World Model Science: Self-Organized Criticality, Weak Chaos, and Metastable Belief Dynamics in Long-Horizon LLM Agents

Xinyuan Song; Zekun Cai

cs.AIJul 12, 2026

World Model Science: Self-Organized Criticality, Weak Chaos, and Metastable Belief Dynamics in Long-Horizon LLM Agents

Authors: Xinyuan Song, Zekun Cai

Organizations: Emory University, Atlanta, GA, USA · The University of Tokyo, Tokyo, Japan · LocationMind, Tokyo, Japan

Abstract

Long-horizon LLM agents must maintain and update a world model over extended sequences of observations, actions, decisions and tool calls. We study the dynamics of long horizon agents from three perspectives: self-organized criticality, weak chaos, and metastable belief dynamics. Through extensive experiments on controlled puzzles, tool-use tasks, embodied environments, multi-hop retrieval, general-assistant reasoning, and Game of Life, we study how world-model errors develop over long interactions. We find that externally increased task pressure can sharply raise the probability of collapse, while naturally occurring pressure is only a modest predictor of later failure. Errors are temporally correlated rather than independent, and their propagation changes with dependency depth while remaining bounded in finite tasks. We also find that locally valid actions can continue after the agent's overall task state has become incorrect, that error structure depends on task topology and horizon, and that belief states in embodied tasks are shallow rather than strongly trapped in incorrect states. Larger horizons allow larger error cascades, but their scale remains constrained by the task and the model's capability. Overall, we observe power-law behavior, bounded divergence consistent with weak chaos, and metastable belief dynamics across long-horizon agent interactions. We call this perspective world model science: the empirical study of self organized criticality in long-horizon agent world models.

Figures & tables

Figure 1. Overview of our framework. We align agent-implied states with benchmark-grounded states and use the resulting trajectories to measure and test the dynamics of long-horizon agent world models.

Related work	State Error	Temporal Dependence	Error Cascades	Structural Propagation	Horizon Scaling	Belief Recovery
World-state evaluation ( Vafa et al., 2024 ; Harang et al., 2025 )	✓	\ding 55	\ding 55	\ding 55	\ding 55	\ding 55
Trajectory diagnosis ( Chung et al., 2026 ; Wang et al., 2026b ; Cuadron et al., 2025 )	✓	✓	\ding 55	\ding 55	✓	\ding 55
Error propagation ( Sinha et al., 2025 ; Xiong et al., 2025 ; Gurram, 2026 )	✓	✓	✓	✓	\ding 55	\ding 55
Hallucination cascades ( Jamshidi et al., 2026 )	\ding 55	✓	✓	\ding 55	\ding 55	\ding 55
Duration and persistence ( Kwa et al., 2025 ; Laban et al., 2025 )	\ding 55	✓	\ding 55	\ding 55	✓	\ding 55
Critical dynamics ( Vock and Meisel, 2025 )	\ding 55	✓	✓	✓	\ding 55	\ding 55

Table 1. Comparison with closely related work.

\checkmark

indicates that the corresponding phenomenon is explicitly studied or measured.

Diagnostic	Stateful Puzzle	$\tau$ -bench Retail	$\tau$ -bench Airline	GAIA L1	ALFWorld	HotpotQA RAG	Game of Life
Small-error trigger and collapse	S	S	$\times$	$\times$	$\times$	$\times$	$\times$
Belief-state persistence and recovery	$\times$	A	$\times$	$\times$	A	$\times$	$\times$
Local action vs. global state	$\times$	$\times$	S	S	$\times$	$\times$	$\times$
Stability under surface perturbations	S	$\times$	$\times$	$\times$	$\times$	S	$\times$
Error accumulation over time	$\times$	S	$\times$	$\times$	$\times$	$\times$	$\times$
Long-range temporal dependence	P	$\times$	$\times$	$\times$	$\times$	S	$\times$

Table 2. Cross-substrate diagnostic matrix. Each cell reports the evidence for a diagnostic on a given substrate: supporting (S), partial (P), boundary (B), alternative (A), or capacity-limited (C).

Diagnostic block	Test	Evidence type	Location
Stress collapse	External stress and avalanche nulls	Supporting evidence	Figure 3
Belief basins	Shallow-basin direction	Alternative explanation	Appendix D.1
Local–global gap	Local validity exceeds global fidelity	Supporting evidence	Figure 4
Surface stability	Macro statistics under surface perturbations	Supporting evidence	Figure 5
Natural stress	Retail precursor prediction	Partial evidence	Figure 12
Temporal memory	Spectral and valid-horizon DFA tests	Supporting evidence	Figure 7

Table 3. Diagnostic map. Each block identifies the test used to characterize long-horizon world-model dynamics, the resulting evidence type, and the corresponding analysis.

Figure 2. Capability boundaries. Schematic input-comprehension boundary and Game-of-Life output-generation boundary.

Figure 3. Collapse under injected stress in StatefulPuzzle-SOC. Collapse rate and mean avalanche size across injected stress levels.

Figure 4. Local–global gap and verification. Left: local–global gap by GAIA Level-1 step type. Right: reward, collapse rate, and recovery time under different verification policies in

\tau

-bench Retail.

Figure 5. Stability under task-preserving changes. KS distance from the identity baseline for prompt and formatting changes in StatefulPuzzle-SOC, HotpotQA-RAG, and

\tau

-bench Retail.

Figure 6. Intervention effects across tasks. Mean reward under verification, exploration, memory, and planning settings in ALFWorld and

\tau

-bench Retail.

Figure 7. Temporal dependence in agent errors. Left: error-autocorrelation exponent across StatefulPuzzle horizons. Right: EM, F1, and error-autocorrelation exponent across HotpotQA-RAG memory and retrieval settings.

Figure 8. Scaling with horizon and grid size. Left: maximum and mean avalanche size across StatefulPuzzle horizons. Right: fractal dimension across Game-of-Life grid sizes.

Figure 9. Error propagation across dependency depths. Left: AIC difference between exponential and power-law fits. Right: fraction of trajectories favoring the power-law fit at each dependency depth.

Figure 10. Error geometry across task structures. Fractal dimension in HotpotQA-RAG, ALFWorld, and Game of Life.

Figure 11. Prompt-regime clustering. Left: silhouette score across numbers of clusters. Right: within-cluster spread, between-cluster distance, and separation ratio.

Figure 12. Retail stress prediction. Top: AUROC under reward-error and concurrent fidelity-drop labels. Bottom: AUROC for individual stress components.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Substrate	$q$	$b$	$c$	$u$	$r$	$m$	$p$	Gold source
StatefulPuzzle	subgoals	attribute table	violations	ambiguity	none	corrupted buffer	next subgoal	simulator log
Retail	subtasks	intent/db belief	policy violations	hedging	env errors	tool history	next action	$\tau$ -bench gold
Airline	subtasks	booking belief	policy violations	ambiguity	tool errors	call history	next action	$\tau$ -bench gold
GAIA	steps	evidence state	none	search uncertainty	none	retrieved docs	next reasoning step	GAIA answer
ALFWorld	subgoals	room/inventory	none	object uncertainty	none	observations	next subgoal	game log
HotpotQA	hops	answer hypothesis	none	retrieval uncertainty	none	passages	next retrieval/reasoning step	supporting facts

Appendix

Table 4. World-state definitions. Each substrate instantiates the agent-implied state with benchmark-observable components and an independently constructed gold state. Unavailable components are omitted from the corresponding state-distance calculation.

Depth	Power beats exponential	$\Delta$ AIC exp-power	Steps to saturation	Mean divergence
1	0.133	-2.754	11.9	0.120
2	0.667	0.648	10.5	1.895
4	0.700	0.558	7.8	1.947
6	0.667	0.136	6.9	1.878
8	0.700	0.533	10.3	1.910

Appendix

Table 5. Additional dependency-depth statistics. Model-fit preference, saturation time, and mean divergence across dependency depths in StatefulPuzzle.

Diagnostic	Main result	Evidence	Takeaway
Avalanche tails	Truncated power law fits better than a pure power law	Partial	Error cascades are heavy-tailed but bounded
Belief-state persistence	$P_{\mathrm{esc}}(\mathrm{wrong})=0.97$ vs. $P_{\mathrm{esc}}(\mathrm{success})=0.82$	Alternative	Incorrect states are shallow rather than strongly trapping
Local–global gap	$\Delta^{LG}=0.857$ on GAIA and $0.62$ – $0.68$ on Airline	Supporting	Locally valid actions can coexist with incorrect global state
Task-preserving changes	KS distances remain between $0.03$ and $0.13$	Supporting	Collapse statistics remain stable under wording and format changes
Natural precursor signal	AUROC $=0.616$	Partial	Naturally observed precursors provide limited predictive value
Temporal dependence	$\alpha_{e}\in[1.38,1.52]$	Supporting	Errors show persistent temporal dependence

Appendix

Table 6. Evidence summary across diagnostics. Each row reports the main quantitative result, its evidence category, and the corresponding interpretation.

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