Guarantees on Dynamical System Distinguishability for LLM Token Generation

# Guarantees on Dynamical System Distinguishability for LLM Token Generation
Source: [https://arxiv.org/html/2607.28667](https://arxiv.org/html/2607.28667)
Mohamed Akrout , Dan Wilson∗ Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville, TN, USA \{mohamed\.akrout,dan\.wilson\}@tennessee\.edu

###### Abstract

Recent work has shown that classifying large language models \(LLMs\)’ responses can be distinguished by modeling token embeddings as trajectories of a black\-box dynamical system \(DS\) and comparing prediction residuals of two DSs\. Despite the empirical success of this dynamical approach, a theoretical understanding of*why*it works,*how well*it scales as a function of the token sequence, and*when*it transfers across embedding models remains lacking\. We address these questions by formalizing the classification task as a binary hypothesis test between two stochastic linear DSs\. We show that the total variation distance between the stationary marginal distributions of the two DSs can be arbitrarily small even when the dynamics differ substantially, which provides a fundamental accuracy floor for any classifier that ignores token dynamics\. We then show that the misclassification probability of DS\-based classification decays exponentially in the sequence lengthLL, with the decay governed by a*dynamical discriminability*quantityδ2\\delta^\{2\}that captures the spectral distance between the two DSs\. We also characterize cross\-embedding generalization by introducing an*approximate intertwining*condition between embedding models and establishing a lower bound on the transferable discriminability in terms of the intertwining map’s smallest singular value\. Together, these results explain the empirical performance of DS\-based classification and motivate further investigation into using DS theory to analyze AI systems, in contrast to the more common approach of using AI to model dynamical systems\.

## 1Introduction

The problem of distinguishing between two dynamical systems from their output trajectories arises in numerous applications, including change\-point detection in control systems\(Xinet al\.,[2025](https://arxiv.org/html/2607.28667#bib.bib1)\), fault diagnosis in engineering\(Sheikhiet al\.,[2025b](https://arxiv.org/html/2607.28667#bib.bib2)\), and, more recently, hallucination detection in Large Language Models \(LLMs\)\(Wilson and Akrout,[2026](https://arxiv.org/html/2607.28667#bib.bib7)\)\. In the LLM context, recent work has demonstrated that factual and hallucinated text can be distinguished by modeling the evolution of token embeddings as trajectories of a black\-box dynamical system and fitting separate Koopman operator approximations for each regime\(Wilson and Akrout,[2026](https://arxiv.org/html/2607.28667#bib.bib7)\)\.

![Refer to caption](https://arxiv.org/html/2607.28667v1/x1.png)\(\(a\)\)Multiple LLM responses
![Refer to caption](https://arxiv.org/html/2607.28667v1/x2.png)\(\(b\)\)token dynamics of one LLM response

Figure 1:The two approaches to analyze the properties of LLM responses: \(a\) multiple responses where tokens are assumed i\.i\.d\. within responses, and \(b\) one single response where the token dynamics is considered\.Existing approaches for classifying LLM responses \(e\.g\., distinguishing safe from unsafe, or correct from hallucinated\) require multiple responses to the same query as shown in Fig\.[1\(a\)](https://arxiv.org/html/2607.28667#S1.F1.sf1)in order to make a reliable decisionLewiset al\.\([2020](https://arxiv.org/html/2607.28667#bib.bib8)\); Manakulet al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib9)\); Zhanget al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib10)\); Dhuliawalaet al\.\([2024](https://arxiv.org/html/2607.28667#bib.bib11)\); Menget al\.\([2022](https://arxiv.org/html/2607.28667#bib.bib12)\)\. This is because they treat the tokens within each generated sentence as independent and identically distributed \(i\.i\.d\.\), and therefore ignore the embedding evolution of the tokens across the response\. As a result, these methods must aggregate statistics over several independent generations to compensate for the loss of token dynamics, which increases both latency and API costs\. By accounting for the rich token dynamics of a single LLM response in an embedding space as shown in Fig\.[1\(b\)](https://arxiv.org/html/2607.28667#S1.F1.sf2)\.

Despite empirical demonstrations that token embedding dynamics enable LLM response classification\(Wilson and Akrout,[2026](https://arxiv.org/html/2607.28667#bib.bib7)\), the theory behind this remains unexplored\. This raises fundamental questions that we address in this work:

1. i\)i\)Why do dynamics matter?If the marginal distributions of token embeddings differ between two response categories, why is a dynamical model necessary?
2. ii\)ii\)How well does the dynamical system method work?What is the classification error as a function of sequence length of the response, and what dynamical system quantities govern the classification performance?
3. iii\)iii\)Are embedding dynamics generalizable?To what extent do token embeddings transfer across embedding models, and what conditions govern their cross\-model generalization? From a DS perspective, embedding transferability is a striking and unexpected finding\. This is because, at a fundimental level, data driven methods for Koopman operator approximation depend on the specific observables used in the model fitting\. Even minor permutations to the ordering of the observable coordinates can ruin the predictive ability; switching to entirely different embedding model represents an even more drastic transformation\. The fact that discriminative information survives such a change requires a theoretical explanation and has practical implications for reducing the cost of fitting models across different embedding backends\.

In this paper, we provide rigorous answers to all three questions\. Our contributions are:

- •Result[1](https://arxiv.org/html/2607.28667#Thmtheorem1)\(Limitation of marginal density analysis\):We show that two stable DSs can have completely different dynamics yet produce stationary distributions that are nearly indistinguishable\. This means that any classifier relying only on marginal statistics has a fundamental accuracy floor and cannot do better than random guessing when the distributions are too similar\. This explains why static features like SVD modes often overlap for different embedding trajectory classes\.
- •Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)\(Dynamical separability\):We prove that a classifier based on prediction residuals can achieve misclassification error that decays exponentially with sequence length, with the rate decay determined by how dynamically distinct the two DSs are and the noise level\. This breaks the accuracy ceiling of classifers with marginal density analysis obtained in Result[1](https://arxiv.org/html/2607.28667#Thmtheorem1)\. We also show that reliable classification with error probabiltyϵ\\epsilonrequires a number of tokensL∗=𝒪​\(log⁡1ϵ\)L^\{\*\}=\\mathcal\{O\}\\big\(\\log\\frac\{1\}\{\\epsilon\}\\big\), which explains why richer embedding models already achieve high accuracy with shorter sequences\.
- •Result[3](https://arxiv.org/html/2607.28667#Thmtheorem3)\(Cross\-embedding transfer\):We prove that dynamical systems fitted on one embedding can be transferred to another, provided the two embedding spaces share similar geometric structure described by an affine transformation\. The lower\-bound on the ability to discriminate between classes is dependent on the singular value of the transformation matrix and the norm of the operator difference of the two DSs\. This explains why cross\-embedding classification performs above chance even without retraining\.

Notation: We use bold lowercase for vectors \(𝐲,𝐳\\mathbf\{y\},\\mathbf\{z\}\), bold uppercase for matrices \(𝐀,𝐏\\mathbf\{A\},\\mathbf\{P\}\), and calligraphic letters for manifolds and sets \(ℳc,ℳh\\mathcal\{M\}\_\{c\},\\mathcal\{M\}\_\{h\}\)\. The operator∥⋅∥\\left\\\|\\cdot\\right\\\|denotes the Euclidean 2\-norm for vectors and the spectral norm for matrices;∥⋅∥F\\left\\\|\\cdot\\right\\\|\_\{F\}denotes the Frobenius norm\. For a positive definite matrix𝐌\\mathbf\{M\}, we write‖𝐱‖𝐌=𝐱⊤​𝐌​𝐱\\left\\\|\\mathbf\{x\}\\right\\\|\_\{\\mathbf\{M\}\}=\\sqrt\{\\mathbf\{x\}^\{\\top\}\\mathbf\{M\}\\,\\mathbf\{x\}\}for the Mahalanobis norm of𝐱\\mathbf\{x\}\. We writetr​\(⋅\)\\mathrm\{tr\}\(\\cdot\)for the trace,λmax​\(⋅\)\\lambda\_\{\\max\}\(\\cdot\)andλmin​\(⋅\)\\lambda\_\{\\min\}\(\\cdot\)for the largest and smallest eigenvalues of a symmetric matrix,σmin​\(⋅\)\\sigma\_\{\\min\}\(\\cdot\)for the smallest singular value, andρ​\(⋅\)\\rho\(\\cdot\)for the spectral radius\. We use𝒪​\(⋅\)\\mathcal\{O\}\(\\cdot\)to denote asymptotic upper bounds andΘ​\(⋅\)\\Theta\(\\cdot\)for asymptotically tight bounds in both directions, both up to constant factors\.

## 2Prior work

Distinguishability of dynamical systems\.The problem of distinguishing between dynamical systems from observed data has a rich history in DS theory and signal processing\. Classical system identification theory\(Ljung,[1999](https://arxiv.org/html/2607.28667#bib.bib13)\)addresses the problem of estimating system parameters from input\-output data, while the related problem of*hypothesis testing*between candidate models has been studied in the context of fault detection and isolation \(Sheikhiet al\.\([2025a](https://arxiv.org/html/2607.28667#bib.bib14)\)\)\. In the linear setting, the question of whether two systems can be distinguished from output trajectories is closely tied to the notion of*observability*\(Kailath \([1980](https://arxiv.org/html/2607.28667#bib.bib5)\)\): two systems with distinct dynamics but identical output distributions are observationally indistinguishable\. Recent work has provided finite\-sample guarantees for learning linear dynamical systems from a single trajectory \(Simchowitzet al\.\([2018](https://arxiv.org/html/2607.28667#bib.bib15)\); Sarkar and Rakhlin \([2019](https://arxiv.org/html/2607.28667#bib.bib16)\)\), establishing that the sample complexity depends on the spectral radius and the signal\-to\-noise ratio of the system\. Online change\-point detection between linear dynamical regimes has been studied in \(Xinet al\.\([2025](https://arxiv.org/html/2607.28667#bib.bib1)\)\), who provide finite\-sample guarantees for detecting switches between system parameters\. Our work differs from these in that we characterize the error as a function of the dynamical discriminability between two dynamical systems\.

Dynamical systems for neural networks\.The perspective of treating neural network inference as a dynamical system has gained traction in recent years\. Neural Ordinary Differential Equations \(Chenet al\.\([2018](https://arxiv.org/html/2607.28667#bib.bib17)\)\) established a formal connection between residual networks and continuous\-time dynamical systems, while Koopman operator theory has been applied to analyze and predict the evolution of nonlinear systems from data \(Budišićet al\.\([2012](https://arxiv.org/html/2607.28667#bib.bib18)\); Mezić \([2013](https://arxiv.org/html/2607.28667#bib.bib19)\); Williamset al\.\([2015](https://arxiv.org/html/2607.28667#bib.bib3)\)\)\. In the LLM context,, token embedding sequences of LLM responses were treated as trajectories of a black\-box dynamical system and fitted to Koopman\-based linear models for factual and hallucinated regimesWilson and Akrout \([2026](https://arxiv.org/html/2607.28667#bib.bib7)\), enabling single\-response classification\. However, the theoretical foundations of this approach remain unclear\. The present work fills this gap\.

## 3Problem setup and outline

### 3\.1Dynamical systems for classification

We consider a discrete\-time system generating a sequence of observables𝐲1,𝐲2,…,𝐲L∈ℝd\\mathbf\{y\}\_\{1\},\\mathbf\{y\}\_\{2\},\\ldots,\\mathbf\{y\}\_\{L\}\\in\\mathbb\{R\}^\{d\}through an embedding map applied to the output of a generative model \(e\.g\., an LLM\)\. The observables are obtained by projecting the raw embeddings onto the topddsingular value decomposition \(SVD\) modes:

𝐲k=𝚽⊤​H​\(𝐱k\)∈ℝd,\\mathbf\{y\}\_\{k\}=\\bm\{\\Phi\}^\{\\top\}H\(\\mathbf\{x\}\_\{k\}\)\\in\\mathbb\{R\}^\{d\},\(1\)where𝐱k\\mathbf\{x\}\_\{k\}is the internal state of the generative model,HHis the composition of the token selection and embedding maps, and𝚽∈ℝM×d\\bm\{\\Phi\}\\in\\mathbb\{R\}^\{M\\times d\}is the SVD projection matrix withMMbeing the raw embedding dimension\. Following the Extended Dynamic Mode Decomposition \(EDMD\) framework\(Williamset al\.,[2015](https://arxiv.org/html/2607.28667#bib.bib3)\), we lift the observables to a higher\-dimensional space:

𝐳k=\[𝐲kflift​\(𝐲k\)\]∈ℝd\+γ,\\mathbf\{z\}\_\{k\}=\\begin\{bmatrix\}\\mathbf\{y\}\_\{k\}\\\\ f\_\{\\text\{lift\}\}\(\\mathbf\{y\}\_\{k\}\)\\end\{bmatrix\}\\in\\mathbb\{R\}^\{d\+\\gamma\},\(2\)whereflift:ℝd→ℝγf\_\{\\text\{lift\}\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{\\gamma\}is a nonlinear lifting function \(e\.g\., polynomial combinations of the observable components\)\. A set of snapshot pairssk=\(𝐳k,𝐳k\+1\)s\_\{k\}=\(\\mathbf\{z\}\_\{k\},\\mathbf\{z\}\_\{k\+1\}\)is collected and arranged into matrices𝑿=\[𝐳1…𝐳q\]\\bm\{X\}=\\begin\{bmatrix\}\\mathbf\{z\}\_\{1\}&\\dots&\\mathbf\{z\}\_\{q\}\\end\{bmatrix\}and𝑿\+=\[𝐳2…𝐳q\+1\]\\bm\{X\}^\{\+\}=\\begin\{bmatrix\}\\mathbf\{z\}\_\{2\}&\\dots&\\mathbf\{z\}\_\{q\+1\}\\end\{bmatrix\}whereqqis the number of snapshot pairs used\. An approximation of the Koopman operator𝐳k\+1=𝐀𝐳k\\mathbf\{z\}\_\{k\+1\}=\\mathbf\{A\}\\mathbf\{z\}\_\{k\}can be obtained according to𝐀=𝐗\+​𝐗†\\mathbf\{A\}=\\mathbf\{X\}^\{\+\}\\mathbf\{X\}^\{\\dagger\}, where†denotes the pseudoinverse\. A prediction for the evolution of the token embeddings can be obtained according to𝐲~k\+1=\[𝑰𝟎\]​𝑨​𝒛k\\tilde\{\\mathbf\{y\}\}\_\{k\+1\}=\\begin\{bmatrix\}\\bm\{I\}&\\bm\{0\}\\end\{bmatrix\}\\bm\{A\}\\bm\{z\}\_\{k\}where𝑰∈ℝM×M\\bm\{I\}\\in\\mathbb\{R\}^\{M\\times M\}denotes the identity matrix,𝟎∈ℝM×γ\\bm\{0\}\\in\\mathbb\{R\}^\{M\\times\\gamma\}is a matrix of zeros\.

In this work, we obtain distinct approximations of the Koopman operator, denoted𝐀1\\mathbf\{A\}\_\{1\}and𝐀2\\mathbf\{A\}\_\{2\}, from data belonging to two different classes\. In the LLM context, and without loss of generality, these correspond to the approximated operators𝐀h\\mathbf\{A\}\_\{h\}and𝐀c\\mathbf\{A\}\_\{c\}which we estimate from hallucinated and correct responses, respectively\. Given an arbitrary LLM response, with sequences of token embeddings𝐳1,…,𝐳L\\mathbf\{z\}\_\{1\},\\ldots,\\mathbf\{z\}\_\{L\}, the prediction errors under each model are:

ϵc,k\\displaystyle\\epsilon\_\{c,k\}≡‖𝐲k\+1−𝐏𝐀c​𝐳k‖,\\displaystyle\\equiv\\left\\\|\\mathbf\{y\}\_\{k\+1\}\-\\mathbf\{P\}\\mathbf\{A\}\_\{c\}\\mathbf\{z\}\_\{k\}\\right\\\|,\(3\)ϵh,k\\displaystyle\\epsilon\_\{h,k\}≡‖𝐲k\+1−𝐏𝐀h​𝐳k‖,\\displaystyle\\equiv\\left\\\|\\mathbf\{y\}\_\{k\+1\}\-\\mathbf\{P\}\\mathbf\{A\}\_\{h\}\\mathbf\{z\}\_\{k\}\\right\\\|,\(4\)where𝐏=\[𝐈d​0d×γ\]\\mathbf\{P\}=\[\\mathbf\{I\}\_\{d\}\\;\\;\\mathbf\{0\}\_\{d\\times\\gamma\}\]extracts the observable components\. The cumulative squared errors are:

Sc=∑k=1L−1ϵc,k2,Sh=∑k=1L−1ϵh,k2,S\_\{c\}=\\sum\_\{k=1\}^\{L\-1\}\\epsilon\_\{c,k\}^\{2\},\\qquad S\_\{h\}=\\sum\_\{k=1\}^\{L\-1\}\\epsilon\_\{h,k\}^\{2\},\(5\)and the differential residual score is:

Δ​ℰ=Sh−Sc\.\\Delta\\mathcal\{E\}=\\sqrt\{S\_\{h\}\}\-\\sqrt\{S\_\{c\}\}\.\(6\)The decision rule classifies the trajectory as hallucinated ifΔ​ℰ<η\\Delta\\mathcal\{E\}<\\etafor a classification thresholdη\\eta\.

### 3\.2Classification as a hypothesis testing problem

We model the classification task as a binary hypothesis test\. Under each hypothesis, the lifted observables evolve according to a stochastic linear dynamical system:

ℋc:𝐳k\+1\\displaystyle\\mathcal\{H\}\_\{c\}:\\quad\\mathbf\{z\}\_\{k\+1\}=𝐀c​𝐳k\+𝝃k\(c\),𝝃k\(c\)∼𝒩​\(𝐦c,𝚺c\),\\displaystyle=\\mathbf\{A\}\_\{c\}\\mathbf\{z\}\_\{k\}\+\\bm\{\\xi\}\_\{k\}^\{\(c\)\},\\qquad\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\sim\\mathcal\{N\}\(\\mathbf\{m\}\_\{c\},\\bm\{\\Sigma\}\_\{c\}\),\(7\)ℋh:𝐳k\+1\\displaystyle\\mathcal\{H\}\_\{h\}:\\quad\\mathbf\{z\}\_\{k\+1\}=𝐀h​𝐳k\+𝝃k\(h\),𝝃k\(h\)∼𝒩​\(𝐦h,𝚺h\),\\displaystyle=\\mathbf\{A\}\_\{h\}\\mathbf\{z\}\_\{k\}\+\\bm\{\\xi\}\_\{k\}^\{\(h\)\},\\qquad\\bm\{\\xi\}\_\{k\}^\{\(h\)\}\\sim\\mathcal\{N\}\(\\mathbf\{m\}\_\{h\},\\bm\{\\Sigma\}\_\{h\}\),\(8\)where𝐀c,𝐀h∈ℝ\(d\+γ\)×\(d\+γ\)\\mathbf\{A\}\_\{c\},\\mathbf\{A\}\_\{h\}\\in\\mathbb\{R\}^\{\(d\+\\gamma\)\\times\(d\+\\gamma\)\}are the finite\-dimensional approximations of the Koopman operator for the correct \(factual\) and hallucinated regimes, respectively\. The noise terms𝝃k\(c\)\\bm\{\\xi\}\_\{k\}^\{\(c\)\}and𝝃k\(h\)\\bm\{\\xi\}\_\{k\}^\{\(h\)\}capture both the approximation error of the finite\-dimensional Koopman truncation and any genuine stochasticity in the generation process\. We allow each regime to have its own noise mean \(𝐦c∈ℝd\+γ\\mathbf\{m\}\_\{c\}\\in\\mathbb\{R\}^\{d\+\\gamma\}or𝐦h∈ℝd\+γ\\mathbf\{m\}\_\{h\}\\in\\mathbb\{R\}^\{d\+\\gamma\}\) and its own noise covariance \(𝚺c\\bm\{\\Sigma\}\_\{c\}or𝚺h\\bm\{\\Sigma\}\_\{h\}\), reflecting the fact that the fidelity of the linear approximation may differ between factual and hallucinated dynamics\.

###### Assumption 1\(Stability\)\.

Both systems are stable:ρ​\(𝐀c\)<1\\rho\(\\mathbf\{A\}\_\{c\}\)<1andρ​\(𝐀h\)<1\\rho\(\\mathbf\{A\}\_\{h\}\)<1\.

###### Assumption 2\(Gaussian noise\)\.

The noise sequences\{𝛏k\(c\)\}\\\{\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\\}and\{𝛏k\(h\)\}\\\{\\bm\{\\xi\}\_\{k\}^\{\(h\)\}\\\}are i\.i\.d\. Gaussian with means𝐦c,𝐦h∈ℝd\+γ\\mathbf\{m\}\_\{c\},\\mathbf\{m\}\_\{h\}\\in\\mathbb\{R\}^\{d\+\\gamma\}and positive definite covariance matrices𝚺c,𝚺h∈ℝ\(d\+γ\)×\(d\+γ\)\\bm\{\\Sigma\}\_\{c\},\\bm\{\\Sigma\}\_\{h\}\\in\\mathbb\{R\}^\{\(d\+\\gamma\)\\times\(d\+\\gamma\)\}, respectively\.

Under Assumptions[1](https://arxiv.org/html/2607.28667#Thmassumption1)and[2](https://arxiv.org/html/2607.28667#Thmassumption2), each system possesses a unique stationary distribution\. The*stationary means*are:

𝝁c=\(𝐈−𝐀c\)−1​𝐦c,𝝁h=\(𝐈−𝐀h\)−1​𝐦h,\\bm\{\\mu\}\_\{c\}=\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{c\}\)^\{\-1\}\\mathbf\{m\}\_\{c\},\\qquad\\bm\{\\mu\}\_\{h\}=\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{h\}\)^\{\-1\}\\mathbf\{m\}\_\{h\},\(9\)where the invertibility of𝐈−𝐀c\\mathbf\{I\}\-\\mathbf\{A\}\_\{c\}and𝐈−𝐀h\\mathbf\{I\}\-\\mathbf\{A\}\_\{h\}is guaranteed byρ​\(𝐀c\)<1\\rho\(\\mathbf\{A\}\_\{c\}\)<1andρ​\(𝐀h\)<1\\rho\(\\mathbf\{A\}\_\{h\}\)<1\. The*stationary covariances*𝐆c\\mathbf\{G\}\_\{c\}and𝐆h\\mathbf\{G\}\_\{h\}are the unique positive definite solutions to the respective discrete Lyapunov equations:

𝐆c=𝐀c​𝐆c​𝐀c⊤\+𝚺c,and​𝐆h=𝐀h​𝐆h​𝐀h⊤\+𝚺h\.\\mathbf\{G\}\_\{c\}=\\mathbf\{A\}\_\{c\}\\mathbf\{G\}\_\{c\}\\mathbf\{A\}\_\{c\}^\{\\top\}\+\\bm\{\\Sigma\}\_\{c\},~\\,\\textrm\{and\}~\\,\\mathbf\{G\}\_\{h\}=\\mathbf\{A\}\_\{h\}\\mathbf\{G\}\_\{h\}\\mathbf\{A\}\_\{h\}^\{\\top\}\+\\bm\{\\Sigma\}\_\{h\}\.\(10\)The covariances in \([10](https://arxiv.org/html/2607.28667#S3.E10)\) are equivalently given by the convergent series𝐆c=∑j=0∞𝐀cj​𝚺c​\(𝐀cj\)⊤\\mathbf\{G\}\_\{c\}=\\sum\_\{j=0\}^\{\\infty\}\\mathbf\{A\}\_\{c\}^\{j\}\\bm\{\\Sigma\}\_\{c\}\(\\mathbf\{A\}\_\{c\}^\{j\}\)^\{\\top\}and𝐆h=∑j=0∞𝐀hj​𝚺h​\(𝐀hj\)⊤\\mathbf\{G\}\_\{h\}=\\sum\_\{j=0\}^\{\\infty\}\\mathbf\{A\}\_\{h\}^\{j\}\\bm\{\\Sigma\}\_\{h\}\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\. Note that the existence, convergence, and uniqueness of these solutions is a classical result in linear systems theory \(Hamilton \([2020](https://arxiv.org/html/2607.28667#bib.bib4)\); Kailath \([1980](https://arxiv.org/html/2607.28667#bib.bib5)\)\)\.

![Refer to caption](https://arxiv.org/html/2607.28667v1/x3.png)Figure 2:Extended DMD is applied to the HaluEval datasetLiet al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib21)\)dataset\. 80,000 samples of𝝃k\(h\)\\bm\{\\xi\}\_\{k\}^\{\(h\)\}and𝝃k\(c\)\\bm\{\\xi\}\_\{k\}^\{\(c\)\}are taken; the distributions are well\-approximated by Gaussian curves \(left\) with mean and variance chosen appropriately\. An estimate of the covariances is also provided \(right\)\.When applied to the HaluEvalLiet al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib21)\)dataset, extended DMD yields predictive models that are well\-approximated by equation[7](https://arxiv.org/html/2607.28667#S3.E7)and equation[8](https://arxiv.org/html/2607.28667#S3.E8)\. After fitting matrices𝐀h\\mathbf\{A\}\_\{h\}and𝐀c\\mathbf\{A\}\_\{c\}with 20% of the dataset, 80,000 samples of𝝃k\(h\)\\bm\{\\xi\}\_\{k\}^\{\(h\)\}and𝝃k\(c\)\\bm\{\\xi\}\_\{k\}^\{\(c\)\}are taken over hallucinated responses, with distributions of the responses shown for the first 3 SVD modes in[2](https://arxiv.org/html/2607.28667#S3.F2)\. Gaussian curves with identical mean and variance are superimposed highlighting that the distribution of errors is well approximated by a normal distribution\. The noise covariance is also computed and included in Fig\.[2](https://arxiv.org/html/2607.28667#S3.F2)\.

### 3\.3Outline

Fig\.[3](https://arxiv.org/html/2607.28667#S3.F3)summarizes the logical dependencies between the key DS quantities already defined and our three main results\.

Dynamical System parameters𝐀c,𝐀h,𝚺c,𝚺h,𝐦c,𝐦h\\mathbf\{A\}\_\{c\},\\mathbf\{A\}\_\{h\},\\bm\{\\Sigma\}\_\{c\},\\bm\{\\Sigma\}\_\{h\},\\mathbf\{m\}\_\{c\},\\mathbf\{m\}\_\{h\}Stationary distributionspc=𝒩​\(𝐏​𝝁c,𝚪c\)p\_\{c\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{c\},\\bm\{\\Gamma\}\_\{c\}\)ph=𝒩​\(𝐏​𝝁h,𝚪h\)p\_\{h\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{h\},\\bm\{\\Gamma\}\_\{h\}\)Dynamical discriminabilityδ2=𝔼​\[‖𝐏​𝚫​𝐳k‖2\]\\delta^\{2\}=\\mathbb\{E\}\[\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\\|^\{2\}\]Intertwining mapG2\(𝜽2\)≈𝐓​G2\(𝜽1\)\+𝒓G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\approx\\mathbf\{T\}\\,G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\+\\bm\{r\}Result[1](https://arxiv.org/html/2607.28667#Thmtheorem1)dTVd\_\{\\operatorname\{TV\}\}can be≈0\\approx 0⇒Perrorstatic≈1/2\\Rightarrow P\_\{\\mathrm\{error\}\}^\{\\mathrm\{static\}\}\\approx 1/2Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)Perror≤e−c​L​δ2/σ2P\_\{\\mathrm\{error\}\}\\leq e^\{\-cL\\delta^\{2\}/\\sigma^\{2\}\}⇒\\Rightarrowaccuracy↑\\uparrowwithLLResult[3](https://arxiv.org/html/2607.28667#Thmtheorem3)δ𝜽1→𝜽22≥σmin2​δ2−𝒪​\(χ2\)\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\\!\\to\\\!\\bm\{\\theta\}\_\{2\}\}\\geq\\sigma\_\{\\min\}^\{2\}\\delta^\{2\}\-\\mathcal\{O\}\(\\chi^\{2\}\)⇒\\Rightarrowtransfer possiblemotivatesextendsFigure 3:Logical dependencies between the key DS variables and the obtained results\.

## 4The Dynamical Discriminability

Given a high\-dimensional embedding token sequence\{𝒛k\}ℓ=1L\\\{\\bm\{z\}\_\{k\}\\\}\_\{\\ell=1\}^\{L\}of lengthLL, we seek to determine whether it originates from dynamical system \(ℋc\\mathcal\{H\}\_\{c\}orℋh\\mathcal\{H\}\_\{h\}\) and characterize how the distinguishability between the two depends onLLand the parameters of the dynamical systems in \([7](https://arxiv.org/html/2607.28667#S3.E7)\) and \([8](https://arxiv.org/html/2607.28667#S3.E8)\)\. Toward this goal, we define the*operator difference*:

𝚫≡𝐀c−𝐀h\.\\bm\{\\Delta\}\\equiv\\mathbf\{A\}\_\{c\}\-\\mathbf\{A\}\_\{h\}\.\(11\)
###### Definition 1\(Dynamical discriminability\)\.

The*dynamical discriminability*of the embedding underℋc\\mathcal\{H\}\_\{c\}is:

δ2≡𝔼​\[‖𝐏​𝚫​𝐳k‖2\]=tr​\(𝐏​𝚫​𝐆c​𝚫⊤​𝐏⊤\)\+‖𝐏​𝚫​𝝁c‖2,\\delta^\{2\}\\equiv\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\\right\]=\\mathrm\{tr\}\\\!\\left\(\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{G\}\_\{c\}\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\right\)\+\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}\\right\\\|^\{2\},\(12\)where𝐆c\\mathbf\{G\}\_\{c\}is the stationary covariance and𝛍c\\bm\{\\mu\}\_\{c\}is the stationary mean underℋc\\mathcal\{H\}\_\{c\}\. The*mean\-bias interaction*is:

βc≡2​𝐦c⊤​𝐏⊤​𝐏​𝚫​𝝁c\.\\beta\_\{c\}\\equiv 2\\,\\mathbf\{m\}\_\{c\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}\.\(13\)

The discriminabilityδ2≥0\\delta^\{2\}\\geq 0measures the expected squared model\-mismatch signal per time step and represents the systematic error that arises when the wrong model𝐀h\\mathbf\{A\}\_\{h\}is used to predict a trajectory generated by𝐀c\\mathbf\{A\}\_\{c\}\. The mean\-bias interactionβc\\beta\_\{c\}\(which can be positive, negative, or zero\) captures the correlation between the noise mean and the model\-mismatch signal evaluated at the stationary operating point\. When the noise has zero mean \(𝐦c=𝟎\\mathbf\{m\}\_\{c\}=\\mathbf\{0\}\), we have𝝁c=𝟎\\bm\{\\mu\}\_\{c\}=\\mathbf\{0\}andβc=0\\beta\_\{c\}=0, and the discriminability reduces toδ2=tr​\(𝐏​𝚫​𝐆c​𝚫⊤​𝐏⊤\)\\delta^\{2\}=\\mathrm\{tr\}\(\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{G\}\_\{c\}\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\)\.

To build intuition, consider what happens when we use the*wrong*model to predict the next observation\. Suppose the true dynamics followℋc\\mathcal\{H\}\_\{c\}, so that𝐳k\+1=𝐀c​𝐳k\+𝝃k\(c\)\\mathbf\{z\}\_\{k\+1\}=\\mathbf\{A\}\_\{c\}\\mathbf\{z\}\_\{k\}\+\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\. If we instead predict using𝐀h\\mathbf\{A\}\_\{h\}, the prediction for the observable component is𝐲^k\+1=𝐏𝐀h​𝐳k\\widehat\{\\mathbf\{y\}\}\_\{k\+1\}=\\mathbf\{P\}\\mathbf\{A\}\_\{h\}\\mathbf\{z\}\_\{k\}, while the true observable evolves as𝐲k\+1=𝐏𝐀c​𝐳k\+𝐏​𝝃k\(c\)\\mathbf\{y\}\_\{k\+1\}=\\mathbf\{P\}\\mathbf\{A\}\_\{c\}\\mathbf\{z\}\_\{k\}\+\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\. The prediction error therefore decomposes as:

𝐲k\+1−𝐲^k\+1=𝐏​\(𝐀c−𝐀h\)​𝐳k\+𝐏​𝝃k\(c\)=𝐏​𝚫​𝐳k⏟model\-mismatch signal\+𝐏​𝝃k\(c\)⏟noise\.\\mathbf\{y\}\_\{k\+1\}\-\\widehat\{\\mathbf\{y\}\}\_\{k\+1\}=\\mathbf\{P\}\(\\mathbf\{A\}\_\{c\}\-\\mathbf\{A\}\_\{h\}\)\\mathbf\{z\}\_\{k\}\+\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}=\\underbrace\{\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\}\_\{\\text\{model\-mismatch signal\}\}\+\\underbrace\{\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\}\_\{\\text\{noise\}\}\.\(14\)The model\-mismatch signal𝐏​𝚫​𝐳k\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}has two components: a random component driven by the stationary covariance𝐆c\\mathbf\{G\}\_\{c\}, and a deterministic offset𝐏​𝚫​𝝁c\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}arising from the stationary mean\. The expected squared magnitude of this signal is:

δ2=𝔼​\[‖𝐏​𝚫​𝐳k‖2\]=\(a\)𝔼​\[𝐳k⊤​𝚫⊤​𝐏⊤​𝐏​𝚫​𝐳k\]\\displaystyle\\delta^\{2\}=\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\\right\]\\stackrel\{\{\\scriptstyle\(a\)\}\}\{\{=\}\}\\mathbb\{E\}\\\!\\left\[\\mathbf\{z\}\_\{k\}^\{\\top\}\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\]=tr​\(𝚫⊤​𝐏⊤​𝐏​𝚫​𝔼​\[𝐳k​𝐳k⊤\]\)\\displaystyle=\\mathrm\{tr\}\\\!\\left\(\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\,\\mathbb\{E\}\[\\mathbf\{z\}\_\{k\}\\mathbf\{z\}\_\{k\}^\{\\top\}\]\\right\)\(15\)=\(b\)tr​\(𝐏​𝚫​\(𝐆c\+𝝁c​𝝁c⊤\)​𝚫⊤​𝐏⊤\),\\displaystyle\\stackrel\{\{\\scriptstyle\(b\)\}\}\{\{=\}\}\\mathrm\{tr\}\\\!\\left\(\\mathbf\{P\}\\bm\{\\Delta\}\(\\mathbf\{G\}\_\{c\}\+\\bm\{\\mu\}\_\{c\}\\bm\{\\mu\}\_\{c\}^\{\\top\}\)\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\right\),where \(a\) follows from the identity𝔼​\[𝐳⊤​𝐌​𝐳\]=tr​\(𝐌​𝔼​\[𝐳​𝐳⊤\]\)\\mathbb\{E\}\[\\mathbf\{z\}^\{\\top\}\\mathbf\{M\}\\,\\mathbf\{z\}\]=\\mathrm\{tr\}\(\\mathbf\{M\}\\,\\mathbb\{E\}\[\\mathbf\{z\}\\,\\mathbf\{z\}^\{\\top\}\]\)and \(b\) uses the decomposition of the second moment𝔼​\[𝐳k​𝐳k⊤\]=𝐆c\+𝝁c​𝝁c⊤\\mathbb\{E\}\[\\mathbf\{z\}\_\{k\}\\mathbf\{z\}\_\{k\}^\{\\top\}\]=\\mathbf\{G\}\_\{c\}\+\\bm\{\\mu\}\_\{c\}\\bm\{\\mu\}\_\{c\}^\{\\top\}into covariance and mean outer product\.

The dynamical discriminabilityδ2\\delta^\{2\}captures two distinct sources of discriminative information\. The first one is the operator difference𝚫=𝐀c−𝐀h\\bm\{\\Delta\}=\\mathbf\{A\}\_\{c\}\-\\mathbf\{A\}\_\{h\}which encodes how differently the two dynamical regimes evolve the state\. If𝐀c=𝐀h\\mathbf\{A\}\_\{c\}=\\mathbf\{A\}\_\{h\}, then𝚫=𝟎\\bm\{\\Delta\}=\\mathbf\{0\}andδ2=0\\delta^\{2\}=0, meaning the two systems are indistinguishable from their dynamics regardless of the embedding\. The second aspect is the state distribution, characterized jointly by the covariance𝐆c\\mathbf\{G\}\_\{c\}and the mean𝝁c\\bm\{\\mu\}\_\{c\}, which determines which regions of the state space the trajectory typically visits\. Even if𝚫\\bm\{\\Delta\}is large in some directions, those directions only contribute toδ2\\delta^\{2\}if the stationary distribution places significant probability mass along them\. However, if𝐆c\\mathbf\{G\}\_\{c\}concentrates the state along directions where𝚫\\bm\{\\Delta\}acts strongly, or if the stationary mean𝝁c\\bm\{\\mu\}\_\{c\}has a large component in a direction where𝚫\\bm\{\\Delta\}acts, thenδ2\\delta^\{2\}will be large\.

A natural question arises: how reliably can a classifier distinguish whether𝐲k\\mathbf\{y\}\_\{k\}is drawn underℋc\\mathcal\{H\}\_\{c\}orℋh\\mathcal\{H\}\_\{h\}? In the next section, we prove theoretical bounds on this DS discrimination task\.

## 5Sub\-optimality of disregarding dynamics

Here we establish that the marginal stationary distributions of the observables𝐲k\\mathbf\{y\}\_\{k\}under the two hypotheses can have bounded divergence even when the dynamics are very different\. This provides a theoretical classification error floor when the dynamic behind the generation of𝐲k\\mathbf\{y\}\_\{k\}is disregarded\.

###### Result 1\(marginal density analysis limitation\)\.

Under Assumptions[1](https://arxiv.org/html/2607.28667#Thmassumption1)and[2](https://arxiv.org/html/2607.28667#Thmassumption2), letpc=𝒩​\(𝐏​𝛍c,𝚪c\)p\_\{c\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{c\},\\bm\{\\Gamma\}\_\{c\}\)andph=𝒩​\(𝐏​𝛍h,𝚪h\)p\_\{h\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{h\},\\bm\{\\Gamma\}\_\{h\}\)be the stationary distributions of the observables𝐲k\\mathbf\{y\}\_\{k\}underℋc\\mathcal\{H\}\_\{c\}andℋh\\mathcal\{H\}\_\{h\}, respectively, where𝚪c=𝐏𝐆c​𝐏⊤\\bm\{\\Gamma\}\_\{c\}=\\mathbf\{P\}\\mathbf\{G\}\_\{c\}\\mathbf\{P\}^\{\\top\}and𝚪h=𝐏𝐆h​𝐏⊤\\bm\{\\Gamma\}\_\{h\}=\\mathbf\{P\}\\mathbf\{G\}\_\{h\}\\mathbf\{P\}^\{\\top\}are the observable\-space stationary covariances\. Defining∥⋅∥\\left\\\|\\cdot\\right\\\|as the Mahalanobis distance, the total variation \(TV\) distancedTV​\(pc,ph\)d\_\{\\operatorname\{TV\}\}\(p\_\{c\},p\_\{h\}\)betweenpcp\_\{c\}andphp\_\{h\}satisfies:

dTV​\(pc,ph\)≤12​‖𝚪c−1/2​\(𝚪c−𝚪h\)​𝚪c−1/2‖F\+12​‖𝐏​\(𝝁c−𝝁h\)‖𝚪h−1,d\_\{\\operatorname\{TV\}\}\(p\_\{c\},p\_\{h\}\)\\leq\\frac\{1\}\{2\}\\left\\\|\\bm\{\\Gamma\}\_\{c\}^\{\-1/2\}\(\\bm\{\\Gamma\}\_\{c\}\-\\bm\{\\Gamma\}\_\{h\}\)\\bm\{\\Gamma\}\_\{c\}^\{\-1/2\}\\right\\\|\_\{F\}\+\\frac\{1\}\{2\}\\left\\\|\\mathbf\{P\}\(\\bm\{\\mu\}\_\{c\}\-\\bm\{\\mu\}\_\{h\}\)\\right\\\|\_\{\\bm\{\\Gamma\}\_\{h\}^\{\-1\}\},\(16\)and the Bayes error of any classifier using only the marginal distribution satisfies:

Perror≥12​\(1−dTV​\(pc,ph\)\)\.P\_\{\\mathrm\{error\}\}\\geq\\frac\{1\}\{2\}\\left\(1\-d\_\{\\operatorname\{TV\}\}\(p\_\{c\},p\_\{h\}\)\\right\)\.\(17\)Moreover, there exist pairs\(𝐀c,𝐀h\)\(\\mathbf\{A\}\_\{c\},\\mathbf\{A\}\_\{h\}\)and noise parameters\(𝐦c,𝚺c,𝐦h,𝚺h\)\(\\mathbf\{m\}\_\{c\},\\bm\{\\Sigma\}\_\{c\},\\mathbf\{m\}\_\{h\},\\bm\{\\Sigma\}\_\{h\}\)with‖𝚫‖F\\left\\\|\\bm\{\\Delta\}\\right\\\|\_\{F\}bounded away from zero such that both‖𝚪c−𝚪h‖F\\left\\\|\\bm\{\\Gamma\}\_\{c\}\-\\bm\{\\Gamma\}\_\{h\}\\right\\\|\_\{F\}and‖𝐏​\(𝛍c−𝛍h\)‖\\left\\\|\\mathbf\{P\}\(\\bm\{\\mu\}\_\{c\}\-\\bm\{\\mu\}\_\{h\}\)\\right\\\|are arbitrarily small\.

This result justifies the situation encountered in practice where the SVD mode distributions \(marginals\) of factual and hallucinated embeddings are nearly indistinguishable, but the transition dynamics differ\. This was observed empirically in\(Wilson and Akrout,[2026](https://arxiv.org/html/2607.28667#bib.bib7)\)where the first two SVD modes of the token embeddings showed distributional differences that are insufficient for hallucination detection\. Note that when the noise means differ \(𝐦c≠𝐦h\\mathbf\{m\}\_\{c\}\\neq\\mathbf\{m\}\_\{h\}\), the second term in \([16](https://arxiv.org/html/2607.28667#S5.E16)\) can be nonzero even for scaled rotations with equal noise covariance, providing limited discriminability through the mean shift alone\.

Rather than classifying based on marginal distributions alone, we ask whether incorporating the prediction error as described in Section[3\.1](https://arxiv.org/html/2607.28667#S3.SS1)leads to a significant improvement in classification performance, which is the focus of the next section\.

## 6Exponential Separability via Dynamical System Prediction

In this section, we prove that the dynamical prediction approach achieves exponentially decaying classification error\.

###### Result 2\(Dynamical separability\)\.

Under Assumptions[1](https://arxiv.org/html/2607.28667#Thmassumption1)and[2](https://arxiv.org/html/2607.28667#Thmassumption2), supposeδ2\+βc\>0\\delta^\{2\}\+\\beta\_\{c\}\>0whereδ2\\delta^\{2\}andβc\\beta\_\{c\}are defined in \([12](https://arxiv.org/html/2607.28667#S4.E12)\) and \([13](https://arxiv.org/html/2607.28667#S4.E13)\)\. Letη∈ℝ\\eta\\in\\mathbb\{R\}be an arbitrary classification threshold and define the mean discriminability gap underℋc\\mathcal\{H\}\_\{c\}as

μL≡𝔼​\[Δ​ℰ∣ℋc\]\>0\.\\mu\_\{L\}\\equiv\\mathbb\{E\}\[\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\]\>0\.\(18\)Then, for any thresholdη<μL\\eta<\\mu\_\{L\}, the misclassification probability satisfies:

ℙ​\(Δ​ℰ​<η∣​ℋc\)≤2​exp⁡\(−\(L−1\)​\(μL−η\)2C1​\(σmax,c2​\(δ2\+\|βc\|\)\+σmax,c2​R2​λmax​\(𝐆c\)\+Pm​R2​λmax​\(𝐆c\)\)\),\\mathbb\{P\}\\\!\\left\(\\Delta\\mathcal\{E\}<\\eta\\mid\\mathcal\{H\}\_\{c\}\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{\(L\-1\)\(\\mu\_\{L\}\-\\eta\)^\{2\}\}\{C\_\{1\}\\left\(\\sigma\_\{\\max,c\}^\{2\}\(\\delta^\{2\}\+\|\\beta\_\{c\}\|\)\+\\sigma\_\{\\max,c\}^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\+P\_\{m\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\\right\)\}\\right\),\(19\)whereσmax,c2=λmax​\(𝐏​𝚺c​𝐏⊤\)\\sigma\_\{\\max,c\}^\{2\}=\\lambda\_\{\\max\}\(\\mathbf\{P\}\\bm\{\\Sigma\}\_\{c\}\\mathbf\{P\}^\{\\top\}\),R=‖𝐏​𝚫‖R=\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\right\\\|,Pm=‖𝐏𝐦c‖2P\_\{m\}=\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}and the constantC1\>0C\_\{1\}\>0\. An similar bound holds underℋh\\mathcal\{H\}\_\{h\}with𝛍c,𝐆c,𝚺c,𝐦c\\bm\{\\mu\}\_\{c\},\\mathbf\{G\}\_\{c\},\\bm\{\\Sigma\}\_\{c\},\\mathbf\{m\}\_\{c\}replaced by𝛍h,𝐆h,𝚺h,𝐦h\\bm\{\\mu\}\_\{h\},\\mathbf\{G\}\_\{h\},\\bm\{\\Sigma\}\_\{h\},\\mathbf\{m\}\_\{h\}, and with the event\{Δ​ℰ≥η\}\\\{\\Delta\\mathcal\{E\}\\geq\\eta\\\}\.

Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)establishes that the misclassification probability decays exponentially in the sequence lengthLL, but the bound involves several variables \(i\.e\.,δ2\\delta^\{2\},βc\\beta\_\{c\},σmax,c2\\sigma\_\{\\max,c\}^\{2\},RR,λmax​\(𝐆c\)\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\) whose interplay may obscure the essential scaling behavior\. To present the result into a single interpretable quantity, we introduce the*separability factor*S​\(L\)S\(L\), defined as the standardized mean of the test statisticΔ​ℰ\\Delta\\mathcal\{E\}underℋc\\mathcal\{H\}\_\{c\}\. This quantity, borrowed from signal detection theory, measures how many standard deviations the expected score is separated from zero: whenS​\(L\)S\(L\)is large, the distribution ofΔ​ℰ\\Delta\\mathcal\{E\}underℋc\\mathcal\{H\}\_\{c\}is concentrated far from the decision boundary, making misclassification unlikely\. The following corollary shows thatS​\(L\)S\(L\)grows asL\\sqrt\{L\}, providing a direct link between sequence length and classification reliability, and yields an explicit formula for the minimum number of tokens needed to achieve a target error rate\.

###### Corollary 1\(Convergence rate and minimum sequence length\)\.

Under the conditions of Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2), the separability factorS​\(L\)S\(L\)ofΔ​ℰ\\Delta\\mathcal\{E\}underℋc\\mathcal\{H\}\_\{c\}grows asL\\sqrt\{L\}:

S​\(L\)≡𝔼​\[Δ​ℰ∣ℋc\]Var​\(Δ​ℰ∣ℋc\)=Θ​\(L⋅\(δ2\+βc\)σmax,c2\)\.S\(L\)\\equiv\\frac\{\\mathbb\{E\}\[\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\]\}\{\\sqrt\{\\mathrm\{Var\}\(\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\)\}\}=\\Theta\\\!\\left\(\\sqrt\{L\}\\cdot\\frac\{\(\\delta^\{2\}\+\\beta\_\{c\}\)\}\{\\sigma\_\{\\max,c\}^\{2\}\}\\right\)\.\(20\)Consequently, for any thresholdη\\etasatisfying\|η\|≤μL/2\|\\eta\|\\leq\\mu\_\{L\}/2, the number of tokens needed for classification with error probability≤ϵ\\leq\\epsilonscales as:

L∗=𝒪​\(σmax,c4\(δ2\+βc\)2​log⁡1ϵ\)\.L^\{\*\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\sigma\_\{\\max,c\}^\{4\}\}\{\(\\delta^\{2\}\+\\beta\_\{c\}\)^\{2\}\}\\log\\frac\{1\}\{\\epsilon\}\\right\)\.\(21\)

###### Proof\.

See Appendix[Appendix C: Proof of Corollary1](https://arxiv.org/html/2607.28667#S3.SSx1)\. ∎

This result explains the empirical observation that larger embedding models \(e\.g\., Llama\-3\) achieve near\-perfect classification even with short sequences, while smaller models \(e\.g\., Jina\-v5 with 30M parameters\) require longer sequences\.

![Refer to caption](https://arxiv.org/html/2607.28667v1/x4.png)Figure 4:Estimates of the separability factor \(solid line\) from equation[20](https://arxiv.org/html/2607.28667#S6.E20)for three different text embedding models for the HaluEvalLiet al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib21)\)dataset\. Dashed lines of the formK​LK\\sqrt\{L\}are fit to the data, whereKKis a constant used for fitting\.Extended DMD is applied to the HaluEvalLiet al\.\([2023](https://arxiv.org/html/2607.28667#bib.bib21)\)dataset to obtain predictive models of the form equation[7](https://arxiv.org/html/2607.28667#S3.E7)and equation[8](https://arxiv.org/html/2607.28667#S3.E8)using token embeddings fromQwen,MistralandLlama\-3\. On the remaining 8000 hallucinated responses not used for model inference, the differential residual scoreΔ​ℰ\\Delta\\mathcal\{E\}from equation[6](https://arxiv.org/html/2607.28667#S3.E6)is computed for 8000 hallucinated responses\.S​\(L\)S\(L\)is computed for responses of different length,LL, and plotted in Fig\.[4](https://arxiv.org/html/2607.28667#S6.F4)\. As predicted by Corollary[1](https://arxiv.org/html/2607.28667#Thmcorollary1), the separability factor is proportional toL\\sqrt\{L\}\.

## 7Cross\-Embedding Generalization

An important practical question is whether the dynamical models fitted using one embedding model can transfer to another\. It has been show in \(Wilson and Akrout \([2026](https://arxiv.org/html/2607.28667#bib.bib7)\)\) that fitting Koopman operators𝐀c\(𝜽1\)\\mathbf\{A\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}and𝐀h\(𝜽1\)\\mathbf\{A\}\_\{h\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}on token embeddings from one model parametrized by𝜽1\\bm\{\\theta\}\_\{1\}then run prediction on token embeddings from a different model parametrized by𝜽2\\bm\{\\theta\}\_\{2\}achieve above\-chance accuracy despite operating in an entirely different observation space\. We now provide a theoretical explanation for this phenomenon\.

Consider two embedding modelsG2\(𝜽1\):𝒬→ℝM1G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}:\\mathcal\{Q\}\\to\\mathbb\{R\}^\{M\_\{1\}\}andG2\(𝜽2\):𝒬→ℝM2G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}:\\mathcal\{Q\}\\to\\mathbb\{R\}^\{M\_\{2\}\}, where𝒬\\mathcal\{Q\}denotes the token space and𝜽1,𝜽2\\bm\{\\theta\}\_\{1\},\\bm\{\\theta\}\_\{2\}parametrize the respective model architectures and weights\. Each embedding model induces its own observation space and, consequently, its own Koopman operator approximations\. We write:

𝐲k\(𝜽i\)\\displaystyle\\mathbf\{y\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}=G2\(𝜽i\)​\(qk\)∈ℝMi,i∈\{1,2\},\\displaystyle=G\_\{2\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}\(q\_\{k\}\)\\in\\mathbb\{R\}^\{M\_\{i\}\},\\qquad i\\in\\\{1,2\\\},\(22\)for the token embeddings under each model, and denote the corresponding fitted Koopman approximations by𝐀c\(𝜽i\),𝐀h\(𝜽i\)\\mathbf\{A\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{i\}\)\},\\mathbf\{A\}\_\{h\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}with operator differences𝚫\(𝜽i\)=𝐀c\(𝜽i\)−𝐀h\(𝜽i\)\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}=\\mathbf\{A\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}\-\\mathbf\{A\}\_\{h\}^\{\(\\bm\{\\theta\}\_\{i\}\)\}\.

The central question is: under what conditions on the pair\(G2\(𝜽1\),G2\(𝜽2\)\)\(G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\},G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\)can dynamical models fitted in the𝜽1\\bm\{\\theta\}\_\{1\}\-observation space provide discriminative information about trajectories observed in the𝜽2\\bm\{\\theta\}\_\{2\}\-observation space?

###### Assumption 3\(Approximate intertwining\)\.

There exists a linear map𝐓∈ℝM2×M1\\mathbf\{T\}\\in\\mathbb\{R\}^\{M\_\{2\}\\times M\_\{1\}\}such that the two embedding models are approximately related by:

G2\(𝜽2\)​\(q\)=𝐓​G2\(𝜽1\)​\(q\)\+𝒓​\(q\),∀q∈𝒬,G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\(q\)=\\mathbf\{T\}\\,G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\(q\)\+\\bm\{r\}\(q\),\\qquad\\forall\\,q\\in\\mathcal\{Q\},\(23\)where𝐫:𝒬→ℝM2\\bm\{r\}:\\mathcal\{Q\}\\to\\mathbb\{R\}^\{M\_\{2\}\}is a residual satisfying𝔼​\[‖𝐫​\(qk\)‖2\]≤χ2\\mathbb\{E\}\[\\left\\\|\\bm\{r\}\(q\_\{k\}\)\\right\\\|^\{2\}\]\\leq\\chi^\{2\}along the token sequences\.

The map𝐓\\mathbf\{T\}acts as a*translation*between the two observation spaces: it captures the extent to which the geometric structure of one embedding is a linear transformation of the other\. The residual𝒓\\bm\{r\}measures the*intertwining defect*, i\.e\., the degree to which this linear relationship breaks down\. Whenχ2=0\\chi^\{2\}=0, the two embeddings are exactly linearly related, and any dynamical structure in one space is perfectly preserved in the other\. Whenχ2\>0\\chi^\{2\}\>0, the transfer is approximate\.

###### Definition 2\(Cross\-embedding discriminability\)\.

The*cross\-embedding discriminability*when fitting onG2\(𝛉1\)G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}and testing onG2\(𝛉2\)G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}is:

δ𝜽1→𝜽22≡𝔼​\[‖𝐏\(𝜽2\)​𝚫\(𝜽1\)​𝐳k\(𝜽2\)‖2\],\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\\equiv\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\right\\\|^\{2\}\\right\],\(24\)where𝐳k\(𝛉2\)\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}is the lifted observable from the𝛉2\\bm\{\\theta\}\_\{2\}\-embedding and𝐏\(𝛉2\)\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}is the corresponding projection onto the observable components\.

###### Result 3\(Cross\-embedding transfer bound\)\.

Under Assumptions[1](https://arxiv.org/html/2607.28667#Thmassumption1),[2](https://arxiv.org/html/2607.28667#Thmassumption2), and[3](https://arxiv.org/html/2607.28667#Thmassumption3), the cross\-embedding discriminability satisfies:

δ𝜽1→𝜽22≥σmin2​\(𝐓\)​δ2​\(𝜽1\)−C2​\(χ2​‖𝚫\(𝜽1\)‖2​λmax​\(𝐆c\(𝜽2\)\)\+χ2​δ2​\(𝜽1\)\),\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\\geq\\sigma\_\{\\min\}^\{2\}\(\\mathbf\{T\}\)\\,\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\-C\_\{2\}\\\!\\left\(\\chi^\{2\}\\left\\\|\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\lambda\_\{\\max\}\\big\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\big\)\+\\chi^\{2\}\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\\right\),\(25\)whereσmin​\(𝐓\)\\sigma\_\{\\min\}\(\\mathbf\{T\}\)is the smallest singular value of𝐓\\mathbf\{T\},C2\>0C\_\{2\}\>0is a constant, andδ2​\(𝛉1\)=tr​\(𝐏\(𝛉1\)​𝚫\(𝛉1\)​𝐆c\(𝛉1\)​\(𝚫\(𝛉1\)\)⊤​\(𝐏\(𝛉1\)\)⊤\)\+‖𝐏\(𝛉1\)​𝚫\(𝛉1\)​𝛍c\(𝛉1\)‖2\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)=\\mathrm\{tr\}\(\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\(\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\)^\{\\top\}\(\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\)^\{\\top\}\)\+\\left\\\|\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\mu\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}is the discriminability under𝛉1\\bm\{\\theta\}\_\{1\}\.

Consequently, the classification accuracy when fitting onG2\(𝛉1\)G\_\{2\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}and testing onG2\(𝛉2\)G\_\{2\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}satisfies:

Accuracy𝜽1→𝜽2\\displaystyle\\mathrm\{Accuracy\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}=12​ℙ​\(Δ​ℰ≥0∣ℋc\)\+12​ℙ​\(Δ​ℰ​<0∣​ℋh\)\\displaystyle=\\frac\{1\}\{2\}\\mathbb\{P\}\(\\Delta\\mathcal\{E\}\\geq 0\\mid\\mathcal\{H\}\_\{c\}\)\+\\frac\{1\}\{2\}\\mathbb\{P\}\(\\Delta\\mathcal\{E\}<0\\mid\\mathcal\{H\}\_\{h\}\)\(26\)≥12\+12​\(1−2​exp⁡\(−\(L−1\)​\(δ𝜽1→𝜽22\)2C1​\(σmax,c2​δ𝜽1→𝜽22\+σmax,c2​R2​λmax​\(𝐆c\(𝜽2\)\)\)\)\)\.\\displaystyle\\geq\\frac\{1\}\{2\}\+\\frac\{1\}\{2\}\\left\(1\-2\\exp\\\!\\left\(\-\\frac\{\(L\-1\)\\left\(\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\\right\)^\{2\}\}\{C\_\{1\}\\big\(\\sigma\_\{\\max,c\}^\{2\}\\,\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\+\\sigma\_\{\\max,c\}^\{2\}\\,R^\{2\}\\,\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\)\\big\)\}\\right\)\\right\)\.

For empirical validation of the bound in \([25](https://arxiv.org/html/2607.28667#S7.E25)\), we select a model pair \(Qwen3,F2LLM\) whose cross\-transfer behavior in\(Wilson and Akrout,[2026](https://arxiv.org/html/2607.28667#bib.bib7)\)was intriguing: fitting the DS onQwen3embedding and testing onF2LLMembeddings led to a cross\-model accuracy of63\.0%63\.0\\%, while doing the opposite yielded an accuracy of50\.5%50\.5\\%\. We then evaluate the intertwining map𝐓\\mathbf\{T\}and defectχ2\\chi^\{2\}between the two embedding spaces\. As depicted in Fig\.[5](https://arxiv.org/html/2607.28667#S7.F5), while both mapping directions exhibit well\-conditioned singular value spectra for𝐓\\mathbf\{T\}across the subspace modes, their reconstruction residual errors differ drastically: the intertwining defect for mappingF2LLM→\\toQwen3\(χ2≈8\.8×106\\chi^\{2\}\\approx 8\.8\\times 10^\{6\}\) is nearly double that ofQwen3→\\toF2LLM\(χ2≈4\.8×106\\chi^\{2\}\\approx 4\.8\\times 10^\{6\}\)\. This aligns directly with our theoretical bound in Result[3](https://arxiv.org/html/2607.28667#Thmtheorem3): when transferring from a lower\-density embedding space \(e\.g\.,F2LLM\) to a denser and compact space \(e\.g\.,Qwen3\), the large intertwining defectχ2\\chi^\{2\}acts as a heavy penalty term that suppresses the transferable discriminabilityδ𝜽1→𝜽22\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}toward zero, causing accuracy to drop to50\.5%50\.5\\%\. However, Transferring fromQwen3toF2LLMincurs a substantially smaller defect, preserving sufficient dynamical separation to maintain an accuracy of63\.0%63\.0\\%without refitting\.

![Refer to caption](https://arxiv.org/html/2607.28667v1/x5.png)Figure 5:Empirical validation of cross\-embedding transfer betweenQwen3andF2LLM\.Left:Singular values of the linear intertwining map𝐓\\mathbf\{T\}across SVD modes\.Right:The reconstruction residual error, i\.e\., intertwining defectχ2\\chi^\{2\}between the two embedding models\.
## 8Conclusion

We have developed a mathematical framework for understanding the distinguishability guarantees of two dynamical systems to classify LLM responses in the embedding space\. First, we establish that classifiers relying solely on the marginal distribution of token embeddings face an irreducible accuracy floor since two dynamical systems can produce nearly identical stationary distributions while having completely different dynamical evolution\. Second, we showed that exploiting temporal dynamics through prediction residuals overcomes this limitation, with classification error decaying exponentially in sequence length at a rate governed by the dynamical distinguishability of the two systems\. This provides practical guidance for choosing the minimum response length needed for reliable detection\. Third, we addressed the surprising phenomenon of cross\-embedding transfer, demonstrating that dynamical models fitted on one embedding can transfer to another when the embedding spaces share similar geometric structure\. The transfer quality depends on affine geometric compatibility, the fitted discriminability space, and the extent of distortion between the fitted and target embedding spaces\. This explains why some embedding pairs transfer well while others do not\. Together, these results establish guarantees on the distinguishability between dynamical systems theory for LLM token generations, shifting the focus from static representations of individual tokens to the dynamical patterns that emerge across the embedding trajectory of a LLM response\.

Limitations and future directions: Our linear Koopman approximation could be extended to nonlinear models \(e\.g\., kernel EDMD or neural network\-based Koopman methods\) to capture richer dynamical structure\. The Gaussian noise assumption which is realistic after testing it on real\-world datasets could be relaxed to heavy tail distributions to broaden the applicability of the obtained bounds\. Future work can also devise a joint optimization over the embedding projection and the Koopman operators to seeking the projection that maximizesδ2\\delta^\{2\}for systematic embedding selection\. Such an algorithm will connect to the classical distinction between variance\-maximizing and discriminability\-maximizing projections\.

## References

- M\. Budišić, R\. Mohr, and I\. Mezić \(2012\)Applied koopmanism\.Chaos: An Interdisciplinary Journal of Nonlinear Science22\(4\)\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p2.1)\.
- R\. T\. Chen, Y\. Rubanova, J\. Bettencourt, and D\. K\. Duvenaud \(2018\)Neural ordinary differential equations\.Advances in neural information processing systems31\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p2.1)\.
- S\. Dhuliawala, M\. Komeili, J\. Xu, R\. Raileanu, X\. Li, A\. Celikyilmaz, and J\. Weston \(2024\)Chain\-of\-verification reduces hallucination in large language models\.InFindings of the association for computational linguistics: ACL 2024,pp\. 3563–3578\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p2.1)\.
- J\. D\. Hamilton \(2020\)Time series analysis\.Princeton university press\.Cited by:[§3\.2](https://arxiv.org/html/2607.28667#S3.SS2.p2.8)\.
- T\. Kailath \(1980\)Linear systems\.Vol\.156,Prentice\-Hall Englewood Cliffs, NJ\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p1.1),[§3\.2](https://arxiv.org/html/2607.28667#S3.SS2.p2.8)\.
- P\. Lewis, E\. Perez, A\. Piktus, F\. Petroni, V\. Karpukhin, N\. Goyal, H\. Küttler, M\. Lewis, W\. Yih, T\. Rocktäschel,et al\.\(2020\)Retrieval\-augmented generation for knowledge\-intensive nlp tasks\.Advances in neural information processing systems33,pp\. 9459–9474\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p2.1)\.
- J\. Li, X\. Cheng, W\. X\. Zhao, J\. Nie, and J\. Wen \(2023\)HaluEval: a large\-scale hallucination evaluation benchmark for large language models\.InProceedings of the 2023 Conference on Empirical Methods in Natural Language Processing,pp\. 6449–6464\.Cited by:[Figure 2](https://arxiv.org/html/2607.28667#S3.F2),[§3\.2](https://arxiv.org/html/2607.28667#S3.SS2.p3.4),[Figure 4](https://arxiv.org/html/2607.28667#S6.F4),[§6](https://arxiv.org/html/2607.28667#S6.p4.4)\.
- L\. Ljung \(1999\)System identification: theory for the user\.2nd edition,Prentice Hall\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p1.1)\.
- P\. Manakul, A\. Liusie, and M\. Gales \(2023\)Selfcheckgpt: zero\-resource black\-box hallucination detection for generative large language models\.InProceedings of the 2023 conference on empirical methods in natural language processing,pp\. 9004–9017\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p2.1)\.
- K\. Meng, D\. Bau, A\. Andonian, and Y\. Belinkov \(2022\)Locating and editing factual associations in gpt\.Advances in neural information processing systems35,pp\. 17359–17372\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p2.1)\.
- I\. Mezić \(2013\)Analysis of fluid flows via spectral properties of the koopman operator\.Annual review of fluid mechanics45\(1\),pp\. 357–378\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p2.1)\.
- M\. Rudelson and R\. Vershynin \(2013\)Hanson\-wright inequality and sub\-gaussian concentration\.Cited by:[Appendix B: Proof of dynamical separability Result](https://arxiv.org/html/2607.28667#S2.SSx1.p4.4)\.
- T\. Sarkar and A\. Rakhlin \(2019\)Near optimal finite time identification of arbitrary linear dynamical systems\.InInternational Conference on Machine Learning,pp\. 5610–5618\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p1.1)\.
- M\. A\. Sheikhi, G\. de Albuquerque Gleizer, P\. M\. Esfahani, and T\. Keviczky \(2025a\)Data\-driven fault isolation in linear time\-invariant systems: a subspace classification approach\.IEEE Control Systems Letters\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p1.1)\.
- M\. A\. Sheikhi, G\. de Albuquerque Gleizer, T\. Keviczky, and P\. Mohajerin Esfahani \(2025b\)Fault diagnosis in dynamical systems: geometric interpretation and tractable algorithms\.Annual Review of Control, Robotics, and Autonomous Systems9\.External Links:[Document](https://dx.doi.org/10.1146/annurev-control-030123-015422)Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p1.1)\.
- M\. Simchowitz, H\. Mania, S\. Tu, M\. I\. Jordan, and B\. Recht \(2018\)Learning without mixing: towards a sharp analysis of linear system identification\.InConference On Learning Theory,pp\. 439–473\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p1.1)\.
- M\. O\. Williams, I\. G\. Kevrekidis, and C\. W\. Rowley \(2015\)A data–driven approximation of the koopman operator: extending dynamic mode decomposition\.Journal of Nonlinear Science25\(6\),pp\. 1307–1346\.Cited by:[§2](https://arxiv.org/html/2607.28667#S2.p2.1),[§3\.1](https://arxiv.org/html/2607.28667#S3.SS1.p1.6)\.
- D\. Wilson and M\. Akrout \(2026\)Low\-cost black\-box detection of llm hallucinations via dynamical system prediction\.arXiv preprint arXiv:2605\.05134\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p1.1),[§1](https://arxiv.org/html/2607.28667#S1.p3.1),[§2](https://arxiv.org/html/2607.28667#S2.p2.1),[§5](https://arxiv.org/html/2607.28667#S5.p2.1),[§7](https://arxiv.org/html/2607.28667#S7.p1.4),[§7](https://arxiv.org/html/2607.28667#S7.p5.13)\.
- L\. Xin, G\. T\. Chiu, and S\. Sundaram \(2025\)Online change points detection for linear dynamical systems with finite sample guarantees\.Automatica171,pp\. 111894\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p1.1),[§2](https://arxiv.org/html/2607.28667#S2.p1.1)\.
- J\. Zhang, Z\. Li, K\. Das, B\. Malin, and S\. Kumar \(2023\)SAC3: reliable hallucination detection in black\-box language models via semantic\-aware cross\-check consistency\.InFindings of the Association for Computational Linguistics: EMNLP 2023,pp\. 15445–15458\.Cited by:[§1](https://arxiv.org/html/2607.28667#S1.p2.1)\.

## Appendices

### Appendix A: Proof of Static analysis limitation

For two multivariate Gaussianspc=𝒩​\(𝐏​𝝁c,𝚪c\)p\_\{c\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{c\},\\bm\{\\Gamma\}\_\{c\}\)andph=𝒩​\(𝐏​𝝁h,𝚪h\)p\_\{h\}=\\mathcal\{N\}\(\\mathbf\{P\}\\bm\{\\mu\}\_\{h\},\\bm\{\\Gamma\}\_\{h\}\), the Kullback\-Leibler divergence is:

DKL​\(pc∥ph\)=12​\[tr​\(𝚪h−1​𝚪c\)−d\+log⁡det𝚪hdet𝚪c\+‖𝐏​\(𝝁c−𝝁h\)‖𝚪h−12\]\.\{D\_\{\\mathrm\{KL\}\}\}\(p\_\{c\}\\\|p\_\{h\}\)=\\frac\{1\}\{2\}\\left\[\\mathrm\{tr\}\(\\bm\{\\Gamma\}\_\{h\}^\{\-1\}\\bm\{\\Gamma\}\_\{c\}\)\-d\+\\log\\frac\{\\det\\bm\{\\Gamma\}\_\{h\}\}\{\\det\\bm\{\\Gamma\}\_\{c\}\}\+\\left\\\|\\mathbf\{P\}\(\\bm\{\\mu\}\_\{c\}\-\\bm\{\\mu\}\_\{h\}\)\\right\\\|\_\{\\bm\{\\Gamma\}\_\{h\}^\{\-1\}\}^\{2\}\\right\]\.\(27\)By Pinsker’s inequality,dTV​\(pc,ph\)≤12​DKL​\(pc∥ph\)d\_\{\\operatorname\{TV\}\}\(p\_\{c\},p\_\{h\}\)\\leq\\sqrt\{\\frac\{1\}\{2\}\{D\_\{\\mathrm\{KL\}\}\}\(p\_\{c\}\\\|p\_\{h\}\)\}\. We bound the covariance and mean contributions separately\. Writing𝚪h=𝚪c\+𝐄\\bm\{\\Gamma\}\_\{h\}=\\bm\{\\Gamma\}\_\{c\}\+\\mathbf\{E\}where𝐄=𝚪h−𝚪c\\mathbf\{E\}=\\bm\{\\Gamma\}\_\{h\}\-\\bm\{\\Gamma\}\_\{c\}, and using the inequalitylog​det\(𝐈\+𝐌\)≤tr​\(𝐌\)\\log\\det\(\\mathbf\{I\}\+\\mathbf\{M\}\)\\leq\\mathrm\{tr\}\(\\mathbf\{M\}\)for‖𝐌‖<1\\left\\\|\\mathbf\{M\}\\right\\\|<1, the covariance\-only part of the KL divergence satisfies:

DKLcov≡12​\[tr​\(𝚪h−1​𝚪c\)−d\+log⁡det𝚪hdet𝚪c\]≤14​‖𝚪c−1/2​𝐄​𝚪c−1/2‖F2\.\{D\_\{\\mathrm\{KL\}\}\}^\{\\hskip\-9\.95863pt\\mathrm\{cov\}\}\\equiv\\frac\{1\}\{2\}\\left\[\\mathrm\{tr\}\(\\bm\{\\Gamma\}\_\{h\}^\{\-1\}\\bm\{\\Gamma\}\_\{c\}\)\-d\+\\log\\frac\{\\det\\bm\{\\Gamma\}\_\{h\}\}\{\\det\\bm\{\\Gamma\}\_\{c\}\}\\right\]\\leq\\frac\{1\}\{4\}\\left\\\|\\bm\{\\Gamma\}\_\{c\}^\{\-1/2\}\\mathbf\{E\}\\bm\{\\Gamma\}\_\{c\}^\{\-1/2\}\\right\\\|\_\{F\}^\{2\}\.\(28\)The mean contribution is the squared Mahalanobis distance:

DKLmean≡12​‖𝐏​\(𝝁c−𝝁h\)‖𝚪h−12\.\{D\_\{\\mathrm\{KL\}\}\}^\{\\hskip\-9\.95863pt\\mathrm\{mean\}\}\\equiv\\frac\{1\}\{2\}\\left\\\|\\mathbf\{P\}\(\\bm\{\\mu\}\_\{c\}\-\\bm\{\\mu\}\_\{h\}\)\\right\\\|\_\{\\bm\{\\Gamma\}\_\{h\}^\{\-1\}\}^\{2\}\.\(29\)UsingdTV≤12​\(DDKLcov\+DKLmean\)≤12​DKLcov\+12​DKLmeand\_\{\\operatorname\{TV\}\}\\leq\\sqrt\{\\frac\{1\}\{2\}\(D\_\{D\_\{\\mathrm\{KL\}\}\}^\{\\mathrm\{cov\}\}\+\{D\_\{\\mathrm\{KL\}\}\}^\{\\hskip\-9\.95863pt\\mathrm\{mean\}\}\)\}\\leq\\sqrt\{\\frac\{1\}\{2\}\{D\_\{\\mathrm\{KL\}\}\}^\{\\hskip\-9\.95863pt\\mathrm\{cov\}\}\}\+\\sqrt\{\\frac\{1\}\{2\}\{D\_\{\\mathrm\{KL\}\}\}^\{\\hskip\-9\.95863pt\\mathrm\{mean\}\}\}, we obtain the bound in \([16](https://arxiv.org/html/2607.28667#S5.E16)\)\.

Recall that the stationary covariances satisfy:

𝚪c\\displaystyle\\bm\{\\Gamma\}\_\{c\}=𝐏𝐆c​𝐏⊤=𝐏​\(∑j=0∞𝐀cj​𝚺c​\(𝐀cj\)⊤\)​𝐏⊤,\\displaystyle=\\mathbf\{P\}\\mathbf\{G\}\_\{c\}\\mathbf\{P\}^\{\\top\}=\\mathbf\{P\}\\left\(\\sum\_\{j=0\}^\{\\infty\}\\mathbf\{A\}\_\{c\}^\{j\}\\bm\{\\Sigma\}\_\{c\}\(\\mathbf\{A\}\_\{c\}^\{j\}\)^\{\\top\}\\right\)\\mathbf\{P\}^\{\\top\},\(30\)𝚪h\\displaystyle\\bm\{\\Gamma\}\_\{h\}=𝐏𝐆h​𝐏⊤=𝐏​\(∑j=0∞𝐀hj​𝚺h​\(𝐀hj\)⊤\)​𝐏⊤,\\displaystyle=\\mathbf\{P\}\\mathbf\{G\}\_\{h\}\\mathbf\{P\}^\{\\top\}=\\mathbf\{P\}\\left\(\\sum\_\{j=0\}^\{\\infty\}\\mathbf\{A\}\_\{h\}^\{j\}\\bm\{\\Sigma\}\_\{h\}\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\\right\)\\mathbf\{P\}^\{\\top\},\(31\)according to which the difference𝐄=𝚪h−𝚪c\\mathbf\{E\}=\\bm\{\\Gamma\}\_\{h\}\-\\bm\{\\Gamma\}\_\{c\}satisfies:

‖𝐄‖F≤∑j=0∞‖𝐏​\(𝐀hj​𝚺h​\(𝐀hj\)⊤−𝐀cj​𝚺c​\(𝐀cj\)⊤\)​𝐏⊤‖F\.\\left\\\|\\mathbf\{E\}\\right\\\|\_\{F\}\\leq\\sum\_\{j=0\}^\{\\infty\}\\left\\\|\\mathbf\{P\}\(\\mathbf\{A\}\_\{h\}^\{j\}\\bm\{\\Sigma\}\_\{h\}\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\-\\mathbf\{A\}\_\{c\}^\{j\}\\bm\{\\Sigma\}\_\{c\}\(\\mathbf\{A\}\_\{c\}^\{j\}\)^\{\\top\}\)\\mathbf\{P\}^\{\\top\}\\right\\\|\_\{F\}\.\(32\)The summand decomposes as𝐀hj​𝚺h​\(𝐀hj\)⊤−𝐀cj​𝚺c​\(𝐀cj\)⊤=\(𝐀hj−𝐀cj\)​𝚺h​\(𝐀hj\)⊤\+𝐀cj​\(𝚺h−𝚺c\)​\(𝐀hj\)⊤\+𝐀cj​𝚺c​\(\(𝐀hj\)⊤−\(𝐀cj\)⊤\)\\mathbf\{A\}\_\{h\}^\{j\}\\bm\{\\Sigma\}\_\{h\}\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\-\\mathbf\{A\}\_\{c\}^\{j\}\\bm\{\\Sigma\}\_\{c\}\(\\mathbf\{A\}\_\{c\}^\{j\}\)^\{\\top\}=\(\\mathbf\{A\}\_\{h\}^\{j\}\-\\mathbf\{A\}\_\{c\}^\{j\}\)\\bm\{\\Sigma\}\_\{h\}\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\+\\mathbf\{A\}\_\{c\}^\{j\}\(\\bm\{\\Sigma\}\_\{h\}\-\\bm\{\\Sigma\}\_\{c\}\)\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\+\\mathbf\{A\}\_\{c\}^\{j\}\\bm\{\\Sigma\}\_\{c\}\(\(\\mathbf\{A\}\_\{h\}^\{j\}\)^\{\\top\}\-\(\\mathbf\{A\}\_\{c\}^\{j\}\)^\{\\top\}\)\. Using the identity𝐀hj−𝐀cj=∑l=0j−1𝐀hj−1−l​𝚫​𝐀cl\\mathbf\{A\}\_\{h\}^\{j\}\-\\mathbf\{A\}\_\{c\}^\{j\}=\\sum\_\{l=0\}^\{j\-1\}\\mathbf\{A\}\_\{h\}^\{j\-1\-l\}\\bm\{\\Delta\}\\mathbf\{A\}\_\{c\}^\{l\}and the submultiplicativity of norms:

‖𝐀hj−𝐀cj‖≤j⋅ρmaxj−1​‖𝚫‖,\\left\\\|\\mathbf\{A\}\_\{h\}^\{j\}\-\\mathbf\{A\}\_\{c\}^\{j\}\\right\\\|\\leq j\\cdot\\rho\_\{\\max\}^\{j\-1\}\\left\\\|\\bm\{\\Delta\}\\right\\\|,\(33\)whereρmax=max⁡\(ρ​\(𝐀c\),ρ​\(𝐀h\)\)\\rho\_\{\\max\}=\\max\(\\rho\(\\mathbf\{A\}\_\{c\}\),\\rho\(\\mathbf\{A\}\_\{h\}\)\)\. After bounding the telescoping sum, we obtain:

‖𝐄‖F≤C​‖𝚫‖​\(‖𝚺c‖\+‖𝚺h‖\)\(1−ρmax\)3\+‖𝚺c−𝚺h‖F1−ρmax2,\\left\\\|\\mathbf\{E\}\\right\\\|\_\{F\}\\leq\\frac\{C\\left\\\|\\bm\{\\Delta\}\\right\\\|\(\\left\\\|\\bm\{\\Sigma\}\_\{c\}\\right\\\|\+\\left\\\|\\bm\{\\Sigma\}\_\{h\}\\right\\\|\)\}\{\(1\-\\rho\_\{\\max\}\)^\{3\}\}\+\\frac\{\\left\\\|\\bm\{\\Sigma\}\_\{c\}\-\\bm\{\\Sigma\}\_\{h\}\\right\\\|\_\{F\}\}\{1\-\\rho\_\{\\max\}^\{2\}\},\(34\)whereC\>0C\>0is a constant\. The first term captures the operator difference contribution and the second captures the noise covariance difference\. For the mean difference,𝐏​\(𝝁c−𝝁h\)=𝐏​\[\(𝐈−𝐀c\)−1​𝐦c−\(𝐈−𝐀h\)−1​𝐦h\]\\mathbf\{P\}\(\\bm\{\\mu\}\_\{c\}\-\\bm\{\\mu\}\_\{h\}\)=\\mathbf\{P\}\[\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{c\}\)^\{\-1\}\\mathbf\{m\}\_\{c\}\-\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{h\}\)^\{\-1\}\\mathbf\{m\}\_\{h\}\], which can be bounded using the resolvent identity\(𝐈−𝐀c\)−1−\(𝐈−𝐀h\)−1=\(𝐈−𝐀c\)−1​𝚫​\(𝐈−𝐀h\)−1\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{c\}\)^\{\-1\}\-\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{h\}\)^\{\-1\}=\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{c\}\)^\{\-1\}\\bm\{\\Delta\}\(\\mathbf\{I\}\-\\mathbf\{A\}\_\{h\}\)^\{\-1\}\.

### Appendix B: Proof of dynamical separability Result

Underℋc\\mathcal\{H\}\_\{c\}, the true dynamics are𝐳k\+1=𝐀c​𝐳k\+𝝃k\(c\)\\mathbf\{z\}\_\{k\+1\}=\\mathbf\{A\}\_\{c\}\\mathbf\{z\}\_\{k\}\+\\bm\{\\xi\}\_\{k\}^\{\(c\)\}with𝝃k\(c\)∼𝒩​\(𝐦c,𝚺c\)\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\sim\\mathcal\{N\}\(\\mathbf\{m\}\_\{c\},\\bm\{\\Sigma\}\_\{c\}\)\. Write𝝃k\(c\)=𝐦c\+𝝃~k\\bm\{\\xi\}\_\{k\}^\{\(c\)\}=\\mathbf\{m\}\_\{c\}\+\\tilde\{\\bm\{\\xi\}\}\_\{k\}where𝝃~k∼𝒩​\(𝟎,𝚺c\)\\tilde\{\\bm\{\\xi\}\}\_\{k\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\bm\{\\Sigma\}\_\{c\}\), and decompose𝐳k=𝝁c\+𝐳~k\\mathbf\{z\}\_\{k\}=\\bm\{\\mu\}\_\{c\}\+\\tilde\{\\mathbf\{z\}\}\_\{k\}where𝐳~k\\tilde\{\\mathbf\{z\}\}\_\{k\}is the zero\-mean centered process satisfying𝐳~k\+1=𝐀c​𝐳~k\+𝝃~k\\tilde\{\\mathbf\{z\}\}\_\{k\+1\}=\\mathbf\{A\}\_\{c\}\\tilde\{\\mathbf\{z\}\}\_\{k\}\+\\tilde\{\\bm\{\\xi\}\}\_\{k\}with𝔼​\[𝐳~k\]=𝟎\\mathbb\{E\}\[\\tilde\{\\mathbf\{z\}\}\_\{k\}\]=\\mathbf\{0\}andCov​\(𝐳~k\)=𝐆c\\mathrm\{Cov\}\(\\tilde\{\\mathbf\{z\}\}\_\{k\}\)=\\mathbf\{G\}\_\{c\}\.

The prediction error using the correct model isϵc,k2=‖𝐏​𝝃k\(c\)‖2\\epsilon\_\{c,k\}^\{2\}=\\left\\\|\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\right\\\|^\{2\}, while the prediction error using the wrong model is:

ϵh,k2=‖𝐏​𝚫​𝐳k\+𝐏​𝝃k\(c\)‖2=‖𝐏​𝝃k\(c\)‖2\+2​\(𝝃k\(c\)\)⊤​𝐏⊤​𝐏​𝚫​𝐳k\+‖𝐏​𝚫​𝐳k‖2\.\\epsilon\_\{h,k\}^\{2\}=\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\+\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\right\\\|^\{2\}=\\left\\\|\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\right\\\|^\{2\}\+2\(\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\)^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\+\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\.\(35\)The difference of cumulative squared errors therefore decomposes as:

Sh−Sc=∑k=1L−1‖𝐏​𝚫​𝐳k‖2⏟≜T1\+2​∑k=1L−1\(𝝃k\(c\)\)⊤​𝐏⊤​𝐏​𝚫​𝐳k⏟≜T2\.S\_\{h\}\-S\_\{c\}=\\underbrace\{\\sum\_\{k=1\}^\{L\-1\}\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\}\_\{\\triangleq\\,T\_\{1\}\}\+\\underbrace\{2\\sum\_\{k=1\}^\{L\-1\}\(\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\)^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\}\_\{\\triangleq\\,T\_\{2\}\}\.\(36\)The signal termT1T\_\{1\}has expectation𝔼​\[T1\]=\(L−1\)​δ2\\mathbb\{E\}\[T\_\{1\}\]=\(L\-1\)\\delta^\{2\}in the stationary regime\. Since𝝃k\(c\)\\bm\{\\xi\}\_\{k\}^\{\(c\)\}is independent of𝐳k\\mathbf\{z\}\_\{k\}, the cross termT2T\_\{2\}has expectation:

𝔼​\[T2\]=2​\(L−1\)​𝐦c⊤​𝐏⊤​𝐏​𝚫​𝝁c=\(L−1\)​βc\.\\mathbb\{E\}\[T\_\{2\}\]=2\(L\-1\)\\mathbf\{m\}\_\{c\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}=\(L\-1\)\\beta\_\{c\}\.\(37\)The total expected gap is therefore𝔼​\[Sh−Sc\]=\(L−1\)​\(δ2\+βc\)\\mathbb\{E\}\[S\_\{h\}\-S\_\{c\}\]=\(L\-1\)\(\\delta^\{2\}\+\\beta\_\{c\}\)\.

To establish concentration, we decomposeT2T\_\{2\}into its mean and centered parts:T2=\(L−1\)​βc\+T~2T\_\{2\}=\(L\-1\)\\beta\_\{c\}\+\\widetilde\{T\}\_\{2\}where

T~2=2​∑k=1L−1\[\(𝝃k\(c\)\)⊤​𝐏⊤​𝐏​𝚫​𝐳k−𝐦c⊤​𝐏⊤​𝐏​𝚫​𝝁c\]=2​∑k=1L−1\[𝝃~k⊤​𝐏⊤​𝐏​𝚫​𝐳k\+𝐦c⊤​𝐏⊤​𝐏​𝚫​𝐳~k\]\.\\widetilde\{T\}\_\{2\}=2\\sum\_\{k=1\}^\{L\-1\}\\left\[\(\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\)^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\-\\mathbf\{m\}\_\{c\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}\\right\]=2\\sum\_\{k=1\}^\{L\-1\}\\left\[\\tilde\{\\bm\{\\xi\}\}\_\{k\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\+\\mathbf\{m\}\_\{c\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\tilde\{\\mathbf\{z\}\}\_\{k\}\\right\]\.\(38\)The centered residualT~2\\widetilde\{T\}\_\{2\}has zero mean\. For the first sum inT~2\\widetilde\{T\}\_\{2\}, conditionally on the trajectory\{𝐳k\}k=1L−1\\\{\\mathbf\{z\}\_\{k\}\\\}\_\{k=1\}^\{L\-1\}, the terms\{𝝃~k⊤​𝐏⊤​𝐏​𝚫​𝐳k\}\\\{\\tilde\{\\bm\{\\xi\}\}\_\{k\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\\}are independent zero\-mean Gaussians with variances‖𝚺c1/2​𝐏⊤​𝐏​𝚫​𝐳k‖2≤σmax,c2​‖𝐏​𝚫​𝐳k‖2\\left\\\|\\bm\{\\Sigma\}\_\{c\}^\{1/2\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\\leq\\sigma\_\{\\max,c\}^\{2\}\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}\. The conditional variance of this part is therefore bounded by4​σmax,c2​T14\\,\\sigma\_\{\\max,c\}^\{2\}\\,T\_\{1\}\. For the second sum,𝐦c⊤​𝐏⊤​𝐏​𝚫​𝐳~k\\mathbf\{m\}\_\{c\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\tilde\{\\mathbf\{z\}\}\_\{k\}is a linear functional of the geometrically mixing process𝐳~k\\tilde\{\\mathbf\{z\}\}\_\{k\}; its partial sums have variance of order𝒪​\(L​‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\)\)\\mathcal\{O\}\(L\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\)\.

Since‖𝐏​𝚫​𝐳k‖2=𝐳k⊤​𝚫⊤​𝐏⊤​𝐏​𝚫​𝐳k\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}=\\mathbf\{z\}\_\{k\}^\{\\top\}\\bm\{\\Delta\}^\{\\top\}\\mathbf\{P\}^\{\\top\}\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}is a quadratic form in the Gaussian vector𝐳k\\mathbf\{z\}\_\{k\}, and the sequence\{𝐳k\}\\\{\\mathbf\{z\}\_\{k\}\\\}has geometrically decaying correlations under the stability assumptionρ​\(𝐀c\)<1\\rho\(\\mathbf\{A\}\_\{c\}\)<1, the Hanson–Wright inequality \(Rudelson and Vershynin \([2013](https://arxiv.org/html/2607.28667#bib.bib20)\)\) applied to blocks of approximately independent iterates yields:

ℙ​\(\|T1−\(L−1\)​δ2\|\>t\)≤2​exp⁡\(−c1​min⁡\{t2\(L−1\)​R4​λmax​\(𝐆c\)2,tR2​λmax​\(𝐆c\)\}\),\\mathbb\{P\}\\\!\\left\(\\left\|T\_\{1\}\-\(L\-1\)\\delta^\{2\}\\right\|\>t\\right\)\\leq 2\\exp\\\!\\left\(\-c\_\{1\}\\min\\\!\\left\\\{\\frac\{t^\{2\}\}\{\(L\-1\)R^\{4\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)^\{2\}\},\\;\\frac\{t\}\{R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\}\\right\\\}\\right\),\(39\)wherec1\>0c\_\{1\}\>0is a constant\. The Gaussian tail bound onT~2\\widetilde\{T\}\_\{2\}, combined with the mixing bound on the linear part, gives:

ℙ​\(\|T~2\|\>t\|\{𝐳k\}\)≤2​exp⁡\(−t28​σmax,c2​T1\+C′​‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\)​\(L−1\)\),\\mathbb\{P\}\\\!\\left\(\|\\widetilde\{T\}\_\{2\}\|\>t\\;\\Big\|\\;\\\{\\mathbf\{z\}\_\{k\}\\\}\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{t^\{2\}\}\{8\\sigma\_\{\\max,c\}^\{2\}T\_\{1\}\+C^\{\\prime\}\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\(L\-1\)\}\\right\),\(40\)for a constantC′\>0C^\{\\prime\}\>0\. Restricting to the high\-probability event whereT1T\_\{1\}is close to its mean, setting the deviationt=\(L−1\)​αt=\(L\-1\)\\alphawithα=\(δ2\+βc\)/4\\alpha=\(\\delta^\{2\}\+\\beta\_\{c\}\)/4, and integrating over the trajectory yields concentration ofSh−ScS\_\{h\}\-S\_\{c\}around\(L−1\)​\(δ2\+βc\)\(L\-1\)\(\\delta^\{2\}\+\\beta\_\{c\}\)\.

To pass fromSh−ScS\_\{h\}\-S\_\{c\}toΔ​ℰ=Sh−Sc\\Delta\\mathcal\{E\}=\\sqrt\{S\_\{h\}\}\-\\sqrt\{S\_\{c\}\}, we use the identity:

Δ​ℰ=Sh−ScSh\+Sc\.\\Delta\\mathcal\{E\}=\\frac\{S\_\{h\}\-S\_\{c\}\}\{\\sqrt\{S\_\{h\}\}\+\\sqrt\{S\_\{c\}\}\}\.\(41\)The denominator is controlled by the concentration ofScS\_\{c\}around its mean\. SinceSc=∑k=1L−1‖𝐏​𝝃k\(c\)‖2S\_\{c\}=\\sum\_\{k=1\}^\{L\-1\}\\left\\\|\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\right\\\|^\{2\}is a sum of i\.i\.d\. terms with𝔼​\[Sc\]=\(L−1\)​\[tr​\(𝐏​𝚺c​𝐏⊤\)\+‖𝐏𝐦c‖2\]\\mathbb\{E\}\[S\_\{c\}\]=\(L\-1\)\[\\mathrm\{tr\}\(\\mathbf\{P\}\\bm\{\\Sigma\}\_\{c\}\\mathbf\{P\}^\{\\top\}\)\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\], standard sub\-exponential concentration givesSc≤2​\(L−1\)​\[d​σmax,c2\+‖𝐏𝐦c‖2\]S\_\{c\}\\leq 2\(L\-1\)\[d\\sigma\_\{\\max,c\}^\{2\}\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\]with high probability\. On this event, together withSh≤Sc\+\(Sh−Sc\)S\_\{h\}\\leq S\_\{c\}\+\(S\_\{h\}\-S\_\{c\}\), the denominator in equation[41](https://arxiv.org/html/2607.28667#S2.E41)satisfies:

Sh\+Sc≤2​Sh≤2​2​\(L−1\)​\(d​σmax,c2\+‖𝐏𝐦c‖2\)\+\(L−1\)​\(δ2\+\|βc\|\),\\sqrt\{S\_\{h\}\}\+\\sqrt\{S\_\{c\}\}\\leq 2\\sqrt\{S\_\{h\}\}\\leq 2\\sqrt\{2\(L\-1\)\(d\\sigma\_\{\\max,c\}^\{2\}\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\)\+\(L\-1\)\(\\delta^\{2\}\+\|\\beta\_\{c\}\|\)\},\(42\)so that:

μL≡𝔼​\[Δ​ℰ∣ℋc\]≥L−1​\(δ2\+βc\)8​2​\(d​σmax,c2\+‖𝐏𝐦c‖2\)\+δ2\+\|βc\|\>0\.\\mu\_\{L\}\\equiv\\mathbb\{E\}\[\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\]\\geq\\frac\{\\sqrt\{L\-1\}\\,\(\\delta^\{2\}\+\\beta\_\{c\}\)\}\{8\\,\\sqrt\{2\(d\\sigma\_\{\\max,c\}^\{2\}\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\)\+\\delta^\{2\}\+\|\\beta\_\{c\}\|\}\}\>0\.\(43\)The same concentration applied to the fluctuations ofΔ​ℰ\\Delta\\mathcal\{E\}aroundμL\\mu\_\{L\}shows that for anyη<μL\\eta<\\mu\_\{L\}:

ℙ​\(Δ​ℰ​<η∣​ℋc\)≤ℙ​\(Δ​ℰ−μL<η−μL\)\\displaystyle\\mathbb\{P\}\\\!\\left\(\\Delta\\mathcal\{E\}<\\eta\\mid\\mathcal\{H\}\_\{c\}\\right\)\\leq\\mathbb\{P\}\\\!\\left\(\\Delta\\mathcal\{E\}\-\\mu\_\{L\}<\\eta\-\\mu\_\{L\}\\right\)\(44\)≤2​exp⁡\(−\(L−1\)​\(μL−η\)2C1​\(σmax,c2​\(δ2\+\|βc\|\)\+σmax,c2​R2​λmax​\(𝐆c\)\+‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\)\)\),\\displaystyle\\leq 2\\exp\\\!\\left\(\-\\frac\{\(L\-1\)\(\\mu\_\{L\}\-\\eta\)^\{2\}\}\{C\_\{1\}\(\\sigma\_\{\\max,c\}^\{2\}\(\\delta^\{2\}\+\|\\beta\_\{c\}\|\)\+\\sigma\_\{\\max,c\}^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\)\}\\right\),whereC1\>0C\_\{1\}\>0is a constant absorbing the numerical factors from the preceding bounds\. This establishes \([19](https://arxiv.org/html/2607.28667#S6.E19)\)\.

### Appendix C: Proof of Corollary[1](https://arxiv.org/html/2607.28667#Thmcorollary1)

We compute the mean and variance ofΔ​ℰ\\Delta\\mathcal\{E\}by working through the auxiliary quantitySh−ScS\_\{h\}\-S\_\{c\}and transferring the results via the identityΔ​ℰ=\(Sh−Sc\)/\(Sh\+Sc\)\\Delta\\mathcal\{E\}=\(S\_\{h\}\-S\_\{c\}\)/\(\\sqrt\{S\_\{h\}\}\+\\sqrt\{S\_\{c\}\}\)\.

From the decomposition \([36](https://arxiv.org/html/2607.28667#S2.E36)\), the mean ofSh−ScS\_\{h\}\-S\_\{c\}underℋc\\mathcal\{H\}\_\{c\}is𝔼​\[Sh−Sc\]=𝔼​\[T1\]\+𝔼​\[T2\]=\(L−1\)​\(δ2\+βc\)\\mathbb\{E\}\[S\_\{h\}\-S\_\{c\}\]=\\mathbb\{E\}\[T\_\{1\}\]\+\\mathbb\{E\}\[T\_\{2\}\]=\(L\-1\)\(\\delta^\{2\}\+\\beta\_\{c\}\)\. For the variance, the centered residualT~2=T2−\(L−1\)​βc\\widetilde\{T\}\_\{2\}=T\_\{2\}\-\(L\-1\)\\beta\_\{c\}gives:

Var​\(Sh−Sc\)=Var​\(T1\)\+Var​\(T~2\)\+2​Cov​\(T1,T~2\)\.\\mathrm\{Var\}\(S\_\{h\}\-S\_\{c\}\)=\\mathrm\{Var\}\(T\_\{1\}\)\+\\mathrm\{Var\}\(\\widetilde\{T\}\_\{2\}\)\+2\\,\\mathrm\{Cov\}\(T\_\{1\},\\widetilde\{T\}\_\{2\}\)\.\(45\)The signal termT1=∑k=1L−1‖𝐏​𝚫​𝐳k‖2T\_\{1\}=\\sum\_\{k=1\}^\{L\-1\}\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\mathbf\{z\}\_\{k\}\\right\\\|^\{2\}is a sum of quadratic forms in the geometrically mixing Gaussian process\{𝐳k\}\\\{\\mathbf\{z\}\_\{k\}\\\}\. Each summand has variance𝒪​\(R4​λmax​\(𝐆c\)2\+R2​‖𝐏​𝚫​𝝁c‖2​λmax​\(𝐆c\)\)\\mathcal\{O\}\(R^\{4\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)^\{2\}\+R^\{2\}\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}\\right\\\|^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\), and the geometric mixing ensures that the effective number of independent blocks isΘ​\(L\)\\Theta\(L\), soVar​\(T1\)=𝒪​\(L⋅\(R4​λmax​\(𝐆c\)2\+R2​‖𝐏​𝚫​𝝁c‖2​λmax​\(𝐆c\)\)\)\\mathrm\{Var\}\(T\_\{1\}\)=\\mathcal\{O\}\\big\(L\\cdot\(R^\{4\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)^\{2\}\+R^\{2\}\\left\\\|\\mathbf\{P\}\\bm\{\\Delta\}\\bm\{\\mu\}\_\{c\}\\right\\\|^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\)\\big\)\. For the second cross\-term, using the law of total variance, we have:

Var​\(T~2\)\\displaystyle\\mathrm\{Var\}\(\\widetilde\{T\}\_\{2\}\)=𝔼​\[Var​\(T~2∣\{𝐳k\}\)\]\+Var​\(𝔼​\[T~2∣\{𝐳k\}\]\)\\displaystyle=\\mathbb\{E\}\[\\mathrm\{Var\}\(\\widetilde\{T\}\_\{2\}\\mid\\\{\\mathbf\{z\}\_\{k\}\\\}\)\]\+\\mathrm\{Var\}\(\\mathbb\{E\}\[\\widetilde\{T\}\_\{2\}\\mid\\\{\\mathbf\{z\}\_\{k\}\\\}\]\)\(46\)≤4​σmax,c2​\(L−1\)​δ2\+C′​\(L−1\)​‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\),\\displaystyle\\leq 4\\,\\sigma\_\{\\max,c\}^\{2\}\(L\-1\)\\delta^\{2\}\+C^\{\\prime\}\(L\-1\)\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\),
where the first term bounds the conditional variance from the zero\-mean noise𝝃~k\\tilde\{\\bm\{\\xi\}\}\_\{k\}, and the second bounds the variance of the linear functional of𝐳~k\\tilde\{\\mathbf\{z\}\}\_\{k\}, andC′\>0C^\{\\prime\}\>0is a constant\. In the regime where the noise\-driven term dominates, we have:

Var​\(Sh−Sc\)=Θ​\(L⋅\[σmax,c2​δ2\+‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\)\]\)\.\\mathrm\{Var\}\(S\_\{h\}\-S\_\{c\}\)=\\Theta\\\!\\left\(L\\cdot\[\\sigma\_\{\\max,c\}^\{2\}\\delta^\{2\}\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\]\\right\)\.\(47\)
To transfer these toΔ​ℰ\\Delta\\mathcal\{E\}, note thatSc=∑k=1L−1‖𝐏​𝝃k\(c\)‖2S\_\{c\}=\\sum\_\{k=1\}^\{L\-1\}\\left\\\|\\mathbf\{P\}\\bm\{\\xi\}\_\{k\}^\{\(c\)\}\\right\\\|^\{2\}concentrates around\(L−1\)​\[tr​\(𝐏​𝚺c​𝐏⊤\)\+‖𝐏𝐦c‖2\]=Θ​\(L​\[σmax,c2​d\+‖𝐏𝐦c‖2\]\)\(L\-1\)\[\\mathrm\{tr\}\(\\mathbf\{P\}\\bm\{\\Sigma\}\_\{c\}\\mathbf\{P\}^\{\\top\}\)\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\]=\\Theta\(L\[\\sigma\_\{\\max,c\}^\{2\}d\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\]\)by the law of large numbers\. The denominator in the identityΔ​ℰ=\(Sh−Sc\)/\(Sh\+Sc\)\\Delta\\mathcal\{E\}=\(S\_\{h\}\-S\_\{c\}\)/\(\\sqrt\{S\_\{h\}\}\+\\sqrt\{S\_\{c\}\}\)therefore satisfies:

Sh\+Sc=Θ​\(L​σmax,c2​d\+‖𝐏𝐦c‖2\),\\sqrt\{S\_\{h\}\}\+\\sqrt\{S\_\{c\}\}=\\Theta\\\!\\left\(\\sqrt\{L\}\\,\\sqrt\{\\sigma\_\{\\max,c\}^\{2\}d\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\}\\right\),\(48\)in the typical regime where the model\-mismatch signal is small relative to the total noise energy\. Applying the delta method to the ratio, the mean and standard deviation ofΔ​ℰ\\Delta\\mathcal\{E\}are:

𝔼​\[Δ​ℰ∣ℋc\]\\displaystyle\\mathbb\{E\}\[\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\]=Θ​\(L⋅δ2\+βcσmax,c2​d\+‖𝐏𝐦c‖2\),\\displaystyle=\\Theta\\\!\\left\(\\sqrt\{L\}\\cdot\\frac\{\\delta^\{2\}\+\\beta\_\{c\}\}\{\\sqrt\{\\sigma\_\{\\max,c\}^\{2\}d\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\}\}\\right\),\(49\)Var​\(Δ​ℰ∣ℋc\)\\displaystyle\\sqrt\{\\mathrm\{Var\}\(\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\)\}=Θ​\(σmax,c2​δ2\+‖𝐏𝐦c‖2​R2​λmax​\(𝐆c\)σmax,c2​d\+‖𝐏𝐦c‖2\)\.\\displaystyle=\\Theta\\\!\\left\(\\frac\{\\sqrt\{\\sigma\_\{\\max,c\}^\{2\}\\delta^\{2\}\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}\)\}\}\{\\sqrt\{\\sigma\_\{\\max,c\}^\{2\}d\+\\left\\\|\\mathbf\{P\}\\mathbf\{m\}\_\{c\}\\right\\\|^\{2\}\}\}\\right\)\.\(50\)Taking the ratio and absorbing the dimension\-dependent constants:

S​\(L\)=𝔼​\[Δ​ℰ∣ℋc\]Var​\(Δ​ℰ∣ℋc\)=Θ​\(L⋅δ2\+βcσmax,c2\),S\(L\)=\\frac\{\\mathbb\{E\}\[\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\]\}\{\\sqrt\{\\mathrm\{Var\}\(\\Delta\\mathcal\{E\}\\mid\\mathcal\{H\}\_\{c\}\)\}\}=\\Theta\\\!\\left\(\\sqrt\{L\}\\cdot\\frac\{\\delta^\{2\}\+\\beta\_\{c\}\}\{\\sigma\_\{\\max,c\}^\{2\}\}\\right\),\(51\)establishing \([20](https://arxiv.org/html/2607.28667#S6.E20)\)\. For the sample complexity, the misclassification probability from Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)decays asexp⁡\(−c​S​\(L\)2\)=exp⁡\(−c​L​\(δ2\+βc\)2/σmax,c4\)\\exp\(\-c\\,S\(L\)^\{2\}\)=\\exp\(\-c\\,L\\,\(\\delta^\{2\}\+\\beta\_\{c\}\)^\{2\}/\\sigma\_\{\\max,c\}^\{4\}\)\. Setting this equal toϵ\\epsilonand solving forLLgives:

L∗=σmax,c4c​\(δ2\+βc\)2​log⁡1ϵ=𝒪​\(σmax,c4\(δ2\+βc\)2​log⁡1ϵ\),L^\{\*\}=\\frac\{\\sigma\_\{\\max,c\}^\{4\}\}\{c\\,\(\\delta^\{2\}\+\\beta\_\{c\}\)^\{2\}\}\\log\\frac\{1\}\{\\epsilon\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\sigma\_\{\\max,c\}^\{4\}\}\{\(\\delta^\{2\}\+\\beta\_\{c\}\)^\{2\}\}\\log\\frac\{1\}\{\\epsilon\}\\right\),\(52\)which corresponds to the desired result in \([21](https://arxiv.org/html/2607.28667#S6.E21)\)\.

### Appendix D: Proof of Cross\-embedding transfer bound

Under the intertwining assumption \([23](https://arxiv.org/html/2607.28667#S7.E23)\), the𝜽2\\bm\{\\theta\}\_\{2\}\-embedding of tokenqkq\_\{k\}can be written as𝐲k\(𝜽2\)=𝐓𝐲k\(𝜽1\)\+𝒓k\\mathbf\{y\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}=\\mathbf\{T\}\\mathbf\{y\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\+\\bm\{r\}\_\{k\}where𝒓k=𝒓​\(qk\)\\bm\{r\}\_\{k\}=\\bm\{r\}\(q\_\{k\}\)\. The prediction error when applying the𝜽1\\bm\{\\theta\}\_\{1\}\-fitted operator𝐀h\(θ1\)\\mathbf\{A\}\_\{h\}^\{\(\\theta\_\{1\}\)\}to the𝜽2\\bm\{\\theta\}\_\{2\}\-trajectory is:

𝐲k\+1\(𝜽2\)−𝐏\(𝜽2\)​𝐀h\(𝜽1\)​𝐳k\(𝜽2\)=𝐓𝐲k\+1\(𝜽1\)\+𝒓k\+1−𝐏\(𝜽2\)​𝐀h\(𝜽1\)​𝐳k\(𝜽2\)\.\\mathbf\{y\}\_\{k\+1\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\-\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\mathbf\{A\}\_\{h\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}=\\mathbf\{T\}\\mathbf\{y\}\_\{k\+1\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\+\\bm\{r\}\_\{k\+1\}\-\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\mathbf\{A\}\_\{h\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\.\(53\)Underℋc\\mathcal\{H\}\_\{c\}, the true dynamics in the𝜽1\\bm\{\\theta\}\_\{1\}\-space satisfy𝐲k\+1\(𝜽1\)=𝐏\(𝜽1\)​𝐀c\(𝜽1\)​𝐳k\(𝜽1\)\+𝐏\(𝜽1\)​𝝃k\(𝜽1\)\\mathbf\{y\}\_\{k\+1\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}=\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{A\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\+\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\xi\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\. Substituting and rearranging, let𝝆k≔𝝆k​\(𝒓k,𝐏\(𝜽1\),𝐀c\(𝜽1\)\)∈ℝn\\bm\{\\rho\}\_\{k\}\\coloneqq\\bm\{\\rho\}\_\{k\}\\\!\\bigl\(\\bm\{r\}\_\{k\},\\,\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\},\\,\\mathbf\{A\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bigr\)\\in\\mathbb\{R\}^\{n\}denote the residual that depends on the observation noise𝒓k\\bm\{r\}\_\{k\}\. The model\-mismatch component of the prediction error takes the form

𝒆k=𝐓​𝐏\(𝜽1\)​𝚫\(𝜽1\)​𝐳k\(𝜽1\)\+𝝆k\.\\bm\{e\}\_\{k\}\\;=\\;\\mathbf\{T\}\\,\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\;\+\\;\\bm\{\\rho\}\_\{k\}\.\(54\)Taking the expected squared norm and applying the reverse triangle inequality‖𝒂\+𝒃‖2≥‖𝒂‖2−2​‖𝒂‖​‖𝒃‖\\left\\\|\\bm\{a\}\+\\bm\{b\}\\right\\\|^\{2\}\\geq\\left\\\|\\bm\{a\}\\right\\\|^\{2\}\-2\\left\\\|\\bm\{a\}\\right\\\|\\left\\\|\\bm\{b\}\\right\\\|yields

δ𝜽1→𝜽22\\displaystyle\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}≥𝔼​\[‖𝐓​𝐏\(𝜽1\)​𝚫\(𝜽1\)​𝐳k\(𝜽1\)‖2\]−2​𝔼​\[‖𝐓​𝐏\(𝜽1\)​𝚫\(𝜽1\)​𝐳k\(𝜽1\)‖​‖𝝆k‖\]\\displaystyle\\;\\geq\\;\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{T\}\\,\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\right\]\\;\-\\;2\\,\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{T\}\\,\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|\\,\\left\\\|\\bm\{\\rho\}\_\{k\}\\right\\\|\\right\]≥σmin2​\(𝐓\)​𝔼​\[‖𝐏\(𝜽1\)​𝚫\(𝜽1\)​𝐳k\(𝜽1\)‖2\]−C2​χ2​\(‖𝚫\(𝜽1\)‖2​λmax​\(𝐆c\(𝜽2\)\)\+δ2​\(𝜽1\)\)\\displaystyle\\;\\geq\\;\\sigma\_\{\\min\}^\{2\}\(\\mathbf\{T\}\)\\;\\,\\mathbb\{E\}\\\!\\left\[\\left\\\|\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\right\]\\;\-\\;C\_\{2\}\\,\\chi^\{2\}\\\!\\left\(\\left\\\|\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\,\\lambda\_\{\\max\}\\\!\\bigl\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\bigr\)\\;\+\\;\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\\right\)=σmin2​\(𝐓\)​δ2​\(𝜽1\)−C2​χ2​\(‖𝚫\(𝜽1\)‖2​λmax​\(𝐆c\(𝜽2\)\)\+δ2​\(𝜽1\)\),\\displaystyle\\;=\\;\\sigma\_\{\\min\}^\{2\}\(\\mathbf\{T\}\)\\;\\,\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\\;\-\\;C\_\{2\}\\,\\chi^\{2\}\\\!\\left\(\\left\\\|\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\,\\lambda\_\{\\max\}\\\!\\bigl\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\bigr\)\\;\+\\;\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\\right\),\(55\)where the second line uses the submultiplicativity‖𝐓​𝒙‖≥σmin​\(𝐓\)​‖𝒙‖\\left\\\|\\mathbf\{T\}\\,\\bm\{x\}\\right\\\|\\geq\\sigma\_\{\\min\}\(\\mathbf\{T\}\)\\,\\left\\\|\\bm\{x\}\\right\\\|together with the bound𝔼​\[‖𝝆k‖2\]≤C2​χ2​\(‖𝚫\(𝜽1\)‖2​λmax​\(𝐆c\(𝜽2\)\)\+δ2​\(𝜽1\)\)\\mathbb\{E\}\\\!\\bigl\[\\left\\\|\\bm\{\\rho\}\_\{k\}\\right\\\|^\{2\}\\bigr\]\\leq C\_\{2\}\\,\\chi^\{2\}\\\!\\bigl\(\\left\\\|\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\,\\lambda\_\{\\max\}\\\!\\bigl\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\\bigr\)\+\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\\bigr\), and the last equality identifies𝔼​\[‖𝐏\(𝜽1\)​𝚫\(𝜽1\)​𝐳k\(𝜽1\)‖2\]=δ2​\(𝜽1\)\\mathbb\{E\}\\\!\\bigl\[\\left\\\|\\mathbf\{P\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\bm\{\\Delta\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\mathbf\{z\}\_\{k\}^\{\(\\bm\{\\theta\}\_\{1\}\)\}\\right\\\|^\{2\}\\bigr\]=\\delta^\{2\}\(\\bm\{\\theta\}\_\{1\}\)\. This establishes \([25](https://arxiv.org/html/2607.28667#S7.E25)\)\.

The accuracy bound \([26](https://arxiv.org/html/2607.28667#S7.E26)\) follows by substitutingδ𝜽1→𝜽22\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}in place ofδ2\\delta^\{2\}in Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)withη=0\\eta=0\. The cross\-embedding setting introduces no structural change to the hypothesis testing framework: the test statisticΔ​ℰ\\Delta\\mathcal\{E\}is still computed from prediction residuals, and the exponential concentration still holds provided the effective discriminabilityδ𝜽1→𝜽22\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}is positive\. The total classification error under equal priors is:

ℙ​\(error\)\\displaystyle\\mathbb\{P\}\(\\text\{error\}\)=12​ℙ​\(Δ​ℰ​<0∣​ℋc\)\+12​ℙ​\(Δ​ℰ≥0∣ℋh\)\\displaystyle=\\frac\{1\}\{2\}\\mathbb\{P\}\(\\Delta\\mathcal\{E\}<0\\mid\\mathcal\{H\}\_\{c\}\)\+\\frac\{1\}\{2\}\\mathbb\{P\}\(\\Delta\\mathcal\{E\}\\geq 0\\mid\\mathcal\{H\}\_\{h\}\)\(56\)≤2​exp⁡\(−\(L−1\)​\(δ𝜽1→𝜽22\)2C1​\(σmax,c2​δ𝜽1→𝜽22\+σmax,c2​R2​λmax​\(𝐆c\(𝜽2\)\)\)\),\\displaystyle\\leq 2\\exp\\\!\\left\(\-\\frac\{\(L\-1\)\\left\(\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\\right\)^\{2\}\}\{C\_\{1\}\\\!\\left\(\\sigma\_\{\\max,c\}^\{2\}\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\+\\sigma\_\{\\max,c\}^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\)\\right\)\}\\right\),where each term is bounded by Result[2](https://arxiv.org/html/2607.28667#Thmtheorem2)applied in the𝜽2\\bm\{\\theta\}\_\{2\}\-observation space with discriminabilityδ𝜽1→𝜽22\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\. Sinceℙ​\(error\)≤1/2\\mathbb\{P\}\(\\text\{error\}\)\\leq 1/2, the accuracy satisfies:

Accuracy𝜽1→𝜽2\\displaystyle\\mathrm\{Accuracy\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}=1−ℙ​\(error\)\\displaystyle=1\-\\mathbb\{P\}\(\\text\{error\}\)\(57\)≥12\+12​\(1−2​exp⁡\(−\(L−1\)​\(δ𝜽1→𝜽22\)2C1​\(σmax,c2​δ𝜽1→𝜽22\+σmax,c2​R2​λmax​\(𝐆c\(𝜽2\)\)\)\)\),\\displaystyle\\geq\\frac\{1\}\{2\}\+\\frac\{1\}\{2\}\\\!\\left\(1\-2\\exp\\\!\\left\(\-\\frac\{\(L\-1\)\\left\(\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\\right\)^\{2\}\}\{C\_\{1\}\\\!\\left\(\\sigma\_\{\\max,c\}^\{2\}\\delta^\{2\}\_\{\\bm\{\\theta\}\_\{1\}\\to\\bm\{\\theta\}\_\{2\}\}\+\\sigma\_\{\\max,c\}^\{2\}R^\{2\}\\lambda\_\{\\max\}\(\\mathbf\{G\}\_\{c\}^\{\(\\bm\{\\theta\}\_\{2\}\)\}\)\\right\)\}\\right\)\\right\),which is precisely \([26](https://arxiv.org/html/2607.28667#S7.E26)\)\.