\leq L\}\(1\-\\phi\_\{kk\}^\{2\}\)withϕk​k\\phi\_\{kk\}the partial autocorrelations\. Second, for FARIMA\(0,d,0\)\(0,d,0\)these are available in closed form,

ϕk​k=dk−d,\\phi\_\{kk\}=\\frac\{d\}\{k\-d\},\(40\)which is the classical fractional identity\([Hosking, 1981](https://arxiv.org/html/2610.08816#bib.bib26);[Inoue, 2002](https://arxiv.org/html/2610.08816#bib.bib29)\)\.

The determinant route gives the same object\. For any purely nondeterministic process with nonsingular finite covariance matrices,VL=DL\+1​\(fX\)/DL​\(fX\)V\_\{L\}=D\_\{L\+1\}\(f\_\{X\}\)/D\_\{L\}\(f\_\{X\}\)for the Toeplitz determinantsDLD\_\{L\}, soDn=∏j<nVjD\_\{n\}=\\prod\_\{j<n\}V\_\{j\}and the finite\-context law is equivalent to a Toeplitz determinant asymptotic\. This is the connection used in App\.[E](https://arxiv.org/html/2610.08816#A5)\.

### C\.6Proof of Corollary[3](https://arxiv.org/html/2610.08816#Thmtheorem3)

###### Proof of Corollary[3](https://arxiv.org/html/2610.08816#Thmtheorem3)\.

Substituting \([40](https://arxiv.org/html/2610.08816#A3.E40)\) into the Durbin–Levinson product and telescoping the Gamma factors gives the exact identity

VLσ2=Γ⁡\(L\+1\)​Γ​\(L\+1−2​d\)Γ​\(L\+1−d\)2,\\frac\{V\_\{L\}\}\{\\sigma^\{2\}\}=\\frac\{\\Gamma\(L\+1\)\\Gamma\(L\+1\-2d\)\}\{\\Gamma\(L\+1\-d\)^\{2\}\},\(41\)since∏k≤L\(1−d2/\(k−d\)2\)=∏k≤Lk⁡\(k−2​d\)/\(k−d\)2\\prod\_\{k\\leq L\}\(1\-d^\{2\}/\(k\-d\)^\{2\}\)=\\prod\_\{k\\leq L\}k\(k\-2d\)/\(k\-d\)^\{2\}andV0=γ⁡\(0\)=σ2​Γ​\(1−2​d\)/Γ​\(1−d\)2V\_\{0\}=\\gamma\(0\)=\\sigma^\{2\}\\Gamma\(1\-2d\)/\\Gamma\(1\-d\)^\{2\}\. Stirling’s formula giveslog⁡\(VL/σ2\)=d2/L\+O⁡\(L−2\)\\log\(V\_\{L\}/\\sigma^\{2\}\)=d^\{2\}/L\+O\(L^\{\-2\}\), soV∞=σ2V\_\{\\infty\}=\\sigma^\{2\}andVL−V∞=V∞​d2/L\+o⁡\(L−1\)V\_\{L\}\-V\_\{\\infty\}=V\_\{\\infty\}d^\{2\}/L\+o\(L^\{\-1\}\)\.

For the last sentence of the corollary, the same computation for a spectrum withQQseparated fractional poles of strengthsd1,…,dQd\_\{1\},\\dots,d\_\{Q\}replacesd2d^\{2\}by∑νdν2\\sum\_\{\\nu\}d\_\{\\nu\}^\{2\}, which is the content of the multiple\-pole result below\. For a real\-valued process the pole set is reflection symmetric, so a nonzero physical frequency contributes its two conjugate singularities separately to this sum\. For a Gaussian process the finite\-past conditional expectation is linear, so \([41](https://arxiv.org/html/2610.08816#A3.E41)\) is also the excess Bayes risk of the best measurable predictor usingLLobservations, not merely the best linear one\. ∎

Equation \([41](https://arxiv.org/html/2610.08816#A3.E41)\) is exact and not asymptotic, which is what makes the numerical panel of Fig\.[1](https://arxiv.org/html/2610.08816#S7.F1)a check of the theorem rather than of an approximation\. The numerically stable route is to evaluatelog⁡\(VL/V∞\)\\log\(V\_\{L\}/V\_\{\\infty\}\)as the cancellation\-free tail sum−∑k\>Llog\(1−d2/\(k−d\)2\)\-\\sum\_\{k\>L\}\\log\(1\-d^\{2\}/\(k\-d\)^\{2\}\)plus a leading\-tail trigamma correction, since differencing the three log\-Gamma terms directly loses several digits at largeLL\.

### C\.7Multiple separated fractional poles

SupposefX​\(ω\)=g⁡\(ω\)​∏ν≤Q\|1−e−i⁡\(ω−ων\)\|−2​dνf\_\{X\}\(\\omega\)=g\(\\omega\)\\prod\_\{\\nu\\leq Q\}\|1\-e^\{\-i\(\\omega\-\\omega\_\{\\nu\}\)\}\|^\{\-2d\_\{\\nu\}\}with distinctων\\omega\_\{\\nu\}\. Two hypotheses onggare used, and they are not the same\. The geometry statement below needs only0<mg≤g≤Mg<∞0<m\_\{g\}\\leq g\\leq M\_\{g\}<\\infty\. The sharp context law additionally needsg=eVg=e^\{V\}withVVreal and smooth enough for the Fisher–Hartwig determinant asymptotics of[Deift et al\. \(2011\)](https://arxiv.org/html/2610.08816#bib.bib14), since boundedness above and below does not by itself place the symbol in the class those asymptotics cover; the pure product of fractional poles,g≡σ2g\\equiv\\sigma^\{2\}, is the simplest admissible case\. Writingℐ𝐝,𝝎\\mathcal\{I\}\_\{\\mathbf\{d\},\\bm\{\\omega\}\}for the composition of the corresponding fractional\-integration factors, the same argument as in Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)gives

mg​‖ℐ𝐝,𝝎​u‖22≤‖u‖pred,fX2≤Mg​‖ℐ𝐝,𝝎​u‖22,m\_\{g\}\\\|\\mathcal\{I\}\_\{\\mathbf\{d\},\\bm\{\\omega\}\}u\\\|\_\{2\}^\{2\}\\leq\\\|u\\\|\_\{\\mathrm\{pred\},f\_\{X\}\}^\{2\}\\leq M\_\{g\}\\\|\\mathcal\{I\}\_\{\\mathbf\{d\},\\bm\{\\omega\}\}u\\\|\_\{2\}^\{2\},\(42\)so each pole contributes one factor to the geometry\. The finite\-context law becomesVL−V∞=V∞​L−1​sf\+o⁡\(L−1\)V\_\{L\}\-V\_\{\\infty\}=V\_\{\\infty\}L^\{\-1\}s\_\{f\}\+o\(L^\{\-1\}\)withsf=∑νdν2s\_\{f\}=\\sum\_\{\\nu\}d\_\{\\nu\}^\{2\}, the strengths adding in squares because separated Fisher–Hartwig singularities contribute additively to the Toeplitz determinant at leading order\. Passing from that determinant asymptotic to the ratio law is not a one\-line step, and it is worth recording why\. Fisher–Hartwig asymptotics givelog⁡Dn=n​log⁡V∞\+sf​log⁡n\+log⁡Cf\+o⁡\(1\)\\log D\_\{n\}=n\\log V\_\{\\infty\}\+s\_\{f\}\\log n\+\\log C\_\{f\}\+o\(1\), while the determinant factorizes aslog⁡Dn=∑j<nlog⁡Vj\\log D\_\{n\}=\\sum\_\{j<n\}\\log V\_\{j\}\. Subtractingn​log⁡V∞n\\log V\_\{\\infty\}and writingrL:=log⁡\(VL/V∞\)r\_\{L\}:=\\log\(V\_\{L\}/V\_\{\\infty\}\)yields∑j<nrj=sf​log⁡n\+C\+o⁡\(1\)\\sum\_\{j<n\}r\_\{j\}=s\_\{f\}\\log n\+C\+o\(1\), but differencing that relation does not giverL=sf/L\+o⁡\(L−1\)r\_\{L\}=s\_\{f\}/L\+o\(L^\{\-1\}\)on its own, because a difference of twoo⁡\(1\)o\(1\)terms need not beo⁡\(L−1\)o\(L^\{\-1\}\)\. What closes the gap is monotonicity: adding a regressor cannot increase least\-squares prediction error, so\(rL\)\(r\_\{L\}\)is nonnegative and nonincreasing, and a monotone sequence whose partial sums ares​log⁡n\+C\+o⁡\(1\)s\\log n\+C\+o\(1\)satisfiesL​rL→sLr\_\{L\}\\to s\.

###### Lemma 18\(Monotone density\)\.

Let\(rn\)n≥1\(r\_\{n\}\)\_\{n\\geq 1\}be nonnegative and nonincreasing with∑k≤nrk=s​log⁡n\+C\+o⁡\(1\)\\sum\_\{k\\leq n\}r\_\{k\}=s\\log n\+C\+o\(1\)for somes\>0s\>0\. Thenn​rn→snr\_\{n\}\\to s\.

###### Proof\.

Fixλ\>1\\lambda\>1and setmn=⌊n/λ⌋m\_\{n\}=\\lfloor n/\\lambda\\rfloor,Nn=⌊λ​n⌋N\_\{n\}=\\lfloor\\lambda n\\rfloor\. Monotonicity gives\(n−mn\)​rn≤∑mn<k≤nrk=s​log⁡\(n/mn\)\+o⁡\(1\)=s​log⁡λ\+o⁡\(1\)\(n\-m\_\{n\}\)r\_\{n\}\\leq\\sum\_\{m\_\{n\}<k\\leq n\}r\_\{k\}=s\\log\(n/m\_\{n\}\)\+o\(1\)=s\\log\\lambda\+o\(1\), solim supnn​rn≤s​λ​log⁡λ/\(λ−1\)\\limsup\_\{n\}nr\_\{n\}\\leq s\\lambda\\log\\lambda/\(\\lambda\-1\)\. In the other direction∑n<k≤Nnrk≤\(Nn−n\)​rn\\sum\_\{n<k\\leq N\_\{n\}\}r\_\{k\}\\leq\(N\_\{n\}\-n\)r\_\{n\}, and the left side iss​log⁡λ\+o⁡\(1\)s\\log\\lambda\+o\(1\), solim infnn​rn≥s​log⁡λ/\(λ−1\)\\liminf\_\{n\}nr\_\{n\}\\geq s\\log\\lambda/\(\\lambda\-1\)\. Both bounds tend tossasλ→1\\lambda\\to 1\. ∎

Applying Lem\.[18](https://arxiv.org/html/2610.08816#Thmtheorem18)to\(rL\)\(r\_\{L\}\)and exponentiating,VL/V∞=erL=1\+sf/L\+o⁡\(L−1\)V\_\{L\}/V\_\{\\infty\}=e^\{r\_\{L\}\}=1\+s\_\{f\}/L\+o\(L^\{\-1\}\), which is the stated law\.

A seasonal fractional model is the degenerate case in which the poles are equally spaced\. There the exact analogue of \([41](https://arxiv.org/html/2610.08816#A3.E41)\) holds withLLreplaced bymL=⌊L/s⌋m\_\{L\}=\\lfloor L/s\\rfloor, givingVL−σ2=σ2​s​d2/L\+o⁡\(L−1\)V\_\{L\}\-\\sigma^\{2\}=\\sigma^\{2\}sd^\{2\}/L\+o\(L^\{\-1\}\), so a seasonal pole structure multiplies the context cost by the period\.

### C\.8Exact exponent misspecification and relative entropy

###### Theorem 19\(Exact risk under exponent misspecification\)\.

For fractional noise with true exponentd∈\(0,1/2\)d\\in\(0,1/2\)and candidated′=d\+Δ∈\(0,1/2\)d^\{\\prime\}=d\+\\Delta\\in\(0,1/2\),

ℰd​\(a⁡\(d′\)\)=σ2​\{Γ⁡\(1\+2​Δ\)Γ​\(1\+Δ\)2−1\},\\mathcal\{E\}\_\{d\}\(a\(d^\{\\prime\}\)\)=\\sigma^\{2\}\\Bigl\\\{\\frac\{\\Gamma\(1\+2\\Delta\)\}\{\\Gamma\(1\+\\Delta\)^\{2\}\}\-1\\Bigr\\\},\(43\)and asΔ→0\\Delta\\to 0this isπ2​σ26​Δ2\+O⁡\(\|Δ\|3\)\\tfrac\{\\pi^\{2\}\\sigma^\{2\}\}\{6\}\\Delta^\{2\}\+O\(\|\\Delta\|^\{3\}\)\.

###### Proof\.

The autoregressive transfer functions areAd​\(z\)=1−\(1−z\)dA\_\{d\}\(z\)=1\-\(1\-z\)^\{d\}, so

Ad′​\(z\)−Ad​\(z\)=\(1−z\)d​\{1−\(1−z\)Δ\}\.A\_\{d^\{\\prime\}\}\(z\)\-A\_\{d\}\(z\)=\(1\-z\)^\{d\}\\bigl\\\{1\-\(1\-z\)^\{\\Delta\}\\bigr\\\}\.Under the true spectral densityfd​\(ω\)=σ2​\|1−e−i​ω\|−2​df\_\{d\}\(\\omega\)=\\sigma^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}the factor\|\(1−e−i​ω\)d\|2\|\(1\-e^\{\-i\\omega\}\)^\{d\}\|^\{2\}cancels the pole exactly, so Prop\.[17](https://arxiv.org/html/2610.08816#Thmtheorem17)gives

ℰd​\(a⁡\(d′\)\)=σ22​π​∫−ππ\|1−\(1−e−i​ω\)Δ\|2​𝑑ω\.\\mathcal\{E\}\_\{d\}\(a\(d^\{\\prime\}\)\)=\\frac\{\\sigma^\{2\}\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\\bigl\|1\-\(1\-e^\{\-i\\omega\}\)^\{\\Delta\}\\bigr\|^\{2\}\\,\\mathrm\{d\}\\omega\.\(44\)The function\(1−z\)Δ\(1\-z\)^\{\\Delta\}is analytic in the open unit disc with integrable boundary values forΔ\>−1\\Delta\>\-1, so its circular mean equals its value atz=0z=0, namely one\. Expanding the modulus in \([44](https://arxiv.org/html/2610.08816#A3.E44)\) therefore contributes11from the constant term,−2\-2from the cross term, andJ⁡\(Δ\)J\(\\Delta\)from the remaining term, where

J⁡\(Δ\):=12​π​∫−ππ\|1−e−i​ω\|2​Δ​𝑑ω\.J\(\\Delta\):=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|1\-e^\{\-i\\omega\}\|^\{2\\Delta\}\\,\\mathrm\{d\}\\omega\.Writing\|1−e−i​ω\|=2​\|sin⁡\(ω/2\)\|\|1\-e^\{\-i\\omega\}\|=2\|\\sin\(\\omega/2\)\|and substitutingx=ω/2x=\\omega/2,

J⁡\(Δ\)=22​Δ\+1π​∫0π/2\(sin⁡x\)2​Δ​𝑑x=22​Δπ​Γ⁡\(Δ\+12\)Γ⁡\(Δ\+1\)=Γ⁡\(1\+2​Δ\)Γ​\(1\+Δ\)2,J\(\\Delta\)=\\frac\{2^\{2\\Delta\+1\}\}\{\\pi\}\\int\_\{0\}^\{\\pi/2\}\(\\sin x\)^\{2\\Delta\}\\,\\mathrm\{d\}x=\\frac\{2^\{2\\Delta\}\}\{\\sqrt\{\\pi\}\}\\frac\{\\Gamma\(\\Delta\+\\tfrac\{1\}\{2\}\)\}\{\\Gamma\(\\Delta\+1\)\}=\\frac\{\\Gamma\(1\+2\\Delta\)\}\{\\Gamma\(1\+\\Delta\)^\{2\}\},the middle step being the Wallis integral and the last the Legendre duplication formula\. Henceℰd​\(a⁡\(d′\)\)=σ2​\{J⁡\(Δ\)−1\}\\mathcal\{E\}\_\{d\}\(a\(d^\{\\prime\}\)\)=\\sigma^\{2\}\\\{J\(\\Delta\)\-1\\\}, which is \([43](https://arxiv.org/html/2610.08816#A3.E43)\)\.

For the expansion,log⁡Γ⁡\(1\+x\)=−γE​x\+∑k≥2\(−1\)k​ζ​\(k\)​xk/k\\log\\Gamma\(1\+x\)=\-\\gamma\_\{\\rm E\}x\+\\sum\_\{k\\geq 2\}\(\-1\)^\{k\}\\zeta\(k\)x^\{k\}/kgives

log⁡Γ⁡\(1\+2​Δ\)−2​log⁡Γ⁡\(1\+Δ\)=∑k≥2\(−1\)k​ζ​\(k\)k​\(2k−2\)​Δk=ζ⁡\(2\)​Δ2\+O⁡\(\|Δ\|3\),\\log\\Gamma\(1\+2\\Delta\)\-2\\log\\Gamma\(1\+\\Delta\)=\\sum\_\{k\\geq 2\}\\frac\{\(\-1\)^\{k\}\\zeta\(k\)\}\{k\}\\bigl\(2^\{k\}\-2\\bigr\)\\Delta^\{k\}=\\zeta\(2\)\\Delta^\{2\}\+O\(\|\\Delta\|^\{3\}\),the Euler constants cancelling, and exponentiating and subtracting one yieldsπ2​Δ2/6\+O⁡\(\|Δ\|3\)\\pi^\{2\}\\Delta^\{2\}/6\+O\(\|\\Delta\|^\{3\}\)\. ∎

Two features are worth noting\. The exact expression depends onΔ\\Deltaalone and not on the truedd, so the local curvatureπ2​σ2/6\\pi^\{2\}\\sigma^\{2\}/6is the same everywhere in the fractional family\. The relative entropy rate is exactly half the risk divided byσ2\\sigma^\{2\}, that isD¯\(d∥d′\)=ℰd\(a\(d′\)\)/\(2σ2\)\\overline\{D\}\(d\\\|d^\{\\prime\}\)=\\mathcal\{E\}\_\{d\}\(a\(d^\{\\prime\}\)\)/\(2\\sigma^\{2\}\), which is the specialization of the duality in App\.[E](https://arxiv.org/html/2610.08816#A5)to this family\.

More generally, if two models share a singular factor and differ only in a regular factor, the shared factor cancels in the likelihood ratio, and the relative entropy rate is again half the normalized forecast risk\. A known common singularity therefore contributes nothing to distinguishability, which is what makes the fractional exponent, rather than the presence of a pole, the estimable quantity\.

### C\.9Fisher information and root\-TTestimation

Since the forecast curvature inddisπ2​σ2/6\\pi^\{2\}\\sigma^\{2\}/6, the Fisher information for the fractional exponent isπ2/6\\pi^\{2\}/6\. If an estimator satisfiesT​\(d^T−d\)⇒N⁡\(0,Vd\)\\sqrt\{T\}\(\\widehat\{d\}\_\{T\}\-d\)\\Rightarrow N\(0,V\_\{d\}\)then

T​ℰd​\(a⁡\(d^T\)\)⇒π2​σ2​Vd6​χ12,T\\,\\mathcal\{E\}\_\{d\}\(a\(\\widehat\{d\}\_\{T\}\)\)\\Rightarrow\\frac\{\\pi^\{2\}\\sigma^\{2\}V\_\{d\}\}\{6\}\\chi^\{2\}\_\{1\},\(45\)by the local quadratic\-risk argument of App\.[E](https://arxiv.org/html/2610.08816#A5)applied to a one\-parameter family\. Standard likelihood theory for the fractional exponent gives an efficient asymptotic varianceVd=6/π2V\_\{d\}=6/\\pi^\{2\}, the reciprocal of the Fisher information, under the regularity conditions for maximum likelihood in this model\([Lieberman & Phillips, 2004](https://arxiv.org/html/2610.08816#bib.bib33);[Hualde & Robinson, 2011](https://arxiv.org/html/2610.08816#bib.bib27)\)\. With that variance the limit isσ2​χ12\\sigma^\{2\}\\chi^\{2\}\_\{1\}and, given uniform integrability, the expected excess risk isσ2/T\+o⁡\(T−1\)\\sigma^\{2\}/T\+o\(T^\{\-1\}\)\. Estimating a fractional exponent therefore costs one parameter’s worth of forecast risk, the same as any smooth scalar, so in this metric and under these regularity conditions long memory does not make the exponent harder to learn\.

Combining estimation with a finite context, on a compact interval\[d−,d\+\]⊂\(0,1/2\)\[d\_\{\-\},d\_\{\+\}\]\\subset\(0,1/2\)and for\|d′−d\|​log⁡L\|d^\{\\prime\}\-d\|\\log Lsmall, one hasℰd​\(a\(L\)​\(d′\)\)≤C⁡\{L−1\+\(d′−d\)2\}\\mathcal\{E\}\_\{d\}\(a^\{\(L\)\}\(d^\{\\prime\}\)\)\\leq C\\\{L^\{\-1\}\+\(d^\{\\prime\}\-d\)^\{2\}\\\}, so

ℰd​\(a\(LT\)​\(d^T\)\)=Oℙ​\(LT−1\+T−1\)\.\\mathcal\{E\}\_\{d\}\\bigl\(a^\{\(L\_\{T\}\)\}\(\\widehat\{d\}\_\{T\}\)\\bigr\)=O\_\{\\mathbb\{P\}\}\\bigl\(L\_\{T\}^\{\-1\}\+T^\{\-1\}\\bigr\)\.\(46\)The two terms are the context law and the parametric law, and the context term dominates unlessLTL\_\{T\}grows at least linearly inTT\.

### C\.10Unknown pole location and cusp geometry

If the location of the pole is unknown the geometry changes character\. For the real two\-pole fractional benchmark, write

Dω,d​\(z\)=\(1−ei​ω​z\)d​\(1−e−i​ω​z\)d,fω,d​\(λ\)=σ2​\|Dω,d​\(e−i​λ\)\|−2,D\_\{\\omega,d\}\(z\)=\(1\-e^\{i\\omega\}z\)^\{d\}\(1\-e^\{\-i\\omega\}z\)^\{d\},\\qquad f\_\{\\omega,d\}\(\\lambda\)=\\sigma^\{2\}\|D\_\{\\omega,d\}\(e^\{\-i\\lambda\}\)\|^\{\-2\},and letAω,d=1−Dω,dA\_\{\\omega,d\}=1\-D\_\{\\omega,d\}be the causal autoregressive whitening transfer function\. Defineℰω,d​\(Δ\)\\mathcal\{E\}\_\{\\omega,d\}\(\\Delta\)as the excess forecast risk underfω,df\_\{\\omega,d\}whenAω\+Δ,dA\_\{\\omega\+\\Delta,d\}is used in place ofAω,dA\_\{\\omega,d\}\.

###### Theorem 20\(Pole\-location misspecification has a cusp\)\.

Fix compact rangesω∈\[ω−,ω\+\]⊂\(0,π\)\\omega\\in\[\\omega\_\{\-\},\\omega\_\{\+\}\]\\subset\(0,\\pi\)andd∈\[d−,d\+\]⊂\(0,1/2\)d\\in\[d\_\{\-\},d\_\{\+\}\]\\subset\(0,1/2\)\. There are0<c<C<∞0<c<C<\\inftyandΔ0\>0\\Delta\_\{0\}\>0such that the excess risk of using pole locationω\+Δ\\omega\+\\Deltain place ofω\\omegasatisfies

c​\|Δ\|≤ℰω,d​\(Δ\)≤C​\|Δ\|\(\|Δ\|≤Δ0\)\.c\|\\Delta\|\\leq\\mathcal\{E\}\_\{\\omega,d\}\(\\Delta\)\\leq C\|\\Delta\|\\qquad\(\|\\Delta\|\\leq\\Delta\_\{0\}\)\.\(47\)

###### Proof\.

Writeλ=ω\+x\\lambda=\\omega\+xnear the positive pole and fix a neighbourhood\|x\|≤η\|x\|\\leq\\etasmall enough to miss the conjugate pole at−ω\-\\omega\. On that neighbourhood the ratio of the two spectral factors is

R\+,Δ​\(x\)​\(1−e−i⁡\(x−Δ\)1−e−i​x\)d,R\_\{\+,\\Delta\}\(x\)\\Bigl\(\\frac\{1\-e^\{\-i\(x\-\\Delta\)\}\}\{1\-e^\{\-ix\}\}\\Bigr\)^\{d\},whereR\+,ΔR\_\{\+,\\Delta\}collects the contribution of the conjugate pole and satisfiesR\+,Δ​\(x\)=1\+O⁡\(\|Δ\|\)R\_\{\+,\\Delta\}\(x\)=1\+O\(\|\\Delta\|\)uniformly on the compact parameter set\. Away from the two pole neighbourhoods every factor is smooth inω\\omega, so that region contributesO⁡\(Δ2\)O\(\\Delta^\{2\}\)and is negligible at first order\.

It therefore suffices to analyse

Jd​\(Δ\):=∫−ηη\|\(1−e−i⁡\(x−Δ\)1−e−i​x\)d−1\|2​𝑑x\.J\_\{d\}\(\\Delta\):=\\int\_\{\-\\eta\}^\{\\eta\}\\Bigl\|\\Bigl\(\\frac\{1\-e^\{\-i\(x\-\\Delta\)\}\}\{1\-e^\{\-ix\}\}\\Bigr\)^\{d\}\-1\\Bigr\|^\{2\}\\,\\mathrm\{d\}x\.Substitutingx=\|Δ\|​yx=\|\\Delta\|ycontributes the Jacobian factor\|Δ\|\|\\Delta\|, and1−e−i​u=i​u\+O⁡\(u2\)1\-e^\{\-iu\}=iu\+O\(u^\{2\}\)turns the integrand into a function ofyyalone at leading order, namely

\|\(y−sgn⁡Δy\)brd−1\|2,\\Bigl\|\\Bigl\(\\frac\{y\-\\operatorname\{sgn\}\\Delta\}\{y\}\\Bigr\)^\{d\}\_\{\\rm br\}\-1\\Bigr\|^\{2\},up to anO⁡\(\|Δ\|\)O\(\|\\Delta\|\)correction\. The subscript records that the power is taken along the continuous boundary branch inherited from the spectral factor, which must be specified because the ratio is negative foryybetween the two singular points andddis not an integer; the branch is the one obtained by continuing the factor\(1−e−i⁡\(x−Δ\)\)d\(1\-e^\{\-i\(x\-\\Delta\)\}\)^\{d\}from the positive axis, and with it the limiting integrand is real, nonnegative, and not identically zero\. Two envelopes make the passage to the limit legitimate\. Neary=0y=0the integrand isO⁡\(\|y\|−2​d\)O\(\|y\|^\{\-2d\}\), which is integrable because2​d<12d<1; for\|y\|→∞\|y\|\\to\\inftyit isO⁡\(\|y\|−2\)O\(\|y\|^\{\-2\}\), since\(1−1/y\)d−1=O⁡\(1/y\)\(1\-1/y\)^\{d\}\-1=O\(1/y\)\. The local Taylor bounds for1−e−i​u1\-e^\{\-iu\}supply an integrable dominating function with exactly these two envelopes, uniformly forddin the compact range and uniformly inΔ\\Deltaover the shrinking window, so dominated convergence applies and givesJd​\(Δ\)=\|Δ\|​\{Cd\+o⁡\(1\)\}J\_\{d\}\(\\Delta\)=\|\\Delta\|\\\{C\_\{d\}\+o\(1\)\\\}with0<Cd<∞0<C\_\{d\}<\\inftybounded above and below on compactdd\-sets\. The conjugate pole contributes a second positive term of the same order, whileR\+,Δ=1\+O⁡\(\|Δ\|\)R\_\{\+,\\Delta\}=1\+O\(\|\\Delta\|\)and the region away from the poles contributeo⁡\(\|Δ\|\)o\(\|\\Delta\|\)\. HenceJd​\(Δ\)≍\|Δ\|J\_\{d\}\(\\Delta\)\\asymp\|\\Delta\|, and inserting this into the spectral risk identity gives \([47](https://arxiv.org/html/2610.08816#A3.E47)\)\. ∎

The risk is locally linear rather than quadratic in the location error, so the pole location is not a regular parameter and the usualT−1T^\{\-1\}forecast cost does not apply to it\. The reason is that misplacing a pole leaves a singularity uncancelled at one frequency and introduces a spurious one at another, and the resulting mismatch is of first order in the displacement\. Estimating a location therefore behaves differently from estimating a strength, and the classical theory for unknown pole location reflects this\([Giraitis et al\., 2001](https://arxiv.org/html/2610.08816#bib.bib20);[Hidalgo, 2005](https://arxiv.org/html/2610.08816#bib.bib25);[Arteche, 2020](https://arxiv.org/html/2610.08816#bib.bib2)\)\.

### C\.11Fractional exponential\-state lower bound

We now transport the state\-complexity theory of App\.[B](https://arxiv.org/html/2610.08816#A2)into the singular geometry\. Let𝐊d\\mathbf\{K\}\_\{d\}be the Hankel operator built from the fractional kernelaj​\(d\)a\_\{j\}\(d\)\.

###### Lemma 21\(Fractional AR memory is a positive exponential mixture\)\.

For0<d<10<d<1,

aj​\(d\)=∫0∞wd​\(t\)​e−\(j−1\)​t​𝑑t,wd​\(t\)=sin⁡\(π​d\)π​e−\(1−d\)​t​\(1−e−t\)d\>0\.a\_\{j\}\(d\)=\\int\_\{0\}^\{\\infty\}w\_\{d\}\(t\)e^\{\-\(j\-1\)t\}\\,\\mathrm\{d\}t,\\qquad w\_\{d\}\(t\)=\\frac\{\\sin\(\\pi d\)\}\{\\pi\}e^\{\-\(1\-d\)t\}\(1\-e^\{\-t\}\)^\{d\}\>0\.\(48\)

###### Proof\.

Substitutex=e−tx=e^\{\-t\}, so thatdt=−dx/x\\,\\mathrm\{d\}t=\-\\,\\mathrm\{d\}x/xande−\(j−1\)​t=xj−1e^\{\-\(j\-1\)t\}=x^\{\\,j\-1\}\. The density becomeswd​\(t\)=sin⁡π​dπ​x1−d​\(1−x\)dw\_\{d\}\(t\)=\\tfrac\{\\sin\\pi d\}\{\\pi\}x^\{1\-d\}\(1\-x\)^\{d\}, so

∫0∞wd​\(t\)​e−\(j−1\)​t​𝑑t=sin⁡π​dπ​∫01xj−d−1​\(1−x\)d​𝑑x=sin⁡π​dπ⋅Γ⁡\(j−d\)​Γ​\(1\+d\)Γ⁡\(j\+1\),\\int\_\{0\}^\{\\infty\}w\_\{d\}\(t\)e^\{\-\(j\-1\)t\}\\,\\mathrm\{d\}t=\\frac\{\\sin\\pi d\}\{\\pi\}\\int\_\{0\}^\{1\}x^\{\\,j\-d\-1\}\(1\-x\)^\{d\}\\,\\mathrm\{d\}x=\\frac\{\\sin\\pi d\}\{\\pi\}\\cdot\\frac\{\\Gamma\(j\-d\)\\Gamma\(1\+d\)\}\{\\Gamma\(j\+1\)\},by the Beta integral, which converges becausej−d\>0j\-d\>0andd\>−1d\>\-1\. Euler’s reflection formulaΓ⁡\(1\+d\)​Γ​\(1−d\)=π​d/sin⁡π​d\\Gamma\(1\+d\)\\Gamma\(1\-d\)=\\pi d/\\sin\\pi dturns the constant intod/Γ⁡\(1−d\)d/\\Gamma\(1\-d\), givingd​Γ​\(j−d\)/\{Γ⁡\(1−d\)​Γ​\(j\+1\)\}=aj​\(d\)d\\Gamma\(j\-d\)/\\\{\\Gamma\(1\-d\)\\Gamma\(j\+1\)\\\}=a\_\{j\}\(d\)as in \([38](https://arxiv.org/html/2610.08816#A3.E38)\)\. ∎

Positivity ofwdw\_\{d\}is what makes the constructive side work, exactly as in Lem\.[10](https://arxiv.org/html/2610.08816#Thmtheorem10), and it also makes𝐊d\\mathbf\{K\}\_\{d\}a moment matrix of a positive measure and hence positive semidefinite\.

###### Lemma 22\(Loewner domination by the power\-law Hankel operator\)\.

For0<d<1/20<d<1/2the operator𝐊d\\mathbf\{K\}\_\{d\}is positive and Hilbert–Schmidt, and in quadratic\-form order

𝐊d⪰cd​Γ​\(1\+d\)​𝐇1\+d,cd=sin⁡π​dπ,\\mathbf\{K\}\_\{d\}\\ \\succeq\\ c\_\{d\}\\,\\Gamma\(1\+d\)\\,\\mathbf\{H\}\_\{1\+d\},\\qquad c\_\{d\}=\\frac\{\\sin\\pi d\}\{\\pi\},\(49\)where𝐇1\+d\\mathbf\{H\}\_\{1\+d\}is the power\-law Hankel operator of exponent1\+d1\+d\. Consequentlysn​\(𝐊d\)≥c​e−Cd​ns\_\{n\}\(\\mathbf\{K\}\_\{d\}\)\\geq c\\,e^\{\-C\_\{d\}\\sqrt\{n\}\}for largenn\.

###### Proof\.

In thexx\-moment representation of Lem\.[21](https://arxiv.org/html/2610.08816#Thmtheorem21)the fractional density carries the factorx−d​\(1−x\)d=\{\(1−x\)/x\}dx^\{\-d\}\(1\-x\)^\{d\}=\\\{\(1\-x\)/x\\\}^\{d\}, while the power\-law kernelj−\(1\+d\)j^\{\-\(1\+d\)\}has moment density proportional to\(−log⁡x\)d\(\-\\log x\)^\{d\}by the substitutionx=e−tx=e^\{\-t\}in Lem\.[10](https://arxiv.org/html/2610.08816#Thmtheorem10)\. For0<x<10<x<1the elementary inequality\(1−x\)/x≥−log⁡x\(1\-x\)/x\\geq\-\\log xholds, sinceu−1≥log⁡uu\-1\\geq\\log uatu=1/xu=1/x\. Raising it to the powerd\>0d\>0givesx−d​\(1−x\)d≥\(−log⁡x\)dx^\{\-d\}\(1\-x\)^\{d\}\\geq\(\-\\log x\)^\{d\}pointwise, so the fractional moment measure dominatescd​Γ​\(1\+d\)c\_\{d\}\\Gamma\(1\+d\)times the power\-law one\. Two moment matrices ordered by their measures are ordered in quadratic\-form order, which is \([49](https://arxiv.org/html/2610.08816#A3.E49)\)\. The min\-max principle then givessn​\(𝐊d\)≥cd​Γ​\(1\+d\)​sn​\(𝐇1\+d\)s\_\{n\}\(\\mathbf\{K\}\_\{d\}\)\\geq c\_\{d\}\\Gamma\(1\+d\)s\_\{n\}\(\\mathbf\{H\}\_\{1\+d\}\), and the Widom scale \([33](https://arxiv.org/html/2610.08816#A2.E33)\) applied at exponent1\+d\>11\+d\>1gives the stated bound\. ∎

###### Lemma 23\(Finite fractional Hankel sections retain the relevant singular value\)\.

For0<d<1/20<d<1/2there isCdC\_\{d\}with‖𝐊d−PN​𝐊d​PN‖op≤Cd​N−d\\\|\\mathbf\{K\}\_\{d\}\-P\_\{N\}\\mathbf\{K\}\_\{d\}P\_\{N\}\\\|\_\{\\rm op\}\\leq C\_\{d\}N^\{\-d\}\. Consequently there areκd,cd′,Cd′\>0\\kappa\_\{d\},c\_\{d\}^\{\\prime\},C\_\{d\}^\{\\prime\}\>0such that, withNr=⌈eκd​r⌉N\_\{r\}=\\lceil e^\{\\kappa\_\{d\}\\sqrt\{r\}\}\\rceil,

sr\+1​\(PNr​𝐊d​PNr\)≥cd′​e−Cd′​rfor all large​r\.s\_\{r\+1\}\\bigl\(P\_\{N\_\{r\}\}\\mathbf\{K\}\_\{d\}P\_\{N\_\{r\}\}\\bigr\)\\ \\geq\\ c\_\{d\}^\{\\prime\}e^\{\-C\_\{d\}^\{\\prime\}\\sqrt\{r\}\}\\qquad\\text\{for all large \}r\.\(50\)

###### Proof\.

Sinceaj​\(d\)≍j−1−da\_\{j\}\(d\)\\asymp j^\{\-1\-d\}by \([38](https://arxiv.org/html/2610.08816#A3.E38)\), the discarded entries obey

‖𝐊d−PN​𝐊d​PN‖op2≤‖𝐊d−PN​𝐊d​PN‖HS2≲∑max⁡\(i,j\)≥N\(i\+j\+1\)−2−2​d≲N−2​d,\\\|\\mathbf\{K\}\_\{d\}\-P\_\{N\}\\mathbf\{K\}\_\{d\}P\_\{N\}\\\|\_\{\\rm op\}^\{2\}\\leq\\\|\\mathbf\{K\}\_\{d\}\-P\_\{N\}\\mathbf\{K\}\_\{d\}P\_\{N\}\\\|\_\{\\rm HS\}^\{2\}\\lesssim\\sum\_\{\\max\(i,j\)\\geq N\}\(i\+j\+1\)^\{\-2\-2d\}\\lesssim N^\{\-2d\},the sum converging because2\+2​d\>22\+2d\>2, which gives the operator bound\. For the singular value, Lem\.[22](https://arxiv.org/html/2610.08816#Thmtheorem22)givessr\+1​\(𝐊d\)≥c​e−Cd​rs\_\{r\+1\}\(\\mathbf\{K\}\_\{d\}\)\\geq c\\,e^\{\-C\_\{d\}\\sqrt\{r\}\}\. Choosingκd\>2​Cd/d\\kappa\_\{d\}\>2C\_\{d\}/dmakesCd​Nr−d≤Cd​e−d​κd​r≤Cd​e−2​Cd​rC\_\{d\}N\_\{r\}^\{\-d\}\\leq C\_\{d\}e^\{\-d\\kappa\_\{d\}\\sqrt\{r\}\}\\leq C\_\{d\}e^\{\-2C\_\{d\}\\sqrt\{r\}\}, which is below half ofc​e−Cd​rc\\,e^\{\-C\_\{d\}\\sqrt\{r\}\}for largerr, and Weyl’s inequalitysr\+1​\(PN​𝐊d​PN\)≥sr\+1​\(𝐊d\)−‖𝐊d−PN​𝐊d​PN‖ops\_\{r\+1\}\(P\_\{N\}\\mathbf\{K\}\_\{d\}P\_\{N\}\)\\geq s\_\{r\+1\}\(\\mathbf\{K\}\_\{d\}\)\-\\\|\\mathbf\{K\}\_\{d\}\-P\_\{N\}\\mathbf\{K\}\_\{d\}P\_\{N\}\\\|\_\{\\rm op\}gives \([50](https://arxiv.org/html/2610.08816#A3.E50)\)\. ∎

The two steps are separated because they do different work\. The Loewner comparison transports the Widom scale of App\.[B](https://arxiv.org/html/2610.08816#A2)without redoing the spectral analysis, and the finite\-section step is what makes the bound usable inside a Hankel block of bounded order\.

###### Theorem 24\(Root\-exponential lower bound in fractional risk\)\.

For fractional noise there arec−,C−\>0c\_\{\-\},C\_\{\-\}\>0such thatinfb∈𝒜rℰd​\(b\)≥c−​e−C−​r\\inf\_\{b\\in\\mathcal\{A\}\_\{r\}\}\\mathcal\{E\}\_\{d\}\(b\)\\geq c\_\{\-\}e^\{\-C\_\{\-\}\\sqrt\{r\}\}for largerr\.

###### Proof\.

Repeat the proof of Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)with𝐇α\\mathbf\{H\}\_\{\\alpha\}replaced by𝐊d\\mathbf\{K\}\_\{d\}, Lem\.[15](https://arxiv.org/html/2610.08816#Thmtheorem15)replaced by Lem\.[23](https://arxiv.org/html/2610.08816#Thmtheorem23), and the Widom scale replaced by Lem\.[22](https://arxiv.org/html/2610.08816#Thmtheorem22)\. The only step that needs the geometry is the final conversion from coefficient error to risk\. For pure fractional noise the regular factor is the constantg≡σ2g\\equiv\\sigma^\{2\}, so the lower half of \([6](https://arxiv.org/html/2610.08816#S4.E6)\) readsℰd​\(u\)≥σ2​‖Id​u‖22\\mathcal\{E\}\_\{d\}\(u\)\\geq\\sigma^\{2\}\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}with no further factor, and since\|1−e−i​ω\|≤2\|1\-e^\{\-i\\omega\}\|\\leq 2on the circle the fractional multiplier obeys\|1−e−i​ω\|−2​d≥2−2​d\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\\geq 2^\{\-2d\}, so Parseval yields‖Id​u‖22≥2−2​d​‖u‖22\\\|I\_\{d\}u\\\|\_\{2\}^\{2\}\\geq 2^\{\-2d\}\\\|u\\\|\_\{2\}^\{2\}and hence

ℰd​\(u\)≥σ2​2−2​d​‖u‖22\.\\mathcal\{E\}\_\{d\}\(u\)\\ \\geq\\ \\sigma^\{2\}\\,2^\{\-2d\}\\,\\\|u\\\|\_\{2\}^\{2\}\.\(51\)That inequality is what converts the coefficient\-level Hankel obstruction into forecast risk\. ∎

### C\.12Fractional exponential\-state upper bound

On the constructive side, Lem\.[21](https://arxiv.org/html/2610.08816#Thmtheorem21)putsa⁡\(d\)a\(d\)in the analytic Laplace\-mixture class of App\.[B](https://arxiv.org/html/2610.08816#A2), so the dyadic Bernstein\-ellipse quadrature applies verbatim\.

###### Corollary 25\(Positive exponential compression of the fractional kernel\)\.

Fix0<d<1/20<d<1/2\. There isCd<∞C\_\{d\}<\\inftysuch that for everyL≥2L\\geq 2and0<η≤e−10<\\eta\\leq e^\{\-1\}there are positivebq,λqb\_\{q\},\\lambda\_\{q\},1≤q≤R1\\leq q\\leq R, with\|aj​\(d\)−∑q≤Rbq​e−λq​\(j−1\)\|≤η​aj​\(d\)\|a\_\{j\}\(d\)\-\\sum\_\{q\\leq R\}b\_\{q\}e^\{\-\\lambda\_\{q\}\(j\-1\)\}\|\\leq\\eta\\,a\_\{j\}\(d\)for1≤j≤L1\\leq j\\leq Land

R≤Cd​\(1\+log⁡L\+log⁡η−1\)2\.R\\leq C\_\{d\}\\bigl\(1\+\\log L\+\\log\\eta^\{\-1\}\\bigr\)^\{2\}\.\(52\)

###### Proof\.

Lem\.[21](https://arxiv.org/html/2610.08816#Thmtheorem21)exhibitsaj​\(d\)a\_\{j\}\(d\)as a positive Laplace mixture, and rewriting it in the form \([34](https://arxiv.org/html/2610.08816#A2.E34)\) with exponentα=1\+d\\alpha=1\+dand density

hd​\(t\)=cd​e−\(1−d\)​t​\(1−e−tt\)d,cd=sin⁡π​dπ,h\_\{d\}\(t\)=c\_\{d\}\\,e^\{\-\(1\-d\)t\}\\Bigl\(\\frac\{1\-e^\{\-t\}\}\{t\}\\Bigr\)^\{d\},\\qquad c\_\{d\}=\\frac\{\\sin\\pi d\}\{\\pi\},shows thathdh\_\{d\}is positive on\[0,∞\)\[0,\\infty\)withhd​\(0\)=cd\>0h\_\{d\}\(0\)=c\_\{d\}\>0\. The remaining two hypotheses of \([34](https://arxiv.org/html/2610.08816#A2.E34)\) are verified rather than assumed\. For analyticity, the only zeros of1−e−z1\-e^\{\-z\}are the points2​π​i​k2\\pi ik, so the quotient\(1−e−z\)/z\(1\-e^\{\-z\}\)/zhas a removable singularity at the origin and is nonzero on a small disk about the origin and on any fixed sector\|arg⁡z\|<ϑ<π/2\|\\arg z\|<\\vartheta<\\pi/2\. It therefore admits a single analytic branch of itsddth power on the neighbourhood required in App\.[B](https://arxiv.org/html/2610.08816#A2), agreeing with the positive one on the real axis; multiplying bye−\(1−d\)​ze^\{\-\(1\-d\)z\}preserves analyticity\. For the decay \([35](https://arxiv.org/html/2610.08816#A2.E35)\), the same quotient is bounded on the sector while\|e−\(1−d\)​z\|=e−\(1−d\)​ℜ⁡z\|e^\{\-\(1\-d\)z\}\|=e^\{\-\(1\-d\)\\Re z\}, so\|hd​\(z\)\|≤Cd​e−μ​ℜ⁡z\|h\_\{d\}\(z\)\|\\leq C\_\{d\}e^\{\-\\mu\\Re z\}withμ=1−d\>1/2\\mu=1\-d\>1/2\. The composite quadrature therefore has both the sectorial and near\-origin holomorphic neighbourhoods it requires\. The analytic Laplace\-mixture compression of App\.[B](https://arxiv.org/html/2610.08816#A2)therefore applies withα=1\+d\\alpha=1\+d, and its head count is \([52](https://arxiv.org/html/2610.08816#A3.E52)\)\. ∎

The constantCdC\_\{d\}here is for fixedddand is not claimed to be uniform asd→0d\\to 0, which is why the short\-memory boundary of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)is proved by a different route\. The relative form of the guarantee is what allows it to be converted in the singular norm\. An absolute bound would be useless here, because the fractional\-integration factor amplifies low\-frequency error and a uniform absolute error need not have small predictive norm\.

### C\.13Proof of Corollary[4](https://arxiv.org/html/2610.08816#Thmtheorem4)

###### Proof of Corollary[4](https://arxiv.org/html/2610.08816#Thmtheorem4)\.

For the upper bound, apply Corollary[25](https://arxiv.org/html/2610.08816#Thmtheorem25)and split the coefficient error into the in\-context parteine^\{\\rm in\}, supported on\{1,…,L\}\\\{1,\\dots,L\\\}, and the omitted tailetaile^\{\\rm tail\}\. The compression guarantee is relative and pointwise,\|ejin\|≤η​aj​\(d\)\|e^\{\\rm in\}\_\{j\}\|\\leq\\eta\\,a\_\{j\}\(d\), so it does not by itself bound a fractionally integratedℓ2\\ell\_\{2\}norm and one further step is needed\. Summing the relative bound gives‖ein‖1≤η​∑j≥1aj​\(d\)=η\\\|e^\{\\rm in\}\\\|\_\{1\}\\leq\\eta\\sum\_\{j\\geq 1\}a\_\{j\}\(d\)=\\eta, the identity because the transfer function satisfiesAd​\(1\)=1−\(1−1\)d=1A\_\{d\}\(1\)=1\-\(1\-1\)^\{d\}=1\. The fractional\-integration kernel of App\.[C](https://arxiv.org/html/2610.08816#A3)hasπk​\(d\)≍kd−1\\pi\_\{k\}\(d\)\\asymp k^\{d\-1\}, hence lies inℓ2\\ell\_\{2\}precisely becaused<1/2d<1/2, so Young’s convolution inequalityℓ1∗ℓ2→ℓ2\\ell\_\{1\}\*\\ell\_\{2\}\\to\\ell\_\{2\}gives‖Id​ein‖2≤‖ein‖1​‖π⁡\(d\)‖2≤Cd​η\\\|I\_\{d\}e^\{\\rm in\}\\\|\_\{2\}\\leq\\\|e^\{\\rm in\}\\\|\_\{1\}\\,\\\|\\pi\(d\)\\\|\_\{2\}\\leq C\_\{d\}\\eta\. This is the step that converts a relative coefficient guarantee into the singular predictive norm; an absolute coefficient bound would not suffice, since fractional integration amplifies low\-frequency error\. The omitted tail is controlled by \([39](https://arxiv.org/html/2610.08816#A3.E39)\), which gives‖Id​etail‖22=Θd​\(L−1\)\\\|I\_\{d\}e^\{\\rm tail\}\\\|\_\{2\}^\{2\}=\\Theta\_\{d\}\(L^\{\-1\}\), and the upper half of \([6](https://arxiv.org/html/2610.08816#S4.E6)\) together with\(u\+v\)2≤2​u2\+2​v2\(u\+v\)^\{2\}\\leq 2u^\{2\}\+2v^\{2\}combines the two intoℰd\(b\)≲dη2\+L−1\\mathcal\{E\}\_\{d\}\(b\)\\lesssim\_\{d\}\\eta^\{2\}\+L^\{\-1\}\. Balancingη2\\eta^\{2\}againstL−1L^\{\-1\}and inverting \([52](https://arxiv.org/html/2610.08816#A3.E52)\) as in Corollary[14](https://arxiv.org/html/2610.08816#Thmtheorem14)givesCd​e−cd​rC\_\{d\}e^\{\-c\_\{d\}\\sqrt\{r\}\}\. The lower bound is Thm\.[24](https://arxiv.org/html/2610.08816#Thmtheorem24)\. The inversion of a two\-sided stretched\-exponential bound with exponent1/21/2givesrd⋆​\(τ\)=Θd​\(log2⁡\(1/τ\)\)r\_\{d\}^\{\\star\}\(\\tau\)=\\Theta\_\{d\}\(\\log^\{2\}\(1/\\tau\)\)\.

The subscript onΘd\\Theta\_\{d\}is not cosmetic, and it is worth saying how it coexists with Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)\. Corollary[4](https://arxiv.org/html/2610.08816#Thmtheorem4)is a fixed\-ddstatement whose constants come from Lemmas[22](https://arxiv.org/html/2610.08816#Thmtheorem22)and[23](https://arxiv.org/html/2610.08816#Thmtheorem23)and from Corollary[25](https://arxiv.org/html/2610.08816#Thmtheorem25), and each of those degrades asd→0d\\to 0\. Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)is a different statement, uniform over0<d≤d00<d\\leq d\_\{0\}, and it is uniform precisely because it never invokes those inputs, compressing the fixed harmonic kernel instead\. The two are therefore complementary rather than in tension, and neither implies the other\. Reading the fixed\-ddconstants as uniform inddis the error that Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)is designed to avoid, and the open problem recorded at the end of this appendix is exactly that no matching lower bound is available withd0d\_\{0\}\-only constants\. ∎

The square\-log law therefore survives the change of geometry, with constants depending ondd\. What does not survive is the context law, which has no counterpart in the regular case, and this asymmetry is the point made in the comparison table of Sec\.[2](https://arxiv.org/html/2610.08816#S2)\.

### C\.14Zero\-memory boundary and Gamma expansion

###### Proposition 26\(Zero\-memory risk\)\.

For fractional noise with0<d<1/20<d<1/2, the zero predictor has

ℰd​\(0\)=σ2​\{Γ⁡\(1−2​d\)Γ​\(1−d\)2−1\}=π2​σ26​d2\+O⁡\(d3\)\.\\mathcal\{E\}\_\{d\}\(0\)=\\sigma^\{2\}\\Bigl\\\{\\frac\{\\Gamma\(1\-2d\)\}\{\\Gamma\(1\-d\)^\{2\}\}\-1\\Bigr\\\}=\\frac\{\\pi^\{2\}\\sigma^\{2\}\}\{6\}d^\{2\}\+O\(d^\{3\}\)\.\(53\)

###### Proof\.

The zero predictor has excess riskγ⁡\(0\)−σ2\\gamma\(0\)\-\\sigma^\{2\}, andγ⁡\(0\)=σ2​Γ​\(1−2​d\)/Γ​\(1−d\)2\\gamma\(0\)=\\sigma^\{2\}\\Gamma\(1\-2d\)/\\Gamma\(1\-d\)^\{2\}for fractional noise, which gives the exact expression\. For the expansion uselog⁡Γ⁡\(1\+x\)=−γE​x\+∑k≥2\(−1\)k​ζ​\(k\)​xk/k\\log\\Gamma\(1\+x\)=\-\\gamma\_\{\\rm E\}x\+\\sum\_\{k\\geq 2\}\(\-1\)^\{k\}\\zeta\(k\)x^\{k\}/k, so that

log⁡Γ⁡\(1−2​d\)−2​log⁡Γ⁡\(1−d\)=∑k≥2ζ⁡\(k\)k​\(2k−2\)​dk=ζ⁡\(2\)​d2\+O⁡\(d3\),\\log\\Gamma\(1\-2d\)\-2\\log\\Gamma\(1\-d\)=\\sum\_\{k\\geq 2\}\\frac\{\\zeta\(k\)\}\{k\}\\bigl\(2^\{k\}\-2\\bigr\)d^\{k\}=\\zeta\(2\)d^\{2\}\+O\(d^\{3\}\),the Euler constants cancelling\. Exponentiating and subtracting one givesζ⁡\(2\)​d2\+O⁡\(d3\)=π2​d2/6\+O⁡\(d3\)\\zeta\(2\)d^\{2\}\+O\(d^\{3\}\)=\\pi^\{2\}d^\{2\}/6\+O\(d^\{3\}\)\. ∎

Consequently, once the target accuracy satisfiesτ≳σ2​d2\\tau\\gtrsim\\sigma^\{2\}d^\{2\}, the zero\-output architecture already attains it and no memory at all is required\. This is the upper edge of the window in Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5), and it is why the constants in Corollary[4](https://arxiv.org/html/2610.08816#Thmtheorem4)cannot be held fixed asd→0d\\to 0\.

### C\.15Harmonic linearization of the fractional coefficients

Expanding \([38](https://arxiv.org/html/2610.08816#A3.E38)\) inddgivesaj​\(d\)=d/j\+O⁡\(d2\)a\_\{j\}\(d\)=d/j\+O\(d^\{2\}\)coefficientwise, and the useful statement is that the remainder is small in the predictive norm uniformly indd\.

###### Lemma 27\(Uniform first\-order linearization\)\.

Fix0<d0<1/20<d\_\{0\}<1/2and lethj=1/jh\_\{j\}=1/j\. There isCd0<∞C\_\{d\_\{0\}\}<\\inftysuch that for every0<d≤d00<d\\leq d\_\{0\},

ℰd​\(d​h\)=σ2​‖Id​\{a⁡\(d\)−d​h\}‖22≤Cd0​σ2​d4\.\\mathcal\{E\}\_\{d\}\(dh\)=\\sigma^\{2\}\\\|I\_\{d\}\\\{a\(d\)\-dh\\\}\\\|\_\{2\}^\{2\}\\leq C\_\{d\_\{0\}\}\\sigma^\{2\}d^\{4\}\.\(54\)

###### Proof\.

WriteH⁡\(z\)=−log⁡\(1−z\)H\(z\)=\-\\log\(1\-z\), so that the fractional autoregressive transfer function isAd​\(z\)=1−\(1−z\)d=1−e−d​H​\(z\)A\_\{d\}\(z\)=1\-\(1\-z\)^\{d\}=1\-e^\{\-dH\(z\)\}and the harmonic surrogate has transfer functiond​H​\(z\)dH\(z\)\. The fractional\-integration multiplier is\(1−z\)−d=ed​H​\(z\)\(1\-z\)^\{\-d\}=e^\{dH\(z\)\}, so on the circle the integrand of‖Id​\{a⁡\(d\)−d​h\}‖22\\\|I\_\{d\}\\\{a\(d\)\-dh\\\}\\\|\_\{2\}^\{2\}is the squared modulus of

ed​H​\(Ad−d​H\)=ed​H−1−d​H​ed​H,e^\{dH\}\\bigl\(A\_\{d\}\-dH\\bigr\)=e^\{dH\}\-1\-dH\\,e^\{dH\},evaluated atz=e−i​ωz=e^\{\-i\\omega\}\. The elementary bound\|ey−1−y​ey\|≤C​\|y\|2​emax⁡\(ℜ⁡y,0\)\|e^\{y\}\-1\-ye^\{y\}\|\\leq C\|y\|^\{2\}e^\{\\max\(\\Re y,0\)\}, valid for all complexyy, applied withy=d​H​\(e−i​ω\)y=dH\(e^\{\-i\\omega\}\)gives a pointwise boundC​d2​\|H\|2​ed​max⁡\(ℜ⁡H,0\)Cd^\{2\}\|H\|^\{2\}e^\{d\\max\(\\Re H,0\)\}\. On the circleℜ⁡H⁡\(e−i​ω\)=−log⁡\|1−e−i​ω\|\\Re H\(e^\{\-i\\omega\}\)=\-\\log\|1\-e^\{\-i\\omega\}\|and\|H⁡\(e−i​ω\)\|≲1\+log⁡\(1/\|ω\|\)\|H\(e^\{\-i\\omega\}\)\|\\lesssim 1\+\\log\(1/\|\\omega\|\), so ford≤d0<1/2d\\leq d\_\{0\}<1/2the exponential factor is at most\|1−e−i​ω\|−d0\|1\-e^\{\-i\\omega\}\|^\{\-d\_\{0\}\}and the whole integrand is dominated byC​d4​\{1\+log⁡\(1/\|ω\|\)\}4​\|ω\|−2​d0Cd^\{4\}\\\{1\+\\log\(1/\|\\omega\|\)\\\}^\{4\}\|\\omega\|^\{\-2d\_\{0\}\}, which is integrable since2​d0<12d\_\{0\}<1\. Integrating gives‖Id​\{a⁡\(d\)−d​h\}‖22≤Cd0​d4\\\|I\_\{d\}\\\{a\(d\)\-dh\\\}\\\|\_\{2\}^\{2\}\\leq C\_\{d\_\{0\}\}d^\{4\}with a constant depending only ond0d\_\{0\}, and multiplying byσ2\\sigma^\{2\}gives \([54](https://arxiv.org/html/2610.08816#A3.E54)\)\. ∎

The exponent four is what sets the lower edge of the intermediate window\. Below accuracyσ2​d4\\sigma^\{2\}d^\{4\}the harmonic surrogate is no longer adequate and the construction of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)does not apply, which is the origin of the open problem recorded below\.

### C\.16Harmonic tail in the fractional predictive norm

###### Lemma 28\(Uniform fractional norm of the harmonic tail\)\.

Fix0<d0<1/20<d\_\{0\}<1/2and lethj\>L=j−1𝟏\{j\>L\}h^\{\>L\}\_\{j\}=j^\{\-1\}\\mathbf\{1\}\\\{j\>L\\\}\. There isCd0<∞C\_\{d\_\{0\}\}<\\inftysuch that for every0≤d≤d00\\leq d\\leq d\_\{0\}andL≥2L\\geq 2,

‖Id​h\>L‖22≤Cd0​L−1\+2​d\.\\\|I\_\{d\}h^\{\>L\}\\\|\_\{2\}^\{2\}\\leq C\_\{d\_\{0\}\}L^\{\-1\+2d\}\.\(55\)

###### Proof\.

The transfer function of the truncated harmonic tail isRL​\(ω\)=∑j\>Le−i​j​ω/jR\_\{L\}\(\\omega\)=\\sum\_\{j\>L\}e^\{\-ij\\omega\}/j, so‖Id​h\>L‖22=\(2​π\)−1​∫−ππ\|RL​\(ω\)\|2​\|1−e−i​ω\|−2​d​𝑑ω\\\|I\_\{d\}h^\{\>L\}\\\|\_\{2\}^\{2\}=\(2\\pi\)^\{\-1\}\\int\_\{\-\\pi\}^\{\\pi\}\|R\_\{L\}\(\\omega\)\|^\{2\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\\,\\mathrm\{d\}\\omega\. Two elementary bounds onRLR\_\{L\}suffice\. For\|ω\|≤L−1\|\\omega\|\\leq L^\{\-1\}, comparing the sum with∫L1/\|ω\|𝑑u/u\\int\_\{L\}^\{1/\|\\omega\|\}\\,\\mathrm\{d\}u/ugives\|RL​\(ω\)\|≲1\+log⁡\{1/\(L​\|ω\|\)\}\|R\_\{L\}\(\\omega\)\|\\lesssim 1\+\\log\\\{1/\(L\|\\omega\|\)\\\}\. For\|ω\|\>L−1\|\\omega\|\>L^\{\-1\}, Abel summation against the partial sums ofe−i​j​ωe^\{\-ij\\omega\}, which are bounded by1/\|sin⁡\(ω/2\)\|≲1/\|ω\|1/\|\\sin\(\\omega/2\)\|\\lesssim 1/\|\\omega\|, gives\|RL​\(ω\)\|≲1/\(L​\|ω\|\)\|R\_\{L\}\(\\omega\)\|\\lesssim 1/\(L\|\\omega\|\)\. Since\|1−e−i​ω\|−2​d≍\|ω\|−2​d\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\\asymp\|\\omega\|^\{\-2d\}and0≤d≤d0<1/20\\leq d\\leq d\_\{0\}<1/2, splitting the integral at\|ω\|=L−1\|\\omega\|=L^\{\-1\}gives

∫\|ω\|≤L−1\{1\+log⁡1L​\|ω\|\}2​\|ω\|−2​d​𝑑ω≲L−1\+2​d,∫\|ω\|\>L−1\|ω\|−2​dL2​ω2​𝑑ω≲L−1\+2​d,\\int\_\{\|\\omega\|\\leq L^\{\-1\}\}\\\!\\\!\\bigl\\\{1\+\\log\\tfrac\{1\}\{L\|\\omega\|\}\\bigr\\\}^\{2\}\|\\omega\|^\{\-2d\}\\,\\mathrm\{d\}\\omega\\ \\lesssim\\ L^\{\-1\+2d\},\\qquad\\int\_\{\|\\omega\|\>L^\{\-1\}\}\\\!\\\!\\frac\{\|\\omega\|^\{\-2d\}\}\{L^\{2\}\\omega^\{2\}\}\\,\\mathrm\{d\}\\omega\\ \\lesssim\\ L^\{\-1\+2d\},the second because2\+2​d\>12\+2d\>1makes the integral converge at infinity of the rescaled variable and the endpoint contributesL−1\+2​dL^\{\-1\+2d\}\. Both constants depend only ond0d\_\{0\}, which gives \([55](https://arxiv.org/html/2610.08816#A3.E55)\)\. ∎

Uniformity indddown to zero is the point\. A bound with a constant blowing up asd→0d\\to 0would be useless in the boundary layer, since that is exactly the regime the theorem is about\.

### C\.17Proof of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)

###### Proof of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)\.

If the displayedτ\\tau\-window is empty there is nothing to prove; for every fixedd0<1/2d\_\{0\}<1/2it is nonempty for all sufficiently smallddafter the constants are fixed as below\. The whole point of the statement is uniformity asd→0d\\to 0, so the compression step must not be applied to the fractional kernel itself\. Corollary[25](https://arxiv.org/html/2610.08816#Thmtheorem25)has a constantCdC\_\{d\}that is not controlled asd→0d\\to 0, and using it here would leave a gap\. Instead we compress the*harmonic*kernelhj=1/jh\_\{j\}=1/j, which is the caseα=1\\alpha=1of Thm\.[12](https://arxiv.org/html/2610.08816#Thmtheorem12)and therefore carries a constant that does not depend onddat all\.

Fix0<d≤d00<d\\leq d\_\{0\}andτ\\tauin the stated windowC0​σ2​d4≤τ≤c0​σ2​d2C\_\{0\}\\sigma^\{2\}d^\{4\}\\leq\\tau\\leq c\_\{0\}\\sigma^\{2\}d^\{2\}, and writeτ¯=τ/σ2\\bar\{\\tau\}=\\tau/\\sigma^\{2\}\. By Thm\.[12](https://arxiv.org/html/2610.08816#Thmtheorem12)withα=1\\alpha=1there is, for everyL≥2L\\geq 2and0<η≤e−10<\\eta\\leq e^\{\-1\}, a positiveRR\-mode exponential sequencesRs\_\{R\}with\|j−1−sR​\(j\)\|≤η​j−1\|j^\{\-1\}\-s\_\{R\}\(j\)\|\\leq\\eta j^\{\-1\}for1≤j≤L1\\leq j\\leq Land

R≤C⁡\(1\+log⁡η−1\)​\(1\+log⁡L\+log⁡η−1\+log⁡log⁡η−1\),R\\leq C\\bigl\(1\+\\log\\eta^\{\-1\}\\bigr\)\\bigl\(1\+\\log L\+\\log\\eta^\{\-1\}\+\\log\\log\\eta^\{\-1\}\\bigr\),the constantCCbeing absolute because the power is fixed\. Putbj=dsR\(j\)𝟏\{j≤L\}b\_\{j\}=d\\,s\_\{R\}\(j\)\\mathbf\{1\}\\\{j\\leq L\\\}, which lies in𝒜R\\mathcal\{A\}\_\{R\}, and decompose

a\(d\)−b=\{a\(d\)−dh\}\+d\{h−sR\}𝟏\{j≤L\}\+dh\>L\.a\(d\)\-b=\\\{a\(d\)\-dh\\\}\+d\\\{h\-s\_\{R\}\\\}\\mathbf\{1\}\\\{j\\leq L\\\}\+d\\,h^\{\>L\}\.\(56\)
The first term is Lem\.[27](https://arxiv.org/html/2610.08816#Thmtheorem27)and contributes at mostCd0​σ2​d4C\_\{d\_\{0\}\}\\sigma^\{2\}d^\{4\}\. For the second, the relative bound gives∥d\(h−sR\)𝟏\{j≤L\}∥1≤Cdηlog\(eL\)\\\|d\(h\-s\_\{R\}\)\\mathbf\{1\}\\\{j\\leq L\\\}\\\|\_\{1\}\\leq Cd\\eta\\log\(eL\), and since the fractional\-integration kernel satisfies

sup0≤d≤d012​π​∫−ππ\|1−e−i​ω\|−2​d​𝑑ω<∞,\\sup\_\{0\\leq d\\leq d\_\{0\}\}\\ \\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\|1\-e^\{\-i\\omega\}\|^\{\-2d\}\\,\\mathrm\{d\}\\omega<\\infty,Young’s inequality for the convolutionℓ1∗ℓ2→ℓ2\\ell\_\{1\}\*\\ell\_\{2\}\\to\\ell\_\{2\}gives∥Id\[d\(h−sR\)𝟏\{j≤L\}\]∥22≤Cd0d2η2log2\(eL\)\\\|I\_\{d\}\[d\(h\-s\_\{R\}\)\\mathbf\{1\}\\\{j\\leq L\\\}\]\\\|\_\{2\}^\{2\}\\leq C\_\{d\_\{0\}\}d^\{2\}\\eta^\{2\}\\log^\{2\}\(eL\)\. The third term is Lem\.[28](https://arxiv.org/html/2610.08816#Thmtheorem28)\. Combining the three through\(x\+y\+z\)2≤3​\(x2\+y2\+z2\)\(x\+y\+z\)^\{2\}\\leq 3\(x^\{2\}\+y^\{2\}\+z^\{2\}\)in the exact geometry of Thm\.[2](https://arxiv.org/html/2610.08816#Thmtheorem2)gives

ℰd​\(b\)≤Cd0​σ2​\{d4\+d2​η2​log2⁡\(e​L\)\+d2​L−1\+2​d\}\.\\mathcal\{E\}\_\{d\}\(b\)\\leq C\_\{d\_\{0\}\}\\sigma^\{2\}\\bigl\\\{d^\{4\}\+d^\{2\}\\eta^\{2\}\\log^\{2\}\(eL\)\+d^\{2\}L^\{\-1\+2d\}\\bigr\\\}\.\(57\)
EnlargingC0C\_\{0\}makes the first term at mostτ/3\\tau/3\. Choosing

L=⌈\(C1​d2/τ¯\)1/\(1−2​d\)⌉,η=c1​τ¯d​log⁡\(e​L\),L=\\Bigl\\lceil\\bigl\(C\_\{1\}d^\{2\}/\\bar\{\\tau\}\\bigr\)^\{1/\(1\-2d\)\}\\Bigr\\rceil,\\qquad\\eta=c\_\{1\}\\frac\{\\sqrt\{\\bar\{\\tau\}\}\}\{d\\log\(eL\)\},makes the remaining two terms at mostτ/3\\tau/3each, and the exponent1/\(1−2​d\)1/\(1\-2d\)is bounded by1/\(1−2​d0\)1/\(1\-2d\_\{0\}\)so thatlog⁡L≤Cd0​log⁡\(σ2​d2/τ\)\\log L\\leq C\_\{d\_\{0\}\}\\log\(\\sigma^\{2\}d^\{2\}/\\tau\)and likewiselog⁡η−1≤Cd0​log⁡\(σ2​d2/τ\)\\log\\eta^\{\-1\}\\leq C\_\{d\_\{0\}\}\\log\(\\sigma^\{2\}d^\{2\}/\\tau\)\. Substituting into the displayed bound onRRgives \([10](https://arxiv.org/html/2610.08816#S4.E10)\)\. By choosingc0,c1c\_\{0\},c\_\{1\}sufficiently small andC0,C1C\_\{0\},C\_\{1\}sufficiently large, the displayed choices satisfyL≥2L\\geq 2and0<η≤e−10<\\eta\\leq e^\{\-1\}throughout the stated window, so Thm\.[12](https://arxiv.org/html/2610.08816#Thmtheorem12)is applicable as invoked\. Every constant that appears depends only ond0d\_\{0\}, since the compression constant is absolute and the two fractional inputs were bounded uniformly over0≤d≤d00\\leq d\\leq d\_\{0\}\. ∎

The mechanism is that the relevant scale is notτ\\taubut the ratioσ2​d2/τ\\sigma^\{2\}d^\{2\}/\\tau, because the memory to be represented has total predictive mass of orderσ2​d2\\sigma^\{2\}d^\{2\}rather than order one\. This is why the bound degenerates to a constant asτ\\tauapproachesσ2​d2\\sigma^\{2\}d^\{2\}from below, matching Prop\.[26](https://arxiv.org/html/2610.08816#Thmtheorem26)\.

### C\.18Remaining double\-scaling problem

Two questions in the boundary layer are open\. Neither concerns the pole location, which is a separate irregularity treated in Thm\.[20](https://arxiv.org/html/2610.08816#Thmtheorem20)and is not a small\-ddphenomenon\.

There is no matching uniform lower law in the window\. The upper bound of Thm\.[5](https://arxiv.org/html/2610.08816#Thmtheorem5)is constructive, and the lower bound of Thm\.[24](https://arxiv.org/html/2610.08816#Thmtheorem24)holds for fixedddwith constants that degrade asd→0d\\to 0, so the two do not currently meet uniformly\. Establishing a lower bound with the samelog2⁡\(σ2​d2/τ\)\\log^\{2\}\(\\sigma^\{2\}d^\{2\}/\\tau\)form would require a Hankel argument uniform in the double limit, and the finite\-section estimate in Lem\.[23](https://arxiv.org/html/2610.08816#Thmtheorem23)is not uniform inddnear zero, since its constants degrade asd→0d\\to 0\.

The deeper regimeτ≪σ2​d4\\tau\\ll\\sigma^\{2\}d^\{4\}is not covered at all\. There the harmonic linearization of Lem\.[27](https://arxiv.org/html/2610.08816#Thmtheorem27)is no longer accurate enough, so the construction must use the exact fractional kernel, and the resulting head count is not known\. We state both gaps explicitly rather than absorbing them into the constants, because the interesting behaviour of the short\-memory boundary lives precisely where they bite\.

## Appendix DNonlinear Contextual Recurrent Memory

### D\.1Model class and stationary solution

LetZtZ\_\{t\}be independent standard Gaussians and consider the contextual recurrence

St=F⁡\(St−1,Zt\),m^t=G⁡\(St,Zt\),S\_\{t\}=F\(S\_\{t\-1\},Z\_\{t\}\),\\qquad\\widehat\{m\}\_\{t\}=G\(S\_\{t\},Z\_\{t\}\),\(58\)withSt∈ℝmS\_\{t\}\\in\\mathbb\{R\}^\{m\}andF,GF,Gdeterministic and measurable\. No linearity in the state is assumed, which is what distinguishes this class from the affine gating benchmark of App\.[B](https://arxiv.org/html/2610.08816#A2)\.

Assume there is a closed forward\-invariant state domainℬ⊂ℝm\\mathcal\{B\}\\subset\\mathbb\{R\}^\{m\}such thatF⁡\(ℬ,z\)⊆ℬF\(\\mathcal\{B\},z\)\\subseteq\\mathcal\{B\}for everyzz\. Under the contraction hypothesis below the maps↦F⁡\(s,z\)s\\mapsto F\(s,z\)is a uniform contraction onℬ\\mathcal\{B\}for everyzz, so for each realization of the input the backward iteration from anys0∈ℬs\_\{0\}\\in\\mathcal\{B\}converges and there is a unique stationary solution inℬ\\mathcal\{B\}

St=limk→∞F\(⋅,Zt\)∘F\(⋅,Zt−1\)∘⋯∘F\(s0,Zt−k\),S\_\{t\}=\\lim\_\{k\\to\\infty\}F\(\\cdot,Z\_\{t\}\)\\circ F\(\\cdot,Z\_\{t\-1\}\)\\circ\\cdots\\circ F\(s\_\{0\},Z\_\{t\-k\}\),\(59\)independent ofs0s\_\{0\}, measurable with respect toσ\(Zs:s≤t\)\\sigma\(Z\_\{s\}:s\\leq t\), and stationary because the input is\. All expectations below are taken under this stationary law\.

### D\.2Differentiability and integrability assumptions

We assumeFFandGGare continuously differentiable in both arguments and impose the derivative budgets of Sec\.[5](https://arxiv.org/html/2610.08816#S5),

‖Ds​F​\(s,z\)‖op≤ρ<1,‖Dz​F​\(s,z\)‖≤K,‖Ds​G​\(s,z\)‖≤M,\\\|D\_\{s\}F\(s,z\)\\\|\_\{\\rm op\}\\leq\\rho<1,\\qquad\\\|D\_\{z\}F\(s,z\)\\\|\\leq K,\\qquad\\\|D\_\{s\}G\(s,z\)\\\|\\leq M,\(60\)uniformly for\(s,z\)∈ℬ×ℝ\(s,z\)\\in\\mathcal\{B\}\\times\\mathbb\{R\}\. We also assume enough integrability for the Gaussian integration by parts below to apply, namely thatm^t\\widehat\{m\}\_\{t\}and the partial derivatives appearing in \([61](https://arxiv.org/html/2610.08816#A4.E61)\) are square integrable and that the interchange of differentiation and expectation is justified by domination on compacts\. This is stronger than strictly necessary and is imposed to keep every interchange explicit\.

The uniform contraction in \([60](https://arxiv.org/html/2610.08816#A4.E60)\) is the substantive hypothesis\. It is what makes the stationary solution exist, and it is also what will force the geometric envelope, so the theorem should be read as a statement about the uniformly contractive regime and not about recurrent models generally\.

### D\.3Gaussian Stein identity

The tool is the Gaussian integration\-by\-parts identity\([Stein, 1981](https://arxiv.org/html/2610.08816#bib.bib40)\)\.

###### Lemma 29\(Gaussian integration by parts\)\.

LetZ∼N⁡\(0,1\)Z\\sim N\(0,1\)and letφ\\varphibe absolutely continuous with𝔼​\|φ′​\(Z\)\|<∞\\mathbb\{E\}\|\\varphi^\{\\prime\}\(Z\)\|<\\infty\. Then𝔼⁡\[Z​φ​\(Z\)\]=𝔼⁡\[φ′​\(Z\)\]\\mathbb\{E\}\[Z\\varphi\(Z\)\]=\\mathbb\{E\}\[\\varphi^\{\\prime\}\(Z\)\]\. The same identity holds coordinatewise for a function of several independent standard Gaussians, differentiating in the coordinate being multiplied\.

###### Proof\.

Integrate by parts against the standard Gaussian densityϕ\\phi, usingϕ′​\(z\)=−z​ϕ​\(z\)\\phi^\{\\prime\}\(z\)=\-z\\phi\(z\)and the vanishing ofφ​ϕ\\varphi\\phiat infinity implied by the integrability hypothesis\. ∎

Applying Lem\.[29](https://arxiv.org/html/2610.08816#Thmtheorem29)to the lag\-jjfirst\-chaos coefficient of \([23](https://arxiv.org/html/2610.08816#A1.E23)\) gives

βj=𝔼⁡\[m^t​Zt\+1−j\]=𝔼⁡\[∂m^t∂Zt\+1−j\],\\beta\_\{j\}=\\mathbb\{E\}\[\\widehat\{m\}\_\{t\}Z\_\{t\+1\-j\}\]=\\mathbb\{E\}\\Bigl\[\\frac\{\\partial\\widehat\{m\}\_\{t\}\}\{\\partial Z\_\{t\+1\-j\}\}\\Bigr\],\(61\)so the first\-chaos memory is an averaged sensitivity of the output to an inputj−1j\-1steps earlier\. This is the step that converts a statement about correlations into a statement about derivatives, and it is where the budgets in \([60](https://arxiv.org/html/2610.08816#A4.E60)\) become usable\.

### D\.4Derivative propagation through the recurrent state

Fixj≥2j\\geq 2and sets=t\+1−js=t\+1\-j\. Causality means thatZsZ\_\{s\}influencesm^t\\widehat\{m\}\_\{t\}only through the state, entering first throughDz​F​\(Ss−1,Zs\)D\_\{z\}F\(S\_\{s\-1\},Z\_\{s\}\)and then propagating fromSsS\_\{s\}toStS\_\{t\}\. Repeated application of the chain rule gives

DZs​m^t=Ds​G​\(St,Zt\)​\{∏u=s\+1tDs​F​\(Su−1,Zu\)\}​Dz​F​\(Ss−1,Zs\),D\_\{Z\_\{s\}\}\\widehat\{m\}\_\{t\}=D\_\{s\}G\(S\_\{t\},Z\_\{t\}\)\\Bigl\\\{\\prod\_\{u=s\+1\}^\{t\}D\_\{s\}F\(S\_\{u\-1\},Z\_\{u\}\)\\Bigr\\\}D\_\{z\}F\(S\_\{s\-1\},Z\_\{s\}\),\(62\)the product being ordered chronologically from right to left\. The indexuuruns froms\+1s\+1tott, so there are exactlyt−s=j−1t\-s=j\-1factors ofDs​FD\_\{s\}F\. Taking operator norms and applying \([60](https://arxiv.org/html/2610.08816#A4.E60)\) to each factor gives the pointwise bound

\|DZs​m^t\|≤M​K​ρj−1\(j≥2\)\.\|D\_\{Z\_\{s\}\}\\widehat\{m\}\_\{t\}\|\\leq MK\\rho^\{\\,j\-1\}\\qquad\(j\\geq 2\)\.\(63\)Each additional lag inserts one more factorDs​FD\_\{s\}F, and each such factor is bounded byρ<1\\rho<1, so the sensitivity decays geometrically at exactly the contraction rate\.

### D\.5Proof of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)

###### Proof of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)\.

Fixj≥2j\\geq 2\. Conditioning on all Gaussian coordinates exceptZsZ\_\{s\}and applying Lem\.[29](https://arxiv.org/html/2610.08816#Thmtheorem29)in that coordinate givesβj=𝔼⁡\[DZs​m^t\]\\beta\_\{j\}=\\mathbb\{E\}\[D\_\{Z\_\{s\}\}\\widehat\{m\}\_\{t\}\], the interchange being justified by the integrability assumed above and by the pointwise bound \([63](https://arxiv.org/html/2610.08816#A4.E63)\)\. Since that bound holds uniformly in the realization, taking expectations preserves it and gives\|βj\|≤M​K​ρj−1\|\\beta\_\{j\}\|\\leq MK\\rho^\{\\,j\-1\}, which is \([11](https://arxiv.org/html/2610.08816#S5.E11)\)\.

The statement is deliberately restricted toj≥2j\\geq 2\. Atj=1j=1the inputZtZ\_\{t\}also enters the readout directly throughGG, and controlling that contribution would require a bound onDz​GD\_\{z\}Gthat is not among the budgets \([60](https://arxiv.org/html/2610.08816#A4.E60)\)\. Nothing downstream needs it, since the algebraic tail argument of Lem\.[30](https://arxiv.org/html/2610.08816#Thmtheorem30)uses only lags beyond a crossing indexJ≥2J\\geq 2\.

The second claim is \([24](https://arxiv.org/html/2610.08816#A1.E24)\) applied to the targetmt⋆=∑j≥1c​j−α​Zt\+1−jm\_\{t\}^\{\\star\}=\\sum\_\{j\\geq 1\}cj^\{\-\\alpha\}Z\_\{t\+1\-j\}withα\>1\\alpha\>1, which lies in the first chaos and is square integrable becauseα\>1/2\\alpha\>1/2\. Orthogonality of the Wiener chaoses givesℰ⁡\(m^\)≥∑j≥1\(c​j−α−βj\)2\\mathcal\{E\}\(\\widehat\{m\}\)\\geq\\sum\_\{j\\geq 1\}\(cj^\{\-\\alpha\}\-\\beta\_\{j\}\)^\{2\}, which is \([12](https://arxiv.org/html/2610.08816#S5.E12)\)\. ∎

The theorem does not assert that the state has finite Hankel rank, and indeed a nonlinear recurrence need not produce a finite\-rank linear memory\. What it asserts is that whatever linear memory the recurrence does produce is dominated by a geometric envelope set by the contraction, and \([24](https://arxiv.org/html/2610.08816#A1.E24)\) makes that sufficient for a risk lower bound\.

### D\.6Algebraic first\-chaos tail lower bound

The envelope and the target cross at a lag determined byρ\\rho, and beyond that lag the target dominates\.

###### Lemma 30\(Tail below the crossing lag\)\.

LetJ≥2J\\geq 2be the smallest integer withM​K​ρj−1≤12​c​j−αMK\\rho^\{\\,j\-1\}\\leq\\tfrac\{1\}\{2\}cj^\{\-\\alpha\}for allj≥Jj\\geq J\. Then

ℰ⁡\(m^\)≥∑j≥J\(c​j−α−βj\)2≥c24​∑j≥Jj−2​α≥c24​\(2​α−1\)​J1−2​α\.\\mathcal\{E\}\(\\widehat\{m\}\)\\ \\geq\\ \\sum\_\{j\\geq J\}\\bigl\(cj^\{\-\\alpha\}\-\\beta\_\{j\}\\bigr\)^\{2\}\\ \\geq\\ \\frac\{c^\{2\}\}\{4\}\\sum\_\{j\\geq J\}j^\{\-2\\alpha\}\\ \\geq\\ \\frac\{c^\{2\}\}\{4\(2\\alpha\-1\)\}J^\{1\-2\\alpha\}\.\(64\)

###### Proof\.

Forj≥Jj\\geq Jthe envelope gives\|βj\|≤12​c​j−α\|\\beta\_\{j\}\|\\leq\\tfrac\{1\}\{2\}cj^\{\-\\alpha\}, hence\|c​j−α−βj\|≥12​c​j−α\|cj^\{\-\\alpha\}\-\\beta\_\{j\}\|\\geq\\tfrac\{1\}\{2\}cj^\{\-\\alpha\}, and squaring and summing gives the middle inequality\. The last step is the integral comparison∑j≥Jj−2​α≥∫J∞x−2​α​𝑑x=J1−2​α/\(2​α−1\)\\sum\_\{j\\geq J\}j^\{\-2\\alpha\}\\geq\\int\_\{J\}^\{\\infty\}x^\{\-2\\alpha\}\\,\\mathrm\{d\}x=J^\{1\-2\\alpha\}/\(2\\alpha\-1\), which converges becauseα\>1/2\\alpha\>1/2\. ∎

Using the entire tail rather than a single coefficient is what produces the exponent2​α−12\\alpha\-1in Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\. A weaker argument that bounds the risk below by the single term at the crossing lag would giveJ−2​αJ^\{\-2\\alpha\}instead ofJ1−2​αJ^\{1\-2\\alpha\}and hence the exponent2​α2\\alphain place of2​α−12\\alpha\-1, which is a strictly weaker necessary condition\.

### D\.7Sharp inversion to the near\-criticality law

Lem\.[30](https://arxiv.org/html/2610.08816#Thmtheorem30)expresses the risk through the crossing lagJJ, so it remains to expressJJthroughδ=−log⁡ρ\\delta=\-\\log\\rho\. The defining inequalityM​K​e−δ⁡\(j−1\)≤12​c​j−αMKe^\{\-\\delta\(j\-1\)\}\\leq\\tfrac\{1\}\{2\}cj^\{\-\\alpha\}holds as soon asδ⁡\(j−1\)≥log⁡\(2​M​K/c\)\+α​log⁡j\\delta\(j\-1\)\\geq\\log\(2MK/c\)\+\\alpha\\log j, and solving for the smallest such integer gives

J≤⌈C0δ​log⁡eδ⌉J\\ \\leq\\ \\left\\lceil\\frac\{C\_\{0\}\}\{\\delta\}\\log\\frac\{e\}\{\\delta\}\\right\\rceil\(65\)for smallδ\\delta, withC0C\_\{0\}depending only on\(α,c,M​K\)\(\\alpha,c,MK\)\. The logarithmic factor is unavoidable and comes from theα​log⁡j\\alpha\\log jterm, that is from the target being algebraic rather than bounded\.

### D\.8Proof of Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)

###### Proof of Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\.

Insert \([65](https://arxiv.org/html/2610.08816#A4.E65)\) into \([64](https://arxiv.org/html/2610.08816#A4.E64)\)\. Since1−2​α<01\-2\\alpha<0and, for smallδ\\delta, the ceiling in \([65](https://arxiv.org/html/2610.08816#A4.E65)\) is at most2​C0​δ−1​log⁡\(e/δ\)2C\_\{0\}\\delta^\{\-1\}\\log\(e/\\delta\), we get

ℰ⁡\(m^\)≥c24​\(2​α−1\)​\(2​C0δ​log⁡eδ\)1−2​α=C​\[δlog⁡\(e/δ\)\]2​α−1,\\mathcal\{E\}\(\\widehat\{m\}\)\\ \\geq\\ \\frac\{c^\{2\}\}\{4\(2\\alpha\-1\)\}\\Bigl\(\\frac\{2C\_\{0\}\}\{\\delta\}\\log\\frac\{e\}\{\\delta\}\\Bigr\)^\{1\-2\\alpha\}\\ =\\ C\\Bigl\[\\frac\{\\delta\}\{\\log\(e/\\delta\)\}\\Bigr\]^\{2\\alpha\-1\},which is \([13](https://arxiv.org/html/2610.08816#S5.E13)\)\. For \([14](https://arxiv.org/html/2610.08816#S5.E14)\), supposeℰ⁡\(m^\)≤τ\\mathcal\{E\}\(\\widehat\{m\}\)\\leq\\tau\. Then\[δ/log⁡\(e/δ\)\]2​α−1≤τ/C\[\\delta/\\log\(e/\\delta\)\]^\{2\\alpha\-1\}\\leq\\tau/C, so witha:=\(τ/C\)1/\(2​α−1\)a:=\(\\tau/C\)^\{1/\(2\\alpha\-1\)\}we haveδ≤a​log⁡\(e/δ\)\\delta\\leq a\\log\(e/\\delta\)\. Split into two cases\. Ifδ≤a\\delta\\leq athe desired bound holds already, sincelog⁡\(e/τ\)≥1\\log\(e/\\tau\)\\geq 1\. Ifδ\>a\\delta\>athenlog⁡\(e/δ\)<log⁡\(e/a\)\\log\(e/\\delta\)<\\log\(e/a\), soδ≤a​log⁡\(e/a\)\\delta\\leq a\\log\(e/a\), and substitutinga=\(τ/C\)1/\(2​α−1\)a=\(\\tau/C\)^\{1/\(2\\alpha\-1\)\}giveslog⁡\(e/a\)≤C′′​log⁡\(e/τ\)\\log\(e/a\)\\leq C^\{\\prime\\prime\}\\log\(e/\\tau\)for smallτ\\tau\. In both casesδ≤C′​τ1/\(2​α−1\)​log⁡\(e/τ\)\\delta\\leq C^\{\\prime\}\\tau^\{1/\(2\\alpha\-1\)\}\\log\(e/\\tau\)\. ∎

### D\.9Why the condition is necessary but not sufficient

Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)is a one\-directional implication\. It says that a uniformly contractive recurrence achieving riskτ\\taumust have contraction margin at most of orderτ1/\(2​α−1\)​log⁡\(e/τ\)\\tau^\{1/\(2\\alpha\-1\)\}\\log\(e/\\tau\), and it says nothing about whether a recurrence with that margin achieves that risk\.

The gap is genuine and has two sources\. The envelope of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)is an upper bound on\|βj\|\|\\beta\_\{j\}\|, so a model can have a small margin and still fail to place its first\-chaos mass where the target does, since nothing forces the sensitivities to have the right signs or the right profile\. Separately, even a model whose first\-chaos memory matches the target exactly can carry higher\-chaos mass, which \([24](https://arxiv.org/html/2610.08816#A1.E24)\) shows can only add to the risk\. The stability sweep of Sec\.[7](https://arxiv.org/html/2610.08816#S7)is consistent with this gap: the diagonal model keeps improving asρ\\rhoapproaches one while the gated model retains a residual floor\. We use that floor only as an illustration and do not attribute it uniquely to first\-chaos profile mismatch, since higher\-chaos variance can contribute as well\.

### D\.10Scope beyond uniform contraction

Three restrictions bound what this section covers\.

The white\-noise input is deliberate: Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)isolates memory generated internally by the recurrence\. A recurrence driven directly by an autocorrelated observed process can inherit long memory through the input stream, so the same innovation\-sensitivity envelope does not follow from this argument\.

The contraction is uniform on the forward\-invariant domainℬ\\mathcal\{B\}in\(s,z\)\(s,z\), so a model whose Jacobian is contractive only on average along the stationary trajectory but not pointwise on such a domain is outside the class\. This is not a vacuous distinction, since selective architectures set their effective decay from the input and can be far from contractive on some inputs while remaining stable in aggregate\.

The budgets in \([60](https://arxiv.org/html/2610.08816#A4.E60)\) are constants, so a model whose derivative bounds grow with width evades the conclusion by a route that has nothing to do with memory\. The experiments in App\.[G](https://arxiv.org/html/2610.08816#A7)enforce width\-independent budgets for exactly this reason\.

Finally the bound uses only the first chaos, so it is insensitive to how the predictor uses higher\-order interactions of the input\. A target that is itself nonlinear in the input would need a different projection, and the argument as written applies to targets in the first chaos\.

## Appendix EInformation Theory for Singular Forecasting

### E\.1Conditional trajectory KL identity

LetPθ,T\(⋅∣ℱ0\)P\_\{\\theta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)denote the law ofX1,…,XTX\_\{1\},\\dots,X\_\{T\}under a Gaussian causal model with memorya⁡\(θ\)a\(\\theta\)and innovation varianceσ2\\sigma^\{2\}, conditioned on the infinite past\.

###### Proof of Prop\.[8](https://arxiv.org/html/2610.08816#Thmtheorem8)\.

UnderPθ,T\(⋅∣ℱ0\)P\_\{\\theta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)the conditional law ofXtX\_\{t\}givenℱt−1\\mathcal\{F\}\_\{t\-1\}isN⁡\(mt−1​\(a⁡\(θ\)\),σ2\)N\(m\_\{t\-1\}\(a\(\\theta\)\),\\sigma^\{2\}\), and likewise forϑ\\vartheta\. The chain rule for relative entropy over the filtration gives

DKL\(Pθ,T\(⋅∣ℱ0\)∥Pϑ,T\(⋅∣ℱ0\)\)\\displaystyle D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{\\theta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)\\,\\\|\\,P\_\{\\vartheta,T\}\(\\cdot\\mid\\mathcal\{F\}\_\{0\}\)\\bigr\)=∑t=1T𝔼θ\[DKL\(N\(mt−1\(a\(θ\)\),σ2\)∥N\(mt−1\(a\(ϑ\)\),σ2\)\)\|ℱ0\]\.\\displaystyle=\\sum\_\{t=1\}^\{T\}\\mathbb\{E\}\_\{\\theta\}\\Bigl\[D\_\{\\mathrm\{KL\}\}\\bigl\(N\(m\_\{t\-1\}\(a\(\\theta\)\),\\sigma^\{2\}\)\\,\\\|\\,N\(m\_\{t\-1\}\(a\(\\vartheta\)\),\\sigma^\{2\}\)\\bigr\)\\Bigm\|\\mathcal\{F\}\_\{0\}\\Bigr\]\.For two Gaussians with common variance the relative entropy is half the squared mean difference divided byσ2\\sigma^\{2\}, so each summand is𝔼θ​\[\(mt−1​\(a⁡\(θ\)\)−mt−1​\(a⁡\(ϑ\)\)\)2∣ℱ0\]/\(2​σ2\)\\mathbb\{E\}\_\{\\theta\}\[\(m\_\{t\-1\}\(a\(\\theta\)\)\-m\_\{t\-1\}\(a\(\\vartheta\)\)\)^\{2\}\\mid\\mathcal\{F\}\_\{0\}\]/\(2\\sigma^\{2\}\)\. Taking the outer expectation and using stationarity makes every term equal toℰθ​\(a⁡\(ϑ\)\)/\(2​σ2\)\\mathcal\{E\}\_\{\\theta\}\(a\(\\vartheta\)\)/\(2\\sigma^\{2\}\)by \([17](https://arxiv.org/html/2610.08816#A1.E17)\), and summing overttgives \([15](https://arxiv.org/html/2610.08816#S6.E15)\)\. ∎

No upper spectral bound is used, only finiteness ofℰθ​\(a​\(ϑ\)\)\\mathcal\{E\}\_\{\\theta\}\(a\(\\vartheta\)\), so the identity holds on the singular spectra of App\.[C](https://arxiv.org/html/2610.08816#A3)as well as on regular ones\. This is why the information statements of Sec\.[6](https://arxiv.org/html/2610.08816#S6)are available in exactly the regime where the norm equivalence of App\.[A](https://arxiv.org/html/2610.08816#A1)fails\.

### E\.2Stationary entropy\-rate identity

If in addition the stationary relative entropy rateD¯\(θ∥ϑ\)=limTT−1DKL\(Pθ\(T\)∥Pϑ\(T\)\)\\overline\{D\}\(\\theta\\\|\\vartheta\)=\\lim\_\{T\}T^\{\-1\}D\_\{\\mathrm\{KL\}\}\(P\_\{\\theta\}^\{\(T\)\}\\\|P\_\{\\vartheta\}^\{\(T\)\}\)exists and coincides with the infinite\-past conditional rate, then dividing \([15](https://arxiv.org/html/2610.08816#S6.E15)\) byTTgives

D¯\(θ∥ϑ\)=12​σ2ℰθ\(a\(ϑ\)\)\.\\overline\{D\}\(\\theta\\\|\\vartheta\)=\\frac\{1\}\{2\\sigma^\{2\}\}\\mathcal\{E\}\_\{\\theta\}\(a\(\\vartheta\)\)\.\(66\)The two hypotheses are separate\. The conditional identity is exact for every finiteTT, while \([66](https://arxiv.org/html/2610.08816#A5.E66)\) additionally requires that conditioning on the infinite past does not change the rate, which is where the finite\-section corrections of the rest of this appendix enter\.

Two consequences are used in App\.[C](https://arxiv.org/html/2610.08816#A3)\. For fractional noise, \([66](https://arxiv.org/html/2610.08816#A5.E66)\) together with Thm\.[19](https://arxiv.org/html/2610.08816#Thmtheorem19)givesD¯\(d∥d\+Δ\)=12\{Γ\(1\+2Δ\)/Γ\(1\+Δ\)2−1\}\\overline\{D\}\(d\\\|d\+\\Delta\)=\\tfrac\{1\}\{2\}\\\{\\Gamma\(1\+2\\Delta\)/\\Gamma\(1\+\\Delta\)^\{2\}\-1\\\}, which isπ2​Δ2/12\+O⁡\(\|Δ\|3\)\\pi^\{2\}\\Delta^\{2\}/12\+O\(\|\\Delta\|^\{3\}\)\. If two models share a singular spectral factor and differ only in a regular one, that factor cancels in the likelihood ratio, so \([66](https://arxiv.org/html/2610.08816#A5.E66)\) applies to the ratio of the regular parts alone and a known common singularity contributes nothing to distinguishability\.

### E\.3Conditional score and predictive curvature

Suppose the predictor isL2L\_\{2\}\-differentiable at an interiorθ0\\theta\_\{0\}, meaningmt−1​\(θ0\+u\)−mt−1​\(θ0\)=u⊤​m˙t−1\+rt−1​\(u\)m\_\{t\-1\}\(\\theta\_\{0\}\+u\)\-m\_\{t\-1\}\(\\theta\_\{0\}\)=u^\{\\top\}\\dot\{m\}\_\{t\-1\}\+r\_\{t\-1\}\(u\)with𝔼​rt−1​\(u\)2=o⁡\(‖u‖2\)\\mathbb\{E\}r\_\{t\-1\}\(u\)^\{2\}=o\(\\\|u\\\|^\{2\}\)\. Differentiating the conditional Gaussian log\-likelihood gives the conditional score

st​\(θ0\)=εtσ2​m˙t−1,s\_\{t\}\(\\theta\_\{0\}\)=\\frac\{\\varepsilon\_\{t\}\}\{\\sigma^\{2\}\}\\dot\{m\}\_\{t\-1\},\(67\)which is a square\-integrable martingale difference becauseεt\\varepsilon\_\{t\}is conditionally centered andm˙t−1\\dot\{m\}\_\{t\-1\}isℱt−1\\mathcal\{F\}\_\{t\-1\}\-measurable\. Its per\-observation information matrix isIθ0cond=Gθ0/σ2I^\{\\rm cond\}\_\{\\theta\_\{0\}\}=G\_\{\\theta\_\{0\}\}/\\sigma^\{2\}withGθ0=𝔼⁡\[m˙t−1​m˙t−1⊤\]G\_\{\\theta\_\{0\}\}=\\mathbb\{E\}\[\\dot\{m\}\_\{t\-1\}\\dot\{m\}\_\{t\-1\}^\{\\top\}\]the forecast curvature\.

The same curvature appears on both sides of the theory, sinceℰθ0​\(a⁡\(θ0\+u\)\)=u⊤​Gθ0​u\+o⁡\(‖u‖2\)\\mathcal\{E\}\_\{\\theta\_\{0\}\}\(a\(\\theta\_\{0\}\+u\)\)=u^\{\\top\}G\_\{\\theta\_\{0\}\}u\+o\(\\\|u\\\|^\{2\}\)by \([17](https://arxiv.org/html/2610.08816#A1.E17)\)\. Information and forecast risk are therefore not merely proportional in the aggregate identity \([66](https://arxiv.org/html/2610.08816#A5.E66)\) but share the same local quadratic form, which is what makes the constant in the next subsection universal\.

### E\.4Local asymptotic minimax transfer

Assume the stationary experiment is locally asymptotically normal atθ0\\theta\_\{0\}with central sequence asymptotically equal to the normalized sum of conditional scores, and that the martingale central limit theorem applies to that sum\. Then the local asymptotic minimax bound for estimatingθ\\theta, in the standard form for locally asymptotically normal experiments\([van der Vaart, 1998](https://arxiv.org/html/2610.08816#bib.bib41), Ch\. 8\), transfers to forecast risk through the local quadratic expansion, and for app\-dimensional smooth parameter the minimax forecast constant isp​σ2/Tp\\sigma^\{2\}/T\.

For the one\-dimensional fractional exponent this gives a concrete number\. The forecast curvature isπ2​σ2/6\\pi^\{2\}\\sigma^\{2\}/6and the efficient asymptotic variance is6/π26/\\pi^\{2\}\([Lieberman & Phillips, 2004](https://arxiv.org/html/2610.08816#bib.bib33);[Hualde & Robinson, 2011](https://arxiv.org/html/2610.08816#bib.bib27)\), so an efficient estimator satisfiesT​ℰd​\(a⁡\(d^T\)\)⇒σ2​χ12T\\mathcal\{E\}\_\{d\}\(a\(\\widehat\{d\}\_\{T\}\)\)\\Rightarrow\\sigma^\{2\}\\chi^\{2\}\_\{1\}and, given uniform integrability,𝔼​ℰd​\(a⁡\(d^T\)\)=σ2/T\+o⁡\(T−1\)\\mathbb\{E\}\\mathcal\{E\}\_\{d\}\(a\(\\widehat\{d\}\_\{T\}\)\)=\\sigma^\{2\}/T\+o\(T^\{\-1\}\)\. The curvature and the variance cancel, which is why the answer does not depend onddand why a long\-memory exponent costs exactly one parameter’s worth of forecast risk\.

### E\.5Exact one\-pole Fisher–Hartwig determinant

By the determinant identity of App\.[C](https://arxiv.org/html/2610.08816#A3),logdetTT\(f\)=∑j<TlogVj\\log\\det T\_\{T\}\(f\)=\\sum\_\{j<T\}\\log V\_\{j\}, so a finite\-section information statement is a Toeplitz determinant asymptotic\. For fractional noise against white noise the symbol has one Fisher–Hartwig singularity of strengthdd, and the classical asymptotic\([Deift et al\., 2011](https://arxiv.org/html/2610.08816#bib.bib14);[Ehrhardt & Silbermann, 1997](https://arxiv.org/html/2610.08816#bib.bib16)\)giveslogdetTT=d2logT\+O\(1\)\\log\\det T\_\{T\}=d^\{2\}\\log T\+O\(1\)\.

###### Theorem 31\(Finite\-section remainder for fractional versus white noise\)\.

Fix0<d0<1/20<d\_\{0\}<1/2\. For every\|d\|≤d0\|d\|\\leq d\_\{0\}andT≥1T\\geq 1,

DKL\(Pd\(T\)∥P0\(T\)\)=TD¯\(d∥0\)−12logdetTT\(φd\),D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{d\}^\{\(T\)\}\\\|P\_\{0\}^\{\(T\)\}\\bigr\)=T\\,\\overline\{D\}\(d\\\|0\)\-\\tfrac\{1\}\{2\}\\log\\det T\_\{T\}\(\\varphi\_\{d\}\),\(68\)withD¯\(d∥0\)=12\{Γ\(1−2d\)/Γ\(1−d\)2−1\}\\overline\{D\}\(d\\\|0\)=\\tfrac\{1\}\{2\}\\\{\\Gamma\(1\-2d\)/\\Gamma\(1\-d\)^\{2\}\-1\\\}, and there isCd0<∞C\_\{d\_\{0\}\}<\\inftywith

0≤TD¯\(d∥0\)−DKL\(Pd\(T\)∥P0\(T\)\)≤Cd0d2log\(eT\)0\\leq T\\,\\overline\{D\}\(d\\\|0\)\-D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{d\}^\{\(T\)\}\\\|P\_\{0\}^\{\(T\)\}\\bigr\)\\leq C\_\{d\_\{0\}\}d^\{2\}\\log\(eT\)\(69\)uniformly over that range\.

###### Proof of the exact identity \([68](https://arxiv.org/html/2610.08816#A5.E68)\)\.

The two laws are centered Gaussians with covariancesσ2​TT​\(φd\)\\sigma^\{2\}T\_\{T\}\(\\varphi\_\{d\}\)andσ2​IT\\sigma^\{2\}I\_\{T\}, so

DKL\(Pd\(T\)∥P0\(T\)\)=12\{trTT\(φd\)−T−logdetTT\(φd\)\}\.D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{d\}^\{\(T\)\}\\\|P\_\{0\}^\{\(T\)\}\\bigr\)=\\tfrac\{1\}\{2\}\\Bigl\\\{\\operatorname\{tr\}T\_\{T\}\(\\varphi\_\{d\}\)\-T\-\\log\\det T\_\{T\}\(\\varphi\_\{d\}\)\\Bigr\\\}\.Every diagonal entry of a Toeplitz matrix is the zeroth Fourier coefficient of its symbol, sotr⁡TT​\(φd\)=T​φ^d​\(0\)\\operatorname\{tr\}T\_\{T\}\(\\varphi\_\{d\}\)=T\\widehat\{\\varphi\}\_\{d\}\(0\)and the first two terms equal

T2\{φ^d\(0\)−1\}=TD¯\(d∥0\)\\frac\{T\}\{2\}\\bigl\\\{\\widehat\{\\varphi\}\_\{d\}\(0\)\-1\\bigr\\\}=T\\,\\overline\{D\}\(d\\\|0\)by the stationary rate identity \([66](https://arxiv.org/html/2610.08816#A5.E66)\)\. The remainder bound \([69](https://arxiv.org/html/2610.08816#A5.E69)\) is proved in App\.[E](https://arxiv.org/html/2610.08816#A5)below\. ∎

The boundary is therefore logarithmic rather than of orderTT, which is the substantive statement\. A singular spectrum could in principle make the finite\-section experiment differ from the stationary rate at leading order, and \([69](https://arxiv.org/html/2610.08816#A5.E69)\) says it does not for a single fractional pole\.

### E\.6Trigamma proof of theO⁡\(d2​log⁡T\)O\(d^\{2\}\\log T\)remainder

We now prove the remainder bound \([69](https://arxiv.org/html/2610.08816#A5.E69)\) of Thm\.[31](https://arxiv.org/html/2610.08816#Thmtheorem31)\. The uniformity inddis what needs an argument, since a bound of the formCd​log⁡TC\_\{d\}\\log TwithCdC\_\{d\}blowing up asd→0d\\to 0would be useless for the local statement below\.

Write the remainder as12​log​detTT​\(φd\)\\tfrac\{1\}\{2\}\\log\\det T\_\{T\}\(\\varphi\_\{d\}\)and use the determinant product together with the exact prediction variances \([41](https://arxiv.org/html/2610.08816#A3.E41)\), which gives

logdetTT\(φd\)=∑j=0T−1logVjσ2=∑j=0T−1\[logΓ\(j\+1\)\+logΓ\(j\+1−2d\)−2logΓ\(j\+1−d\)\]\.\\log\\det T\_\{T\}\(\\varphi\_\{d\}\)=\\sum\_\{j=0\}^\{T\-1\}\\log\\frac\{V\_\{j\}\}\{\\sigma^\{2\}\}=\\sum\_\{j=0\}^\{T\-1\}\\bigl\[\\log\\Gamma\(j\+1\)\+\\log\\Gamma\(j\+1\-2d\)\-2\\log\\Gamma\(j\+1\-d\)\\bigr\]\.Each summand is a second difference oflog⁡Γ\\log\\Gammain the parameter, so by Taylor expansion with integral remainder it equalsd2​ψ′​\(j\+1−ξj\)d^\{2\}\\psi^\{\\prime\}\(j\+1\-\\xi\_\{j\}\)for someξj\\xi\_\{j\}lying between00and2​d2d\(in either order\), hence with\|ξj\|≤2​d0\|\\xi\_\{j\}\|\\leq 2d\_\{0\}, whereψ′\\psi^\{\\prime\}is the trigamma function\. Forj≥1j\\geq 1the argument satisfiesj\+1−ξj≥j\+1−2​d0≥12j\+1\-\\xi\_\{j\}\\geq j\+1\-2d\_\{0\}\\geq\\tfrac\{1\}\{2\}, whereψ′​\(x\)≤C/x\\psi^\{\\prime\}\(x\)\\leq C/x, so those terms contribute at mostC​d2​∑1≤j<T\(j\+1−2​d0\)−1≤C​d2​log⁡\(e​T\)Cd^\{2\}\\sum\_\{1\\leq j<T\}\(j\+1\-2d\_\{0\}\)^\{\-1\}\\leq Cd^\{2\}\\log\(eT\)\. The termj=0j=0needs separate treatment, because its argument1−ξ01\-\\xi\_\{0\}can be as small as1−2​d01\-2d\_\{0\}andψ′​\(x\)≍x−2\\psi^\{\\prime\}\(x\)\\asymp x^\{\-2\}near the origin, so the boundψ′​\(x\)≤C/x\\psi^\{\\prime\}\(x\)\\leq C/xis unavailable there\. Sinced≤d0<1/2d\\leq d\_\{0\}<1/2gives1−ξ0≥1−2​d0\>01\-\\xi\_\{0\}\\geq 1\-2d\_\{0\}\>0, monotonicity ofψ′\\psi^\{\\prime\}bounds that single term byd2​ψ′​\(1−2​d0\)d^\{2\}\\psi^\{\\prime\}\(1\-2d\_\{0\}\), a constant depending only ond0d\_\{0\}, and adding the two contributions gives at mostCd0​d2​log⁡\(e​T\)C\_\{d\_\{0\}\}d^\{2\}\\log\(eT\)\. It is nonnegative becauselog⁡Γ\\log\\Gammais convex, so every second difference is nonnegative\. The constant depends only ond0d\_\{0\}through the range ofξj\\xi\_\{j\}, which gives the uniformity and covers negativeddas well\.

### E\.7Direct local information atd=0d=0

###### Corollary 32\(Local finite\-section information\)\.

For fixedh∈ℝh\\in\\mathbb\{R\}letdT=h/Td\_\{T\}=h/\\sqrt\{T\}\. ThenDKL\(PdT\(T\)∥P0\(T\)\)=π2h2/12\+o\(1\)D\_\{\\mathrm\{KL\}\}\(P\_\{d\_\{T\}\}^\{\(T\)\}\\\|P\_\{0\}^\{\(T\)\}\)=\\pi^\{2\}h^\{2\}/12\+o\(1\), so the finite stationary experiment recovers the Fisher informationI⁡\(0\)=π2/6I\(0\)=\\pi^\{2\}/6\.

###### Proof\.

InsertdT=h/Td\_\{T\}=h/\\sqrt\{T\}into \([68](https://arxiv.org/html/2610.08816#A5.E68)\)\. The leading term isTD¯\(dT∥0\)=T\{π2dT2/12\+O\(dT3\)\}=π2h2/12\+O\(T−1/2\)T\\overline\{D\}\(d\_\{T\}\\\|0\)=T\\\{\\pi^\{2\}d\_\{T\}^\{2\}/12\+O\(d\_\{T\}^\{3\}\)\\\}=\\pi^\{2\}h^\{2\}/12\+O\(T^\{\-1/2\}\), and the remainder is at mostCd0​dT2​log⁡\(e​T\)=O⁡\(T−1​log⁡T\)C\_\{d\_\{0\}\}d\_\{T\}^\{2\}\\log\(eT\)=O\(T^\{\-1\}\\log T\)by \([69](https://arxiv.org/html/2610.08816#A5.E69)\)\. ∎

The local scaled=h/Td=h/\\sqrt\{T\}is exactly the one on which the logarithmic remainder is negligible, so the finite stationary block has the same local relative\-entropy constant as the stationary conditional experiment\. We do not infer local asymptotic normality from this, since that is a stochastic likelihood\-ratio expansion and a limit for the relative entropy alone does not supply a central sequence\. What the calculation shows is that the Fisher–Hartwig boundary does not alter the local information constant on theT−1/2T^\{\-1/2\}scale\.

### E\.8Fixed\-QQmoving and merging fractional poles

The single\-pole statement of Thm\.[31](https://arxiv.org/html/2610.08816#Thmtheorem31)and Corollary[32](https://arxiv.org/html/2610.08816#Thmtheorem32)extends to a fixed number of poles whose locations are allowed to depend onTTand to merge\.

To avoid a collision with the exponential mixing densitywd​\(t\)w\_\{d\}\(t\)of App\.[C](https://arxiv.org/html/2610.08816#A3), we write the Toeplitz spectral symbols as

φd​\(ei​ω\)=\|1−ei​ω\|−2​d,φ𝐝,𝐭​\(ei​ω\)=∏q=1Q\|ei​ω−ei​tq\|−2​dq\.\\varphi\_\{d\}\(e^\{i\\omega\}\)=\|1\-e^\{i\\omega\}\|^\{\-2d\},\\qquad\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\(e^\{i\\omega\}\)=\\prod\_\{q=1\}^\{Q\}\|e^\{i\\omega\}\-e^\{it\_\{q\}\}\|^\{\-2d\_\{q\}\}\.\(70\)For real\-valued processes we restrict to reflection\-symmetric configurations: whenevertq∉\{0,π\}t\_\{q\}\\notin\\\{0,\\pi\\\}occurs with strengthdqd\_\{q\}, the conjugate location2​π−tq2\\pi\-t\_\{q\}occurs with the same strength\. Thenφ𝐝,𝐭\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}is even andTT​\(φ𝐝,𝐭\)T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)is real symmetric\. We setΣ𝐝,𝐭,T=σ2​TT​\(φ𝐝,𝐭\)\\Sigma\_\{\\mathbf\{d\},\\mathbf\{t\},T\}=\\sigma^\{2\}T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)andP𝐝,𝐭\(T\)=N⁡\(0,Σ𝐝,𝐭,T\)P^\{\(T\)\}\_\{\\mathbf\{d\},\\mathbf\{t\}\}=N\(0,\\Sigma\_\{\\mathbf\{d\},\\mathbf\{t\},T\}\), withP0\(T\)=N⁡\(0,σ2​IT\)P\_\{0\}^\{\(T\)\}=N\(0,\\sigma^\{2\}I\_\{T\}\)the white\-noise law\.

###### Theorem 33\(Uniform finite\-section expansion forQQmoving poles\)\.

FixQQand0<d⋆<1/20<d\_\{\\star\}<1/2, and fix a strength vector𝐝=\(d1,…,dQ\)\\mathbf\{d\}=\(d\_\{1\},\\dots,d\_\{Q\}\)with

dq≥0,∑q=1Qdq≤d⋆<12\.d\_\{q\}\\geq 0,\\qquad\\sum\_\{q=1\}^\{Q\}d\_\{q\}\\leq d\_\{\\star\}<\\tfrac\{1\}\{2\}\.\(71\)Uniformly over distinct reflection\-symmetric locations0≤t1<⋯<tQ<2​π0\\leq t\_\{1\}<\\cdots<t\_\{Q\}<2\\pi, asT→∞T\\to\\infty,

DKL\(P𝐝,𝐭\(T\)∥P0\(T\)\)\\displaystyle D\_\{\\mathrm\{KL\}\}\\bigl\(P^\{\(T\)\}\_\{\\mathbf\{d\},\\mathbf\{t\}\}\\\|P\_\{0\}^\{\(T\)\}\\bigr\)=TD¯\(𝐝,𝐭∥0\)−12logdetTT\(φ𝐝,𝐭\),\\displaystyle=T\\,\\overline\{D\}\(\\mathbf\{d\},\\mathbf\{t\}\\\|0\)\-\\tfrac\{1\}\{2\}\\log\\det T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\),\(72\)logdetTT\(φ𝐝,𝐭\)\\displaystyle\\log\\det T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)=\(∑qdq2\)​log⁡T\+2​∑q<ℓdq​dℓ​log⁡1sin⁡\(\|tq−tℓ\|/2\)\+T−1\+OQ,𝐝​\(1\)\.\\displaystyle=\\Bigl\(\\sum\_\{q\}d\_\{q\}^\{2\}\\Bigr\)\\log T\+2\\sum\_\{q<\\ell\}d\_\{q\}d\_\{\\ell\}\\log\\frac\{1\}\{\\sin\(\|t\_\{q\}\-t\_\{\\ell\}\|/2\)\+T^\{\-1\}\}\+O\_\{Q,\\mathbf\{d\}\}\(1\)\.\(73\)

The second term is the interaction between poles, and the regularization byT−1T^\{\-1\}inside the logarithm is what makes the expansion survive merging\. Two poles at separation much larger thanT−1T^\{\-1\}interact throughlog⁡\(1/sin⁡\(\|tq−tℓ\|/2\)\)\\log\(1/\\sin\(\|t\_\{q\}\-t\_\{\\ell\}\|/2\)\), while two poles at separation much smaller thanT−1T^\{\-1\}behave as a single pole of combined strength, and the crossover is at separation of orderT−1T^\{\-1\}\. This is the same resolution scale that appears in the block prediction information, where the cumulative log\-variance over a dyadic block of contexts tends to4​d2​log⁡24d^\{2\}\\log 2when the separation is below the scale and to2​d2​log⁡22d^\{2\}\\log 2when it is above\.

### E\.9Uniform Fisher–Hartwig input and the role of the integrability constraint

Thm\.[33](https://arxiv.org/html/2610.08816#Thmtheorem33)consumes a Toeplitz asymptotic that is uniform in the singularity locations rather than a pointwise one, since the pole locations move withTT\. The input is[Fahs \(2021\)](https://arxiv.org/html/2610.08816#bib.bib17), together with the two\-singularity merging analysis of[Claeys & Krasovsky \(2015\)](https://arxiv.org/html/2610.08816#bib.bib11)\. It is worth stating that input precisely, because the scope of its uniformity is narrower than the phrase*uniform Fisher–Hartwig*might suggest\.

###### Proof of Thm\.[33](https://arxiv.org/html/2610.08816#Thmtheorem33)\.

For the exact identity,Σ𝐝,𝐭,T=σ2​TT​\(φ𝐝,𝐭\)\\Sigma\_\{\\mathbf\{d\},\\mathbf\{t\},T\}=\\sigma^\{2\}T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)andP0\(T\)=N⁡\(0,σ2​IT\)P\_\{0\}^\{\(T\)\}=N\(0,\\sigma^\{2\}I\_\{T\}\), so the finite\-dimensional Gaussian relative entropy between two centered laws gives

DKL\(P𝐝,𝐭\(T\)∥P0\(T\)\)=12\{trTT\(φ𝐝,𝐭\)−T−logdetTT\(φ𝐝,𝐭\)\}\.D\_\{\\mathrm\{KL\}\}\\bigl\(P^\{\(T\)\}\_\{\\mathbf\{d\},\\mathbf\{t\}\}\\\|P\_\{0\}^\{\(T\)\}\\bigr\)=\\tfrac\{1\}\{2\}\\Bigl\\\{\\operatorname\{tr\}T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)\-T\-\\log\\det T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)\\Bigr\\\}\.Every diagonal entry ofTT​\(φ𝐝,𝐭\)T\_\{T\}\(\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\)equalsφ^𝐝,𝐭​\(0\)=\(2​π\)−1​∫−ππφ𝐝,𝐭​\(ei​ω\)​𝑑ω\\widehat\{\\varphi\}\_\{\\mathbf\{d\},\\mathbf\{t\}\}\(0\)=\(2\\pi\)^\{\-1\}\\int\_\{\-\\pi\}^\{\\pi\}\\varphi\_\{\\mathbf\{d\},\\mathbf\{t\}\}\(e^\{i\\omega\}\)\\,\\mathrm\{d\}\\omega, sotr⁡TT=T​φ^𝐝,𝐭​\(0\)\\operatorname\{tr\}T\_\{T\}=T\\widehat\{\\varphi\}\_\{\\mathbf\{d\},\\mathbf\{t\}\}\(0\)and the first two terms areTD¯\(𝐝,𝐭∥0\)T\\,\\overline\{D\}\(\\mathbf\{d\},\\mathbf\{t\}\\\|0\), which is \([72](https://arxiv.org/html/2610.08816#A5.E72)\)\.

For the determinant, apply Theorem 1\.1 and Remark 1\.2 of[Fahs \(2021\)](https://arxiv.org/html/2610.08816#bib.bib17)with smooth partV≡0V\\equiv 0, jump exponentsβq=0\\beta\_\{q\}=0, and root exponentsαq=−dq\\alpha\_\{q\}=\-d\_\{q\}\. Condition \([71](https://arxiv.org/html/2610.08816#A5.E71)\) implies the same bound for every merging cluster because alldqd\_\{q\}are nonnegative, so the sum of root exponents in each cluster is at least−d⋆\>−1/2\-d\_\{\\star\}\>\-1/2and the negative\-exponent extension applies uniformly over the locations\. Sinceαq2=dq2\\alpha\_\{q\}^\{2\}=d\_\{q\}^\{2\}and2​αq​αℓ=2​dq​dℓ2\\alpha\_\{q\}\\alpha\_\{\\ell\}=2d\_\{q\}d\_\{\\ell\}, the two signs cancel in both explicit terms and Fahs’s formula is exactly \([73](https://arxiv.org/html/2610.08816#A5.E73)\)\. ∎

The mapping deserves a word, because the sign is easy to get wrong\.[Fahs \(2021\)](https://arxiv.org/html/2610.08816#bib.bib17)considers non\-negative symbolsf⁡\(z\)=eV⁡\(z\)​∏j≤mωαj,βj​\(z/zj\)f\(z\)=e^\{V\(z\)\}\\prod\_\{j\\leq m\}\\omega\_\{\\alpha\_\{j\},\\beta\_\{j\}\}\(z/z\_\{j\}\)withωα,β​\(z\)=\(z/eπ​i\)β​\|z−1\|2​α\\omega\_\{\\alpha,\\beta\}\(z\)=\(z/e^\{\\pi i\}\)^\{\\beta\}\|z\-1\|^\{2\\alpha\}, whereVVis real valued and analytic on a neighbourhood of the unit circle,zj=ei​tjz\_\{j\}=e^\{it\_\{j\}\}with0≤t1<⋯<tm<2​π0\\leq t\_\{1\}<\\cdots<t\_\{m\}<2\\pi, andαj≥0\\alpha\_\{j\}\\geq 0withℜ⁡βj=0\\Re\\beta\_\{j\}=0\. For a fixed numbermmof such singularities the result gives

log⁡Dn​\(f\)=n​V0\+∑j≤m\(αj2−βj2\)​log⁡n\+∑j<k2​\(αj​αk−βj​βk\)​log⁡1sin⁡\|tj−tk\|/2\+n−1\+O⁡\(1\),\\log D\_\{n\}\(f\)=nV\_\{0\}\+\\sum\_\{j\\leq m\}\(\\alpha\_\{j\}^\{2\}\-\\beta\_\{j\}^\{2\}\)\\log n\+\\sum\_\{j<k\}2\(\\alpha\_\{j\}\\alpha\_\{k\}\-\\beta\_\{j\}\\beta\_\{k\}\)\\log\\frac\{1\}\{\\sin\|t\_\{j\}\-t\_\{k\}\|/2\+n^\{\-1\}\}\+O\(1\),with the error term uniform in the locationst1,…,tmt\_\{1\},\\dots,t\_\{m\}\. The uniformity asserted there is in the locations\. The exponents are fixed parameters of the symbol, and we do not attribute to that result any uniformity jointly in the strengths\.

Matching\|ei​ω−ei​tq\|−2​dq\|e^\{i\\omega\}\-e^\{it\_\{q\}\}\|^\{\-2d\_\{q\}\}against\|z/zq−1\|2​αq\|z/z\_\{q\}\-1\|^\{2\\alpha\_\{q\}\}forces2​αq=−2​dq2\\alpha\_\{q\}=\-2d\_\{q\}, so a long\-memory pole of strengthdqd\_\{q\}is a Fisher–Hartwig root singularity of exponent

αq=−dq≤0,βq=0\.\\alpha\_\{q\}=\-d\_\{q\}\\leq 0,\\qquad\\beta\_\{q\}=0\.\(74\)Theorem 1\.1 of that paper is stated for non\-negative root exponents, while Remark 1\.2 extends the same proof to negative root exponents provided every combination of merging singularities has total exponent strictly greater than−1/2\-1/2\. Under \([74](https://arxiv.org/html/2610.08816#A5.E74)\) that requirement reads∑q∈Sdq<1/2\\sum\_\{q\\in S\}d\_\{q\}<1/2for every merging clusterSS, and the global bound \([71](https://arxiv.org/html/2610.08816#A5.E71)\) implies all of these inequalities becausedq≥0d\_\{q\}\\geq 0\.

The margin therefore plays two roles at once, and we do not attribute it to only one of them\. It is the integrability condition that keeps each pole, and each limit of a collapsing cluster, inside the stationary fractional class on which the predictive geometry of Sec\.[4](https://arxiv.org/html/2610.08816#S4)is defined\. It is simultaneously the applicability condition for the negative\-exponent Fisher–Hartwig expansion\. What we use from[Fahs \(2021\)](https://arxiv.org/html/2610.08816#bib.bib17)is the uniformity in the pole locations, with the strength vector held fixed, and we make no claim of uniformity over strengths\.

### E\.10What is not covered for growingQQor vanishing pole strengths

Two regimes are outside the theory as stated\.

The number of poles is fixed\. The constant in \([73](https://arxiv.org/html/2610.08816#A5.E73)\) depends onQQ, and the available uniform Fisher–Hartwig input does not control that dependence, so a spectrum whose number of singularities grows withTTis not covered\. This matters because a seasonal fractional model with a period growing inTTis exactly such a spectrum, and its finite\-section information is unknown\.

TheO⁡\(1\)O\(1\)remainder in \([73](https://arxiv.org/html/2610.08816#A5.E73)\) is uniform over the moving locations, which is the uniformity the external input supplies, and it is stated for a fixed vector of strengths\. What we do not claim is any uniformity in the strengths, nor a strength\-sensitive refinement of that remainder, for instance one quadratic in𝐝\\mathbf\{d\}, for triangular arrays of vanishing strengths coupled to merging locations\. Such a refinement would be the natural analogue of theO⁡\(d2​log⁡T\)O\(d^\{2\}\\log T\)single\-pole bound, and neither the trigamma argument nor the fixed\-QQexpansion supplies it\. We record both gaps rather than absorbing them, because Sec\.[6](https://arxiv.org/html/2610.08816#S6)states the fixed\-QQresult and should not be read as covering either\.

## Appendix FMultivariate Predictive Memory

The main text is stated for a scalar target\. This appendix records the multivariate extension referred to in Sec\.[8](https://arxiv.org/html/2610.08816#S8)\. It is a complete but secondary development, and none of the three resource laws of Table[1](https://arxiv.org/html/2610.08816#S2.T1)depends on it\.

Four symbols are reused here with a different meaning from the one they carry in the main text, and the reuse is local to this appendix\. Capital letters denote matrix\-valued coefficient arrays rather than the scalar transfer functions of Sec\.[2](https://arxiv.org/html/2610.08816#S2), soA⋆A^\{\\star\}below is a matrix memory and not the transfer function ofa⋆a^\{\\star\}\. The letterddis the ambient dimension of the observed vector, not the fractional memory strength of Sec\.[4](https://arxiv.org/html/2610.08816#S4); the fractional model plays no role in this appendix\. The letterBBdenotes a matrix array and not the backshift operator, andΔ\\Deltadenotes a coefficient error and not a pole\-location offset\.

### F\.1Matrix\-valued predictive geometry

Let\(Xt\)\(X\_\{t\}\)be a centered stationaryℝd\\mathbb\{R\}^\{d\}\-valued process with matrix predictive memoryA⋆=\(Aj⋆\)j≥1A^\{\\star\}=\(A\_\{j\}^\{\\star\}\)\_\{j\\geq 1\}, so thatmt⋆=∑j≥1Aj⋆​Xt\+1−jm\_\{t\}^\{\\star\}=\\sum\_\{j\\geq 1\}A\_\{j\}^\{\\star\}X\_\{t\+1\-j\}, and letFX​\(ω\)F\_\{X\}\(\\omega\)be the spectral density matrix\.

###### Theorem 34\(Multivariate excess forecast risk\)\.

Assume‖FX‖op\\\|F\_\{X\}\\\|\_\{\\rm op\}essentially bounded and∑j‖Δj‖F2<∞\\sum\_\{j\}\\\|\\Delta\_\{j\}\\\|\_\{F\}^\{2\}<\\inftywithΔj=Aj−Aj⋆\\Delta\_\{j\}=A\_\{j\}\-A\_\{j\}^\{\\star\}\. Then the prediction\-error filter converges inL2L\_\{2\}and

ℰ⁡\(A\)=12​π​∫−ππtr⁡\[Δ⁡\(e−i​ω\)​FX​\(ω\)​Δ​\(e−i​ω\)∗\]​𝑑ω\.\\mathcal\{E\}\(A\)=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\\operatorname\{tr\}\\bigl\[\\Delta\(e^\{\-i\\omega\}\)F\_\{X\}\(\\omega\)\\Delta\(e^\{\-i\\omega\}\)^\{\*\}\\bigr\]\\,\\mathrm\{d\}\\omega\.\(75\)If in addition0<m​Id⪯FX⪯M​Id0<mI\_\{d\}\\preceq F\_\{X\}\\preceq MI\_\{d\}almost everywhere, thenm​∑j‖Δj‖F2≤ℰ⁡\(A\)≤M​∑j‖Δj‖F2m\\sum\_\{j\}\\\|\\Delta\_\{j\}\\\|\_\{F\}^\{2\}\\leq\\mathcal\{E\}\(A\)\\leq M\\sum\_\{j\}\\\|\\Delta\_\{j\}\\\|\_\{F\}^\{2\}\.

###### Proof\.

PutUt=∑j≥1Δj​Xt\+1−jU\_\{t\}=\\sum\_\{j\\geq 1\}\\Delta\_\{j\}X\_\{t\+1\-j\}, which converges inL2L\_\{2\}because‖FX‖op\\\|F\_\{X\}\\\|\_\{\\rm op\}is essentially bounded and∑j‖Δj‖F2<∞\\sum\_\{j\}\\\|\\Delta\_\{j\}\\\|\_\{F\}^\{2\}<\\infty, by the same Cauchy argument as in App\.[A](https://arxiv.org/html/2610.08816#A1)applied coordinatewise\. The limit isℱt\\mathcal\{F\}\_\{t\}\-measurable, so it is orthogonal to the innovation, and expanding the squared error givesℰ⁡\(A\)=𝔼​‖Ut‖2\\mathcal\{E\}\(A\)=\\mathbb\{E\}\\\|U\_\{t\}\\\|^\{2\}\. The multivariate spectral representation turns this into

𝔼​‖Ut‖2=12​π​∫−ππtr⁡\[Δ⁡\(e−i​ω\)​FX​\(ω\)​Δ​\(e−i​ω\)∗\]​𝑑ω,\\mathbb\{E\}\\\|U\_\{t\}\\\|^\{2\}=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\\operatorname\{tr\}\\bigl\[\\Delta\(e^\{\-i\\omega\}\)F\_\{X\}\(\\omega\)\\Delta\(e^\{\-i\\omega\}\)^\{\*\}\\bigr\]\\,\\mathrm\{d\}\\omega,which is \([75](https://arxiv.org/html/2610.08816#A6.E75)\)\. For the two\-sided bound,m​Id⪯FX⪯M​IdmI\_\{d\}\\preceq F\_\{X\}\\preceq MI\_\{d\}givesm​tr⁡\(Δ​Δ∗\)≤tr⁡\(Δ​FX​Δ∗\)≤M​tr⁡\(Δ​Δ∗\)m\\operatorname\{tr\}\(\\Delta\\Delta^\{\*\}\)\\leq\\operatorname\{tr\}\(\\Delta F\_\{X\}\\Delta^\{\*\}\)\\leq M\\operatorname\{tr\}\(\\Delta\\Delta^\{\*\}\)pointwise inω\\omega, and Parseval applied entrywise turns∫tr⁡\(Δ​Δ∗\)\\int\\operatorname\{tr\}\(\\Delta\\Delta^\{\*\}\)into∑j‖Δj‖F2\\sum\_\{j\}\\\|\\Delta\_\{j\}\\\|\_\{F\}^\{2\}\. ∎

Everything downstream uses only \([75](https://arxiv.org/html/2610.08816#A6.E75)\) and the two\-sided bound, so the scalar and matrix theories differ in bookkeeping rather than in mechanism up to this point\.

### F\.2Shared row/column factor memory

The architecturally interesting classes are factorized\. Write

Aj=∑q=1rkθq​\(j\)​Bq,Bq∈ℝd×d,A\_\{j\}=\\sum\_\{q=1\}^\{r\}k\_\{\\theta\_\{q\}\}\(j\)B\_\{q\},\\qquad B\_\{q\}\\in\\mathbb\{R\}^\{d\\times d\},\(76\)so thatrrtemporal laws are shared across all variable pairs and the cross\-variable structure lives entirely in the matricesBqB\_\{q\}\. Ifrank⁡\(Bq\)≤s\\operatorname\{rank\}\(B\_\{q\}\)\\leq sand we factorBq=Oq​VqB\_\{q\}=O\_\{q\}V\_\{q\}withOq∈ℝd×sO\_\{q\}\\in\\mathbb\{R\}^\{d\\times s\}andVq∈ℝs×dV\_\{q\}\\in\\mathbb\{R\}^\{s\\times d\}, then a causal attention head with temporal weights proportional tokθqk\_\{\\theta\_\{q\}\}, value mapVqV\_\{q\}, output mapOqO\_\{q\}, and the output multiplier that cancels the softmax normalizer produces exactly∑j≤Lkθq​\(j\)​Bq​Xt\+1−j\\sum\_\{j\\leq L\}k\_\{\\theta\_\{q\}\}\(j\)B\_\{q\}X\_\{t\+1\-j\}, and summingrrheads realizes \([76](https://arxiv.org/html/2610.08816#A6.E76)\)\. This is the multivariate analogue of the realization statement in App\.[A](https://arxiv.org/html/2610.08816#A1), and it is what makesrrthe number of heads andssthe head width\.

The most restrictive and most useful case shares the subspaces as well,

Bq=U​Cq​V⊤,U,V∈ℝd×s,B\_\{q\}=UC\_\{q\}V^\{\\top\},\\qquad U,V\\in\\mathbb\{R\}^\{d\\times s\},\(77\)so that all heads read from oness\-dimensional input subspace and write into oness\-dimensional output subspace, with the coresCq∈ℝs×sC\_\{q\}\\in\\mathbb\{R\}^\{s\\times s\}carrying the head\-specific cross\-variable mixing\.

### F\.3Fixed temporal heads

When the temporal laws are known and only the matrices are estimated, the problem is linear\. Stacking therrfiltered regressors intoZt∈ℝr​dZ\_\{t\}\\in\\mathbb\{R\}^\{rd\}and writingΘ⋆\\Theta^\{\\star\}for the true coefficient matrix, ordinary least squares satisfies

T​vec⁡\(Θ^T−Θ⋆\)⇒N⁡\(0,σ2​\(ΓZ−1⊗Id\)\)\\sqrt\{T\}\\operatorname\{vec\}\(\\widehat\{\\Theta\}\_\{T\}\-\\Theta^\{\\star\}\)\\Rightarrow N\\bigl\(0,\\sigma^\{2\}\(\\Gamma\_\{Z\}^\{\-1\}\\otimes I\_\{d\}\)\\bigr\)\(78\)under stationarity, ergodicity,ΓZ≻0\\Gamma\_\{Z\}\\succ 0, and the usual martingale Lindeberg condition\. The last three hold for a stable Gaussian vector autoregression with fixed predictively nondegenerate filters, the nondegeneracy being needed because repeated or linearly dependent temporal filters make the regressor Gram matrix singular\. Combining \([78](https://arxiv.org/html/2610.08816#A6.E78)\) with the local expansion of \([75](https://arxiv.org/html/2610.08816#A6.E75)\) gives excess forecast risk of orderr​d2​σ2/Trd^\{2\}\\sigma^\{2\}/T, which is the multivariate counterpart of the local parametric law of App\.[E](https://arxiv.org/html/2610.08816#A5)\.

More generally, for app\-parameter family that is twice differentiable with ano⁡\(‖h‖2\)o\(\\\|h\\\|^\{2\}\)remainder and an estimator obeyingT​\(η^T−η⋆\)⇒N⁡\(0,σ2​G⋆−1\)\\sqrt\{T\}\(\\widehat\{\\eta\}\_\{T\}\-\\eta\_\{\\star\}\)\\Rightarrow N\(0,\\sigma^\{2\}G\_\{\\star\}^\{\-1\}\), one getsT​ℰ​\(A⁡\(η^T\)\)⇒σ2​χp2T\\mathcal\{E\}\(A\(\\widehat\{\\eta\}\_\{T\}\)\)\\Rightarrow\\sigma^\{2\}\\chi^\{2\}\_\{p\}and, under uniform integrability of that sequence,𝔼​ℰ=p​σ2/T\+o⁡\(T−1\)\\mathbb\{E\}\\mathcal\{E\}=p\\sigma^\{2\}/T\+o\(T^\{\-1\}\)\. The whole content of the multivariate theory is therefore whichppapplies to which class\.

### F\.4Growing core and shared\-subspace lower bounds

The dimension counts are not merely upper bookkeeping\. For the shared\-factor class with transfer\-function stability marginρ\\rho, two separate packings give matching lower scales\.

Write the minimax excess forecast risk over the shared\-factor class asinfA^supA⋆𝔼​ℰ​\(A^\)\\inf\_\{\\widehat\{A\}\}\\sup\_\{A^\{\\star\}\}\\mathbb\{E\}\\,\\mathcal\{E\}\(\\widehat\{A\}\), the infimum being over estimators measurable with respect to the observed trajectory\.

###### Lemma 35\(Multivariate trajectory KL under transfer stability\)\.

Fix0<ρ<10<\\rho<1and letA,BA,Bgenerate stationary Gaussian vector autoregressions with innovationsN⁡\(0,σ2​Id\)N\(0,\\sigma^\{2\}I\_\{d\}\)whose transfer functions satisfyess​supω⁡‖𝖠⁡\(ω\)‖op≤ρ\\operatorname\{ess\\,sup\}\_\{\\omega\}\\\|\\mathsf\{A\}\(\\omega\)\\\|\_\{\\rm op\}\\leq\\rhoand likewise forBB\. Then the spectral densities obeyσ2​\(1\+ρ\)−2​Id⪯FA,FB⪯σ2​\(1−ρ\)−2​Id\\sigma^\{2\}\(1\+\\rho\)^\{\-2\}I\_\{d\}\\preceq F\_\{A\},F\_\{B\}\\preceq\\sigma^\{2\}\(1\-\\rho\)^\{\-2\}I\_\{d\}almost everywhere, and there isCρ<∞C\_\{\\rho\}<\\infty, independent ofTT,ddandσ\\sigma, with

DKL\(PA\(T\)∥PB\(T\)\)≤CρT∑j≥1∥Aj−Bj∥F2\.D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{A\}^\{\(T\)\}\\\|P\_\{B\}^\{\(T\)\}\\bigr\)\\ \\leq\\ C\_\{\\rho\}\\,T\\sum\_\{j\\geq 1\}\\\|A\_\{j\}\-B\_\{j\}\\\|\_\{F\}^\{2\}\.\(79\)The bound carries no factor ofσ\\sigma, which is forced: rescalingXt↦Xt/σX\_\{t\}\\mapsto X\_\{t\}/\\sigmaleaves the coefficients unchanged and pushes both trajectory laws forward by one common invertible map, so their relative entropy is unchanged while anyσ\\sigma\-dependent right\-hand side would not be\.

###### Proof\.

WriteDA​\(ω\)=I−𝖠⁡\(ω\)D\_\{A\}\(\\omega\)=I\-\\mathsf\{A\}\(\\omega\)\. Stability gives‖DA‖op≤1\+ρ\\\|D\_\{A\}\\\|\_\{\\rm op\}\\leq 1\+\\rhoandsmin​\(DA\)≥1−ρs\_\{\\min\}\(D\_\{A\}\)\\geq 1\-\\rho, hence‖DA−1‖op≤\(1−ρ\)−1\\\|D\_\{A\}^\{\-1\}\\\|\_\{\\rm op\}\\leq\(1\-\\rho\)^\{\-1\}, and the same forDBD\_\{B\}\. SinceFA=σ2​DA−1​DA−⁣∗F\_\{A\}=\\sigma^\{2\}D\_\{A\}^\{\-1\}D\_\{A\}^\{\-\*\}, these singular\-value bounds give the stated two\-sided spectral bounds\.

LetΣA,T\\Sigma\_\{A,T\}andΣB,T\\Sigma\_\{B,T\}be thed​T×d​TdT\\times dTblock\-Toeplitz covariance matrices of the two length\-TTtrajectories\. Matrix Parseval and the spectral bounds givemρ​I⪯ΣA,T,ΣB,T⪯Mρ​Im\_\{\\rho\}I\\preceq\\Sigma\_\{A,T\},\\Sigma\_\{B,T\}\\preceq M\_\{\\rho\}Iwithmρ=σ2​\(1\+ρ\)−2m\_\{\\rho\}=\\sigma^\{2\}\(1\+\\rho\)^\{\-2\}andMρ=σ2​\(1−ρ\)−2M\_\{\\rho\}=\\sigma^\{2\}\(1\-\\rho\)^\{\-2\}\. PutR=ΣB,T−1/2ΣA,TΣB,T−1/2R=\\Sigma\_\{B,T\}^\{\-1/2\}\\Sigma\_\{A,T\}\\Sigma\_\{B,T\}^\{\-1/2\}, whose eigenvalues lie in the fixed compact interval\[mρ/Mρ,Mρ/mρ\]\[m\_\{\\rho\}/M\_\{\\rho\},\\,M\_\{\\rho\}/m\_\{\\rho\}\], an interval depending onρ\\rhoalone\. On such an interval the centered Gaussian relative entropy12​∑i\(λi−1−log⁡λi\)\\tfrac\{1\}\{2\}\\sum\_\{i\}\(\\lambda\_\{i\}\-1\-\\log\\lambda\_\{i\}\)is bounded by a constant multiple of∑i\(λi−1\)2=‖R−I‖F2\\sum\_\{i\}\(\\lambda\_\{i\}\-1\)^\{2\}=\\\|R\-I\\\|\_\{F\}^\{2\}, so

DKL\(PA\(T\)∥PB\(T\)\)≤Cρ∥ΣB,T−1/2\(ΣA,T−ΣB,T\)ΣB,T−1/2∥F2≤Cρmρ2∥ΣA,T−ΣB,T∥F2\.D\_\{\\mathrm\{KL\}\}\\bigl\(P\_\{A\}^\{\(T\)\}\\\|P\_\{B\}^\{\(T\)\}\\bigr\)\\ \\leq\\ C\_\{\\rho\}\\,\\bigl\\\|\\Sigma\_\{B,T\}^\{\-1/2\}\(\\Sigma\_\{A,T\}\-\\Sigma\_\{B,T\}\)\\Sigma\_\{B,T\}^\{\-1/2\}\\bigr\\\|\_\{F\}^\{2\}\\ \\leq\\ \\frac\{C\_\{\\rho\}\}\{m\_\{\\rho\}^\{2\}\}\\,\\\|\\Sigma\_\{A,T\}\-\\Sigma\_\{B,T\}\\\|\_\{F\}^\{2\}\.This is where the two powers ofσ\\sigmacancel\. The spectral\-density difference satisfies‖FA−FB‖F≤Cρ​σ2​‖𝖠−𝖡‖F\\\|F\_\{A\}\-F\_\{B\}\\\|\_\{F\}\\leq C\_\{\\rho\}\\sigma^\{2\}\\\|\\mathsf\{A\}\-\\mathsf\{B\}\\\|\_\{F\}pointwise, by writingFA−FB=σ2​\{DA−1​\(DB−DA\)​DB−1​DB−⁣∗\+DA−1​DA−⁣∗​\(DB−DA\)∗​DB−⁣∗\}F\_\{A\}\-F\_\{B\}=\\sigma^\{2\}\\\{D\_\{A\}^\{\-1\}\(D\_\{B\}\-D\_\{A\}\)D\_\{B\}^\{\-1\}D\_\{B\}^\{\-\*\}\+D\_\{A\}^\{\-1\}D\_\{A\}^\{\-\*\}\(D\_\{B\}\-D\_\{A\}\)^\{\*\}D\_\{B\}^\{\-\*\}\\\}and using the uniform inverse bounds, so Toeplitz Frobenius comparison and Parseval give‖ΣA,T−ΣB,T‖F2≤Cρ​T​σ4​∑j‖Aj−Bj‖F2\\\|\\Sigma\_\{A,T\}\-\\Sigma\_\{B,T\}\\\|\_\{F\}^\{2\}\\leq C\_\{\\rho\}T\\sigma^\{4\}\\sum\_\{j\}\\\|A\_\{j\}\-B\_\{j\}\\\|\_\{F\}^\{2\}\. Dividing bymρ2≍σ4m\_\{\\rho\}^\{2\}\\asymp\\sigma^\{4\}cancels theσ4\\sigma^\{4\}exactly and leaves \([79](https://arxiv.org/html/2610.08816#A6.E79)\)\. ∎

###### Lemma 36\(Flat core packing\)\.

Fixr,s≥1r,s\\geq 1and letp0=r​s2p\_\{0\}=rs^\{2\}\. There is a subset𝒱⊂\{−1,1\}p0\\mathcal\{V\}\\subset\\\{\-1,1\\\}^\{p\_\{0\}\}, reshaped into sign cores\(S1\(v\),…,Sr\(v\)\)\(S\_\{1\}^\{\(v\)\},\\dots,S\_\{r\}^\{\(v\)\}\)withSq\(v\)∈\{−1,1\}s×sS\_\{q\}^\{\(v\)\}\\in\\\{\-1,1\\\}^\{s\\times s\}, such that

log\|𝒱\|≥cp0,dH\(u,v\)≥p0/4\(u≠v\),\\log\|\\mathcal\{V\}\|\\ \\geq\\ c\\,p\_\{0\},\\qquad d\_\{H\}\(u,v\)\\ \\geq\\ p\_\{0\}/4\\quad\(u\\neq v\),\(80\)and everyv∈𝒱v\\in\\mathcal\{V\}satisfies the flatness bound

supω‖∑q≤rSq\(v\)​e−i​q​ω‖op≤C​r​\{s\+log⁡\(e​r​s\)\}\.\\sup\_\{\\omega\}\\Bigl\\\|\\sum\_\{q\\leq r\}S\_\{q\}^\{\(v\)\}e^\{\-iq\\omega\}\\Bigr\\\|\_\{\\rm op\}\\ \\leq\\ C\\sqrt\{r\\\{s\+\\log\(ers\)\\\}\}\.\(81\)

###### Proof\.

Draw allp0p\_\{0\}signs independently and uniformly, and writeP⁡\(ω\)=∑q≤rSq​e−i​q​ωP\(\\omega\)=\\sum\_\{q\\leq r\}S\_\{q\}e^\{\-iq\\omega\}\. Fix unit vectorsx,y∈ℝsx,y\\in\\mathbb\{R\}^\{s\}\. The real and imaginary parts ofx⊤​P​\(ω\)​yx^\{\\top\}P\(\\omega\)yare sums of independent centered terms with squared coefficient sums at mostrr, socosh⁡t≤et2/2\\cosh t\\leq e^\{t^\{2\}/2\}and Chernoff givePr\{\|x⊤P\(ω\)y\|\>u\}≤4e−u2/\(8r\)\\Pr\\\{\|x^\{\\top\}P\(\\omega\)y\|\>u\\\}\\leq 4e^\{\-u^\{2\}/\(8r\)\}\. A1/41/4\-net of the sphere inℝs\\mathbb\{R\}^\{s\}has at most9s9^\{s\}points and converts the bilinear bound into an operator\-norm bound at the cost of a factor22and a union over81s81^\{s\}pairs\.

Uniformity inω\\omeganeeds the derivative, and the mesh must be chosen against it rather than guessed\. Since‖dd​ω​P​\(ω\)‖op≤∑q≤rq​‖Sq‖op≤s​r2\\\|\\tfrac\{d\}\{d\\omega\}P\(\\omega\)\\\|\_\{\\rm op\}\\leq\\sum\_\{q\\leq r\}q\\\|S\_\{q\}\\\|\_\{\\rm op\}\\leq sr^\{2\}, a uniform grid of meshh=u/\(2​s​r2\)h=u/\(2sr^\{2\}\)has at most1\+C​s​r2/u1\+Csr^\{2\}/upoints, and ifsupω‖P‖op\>2​u\\sup\_\{\\omega\}\\\|P\\\|\_\{\\rm op\}\>2uthen some grid value exceedsuu\. A union bound over the grid and the net gives

Pr\{supω∥P\(ω\)∥op\>2u\}≤C\(1\+s​r2u\)81se−u2/\(8r\)\.\\Pr\\Bigl\\\{\\sup\_\{\\omega\}\\\|P\(\\omega\)\\\|\_\{\\rm op\}\>2u\\Bigr\\\}\\leq C\\Bigl\(1\+\\frac\{sr^\{2\}\}\{u\}\\Bigr\)81^\{s\}e^\{\-u^\{2\}/\(8r\)\}\.Takingu=C0​r​\{s\+log⁡\(e​r​s\)\}u=C\_\{0\}\\sqrt\{r\\\{s\+\\log\(ers\)\\\}\}withC0C\_\{0\}large makes the right side at most1/41/4\.

Hence at least three quarters of the2p02^\{p\_\{0\}\}sign vectors satisfy \([81](https://arxiv.org/html/2610.08816#A6.E81)\)\. The packing must now be built inside that good subset, since a code chosen first need not be flat\. Apply greedy Hamming packing within the flat subset, repeatedly selecting a flat vector and deleting all flat vectors within Hamming radiusp0/4p\_\{0\}/4\. Each deletion removes at most∑i≤p0/4\(p0i\)≤2H⁡\(1/4\)​p0\\sum\_\{i\\leq p\_\{0\}/4\}\\binom\{p\_\{0\}\}\{i\}\\leq 2^\{H\(1/4\)p\_\{0\}\}vectors withH⁡\(1/4\)<1H\(1/4\)<1, while the flat subset has at least34​2p0\\tfrac\{3\}\{4\}2^\{p\_\{0\}\}, so exponentially many selections remain and \([80](https://arxiv.org/html/2610.08816#A6.E80)\) holds\. ∎

The two lemmas calibrate against each other, which is what turns the packing into a rate\. Give each core amplitudeϵ\\epsilonwith

ϵ2=cϵT,\\epsilon^\{2\}=\\frac\{c\_\{\\epsilon\}\}\{T\},\(82\)and setAq\(v\)=ϵ​U0​Sq\(v\)​V0⊤A\_\{q\}^\{\(v\)\}=\\epsilon\\,U\_\{0\}S\_\{q\}^\{\(v\)\}V\_\{0\}^\{\\top\}for fixed orthonormalU0,V0U\_\{0\},V\_\{0\}\. The flatness bound \([81](https://arxiv.org/html/2610.08816#A6.E81)\) makes the transfer function of every packing point at mostC​ϵ​r​\{s\+log⁡\(e​r​s\)\}C\\epsilon\\sqrt\{r\\\{s\+\\log\(ers\)\\\}\}in operator norm, a quantity free ofσ\\sigma, so choosingcϵc\_\{\\epsilon\}small keeps the whole packing inside the stability class once

T≥C​r​\{s\+log⁡\(e​r​s\)\}\.T\\ \\geq\\ C\\,r\\\{s\+\\log\(ers\)\\\}\.\(83\)The scale chain is worth following, sinceσ\\sigmaenters in exactly one place\. The Hamming separation in \([80](https://arxiv.org/html/2610.08816#A6.E80)\) gives coefficient separation∑q‖Aq\(u\)−Aq\(v\)‖F2≥4​ϵ2​p0/4=ϵ2​r​s2≍r​s2/T\\sum\_\{q\}\\\|A\_\{q\}^\{\(u\)\}\-A\_\{q\}^\{\(v\)\}\\\|\_\{F\}^\{2\}\\geq 4\\epsilon^\{2\}p\_\{0\}/4=\\epsilon^\{2\}rs^\{2\}\\asymp rs^\{2\}/T, which carries noσ\\sigma\. The lower spectral boundFX⪰σ2​\(1\+ρ\)−2​IF\_\{X\}\\succeq\\sigma^\{2\}\(1\+\\rho\)^\{\-2\}Iin Thm\.[34](https://arxiv.org/html/2610.08816#Thmtheorem34)is what converts that into predictive separation of orderσ2​r​s2/T\\sigma^\{2\}rs^\{2\}/T, so the innovation scale enters through the geometry rather than through the construction\. On the other side, Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)bounds the relative entropy between any packing point and the centre byCρ​T​ϵ2​r​s2=Cρ​cϵ​r​s2C\_\{\\rho\}T\\epsilon^\{2\}rs^\{2\}=C\_\{\\rho\}c\_\{\\epsilon\}rs^\{2\}, which is at most a small multiple oflog⁡\|𝒱\|≥c​r​s2\\log\|\\mathcal\{V\}\|\\geq crs^\{2\}oncecϵc\_\{\\epsilon\}is small\. Fano’s inequality therefore gives

infA^supA⋆𝔼​ℰ​\(A^\)≥c​σ2​r​s2T\.\\inf\_\{\\widehat\{A\}\}\\sup\_\{A^\{\\star\}\}\\mathbb\{E\}\\,\\mathcal\{E\}\(\\widehat\{A\}\)\\ \\geq\\ c\\sigma^\{2\}\\frac\{rs^\{2\}\}\{T\}\.\(84\)
For the shared subspace we use a local perturbation rather than an external Grassmann packing, which keeps the construction self\-contained and makes the calibration explicit\.

###### Theorem 37\(Shared\-subspace lower bound\)\.

Assume1≤s<d1\\leq s<d, with the supremum ranging over the stable shared\-subspace class of App\.[F](https://arxiv.org/html/2610.08816#A6)at transfer stability constantρ\\rho\. There arec,C\>0c,C\>0depending only onρ\\rhosuch that ifT≥C​s​\(d−s\)T\\geq Cs\(d\-s\)then

infA^supA⋆𝔼​ℰ​\(A^\)≥c​σ2​s⁡\(d−s\)T\.\\inf\_\{\\widehat\{A\}\}\\sup\_\{A^\{\\star\}\}\\mathbb\{E\}\\,\\mathcal\{E\}\(\\widehat\{A\}\)\\ \\geq\\ c\\sigma^\{2\}\\frac\{s\(d\-s\)\}\{T\}\.\(85\)

###### Proof\.

Fix orthonormalU0∈ℝd×sU\_\{0\}\\in\\mathbb\{R\}^\{d\\times s\},U⟂∈ℝd×\(d−s\)U\_\{\\perp\}\\in\\mathbb\{R\}^\{d\\times\(d\-s\)\}withU0⊤​U⟂=0U\_\{0\}^\{\\top\}U\_\{\\perp\}=0, andV0∈ℝd×sV\_\{0\}\\in\\mathbb\{R\}^\{d\\times s\}, and putp1=s⁡\(d−s\)p\_\{1\}=s\(d\-s\)\. ForX∈ℝ\(d−s\)×sX\\in\\mathbb\{R\}^\{\(d\-s\)\\times s\}setU⁡\(X\)=U0\+U⟂​XU\(X\)=U\_\{0\}\+U\_\{\\perp\}X, which has full column rank sinceU​\(X\)⊤​U​\(X\)=Is\+X⊤​X≻0U\(X\)^\{\\top\}U\(X\)=I\_\{s\}\+X^\{\\top\}X\\succ 0\. Fix0<γ<ρ/40<\\gamma<\\rho/4and use the one\-active\-lag subclassA1​\(X\)=γ​U​\(X\)​V0⊤A\_\{1\}\(X\)=\\gamma U\(X\)V\_\{0\}^\{\\top\}withAj​\(X\)=0A\_\{j\}\(X\)=0forj≥2j\\geq 2\.

Let𝒱p1⊂\{−1,1\}p1\\mathcal\{V\}\_\{p\_\{1\}\}\\subset\\\{\-1,1\\\}^\{p\_\{1\}\}be a Gilbert–Varshamov packing withlog⁡\|𝒱p1\|≥c​p1\\log\|\\mathcal\{V\}\_\{p\_\{1\}\}\|\\geq cp\_\{1\}and pairwise Hamming distance at leastp1/4p\_\{1\}/4, reshapevvintoS\(v\)∈\{−1,1\}\(d−s\)×sS^\{\(v\)\}\\in\\\{\-1,1\\\}^\{\(d\-s\)\\times s\}, and setX\(v\)=δ​S\(v\)X^\{\(v\)\}=\\delta S^\{\(v\)\}withδ2=cδ/T\\delta^\{2\}=c\_\{\\delta\}/T\. UnderT≥C​s​\(d−s\)T\\geq Cs\(d\-s\)we have‖X\(v\)‖op≤‖X\(v\)‖F=δ​p1\\\|X^\{\(v\)\}\\\|\_\{\\rm op\}\\leq\\\|X^\{\(v\)\}\\\|\_\{F\}=\\delta\\sqrt\{p\_\{1\}\}uniformly small, so every member stays inside the stability class and has one active lag, hence transfer norm at most2​γ<ρ2\\gamma<\\rho\.

BecauseU⟂U\_\{\\perp\}has orthonormal columns andV0V\_\{0\}is fixed,

‖A1​\(X\(u\)\)−A1​\(X\(v\)\)‖F2=γ2​‖U⟂​\(X\(u\)−X\(v\)\)​V0⊤‖F2=γ2​δ2⋅4​dH​\(u,v\)≥γ2​δ2​p1,\\bigl\\\|A\_\{1\}\(X^\{\(u\)\}\)\-A\_\{1\}\(X^\{\(v\)\}\)\\bigr\\\|\_\{F\}^\{2\}=\\gamma^\{2\}\\bigl\\\|U\_\{\\perp\}\(X^\{\(u\)\}\-X^\{\(v\)\}\)V\_\{0\}^\{\\top\}\\bigr\\\|\_\{F\}^\{2\}=\\gamma^\{2\}\\delta^\{2\}\\cdot 4\\,d\_\{H\}\(u,v\)\\ \\geq\\ \\gamma^\{2\}\\delta^\{2\}p\_\{1\},which is of orders⁡\(d−s\)/Ts\(d\-s\)/T, and Thm\.[34](https://arxiv.org/html/2610.08816#Thmtheorem34)converts it into predictive separation of the same order\. Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)givesDKL≤Cρ​T​γ2​δ2​p1=Cρ​γ2​cδ​p1D\_\{\\mathrm\{KL\}\}\\leq C\_\{\\rho\}T\\gamma^\{2\}\\delta^\{2\}p\_\{1\}=C\_\{\\rho\}\\gamma^\{2\}c\_\{\\delta\}p\_\{1\}, at most a small multiple oflog⁡\|𝒱p1\|≥c​p1\\log\|\\mathcal\{V\}\_\{p\_\{1\}\}\|\\geq cp\_\{1\}oncecδc\_\{\\delta\}is small\. Fano’s inequality gives \([85](https://arxiv.org/html/2610.08816#A6.E85)\)\. ∎

The same local construction with the roles of the left and right subspaces interchanged, perturbingVVwhile holdingUUfixed, gives the same lower bound \([85](https://arxiv.org/html/2610.08816#A6.E85)\) for the input subspace\. Thus the core, output\-subspace, and input\-subspace packings give three valid lower bounds of respective ordersr​s2/Trs^\{2\}/T,s⁡\(d−s\)/Ts\(d\-s\)/T, ands⁡\(d−s\)/Ts\(d\-s\)/T\. Since a maximum dominates a third of their sum, under

T≳r⁡\{s\+log⁡\(e​r​s\)\}\+s⁡\(d−s\)T\\ \\gtrsim\\ r\\\{s\+\\log\(ers\)\\\}\+s\(d\-s\)\(86\)we obtain

infA^supA⋆𝔼​ℰ​\(A^\)≥c​σ2​2​s​\(d−s\)\+r​s2T\.\\inf\_\{\\widehat\{A\}\}\\sup\_\{A^\{\\star\}\}\\mathbb\{E\}\\,\\mathcal\{E\}\(\\widehat\{A\}\)\\ \\geq\\ c\\sigma^\{2\}\\frac\{2s\(d\-s\)\+rs^\{2\}\}\{T\}\.\(87\)No independent packing over the temporal parameter space is used here, so \([87](https://arxiv.org/html/2610.08816#A6.E87)\) does not claim an additionalr​pθrp\_\{\\theta\}lower\-bound term\. That term enters the regular\-chart parameter count below; whenpθp\_\{\\theta\}is fixed ands≥1s\\geq 1, it is bounded by a fixed multiple ofr​s2rs^\{2\}and therefore does not change the order comparison\. Condition \([86](https://arxiv.org/html/2610.08816#A6.E86)\) is the sharper of the thresholds the three packings need, and it is what the flat construction of Lem\.[36](https://arxiv.org/html/2610.08816#Thmtheorem36)buys relative to a naive bound of orderr2​s2r^\{2\}s^\{2\}\. These are nonasymptotic same\-realization statements in whichdd,rr, andssare allowed to grow\.

### F\.5Learned smooth temporal heads on a regular chart

When the temporal laws are learned rather than fixed, the matching upper bound is available on charts that are uniformly well conditioned\.

Write𝒱Φ\\mathcal\{V\}\_\{\\Phi\}for the normalized two\-secant set of the chart and𝒱Φ0:=𝒱Φ∪\{0\}\\mathcal\{V\}\_\{\\Phi\}^\{0\}:=\\mathcal\{V\}\_\{\\Phi\}\\cup\\\{0\\\}\. The chaining proof also needs normalized differences of such directions, so define the normalized increment base

𝒮Φ:=\{u−vℰ⁡\(u−v\):u,v∈𝒱Φ0,u≠v\},𝒮Φ0:=𝒮Φ∪\{0\}\.\\mathcal\{S\}\_\{\\Phi\}:=\\left\\\{\\frac\{u\-v\}\{\\sqrt\{\\mathcal\{E\}\(u\-v\)\}\}:\\ u,v\\in\\mathcal\{V\}\_\{\\Phi\}^\{0\},\\ u\\neq v\\right\\\},\\qquad\\mathcal\{S\}\_\{\\Phi\}^\{0\}:=\\mathcal\{S\}\_\{\\Phi\}\\cup\\\{0\\\}\.In particular𝒱Φ⊂𝒮Φ\\mathcal\{V\}\_\{\\Phi\}\\subset\\mathcal\{S\}\_\{\\Phi\}by taking one endpoint to be the adjoined origin\. Call\(Φ,Θ\)\(\\Phi,\\Theta\)a uniformly regular predictive chart if the normalized increment base has entropy

log⁡N⁡\(𝒮Φ,ϵ\)≤C0​p​log⁡\(C0/ϵ\)\\log N\(\\mathcal\{S\}\_\{\\Phi\},\\epsilon\)\\leq C\_\{0\}p\\log\(C\_\{0\}/\\epsilon\)in the predictive metric and the transfer envelope holds on one further difference closure,

∑j≤L‖Wj‖op≤Λσ​ℰ⁡\(W\)for every nonzero​W∈cone⁡\(𝒮Φ0−𝒮Φ0\)\.\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{\\rm op\}\\leq\\frac\{\\Lambda\}\{\\sigma\}\\sqrt\{\\mathcal\{E\}\(W\)\}\\qquad\\text\{for every nonzero \}W\\in\\operatorname\{cone\}\(\\mathcal\{S\}\_\{\\Phi\}^\{0\}\-\\mathcal\{S\}\_\{\\Phi\}^\{0\}\)\.\(88\)The entropy assumption is therefore imposed directly on the normalized base consumed by the quadratic concentration argument; it is not inferred by normalizing an unnormalized difference set\. The extra difference closure in \([88](https://arxiv.org/html/2610.08816#A6.E88)\) supplies the operator\-metric Lipschitz bound between two points of𝒮Φ\\mathcal\{S\}\_\{\\Phi\}\. Both conditions are imposed on the image geometry rather than on raw coordinates, so they are invariant under reparameterization and separate genuine statistical difficulty from a poor choice of head coordinates\.

The envelope constant records any context\-length dependence rather than hiding it in the theorem\. Indeed, under the transfer stability of Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35), the spectral lower bound and Cauchy–Schwarz give, for every length\-LLperturbation,

∑j≤L‖Wj‖op≤L​\(∑j≤L‖Wj‖F2\)1/2≤\(1\+ρ\)​Lσ​ℰ⁡\(W\),\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{\\rm op\}\\leq\\sqrt\{L\}\\left\(\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{F\}^\{2\}\\right\)^\{1/2\}\\leq\\frac\{\(1\+\\rho\)\\sqrt\{L\}\}\{\\sigma\}\\sqrt\{\\mathcal\{E\}\(W\)\},so one may always takeΛ≤\(1\+ρ\)​L\\Lambda\\leq\(1\+\\rho\)\\sqrt\{L\}on a finite context\. Thus the generic sample condition below can inflate to orderL⁡\(p\+x\)L\(p\+x\)\. For structured temporal spans whose normalized secant differences have a uniformly boundedℓ1/ℓ2\\ell\_\{1\}/\\ell\_\{2\}ratio, including uniformly exponentially localized spans,Λ=O⁡\(1\)\\Lambda=O\(1\); for long\-memory or power\-law heads it may grow withLL, and the theorem retains that dependence explicitly\.

Writedt​\(D\)=∑j≤LDj​Xt\+1−jd\_\{t\}\(D\)=\\sum\_\{j\\leq L\}D\_\{j\}X\_\{t\+1\-j\}fort=L,…,T−1t=L,\\ldots,T\-1and putn=T−Ln=T\-L\. Thendt​\(D\)d\_\{t\}\(D\)isℱt\\mathcal\{F\}\_\{t\}\-measurable and the innovation paired with it in the empirical one\-step loss isεt\+1\\varepsilon\_\{t\+1\}\.

###### Lemma 38\(Uniform same\-realization concentration on a regular chart\)\.

Assume the stable Gaussian vector autoregression of Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)together with the entropy and envelope conditions defining a uniformly regular chart\. There areC,cC,cdepending only onρ\\rhoandC0C\_\{0\}such that, for everyx≥1x\\geq 1, ifn≥C​Λ2​\(p\+x\)n\\geq C\\Lambda^\{2\}\(p\+x\), then with probability at least1−C​e−c​x1\-Ce^\{\-cx\}the following hold simultaneously for everyD∈cone⁡\(𝒮Φ\)D\\in\\operatorname\{cone\}\(\\mathcal\{S\}\_\{\\Phi\}\)and everyAAin the chart:

12​n​ℰ​\(D\)≤∑t=LT−1‖dt​\(D\)‖2≤32​n​ℰ​\(D\),\\tfrac\{1\}\{2\}n\\,\\mathcal\{E\}\(D\)\\ \\leq\\ \\sum\_\{t=L\}^\{T\-1\}\\\|d\_\{t\}\(D\)\\\|^\{2\}\\ \\leq\\ \\tfrac\{3\}\{2\}n\\,\\mathcal\{E\}\(D\),\(89\)2​\|∑t=LT−1⟨εt\+1,dt​\(A−A⋆\)⟩\|≤14​n​ℰ​\(A\)\+C​σ2​\(p\+x\)\.2\\Bigl\|\\sum\_\{t=L\}^\{T\-1\}\\langle\\varepsilon\_\{t\+1\},d\_\{t\}\(A\-A^\{\\star\}\)\\rangle\\Bigr\|\\ \\leq\\ \\tfrac\{1\}\{4\}n\\,\\mathcal\{E\}\(A\)\+C\\sigma^\{2\}\(p\+x\)\.\(90\)

###### Proof\.

No truncation of the infinite innovation sequence is needed for the curvature term, because the observed block is already finite dimensional and can be whitened directly\. Stack the observations intoXblock∈ℝd​TX\_\{\\rm block\}\\in\\mathbb\{R\}^\{dT\}and writeXblock=ΣA⋆,T1/2​ξX\_\{\\rm block\}=\\Sigma\_\{A^\{\\star\},T\}^\{1/2\}\\xiwithξ∼N⁡\(0,Id​T\)\\xi\\sim N\(0,I\_\{dT\}\), which is legitimate sinceΣA⋆,T≻0\\Sigma\_\{A^\{\\star\},T\}\\succ 0by Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)\. For a perturbationDDlet𝒯D\\mathcal\{T\}\_\{D\}be the linear map sending the block to the stacked residual differences, so that𝐝⁡\(D\)=𝒯D​ΣA⋆,T1/2​ξ\\mathbf\{d\}\(D\)=\\mathcal\{T\}\_\{D\}\\Sigma\_\{A^\{\\star\},T\}^\{1/2\}\\xiand

∑t=LT−1‖dt​\(D\)‖2=‖BD​ξ‖2,BD:=𝒯D​ΣA⋆,T1/2\.\\sum\_\{t=L\}^\{T\-1\}\\\|d\_\{t\}\(D\)\\\|^\{2\}=\\bigl\\\|B\_\{D\}\\,\\xi\\bigr\\\|^\{2\},\\qquad B\_\{D\}:=\\mathcal\{T\}\_\{D\}\\Sigma\_\{A^\{\\star\},T\}^\{1/2\}\.This is a finite\-dimensional Gaussian chaos family\. ForD∈𝒮ΦD\\in\\mathcal\{S\}\_\{\\Phi\}, stationarity gives‖BD‖F2=n\\\|B\_\{D\}\\\|\_\{F\}^\{2\}=n, while forD,D′∈𝒮ΦD,D^\{\\prime\}\\in\\mathcal\{S\}\_\{\\Phi\}the differenceD−D′D\-D^\{\\prime\}lies in𝒮Φ0−𝒮Φ0\\mathcal\{S\}\_\{\\Phi\}^\{0\}\-\\mathcal\{S\}\_\{\\Phi\}^\{0\}and \([88](https://arxiv.org/html/2610.08816#A6.E88)\) gives

‖BD−BD′‖op≤\(1−ρ\)−1​Λ​ℰ⁡\(D−D′\)\.\\\|B\_\{D\}\-B\_\{D^\{\\prime\}\}\\\|\_\{\\rm op\}\\leq\(1\-\\rho\)^\{\-1\}\\Lambda\\sqrt\{\\mathcal\{E\}\(D\-D^\{\\prime\}\)\}\.The entropy hypothesis is already stated on𝒮Φ\\mathcal\{S\}\_\{\\Phi\}, so Dudley’s integral givesγ2≲\(1−ρ\)−1​Λ​p\\gamma\_\{2\}\\lesssim\(1\-\\rho\)^\{\-1\}\\Lambda\\sqrt\{p\}and the operator diameter isOρ​\(Λ\)O\_\{\\rho\}\(\\Lambda\)\. Substituting these quantities and Frobenius radiusn\\sqrt\{n\}into the suprema\-of\-chaos bound of[Krahmer et al\. \(2014\)](https://arxiv.org/html/2610.08816#bib.bib31)shows that the relative deviation is at most1/21/2on𝒮Φ\\mathcal\{S\}\_\{\\Phi\}oncen≥C​Λ2​\(p\+x\)n\\geq C\\Lambda^\{2\}\(p\+x\), with failure probability at mostC​e−c​xCe^\{\-cx\}\. Homogeneity then gives \([89](https://arxiv.org/html/2610.08816#A6.E89)\) oncone⁡\(𝒮Φ\)\\operatorname\{cone\}\(\\mathcal\{S\}\_\{\\Phi\}\)\.

For \([90](https://arxiv.org/html/2610.08816#A6.E90)\), the corrected one\-step indexing makes the summands⟨εt\+1,dt​\(A−A⋆\)⟩\\langle\\varepsilon\_\{t\+1\},d\_\{t\}\(A\-A^\{\\star\}\)\\ranglea martingale difference sequence:dtd\_\{t\}isℱt\\mathcal\{F\}\_\{t\}\-measurable and𝔼⁡\[εt\+1∣ℱt\]=0\\mathbb\{E\}\[\\varepsilon\_\{t\+1\}\\mid\\mathcal\{F\}\_\{t\}\]=0with conditional covarianceσ2​Id\\sigma^\{2\}I\_\{d\}\. The self\-normalized inequality of[de la Peña et al\. \(2009\)](https://arxiv.org/html/2610.08816#bib.bib13)bounds a fixed direction by its predictable quadratic variationσ2​∑t‖dt‖2\\sigma^\{2\}\\sum\_\{t\}\\\|d\_\{t\}\\\|^\{2\}\. To make this uniform, chain only over the normalized two\-secant set𝒱Φ⊂𝒮Φ\\mathcal\{V\}\_\{\\Phi\}\\subset\\mathcal\{S\}\_\{\\Phi\}\. Fors≥0s\\geq 0let𝒩s\\mathcal\{N\}\_\{s\}be a2−s2^\{\-s\}\-net of𝒱Φ\\mathcal\{V\}\_\{\\Phi\}in the predictive metric; since𝒱Φ⊂𝒮Φ\\mathcal\{V\}\_\{\\Phi\}\\subset\\mathcal\{S\}\_\{\\Phi\}, the entropy bound giveslog⁡\|𝒩s\|≤C​p​\(s\+1\)\\log\|\\mathcal\{N\}\_\{s\}\|\\leq Cp\(s\+1\)\. For a normalized directionWW, choose nearest pointsπs​\(W\)∈𝒩s\\pi\_\{s\}\(W\)\\in\\mathcal\{N\}\_\{s\}\. Each linkπs​\(W\)−πs−1​\(W\)\\pi\_\{s\}\(W\)\-\\pi\_\{s\-1\}\(W\)is a difference of two members of𝒱Φ0\\mathcal\{V\}\_\{\\Phi\}^\{0\}; after normalization it belongs to𝒮Φ\\mathcal\{S\}\_\{\\Phi\}by definition, so the curvature event \([89](https://arxiv.org/html/2610.08816#A6.E89)\) applies to every link by homogeneity and bounds its predictable quadratic variation byC​σ2​n​4−sC\\sigma^\{2\}n4^\{\-s\}\. Assigning thresholdC​σ​n​2−s​p⁡\(s\+1\)\+x\+sC\\sigma\\sqrt\{n\}\\,2^\{\-s\}\\sqrt\{p\(s\+1\)\+x\+s\}at levelss, the self\-normalized inequality followed by a union bound over the at most\|𝒩s\|​\|𝒩s−1\|\|\\mathcal\{N\}\_\{s\}\|\|\\mathcal\{N\}\_\{s\-1\}\|links leaves failure probabilityC​e−c​x−c​sCe^\{\-cx\-cs\}\. Summing overssand using∑s2−s​s\+1<∞\\sum\_\{s\}2^\{\-s\}\\sqrt\{s\+1\}<\\inftygives

supW∈𝒱Φ\|∑t=LT−1⟨εt\+1,dt​\(W\)⟩\|≲σ​n⁡\(p\+x\)\.\\sup\_\{W\\in\\mathcal\{V\}\_\{\\Phi\}\}\\left\|\\sum\_\{t=L\}^\{T\-1\}\\langle\\varepsilon\_\{t\+1\},d\_\{t\}\(W\)\\rangle\\right\|\\ \\lesssim\\ \\sigma\\sqrt\{n\(p\+x\)\}\.RescalingW=\(A−A⋆\)/ℰ⁡\(A\)W=\(A\-A^\{\\star\}\)/\\sqrt\{\\mathcal\{E\}\(A\)\}gives a uniform bound of orderσ2​n​ℰ​\(A\)​\(p\+x\)\\sqrt\{\\sigma^\{2\}n\\,\\mathcal\{E\}\(A\)\(p\+x\)\}, and2​P​Q≤14​P\+4​Q2\\sqrt\{PQ\}\\leq\\tfrac\{1\}\{4\}P\+4QwithP=n​ℰ​\(A\)P=n\\mathcal\{E\}\(A\)yields \([90](https://arxiv.org/html/2610.08816#A6.E90)\)\. ∎

###### Theorem 39\(Nonasymptotic ERM on a regular chart\)\.

Under stable Gaussian vector autoregression and true transfer stability, letη^\\widehat\{\\eta\}minimize the empirical criterion over a uniformly regular chart containingA⋆A^\{\\star\}, putA^=A⁡\(η^\)\\widehat\{A\}=A\(\\widehat\{\\eta\}\), and putn=T−Ln=T\-L\. There arec,Cc,Cdepending only on the stability margin andC0C\_\{0\}such that, for everyx≥1x\\geq 1, ifn≥C​Λ2​\(p\+x\)n\\geq C\\Lambda^\{2\}\(p\+x\)then, with probability at least1−C​e−c​x1\-Ce^\{\-cx\},

ℰ⁡\(A^\)≤C​σ2​p\+xn\.\\mathcal\{E\}\(\\widehat\{A\}\)\\leq C\\sigma^\{2\}\\frac\{p\+x\}\{n\}\.\(91\)

###### Proof\.

The proof now uses exactly the two events of Lem\.[38](https://arxiv.org/html/2610.08816#Thmtheorem38)\. BecauseA⋆A^\{\\star\}andA^\\widehat\{A\}lie in the chart, the normalized direction\(A^−A⋆\)/ℰ⁡\(A^\)\(\\widehat\{A\}\-A^\{\\star\}\)/\\sqrt\{\\mathcal\{E\}\(\\widehat\{A\}\)\}belongs to𝒱Φ⊂𝒮Φ\\mathcal\{V\}\_\{\\Phi\}\\subset\\mathcal\{S\}\_\{\\Phi\}, so \([89](https://arxiv.org/html/2610.08816#A6.E89)\) applies to the empirical quadratic term\. The basic inequality for the empirical minimizer, with one\-step residuals indexed byt=L,…,T−1t=L,\\ldots,T\-1, has cross term2​∑t⟨εt\+1,dt​\(A^−A⋆\)⟩2\\sum\_\{t\}\\langle\\varepsilon\_\{t\+1\},d\_\{t\}\(\\widehat\{A\}\-A^\{\\star\}\)\\rangle, which is controlled by \([90](https://arxiv.org/html/2610.08816#A6.E90)\) on the same event\. Absorbing the resulting quarter ofn​ℰ​\(A^\)n\\mathcal\{E\}\(\\widehat\{A\}\)into the curvature lower bound gives \([91](https://arxiv.org/html/2610.08816#A6.E91)\)\. ∎

The structural reason the strengthened regularity assumptions remain comparable to the original shared\-factor class is a constant\-factor rank count\. A normalized two\-secant direction in𝒱Φ\\mathcal\{V\}\_\{\\Phi\}has Tucker multilinear rank at most\(2​s,2​s,2​r\)\(2s,2s,2r\)in the two spatial modes and the temporal mode\. A normalized increment in𝒮Φ\\mathcal\{S\}\_\{\\Phi\}has rank at most\(4​s,4​s,4​r\)\(4s,4s,4r\), while the further differences consumed by \([88](https://arxiv.org/html/2610.08816#A6.E88)\) have rank at most\(8​s,8​s,8​r\)\(8s,8s,8r\)\. Thus the entropy condition is imposed on the normalized four\-secant base actually used by the proof and the transfer envelope on its eight\-secant differences, rather than on an uncontrolled ambient tensor class\.

The regular\-chart conditions can be verified for a concrete sequence\-model class rather than left only as abstract hypotheses\. We record the verification for bounded\-head diagonal exponential memories, which are the temporal laws of a diagonal linear state\-space model\.

###### Lemma 40\(Stable exponential sequences have a uniformℓ1/ℓ2\\ell\_\{1\}/\\ell\_\{2\}bound\)\.

Fix an integerm≥1m\\geq 1and0<ρh<10<\\rho\_\{\\rm h\}<1\. There isCm,ρh<∞C\_\{m,\\rho\_\{\\rm h\}\}<\\inftysuch that, for every Hilbert space𝖧\\mathsf\{H\}, every

xj=∑q=1mvq​λqj−1,vq∈𝖧,\|λq\|≤ρh,x\_\{j\}=\\sum\_\{q=1\}^\{m\}v\_\{q\}\\lambda\_\{q\}^\{j\-1\},\\qquad v\_\{q\}\\in\\mathsf\{H\},\\qquad\|\\lambda\_\{q\}\|\\leq\\rho\_\{\\rm h\},and everyL≥1L\\geq 1,

∑j=1L‖xj‖𝖧≤Cm,ρh​\(∑j=1L‖xj‖𝖧2\)1/2\.\\sum\_\{j=1\}^\{L\}\\\|x\_\{j\}\\\|\_\{\\mathsf\{H\}\}\\leq C\_\{m,\\rho\_\{\\rm h\}\}\\left\(\\sum\_\{j=1\}^\{L\}\\\|x\_\{j\}\\\|\_\{\\mathsf\{H\}\}^\{2\}\\right\)^\{1/2\}\.\(92\)The constant is independent ofLL, of𝖧\\mathsf\{H\}, and of the coefficientsvqv\_\{q\}\.

###### Proof\.

IfL≤mL\\leq m, Cauchy–Schwarz gives the claim after enlarging the constant bym\\sqrt\{m\}\. AssumeL\>mL\>mand letq⁡\(z\)=∏q=1m\(z−λq\)=zm\+a1​zm−1\+⋯\+amq\(z\)=\\prod\_\{q=1\}^\{m\}\(z\-\\lambda\_\{q\}\)=z^\{m\}\+a\_\{1\}z^\{m\-1\}\+\\cdots\+a\_\{m\}\. Then\(xj\)\(x\_\{j\}\)obeys the order\-mmrecurrencexj\+m\+a1​xj\+m−1\+⋯\+am​xj=0x\_\{j\+m\}\+a\_\{1\}x\_\{j\+m\-1\}\+\\cdots\+a\_\{m\}x\_\{j\}=0\. LetC⁡\(a\)C\(a\)be its companion matrix acting on𝖧m\\mathsf\{H\}^\{m\}\. The coefficient vectorsaaobtained from roots in the closed disk\|z\|≤ρh\|z\|\\leq\\rho\_\{\\rm h\}form a compact set, and every corresponding companion matrix has spectral radius at mostρh\\rho\_\{\\rm h\}\. Fixρ~∈\(ρh,1\)\\widetilde\{\\rho\}\\in\(\\rho\_\{\\rm h\},1\)\. Compactness and the resolvent formula on\|z\|=ρ~\|z\|=\\widetilde\{\\rho\}give

supa‖C​\(a\)n‖op≤Cm,ρh,ρ~​ρ~n\(n≥0\)\.\\sup\_\{a\}\\\|C\(a\)^\{n\}\\\|\_\{\\rm op\}\\leq C\_\{m,\\rho\_\{\\rm h\},\\widetilde\{\\rho\}\}\\widetilde\{\\rho\}^\{n\}\\qquad\(n\\geq 0\)\.Hence, withY1=\(xm,…,x1\)Y\_\{1\}=\(x\_\{m\},\\ldots,x\_\{1\}\),

‖xm\+n‖𝖧≤C​ρ~n​‖Y1‖𝖧m≤C​ρ~n​\(∑j=1m‖xj‖𝖧2\)1/2\.\\\|x\_\{m\+n\}\\\|\_\{\\mathsf\{H\}\}\\leq C\\widetilde\{\\rho\}^\{n\}\\\|Y\_\{1\}\\\|\_\{\\mathsf\{H\}^\{m\}\}\\leq C\\widetilde\{\\rho\}^\{n\}\\left\(\\sum\_\{j=1\}^\{m\}\\\|x\_\{j\}\\\|\_\{\\mathsf\{H\}\}^\{2\}\\right\)^\{1/2\}\.Summing the geometric tail and adding the firstmmterms proves \([92](https://arxiv.org/html/2610.08816#A6.E92)\)\. ∎

Fix now a head capr¯≥1\\bar\{r\}\\geq 1and0<ρh<10<\\rho\_\{\\rm h\}<1\. Forr≤r¯r\\leq\\bar\{r\}, let𝒜exp​\(r,s,ρh\)\\mathcal\{A\}\_\{\\exp\}\(r,s,\\rho\_\{\\rm h\}\)denote the length\-LLshared\-factor arrays

Aj=U⁡\(∑q=1rCq​λqj−1\)​V⊤,\|λq\|≤ρh,1≤j≤L,A\_\{j\}=U\\left\(\\sum\_\{q=1\}^\{r\}C\_\{q\}\\lambda\_\{q\}^\{j\-1\}\\right\)V^\{\\top\},\\qquad\|\\lambda\_\{q\}\|\\leq\\rho\_\{\\rm h\},\\qquad 1\\leq j\\leq L,\(93\)withU,V∈ℝd×sU,V\\in\\mathbb\{R\}^\{d\\times s\}, intersected with the transfer\-stable class of Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)\. No separation between the learned rates is imposed in the statement below\.

###### Proposition 41\(Bounded\-head exponential charts are uniformly regular\)\.

Fixr¯\\bar\{r\},ρh<1\\rho\_\{\\rm h\}<1, and the transfer\-stability marginρ<1\\rho<1\. For everyr≤r¯r\\leq\\bar\{r\}, every predictive chart contained in𝒜exp​\(r,s,ρh\)\\mathcal\{A\}\_\{\\exp\}\(r,s,\\rho\_\{\\rm h\}\)satisfies the entropy and envelope conditions above with

pexp=2​s​\(d−s\)\+r​s2\+r,log⁡N⁡\(𝒮Φ,ϵ\)≤C​pexp​log⁡\(C/ϵ\),Λ≤C,p\_\{\\exp\}=2s\(d\-s\)\+rs^\{2\}\+r,\\qquad\\log N\(\\mathcal\{S\}\_\{\\Phi\},\\epsilon\)\\leq Cp\_\{\\exp\}\\log\(C/\\epsilon\),\\qquad\\Lambda\\leq C,\(94\)whereCCdepends only onr¯\\bar\{r\},ρh\\rho\_\{\\rm h\}, andρ\\rho, and is independent ofL,T,d,sL,T,d,s, andσ\\sigma\.

###### Proof\.

We verify the two conditions separately\. Every element of𝒱Φ\\mathcal\{V\}\_\{\\Phi\}is a normalized difference of two model points, and every element of𝒮Φ\\mathcal\{S\}\_\{\\Phi\}is a normalized difference of two such directions\. Consequently eachD∈𝒮ΦD\\in\\mathcal\{S\}\_\{\\Phi\}is a linear combination of at most four members of \([93](https://arxiv.org/html/2610.08816#A6.E93)\)\. Its row and column spaces therefore have dimensions at most4​s4s, while each scalar core coordinate is a sum of at most4​r4rreal exponentials with rates in\[−ρh,ρh\]\[\-\\rho\_\{\\rm h\},\\rho\_\{\\rm h\}\]\. After padding by zero roots if necessary, every such coordinate obeys an order\-4​r4rrecurrence whose characteristic roots lie in that interval\. Thus one may write

Dj=U~​Gj​V~⊤,U~,V~∈ℝd×k,k≤min⁡\(4​s,d\),D\_\{j\}=\\widetilde\{U\}G\_\{j\}\\widetilde\{V\}^\{\\top\},\\qquad\\widetilde\{U\},\\widetilde\{V\}\\in\\mathbb\{R\}^\{d\\times k\},\\qquad k\\leq\\min\(4s,d\),where\(Gj\)\(G\_\{j\}\)is determined by an order\-4​r4rstable recurrence and its first4​r4rmatrix values\.

BecauseDDis predictive\-normalized, the lower spectral bound of Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35)gives

∑j≤L‖Dj‖F2≤\(1\+ρ\)2σ2\.\\sum\_\{j\\leq L\}\\\|D\_\{j\}\\\|\_\{F\}^\{2\}\\leq\\frac\{\(1\+\\rho\)^\{2\}\}\{\\sigma^\{2\}\}\.The recurrence coefficients range over a fixed compact subset ofℝ4​r\\mathbb\{R\}^\{4r\}\. The proof of Lem\.[40](https://arxiv.org/html/2610.08816#Thmtheorem40), together with the telescoping identityCn−C~n=∑k=0n−1Ck​\(C−C~\)​C~n−1−kC^\{n\}\-\\widetilde\{C\}^\{n\}=\\sum\_\{k=0\}^\{n\-1\}C^\{k\}\(C\-\\widetilde\{C\}\)\\widetilde\{C\}^\{n\-1\-k\}, shows that the map from the recurrence coefficients and the first4​r4rcore matrices to the whole coefficient sequence is Lipschitz inℓ2\\ell\_\{2\}, uniformly inLL\. Standard volumetric nets for the twokk\-dimensional spatial subspaces, for the compact recurrence\-coefficient set, and for the first4​r4rcore matrices therefore give

log⁡N⁡\(𝒮Φ,ϵ\)≤C⁡\{s​d\+r​s2\+r\}​log⁡\(C/ϵ\)\.\\log N\(\\mathcal\{S\}\_\{\\Phi\},\\epsilon\)\\leq C\\\{sd\+rs^\{2\}\+r\\\}\\log\(C/\\epsilon\)\.Here the factors ofσ\\sigmacancel because both the radius of the normalized coefficient set and the coefficient accuracy corresponding to predictive accuracyϵ\\epsilonscale asσ−1\\sigma^\{\-1\}\. Sinces​d≤s⁡\(d−s\)\+s2sd\\leq s\(d\-s\)\+s^\{2\}ands2≤r​s2s^\{2\}\\leq rs^\{2\}forr≥1r\\geq 1, the displayed exponent is at most a constant multiple ofpexpp\_\{\\exp\}, proving the entropy bound in \([94](https://arxiv.org/html/2610.08816#A6.E94)\)\.

For the envelope, anyW∈cone⁡\(𝒮Φ0−𝒮Φ0\)W\\in\\operatorname\{cone\}\(\\mathcal\{S\}\_\{\\Phi\}^\{0\}\-\\mathcal\{S\}\_\{\\Phi\}^\{0\}\)is a scalar multiple of a difference of two elements of𝒮Φ\\mathcal\{S\}\_\{\\Phi\}and hence is a linear combination of at most eight model points\. Thus the matrix\-valued sequence\(Wj\)\(W\_\{j\}\)is a sum of at most8​r≤8​r¯8r\\leq 8\\bar\{r\}exponentials with rates bounded byρh\\rho\_\{\\rm h\}\. Apply Lem\.[40](https://arxiv.org/html/2610.08816#Thmtheorem40)in the Hilbert space ofd×dd\\times dmatrices with Frobenius norm:

∑j≤L‖Wj‖op≤∑j≤L‖Wj‖F≤C8​r¯,ρh​\(∑j≤L‖Wj‖F2\)1/2\.\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{\\rm op\}\\leq\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{F\}\\leq C\_\{8\\bar\{r\},\\rho\_\{\\rm h\}\}\\left\(\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{F\}^\{2\}\\right\)^\{1/2\}\.The same lower spectral bound then gives

∑j≤L‖Wj‖op≤\(1\+ρ\)​C8​r¯,ρhσ​ℰ⁡\(W\),\\sum\_\{j\\leq L\}\\\|W\_\{j\}\\\|\_\{\\rm op\}\\leq\\frac\{\(1\+\\rho\)C\_\{8\\bar\{r\},\\rho\_\{\\rm h\}\}\}\{\\sigma\}\\sqrt\{\\mathcal\{E\}\(W\)\},which is \([88](https://arxiv.org/html/2610.08816#A6.E88)\) with a constant independent ofLL\. ∎

###### Corollary 42\(Parametric forecast risk for bounded\-head diagonal SSM charts\)\.

Under the assumptions of Prop\.[41](https://arxiv.org/html/2610.08816#Thmtheorem41), letA^\\widehat\{A\}be the empirical\-risk minimizer over a predictive chart contained in𝒜exp​\(r,s,ρh\)\\mathcal\{A\}\_\{\\exp\}\(r,s,\\rho\_\{\\rm h\}\)and containingA⋆A^\{\\star\}\. There arec,Cc,Cdepending only onr¯\\bar\{r\},ρh\\rho\_\{\\rm h\}, and the transfer\-stability margin such that, for everyx≥1x\\geq 1, if

n=T−L≥C⁡\(pexp\+x\),n=T\-L\\geq C\(p\_\{\\exp\}\+x\),then, with probability at least1−C​e−c​x1\-Ce^\{\-cx\},

ℰ⁡\(A^\)≤C​σ2​pexp\+xn\.\\mathcal\{E\}\(\\widehat\{A\}\)\\leq C\\sigma^\{2\}\\frac\{p\_\{\\exp\}\+x\}\{n\}\.\(95\)In particular there is no additional context\-length factor in the sample condition for this bounded\-head exponentially stable class\.

###### Proof\.

Prop\.[41](https://arxiv.org/html/2610.08816#Thmtheorem41)gives the hypotheses of Thm\.[39](https://arxiv.org/html/2610.08816#Thmtheorem39)withppa constant multiple ofpexpp\_\{\\exp\}andΛ=O⁡\(1\)\\Lambda=O\(1\)\. ∎

The same class also contains a patch on which thepexp/Tp\_\{\\exp\}/Tscale is necessary up to constants\. This is useful because it shows that the concrete upper law is not only a consequence of an oversized entropy bound\.

###### Proposition 43\(Matching lower scale on an exponential patch\)\.

Fixr≤r¯r\\leq\\bar\{r\}and distinct ratesλ1,…,λr∈\(−ρh,ρh\)\\lambda\_\{1\},\\ldots,\\lambda\_\{r\}\\in\(\-\\rho\_\{\\rm h\},\\rho\_\{\\rm h\}\)\. ForL≥rL\\geq r, the stable class \([93](https://arxiv.org/html/2610.08816#A6.E93)\) contains a neighbourhood for which, whenever

T≥C⁡\{r​s2\+s⁡\(d−s\)\},T\\geq C\\\{rs^\{2\}\+s\(d\-s\)\\\},one has

infA^supA⋆𝔼​ℰ​\(A^\)≥c​σ2​2​s​\(d−s\)\+r​s2T,\\inf\_\{\\widehat\{A\}\}\\sup\_\{A^\{\\star\}\}\\mathbb\{E\}\\,\\mathcal\{E\}\(\\widehat\{A\}\)\\geq c\\sigma^\{2\}\\frac\{2s\(d\-s\)\+rs^\{2\}\}\{T\},\(96\)wherec,Cc,Cmay depend onr¯\\bar\{r\},ρh\\rho\_\{\\rm h\}, the fixed rates, and the transfer\-stability margin, but not onT,d,sT,d,s, orσ\\sigma\. Sincer≤r​s2r\\leq rs^\{2\}fors≥1s\\geq 1, the numerator in \([96](https://arxiv.org/html/2610.08816#A6.E96)\) is equivalent up to constants topexpp\_\{\\exp\}\.

###### Proof\.

For the core term, letKL∈ℝL×rK\_\{L\}\\in\\mathbb\{R\}^\{L\\times r\}have entries\(KL\)j​q=λqj−1\(K\_\{L\}\)\_\{jq\}=\\lambda\_\{q\}^\{j\-1\}\. Distinctness of the fixed rates implies thatKrK\_\{r\}is a nonsingular Vandermonde matrix, hence

0<cλ​Ir⪯KL⊤​KL⪯Cλ​Ir\(L≥r\),0<c\_\{\\lambda\}I\_\{r\}\\preceq K\_\{L\}^\{\\top\}K\_\{L\}\\preceq C\_\{\\lambda\}I\_\{r\}\\qquad\(L\\geq r\),where the lower bound follows fromKL⊤​KL⪰Kr⊤​KrK\_\{L\}^\{\\top\}K\_\{L\}\\succeq K\_\{r\}^\{\\top\}K\_\{r\}and the upper bound from the convergent infinite geometric Gram matrix\. Use the Hamming packing of Lem\.[36](https://arxiv.org/html/2610.08816#Thmtheorem36), but now set

Aj\(v\)=ϵ​U0​\(∑q=1rSq\(v\)​λqj−1\)​V0⊤,ϵ2=cϵ/T,A\_\{j\}^\{\(v\)\}=\\epsilon U\_\{0\}\\left\(\\sum\_\{q=1\}^\{r\}S\_\{q\}^\{\(v\)\}\\lambda\_\{q\}^\{j\-1\}\\right\)V\_\{0\}^\{\\top\},\\qquad\\epsilon^\{2\}=c\_\{\\epsilon\}/T,for fixed orthonormalU0,V0U\_\{0\},V\_\{0\}\. The Gram bound gives pairwise coefficient separation of orderϵ2​r​s2\\epsilon^\{2\}rs^\{2\}\. Moreover

supω‖𝖠\(v\)​\(ω\)‖op≤ϵ1−ρh​∑q≤r‖Sq\(v\)‖op≤ϵ​r​s1−ρh\.\\sup\_\{\\omega\}\\\|\\mathsf\{A\}^\{\(v\)\}\(\\omega\)\\\|\_\{\\rm op\}\\leq\\frac\{\\epsilon\}\{1\-\\rho\_\{\\rm h\}\}\\sum\_\{q\\leq r\}\\\|S\_\{q\}^\{\(v\)\}\\\|\_\{\\rm op\}\\leq\\frac\{\\epsilon rs\}\{1\-\\rho\_\{\\rm h\}\}\.Becauser≤r¯r\\leq\\bar\{r\}, the stated sample condition and a sufficiently smallcϵc\_\{\\epsilon\}keep the packing inside the transfer\-stable class\. Lem\.[35](https://arxiv.org/html/2610.08816#Thmtheorem35), the regular spectral lower bound, and Fano’s inequality then give a lower boundc​σ2​r​s2/Tc\\sigma^\{2\}rs^\{2\}/Texactly as in \([84](https://arxiv.org/html/2610.08816#A6.E84)\)\.

For the output\-subspace term, fix one rateλ0∈\(−ρh,ρh\)\\lambda\_\{0\}\\in\(\-\\rho\_\{\\rm h\},\\rho\_\{\\rm h\}\), fixedV0V\_\{0\}, and a small constantγ\>0\\gamma\>0, and replace the one\-lag subclass in Thm\.[37](https://arxiv.org/html/2610.08816#Thmtheorem37)by

Aj​\(X\)=γ​λ0j−1​U​\(X\)​V0⊤,U⁡\(X\)=U0\+U⟂​X\.A\_\{j\}\(X\)=\\gamma\\lambda\_\{0\}^\{j\-1\}U\(X\)V\_\{0\}^\{\\top\},\\qquad U\(X\)=U\_\{0\}\+U\_\{\\perp\}X\.For the same local Hamming packing ofXX, the squared coefficient separation is the one\-lag separation multiplied by∑j=1L\|λ0\|2​\(j−1\)\\sum\_\{j=1\}^\{L\}\|\\lambda\_\{0\}\|^\{2\(j\-1\)\}, which lies between11and\(1−ρh2\)−1\(1\-\\rho\_\{\\rm h\}^\{2\}\)^\{\-1\}\. The transfer norm is at mostγ​‖U⁡\(X\)‖op/\(1−ρh\)\\gamma\\\|U\(X\)\\\|\_\{\\rm op\}/\(1\-\\rho\_\{\\rm h\}\), so choosingγ\\gammasmall and the perturbation local keeps the whole packing stable\. The proof of Thm\.[37](https://arxiv.org/html/2610.08816#Thmtheorem37)therefore givesc​σ2​s​\(d−s\)/Tc\\sigma^\{2\}s\(d\-s\)/Tinside the exponential class\. Interchanging the left and right subspaces gives the same lower bound for the input subspace\. Taking the maximum of the three packings and using that a maximum dominates one third of their sum yields \([96](https://arxiv.org/html/2610.08816#A6.E96)\)\. ∎

For boundedrrandn≍Tn\\asymp T, Cor\.[42](https://arxiv.org/html/2610.08816#Thmtheorem42)and Prop\.[43](https://arxiv.org/html/2610.08816#Thmtheorem43)therefore give matching parametric forecast\-risk order on a nondegenerate exponential shared\-factor patch\. The constants are uniform in the context length, but no uniformity asr→∞r\\to\\inftyis claimed; that distinction is essential in view of the head\-crowding phenomenon below\.

### F\.6Degrees of freedom2​s​\(d−s\)\+r​s2\+r​pθ2s\(d\-s\)\+rs^\{2\}\+rp\_\{\\theta\}

Collecting the counts, at locally identifiable regular points the parameter dimension isr​d2\+r​pθrd^\{2\}\+rp\_\{\\theta\}for unrestricted matrices,r⁡\{s⁡\(2​d−s\)\+pθ\}r\\\{s\(2d\-s\)\+p\_\{\\theta\}\\\}for independent rank\-ssmatrices, and

psm=2​s​\(d−s\)\+r​s2\+r​pθp\_\{\\rm sm\}=2s\(d\-s\)\+rs^\{2\}\+rp\_\{\\theta\}\(97\)for the shared\-subspace class \([77](https://arxiv.org/html/2610.08816#A6.E77)\) withpθp\_\{\\theta\}learned temporal parameters per head\. The three terms count the two Grassmannians, the cores, and the temporal shapes, and the Grassmannian term appears once for the input subspace and once for the output subspace, which is where the factor two comes from\.

For fixedpθp\_\{\\theta\}ands≥1s\\geq 1, ther​pθrp\_\{\\theta\}term is at most a fixed multiple ofr​s2rs^\{2\}, so the intrinsic count \([97](https://arxiv.org/html/2610.08816#A6.E97)\) and the lower bound \([87](https://arxiv.org/html/2610.08816#A6.E87)\) agree up to constants in their dependence ondd,rr, andss\. When in additionΛ=O⁡\(1\)\\Lambda=O\(1\)and the sample thresholds of \([86](https://arxiv.org/html/2610.08816#A6.E86)\) and Thm\.[39](https://arxiv.org/html/2610.08816#Thmtheorem39)hold, the upper and lower rates therefore match in order on regular charts\. No independent minimax packing over the temporal parameters is claimed\. In this sense \([97](https://arxiv.org/html/2610.08816#A6.E97)\) is the intrinsic dimension governing the regular\-chart upper law, while singular temporal parameterizations are treated separately below\.

### F\.7Whitening temporal spans and raw Gram crowding

The conditioning that Thm\.[39](https://arxiv.org/html/2610.08816#Thmtheorem39)requires is a statement about the temporal span and not about the chosen coordinates\. Suppose the temporal Gram matrixGrG\_\{r\}is positive definite, which is exactly what fails at a head collision\. Writingk⁡\(j\)=\(k1​\(j\),…,kr​\(j\)\)⊤k\(j\)=\(k\_\{1\}\(j\),\\dots,k\_\{r\}\(j\)\)^\{\\top\}andψ\(j\)=Gr−1/2k\(j\)\\psi\(j\)=G\_\{r\}^\{\-1/2\}k\(j\), the sequencesψ1,…,ψr\\psi\_\{1\},\\dots,\\psi\_\{r\}are orthonormal, every core combination can be rewritten withC~=Gr1/2​C\\widetilde\{C\}=G\_\{r\}^\{1/2\}C, and the resulting map is an isometry\. The apparent ill\-conditioning of a raw head basis therefore disappears under whitening, and the statistically meaningful quantity is the frequency leverage of the orthonormal span rather than the condition number ofGrG\_\{r\}\.

That said, crowding is unavoidable once many smooth heads are used\. The bounded\-head constants in Prop\.[41](https://arxiv.org/html/2610.08816#Thmtheorem41)are not asserted to remain uniform asr→∞r\\to\\infty\. If the temporal parameter set is compact with covering numbersε−pθ\\varepsilon^\{\-p\_\{\\theta\}\}and the normalized kernel is Lipschitz in the parameter, then for anyrrparameter values the normalized Gram matrixG~r\\widetilde\{G\}\_\{r\}has smallest eigenvalue tending to zero asrrgrows, at a rate governed bypθp\_\{\\theta\}\. Two smooth heads with nearby parameters produce nearly parallel temporal profiles, so a raw parameterization must become singular, and this is why the regularity in Thm\.[39](https://arxiv.org/html/2610.08816#Thmtheorem39)is imposed on the image\.

### F\.8Head collisions, vanishing cores, and singular strata

Three degeneracies leave the regular charts\. Two heads can collide,θq→θq′\\theta\_\{q\}\\to\\theta\_\{q^\{\\prime\}\}, in which case the temporal profiles become parallel and the divided\-difference basis of App\.[B](https://arxiv.org/html/2610.08816#A2)would be needed to keep a well\-conditioned chart\. A core can vanish,Cq→0C\_\{q\}\\to 0, in which case the corresponding temporal parameter becomes unidentifiable\. The predictive Jacobian can lose rank for gauge reasons, since the factorization \([77](https://arxiv.org/html/2610.08816#A6.E77)\) is invariant underU↦U​OU\\mapsto UO,V↦V​O′V\\mapsto VO^\{\\prime\},Cq↦O⊤​Cq​O′C\_\{q\}\\mapsto O^\{\\top\}C\_\{q\}O^\{\\prime\}for orthogonalO,O′O,O^\{\\prime\}, and a chart must fix that symmetry\.

Thm\.[39](https://arxiv.org/html/2610.08816#Thmtheorem39)assumes uniform separation from all three, so the minimax behaviour on the singular strata is open\. This is not a technical gap that better constants would close, since the intrinsic dimension itself changes on those strata, and \([97](https://arxiv.org/html/2610.08816#A6.E97)\) is not the right count there\. We flag it because a trained multi\-head model is not guaranteed to stay away from collisions, and the crowding statement above says that with many heads it cannot\.

### F\.9Relation to low\-rank VAR and Tucker time\-series models

The class \([77](https://arxiv.org/html/2610.08816#A6.E77)\) is a Tucker factorization of the coefficient tensor\(Aj\)\(A\_\{j\}\), with the two Grassmannians as the cross\-variable factors and the temporal laws as the third mode\. Tensor\-structured and supervised\-factor vector autoregressions study closely related coefficient\-tensor restrictions from an estimation standpoint\([Wang et al\., 2022](https://arxiv.org/html/2610.08816#bib.bib42);[Huang et al\., 2025](https://arxiv.org/html/2610.08816#bib.bib28)\), and low\-rank tensor recovery supplies related geometric and restricted\-isometry tools\([Rauhut et al\., 2017](https://arxiv.org/html/2610.08816#bib.bib37)\)\. General likelihood theory for multidimensional causal time\-series models is developed by[Bardet & Wintenberger \(2009\)](https://arxiv.org/html/2610.08816#bib.bib4)\. The singular strata generated by temporal\-head collisions are specific to the shared\-head parameterization introduced here and are not treated by any of these\.

Two mismatch statements complete the picture and mirror the scalar Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)\. Using too few temporal heads leaves a residual determined by the singular tail of the mode\-three unfolding of the true coefficient tensor, while using too small a shared spatial subspace leaves residuals from the mode\-one and mode\-two unfoldings\. Both are proved by the same low\-rank comparison used in App\.[A](https://arxiv.org/html/2610.08816#A1), applied to the appropriate unfolding, and both hold under the matrix regular\-spectrum condition of Thm\.[34](https://arxiv.org/html/2610.08816#Thmtheorem34)\. The contribution here is therefore not the tensor model but the identification of which unfolding controls which architectural resource in forecast risk\.

## Appendix GExperimental Details

### G\.1Architecture definitions

Three model families appear in Fig\.[2](https://arxiv.org/html/2610.08816#S7.F2), all on the same benchmark\. The benchmark target isyt=∑j=1LKj​Zt\+1−jy\_\{t\}=\\sum\_\{j=1\}^\{L\}K\_\{j\}Z\_\{t\+1\-j\}withZZindependent standard Gaussians andL=64L=64, and the target kernelKKis one of the three described below\. The target carries no additive noise, so the memory shape is identified by the data and the fitted first\-chaos decay can be compared directly with the truth\.

The diagonal state\-space model hasmmmodes with eigenvaluesλi=ρ​σ​\(θi\)\\lambda\_\{i\}=\\rho\\,\\sigma\(\\theta\_\{i\}\)for a learnedθ\\theta, an input mapBB, and a readoutCC, so its impulse response is∑iCi​Bi​λij−1\\sum\_\{i\}C\_\{i\}B\_\{i\}\\lambda\_\{i\}^\{\\,j\-1\}, whereBiB\_\{i\}andCiC\_\{i\}denote theiith components of those vectors, and the model is linear in the state\. Since the eigenvalues areρ\\rhotimes a sigmoid they lie strictly inside\(0,ρ\)\(0,\\rho\), which matters for the boundary\-matched control below\. The gated recurrence isSt=g⁡\(Zt\)⊙St−1\+v⁡\(Zt\)S\_\{t\}=g\(Z\_\{t\}\)\\odot S\_\{t\-1\}\+v\(Z\_\{t\}\)with a bounded data\-dependent gategg, a bounded injectionvv, and a linear readout\. It is nonlinear in and gated by the current input while remaining affine in the previous state, so it is a contextual affine\-in\-state recurrence rather than a generalF⁡\(S,Z\)F\(S,Z\)\. This is a legitimate subclass of the class treated by Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6): bounded injection and contraction produce a bounded forward\-invariant state domain on which the derivative budgets can be verified at each fixedρ\\rho\. The experiment should not be read as testing a genuinely nonlinear\-in\-state recurrence, which we do not implement\. The attention model is a two\-layer causal Transformer with model width3232, four heads, feedforward width6464, and learned relative positional biases\.

### G\.2Width\-independent derivative and amplitude budgets

Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)assumes uniform bounds on the three derivative blocks, so a model whose bounds grow with width would evade the conclusion for reasons unrelated to memory\. Every recurrent model is therefore constrained to width\-independent budgets, enforced through the effective parameterization rather than by a penalty or by projecting raw parameters after each optimizer step\. Each budgeted parameter block is passed through a fixedℓ2\\ell\_\{2\}cap before use, so the effective weights satisfy the budget at every step by construction\. The budgets used are‖B‖≤2\\\|B\\\|\\leq 2,‖C‖≤2\\\|C\\\|\\leq 2, gate norm at most22, value norm at most22, and input scale at most44, all independent ofmm\. The contractionρ\\rhois fixed by construction rather than learned, so it is an experimental variable and not an outcome\.

This is the single most important protocol choice in the experiment\. Without it, a width sweep would confound two effects, since a wider model could reduce error either by using more modes or by using larger weights, and only the first is what Thm\.[1](https://arxiv.org/html/2610.08816#Thmtheorem1)and Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)are about\.

### G\.3Bounded nonlinear injection

For the gated cell,‖g⁡\(z\)‖∞≤ρ<1\\\|g\(z\)\\\|\_\{\\infty\}\\leq\\rho<1and bounded injection‖v⁡\(z\)‖∞≤Vmax\\\|v\(z\)\\\|\_\{\\infty\}\\leq V\_\{\\max\}make the boxℬρ=\{s:‖s‖∞≤Vmax/\(1−ρ\)\}\\mathcal\{B\}\_\{\\rho\}=\\\{s:\\\|s\\\|\_\{\\infty\}\\leq V\_\{\\max\}/\(1\-\\rho\)\\\}forward invariant\. The input derivative isDz​F​\(s,z\)=Dz​g​\(z\)⊙s\+Dz​v​\(z\)D\_\{z\}F\(s,z\)=D\_\{z\}g\(z\)\\odot s\+D\_\{z\}v\(z\), so the fixed parameter caps give a finite width\-independent bound‖Dz​F‖≤Kρ\\\|D\_\{z\}F\\\|\\leq K\_\{\\rho\}uniformly onℬρ×ℝ\\mathcal\{B\}\_\{\\rho\}\\times\\mathbb\{R\}for each fixedρ\\rho, rather than on all ofℝm×ℝ\\mathbb\{R\}^\{m\}\\times\\mathbb\{R\}\. The gate is likewise bounded, so the state Jacobian norm is at mostρ\\rhopointwise on this domain and not merely in expectation, which is the uniform contraction the theorem requires\. BecauseKρK\_\{\\rho\}may vary withρ\\rho, the stability sweep is interpreted qualitatively rather than as a numerical test of the exact asymptotic constant in Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\.

### G\.4Training, validation, and seed protocol

Each configuration uses12,00012\{,\}000training,3,0003\{,\}000validation, and6,0006\{,\}000test windows, generated with distinct fixed data seeds so that the three splits are independent draws rather than a partition of one sample\. The state\-space model is trained for15001500steps at learning rate3×10−23\\times 10^\{\-2\}, the gated model for900900steps at2×10−22\\times 10^\{\-2\}, and the attention model for10001000steps at3×10−33\\times 10^\{\-3\}, all with batch size128128\. Model selection uses the validation split and all reported numbers are on the test split\.

Seeds are set before model construction so that initialization and data order are both controlled\. The protocol is versioned and the identifier is recorded in the summary file shipped with the archive, which is what ties the reported numbers to a specific configuration\. The width and stability sweeps use seeds00through33, while the attention reference and each exponential control use seeds00through22\. Attention checkpoints are cached under a hash of the full configuration including the seed and the step count, so changing any protocol field forces retraining rather than silently reusing an earlier run\.

### G\.5Exact SSM impulse response and first\-chaos estimation

The first\-chaos coefficientsβj=𝔼⁡\[m^t​Zt\+1−j\]\\beta\_\{j\}=\\mathbb\{E\}\[\\widehat\{m\}\_\{t\}Z\_\{t\+1\-j\}\]are obtained differently for linear and nonlinear models, and the distinction is worth stating because it removes estimation noise from one of the three curves in panel \(a\)\. The state\-space model is linear in the state, so its first\-chaos memory is exactly its impulse response and is read off the parameters in closed form with no sampling at all\. The gated and attention models are nonlinear, so their coefficients are estimated by Monte Carlo on an independent diagnostic sample of200,000200\{,\}000standard Gaussian windows, using the empirical correlation between the output and each lagged input\.

The fitted slopes are obtained by regressinglog⁡\|βj\|\\log\|\\beta\_\{j\}\|onlog⁡j\\log jover lags33to3030, which is the range where the target is well above the diagnostic noise floor and below the truncation effects atjjnearLL\. They are descriptive finite\-window shape summaries only: an exponentially decaying sequence is not expected to have a constant log\-log slope\.

### G\.6Algebraic target results

The algebraic target isKj=j−1\.1K\_\{j\}=j^\{\-1\.1\}, soα=1\.1\>1\\alpha=1\.1\>1as Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)requires, and its variance on theL=64L=64benchmark is∑j=1LKj2=1\.485\\sum\_\{j=1\}^\{L\}K\_\{j\}^\{2\}=1\.485\. The fitted first\-chaos slopes are−1\.110\-1\.110for attention against a true exponent of−1\.1\-1\.1,−2\.26\-2\.26for the diagonal state\-space model, and−2\.56\-2\.56for the gated recurrence\. Attention therefore recovers the algebraic law almost exactly, while both contractive recurrences decay far faster than the target, which is the qualitative content of the envelope in \([11](https://arxiv.org/html/2610.08816#S5.E11)\)\.

The recovery by attention here should not be read across to a noisy forecasting setting\. The target in this benchmark is exactly a linear functional of the observed lag vector, so the coefficient law is identified by the data, which is not the case when the predictor sits near an innovation floor\.

### G\.7Matched exponential\-target specificity control

Two exponential controls are run with the architecture, budgets, optimizer, and data splits held fixed and only the target changed\. The controls use the matched seeds00,11, and22, while the algebraic sweep additionally includes seed33, so the comparison is seed matched on the first three and not on the fourth\. The first is boundary matched withβ=0\.8\\beta=0\.8equal to the contractionρ=0\.8\\rho=0\.8, and since the diagonal eigenvalues areρ\\rhotimes a sigmoid this target lies on the closure of the model class rather than in its interior\. Hereβ\\betadenotes only the exponential\-control decay used in the figure labels and is unrelated to the first\-chaos sequence\(βj\)\(\\beta\_\{j\}\)of Sec\.[5](https://arxiv.org/html/2610.08816#S5)\. The second hasβ=0\.9\>ρ\\beta=0\.9\>\\rho, so it decays more slowly than the envelope of Thm\.[6](https://arxiv.org/html/2610.08816#Thmtheorem6)and its tail leaves what aρ\\rho\-contractive state can carry, while remaining exponential\.

The three targets have different variances, namely1\.4851\.485,2\.7782\.778, and5\.2635\.263for the algebraic, boundary\-matched, and slower exponential targets, so raw mean squared error is not comparable across them\. All reported numbers and both sweeps therefore use the unexplained fraction of target variance, that is held\-out error divided by the variance of the target\. This normalization is not cosmetic, since the raw comparison would understate the boundary\-matched advantage and overstate the slower\-exponential gap\.

### G\.8Width sweep

The width sweep fixesρ=0\.8\\rho=0\.8and variesmmover\{2,4,8,16,32,64,128\}\\\{2,4,8,16,32,64,128\\\}\. At the largest width the diagonal state\-space model attains an unexplained variance fraction of6\.8×10−66\.8\\times 10^\{\-6\}on the boundary\-matched control,1\.1×10−21\.1\\times 10^\{\-2\}on the algebraic target, and4\.5×10−24\.5\\times 10^\{\-2\}on the slower exponential\. The gated recurrence shows the same ordering with a much smaller gap on the boundary\-matched control, reaching2\.0×10−32\.0\\times 10^\{\-3\}there against1\.4×10−21\.4\\times 10^\{\-2\}on the algebraic target\. The attention reference is at5\.6×10−45\.6\\times 10^\{\-4\}on the algebraic target\.

Two features matter more than the individual values\. For the diagonal state\-space model the slower\-exponential floor is essentially width independent, varying by about one percent fromm=2m=2tom=128m=128\. The algebraic curve for the same model, and both gated curves, improve appreciably at small widths and then plateau well above zero, the largest such improvement being about forty percent for the gated slower\-exponential control\. The shape of the evidence is therefore an early\-width gain followed by a plateau, and what it supports is that increasing width alone does not remove the floor under a fixed contraction and fixed derivative budgets, which is what an envelope obstruction rather than a capacity limit should look like\. An exponential target is simultaneously the easiest and the hardest of the three, which removes the distinction between power\-law and exponential memory as the explanation and places the floor in the relation between the target tail and the contraction\.

### G\.9Stability sweep

The stability sweep fixesm=16m=16and variesρ\\rhoover\{0\.5,0\.7,0\.8,0\.9,0\.95,0\.99,0\.995\}\\\{0\.5,0\.7,0\.8,0\.9,0\.95,0\.99,0\.995\\\}on the algebraic target\. All numbers here are again the unexplained fraction of target variance, the same metric plotted in panel \(c\) and used in the width sweep\. The diagonal state\-space model improves monotonically from6\.2×10−26\.2\\times 10^\{\-2\}to7\.5×10−67\.5\\times 10^\{\-6\}across that range, a drop of nearly four orders of magnitude\. The gated recurrence improves from6\.8×10−26\.8\\times 10^\{\-2\}to about2\.0×10−32\.0\\times 10^\{\-3\}and then stops, with the values atρ=0\.95\\rho=0\.95,0\.990\.99, and0\.9950\.995all near that level\.

The upper axis of panel \(c\) is reparameterized by the marginδ=−log⁡ρ\\delta=\-\\log\\rho, which is the quantity appearing in Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\. We read the sweep as removal of the fixed\-contraction obstruction at this finite context and not as evidence that near\-criticality suffices\. At the largestρ\\rho, the contraction timescale exceedsL=64L=64, so those points do not probe the untruncated asymptotic algebraic tail required for a direct numerical test of Corollary[7](https://arxiv.org/html/2610.08816#Thmtheorem7)\. The gated residual floor can also include higher\-chaos variance, as discussed in App\.[D](https://arxiv.org/html/2610.08816#A4), and is not attributed uniquely to first\-chaos tail mismatch\. The effect spans orders of magnitude, and means with standard errors are reported over four independent training seeds\.

### G\.10Complete numerical tables

The supplementary material contains the code and numerical summaries sufficient to regenerate both figures, including the archived per\-configuration summary behind Fig\.[2](https://arxiv.org/html/2610.08816#S7.F2); the panels of Fig\.[1](https://arxiv.org/html/2610.08816#S7.F1)are recomputed from the closed\-form and quadrature routines rather than read from a stored table\. For Fig\.[1](https://arxiv.org/html/2610.08816#S7.F1)these are deterministic, and the reported items are the constructive upper bound and the Hankel lower bound againstrrwith the fitted slopes againstr\\sqrt\{r\}and their coefficients of determination, the exact fractional prediction gap againstLLfor eachddwith the fitted log\-log slopes, and the exact zero\-memory risk againstddwith its ratio to\(π2/6\)​d2\(\\pi^\{2\}/6\)d^\{2\}\.

Two numerical safeguards are recorded with those tables\. The lower bound in panel \(a\) is computed from the singular values of a finite Hankel section of order12001200\. Singular values below10310^\{3\}times machine epsilon times the leading singular value are discarded as noise, so the curve is truncated at that index rather than continued into the floating\-point floor\. The fractional prediction gap is computed as a cancellation\-free suffix sum of−log⁡\(1−d2/\(k−d\)2\)\-\\log\(1\-d^\{2\}/\(k\-d\)^\{2\}\)with a leading\-tail trigamma correction rather than by differencing log\-Gamma values, which would lose several digits at the largest contexts\. The correction uses−log⁡\(1−u\)=u\+O⁡\(u2\)\-\\log\(1\-u\)=u\+O\(u^\{2\}\)on the tail beyond a cutoffKK, so it capturesd2​ψ1​\(K\+1−d\)d^\{2\}\\psi\_\{1\}\(K\+1\-d\)and omitsO⁡\(d4​∑k\>Kk−4\)=O⁡\(d4​K−3\)O\(d^\{4\}\\sum\_\{k\>K\}k^\{\-4\}\)=O\(d^\{4\}K^\{\-3\}\); at the cutoffK=220K=2^\{20\}used here that residual is far below the plotted scale, but the term is a leading\-tail correction and not an exact remainder\.

For Fig\.[2](https://arxiv.org/html/2610.08816#S7.F2)the archived summary records, for every model, target, width, and contraction, the seed mean and standard error that are plotted, together with the fitted first\-chaos slopes, the target variances used for normalization, and the seed lists\. Because the exponential controls use three seeds and the main sweeps use four, the seed list rather than a single global count is what makes each entry interpretable\.