Spectral Origins of the Self-Correction Blind Spot in Autoregressive Generation

# Spectral Origins of the Self-Correction Blind Spot in Autoregressive Generation
Source: [https://arxiv.org/html/2607.09803](https://arxiv.org/html/2607.09803)
###### Abstract

Large autoregressive language models exhibit a self\-correction blind spot: they reliably fix identical errors when attributed to an external source yet fail to fix the same errors in their own outputs\. Prior work has documented this phenomenon empirically, through controlled error injection, error\-depth decompositions, RL\-based verifier–corrector training, and intrinsic self\-verification, but offers no formal model of why generating a token suppresses the ability to detect its error, no quantitative activation condition for correction markers, and no convergence guarantee for reinforcement\-learning\-based self\-correction\. We close these gaps with SPARC, a spectral\-algebraic theory of self\-correction in autoregressive generation\. We define the error\-propagation operator as the product of per\-step attention Jacobians on the residual stream and prove that the blind spot arises if and only if the spectral radius of this operator is at least one\. We derive a sharp activation threshold, given as a function of the spectral radius, that a correction marker must exceed, recovering the 89\.3% blind\-spot reduction observed with a simple “Wait” marker\. We further prove that RL\-based verifier–corrector training converges at a rate proportional to the squared coupling strength over the square root of the number of samples if and only if the verifier–corrector coupling matrix has spectral norm below one, and that this criterion is invariant across residual\-stream autoregressive modalities, unifying text LLMs and autoregressive image and video generation\. Experiments across four backbones and a visual autoregressive probe validate every theorem, with spectral predictions matching measured blind\-spot rates within 3\.2% RMSE\.

## 1Introduction

Large language models \(LLMs\) have become transformative across natural language processing, yet they remain unreliable: they make mistakes, follow unproductive reasoning paths, and fail to revise errors in their own outputs\(Tsui,[2025](https://arxiv.org/html/2607.09803#bib.bib1); Kumaret al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib11)\)\. Self\-correction — the ability of a model to detect and revise errors in its own generation — is a prerequisite for trustworthy deployment\. A striking empirical finding, however, is the*Self\-Correction Blind Spot*: LLMs can identify and correct an error when it is attributed to an external source \(a user, a tool\) but systematically fail to correct the identical error when it appears in their own output\(Tsui,[2025](https://arxiv.org/html/2607.09803#bib.bib1)\)\. Across fourteen open\-source models the blind\-spot rate averages 64\.5%, and even a simple prompt\-level intervention — appending the token “Wait” — reduces it by 89\.3%, suggesting the capability is latent but suppressed during standard decoding\.

A wave of recent work has refined this picture\.Li \([2025](https://arxiv.org/html/2607.09803#bib.bib3)\)decompose self\-correction into detectability and correctability and propose the Error Depth Hypothesis, observing that correction success is non\-monotonic in how deeply an error is embedded in the reasoning trace\.Maet al\.\([2025](https://arxiv.org/html/2607.09803#bib.bib4)\)show that reinforcement learning \(RL\) — not supervised fine\-tuning \(SFT\) — is what installs usable self\-correction, via a two\-stage verifier–corrector pipeline \(S2R\)\.Leeet al\.\([2025](https://arxiv.org/html/2607.09803#bib.bib2)\)train an intrinsic verifier \(ReVISE\) and expose a refinement\-instability regime in which self\-refinement degrades output quality\. Iterative self\-feedback\(Madaanet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib18); Shinnet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib23)\), RL\-based self\-correction training\(Kumaret al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib10)\), and confidence\-aware verification\(Liet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib12); Liuet al\.,[2024a](https://arxiv.org/html/2607.09803#bib.bib16)\)all improve self\-correction empirically but rest on the same foundational question that none of them answers formally:*why does generating a token suppress the ability to detect its error?*

#### Limitations of prior work\.

All existing accounts are empirical or phenomenological\. There is \(i\) no formal model tying the decoding dynamics to the suppression of error detection; \(ii\) no quantitative condition predicting when a correction marker will succeed or fail; \(iii\) no convergence guarantee or sample\-complexity bound for RL\-based verifier–corrector training; and \(iv\) no theory that connects the blind spot in text LLMs to the structurally identical error\-propagation dynamics of autoregressive image and video generation, where a large body of recent work confronts the same error\-propagation problem without a unifying account\(Yuet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib31); Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39);[b](https://arxiv.org/html/2607.09803#bib.bib40); Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41); Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42); Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43); Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44); Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45); Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46);[2025](https://arxiv.org/html/2607.09803#bib.bib47);[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)\.

#### Our approach\.

We buildSPARC\(Spectral Propagation Analysis for Reasoning Correction\), a spectral\-algebraic theory that formalizes the blind spot as a property of the*error\-propagation operator*FTF\_\{T\}, defined as the product of per\-step attention Jacobians on the residual stream\. The theory is organized around three objects:FTF\_\{T\}, whose spectral radiusρ​\(FT\)\\rho\(F\_\{T\}\)governs error amplification; the*correction sensitivity*κ\\kappaof a marker; and the*verifier–corrector coupling matrix*CC, whose spectral norm governs RL stability\.

#### Contributions\.

- •Blind\-spot spectral equivalence \(Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)\)\.We prove that the self\-correction blind spot occurs*if and only if*ρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1, giving the first formal account of why token ownership suppresses error detection\.
- •Sharp activation threshold \(Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)\)\.A correction marker of sensitivityκ\\kappaactivates self\-correction iffκ\>κ⋆​\(ρ\)=\(1−ρ\)/\(1\+ρ\)\\kappa\>\\kappa^\{\\star\}\(\\rho\)=\(1\-\\rho\)/\(1\+\\rho\); this threshold is tight and recovers the 89\.3% “Wait” reduction\.
- •RL convergence guarantee \(Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)\)\.Verifier–corrector RL converges at rateO​\(‖C‖22/n\)O\(\\\|C\\\|\_\{2\}^\{2\}/\\sqrt\{n\}\)iff‖C‖2<1\\\|C\\\|\_\{2\}<1, with SFT provably unable to install the coupling — formalizing why RL, not SFT, is necessary\.
- •Cross\-modal invariance \(Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\)\.The criterionρ​\(FT\)<1\\rho\(F\_\{T\}\)<1is invariant over residual\-stream autoregressive architectures, unifying text LLMs and autoregressive visual generation under one theory\.

#### Organization\.

Section[2](https://arxiv.org/html/2607.09803#S2)surveys related work; Section[3](https://arxiv.org/html/2607.09803#S3)develops the SPARC theory, starting with the problem formulation \(Section[3\.1](https://arxiv.org/html/2607.09803#S3.SS1)\) and corollaries \(Section[3\.7](https://arxiv.org/html/2607.09803#S3.SS7)\); Section[4](https://arxiv.org/html/2607.09803#S4)reports experiments including ablations \(Section[4\.4](https://arxiv.org/html/2607.09803#S4.SS4)\); Section[5](https://arxiv.org/html/2607.09803#S5)concludes\. Lemma proofs are deferred to Appendix[A](https://arxiv.org/html/2607.09803#A1)\.

## 2Related Work

### 2\.1Self\-Correction and Self\-Refinement in LLMs

The empirical study of LLM self\-correction begins with iterative self\-feedback: Self\-Refine\(Madaanet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib18)\)iterates a critic\-and\-revise loop, while Reflexion\(Shinnet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib23)\)augments agents with verbal reinforcement\. Confidence\-aware analyses\(Liet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib12); Liuet al\.,[2024a](https://arxiv.org/html/2607.09803#bib.bib16)\)show intrinsic self\-correction is bounded by the model’s self\-knowledge, and alignment\-based reasoners\(Wanget al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib25)\)improve step\-level correction\. The Self\-Correction Bench ofTsui \([2025](https://arxiv.org/html/2607.09803#bib.bib1)\)isolates the*blind spot*— the gap between external and internal error correction — through controlled error injection\.Li \([2025](https://arxiv.org/html/2607.09803#bib.bib3)\)decompose the accuracy\-correction paradox via the Error Depth Hypothesis, and Socratic self\-refinement\(Shiet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib22)\)and synergistic training\-inference co\-design\(Zenget al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib33)\)push activation further\. Our work departs from these empirical accounts by providing the first formal spectral model of*why*the blind spot arises and*when*activation succeeds\.

### 2\.2Reinforcement Learning for Self\-Correction

A second line installs self\-correction via RL\. SCoRe\(Kumaret al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib10)\)trains self\-correction through multi\-turn RL, and S2R\(Maet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib4)\)decouples a verifier from a corrector with a two\-stage RL pipeline, demonstrating that RL — not SFT — is what makes self\-correction usable\. ReVISE\(Leeet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib2)\)learns an intrinsic verifier and exposes a refinement\-instability regime\. Surveys of RL for LLMs\(Liuet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib17); Kumaret al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib11)\)and weakness\-driven problem synthesis\(Lianget al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib13)\)confirm that outcome feedback is the active ingredient, echoing the training\-data analysis ofTsui \([2025](https://arxiv.org/html/2607.09803#bib.bib1)\)\. Value\-bonus exploration\(Wahabet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib24)\)and intrinsic verifier training\(Leeet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib2)\)further refine the RL toolbox\. None of these works provides convergence rates, sample\-complexity bounds, or a sharp stability condition; Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)fills this gap\. RLHF foundations\(Ouyanget al\.,[2022](https://arxiv.org/html/2607.09803#bib.bib19); Baiet al\.,[2022](https://arxiv.org/html/2607.09803#bib.bib38)\)and direct preference optimization\(Rafailovet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib21)\)provide the policy\-gradient substrate our analysis builds on\.

### 2\.3Autoregressive Generation and Error Propagation

A third line studies error propagation in autoregressive generation, both textual and visual\. Local and global decoding\(Gareevet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib7)\)and hierarchical skip decoding\(Zhuet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib37)\)analyze how decoding choices affect error propagation in text\. For visual autoregressive generation, randomized generation\(Yuet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib31)\), speculative decoding\(Yueet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib32); Xianget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib26); Zhouet al\.,[2026a](https://arxiv.org/html/2607.09803#bib.bib36)\), diffusion\-correction hybrids\(Chunget al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib6)\), multi\-resolution models\(Liuet al\.,[2024b](https://arxiv.org/html/2607.09803#bib.bib15)\), retrieval augmentation\(Qiet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib20)\), and non\-autoregressive variants\(Yanget al\.,[2019](https://arxiv.org/html/2607.09803#bib.bib27); Zhaoet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib34)\)all confront the same token\-level error\-propagation dynamics\. Vision\-representation compression for video generation\(Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39)\), multi\-subject in\-context image generation\(Zhouet al\.,[2026b](https://arxiv.org/html/2607.09803#bib.bib40)\), entropy\-guided RPO for autoregressive image generation\(Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41)\), holistic complex\-instruction image\-generation benchmarks\(Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42)\), training acceleration for autoregressive video generation\(Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43)\), condition\-error refinement in autoregressive image generation with diffusion loss\(Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44)\), cross\-modal alignment for text\-guided inpainting\(Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45)\), locality\-aware dynamic rescue for text\-to\-image dLLMs\(Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46)\), complex\-instruction image\-editing benchmarks\(Wanget al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib47)\), and memory\-augmented state\-space defect recognition\(Wanget al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)all face error propagation in autoregressive visual decoding but lack a unifying theoretical account\. Multimodal LLM surveys\(Yinet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib30); Yanget al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib28)\)and visual token pruning\(Huanget al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib8)\)provide architectural context\.

### 2\.4Theoretical Analyses of Attention and Decoding

A growing theoretical literature analyzes attention dynamics: thermodynamic isomorphisms\(Kim,[2026](https://arxiv.org/html/2607.09803#bib.bib9)\), gated attention with forget gates\(Linet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib14)\), and self\-cognition studies\(Chenet al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib5)\)examine structural properties of transformers\. Code\-based self\-verification\(Zhouet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib35)\)and thought\-template distillation\(Yanget al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib29)\)give empirical proxies for the theoretical quantities we formalize\. SPARC complements these by giving the first spectral criterion —ρ​\(FT\)<1\\rho\(F\_\{T\}\)<1— that certifies self\-correction capability, made predictively checkable via the Hutchinson estimator \(Corollary[12](https://arxiv.org/html/2607.09803#Thmtheorem12)\)\.

## 3The SPARC Theory

### 3\.1Problem Formulation

LetGθG\_\{\\theta\}be an autoregressive generator emitting tokensx1,…,xTx\_\{1\},\\ldots,x\_\{T\}by samplingxt∼pθ\(⋅∣x<t\)x\_\{t\}\\sim p\_\{\\theta\}\(\\cdot\\mid x\_\{<t\}\)with residual\-stream hidden statesht∈ℝdh\_\{t\}\\in\\mathbb\{R\}^\{d\}\. A*self\-correction event*at stepτ\\tauis a transitionhτ↦hτ\+Δτh\_\{\\tau\}\\mapsto h\_\{\\tau\}\+\\Delta\_\{\\tau\}induced by a correction marker \(e\.g\., the token “Wait”\) intended to let the model detect and revise an error in\{xt\}t≤τ\\\{x\_\{t\}\\\}\_\{t\\leq\\tau\}\. The Self\-Correction Blind Spot\(Tsui,[2025](https://arxiv.org/html/2607.09803#bib.bib1)\)is the empirical fact that, under standard SFT decoding,GθG\_\{\\theta\}fails to perform this revision on its own outputs while succeeding on identical external inputs\.

We seek formal answers to four questions:\(Q1\)Under what spectral condition does the blind spot arise?\(Q2\)What is the sharp threshold a correction marker must exceed?\(Q3\)Under what condition does RL\-based verifier–corrector training provably install self\-correction, and at what rate?\(Q4\)Is the criterion invariant across autoregressive modalities?

### 3\.2Preliminaries and Notation

Modern autoregressive generators — text LLMs and autoregressive image/video models — share the residual\-stream updateht=ht−1\+fθ​\(ht−1,x<t\)h\_\{t\}=h\_\{t\-1\}\+f\_\{\\theta\}\(h\_\{t\-1\},x\_\{<t\}\), wherefθf\_\{\\theta\}is a stack of self\-attention and MLP blocks with a residual connection\. This residual form is the structural basis of cross\-modal invariance \(Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\)\. For a matrixAAwith eigenvalues\{λi\}\\\{\\lambda\_\{i\}\\\},ρ​\(A\)=maxi⁡\|λi\|\\rho\(A\)=\\max\_\{i\}\|\\lambda\_\{i\}\|is the spectral radius; by the Gelfand formulaρ​\(A\)=limk→∞‖Ak‖1/k\\rho\(A\)=\\lim\_\{k\\to\\infty\}\\\|A^\{k\}\\\|^\{1/k\}, soρ​\(A\)<1\\rho\(A\)<1iffAk→0A^\{k\}\\to 0, i\.e\., the linear map is asymptotically contractive\.

### 3\.3Definitions

###### Definition 1\(Error\-Propagation Operator\)\.

For an autoregressive generatorGθG\_\{\\theta\}with residual states\{ht\}t=1T\\\{h\_\{t\}\\\}\_\{t=1\}^\{T\}, the*error\-propagation operator*over horizonTTis

FT=∏t=1TJt,Jt=∂ht∂ht−1=I\+∂fθ∂ht−1\.F\_\{T\}\\;=\\;\\prod\_\{t=1\}^\{T\}J\_\{t\},\\qquad J\_\{t\}\\;=\\;\\frac\{\\partial h\_\{t\}\}\{\\partial h\_\{t\-1\}\}\\;=\\;I\+\\frac\{\\partial f\_\{\\theta\}\}\{\\partial h\_\{t\-1\}\}\.\(1\)FTF\_\{T\}maps a perturbationδ​h0\\delta h\_\{0\}at step0to the induced perturbationδ​hT=FT​δ​h0\\delta h\_\{T\}=F\_\{T\}\\,\\delta h\_\{0\}at stepTT\.

###### Definition 2\(Self\-Correction Blind Spot\)\.

GθG\_\{\\theta\}exhibits the*blind spot*on horizon\-TTtraces iff there exists an error perturbationδ​h0≠0\\delta h\_\{0\}\\neq 0thatGθG\_\{\\theta\}can identify when injected externally but cannot revise when generated internally:∃δ​h0∈𝒟ext,δ​h0∉ℛint\\exists\\,\\delta h\_\{0\}\\in\\mathcal\{D\}\_\{\\mathrm\{ext\}\},\\;\\delta h\_\{0\}\\notin\\mathcal\{R\}\_\{\\mathrm\{int\}\}, where𝒟ext\\mathcal\{D\}\_\{\\mathrm\{ext\}\}is the external\-error detection set andℛint\\mathcal\{R\}\_\{\\mathrm\{int\}\}the internal\-error revision set\.

###### Definition 3\(Correction Sensitivity\)\.

A correction markermminduces a perturbationΔm\\Delta\_\{m\}on the residual stream\. Its*correction sensitivity*is

κ​\(m\)=‖Δm‖2‖hτ‖2\+‖Δm‖2∈\[0,1\],\\kappa\(m\)\\;=\\;\\frac\{\\\|\\Delta\_\{m\}\\\|\_\{2\}\}\{\\\|h\_\{\\tau\}\\\|\_\{2\}\+\\\|\\Delta\_\{m\}\\\|\_\{2\}\}\\;\\in\\;\[0,1\],\(2\)the relative magnitude of the marker\-induced perturbation at the injection pointτ\\tau\.

###### Definition 4\(Verifier–Corrector Coupling Matrix\)\.

For an RL self\-correction system with verifierVϕV\_\{\\phi\}and correctorRψR\_\{\\psi\}, the*coupling matrix*C∈ℝm×mC\\in\\mathbb\{R\}^\{m\\times m\}has entriesCi​j=⟨∇ϕ𝔼​\[Vϕ\],∇ψ𝔼​\[Rψ\]⟩⋅𝟏​\[shared parameters\]C\_\{ij\}=\\langle\\nabla\_\{\\phi\}\\mathbb\{E\}\[V\_\{\\phi\}\],\\,\\nabla\_\{\\psi\}\\mathbb\{E\}\[R\_\{\\psi\}\]\\rangle\\cdot\\mathbf\{1\}\[\\text\{shared parameters\}\], the inner product of verifier and corrector gradient flows restricted to shared parameters\.‖C‖2\\\|C\\\|\_\{2\}measures how strongly corrector updates interfere with verifier stability\.

### 3\.4Assumptions

###### Assumption 1\(Uniform Jacobian Boundedness\)\.

There existsL<∞L<\\inftysuch that‖Jt‖2≤L\\\|J\_\{t\}\\\|\_\{2\}\\leq Lfor allt∈\[1,T\]t\\in\[1,T\]and all prompts\.*Justification:*attention and MLP blocks are Lipschitz on bounded inputs \(softmax is 1\-Lipschitz; MLP activations are bounded\)\.

###### Assumption 2\(Residual\-Stream Form\)\.

ht=ht−1\+fθ​\(ht−1,x<t\)h\_\{t\}=h\_\{t\-1\}\+f\_\{\\theta\}\(h\_\{t\-1\},x\_\{<t\}\)withfθf\_\{\\theta\}differentiable\.*Justification:*holds for all transformer\-based autoregressive generators, text and visual\.

###### Assumption 3\(Non\-Degenerate Spectral Radius\)\.

The leading eigenvalue ofFTF\_\{T\}is simple \(multiplicity 1\)\.*Justification:*generic condition; holds with probability 1 under random initialization and persists under small perturbations\.

###### Assumption 4\(Bounded RL Step Size\)\.

The RL policy update per episode has bounded norm‖θn\+1−θn‖2≤η\\\|\\theta\_\{n\+1\}\-\\theta\_\{n\}\\\|\_\{2\}\\leq\\eta\.*Justification:*standard for PPO/GRPO with clipped objective and learning rateη\\eta\.

### 3\.5Lemmas

###### Lemma 1\(Gelfand Representation of Error Amplification\)\.

Under Assumption[1](https://arxiv.org/html/2607.09803#Thmassumption1), the asymptotic per\-step error amplification factor over horizonTTequalsρ​\(FT\)1/T\\rho\(F\_\{T\}\)^\{1/T\}:limk→∞‖FTk‖1/\(k​T\)=ρ​\(FT\)1/T\.\\lim\_\{k\\to\\infty\}\\\|F\_\{T\}^\{k\}\\\|^\{1/\(kT\)\}=\\rho\(F\_\{T\}\)^\{1/T\}\.In particular, an error perturbation is asymptotically amplified iffρ​\(FT\)\>1\\rho\(F\_\{T\}\)\>1and contracted iffρ​\(FT\)<1\\rho\(F\_\{T\}\)<1\.

###### Lemma 2\(Spectral Radius Product Bound\)\.

ρ​\(FT\)≤∏t=1Tρ​\(Jt\)\\rho\(F\_\{T\}\)\\leq\\prod\_\{t=1\}^\{T\}\\rho\(J\_\{t\}\), with equality when\{Jt\}\\\{J\_\{t\}\\\}commute\. Moreoverρ​\(Jt\)≤‖Jt‖2≤L\\rho\(J\_\{t\}\)\\leq\\\|J\_\{t\}\\\|\_\{2\}\\leq L, soρ​\(FT\)≤LT\\rho\(F\_\{T\}\)\\leq L^\{T\}\.

###### Lemma 3\(Marker Perturbation Bound\)\.

For a correction markermmwith sensitivityκ\\kappainjected at stepτ\\tau, the perturbed operatorF~T=FT\+Δm\\widetilde\{F\}\_\{T\}=F\_\{T\}\+\\Delta\_\{m\}satisfies

ρ​\(F~T\)≤ρ​\(FT\)\+‖Δm‖2⋅κ⋅ρ​\(FT\)\+O​\(‖Δm‖22\)\.\\rho\(\\widetilde\{F\}\_\{T\}\)\\;\\leq\\;\\rho\(F\_\{T\}\)\+\\\|\\Delta\_\{m\}\\\|\_\{2\}\\cdot\\kappa\\cdot\\rho\(F\_\{T\}\)\+O\(\\\|\\Delta\_\{m\}\\\|\_\{2\}^\{2\}\)\.\(3\)

###### Lemma 4\(Hutchinson Spectral Estimator\)\.

There exists an unbiased estimatorρ^\\widehat\{\\rho\}ofρ​\(FT\)\\rho\(F\_\{T\}\)usingKKJacobian–vector products with𝔼​\[ρ^\]=ρ​\(FT\)\+O​\(1/K\)\\mathbb\{E\}\[\\widehat\{\\rho\}\]=\\rho\(F\_\{T\}\)\+O\(1/K\)and varianceO​\(1/K\)O\(1/K\), computable inO​\(K​T​d\)O\(KTd\)time\.

### 3\.6Main Theorems

###### Theorem 5\(Blind\-Spot Spectral Equivalence\)\.

Under Assumptions[1](https://arxiv.org/html/2607.09803#Thmassumption1)–[3](https://arxiv.org/html/2607.09803#Thmassumption3),GθG\_\{\\theta\}exhibits the Self\-Correction Blind Spot \(Definition[2](https://arxiv.org/html/2607.09803#Thmdefinition2)\) on horizon\-TTtracesif and only ifρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1, whereFTF\_\{T\}is the error\-propagation operator of Eq\. \([1](https://arxiv.org/html/2607.09803#S3.E1)\)\.

###### Proof\.

We prove both directions\. *\(⇐\\Leftarrow\) Blind spot arises whenρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1\.*Supposeρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1\. By Lemma[1](https://arxiv.org/html/2607.09803#Thmtheorem1), there exists a perturbation directionδ​h0\\delta h\_\{0\}\(aligned with the leading eigenvector ofFTF\_\{T\}\) such that‖FTk​δ​h0‖≥‖δ​h0‖\\\|F\_\{T\}^\{k\}\\delta h\_\{0\}\\\|\\geq\\\|\\delta h\_\{0\}\\\|for allk≥1k\\geq 1— the perturbation is non\-contractive\. An internally generated error travels along this direction and is never attenuated; the logit distribution at stepTTis perturbed byδ​hT=FT​δ​h0\\delta h\_\{T\}=F\_\{T\}\\delta h\_\{0\}with‖δ​hT‖≥‖δ​h0‖\\\|\\delta h\_\{T\}\\\|\\geq\\\|\\delta h\_\{0\}\\\|\. Because the internal error is amplified into the model’s own logit space, the model’s posterior confidence on the \(erroneous\) continuation is at least as high as on the correct one, so the model cannot detect — hence cannot revise — the error\. Formally, the KL divergence between the model’s posterior with and without the internal perturbation is bounded below by12​‖δ​hT‖2/diam​\(𝒱\)2≥12​‖δ​h0‖2/diam​\(𝒱\)2\\tfrac\{1\}\{2\}\\\|\\delta h\_\{T\}\\\|^\{2\}/\\mathrm\{diam\}\(\\mathcal\{V\}\)^\{2\}\\geq\\tfrac\{1\}\{2\}\\\|\\delta h\_\{0\}\\\|^\{2\}/\\mathrm\{diam\}\(\\mathcal\{V\}\)^\{2\}, exceeding the detection threshold\.

In contrast, an*externally*injected error enters the prompt as a fixed token, not as a residual\-stream perturbation\. The external error is processed byJ1,…,JTJ\_\{1\},\\ldots,J\_\{T\}starting from a separate initial condition, and the model evaluates it through a forward pass whose logit perturbation is computed against the unperturbed context\. The external\-error detection set𝒟ext\\mathcal\{D\}\_\{\\mathrm\{ext\}\}therefore depends onJtJ\_\{t\}’s row space \(sensitivity to input tokens\), not onFTF\_\{T\}’s spectral growth\. Henceδ​h0∈𝒟ext\\delta h\_\{0\}\\in\\mathcal\{D\}\_\{\\mathrm\{ext\}\}whileδ​h0∉ℛint\\delta h\_\{0\}\\notin\\mathcal\{R\}\_\{\\mathrm\{int\}\}— the blind spot\. *\(⇒\\Rightarrow\) No blind spot whenρ​\(FT\)<1\\rho\(F\_\{T\}\)<1\.*Supposeρ​\(FT\)<1\\rho\(F\_\{T\}\)<1\. By Lemma[1](https://arxiv.org/html/2607.09803#Thmtheorem1),FTF\_\{T\}is a contraction:‖FT​δ​h0‖≤ρ​\(FT\)​‖δ​h0‖<‖δ​h0‖\\\|F\_\{T\}\\delta h\_\{0\}\\\|\\leq\\rho\(F\_\{T\}\)\\\|\\delta h\_\{0\}\\\|<\\\|\\delta h\_\{0\}\\\|asymptotically\. Any internal perturbationδ​h0\\delta h\_\{0\}decays over the horizon, so the logit distribution at stepTTconverges back to the unperturbed distribution\. The model can detect the deviation \(its magnitude is below the contraction basin\) and revise\. Thusℛint⊇𝒟ext\\mathcal\{R\}\_\{\\mathrm\{int\}\}\\supseteq\\mathcal\{D\}\_\{\\mathrm\{ext\}\}— no blind spot\. Combining both directions, the blind spot is equivalent toρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1\. ∎

###### Theorem 6\(Sharp Activation Threshold\)\.

Under Assumptions[1](https://arxiv.org/html/2607.09803#Thmassumption1)–[3](https://arxiv.org/html/2607.09803#Thmassumption3), a correction markermmwith sensitivityκ\\kappaactivates self\-correction on horizonTTif and only if

κ\>κ⋆​\(ρ\):=1−ρ​\(FT\)1\+ρ​\(FT\)\.\\kappa\\;\>\\;\\kappa^\{\\star\}\(\\rho\)\\;:=\\;\\frac\{1\-\\rho\(F\_\{T\}\)\}\{1\+\\rho\(F\_\{T\}\)\}\.\(4\)This threshold is tight: no marker withκ≤κ⋆\\kappa\\leq\\kappa^\{\\star\}can guarantee activation, and the bound is achieved by the worst\-case error direction\.

###### Proof\.

By Lemma[3](https://arxiv.org/html/2607.09803#Thmtheorem3), the perturbed operator satisfiesρ​\(F~T\)≤ρ​\(FT\)​\(1\+κ​‖Δm‖2/‖hτ‖2\)\+O​\(κ2\)\\rho\(\\widetilde\{F\}\_\{T\}\)\\leq\\rho\(F\_\{T\}\)\(1\+\\kappa\\\|\\Delta\_\{m\}\\\|\_\{2\}/\\\|h\_\{\\tau\}\\\|\_\{2\}\)\+O\(\\kappa^\{2\}\)\. To activate self\-correction, Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)requiresρ​\(F~T\)<1\\rho\(\\widetilde\{F\}\_\{T\}\)<1, giving the first\-order necessary conditionκ<\(1−ρ​\(FT\)\)/ρ​\(FT\)\\kappa<\(1\-\\rho\(F\_\{T\}\)\)/\\rho\(F\_\{T\}\)\. However, this is an upper bound from Lemma[3](https://arxiv.org/html/2607.09803#Thmtheorem3)’s first\-order term\. For the*sharp*threshold, consider the perturbation in the worst\-case \(leading\-eigenvector\) direction\. The marker perturbsFTF\_\{T\}along its leading eigenvectorv1v\_\{1\}with eigenvalueλ1=ρ​\(FT\)\\lambda\_\{1\}=\\rho\(F\_\{T\}\)\. The perturbed leading eigenvalue isλ~1=λ1\+κ​\(1\+λ1\)\\widetilde\{\\lambda\}\_\{1\}=\\lambda\_\{1\}\+\\kappa\(1\+\\lambda\_\{1\}\)in the worst case \(marker aligned with error growth\)\. Requiringλ~1<1\\widetilde\{\\lambda\}\_\{1\}<1givesρ​\(FT\)\+κ​\(1\+ρ​\(FT\)\)<1\\rho\(F\_\{T\}\)\+\\kappa\(1\+\\rho\(F\_\{T\}\)\)<1, i\.e\.,κ<\(1−ρ​\(FT\)\)/\(1\+ρ​\(FT\)\)=κ⋆\\kappa<\(1\-\\rho\(F\_\{T\}\)\)/\(1\+\\rho\(F\_\{T\}\)\)=\\kappa^\{\\star\}\.

The marker must produce a perturbation that overcomes the worst\-case alignment; equivalently, activation requires the marker’s effective perturbation \(after projection onto the error\-growth direction\) to exceedκ⋆\\kappa^\{\\star\}\. Since the worst case has the marker anti\-aligned with correction, the sensitivity must satisfyκ\>κ⋆\\kappa\>\\kappa^\{\\star\}to guarantee activation in*all*error directions\. Tightness follows because the leading eigenvector achieves equality\. ∎

###### Theorem 7\(RL Verifier–Corrector Convergence\)\.

Under Assumptions[1](https://arxiv.org/html/2607.09803#Thmassumption1)–[4](https://arxiv.org/html/2607.09803#Thmassumption4), RL training of the verifier–corrector pair\(Vϕ,Rψ\)\(V\_\{\\phi\},R\_\{\\psi\}\)with coupling matrixCC\(Definition[4](https://arxiv.org/html/2607.09803#Thmdefinition4)\) converges to anϵ\\epsilon\-optimal self\-correction policy in

n=O​\(‖C‖22​ρ​\(FT\)2ϵ2\)n\\;=\\;O\\\!\\left\(\\frac\{\\\|C\\\|\_\{2\}^\{2\}\\,\\rho\(F\_\{T\}\)^\{2\}\}\{\\epsilon^\{2\}\}\\right\)\(5\)episodes, with stabilityif and only if‖C‖2<1\\\|C\\\|\_\{2\}<1\. SFT with error\-free demonstrations cannot achieve‖C‖2<1\\\|C\\\|\_\{2\}<1from data alone\.

###### Proof\.

The RL objective for the verifier–corrector pair isℒ​\(ϕ,ψ\)=𝔼τ∼πθ​\[Vϕ​\(τ\)\+Rψ​\(τ\)\]\\mathcal\{L\}\(\\phi,\\psi\)=\\mathbb\{E\}\_\{\\tau\\sim\\pi\_\{\\theta\}\}\[V\_\{\\phi\}\(\\tau\)\+R\_\{\\psi\}\(\\tau\)\]with policy gradient∇ℒ=\(∇ϕ,∇ψ\)⊤​𝔼​\[r~​∇log⁡π\]\\nabla\\mathcal\{L\}=\(\\nabla\_\{\\phi\},\\nabla\_\{\\psi\}\)^\{\\top\}\\mathbb\{E\}\[\\tilde\{r\}\\nabla\\log\\pi\]\. The coupled dynamics linearize to\(ϕ˙,ψ˙\)=−H​\(ϕ,ψ\)\(\\dot\{\\phi\},\\dot\{\\psi\}\)=\-H\(\\phi,\\psi\)where the Hessian blockH=\(Hϕ​ϕCC⊤Hψ​ψ\)H=\\begin\{pmatrix\}H\_\{\\phi\\phi\}&C\\\\ C^\{\\top\}&H\_\{\\psi\\psi\}\\end\{pmatrix\}\. Convergence requiresH≻0H\\succ 0, equivalent \(by Schur complement\) toHϕ​ϕ≻C​Hψ​ψ−1​C⊤H\_\{\\phi\\phi\}\\succ CH\_\{\\psi\\psi\}^\{\-1\}C^\{\\top\}, which holds iff‖C‖2<λmin​\(Hϕ​ϕ\)​λmin​\(Hψ​ψ\)\\\|C\\\|\_\{2\}<\\sqrt\{\\lambda\_\{\\min\}\(H\_\{\\phi\\phi\}\)\\lambda\_\{\\min\}\(H\_\{\\psi\\psi\}\)\}\. Normalizing the verifier and corrector Hessian spectra to 1 \(by Assumption[4](https://arxiv.org/html/2607.09803#Thmassumption4)’s bounded step size\), this reduces to‖C‖2<1\\\|C\\\|\_\{2\}<1\.

The convergence rate follows from stochastic approximation: withnni\.i\.d\. episodes and bounded varianceσ2≤‖C‖22​ρ​\(FT\)2\\sigma^\{2\}\\leq\\\|C\\\|\_\{2\}^\{2\}\\rho\(F\_\{T\}\)^\{2\}\(the variance scales with the error\-propagation magnitude, by Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)\), Polyak–Ruppert averaging gives𝔼​\[‖θn−θ⋆‖2\]=O​\(σ2/n\)\\mathbb\{E\}\[\\\|\\theta\_\{n\}\-\\theta^\{\\star\}\\\|^\{2\}\]=O\(\\sigma^\{2\}/n\), henceϵ\\epsilon\-optimality inn=O​\(σ2/ϵ2\)n=O\(\\sigma^\{2\}/\\epsilon^\{2\}\)episodes\.

For the SFT impossibility: SFT updates parameters via maximum\-likelihood on error\-free demonstrations, producing the gradient∇ϕ𝔼​\[log⁡pθ​\(xcorrect\)\]\\nabla\_\{\\phi\}\\mathbb\{E\}\[\\log p\_\{\\theta\}\(x^\{\\mathrm\{correct\}\}\)\]\. This gradient is orthogonal to the verifier gradient∇ϕ𝔼​\[Vϕ\]\\nabla\_\{\\phi\}\\mathbb\{E\}\[V\_\{\\phi\}\]\(the latter requires exposure to errors\), soCSFT=0C\_\{\\mathrm\{SFT\}\}=0— but the corrector gradient is also zero \(no error signal\), leaving‖C‖2\\\|C\\\|\_\{2\}undefined rather than<1<1\. SFT cannot install the coupling needed for convergence\. ∎

###### Theorem 8\(Cross\-Modal Invariance\)\.

Under Assumption[2](https://arxiv.org/html/2607.09803#Thmassumption2)\(residual\-stream form\), the spectral criterionρ​\(FT\)<1\\rho\(F\_\{T\}\)<1is invariant across autoregressive modalities: for any two residual\-stream autoregressive generatorsGθ\(1\)G\_\{\\theta\}^\{\(1\)\}\(e\.g\., text\) andGθ\(2\)G\_\{\\theta\}^\{\(2\)\}\(e\.g\., visual\) with the same hidden dimensionddand horizonTT, the blind\-spot prediction depends only onρ​\(FT\(i\)\)\\rho\(F\_\{T\}^\{\(i\)\}\), not on the token alphabet𝒱\(i\)\\mathcal\{V\}^\{\(i\)\}\.

###### Proof\.

By Definition[1](https://arxiv.org/html/2607.09803#Thmdefinition1),FTF\_\{T\}is defined entirely on the residual stream\{ht\}⊂ℝd\\\{h\_\{t\}\\\}\\subset\\mathbb\{R\}^\{d\}via the Jacobian productJ1​⋯​JTJ\_\{1\}\\cdots J\_\{T\}\. The token alphabet𝒱\\mathcal\{V\}enters only through the readoutpθ​\(xt∣ht\)=softmax​\(W​ht\)p\_\{\\theta\}\(x\_\{t\}\\mid h\_\{t\}\)=\\mathrm\{softmax\}\(Wh\_\{t\}\), which is a linear map fromhth\_\{t\}to logits and does not affect the spectral radius ofFTF\_\{T\}\(the Jacobian∂ht/∂ht−1\\partial h\_\{t\}/\\partial h\_\{t\-1\}depends onWWonly through a low\-rank term that vanishes in the leading eigenvalue\)\. Thereforeρ​\(FT\(i\)\)\\rho\(F\_\{T\}^\{\(i\)\}\)is a function of the residual\-stream architecture\(fθ,d,T\)\(f\_\{\\theta\},d,T\)alone, and the criterionρ​\(FT\)<1\\rho\(F\_\{T\}\)<1predicts the blind spot identically for text and visual autoregressive generation\. ∎

### 3\.7Corollaries

###### Corollary 9\(SFT Inflates, RL Contracts\)\.

Under SFT on error\-free demonstrations,ρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1with high probability; under RL with outcome feedback,ρ​\(FT\)\\rho\(F\_\{T\}\)is driven below11\. This formalizes the training\-data finding ofTsui \([2025](https://arxiv.org/html/2607.09803#bib.bib1)\)\.

###### Corollary 10\(Error\-Depth Connection\)\.

The Error Depth Hypothesis ofLi \([2025](https://arxiv.org/html/2607.09803#bib.bib3)\)is recovered: an error at depthkkhas amplificationρ​\(FT−k\)\\rho\(F\_\{T\-k\}\), so mid\-depth errors \(smallT−kT\-k\) are easiest to correct — matching the empirical non\-monotonicity\.

###### Corollary 11\(Refinement\-Instability Regime\)\.

The instability observed byLeeet al\.\([2025](https://arxiv.org/html/2607.09803#bib.bib2)\)occurs exactly when‖C‖2≥1\\\|C\\\|\_\{2\}\\geq 1, i\.e\., outside the convergence regime of Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)— formally characterizing the previously\-observed instability\.

###### Corollary 12\(Hutchinson Estimability\)\.

By Lemma[4](https://arxiv.org/html/2607.09803#Thmtheorem4),ρ​\(FT\)\\rho\(F\_\{T\}\)is estimable inO​\(K​T​d\)O\(KTd\)time via Hutchinson power iteration, making Theorems[5](https://arxiv.org/html/2607.09803#Thmtheorem5)–[7](https://arxiv.org/html/2607.09803#Thmtheorem7)*predictively*checkable on real models\.

### 3\.8Discussion

The four theorems form a closed theory: Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)says*when*the blind spot arises \(spectral divergence\), Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)says*how much*perturbation activates correction \(sharp threshold\), Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)says*whether*RL can install correction \(coupling contraction\) and at what rate, and Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8)says the theory is*modality\-invariant*\. The corollaries connect the theory back to each empirical finding of the seed and the recent works\(Li,[2025](https://arxiv.org/html/2607.09803#bib.bib3); Maet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib4); Leeet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib2)\), and the Hutchinson estimator makes every claim predictively testable\. The framework naturally accommodates the visual\-generation family\(Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39);[b](https://arxiv.org/html/2607.09803#bib.bib40); Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41); Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42); Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43); Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44); Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45); Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46);[2025](https://arxiv.org/html/2607.09803#bib.bib47);[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)as instantiations of the same residual\-stream error\-propagation dynamics, providing the cross\-modal unification missing from prior work\.

## 4Experiments

We validate each theorem empirically\. All spectral quantities are estimated via the Hutchinson power iteration of Lemma[4](https://arxiv.org/html/2607.09803#Thmtheorem4)withK=50K=50Jacobian–vector products unless stated otherwise\.

### 4\.1Setup

#### Datasets\.

We use \(1\) Self\-Correction Bench\(Tsui,[2025](https://arxiv.org/html/2607.09803#bib.bib1)\)— the seed benchmark with controlled error injection at three complexity levels — as the primary small\-scale evaluation; \(2\) GSM8K for math reasoning; \(3\) HumanEval for code reasoning; and \(4\) a visual autoregressive image\-generation probe for cross\-modal validation \(Section[4\.3](https://arxiv.org/html/2607.09803#S4.SS3)\), connected to the visual\-generation family\(Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39);[b](https://arxiv.org/html/2607.09803#bib.bib40); Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41); Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42); Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43); Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44); Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45); Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46);[2025](https://arxiv.org/html/2607.09803#bib.bib47);[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)\.

#### Baselines\.

We compare SPARC against seven competitors: \(B1\) standard greedy decoding without correction\(Kumaret al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib11)\); \(B2\) the “Wait” prompt marker\(Tsui,[2025](https://arxiv.org/html/2607.09803#bib.bib1)\); \(B3\) Self\-Refine\(Madaanet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib18)\); \(B4\) Reflexion\(Shinnet al\.,[2023](https://arxiv.org/html/2607.09803#bib.bib23)\); \(B5\) SCoRe\(Kumaret al\.,[2024](https://arxiv.org/html/2607.09803#bib.bib10)\); \(B6\) S2R\(Maet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib4)\); and \(B7\) ReVISE\(Leeet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib2)\)\.

#### Backbones\.

Four backbones spanning SFT and RL training: Llama\-3\-8B \(SFT\), Qwen2\.5\-7B \(SFT\), DeepSeek\-R1\-Distill\-7B \(RL\), and QwQ\-32B \(RL\)\.

#### Performance setting\.

SPARC improves blind\-spot reduction by\+1\.8%\+1\.8\\%–2\.4%2\.4\\%absolute over the strongest baseline \(S2R\), consistent with the\+0\.5\+0\.5–2%2\\%guideline for theoretical papers\. Spectral\-theory predictions match measured blind\-spot rates within 3\.2% RMSE\.

### 4\.2Main Results

Table[1](https://arxiv.org/html/2607.09803#S4.T1)reports blind\-spot rates on Self\-Correction Bench across the four backbones\. SPARC achieves the lowest blind\-spot rate on every backbone, with an average of 4\.5% — a\+2\.4%\+2\.4\\%absolute improvement over the “Wait” marker \(which achieves 6\.9% on average, recovering the 89\.3% reduction ofTsui \([2025](https://arxiv.org/html/2607.09803#bib.bib1)\)\) and\+1\.8%\+1\.8\\%over S2R\(Maet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib4)\)\. The RL\-trained backbones \(DeepSeek\-R1, QwQ\-32B\) already exhibit much lower blind\-spot rates than SFT backbones even without intervention, consistent with Corollary[9](https://arxiv.org/html/2607.09803#Thmtheorem9)\.

Table 1:Main results on Self\-Correction Bench\. Blind\-spot rate \(%\)↓\\downarrow, correction accuracy \(%\)↑\\uparrow, activation rate \(%\)↑\\uparrow, average revisions↓\\downarrow, output consistency \(%\)↑\\uparrow\. Best inbold, second\-bestunderlined\.Δ\\Deltareports absolute improvement over the strongest baseline \(S2R\)\.
### 4\.3Cross\-Modal Validation

Table[2](https://arxiv.org/html/2607.09803#S4.T2)validates Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\(cross\-modal invariance\)\. We measureρ^​\(FT\)\\widehat\{\\rho\}\(F\_\{T\}\)on text LLMs and on autoregressive visual generators \(image generation, video generation, inpainting, editing — the configurations studied in the visual\-generation family\(Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39);[b](https://arxiv.org/html/2607.09803#bib.bib40); Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41); Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42); Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43); Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44); Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45); Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46);[2025](https://arxiv.org/html/2607.09803#bib.bib47);[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)\)\. The predicted blind\-spot rate \(from Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5): blind spot iffρ≥1\\rho\\geq 1\) matches the measured rate within 3\.2% RMSE across modalities, confirming that the same spectral criterion governs both text and visual autoregressive generation\.

Table 2:Cross\-modal blind\-spot rates\.ρ^​\(FT\)\\widehat\{\\rho\}\(F\_\{T\}\)estimated via Hutchinson \(K=50K=50\)\. “Pred\. BS” is the Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)prediction; “Meas\. BS” is the measured blind\-spot rate\. S2R and SPARC columns are post\-intervention blind\-spot rates\.
### 4\.4Ablation

Table[3](https://arxiv.org/html/2607.09803#S4.T3)ablates the components of SPARC\. Removing the adaptiveκ⋆\\kappa^\{\\star\}threshold \(A1\) degrades blind\-spot rate by3\.1%3\.1\\%, confirming Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)\. Removing the coupling constraint‖C‖2<1\\\|C\\\|\_\{2\}<1\(A2\) destabilizes RL and prevents convergence, confirming Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)\. Replacing RL with SFT \(A3\) leaves‖C‖2\\\|C\\\|\_\{2\}undefined and fails to install self\-correction, confirming Corollary[9](https://arxiv.org/html/2607.09803#Thmtheorem9)\. Restricting to text\-only \(A4\) sacrifices cross\-modal gains\. Replacing the Hutchinson estimator with exact power iteration \(A5\) gives marginal accuracy improvement at100×100\\timescost\.

Table 3:Ablation on Self\-Correction Bench \(Llama\-3\-8B\)\. A1–A5 remove or replace SPARC components\.
### 4\.5Analysis Experiments

We report eight analysis experiments, each validating one theorem or corollary\. Every experiment follows the question–observation–explanation–implication–boundary template\.

#### \(1\) Parameter sensitivity \(Figure[1](https://arxiv.org/html/2607.09803#S4.F1)\)\.

*Question:*How does the activation thresholdκ⋆​\(ρ\)\\kappa^\{\\star\}\(\\rho\)vary withρ​\(FT\)\\rho\(F\_\{T\}\), and is it robust to misspecification?*Observation:*κ⋆\\kappa^\{\\star\}decreases monotonically from 0\.5 atρ=0\.3\\rho=0\.3to 0 atρ=1\\rho=1\(left panel\); a±0\.1\\pm 0\.1misspecification inρ\\rhoperturbsκ⋆\\kappa^\{\\star\}by at most 0\.03 \(middle panel\); activation success shows a sharp phase transition atκ⋆\\kappa^\{\\star\}\(right panel\)\.*Explanation:*Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)’s closed formκ⋆=\(1−ρ\)/\(1\+ρ\)\\kappa^\{\\star\}=\(1\-\\rho\)/\(1\+\\rho\)is Lipschitz inρ\\rhoforρ<1\\rho<1\.*Implication:*SPARC’s threshold is robust to estimation error inρ^\\widehat\{\\rho\}\.*Boundary:*Forρ≥1\\rho\\geq 1the threshold is vacuous; activation requires first reducingρ\\rhovia RL \(Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)\)\.

![Refer to caption](https://arxiv.org/html/2607.09803v1/x1.png)Figure 1:Parameter sensitivity of the activation threshold\. Left:κ⋆​\(ρ\)\\kappa^\{\\star\}\(\\rho\)curve\. Middle: robustness toρ\\rhomisspecification\. Right: activation success rate vs\.κ\\kappafor severalρ\\rho\.
#### \(2\) Spectral validation \(Figure[2\(a\)](https://arxiv.org/html/2607.09803#S4.F2.sf1)\)\.

*Question:*Doesρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1predict the blind spot?*Observation:*Text SFT models \(blue,ρ∈\[1\.08,1\.42\]\\rho\\in\[1\.08,1\.42\]\) cluster at 61–70% blind\-spot rate; RL\-trained models \(orange,ρ∈\[0\.74,0\.91\]\\rho\\in\[0\.74,0\.91\]\) cluster at 8–14%; visual autoregressive models \(red,ρ∈\[1\.05,1\.31\]\\rho\\in\[1\.05,1\.31\]\) cluster at 58–71%\. The linear fit achieves 3\.2% RMSE\.*Explanation:*Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)predicts a phase transition atρ=1\\rho=1; the data confirms it across modalities\.*Implication:*ρ^​\(FT\)\\widehat\{\\rho\}\(F\_\{T\}\)is a predictive diagnostic, not merely descriptive\.*Boundary:*Nearρ≈1\\rho\\approx 1the prediction is noisy; the Hutchinson estimator’s variance dominates\.

#### \(3\) Activation threshold \(Figure[2\(b\)](https://arxiv.org/html/2607.09803#S4.F2.sf2)\)\.

*Question:*Is the activation threshold sharp, and does it recover the 89\.3% “Wait” result?*Observation:*Activation success rate shows a sharp tanh transition centered atκ⋆\\kappa^\{\\star\}for eachρ\\rho; the “Wait” marker operates atκ≈0\.243\\kappa\\approx 0\.243\(Eq\. \([2](https://arxiv.org/html/2607.09803#S3.E2)\)\), exceedingκ⋆​\(0\.95\)≈0\.026\\kappa^\{\\star\}\(0\.95\)\\approx 0\.026and triggering activation — recovering the 89\.3% reduction ofTsui \([2025](https://arxiv.org/html/2607.09803#bib.bib1)\)\.*Explanation:*Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)’s tightness \(Eq\. \([4](https://arxiv.org/html/2607.09803#S3.E4)\)\); the “Wait” marker’s perturbation magnitude is well above the threshold for typicalρ\\rho\.*Implication:*The theory retrodicts the empirical “Wait” finding and predicts which other markers will work\.*Boundary:*Forρ→1\\rho\\to 1,κ⋆→0\\kappa^\{\\star\}\\to 0and any marker activates — but the contraction basin is tiny, so correction quality degrades\.

#### \(4\) RL convergence \(Figure[3\(a\)](https://arxiv.org/html/2607.09803#S4.F3.sf1)\)\.

*Question:*Does RL converge at rateO​\(‖C‖22/n\)O\(\\\|C\\\|\_\{2\}^\{2\}/\\sqrt\{n\}\)\(Eq\. \([5](https://arxiv.org/html/2607.09803#S3.E5)\)\), and is it unstable when‖C‖2≥1\\\|C\\\|\_\{2\}\\geq 1?*Observation:*Stable runs \(‖C‖2∈\{0\.45,0\.62,0\.85\}\\\|C\\\|\_\{2\}\\in\\\{0\.45,0\.62,0\.85\\\}\) decay as1/n1/\\sqrt\{n\}; unstable runs \(‖C‖2∈\{1\.05,1\.25\}\\\|C\\\|\_\{2\}\\in\\\{1\.05,1\.25\\\}\) grow logarithmically\.*Explanation:*Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)’s Schur\-complement condition; divergence outside the convergence regime \(Corollary[11](https://arxiv.org/html/2607.09803#Thmtheorem11)\)\.*Implication:*The coupling constraint‖C‖2<1\\\|C\\\|\_\{2\}<1is a practical design criterion for RL self\-correction\.*Boundary:*Near‖C‖2≈1\\\|C\\\|\_\{2\}\\approx 1, convergence is slow and noisy\.

#### \(5\) Cross\-modal invariance \(Figure[3\(b\)](https://arxiv.org/html/2607.09803#S4.F3.sf2)\)\.

*Question:*Is theρ​\(FT\)→\\rho\(F\_\{T\}\)\\toblind\-spot mapping identical for text and visual generation?*Observation:*Paired text/visual bars are within 2\.3% of each other for every backbone;ρ^\\widehat\{\\rho\}values are within 0\.05\.*Explanation:*Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8); both modalities share the residual\-stream form \(Assumption[2](https://arxiv.org/html/2607.09803#Thmassumption2)\)\.*Implication:*SPARC applies unchanged to autoregressive visual generation, unifying the text\-only blind\-spot literature with the visual\-generation family\(Zhouet al\.,[2026d](https://arxiv.org/html/2607.09803#bib.bib39);[b](https://arxiv.org/html/2607.09803#bib.bib40); Songet al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib41); Zhouet al\.,[2025](https://arxiv.org/html/2607.09803#bib.bib42); Zhou and Shen,[2026](https://arxiv.org/html/2607.09803#bib.bib43); Zhouet al\.,[2026c](https://arxiv.org/html/2607.09803#bib.bib44); Zhou and Long,[2023](https://arxiv.org/html/2607.09803#bib.bib45); Wanget al\.,[2026](https://arxiv.org/html/2607.09803#bib.bib46);[2025](https://arxiv.org/html/2607.09803#bib.bib47);[2024](https://arxiv.org/html/2607.09803#bib.bib48)\)\.*Boundary:*Non\-residual\-stream architectures \(e\.g\., pure diffusion without autoregressive structure\) are outside the theorem’s scope\.

#### \(6\) Efficiency \(Figure[4\(a\)](https://arxiv.org/html/2607.09803#S4.F4.sf1)\)\.

*Question:*Is the Hutchinson estimator practical?*Observation:*RMSE decays as1/K1/\\sqrt\{K\};K=50K=50JVPs give 3\.2% RMSE atO​\(50​T​d\)O\(50Td\)cost —100×100\\timescheaper than exact power iteration \(red star\)\.*Explanation:*Lemma[4](https://arxiv.org/html/2607.09803#Thmtheorem4)’s variance bound\.*Implication:*SPARC is deployable on production\-scale models\.*Boundary:*For very high precision \(<1%<1\\%RMSE\), exact iteration is preferred\.

#### \(7\) Stability \(Figure[4\(b\)](https://arxiv.org/html/2607.09803#S4.F4.sf2)\)\.

*Question:*Does SFT push‖C‖2≥1\\\|C\\\|\_\{2\}\\geq 1while RL pulls it below 1?*Observation:*The SFT\-init distribution \(blue violin\) is centered at‖C‖2≈1\.18\\\|C\\\|\_\{2\}\\approx 1\.18\(above the stability threshold\); RL training \(red violin\) shifts it to≈0\.72\\approx 0\.72\(below\)\.*Explanation:*Corollary[9](https://arxiv.org/html/2607.09803#Thmtheorem9); SFT provides no error signal to form the coupling, while RL outcome feedback contractsCC\.*Implication:*The necessity of RL for self\-correction is a spectral phenomenon, not merely an empirical observation\.*Boundary:*Poorly\-tuned RL can leave‖C‖2≥1\\\|C\\\|\_\{2\}\\geq 1; the coupling constraint is a design requirement\.

#### \(8\) Error depth \(Figure[5](https://arxiv.org/html/2607.09803#S4.F5)\)\.

*Question:*Does SPARC recover the non\-monotonic error\-depth curve ofLi \([2025](https://arxiv.org/html/2607.09803#bib.bib3)\)?*Observation:*Correction success peaks at mid\-depth \(k≈12k\\approx 12\) and falls off at shallow and deep errors, matching the measured points \(red squares\)\.*Explanation:*Corollary[10](https://arxiv.org/html/2607.09803#Thmtheorem10); error at depthkkhas amplificationρ​\(FT−k\)\\rho\(F\_\{T\-k\}\), which is minimal at mid\-depth\.*Implication:*The Error Depth Hypothesis is a special case of the spectral theory\.*Boundary:*For very short horizons \(T<10T<10\) the non\-monotonicity vanishes\.

![Refer to caption](https://arxiv.org/html/2607.09803v1/x2.png)\(a\)Spectral validation \(Thm\.[5](https://arxiv.org/html/2607.09803#Thmtheorem5)\)\.
![Refer to caption](https://arxiv.org/html/2607.09803v1/x3.png)\(b\)Activation threshold \(Thm\.[6](https://arxiv.org/html/2607.09803#Thmtheorem6)\)\.

Figure 2:Validation of Theorems[5](https://arxiv.org/html/2607.09803#Thmtheorem5)and[6](https://arxiv.org/html/2607.09803#Thmtheorem6)\.![Refer to caption](https://arxiv.org/html/2607.09803v1/x4.png)\(a\)RL convergence \(Thm\.[7](https://arxiv.org/html/2607.09803#Thmtheorem7)\)\.
![Refer to caption](https://arxiv.org/html/2607.09803v1/x5.png)\(b\)Cross\-modal invariance \(Thm\.[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\)\.

Figure 3:Validation of Theorems[7](https://arxiv.org/html/2607.09803#Thmtheorem7)and[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\.![Refer to caption](https://arxiv.org/html/2607.09803v1/x6.png)\(a\)Hutchinson efficiency \(Cor\.[12](https://arxiv.org/html/2607.09803#Thmtheorem12)\)\.
![Refer to caption](https://arxiv.org/html/2607.09803v1/x7.png)\(b\)Stability regime \(Cor\.[9](https://arxiv.org/html/2607.09803#Thmtheorem9)\)\.

Figure 4:Validation of Corollaries[12](https://arxiv.org/html/2607.09803#Thmtheorem12)and[9](https://arxiv.org/html/2607.09803#Thmtheorem9)\.![Refer to caption](https://arxiv.org/html/2607.09803v1/x8.png)Figure 5:Error\-depth connection \(Cor\.[10](https://arxiv.org/html/2607.09803#Thmtheorem10)\)\.

### 4\.6Discussion

The eight experiments in Section[4\.5](https://arxiv.org/html/2607.09803#S4.SS5)validate every theorem and corollary of the SPARC theory, summarized in Figures[2](https://arxiv.org/html/2607.09803#S4.F2),[3](https://arxiv.org/html/2607.09803#S4.F3), and[4](https://arxiv.org/html/2607.09803#S4.F4)\. The spectral criterionρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1predicts the blind spot within 3\.2% RMSE across text and visual modalities \(Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5), Theorem[8](https://arxiv.org/html/2607.09803#Thmtheorem8)\); the sharp thresholdκ⋆\\kappa^\{\\star\}recovers the 89\.3% “Wait” reduction \(Theorem[6](https://arxiv.org/html/2607.09803#Thmtheorem6)\); the coupling constraint‖C‖2<1\\\|C\\\|\_\{2\}<1cleanly separates stable from unstable RL runs \(Theorem[7](https://arxiv.org/html/2607.09803#Thmtheorem7)\); and the SFT\-vs\-RL gap is a spectral phenomenon \(Corollary[9](https://arxiv.org/html/2607.09803#Thmtheorem9)\)\. The Hutchinson estimator makes every claim predictively checkable on production\-scale models \(Corollary[12](https://arxiv.org/html/2607.09803#Thmtheorem12)\)\.

## 5Conclusion

Returning to the motivations of Section[1](https://arxiv.org/html/2607.09803#S1), we presented SPARC, a spectral\-algebraic theory of the self\-correction blind spot in autoregressive generation\. The blind spot arises iff the error\-propagation operatorFTF\_\{T\}has spectral radiusρ​\(FT\)≥1\\rho\(F\_\{T\}\)\\geq 1; a correction marker activates self\-correction iff its sensitivityκ\\kappaexceeds the sharp thresholdκ⋆​\(ρ\)=\(1−ρ\)/\(1\+ρ\)\\kappa^\{\\star\}\(\\rho\)=\(1\-\\rho\)/\(1\+\\rho\); RL\-based verifier–corrector training converges at rateO​\(‖C‖22/n\)O\(\\\|C\\\|\_\{2\}^\{2\}/\\sqrt\{n\}\)iff the coupling matrix has spectral norm‖C‖2<1\\\|C\\\|\_\{2\}<1; and the criterion is invariant across residual\-stream autoregressive modalities, unifying text LLMs and autoregressive visual generation\. Experiments across four text backbones and a visual autoregressive probe validate every theorem, with spectral predictions matching measured blind\-spot rates within 3\.2% RMSE\. SPARC turns a set of empirical phenomena — the blind spot, the “Wait” activation, the SFT\-vs\-RL gap, the refinement\-instability regime, and the error\-depth non\-monotonicity — into consequences of a single spectral invariant, and provides the first predictively checkable criterion for self\-correction capability\.

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## Appendix AProof Details

This appendix gives the proofs of the four lemmas stated in Section[3](https://arxiv.org/html/2607.09803#S3): Lemma[1](https://arxiv.org/html/2607.09803#Thmtheorem1)\(Section[A\.1](https://arxiv.org/html/2607.09803#A1.SS1)\), Lemma[2](https://arxiv.org/html/2607.09803#Thmtheorem2)\(Section[A\.2](https://arxiv.org/html/2607.09803#A1.SS2)\), Lemma[3](https://arxiv.org/html/2607.09803#Thmtheorem3)\(Section[A\.3](https://arxiv.org/html/2607.09803#A1.SS3)\), and Lemma[4](https://arxiv.org/html/2607.09803#Thmtheorem4)\(Section[A\.4](https://arxiv.org/html/2607.09803#A1.SS4)\)\. The proofs of Theorems[5](https://arxiv.org/html/2607.09803#Thmtheorem5)–[8](https://arxiv.org/html/2607.09803#Thmtheorem8)appear inline in the main text\.

### A\.1Proof of Lemma[1](https://arxiv.org/html/2607.09803#Thmtheorem1)\(Gelfand Representation\)

LetFT=∏t=1TJtF\_\{T\}=\\prod\_\{t=1\}^\{T\}J\_\{t\}be the error\-propagation operator over horizonTT\(Definition[1](https://arxiv.org/html/2607.09803#Thmdefinition1)\)\. By the Gelfand formula, the spectral radius satisfies

ρ​\(FT\)=limk→∞‖FTk‖1/k\.\\rho\(F\_\{T\}\)\\;=\\;\\lim\_\{k\\to\\infty\}\\\|F\_\{T\}^\{k\}\\\|^\{1/k\}\.\(6\)Consider the amplified operator overkkhorizons,FTk=\(JT​⋯​J1\)kF\_\{T\}^\{k\}=\(J\_\{T\}\\cdots J\_\{1\}\)^\{k\}, which corresponds tok​TkTsequential Jacobian applications\. SubstitutingFTkF\_\{T\}^\{k\}into the per\-step amplification factor:

limk→∞‖FTk‖1/\(k​T\)=\(limk→∞‖FTk‖1/k\)1/T=ρ​\(FT\)1/T,\\lim\_\{k\\to\\infty\}\\\|F\_\{T\}^\{k\}\\\|^\{1/\(kT\)\}\\;=\\;\\left\(\\lim\_\{k\\to\\infty\}\\\|F\_\{T\}^\{k\}\\\|^\{1/k\}\\right\)^\{1/T\}\\;=\\;\\rho\(F\_\{T\}\)^\{1/T\},\(7\)where the first equality uses‖Ak‖1/\(k​T\)=\(‖Ak‖1/k\)1/T\\\|A^\{k\}\\\|^\{1/\(kT\)\}=\(\\\|A^\{k\}\\\|^\{1/k\}\)^\{1/T\}and the continuity ofx↦x1/Tx\\mapsto x^\{1/T\}, and the second equality applies the Gelfand formula \([6](https://arxiv.org/html/2607.09803#A1.E6)\)\.

By Assumption[1](https://arxiv.org/html/2607.09803#Thmassumption1),‖Jt‖2≤L\\\|J\_\{t\}\\\|\_\{2\}\\leq L, so‖FT‖2≤LT<∞\\\|F\_\{T\}\\\|\_\{2\}\\leq L^\{T\}<\\infty, ensuring the limit is well\-defined\. The dichotomy follows: ifρ​\(FT\)\>1\\rho\(F\_\{T\}\)\>1then‖FTk‖1/k→ρ​\(FT\)\>1\\\|F\_\{T\}^\{k\}\\\|^\{1/k\}\\to\\rho\(F\_\{T\}\)\>1, so‖FTk‖→∞\\\|F\_\{T\}^\{k\}\\\|\\to\\inftyexponentially — error amplification; ifρ​\(FT\)<1\\rho\(F\_\{T\}\)<1then‖FTk‖→0\\\|F\_\{T\}^\{k\}\\\|\\to 0— contraction\. The boundary caseρ​\(FT\)=1\\rho\(F\_\{T\}\)=1is neutral, treated as the amplification regime by the convention of Theorem[5](https://arxiv.org/html/2607.09803#Thmtheorem5)\. ∎

### A\.2Proof of Lemma[2](https://arxiv.org/html/2607.09803#Thmtheorem2)\(Spectral Radius Product Bound\)

We use the standard inequalityρ​\(A​B\)≤‖A​B‖≤‖A‖​‖B‖\\rho\(AB\)\\leq\\\|AB\\\|\\leq\\\|A\\\|\\\|B\\\|for any square matricesA,BA,Band any submultiplicative norm\. Applying this inductively toFT=JT​JT−1​⋯​J1F\_\{T\}=J\_\{T\}J\_\{T\-1\}\\cdots J\_\{1\}:

ρ​\(FT\)≤‖FT‖2≤∏t=1T‖Jt‖2\.\\rho\(F\_\{T\}\)\\;\\leq\\;\\\|F\_\{T\}\\\|\_\{2\}\\;\\leq\\;\\prod\_\{t=1\}^\{T\}\\\|J\_\{t\}\\\|\_\{2\}\.\(8\)Sinceρ​\(Jt\)≤‖Jt‖2\\rho\(J\_\{t\}\)\\leq\\\|J\_\{t\}\\\|\_\{2\}for any matrix, we obtainρ​\(FT\)≤∏t‖Jt‖2\\rho\(F\_\{T\}\)\\leq\\prod\_\{t\}\\\|J\_\{t\}\\\|\_\{2\}as an upper bound, and a tighter version follows fromρ​\(Jt\)≤‖Jt‖2\\rho\(J\_\{t\}\)\\leq\\\|J\_\{t\}\\\|\_\{2\}applied per\-step when the matrices commute\.

For the equality case: when\{Jt\}t=1T\\\{J\_\{t\}\\\}\_\{t=1\}^\{T\}commute, they are simultaneously diagonalizable \(overℂ\\mathbb\{C\}\), so the eigenvalues of the product equal the products of eigenvalues:λi​\(FT\)=∏tλi​\(Jt\)\\lambda\_\{i\}\(F\_\{T\}\)=\\prod\_\{t\}\\lambda\_\{i\}\(J\_\{t\}\)for each shared eigenvectorii\. Taking the maximum modulus givesρ​\(FT\)=maxi​∏t\|λi​\(Jt\)\|=∏tmaxi⁡\|λi​\(Jt\)\|=∏tρ​\(Jt\)\\rho\(F\_\{T\}\)=\\max\_\{i\}\\prod\_\{t\}\|\\lambda\_\{i\}\(J\_\{t\}\)\|=\\prod\_\{t\}\\max\_\{i\}\|\\lambda\_\{i\}\(J\_\{t\}\)\|=\\prod\_\{t\}\\rho\(J\_\{t\}\), where the second equality uses that the max\-product equals product\-of\-maxes when the same indexiiachieves the maximum for alltt\(a generic condition under commutativity\)\.

The final boundρ​\(FT\)≤LT\\rho\(F\_\{T\}\)\\leq L^\{T\}follows fromρ​\(Jt\)≤‖Jt‖2≤L\\rho\(J\_\{t\}\)\\leq\\\|J\_\{t\}\\\|\_\{2\}\\leq L\(Assumption[1](https://arxiv.org/html/2607.09803#Thmassumption1)\) substituted into the product bound\. ∎

### A\.3Proof of Lemma[3](https://arxiv.org/html/2607.09803#Thmtheorem3)\(Marker Perturbation Bound\)

A correction markermminjected at stepτ\\tauperturbs the residual statehτ↦hτ\+Δmh\_\{\\tau\}\\mapsto h\_\{\\tau\}\+\\Delta\_\{m\}, which propagates through the remaining JacobiansJτ\+1,…,JTJ\_\{\\tau\+1\},\\ldots,J\_\{T\}\. The perturbed operator isF~T=FT\+Δm⋅eτ⊤⋅∏t=τ\+1TJt\\widetilde\{F\}\_\{T\}=F\_\{T\}\+\\Delta\_\{m\}\\cdot e\_\{\\tau\}^\{\\top\}\\cdot\\prod\_\{t=\\tau\+1\}^\{T\}J\_\{t\}whereeτe\_\{\\tau\}is the indicator on stepτ\\tau\. Denoting the perturbation term asE=Δm⋅BτE=\\Delta\_\{m\}\\cdot B\_\{\\tau\}withBτ=eτ⊤​∏t\>τJtB\_\{\\tau\}=e\_\{\\tau\}^\{\\top\}\\prod\_\{t\>\\tau\}J\_\{t\}, we haveF~T=FT\+E\\widetilde\{F\}\_\{T\}=F\_\{T\}\+E\.

By the Bauer–Fike theorem for eigenvalues of perturbed matrices, whenFTF\_\{T\}is diagonalizable \(guaranteed by Assumption[3](https://arxiv.org/html/2607.09803#Thmassumption3), which ensures the leading eigenvalue is simple\), each eigenvalueλ~i\\widetilde\{\\lambda\}\_\{i\}ofF~T\\widetilde\{F\}\_\{T\}satisfies

\|λ~i−λi​\(FT\)\|≤κV​‖E‖2,\|\\widetilde\{\\lambda\}\_\{i\}\-\\lambda\_\{i\}\(F\_\{T\}\)\|\\;\\leq\\;\\kappa\_\{V\}\\\|E\\\|\_\{2\},\(9\)whereκV\\kappa\_\{V\}is the condition number of the eigenvector matrixVVofFTF\_\{T\}\. Under Assumption[3](https://arxiv.org/html/2607.09803#Thmassumption3),κV=O​\(1\)\\kappa\_\{V\}=O\(1\)near the leading eigenvalue\. The perturbationEEhas norm‖E‖2≤‖Δm‖2​‖Bτ‖2≤‖Δm‖2⋅ρ​\(FT\)\\\|E\\\|\_\{2\}\\leq\\\|\\Delta\_\{m\}\\\|\_\{2\}\\\|B\_\{\\tau\}\\\|\_\{2\}\\leq\\\|\\Delta\_\{m\}\\\|\_\{2\}\\cdot\\rho\(F\_\{T\}\)\(using Lemma[2](https://arxiv.org/html/2607.09803#Thmtheorem2)on the tail product\)\.

By Definition[3](https://arxiv.org/html/2607.09803#Thmdefinition3),κ=‖Δm‖2/\(‖hτ‖2\+‖Δm‖2\)\\kappa=\\\|\\Delta\_\{m\}\\\|\_\{2\}/\(\\\|h\_\{\\tau\}\\\|\_\{2\}\+\\\|\\Delta\_\{m\}\\\|\_\{2\}\), so‖Δm‖2=κ​‖hτ‖2/\(1−κ\)\\\|\\Delta\_\{m\}\\\|\_\{2\}=\\kappa\\\|h\_\{\\tau\}\\\|\_\{2\}/\(1\-\\kappa\)\. Substituting and normalizing‖hτ‖2=1\\\|h\_\{\\tau\}\\\|\_\{2\}=1\(without loss of generality, sinceρ\\rhois scale\-invariant\):

\|λ~i−λi\|≤κ​ρ​\(FT\)\+O​\(κ2\),\|\\widetilde\{\\lambda\}\_\{i\}\-\\lambda\_\{i\}\|\\;\\leq\\;\\kappa\\,\\rho\(F\_\{T\}\)\+O\(\\kappa^\{2\}\),\(10\)where theO​\(κ2\)O\(\\kappa^\{2\}\)term comes from the higher\-order expansion of‖Δm‖2/\(1−κ\)\\\|\\Delta\_\{m\}\\\|\_\{2\}/\(1\-\\kappa\)\. Taking the maximum overiigivesρ​\(F~T\)≤ρ​\(FT\)\+‖Δm‖2​κ​ρ​\(FT\)\+O​\(‖Δm‖22\)\\rho\(\\widetilde\{F\}\_\{T\}\)\\leq\\rho\(F\_\{T\}\)\+\\\|\\Delta\_\{m\}\\\|\_\{2\}\\kappa\\rho\(F\_\{T\}\)\+O\(\\\|\\Delta\_\{m\}\\\|\_\{2\}^\{2\}\), which is the claimed bound \([3](https://arxiv.org/html/2607.09803#S3.E3)\)\. ∎

### A\.4Proof of Lemma[4](https://arxiv.org/html/2607.09803#Thmtheorem4)\(Hutchinson Spectral Estimator\)

We estimateρ​\(FT\)\\rho\(F\_\{T\}\)via power iteration onFTF\_\{T\}with Hutchinson\-type Jacobian–vector products\. The standard power iteration computesvk\+1=FT​vk/‖FT​vk‖v\_\{k\+1\}=F\_\{T\}v\_\{k\}/\\\|F\_\{T\}v\_\{k\}\\\|and approximatesρ​\(FT\)≈‖FT​vk‖/‖vk‖\\rho\(F\_\{T\}\)\\approx\\\|F\_\{T\}v\_\{k\}\\\|/\\\|v\_\{k\}\\\|for largekk\.

The matrix–vector productFT​v=JT​JT−1​⋯​J1​vF\_\{T\}v=J\_\{T\}J\_\{T\-1\}\\cdots J\_\{1\}vis computed via reverse\-mode Jacobian–vector products \(JVPs\): eachJt⊤​wJ\_\{t\}^\{\\top\}wrequires one backward pass through blocktt, costingO​\(d\)O\(d\)time and memory for the residual\-stream dimensiondd\. A single power\-iteration step costsO​\(T​d\)O\(Td\);KKsteps costO​\(K​T​d\)O\(KTd\)\.

Bias\.Power iteration converges to the leading eigenvalue from any initial vector not orthogonal to the leading eigenvector\. AfterKKsteps, the relative bias isO​\(\(λ2/λ1\)K\)O\(\(\\lambda\_\{2\}/\\lambda\_\{1\}\)^\{K\}\)whereλ2\\lambda\_\{2\}is the second\-largest eigenvalue magnitude\. Under Assumption[3](https://arxiv.org/html/2607.09803#Thmassumption3)\(simple leading eigenvalue\),λ2/λ1<1\\lambda\_\{2\}/\\lambda\_\{1\}<1, so the bias decays geometrically; forK=O​\(log⁡\(1/ϵ\)\)K=O\(\\log\(1/\\epsilon\)\)steps the bias isO​\(ϵ\)O\(\\epsilon\)\. Averaging overKKindependent restarts with random initialv0∼𝒩​\(0,Id/d\)v\_\{0\}\\sim\\mathcal\{N\}\(0,I\_\{d\}/d\)\(Hutchinson\) reduces variance toO​\(1/K\)O\(1/K\)while keeping biasO​\(1/K\)O\(1/K\)via the geometric\-decay bound\.

Variance\.Each Hutchinson sampleρ^\(k\)=‖FT​v\(k\)‖/‖v\(k\)‖\\widehat\{\\rho\}^\{\(k\)\}=\\\|F\_\{T\}v^\{\(k\)\}\\\|/\\\|v^\{\(k\)\}\\\|is an independent estimator with bounded varianceVar​\[ρ^\(k\)\]≤ρ​\(FT\)2≤L2​T\\mathrm\{Var\}\[\\widehat\{\\rho\}^\{\(k\)\}\]\\leq\\rho\(F\_\{T\}\)^\{2\}\\leq L^\{2T\}\(using Lemma[2](https://arxiv.org/html/2607.09803#Thmtheorem2)\)\. AveragingKKsamples givesVar​\[ρ^\]=O​\(ρ​\(FT\)2/K\)=O​\(1/K\)\\mathrm\{Var\}\[\\widehat\{\\rho\}\]=O\(\\rho\(F\_\{T\}\)^\{2\}/K\)=O\(1/K\)after normalization\. The expectation is𝔼​\[ρ^\]=ρ​\(FT\)\+O​\(1/K\)\\mathbb\{E\}\[\\widehat\{\\rho\}\]=\\rho\(F\_\{T\}\)\+O\(1/K\)\(theO​\(1/K\)O\(1/K\)bias from finite power\-iteration depth\)\.

Time complexity\.KKHutchinson samples, each requiring oneO​\(T​d\)O\(Td\)JVP chain, gives total timeO​\(K​T​d\)O\(KTd\)and memoryO​\(d\)O\(d\)\(independent ofTTvia reverse\-mode\)\. ∎