Repair the Amplifier, Not the Symptom: Stable World-Model Correction for Agent Rollouts

# Repair the Amplifier, Not the Symptom: Stable World-Model Correction for Agent Rollouts
Source: [https://arxiv.org/html/2607.01767](https://arxiv.org/html/2607.01767)
Xinyuan Song1Zekun Cai2,3 1Emory University, Atlanta, GA, USA2The University of Tokyo, Tokyo, Japan 3LocationMind, Tokyo, Japan xinyuan\.song@emory\.edu, caizekun@csis\.u\-tokyo\.ac\.jp

###### Abstract

As agent planning moves from short tool chains toward persistent workflows with thousands or tens of thousands of steps, failures will occur inside large planning graphs rather than in isolated predictions\. Replanning the entire graph after every mistake is neither computationally realistic nor desirable: full\-graph replay consumes large context budgets, exposes the LLM to many irrelevant symptoms, and can degrade long\-context retrieval\. This paper studies the missing component in such systems: a world\-model corrector that repairs the failed planning graph in place\. We compare two families of correctors\. The first is the common engineering approach: scan nodes and edges, choose a suspicious local region, and ask an LLM to repair it\. We implement strong engineering LLM correctors and find that they can help, especially when given very large contexts\. The second family is our approach,WM\-SAR\(World\-ModelSubgraphAmplificationRepair\): instead of scanning for visible symptoms, it works backward from subgraph amplification, identifies the nodes and edges that keep re\-amplifying error, and sends only that causal subgraph to the LLM\. Across graph simulations and LLM repair experiments,WM\-SARsubstantially outperforms engineering correctors under realistic token budgets, achieves near\-whole\-graph stabilization with a compact region, and gives the LLM a cleaner repair target\. Our code is available at:[https://github\.com/Hik289/world\-model\-corrector\.git](https://github.com/Hik289/world-model-corrector.git)

Repair the Amplifier, Not the Symptom: Stable World\-Model Correction for Agent Rollouts

Xinyuan Song1Zekun Cai2,31Emory University, Atlanta, GA, USA2The University of Tokyo, Tokyo, Japan3LocationMind, Tokyo, Japanxinyuan\.song@emory\.edu, caizekun@csis\.u\-tokyo\.ac\.jp

## 1Introduction

#### World models are becoming planning infrastructure\.

World models were introduced as learned simulators that let agents imagine future states before acting\(Ha and Schmidhuber,[2018](https://arxiv.org/html/2607.01767#bib.bib1); Sutton,[1991](https://arxiv.org/html/2607.01767#bib.bib3)\)\. Modern variants support latent rollouts, search, model\-based control, and relational state prediction\(Chuaet al\.,[2018](https://arxiv.org/html/2607.01767#bib.bib36); Hafneret al\.,[2019](https://arxiv.org/html/2607.01767#bib.bib25); Schrittwieseret al\.,[2020](https://arxiv.org/html/2607.01767#bib.bib4); Hafneret al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib2); Zhanget al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib33); Fenget al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib34)\)\. In parallel, LLM agents are moving from single\-turn prediction toward tool use, web navigation, software engineering, and persistent exploration\(Yaoet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib5); Zhouet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib46); Jimenezet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib45); Yanget al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib47); Liuet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib16); Shenet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib19); Wanget al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib40)\)\. These two trends point to the same future system: an agent maintains a working world model, plans through it, updates it with tool feedback, and reuses it over many decisions\.

#### Long planning naturally becomes a graph\.

In a short task, a rollout can be treated as a list of steps\. In a long task, that view breaks down\. A planner spawns tool calls; validators gate later actions; memory writes are reused by distant descendants; exceptions trigger recovery branches; final answers depend on several earlier sub\-trajectories\. The resulting object is a planning graph, not a flat transcript\. As agents are deployed for longer workflows, this graph may contain thousands or tens of thousands of nodes\. Replanning the whole graph after each error is therefore not a viable maintenance strategy\. It consumes the context budget, forces the LLM to retrieve the relevant cause from a sea of irrelevant nodes, and is especially risky because long\-context models can fail to use information placed away from the prompt boundaries\(Liuet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib39)\)\.

#### This creates the need for world\-model correctors\.

A corrector is a maintenance robot for a planning graph\. When a failure appears, the corrector should decide which part of the graph to revise, pass that region to an LLM or another repair module, and keep the rest of the world model intact\. This is different from ordinary replanning: the corrector must identify a small causal region inside a much larger graph\. If it chooses only the visible symptom, the next rollout can recreate the failure; if it chooses too much of the graph, the method becomes a costly full replay\.

#### A strong default is an engineering LLM corrector\.

The most direct implementation is to scan the planning graph with engineering rules\. One can rank nodes by error, take the top\-KKsuspicious nodes, inspect high\-error edges, choose a temporal window around the failure, expand akk\-hop neighbourhood, run a cascade scan, or serialize a large region for an LLM root\-cause prompt\. This family is attractive because it is simple, model\-agnostic, and compatible with feedback, self\-refinement, and replanning methods\(Yaoet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib18); Madaanet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib24); Shinnet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib7); Zhouet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib8); Cobbeet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib37); Lightmanet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib38)\)\. We implement this engineering family carefully, including LLM\-based versions\. The result is a meaningful baseline: engineering correctors can work, and full\-graph LLM repair can be strong on small graphs\. Their limitation is the same limitation that motivates the problem\. They spend budget by scanning visible evidence, so they may miss the small subgraph that re\-amplifies error, and their best variant relies on contexts that become expensive and noisy as the planning graph grows\.

#### Our corrector works backward from amplification\.

We proposeWM\-SAR\(World\-ModelSubgraphAmplificationRepair\)\. Instead of scanning for the node that looks most wrong,WM\-SARasks which connected subgraph keeps the residual world model unstable\. We represent a failed rollout as a typed graph and define a residual node\-edge error operator\. Its spectral radius determines whether remaining errors decay or amplify in the next rollout; proofs are in the appendix\.WM\-SARuses this quantity to work backward from amplification: it scores local amplification, estimates node\-edge coupling, grows a connected region by marginal spectral relief, and then sends only that region to an LLM\. Figure[1](https://arxiv.org/html/2607.01767#S1.F1)shows why engineering repair and amplification repair lead to different outcomes\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/intuition.png)Figure 1:World\-model correction should target the amplifier rather than the most visible symptom\. Greedy repair fixes the highest\-error node but leaves the causal path active\. A fixed local neighbourhood can still miss the node\-edge coupling that recreates the failure\.WM\-SARselects the compact connected path whose removal lowers residual amplification, which flattens the empirical NodeMSE curve over horizonHH\.
#### Contributions\.

1. ∙\\bulletProblem formulation: we formulate world\-model correction as maintaining a failed planning graph under a limited repair budget, rather than re\-planning the full graph\.
2. ∙\\bulletEngineering correctors: we implement a strong family of LLM and non\-LLM engineering correctors based on node scans, windows, neighbourhoods, cascades, and full\-graph prompting\.
3. ∙\\bulletAmplification corrector: we introduceWM\-SAR, which infers the amplifying subgraph using GEAF, node\-edge coupling, and marginal residual\-spectral relief before invoking an LLM\.
4. ∙\\bulletTheory and evidence: we prove that residual spectral radius controls rollout error and planning regret, and show empirically thatWM\-SARoutperforms engineering correctors while using substantially smaller repair contexts\.

## 2Related Work

#### World models for planning\.

World\-model planning traces back to architectures that interleave learning, planning, and acting\(Sutton,[1991](https://arxiv.org/html/2607.01767#bib.bib3)\); modern learned world models simulate future states in latent space\(Ha and Schmidhuber,[2018](https://arxiv.org/html/2607.01767#bib.bib1); Hafneret al\.,[2019](https://arxiv.org/html/2607.01767#bib.bib25),[2023](https://arxiv.org/html/2607.01767#bib.bib2)\)and can be coupled with search\(Schrittwieseret al\.,[2020](https://arxiv.org/html/2607.01767#bib.bib4)\)\. Probabilistic dynamics models propagate uncertainty through imagined rollouts\(Chuaet al\.,[2018](https://arxiv.org/html/2607.01767#bib.bib36)\), while model\-based policy optimization studies when short learned rollouts remain trustworthy under accumulated model bias\(Janneret al\.,[2019](https://arxiv.org/html/2607.01767#bib.bib41)\)\.Asadiet al\.\([2018](https://arxiv.org/html/2607.01767#bib.bib35)\)establish Lipschitz continuity as a sufficient condition for bounded rollout error—a scalar criterion that our graph operator view lifts to structured agent plans\. Graph\-structured world models\(Zhanget al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib33); Fenget al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib34); Battagliaet al\.,[2018](https://arxiv.org/html/2607.01767#bib.bib26); Hafneret al\.,[2019](https://arxiv.org/html/2607.01767#bib.bib25)\)extend this to relational state spaces, where message passing predicts how entities and dependencies evolve together\.Anokhinet al\.\([2025](https://arxiv.org/html/2607.01767#bib.bib42)\)build episodic graph memories for agent planning\. Recent LLM\-agent systems also use world knowledge or environment dynamics to guide planning\(Qiaoet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib44); Chaeet al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib43)\)\. Our work addresses the previously unstudied problem of*how to repair*a failed GWM rollout: which subgraph to target and why\.

#### LLM agents and failure repair\.

ReAct\(Yaoet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib5)\)interleaves reasoning and acting; Toolformer\(Schicket al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib6)\), Gorilla\(Patilet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib10)\), RestGPT\(Songet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib11)\), and ToolLLM\(Qinet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib9)\)teach LLMs to invoke tools or APIs\. Self\-Refine\(Madaanet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib24)\)applies iterative self\-feedback to improve generation quality\. Reflexion\(Shinnet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib7)\)stores verbal feedback in memory, while LATS\(Zhouet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib8)\)combines language\-agent planning with tree search\. Verifier\-based and process\-supervised approaches provide more granular feedback for multi\-step reasoning\(Cobbeet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib37); Lightmanet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib38)\)\. Retroformer\(Yaoet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib18)\)uses policy gradient retrospection to correct failed ALFWorld\(Shridharet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib17)\)trajectories\. AgentBench\(Liuet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib16)\)and AgentBoard\(Maet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib20)\)benchmark LLM agents across diverse tasks; TaskBench\(Shenet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib19)\)evaluates tool\-chaining task automation\. SWE\-bench\(Jimenezet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib45)\)and SWE\-agent\(Yanget al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib47)\)benchmark code repository repair; WebArena\(Zhouet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib46)\)benchmarks realistic web navigation agents\. Section[7\.7](https://arxiv.org/html/2607.01767#S7.SS7)simulates benchmark\-inspired topologies to test whetherWM\-SARdepends on the synthetic calling\-tree shape\.Zhanget al\.\([2025](https://arxiv.org/html/2607.01767#bib.bib21)\)attribute task failures to specific agents in multi\-agent LLM systems—a complementary problem to ours \(attribution vs\. repair\)\.Luet al\.\([2025](https://arxiv.org/html/2607.01767#bib.bib22)\)visualise LLM agent behaviour for interactive analysis\. All these methods rely on linear trace scanning or full\-trajectory replay; none uses graph spectral analysis to select a*minimal causal subgraph*for targeted LLM repair\.

#### Graph\-theoretic error analysis\.

Spectral graph theory\(Chung,[1997](https://arxiv.org/html/2607.01767#bib.bib29)\)links graph eigenvalues to diffusion and propagation phenomena\.Toppinget al\.\([2022](https://arxiv.org/html/2607.01767#bib.bib28)\)study over\-squashing in GNNs via Ricci curvature, showing that bottleneck edges cause information loss—ourκv\\kappa\_\{v\}coupling factor identifies the analogous*amplification bottlenecks*\.Gilmeret al\.\([2017](https://arxiv.org/html/2607.01767#bib.bib27)\)establish the message\-passing neural network framework that underlies most GWM architectures\. Graph anomaly detection\(Niuet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib30)\)and fault localisation\(Louet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib31)\)similarly seek to identify critical subgraphs, but in static graphs without a planning\-horizon regret objective\. Our GEAF \(Graph Error Amplification Field\) is the first per\-node quantity derived from rollout error theory that enables*prospective*identification of amplification sources before invoking an LLM\.

#### Subgraph selection and PageRank\.

PageRank\(Pageet al\.,[1999](https://arxiv.org/html/2607.01767#bib.bib32)\)scores nodes by global influence; TopK selection andkk\-hop expansion are standard heuristics in graph analysis\.Gleave and Irving \([2022](https://arxiv.org/html/2607.01767#bib.bib23)\)catalogue failure modes of LLM\-based reward models, noting that local corrections can leave global pathology intact—consistent with our finding that node\-level repair does not reduceρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\. WM\-SAR instead grows a*connected*region greedily under aΔ​ρ\\Delta\\rho\-relief criterion, directly targeting the spectral quantity that controls the planning\-regret bound\.

## 3Rollout Error Theory

We now state why a corrector should optimize amplification rather than visible error\. In a large planning graph, the immediate symptom can be far downstream from the node or edge that will recreate the failure in the next rollout\. A repair is therefore useful only if it lowers the operator that propagates both node\-state and edge\-state errors\. Proofs are deferred to the appendix\.

###### Definition 1\(Node\-state rollout error\)\.

For a graph world model with true node statesXkX\_\{k\}and predicted node statesX^k\\hat\{X\}\_\{k\}at rollout stepkk, defineekX=‖X^k−Xk‖Fe^\{X\}\_\{k\}=\\\|\\hat\{X\}\_\{k\}\-X\_\{k\}\\\|\_\{F\}\. LetϵX\\epsilon\_\{X\}denote the one\-step model approximation error\.

###### Assumption 1\(Fixed\-edge Lipschitz rollout\)\.

For a fixed graph with adjacency matrixAA, the message\-passing update is globally Lipschitz in the node state with constant

LX=Lσ​ρ​\(A\)​∏ℓ‖Wℓ‖2,L\_\{X\}=L\_\{\\sigma\}\\rho\(A\)\\prod\_\{\\ell\}\\\|W\_\{\\ell\}\\\|\_\{2\},\(1\)whereLσL\_\{\\sigma\}is the activation Lipschitz constant andWℓW\_\{\\ell\}are the message\-passing layer matrices\.

###### Theorem 1\(Fixed\-edge node\-error growth\)\.

Under Assumption[1](https://arxiv.org/html/2607.01767#Thmassumption1), ifek\+1X≤LX​ekX\+ϵXe^\{X\}\_\{k\+1\}\\leq L\_\{X\}e^\{X\}\_\{k\}\+\\epsilon\_\{X\}, then for everyk≥1k\\geq 1,

ekX≤LXk​e0X\+ϵX​LXk−1LX−1\.e^\{X\}\_\{k\}\\leq L\_\{X\}^\{k\}e^\{X\}\_\{0\}\+\\epsilon\_\{X\}\\frac\{L\_\{X\}^\{k\}\-1\}\{L\_\{X\}\-1\}\.\(2\)ForLX=1L\_\{X\}=1, the second term isk​ϵXk\\epsilon\_\{X\}\.

The theorem separates two sources of risk: one\-step approximation error and structural amplification\. Engineering correctors mostly observe the first term throughe​\(v\)e\(v\)\.WM\-SARalso estimates the second term withGEAF^v=e​\(v\)​ρ​\(Av\)​wH\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}=e\(v\)\\rho\(A\_\{v\}\)w^\{H\}, whereρ​\(Av\)​wH\\rho\(A\_\{v\}\)w^\{H\}is the local capacity to amplify the observed error\.

###### Assumption 2\(Coupled node\-edge channel\)\.

Letzk=\(ekX,ekA\)⊤z\_\{k\}=\(e^\{X\}\_\{k\},e^\{A\}\_\{k\}\)^\{\\top\}collect node\-state and edge\-state errors, and letε=\(ϵX,ϵA\)⊤\\varepsilon=\(\\epsilon\_\{X\},\\epsilon\_\{A\}\)^\{\\top\}\. Their one\-step linearized upper envelope is

zk\+1\\displaystyle z\_\{k\+1\}⪯𝐁​zk\+ε,\\displaystyle\\preceq\\mathbf\{B\}z\_\{k\}\+\\varepsilon,\(3\)𝐁=\(LXLAMXMA\),\\displaystyle\\mathbf\{B\}=\\begin\{pmatrix\}L\_\{X\}&L\_\{A\}\\\\ M\_\{X\}&M\_\{A\}\\end\{pmatrix\},where all four entries are nonnegative\. Here⪯\\preceqdenotes elementwise inequality\.

###### Theorem 2\(Coupled node\-edge amplification\)\.

Under Assumption[2](https://arxiv.org/html/2607.01767#Thmassumption2), the spectral radius of the coupled operator is

ΔB\\displaystyle\\Delta\_\{B\}=\(LX−MA\)2\+4​LA​MX,\\displaystyle=\\sqrt\{\(L\_\{X\}\-M\_\{A\}\)^\{2\}\+4L\_\{A\}M\_\{X\}\},ρ​\(𝐁\)\\displaystyle\\rho\(\\mathbf\{B\}\)=LX\+MA\+ΔB2\.\\displaystyle=\\frac\{L\_\{X\}\+M\_\{A\}\+\\Delta\_\{B\}\}\{2\}\.\(4\)IfLA​MX\>0L\_\{A\}M\_\{X\}\>0, thenρ​\(𝐁\)\>max⁡\(LX,MA\)\\rho\(\\mathbf\{B\}\)\>\\max\(L\_\{X\},M\_\{A\}\)\.

The local coupling scoreκv=LA​\(v\)​MX​\(v\)\\kappa\_\{v\}=L\_\{A\}\(v\)M\_\{X\}\(v\)estimates how strongly nodevvparticipates in this super\-additive node\-edge channel\. A node can therefore be dangerous even when its observed error is not the largest: it may sit on a node\-edge channel that increases the Perron root of the residual operator\.

###### Assumption 3\(Value sensitivity\)\.

For any policyπ\\pi, the reward isLRL\_\{R\}\-Lipschitz in rollout state, the policy selection map has sensitivityκ\\kappa, and the reward approximation error per step is bounded byϵR\\epsilon\_\{R\}\. The coupled model error satisfies‖zk‖≤ϵ​ρ​\(𝐁\)k\\\|z\_\{k\}\\\|\\leq\\epsilon\\,\\rho\(\\mathbf\{B\}\)^\{k\}forzk=\(ekX,ekA\)z\_\{k\}=\(e^\{X\}\_\{k\},e^\{A\}\_\{k\}\)\.

###### Theorem 3\(Planning\-regret bound\)\.

Letπ∗\\pi^\{\*\}be the optimal policy under the true rollout dynamics andπ^\\hat\{\\pi\}the policy selected under the learned GWM\. Under Assumption[3](https://arxiv.org/html/2607.01767#Thmassumption3),

J​\(π∗\)−J​\(π^\)≤2​LR​κ​ϵ​ΦH​\(γ,ρ​\(𝐁\)\)\+2​ϵR​H,J\(\\pi^\{\*\}\)\-J\(\\hat\{\\pi\}\)\\leq 2L\_\{R\}\\kappa\\epsilon\\,\\Phi\_\{H\}\(\\gamma,\\rho\(\\mathbf\{B\}\)\)\+2\\epsilon\_\{R\}H,\(5\)where

ΦH​\(γ,ρ\)=∑t=0H−1\(γ​ρ\)t\.\\Phi\_\{H\}\(\\gamma,\\rho\)=\\sum\_\{t=0\}^\{H\-1\}\(\\gamma\\rho\)^\{t\}\.\(6\)

The bound makes the repair criterion explicit\. For a repaired regionRR, the future rollout is controlled by the residual spectral radiusρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\. A scanning corrector can repair high\-error nodes while leaving this quantity large\.WM\-SARinstead chooses the connected subgraph whose correction lowers the residual operator, which is why it can outperform engineering correctors even when they see comparable local evidence\.

## 4Problem Formulation

The repair problem is observable after a failure: we have the failed trace, typed dependencies, node\-level error estimates, and a budget for re\-running or revising part of the trace\. The future deployment constraint is that this budget is small relative to the full planning graph\. A corrector cannot assume that it can serialize thousands of nodes into an LLM prompt after every failure\. The target is therefore not the largest error node, but the smallest connected region whose correction makes the remaining world model stable\.

###### Definition 2\(Failure graph\)\.

A failed rolloutτ\\tauis represented asGf=\(V,E,𝐗,t⋆\)G\_\{f\}=\(V,E,\\mathbf\{X\},t^\{\\star\}\), where nodes are agent calls and edges are causal dependencies\. Each nodev∈Vv\\in Vcarries an observed error magnitudee​\(v\)e\(v\), uncertaintyu​\(v\)u\(v\), repair costc​\(v\)c\(v\), and a node type \(planner, executor, validator, checker, aggregator, reporter, logger, error\_handler, or final\_answer\)\. Edge types encode calls, validation, reporting, error routing, triggering, and logging\. The sinkt⋆t^\{\\star\}is the final\-answer node, andℛ∗\\mathcal\{R\}^\{\*\}denotes the ground\-truth corrupted region used only for evaluation\.

###### Definition 3\(Repair objective\)\.

Given budgetKmaxK\_\{\\max\}, the spectral repair region is

R∗=arg⁡minR⊆V:Gf​\[R\]​connected,\|R\|≤Kmax⁡ρ​\(𝐁Gf∖R\)\.R^\{\*\}=\\arg\\min\_\{R\\subseteq V:\\;G\_\{f\}\[R\]\\text\{ connected\},\\,\|R\|\\leq K\_\{\\max\}\}\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\.\(7\)

Exactly minimizingρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)over connected subsets with bounded size is NP\-hard in general because it contains minimum spectral cut as a special case\.WM\-SARis the greedy approximation used in our experiments\.

Engineering baselines instead optimize variants of∑v∈Re​\(v\)\\sum\_\{v\\in R\}e\(v\)\. This is a useful attribution signal, but it is not a stability objective: removing a large downstream symptom may leave the residual Perron root unchanged\.

## 5WM\-SAR Algorithm

![Refer to caption](https://arxiv.org/html/2607.01767v1/pipeline.png)Figure 2:WM\-SARworkflow\. Region selection first scores every node by local amplification and node\-edge coupling, seeds compact candidate regions, expands them by marginal residual\-spectral relief, and prunes redundant nodes\. Only the final connected region is serialized for LLM root\-cause repair, keeping the language\-model context focused on the amplifying subgraph\.We study two kinds of world\-model correctors\. An*engineering corrector*scans the planning graph directly: it chooses the largest\-error node, a TopK set, a temporal window, a local neighbourhood, or a large serialized trace for an LLM\. This is the default design pattern because it is easy to implement and often works when the graph is small\.WM\-SARis an*amplification\-based corrector*\. It first estimates which subgraph keeps error alive in the residual world model, then asks the LLM to repair only that subgraph\. Figure[2](https://arxiv.org/html/2607.01767#S5.F2)summarizes this division\. The graph phase uses the spectral\-graph principle that eigenvalues govern propagation\(Chung,[1997](https://arxiv.org/html/2607.01767#bib.bib29)\); the LLM phase uses structured tool\-context prompting\(Yaoet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib5); Schicket al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib6); Patilet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib10); Songet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib11); Qinet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib9)\)\. The goal is not to avoid LLM reasoning, but to give the LLM the right graph slice before its context budget is spent\.

### 5\.1Phase 1: Finding the Repair Region

#### Step 1 \- Per\-node GEAF score\.

Engineering correctors begin with observed error\.WM\-SARbegins with observed error multiplied by the local ability to amplify it\. The fixed\-edge theorem shows that rollout error grows with the local Lipschitz\-amplification scale, so we useGEAF^v\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}as a graph\-only proxy:

GEAF^v=e​\(v\)​ρ​\(Av\)​wH,\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}=e\(v\)\\,\\rho\(A\_\{v\}\)\\,w^\{H\},\(8\)wheree​\(v\)e\(v\)is the observed error atvv;AvA\_\{v\}is thenv×nvn\_\{v\}\\\!\\times\\\!n\_\{v\}adjacency matrix ofvv’sHH\-hop subgraphGvG\_\{v\};ρ​\(Av\)\\rho\(A\_\{v\}\)is its spectral radius \(how muchvv’s neighbourhood amplifies signals\); andw∈\(0,1\)w\\in\(0,1\)is a horizon discount factor\. All default hyperparameters are listed in Appendix[C](https://arxiv.org/html/2607.01767#A3)\. We use this spectral\-radius proxy rather than a walk\-count alternative \(∑k=1H\(Ak​𝟏\)v\\sum\_\{k=1\}^\{H\}\(A^\{k\}\\mathbf\{1\}\)\_\{v\}\); both are monotonic in structural amplification\. This is analogous in spirit to influence scores such as PageRank, but it is local to a repair candidate and tied to rollout stability\(Pageet al\.,[1999](https://arxiv.org/html/2607.01767#bib.bib32); Chung,[1997](https://arxiv.org/html/2607.01767#bib.bib29)\)\.

Intuitively,ρ​\(Av\)\\rho\(A\_\{v\}\)measures*structural*amplification—how many independent paths carry error out ofvv—whilee​\(v\)e\(v\)measures the*observed*error magnitude\. A node with highe​\(v\)e\(v\)but lowρ​\(Av\)\\rho\(A\_\{v\}\)is a dead\-end symptom node; a node with lowe​\(v\)e\(v\)but highρ​\(Av\)\\rho\(A\_\{v\}\)is a latent amplifier\.WM\-SARtargets nodes where both are high\.

#### Step 2 \- Coupling factor\.

From the coupled node\-edge theorem, thecoupling factorκv=LA​\(v\)⋅MX​\(v\)\\kappa\_\{v\}=L\_\{A\}\(v\)\\cdot M\_\{X\}\(v\)captures the super\-additive interaction betweenvv’s input and output channels:

LA​\(v\)\\displaystyle L\_\{A\}\(v\)=w​βA​d¯in​\(v\)⋅e¯​\(v\),\\displaystyle=w\\,\\beta\_\{A\}\\,\\bar\{d\}\_\{\\rm in\}\(v\)\\cdot\\bar\{e\}\(v\),MX​\(v\)\\displaystyle M\_\{X\}\(v\)=w​βX​d¯out​\(v\)⋅e¯​\(v\),\\displaystyle=w\\,\\beta\_\{X\}\\,\\bar\{d\}\_\{\\rm out\}\(v\)\\cdot\\bar\{e\}\(v\),\(9\)whered¯in​\(v\)\\bar\{d\}\_\{\\rm in\}\(v\)/d¯out​\(v\)\\bar\{d\}\_\{\\rm out\}\(v\)are the average in/out edge\-type diversity ofvv,e¯​\(v\)\\bar\{e\}\(v\)is the mean error invv’s neighbourhood, andβA,βX\\beta\_\{A\},\\beta\_\{X\}are calibration constants described in Appendix[C](https://arxiv.org/html/2607.01767#A3)\. Nodes with highκv\\kappa\_\{v\}lie on*cross\-coupling paths*: repairing them drives downρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)through both state and edge channels\.

#### Step 3 \- Seed selection\.

We rank every node by a combined seed score:

scoreseed​\(v\)\\displaystyle\\mathrm\{score\}\_\{\\rm seed\}\(v\)=e​\(v\)​GEAF^v​\(1\+κv\)\.\\displaystyle=e\(v\)\\,\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}\\,\(1\+\\kappa\_\{v\}\)\.\(10\)and select the top\-k0k\_\{0\}nodes as candidate seeds\. Each seedssinitiates an independent growing process\.

#### Step 4 \-ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-relief growing\.

This is whereWM\-SARdeparts most sharply from engineering scanning\. Starting fromR=\{s\}R=\\\{s\\\}, it greedily expandsRRby adding the boundary nodeu∗u^\{\*\}that maximally reduces the residual spectral coupling:

u∗\\displaystyle u^\{\*\}=arg⁡maxu∈∂R⁡gR​\(u\),\\displaystyle=\\arg\\max\_\{u\\in\\partial R\}g\_\{R\}\(u\),\(11\)gR​\(u\)\\displaystyle g\_\{R\}\(u\)=λ1​e​\(u\)​\(1\+κu\)\+λ2​Δ​ρrel​\(u\)−λ3​c​\(u\)\.\\displaystyle=\\lambda\_\{1\}e\(u\)\(1\+\\kappa\_\{u\}\)\+\\lambda\_\{2\}\\Delta\\rho\_\{\\rm rel\}\(u\)\-\\lambda\_\{3\}c\(u\)\.where∂R\\partial Ris the frontier of nodes adjacent toRRinGfG\_\{f\};

Δ​ρrel​\(u\)\\displaystyle\\Delta\\rho\_\{\\rm rel\}\(u\)=ρ​\(𝐁Gf∖R\)−ρ​\(𝐁Gf∖\(R∪\{u\}\)\),\\displaystyle=\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\-\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus\(R\\cup\\\{u\\\}\)\}\),\(12\)c​\(u\)\\displaystyle c\(u\)=1\+\|N​\(u\)∩R\|\|R\|\.\\displaystyle=1\+\\frac\{\|N\(u\)\\cap R\|\}\{\|R\|\}\.\(13\)HereΔ​ρrel​\(u\)\\Delta\\rho\_\{\\rm rel\}\(u\)is the marginalρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction gained by addinguu, andc​\(u\)c\(u\)penalises redundant already\-covered nodes\.λ1,λ2,λ3\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}balance error coverage, spectral relief, and cost penalty \(values in Appendix[C](https://arxiv.org/html/2607.01767#A3)\)\. Growing stops when no node provides positive gain or the budgetKmaxK\_\{\\max\}is reached\.

Why this beats engineering baselines\.TopK selects high\-error nodes independently, so the chosen nodes can be disconnected symptoms\. Window repair assumes temporal adjacency, so it can miss a root cause that is far upstream\. LocalRepair expands by graph distance, so it may spend budget on harmless neighbours while missing a high\-gain coupling edge\. Full\-graph LLM repair can recover the answer on small graphs, but its cost grows with the whole planning graph and long contexts are not reliably used by current LLMs\(Liuet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib39)\)\.WM\-SARinstead usesΔ​ρrel\\Delta\\rho\_\{\\rm rel\}to ask how much each candidate node or edge lowers the residual amplifier, the same quantity that appears in the planning\-regret bound\. This makes the corrector closer to fault\-localization and critical\-subgraph methods\(Louet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib31); Niuet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib30)\)than to generic trace scanning\.

#### Step 5 \- Pruning and selection\.

After growing, we prune: remove anyv∈Rv\\\!\\in\\\!Rsuch thatρ​\(𝐁Gf∖\(R∖\{v\}\)\)≤ρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus\(R\\setminus\\\{v\\\}\)\}\)\\leq\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\(removingvvdoes not increase residual amplification\)\. Among allk0k\_\{0\}candidate regions, we return the one maximising

Score​\(R\)\\displaystyle\\mathrm\{Score\}\(R\)=\(∑v∈Re​\(v\)​\(1\+κv\)\)​Δ​ρrel​\(R\)1\+\|R\|\.\\displaystyle=\\frac\{\\left\(\\sum\_\{v\\in R\}e\(v\)\(1\+\\kappa\_\{v\}\)\\right\)\\Delta\\rho\_\{\\rm rel\}\(R\)\}\{1\+\|R\|\}\.\(14\)
This score is deliberately residual: it rewards regions that lowerρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\), not regions that merely contain high\-error nodes\. Uniform uncertainty in this dataset carries no ranking signal, so the uncertainty term is inactive at the default hyperparameters; the ablation in Table[2](https://arxiv.org/html/2607.01767#S7.T2)verifies that removing it has no effect\.

The full procedure is summarised in Algorithm[1](https://arxiv.org/html/2607.01767#alg1)\.

Algorithm 1WM\-SARRegion Selection \(greedy approximation to Definition[3](https://arxiv.org/html/2607.01767#Thmdefinition3)\)1:Failure graph

GfG\_\{f\}, budget

KmaxK\_\{\\max\}, parameters

\(H,w,k0,λ1,λ2,λ3\)\(H,w,k\_\{0\},\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}\)
2:Connected repair region

R∗R^\{\*\}
3:Compute

GEAF^v←e​\(v\)⋅ρ​\(Av\)⋅wH\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}\\leftarrow e\(v\)\\cdot\\rho\(A\_\{v\}\)\\cdot w^\{H\}and

κv←LA​\(v\)⋅MX​\(v\)\\kappa\_\{v\}\\leftarrow L\_\{A\}\(v\)\\cdot M\_\{X\}\(v\)for every

v∈Gfv\\in G\_\{f\}
4:

𝒮←top\-​k0\\mathcal\{S\}\\leftarrow\\text\{top\-\}k\_\{0\}nodes by

e​\(v\)⋅GEAF^v⋅\(1\+κv\)e\(v\)\\cdot\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}\\cdot\(1\{\+\}\\kappa\_\{v\}\)
5:foreach seed

s∈𝒮s\\in\\mathcal\{S\}do

6:

R←\{s\}R\\leftarrow\\\{s\\\}
7:while

\|R\|<Kmax\|R\|<K\_\{\\max\}do

8:

u∗←arg⁡maxu∈∂R⁡λ1​e​\(u\)​\(1\+κu\)\+λ2​Δ​ρrel​\(u\)−λ3​c​\(u\)u^\{\*\}\\leftarrow\\arg\\max\_\{u\\in\\partial R\}\\lambda\_\{1\}e\(u\)\(1\{\+\}\\kappa\_\{u\}\)\+\\lambda\_\{2\}\\Delta\\rho\_\{\\rm rel\}\(u\)\-\\lambda\_\{3\}c\(u\)
9:if

Δ​ρrel​\(u∗\)\>0\\Delta\\rho\_\{\\rm rel\}\(u^\{\*\}\)\>0then

10:

R←R∪\{u∗\}R\\leftarrow R\\cup\\\{u^\{\*\}\\\}
11:elsebreak

12:endif

13:endwhile

14:Remove

vvfrom

RRif

ρ​\(𝐁Gf∖\(R∖\{v\}\)\)≤ρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus\(R\\setminus\\\{v\\\}\)\}\)\\leq\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)
15:endfor

16:return

R∗←arg⁡maxR⁡Score​\(R\)R^\{\*\}\\leftarrow\\arg\\max\_\{R\}\\mathrm\{Score\}\(R\)where

Score​\(R\)=∑ve​\(v\)​\(1\+κv\)⋅Δ​ρrel​\(R\)1\+\|R\|\\mathrm\{Score\}\(R\)=\\frac\{\\sum\_\{v\}e\(v\)\(1\+\\kappa\_\{v\}\)\\cdot\\Delta\\rho\_\{\\rm rel\}\(R\)\}\{1\+\|R\|\}

### 5\.2Phase 2: LLM\-Based Repair

OnceWM\-SARselectsR∗R^\{\*\}, the LLM stage is deliberately simple\. We serialize only the selected subgraph, preserving node types, local state features, and incoming/outgoing boundary edges\. The LLM is then asked to identify the root\-cause node\(s\) and produce a structured repair decision\. In our simulated evaluation, the repair operator zeroes the selected node errors and re\-runs the rollout; in a deployed agent, the same decision would trigger a targeted re\-invocation of the corresponding planner, tool call, validator, or memory update\. Prompt and serialization details are in Appendix[E](https://arxiv.org/html/2607.01767#A5)\.

This stage is where token efficiency matters\. A full\-graph LLM corrector can be competitive on small graphs, but it pays for every irrelevant node and asks the model to retrieve the causal path from a long context\.WM\-SARinstead spends the LLM budget on the subgraph most likely to change the next rollout\. Figure[4](https://arxiv.org/html/2607.01767#S7.F4)and Table[4](https://arxiv.org/html/2607.01767#S7.T4)report the resulting token–quality tradeoff\.

## 6Dataset: Agent Calling\-Tree Testbed

We construct a synthetic heterogeneous calling\-tree testbed that mimics the typed call structure of planner\-executor\-validator agents\.

#### Graph structure\.

Each instance is a medium\-size typed calling tree with agent roles such as planner, executor, validator, checker, and final\-answer nodes\. Edges encode calls, validation, reporting, error routing, triggering, and logging\. Each node stores the state features needed to distinguish a root error from a downstream cascade victim; full statistics are in Appendix[D](https://arxiv.org/html/2607.01767#A4)\.

#### Failure injection\.

Failures originate at an upstream planner or executor and then propagate along the typed dependency graph\. This creates the intended ambiguity: the largest observed error is often downstream of the node that introduced the mistake\. The ground\-truth corrupted region is used only for evaluation\.

#### Multi\-step error metrics\.

After zeroing errors inRR\(simulated repair\), remaining errors propagate via the fixed\-edge recursion in Equation[2](https://arxiv.org/html/2607.01767#S3.E2):ek\+1​\(v\)=LX⋅ek​\(v\)\+LA​∑u→vek​\(u\)e\_\{k\+1\}\(v\)=L\_\{X\}\\cdot e\_\{k\}\(v\)\+L\_\{A\}\\\!\\sum\_\{u\\to v\}e\_\{k\}\(u\)\. We report long\-horizon NodeMSE, GrowthSlope, and residual spectral\-radius reduction; these metrics separate immediate error removal from true rollout stabilization\.

## 7Experiments

### 7\.1Baselines: Engineering Correctors

All baselines use the same simulated repair operator: selected node errors are zeroed and the rollout is re\-simulated\. The only variable is the corrector’s choice of repair region\. This isolates the central question: does success come from engineering scan rules, or from selecting the subgraph that actually controls residual amplification?

- •Greedy\-Point\(K=1\): top\-1 by error\.
- •TopK\-Point\(K=3/5\): top\-KKby error \(may be disconnected\)\.
- •Window\-2/4/8\-Point: sliding window ofkksteps; best\-mean\-error window\.
- •LocalRepair\-2/3Hop:kk\-hop neighbourhood of the top\-error node \(the larger neighbourhood is close to whole\-graph repair in this testbed\)\.
- •CascadeRepair: topological scan until error drops below threshold\.
- •Oracle: exact GT regionℛ∗\\mathcal\{R\}^\{\*\}\.
- •LLM engineering correctors: serialize regions selected by the above rules, including full\-graph variants, and ask the same LLM to identify the root cause\.
- •WM\-SAR: amplification\-based corrector using Definition[3](https://arxiv.org/html/2607.01767#Thmdefinition3)and Algorithm[1](https://arxiv.org/html/2607.01767#alg1)\.

### 7\.2Main Results

Table[1](https://arxiv.org/html/2607.01767#S7.T1)gives the main comparison\. The engineering correctors are not weak strawmen: neighbourhood and cascade repair often find plausible corrupted regions, and large LLM contexts improve root\-cause recovery\. The failure is more specific\. They do not directly optimize the residual amplifier, so under a compact repair budget they often leave a graph that will recreate the same error\.

Table 1:Main repair comparison on the agent calling\-tree testbed\. Every method uses the same state\-correction operator, so differences come only from the corrector’s region choice\. Engineering correctors repair visible symptoms or broad local neighbourhoods;WM\-SARrepairs the subgraph that lowers residual amplification\. It reaches near\-whole\-neighbourhood stabilization with a compact region and keeps the post\-repair slope close to flat\. The Oracle repairs the injected corrupted region; it is an attribution upper bound rather than a spectral\-relief optimizer\.Size = mean region size\. Conn = % connected instances\.ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-red =ρ​\(𝐁Gf\)−ρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\}\)\-\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\. Slope = GrowthSlope×100\\times 10^\{0\}\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_rho_reduction.png)Figure 3:Spectral relief per repaired region\. Engineering correctors can reduce error, but many spend budget on nodes that do little to lower the residual amplifier\.WM\-SARreaches the high\-relief regime with a small connected subgraph, whereas fixed\-hop repair obtains similar relief mainly by expanding toward whole\-graph replay\.#### Finding 1: engineering scans repair symptoms but often leave the amplifier\.

Greedy, TopK, and window correctors repair nodes that look bad in the observed trace, and this can reduce short\-horizon error\. Figure[3](https://arxiv.org/html/2607.01767#S7.F3)shows why the improvement does not last: these methods leave most residual spectral mass in place\. The symptom has been patched, but the causal channel that recreates the symptom is still present\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_tradeoff.png)Figure 4:LLM repair tradeoff after graph\-based context selection\. Each point is a corrector whose selected region is serialized for the same LLM root\-cause prompt; point size indicates token cost\. Full\-graph engineering repair can be competitive but is token\-heavy\.WM\-SARsits on the efficient frontier because it preserves the causal bridge while omitting downstream symptoms\.
#### Finding 2: amplification repair turns correction into stabilization\.

WM\-SARselects about eight nodes on average, but those nodes are the ones that control the residual operator\. After they are repaired, the rollout no longer behaves like a delayed failure: the error curve flattens, matching the stable regime predicted by the fixed\-edge theorem\. The horizon curve in Figure[6](https://arxiv.org/html/2607.01767#S7.F6)makes this distinction visible:WM\-SARdoes not merely reduce the first repaired step; it prevents error from re\-growing\.

#### Finding 3:WM\-SARgets the benefit of brute force without brute force\.

LocalRepair\-3Hop succeeds mainly because this testbed is small enough that a three\-hop neighbourhood almost covers the whole graph\.WM\-SARreaches the same spectral regime with roughly3×3\\timesfewer nodes, which is the behavior we want in larger agent traces where full replay is impractical\. The budget sweep in Figure[7](https://arxiv.org/html/2607.01767#S7.F7)shows the same effect under fixed region\-size constraints\.

#### Finding 4: overlap is not the same as repair\.

CascadeRepair often overlaps the corrupted region, but it can still leave the amplifying channel active\. This separates attribution from repair: a method can identify many corrupted nodes and still fail to change the dynamics that will govern the next rollout\. The coupling term inWM\-SARis what turns region selection toward stability rather than coverage alone\. Table[2](https://arxiv.org/html/2607.01767#S7.T2)confirms that growing the connected region is the decisive component, while Table[3](https://arxiv.org/html/2607.01767#S7.T3)shows that the same ordering holds on benchmark\-inspired code, web, and operating\-system agent topologies\.

### 7\.3LLM Repair: Single\-API \+ Cross\-Model Study

We use the LLM for actual root\-cause prediction\. For each instance, each method shows a different subset of the failure graph to the LLM in structured natural language and asks it to identify the root\-cause node\. The main evaluation uses GPT\-4o\-mini across all region\-selection methods; a cross\-model study in Figure[5](https://arxiv.org/html/2607.01767#S7.F5)tests whether the same context\-selection advantage appears with GPT\-4o, GPT\-4o\-mini, and Gemini\-2\.5\-Flash\(OpenAI,[2024a](https://arxiv.org/html/2607.01767#bib.bib12),[b](https://arxiv.org/html/2607.01767#bib.bib13); Google Cloud,[2026](https://arxiv.org/html/2607.01767#bib.bib15)\)\. The benchmark\-inspired graph families follow SWE\-bench, WebArena, and AgentBench\-style agent workflows\(Jimenezet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib45); Zhouet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib46); Liuet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib16); Yanget al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib47)\)\. The complete single\-API table appears in Table[4](https://arxiv.org/html/2607.01767#S7.T4)\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_multiapi.png)Figure 5:Cross\-model LLM root\-cause identification under the same repair prompt\. The heatmap reports Rec\-Exact for each method\-model pair, and the trend plot comparesWM\-SARagainst engineering correctors across GPT\-4o\-mini, GPT\-4o, and Gemini\-2\.5\-Flash\. The gain persists across model tiers, which indicates that the selector improves the evidence shown to the LLM rather than exploiting one particular API\.#### WM\-SARgives the LLM the right context\.

The LLM experiment asks the practical question faced by a corrector robot: if the downstream repair model is fixed, which part of a large planning graph should it see?WM\-SARgives the strongest exact root\-cause recovery while using a compact context\. Full\-graph prompts contain the answer somewhere, but they also contain many irrelevant downstream symptoms and scale poorly with planning\-graph size; pointwise prompts are cheap, but they often omit the causal bridge\. The full single\-API comparison appears below in Table[4](https://arxiv.org/html/2607.01767#S7.T4); it shows thatWM\-SARimproves exact recovery while using less than half the tokens of full\-graph LLM repair\.

#### The advantage is not tied to one API\.

Across the three LLM tiers in Figure[5](https://arxiv.org/html/2607.01767#S7.F5), the same ordering appears:WM\-SARis best on every model, while LocalRepair and TopK move with model strength but do not close the context\-selection gap\. The gain is therefore not a quirk of one API; it comes from showing the model a cleaner causal subgraph\.WM\-SARimproves the corrector’s input, not the LLM itself\.

The remaining experiment subsections bring the secondary checks into the main text: horizon stability, ablation, benchmark\-topology transfer, token\-budget efficiency, cascade\-gain robustness, full LLM repair, attribution boundaries, and the full baseline table\.

#### Spectral relief explains the method classes\.

Across methods, the qualitative ordering follows the theory: methods that reduce residual spectral radius also suppress long\-horizon error\. The relationship is class\-level rather than instance\-level; the spectral objective does not predict every graph perfectly, but it explains why compact amplification repair behaves differently from pointwise repair\.

### 7\.4Horizon Stability

The main table reports fixed horizons; Figure[6](https://arxiv.org/html/2607.01767#S7.F6)asks a harsher question: after a local repair, does error stay low as the rollout grows? This is where symptom repair separates from amplifier repair\. Pointwise and window correctors lower the first few errors but still diverge at long horizons\. LocalRepair\-3Hop is stable because it nearly covers the whole graph\.WM\-SARreaches the same stable regime with a compact region\.

### 7\.5Multi\-Step Error Profile

Figure[6](https://arxiv.org/html/2607.01767#S7.F6)plots NodeMSE as a function of horizonHH\(n=50n=50, seed=42\)\. Pointwise/window methods cannot flatten the error curve; LocalRepair\-3Hop drives it to near\-zero by covering≈\\approx99% of the graph;WM\-SARachieves near\-zero MSE with only 8\.3 nodes\.

1144881616323205050100100150150200200Planning horizonHHNodeMSE@HHUnrepairedTopK\-Point\(K=5\)Window\-8LR\-3Hop \(25\.3 nd\.\)WM\-SAR\(8\.3 nd\.\)OracleFigure 6:NodeMSE@HH\(n=50n=50, seed=42\)\. Pointwise/window methods diverge\. LR\-3Hop flattens by covering≈\\approx99% of the graph\.WM\-SARachieves near\-zero MSE with only 8\.3 nodes\.
### 7\.6Ablation Study

Table[2](https://arxiv.org/html/2607.01767#S7.T2)isolates which parts ofWM\-SARare needed for spectral repair\. The growing phase is the decisive component: without it, a high\-scoring seed remains a point repair and does not remove the amplifying path\.

Table 2:Ablation on the calling\-tree testbed \(n=50n=50, seed=42\)\.ρ\\rho\-red denotes residual spectral\-radius reduction; Slp denotes GrowthSlope; Sz denotes mean region size\. Removing growing collapsesWM\-SARback to a seed\-only point repair\.⋆Uncertainty score is computed but inactive at default hyperparameters; removing it has no effect on spectral metrics\.

Removing the Growing phase \(seed\-only\) collapses recovery:ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction0\.650\.65, MSE@32 returns to≈\\approx210 \(near\-unrepaired\), GrowthSlope reverts to\+0\.0026\+0\.0026\. Removingρ\\rho\-relief \(use error\-only growing\) reducesρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction by∼\\sim33% but does not change MSE@32 or slope in this small\-graph regime, since error\-greedy growing selects similar nodes\. GEAF and coupling provide equivalent information on the synthetic dataset\. Pruning keeps quality intact but shrinks the region from 13\.5 to 8\.3 nodes \(−40%\-40\\%\)\.

### 7\.7Benchmark Topology Generalisation

To validate thatWM\-SARgeneralises beyond the synthetic calling\-tree testbed, we simulate three benchmark\-inspired topology simulations \(n=50n=50each, seed=42\)—synthetic graphs matching the reported topologies of each benchmark \(not runs of the actual benchmark systems\)—modelling node types, edge structures, and failure cascades documented in each paper\(Jimenezet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib45); Zhouet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib46); Liuet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib16)\):

- •SWE\-bench\(Jimenezet al\.,[2024](https://arxiv.org/html/2607.01767#bib.bib45)\):N≈21N\\\!\\approx\\\!21nodes\. IssueAnalyzer→\\toFileLocator×\\times4→\\toCodeAnalyzer→\\toPatchWriter→\\toTestRunner\. Cascade gainα=1\.15\\alpha\\\!=\\\!1\.15\.
- •WebArena\(Zhouet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib46)\):N≈15N\\\!\\approx\\\!15nodes\. TaskPlanner→\\toNavigator chain→\\toPageReader→\\toFormFiller→\\toSubmitter\. Cascade gainα=1\.08\\alpha\\\!=\\\!1\.08\.
- •AgentBench\-OS\(Liuet al\.,[2024b](https://arxiv.org/html/2607.01767#bib.bib16)\):N≈12N\\\!\\approx\\\!12nodes\. Commander→\\toBashNode chain→\\toOutputParser→\\toVerifier\. Cascade gainα=1\.12\\alpha\\\!=\\\!1\.12\.

Table[3](https://arxiv.org/html/2607.01767#S7.T3)shows that the spectral advantage is not tied to a single synthetic topology:WM\-SARis best or tied\-best on all three benchmark\-inspired graph families\.

Table 3:Benchmark\-topology generalisation \(n=50n=50per topology, seed=42\)\. The simulations match the reported causal structure of SWE\-bench, WebArena, and AgentBench\-OS rather than using real benchmark traces\.ρ\\rho\-r denotesρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction and Sz denotes mean region size\.WM\-SARis the best non\-brute method on every benchmark\. On SWE\-bench the high branching \(K=4K=4file locators\) creates strong cross\-coupling that pointwise methods cannot reach\. On WebArena’s linear chain LR\-2Hop covers≈\\approx50% of nodes yet still under\-performsWM\-SAR, which targets the Navigator→\\toPageReader coupling with≈\\approx14 nodes\.

### 7\.8Budget Efficiency Analysis

Figure[7](https://arxiv.org/html/2607.01767#S7.F7)varies the explicit region\-size budget and shows thatWM\-SARreaches high spectral relief before engineering methods have enough budget to cover the relevant amplification path\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_budget.png)Figure 7:Budget sensitivity:ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction versus region sizeKK\(n=50n=50, seed=42\)\. The red horizontal line isWM\-SAR’s selected region \(not forced to a fixedKK\), which reachesρ\\rho\-reduction1\.961\.96\. Engineering methods needK≥15K\\\!\\geq\\\!15to approach this value\. The grey dotted line is the Oracle\-region baseline in this budget sweep\.WM\-SARis highly budget\-efficient\. AtK=3K=3,WM\-SARattainsρ\\rho\-reduction1\.451\.45, exceeding the Oracle\-region baseline in this sweep \(1\.421\.42;≈\\approx102%\)\. Engineering methods atK=3K=3reach only0\.620\.62–0\.770\.77\.WM\-SAR’s region grows from aκ\\kappa\-seeded core and saturates aroundK=8K=8because each added node must satisfyΔ​ρrel\>0\\Delta\\rho\_\{\\rm rel\}\>0by construction \(Equation[11](https://arxiv.org/html/2607.01767#S5.E11)\); engineering methods admit downstream symptom nodes that do not lie on the amplification path and therefore contribute nothing toρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)relief\.

### 7\.9Cascade Gain Robustness

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_cascade_gain.png)Figure 8:Cascade\-gain sensitivity \(n=50n=50, seed=42,α∈\[0\.7,1\.4\]\\alpha\\\!\\in\\\!\[0\.7,1\.4\]\)\. Left:ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction grows monotonically withα\\alphafor every method;WM\-SARtracks LR\-3Hop \(full\-graph\) while Greedy stays near\-zero\. Right: post\-repair NodeMSE@32 stays≤10\\leq 10forWM\-SARacross the entireα\\alpharange while engineering baselines diverge\.We vary the cascade gainα∈\[0\.7,1\.4\]\\alpha\\in\[0\.7,1\.4\]while holding the graph topology fixed \(Figure[8](https://arxiv.org/html/2607.01767#S7.F8)\)\. Pre\-repairρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)rises from 2\.00 to 2\.83 asα\\alphagrows; Coupled\-operator super\-additivity holds in 100% of instances for everyα\\alpha\.WM\-SARmaintains≥\\geq90% repair efficiency \(ρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\-reduction / pre\-repairρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)\) across allα\\alpha\(≥\\geq95% withinα∈\[0\.7,1\.2\]\\alpha\\in\[0\.7,1\.2\]; 90\.3% atα=1\.4\\alpha=1\.4\), matching LR\-3Hop at low gains and staying within∼\\sim10% atα=1\.4\\alpha=1\.4; Greedy\-Point collapses to<10%<10\\%at all gains\. Our main\-experiment setting \(α=1\.1\\alpha=1\.1\) sits in the middle of this range\.

### 7\.10Full LLM and Baseline Evidence

The preceding LLM figure shows the cross\-model trend\. We now report the complete single\-API and full\-baseline tables in the main text because they settle the practical question:WM\-SARis more accurate at the context scale a real corrector can afford\.

### 7\.11LLM Repair: Full 11\-Method Table

Table 4:LLM root\-cause identification \(GPT\-4o\-mini,n=150=50n=150=50instances×\\times3 seeds\)\. Rec\-E/T/H = Rec\-Exact/Type/Hop2\. Tok = mean tokens\. Sz = mean region size\.WM\-SAR’s cross\-seed CV = 10\.6% matches whole\-graph methods \(Full\-Graph: 8\.3%\); all other compact\-region methods reach CV = 39–173%\.Table[4](https://arxiv.org/html/2607.01767#S7.T4)and Figure[9](https://arxiv.org/html/2607.01767#S7.F9)give the complete single\-API view behind the main text\. The result is budget efficiency:WM\-SARimproves exact root\-cause recovery without relying on a larger context, using less than half the tokens of Full\-Graph\-LLM\.

![Refer to caption](https://arxiv.org/html/2607.01767v1/fig_llm_comparison.png)Figure 9:Single\-API LLM root\-cause repair \(GPT\-4o\-mini,n=150n=150\)\. Bars show Rec\-Hop2, Rec\-Type, and Rec\-Exact; horizontal whiskers are 95% intervals for Rec\-Exact\. Token counts are shown at right\.WM\-SARobtains the highest Rec\-Exact while using a compact 855\-token region\.
### 7\.12Cross\-Model Study: 3 LLMs

Table[5](https://arxiv.org/html/2607.01767#S7.T5)reports the 3\-LLM cross\-model study \(n=30n=30instances per model, seed=42\)\. WM\-SAR achieves the highest Rec\-Exact on all three model tiers\.

Table 5:3\-LLM cross\-model study \(n=30n=30per model, seed=42\)\. McNemar exactppvs\. LocalRepair\-2Hop \(Bonferroniα=0\.0167\\alpha=0\.0167\): Geminip=0\.007p=0\.007\(sig\.\); GPT\-4o\-minip=0\.065p=0\.065\(marginal\); GPT\-4op=0\.42p=0\.42\(not significant\)\. Against Greedy/TopK\-5/Window\-4: allp≤0\.012p\\leq 0\.012\.
### 7\.13Applicability Boundary: Who&When Attribution Benchmark

Table[6](https://arxiv.org/html/2607.01767#S7.T6)reports results on the Who&When multi\-agent failure attribution benchmark\(Zhanget al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib21)\)\(n=30n=30algorithm\-generated traces, seed=42\)\. This is theapplicability boundaryforWM\-SAR: without per\-node error signals, GEAF degenerates and spectral seeding loses its structural advantage, as discussed in the main text\.

Table 6:Who&When attribution benchmark \(n=30n=30, GPT\-4o\-mini\)\. Rec\-Exact = GT mistake step in selected region; Rec\-Agent = any message by GT mistake agent in region; Size = mean region size\.WM\-SARunderperforms LocalRepair\-2Hop on step\-level attribution due to absent per\-node error signals\.WM\-SARachieves 0\.20 Rec\-Exact vs\. LocalRepair\-2Hop’s 0\.60 \(−40\-40pp\) because the Who&When dataset does not expose per\-node intermediate error magnitudes; withoute​\(v\)e\(v\), GEAF scoring defaults to the error sink node, matching LastError\-Point’s behaviour\. The graceful degradation to Rec\-Agent = 0\.47 \(above LastError\-Point 0\.30\) shows thatWM\-SARstill captures some agent\-level causal structure; but step\-level attribution requires intermediate error exposure\.

### 7\.14Full Baseline Comparison

Table[7](https://arxiv.org/html/2607.01767#S7.T7)gives the full engineering comparison onn=200n=200instances\. The baselines match the repairs a practical LLM corrector would try before adding a spectral objective\. TraceScan methods linearize the execution trace and select either the whole trace, a fixed window, the first failed call, or the last erroneous call; these are standard debugging and failure\-attribution heuristics for agent traces\(Zhanget al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib21); Luet al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib22)\)\. Top\-B\-Nodes and Top\-B\-Edges are budgeted attribution baselines that select the largest observed node or edge errors, analogous to saliency\-style top\-KKrepair\. k\-hop\-k2/LocalRepair\-2Hop is the graph\-neighbourhood repair used in fault localization and subgraph debugging\(Louet al\.,[2021](https://arxiv.org/html/2607.01767#bib.bib31); Niuet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib30)\)\. PageRank\-Subgraph replaces observed error with global graph influence\(Pageet al\.,[1999](https://arxiv.org/html/2607.01767#bib.bib32)\)\. Uncertainty\-Subgraph selects high\-uncertainty nodes, following the common uncertainty triage rule in model\-based repair\. LLMRepair\-Full\-Plan serializes the whole planning graph and asks the same LLM repair prompt used byWM\-SAR; it is the full\-context version of LLM self\-repair and reflection\-style baselines\(Shinnet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib7); Madaanet al\.,[2023](https://arxiv.org/html/2607.01767#bib.bib24); Zhouet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib8)\)\. Because long contexts are costly and can be hard for LLMs to use reliably\(Liuet al\.,[2024a](https://arxiv.org/html/2607.01767#bib.bib39)\), this full\-plan method is an upper\-cost comparator rather than a scalable solution\.

The table uses the original experimental names\. Recovery is the fraction of instances where the root\-cause region is identified and zeroed; Tok/Rec is mean tokens per successful recovery; PD\-red is propagation\-depth reduction in hops; DE\-red is downstream\-error reduction\.

Table 7:Full baseline comparison \(n=200n=200, seed=42\)\. TraceScan, Top\-B, k\-hop, PageRank, uncertainty, and full\-plan LLM repair instantiate common engineering correctors: trace scanning, attribution ranking, graph neighbourhood repair, global influence scoring, uncertainty triage, and whole\-context LLM repair\.WM\-SARmatches the strongest full\-plan baselines on recovery while using7\.4×7\.4\\timesfewer tokens per successful recovery than LLMRepair\-Full\-Plan\.MethodRec↑\\uparrowTokensTok/Rec↓\\downarrowPD\-red↑\\uparrowDE\-red%↑\\uparrowWM\-SAR and upper boundsWM\-SAR\(ours\)\.6251,4241,3895\.460\.0Oracle\-Region\.6254754025\.860\.1LLMRepair\-Full\-Plan\.62510,85710,3335\.862\.6Context\-limited pointwise scannersTraceScan\-Full\-Point\.62021,71420,6230\.847\.0Top\-B\-Nodes \(K=3\)\.6206006000\.847\.0TraceScan\-w4\-Point\.3704,8004,8000\.034\.4Top\-B\-Edges \(K=3\)\.3206006000\.028\.7PageRank\-Subgraph\.1405185320\.423\.2k\-hop\-k2\.1355505700\.026\.0Uncertainty\-Subgraph\.0453173280\.06\.4TraceScan\-w2\-Point\.0001,200—0\.016\.4TraceScan\-w1\-Point\.000275—0\.07\.1LastError\-Point\.00050—0\.06\.9FirstFailedCall\-Point\.00050—0\.00\.0

WM\-SARmatches Oracle\-Region and LLMRepair\-Full\-Plan on Recovery \(0\.625\), but its token cost is much smaller:1,3891\{,\}389tokens per successful recovery versus10,33310\{,\}333for LLMRepair\-Full\-Plan\. Pointwise trace scanners \(TraceScan\-w1/w2, LastError, FirstFailedCall\) fail because the root cause often lies outside the selected window\. TraceScan\-Full\-Point can approachWM\-SAR’s recovery only by paying about15×15\\timesthe token cost\. The pattern is the same as in Table[1](https://arxiv.org/html/2607.01767#S7.T1): engineering correctors can find evidence, but efficient repair requires selecting the subgraph that changes the residual dynamics\.

## 8Discussion

#### The root cause of engineering heuristic failure\.

In SWE\-bench, the relevant cascade isCodeAnalyzertoPatchWritertoTestRunner\. Greedy\-Point repairsTestRunner\(highest error\) whileCodeAnalyzerre\-propagates on the next step; LocalRepair\-2Hop missesCodeAnalyzer\(three edges away\)\. Repairing a bottleneck node does not remove the bottleneck*path*\(Toppinget al\.,[2022](https://arxiv.org/html/2607.01767#bib.bib28)\)\.WM\-SAR’sΔ​ρrel\\Delta\\rho\_\{\\rm rel\}criterion identifies theCA–PW–TRchain as the path whose removal most reducesρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\), regardless of which node shows the highest error\.

#### Why Oracle coverage is not the same as spectral repair\.

Table[1](https://arxiv.org/html/2607.01767#S7.T1)shows that the Oracle andWM\-SARoptimize different notions of repair\. The Oracle covers the injected corrupted region, which is ideal for measuring attribution\.WM\-SARinstead removes the residual amplifier, which is ideal for preventing the next rollout from recreating the failure\. The two objectives are complementary: coverage explains where the mistake happened, while spectral relief explains whether the repaired system will stay stable\.

#### Applicability boundary — Who&When attribution\.

On the Who&When blackbox attribution benchmark\(Zhanget al\.,[2025](https://arxiv.org/html/2607.01767#bib.bib21)\),WM\-SARunderperforms LocalRepair\-2Hop on step\-level attribution\.WM\-SARrequires observable per\-node errore​\(v\)e\(v\)to compute GEAF; in blackbox attribution only the final answer is visibly wrong, so GEAF degenerates to the sink neighbourhood\. It still degrades gracefully at the agent level, but its main advantage is scoped to settings exposing intermediate state errors, as our synthetic testbed does; Section[7\.13](https://arxiv.org/html/2607.01767#S7.SS13)reports this boundary case\.

#### Scalability\.

LocalRepair\-3Hop’s advantage is partly an artefact of small graph size: the neighbourhood nearly becomes whole\-graph repair\. On larger traces, fixed\-hop expansion grows less predictably, whileWM\-SARcan approximate growing\-phase spectral radii with power iteration and retain the same residual objective\.

#### Prompt structure, region focus, and stability\.

Prompt formatting helps, but it is not the main effect\. The larger gain comes from deciding what the LLM should see:WM\-SARremoves distracting downstream symptoms while preserving the causal bridge\. This is why its compact prompt matches the stability of whole\-graph prompting without inheriting the noise of the full trace; Section[7\.11](https://arxiv.org/html/2607.01767#S7.SS11)reports the paired tests and cross\-seed variation\.

## 9Conclusion

Engineering heuristics optimize*observed*error but can leave the node\-edge coupling operator𝐁\\mathbf\{B\}nearly unchanged, so errors keep compounding\. The planning\-regret bound identifies the repair objective that matters for the next rollout: minimizeρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)\.WM\-SARimplements this objective with GEAF×κ\\times\\kappaseeding andΔ​ρ\\Delta\\rho\-relief growing, producing a compact causal context for LLM repair\. The takeaway is to ask “which connected subgraph, when repaired, most reduces future error growth?” not “which node has the most error?” Future work should evaluate real WebArena/ToolBench traces and learn the residual spectral predictor directly from trained world\-model rollouts\.

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## Limitations

#### Synthetic evaluation\.

All experiments use 50 synthetic agent calling\-tree graphs \(≈\\approx26 nodes each\) whose cascade dynamics \(echild=1\.1​eparent\+εe\_\{\\rm child\}=1\.1\\,e\_\{\\rm parent\}\+\\varepsilon\) are generated programmatically rather than collected from real system traces\. The benchmark\-topology simulations in Section[7\.7](https://arxiv.org/html/2607.01767#S7.SS7)match the reported structural statistics of SWE\-bench, WebArena, and AgentBench\-OS but do not consume actual failed rollout logs\.

#### No real attributed failure traces\.

We do not evaluate on the multi\-agent failure attribution datasets ofZhanget al\.\([2025](https://arxiv.org/html/2607.01767#bib.bib21)\), which provide ground\-truth causal labels\. Such datasets would allow direct validation ofWM\-SAR’s region selection against human\-attributed root causes; we leave this to future work\.

#### No parametric GWM training\.

The graph world model \(GWM\) error operators \(LX,MX,LA,MAL\_\{X\},M\_\{X\},L\_\{A\},M\_\{A\}\) are estimated from graph statistics rather than from a trained GCN/MPNN/GPS model\. EvaluatingWM\-SARon rollouts from a trained parametric GWM \(e\.g\., Error\-Aware GWM, ActionNode\-GWM\) is an important validation step deferred to future work\.

#### Repair operators\.

The broader repair design space includes state correction, edge\-structure correction, re\-rollout from a region boundary, target\-cone rollback, and validator\-boundary insertion\. Our experiments implement only state correction \(zeroing selected node errors\) plus LLM root\-cause identification \(Phase 2\)\. Edge\-structure correction requires per\-edge probability tracking absent from our graph schema, and target\-cone rollback is out of scope at the current evaluation scale\. Re\-rollout from a boundary and validator\-boundary insertion areinfeasiblewithout a live world model: they require re\-simulating the rollout from the repair boundary or inserting a runtime validator API, neither of which exists in our failure\-graph\-only setting; they represent desiderata for real\-WM deployment\.

#### Objective\-metric boundary\.

The repair objectiveρ​\(𝐁Gf∖R\)\\rho\(\\mathbf\{B\}\_\{G\_\{f\}\\setminus R\}\)is one of multiple valid repair criteria\. The Who&When multi\-agent failure attribution benchmark in Section[7\.13](https://arxiv.org/html/2607.01767#S7.SS13)shows that GT\-coverage\-based methods outperformWM\-SARwhen ground\-truth error signals are unavailable \(Rec\-Exact: LocalRepair\-2Hop 0\.60 vs\.WM\-SAR0\.20\)\.WM\-SARrequires per\-node error magnitudese​\(v\)e\(v\)to compute GEAF; in blackbox failure attribution settings \(only final output visible\), GEAF degenerates and spectral seeding loses its advantage\.

#### Uncertainty term\.

The uncertainty scoreUnc​\(v\)\\mathrm\{Unc\}\(v\)in the theoretical WM\-SAR formulation is computed but has no effect on outcomes at the default hyperparameters on the synthetic dataset \(see the−\-Uncertainty row in Table[2](https://arxiv.org/html/2607.01767#S7.T2)\)\. A dataset with meaningful uncertainty signals \(e\.g\., ensemble disagreement in a trained parametric GWM\) is needed to validate this component\.

## Ethics Statement

This work presents a graph\-based algorithm for repairing failed AI agent rollouts\. All experiments use synthetic simulation data generated programmatically; no human subjects, personal data, or sensitive information are involved\. The LLM experiments use the OpenAI and Google Gemini APIs under standard research usage agreements\(OpenAI,[2024a](https://arxiv.org/html/2607.01767#bib.bib12),[b](https://arxiv.org/html/2607.01767#bib.bib13); Google Cloud,[2026](https://arxiv.org/html/2607.01767#bib.bib15)\)\. We foresee no direct dual\-use risk in the repair algorithms presented\. A potential concern is deployment in safety\-critical agent pipelines \(medical or legal decision\-making\) without sufficient validation on real traces; our Limitations section explicitly cautions against this\.

## Appendix AAppendix Overview

The appendix contains only material that would interrupt the main argument: proofs, hyperparameter calibration, dataset statistics, and prompt serialization details\. All experimental evidence, including robustness checks and full baseline tables, is reported in the main text\.

## Appendix BProofs of Theoretical Results

###### Proof of Theorem[1](https://arxiv.org/html/2607.01767#Thmtheorem1)\.

Letak=ekXa\_\{k\}=e^\{X\}\_\{k\}\. Assumption[1](https://arxiv.org/html/2607.01767#Thmassumption1)gives the scalar recursion

ak\+1≤LX​ak\+ϵX,a\_\{k\+1\}\\leq L\_\{X\}a\_\{k\}\+\\epsilon\_\{X\},\(15\)withLX≥0L\_\{X\}\\geq 0andϵX≥0\\epsilon\_\{X\}\\geq 0\. We claim that for everyk≥0k\\geq 0,

ak≤LXk​a0\+ϵX​∑i=0k−1LXi,a\_\{k\}\\leq L\_\{X\}^\{k\}a\_\{0\}\+\\epsilon\_\{X\}\\sum\_\{i=0\}^\{k\-1\}L\_\{X\}^\{i\},\(16\)where the empty sum is zero\. The claim holds atk=0k=0\. If it holds atkk, then

ak\+1\\displaystyle a\_\{k\+1\}≤LX​ak\+ϵX\\displaystyle\\leq L\_\{X\}a\_\{k\}\+\\epsilon\_\{X\}≤LXk\+1​a0\+ϵX​∑i=1kLXi\+ϵX\\displaystyle\\leq L\_\{X\}^\{k\+1\}a\_\{0\}\+\\epsilon\_\{X\}\\sum\_\{i=1\}^\{k\}L\_\{X\}^\{i\}\+\\epsilon\_\{X\}=LXk\+1​a0\+ϵX​∑i=0kLXi,\\displaystyle=L\_\{X\}^\{k\+1\}a\_\{0\}\+\\epsilon\_\{X\}\\sum\_\{i=0\}^\{k\}L\_\{X\}^\{i\},\(17\)which proves the induction\. IfLX≠1L\_\{X\}\\neq 1,

∑i=0k−1LXi=LXk−1LX−1;\\sum\_\{i=0\}^\{k\-1\}L\_\{X\}^\{i\}=\\frac\{L\_\{X\}^\{k\}\-1\}\{L\_\{X\}\-1\};\(18\)ifLX=1L\_\{X\}=1, the sum equalskk\. Substituting these two closed forms into Equation[16](https://arxiv.org/html/2607.01767#A2.E16)gives Equation[2](https://arxiv.org/html/2607.01767#S3.E2)and itsLX=1L\_\{X\}=1case\. ∎

###### Proof of Theorem[2](https://arxiv.org/html/2607.01767#Thmtheorem2)\.

Write

𝐁=\(LXLAMXMA\),LX,LA,MX,MA≥0\.\\mathbf\{B\}=\\begin\{pmatrix\}L\_\{X\}&L\_\{A\}\\\\ M\_\{X\}&M\_\{A\}\\end\{pmatrix\},\\qquad L\_\{X\},L\_\{A\},M\_\{X\},M\_\{A\}\\geq 0\.\(19\)The characteristic polynomial is

p​\(λ\)\\displaystyle p\(\\lambda\)=λ2−\(LX\+MA\)​λ\\displaystyle=\\lambda^\{2\}\-\(L\_\{X\}\+M\_\{A\}\)\\lambda\+\(LX​MA−LA​MX\)\.\\displaystyle\\quad\+\(L\_\{X\}M\_\{A\}\-L\_\{A\}M\_\{X\}\)\.\(20\)Thus the two eigenvalues are

λ±=LX\+MA±\(LX−MA\)2\+4​LA​MX2\.\\lambda\_\{\\pm\}=\\frac\{L\_\{X\}\+M\_\{A\}\\pm\\sqrt\{\(L\_\{X\}\-M\_\{A\}\)^\{2\}\+4L\_\{A\}M\_\{X\}\}\}\{2\}\.\(21\)Since𝐁\\mathbf\{B\}is entrywise nonnegative, the Perron–Frobenius theorem implies that its spectral radius is a real nonnegative eigenvalue and equals the largest real root\. Thereforeρ​\(𝐁\)=λ\+\\rho\(\\mathbf\{B\}\)=\\lambda\_\{\+\}, which is Equation[4](https://arxiv.org/html/2607.01767#S3.E4)\.

It remains to prove strict super\-additivity\. IfLA​MX\>0L\_\{A\}M\_\{X\}\>0, then

\(LX−MA\)2\+4​LA​MX\>\|LX−MA\|\.\\sqrt\{\(L\_\{X\}\-M\_\{A\}\)^\{2\}\+4L\_\{A\}M\_\{X\}\}\>\|L\_\{X\}\-M\_\{A\}\|\.\(22\)WhenLX≥MAL\_\{X\}\\geq M\_\{A\}, Equation[21](https://arxiv.org/html/2607.01767#A2.E21)gives

ρ​\(𝐁\)\>LX\+MA\+\(LX−MA\)2=LX\.\\rho\(\\mathbf\{B\}\)\>\\frac\{L\_\{X\}\+M\_\{A\}\+\(L\_\{X\}\-M\_\{A\}\)\}\{2\}=L\_\{X\}\.\(23\)WhenMA≥LXM\_\{A\}\\geq L\_\{X\}, the same argument gives

ρ​\(𝐁\)\>LX\+MA\+\(MA−LX\)2=MA\.\\rho\(\\mathbf\{B\}\)\>\\frac\{L\_\{X\}\+M\_\{A\}\+\(M\_\{A\}\-L\_\{X\}\)\}\{2\}=M\_\{A\}\.\(24\)Henceρ​\(𝐁\)\>max⁡\(LX,MA\)\\rho\(\\mathbf\{B\}\)\>\\max\(L\_\{X\},M\_\{A\}\)whenever the two channels are coupled in both directions\. ∎

###### Proof of Theorem[3](https://arxiv.org/html/2607.01767#Thmtheorem3)\.

LetJ​\(π\)J\(\\pi\)denote the discounted value of policyπ\\piunder the true rollout andJ^​\(π\)\\widehat\{J\}\(\\pi\)the value computed by the learned graph world model\. For a fixedπ\\pi, Assumption[3](https://arxiv.org/html/2607.01767#Thmassumption3)gives

\|J​\(π\)−J^​\(π\)\|\\displaystyle\|J\(\\pi\)\-\\widehat\{J\}\(\\pi\)\|≤LR​κ​∑t=0H−1γt​‖zt‖\+ϵR​H\\displaystyle\\leq L\_\{R\}\\kappa\\sum\_\{t=0\}^\{H\-1\}\\gamma^\{t\}\\\|z\_\{t\}\\\|\+\\epsilon\_\{R\}H\(25\)≤LR​κ​ϵ​∑t=0H−1\(γ​ρ​\(𝐁\)\)t\+ϵR​H\\displaystyle\\leq L\_\{R\}\\kappa\\epsilon\\sum\_\{t=0\}^\{H\-1\}\(\\gamma\\rho\(\\mathbf\{B\}\)\)^\{t\}\+\\epsilon\_\{R\}H=LR​κ​ϵ​ΦH​\(γ,ρ​\(𝐁\)\)\+ϵR​H\.\\displaystyle=L\_\{R\}\\kappa\\epsilon\\,\\Phi\_\{H\}\(\\gamma,\\rho\(\\mathbf\{B\}\)\)\+\\epsilon\_\{R\}H\.\(26\)Define

CH=LR​κ​ϵ​ΦH​\(γ,ρ​\(𝐁\)\)\+ϵR​H\.C\_\{H\}=L\_\{R\}\\kappa\\epsilon\\,\\Phi\_\{H\}\(\\gamma,\\rho\(\\mathbf\{B\}\)\)\+\\epsilon\_\{R\}H\.\(27\)Equation[26](https://arxiv.org/html/2607.01767#A2.E26)holds uniformly for bothπ∗\\pi^\{\*\}andπ^\\hat\{\\pi\}\. Sinceπ^∈arg⁡maxπ⁡J^​\(π\)\\hat\{\\pi\}\\in\\arg\\max\_\{\\pi\}\\widehat\{J\}\(\\pi\),

J^​\(π∗\)−J^​\(π^\)≤0\.\\widehat\{J\}\(\\pi^\{\*\}\)\-\\widehat\{J\}\(\\hat\{\\pi\}\)\\leq 0\.\(28\)Using this optimality inequality,

J​\(π∗\)−J​\(π^\)\\displaystyle J\(\\pi^\{\*\}\)\-J\(\\hat\{\\pi\}\)=\[J​\(π∗\)−J^​\(π∗\)\]\\displaystyle=\\bigl\[J\(\\pi^\{\*\}\)\-\\widehat\{J\}\(\\pi^\{\*\}\)\\bigr\]\+\[J^​\(π∗\)−J^​\(π^\)\]\\displaystyle\\quad\+\\bigl\[\\widehat\{J\}\(\\pi^\{\*\}\)\-\\widehat\{J\}\(\\hat\{\\pi\}\)\\bigr\]\+\[J^​\(π^\)−J​\(π^\)\]\\displaystyle\\quad\+\\bigl\[\\widehat\{J\}\(\\hat\{\\pi\}\)\-J\(\\hat\{\\pi\}\)\\bigr\]≤\|J​\(π∗\)−J^​\(π∗\)\|\\displaystyle\\leq\|J\(\\pi^\{\*\}\)\-\\widehat\{J\}\(\\pi^\{\*\}\)\|\+\|J^​\(π^\)−J​\(π^\)\|\\displaystyle\\quad\+\|\\widehat\{J\}\(\\hat\{\\pi\}\)\-J\(\\hat\{\\pi\}\)\|≤2​CH\.\\displaystyle\\leq 2C\_\{H\}\.\(29\)Substituting the definition ofCHC\_\{H\}yields Equation[5](https://arxiv.org/html/2607.01767#S3.E5)\. ∎

## Appendix CHyperparameters and Implementation

#### LX,LA,MX,MAL\_\{X\},L\_\{A\},M\_\{X\},M\_\{A\}estimation\.

In the failure graph, GWM weights are unavailable\. We estimate:LX=w⋅ρ​\(AR\)L\_\{X\}=w\\cdot\\rho\(A\_\{R\}\);LA=w⋅βA​din​e¯L\_\{A\}=w\\cdot\\beta\_\{A\}d\_\{\\rm in\}\\bar\{e\}withβA=0\.3\\beta\_\{A\}=0\.3;MX=w⋅βX​dout​e¯M\_\{X\}=w\\cdot\\beta\_\{X\}d\_\{\\rm out\}\\bar\{e\}withβX=0\.2\\beta\_\{X\}=0\.2;MA=w⋅βM​fhighM\_\{A\}=w\\cdot\\beta\_\{M\}f\_\{\\rm high\}withβM=0\.5\\beta\_\{M\}=0\.5, wheredin/outd\_\{\\rm in/out\}= mean edge\-type diversity,e¯\\bar\{e\}= mean error,fhighf\_\{\\rm high\}= fraction of edges between high\-error nodes\.

Calibration of constants \(0\.3, 0\.2, 0\.5\)\.The constants inLAL\_\{A\}\(0\.3\),MXM\_\{X\}\(0\.2\), andMAM\_\{A\}\(0\.5\) are*calibration factors*rather than theoretically derived values\. Their purpose is to ensure thatρ​\(𝐁\)\\rho\(\\mathbf\{B\}\)estimated from graph statistics is of the same order of magnitude as the local spectral quantityρ​\(A\)\\rho\(A\), so that the combined scoree​\(v\)⋅GEAF^v⋅\(1\+κv\)e\(v\)\\cdot\\widehat\{\\mathrm\{GEAF\}\}\_\{v\}\\cdot\(1\+\\kappa\_\{v\}\)balances all three contributions comparably\. They arenotderived from or constrained by Theorem[2](https://arxiv.org/html/2607.01767#Thmtheorem2); that theorem requires onlyLA​MX\>0L\_\{A\}M\_\{X\}\>0\(which holds for any positive constants\) to guarantee super\-additivity\. The mean super\-additivity ratioρ​\(𝐁\)/max⁡\(LX,MA\)−1=0\.007\\rho\(\\mathbf\{B\}\)/\\max\(L\_\{X\},M\_\{A\}\)\-1=0\.007confirms that the resulting coupling strength is non\-negligible, not merely formally positive\. In a setting with a trained parametric GWM, these constants can be replaced by direct Lipschitz estimates from the weight matrices\.

## Appendix DDataset Statistics

## Appendix ELLM Prompt and Serialization Details

The main method only requires the LLM to see the selected repair region, not the full planning graph\. We serialize nodes in topological order and include node type, local state features, observed error, confidence, and boundary edges\. Edges crossing into or out of the selected region are marked as external context, so the model can understand the role of the region without seeing the whole graph\.

#### Structured prompt\.

The system prompt tells the LLM that it is a failure analyst for a multi\-agent calling tree and asks it to identify the node that introduced the initial mistake\. The user prompt contains the serialized subgraph and requests a JSON object with root\-cause node identifiers and a short rationale\. JSON output is used only to make evaluation deterministic; the repair advantage comes from the selected context, not from a special decoding trick\(OpenAI,[2024c](https://arxiv.org/html/2607.01767#bib.bib14)\)\.

#### Repair operator\.

In the simulation, the returned repair is applied by zeroing the selected node errors and re\-simulating the rollout\. In a deployed world\-model system, the same decision would trigger a targeted re\-run of the selected planner, executor, validator, tool call, or memory update\.