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

Abstract: Long-horizon LLM agents must maintain task state across extended sequences of observations, actions, tool calls, and intermediate beliefs. We study these trajectories through three dynamical views: self-organized criticality, weak chaos, and metastable belief dynamics. Our framework aligns agent-implied states with benchmark-grounded states and measures stress accumulation, error avalanches, temporal dependence, local--global mismatch, bounded divergence, belief-basin transitions, and finite-size scaling under explicit null models. Across 22 experiments spanning controlled puzzles, tool use, embodied tasks, multi-hop retrieval, general-assistant reasoning, and Game of Life, we find that locally valid actions can persist after global state fidelity fails, stress can trigger abrupt collapse, error sequences exhibit long memory, dependency depth changes the propagation regime, and larger horizons support larger avalanches. At the same time, divergence remains bounded, belief states show metastable rather than fully chaotic behavior, and stronger claims of universal power laws, critical points, or shared intervention optima are not supported. These results suggest a science of agent world models based on trajectory-level dynamical diagnostics rather than terminal reward alone.

• The paper suggests analyzing the trajectories of long-horizon LLM agents through a finite dynamical system framework that detects stress-sensitive collapses and structural conditionings without imposing classical SOC requirements.
• The analysis assumes that each LLM agent experiences propagating stress even though in this distributed context it does not claim SOC phenomena anything more than the claim of the criticality dynamics in the given settings.
• Flow chart experiments show that these stress-sensitive behavior signifiers in LLM agents perform weaker than the classical SOC.

Research objective and conceptual framework

The paper proposes a trajectory-level framework for analyzing long-horizon LLM agents as finite dynamical systems. Its central claim is deliberately narrower than the physical claim suggested by self-organized criticality (SOC): agent trajectories can exhibit measurable signatures of correlated, stress-sensitive, structurally propagated collapse without thereby establishing physical SOC, a universal critical point, or universal power-law behavior. The framework is intended to distinguish these signatures from independent per-step errors and from ordinary persistence caused by Markovian dependence (2609.17419).

The proposed analogy treats unresolved uncertainty, contradictions, retrieval conflicts, unverified assumptions, and tool-error debt as accumulated stress. A threshold-crossing event corresponds to degradation in the agent’s benchmark-grounded world state, while an avalanche is a temporally clustered sequence of state errors. This framing extends established diagnostics from SOC and crackling-noise systems—event-size distributions, temporal dependence, finite-size scaling, and invariance under surface perturbations—without assuming that LLM agents instantiate the underlying physical mechanisms.

Figure 1: SOC-inspired diagnostic framing in which stress accumulation, threshold crossing, and cascade release are used as diagnostic signatures of correlated agent collapse.

The measured object is not a hidden activation trajectory. Instead, the authors define a substrate-specific state extractor over logged interactions. The resulting state vector contains progress, belief, constraints, uncertainty, risk or tool debt, memory or retrieval context, and plan intention. A corresponding gold-state extractor is obtained from simulator state, database state, supporting facts, audit logs, or benchmark annotations. World-state fidelity is then compared with local action validity. The distinction is essential: an action may be syntactically valid, immediately admissible, and environmentally executable even when the agent’s latent task state has already diverged.

The paper formalizes this distinction through the local-global gap, defined as local validity minus global world-state fidelity. A proposition establishes that local-validity traces cannot, in general, identify global fidelity: two histories can generate identical sequences of locally admissible actions while differing in hidden constraints or task state. This is not merely a measurement preference but an identifiability limitation. Evaluations based only on action validity therefore cannot recover the state trajectory relevant to long-horizon correctness.

Measurement protocol and diagnostic criteria

The empirical program comprises 22 experiments over StatefulPuzzle-SOC, $\tau$-bench Retail and Airline, GAIA Level 1, ALFWorld, HotpotQA-RAG, and Game of Life. The experiments use a fixed model interface, temperature zero, and fixed seeds. The authors freeze state extractors, coordinate distances, fidelity weights, and stress weights before outcome analysis. This precommitment is intended to constrain researcher degrees of freedom, although the validity of each extractor remains substrate-dependent.

The diagnostic pipeline maps interaction traces into state estimates, stress components, thresholded error events, and statistical summaries. It compares observed trajectories with several null models: independent Bernoulli errors, Markov persistence, shuffled spectra, task-difficulty predictors, and surface-perturbation controls.

Figure 2: Measurement pipeline separating local action validity from global world-state fidelity and comparing observed collapse with independent-error and persistence nulls.

The paper defines finite world-model SOC through four jointly required properties:

1. Temporal dependence: thresholded state errors are correlated beyond an independent-error null.
2. Stress response: collapse probability or magnitude increases with accumulated or externally injected stress.
3. Finite-size scaling: cascade scale depends on a finite system-size variable such as trajectory horizon.
4. Structural conditioning: propagation depends on dependency depth or task-graph topology.

This definition intentionally excludes stronger requirements often associated with SOC. A universal exponent, exact critical point, asymptotic scale-free behavior, and shared intervention optimum are not necessary. The resulting construct is therefore an operational finite-system diagnostic rather than a claim that LLM agents reproduce classical SOC mechanisms.

Figure 3: Cross-substrate evidence matrix distinguishing supporting, partial, boundary, alternative, capacity-limited, and untested diagnostic results.

Stress-sensitive collapse and the local-global mismatch

The strongest causal result comes from the controlled StatefulPuzzle-SOC intervention. Exogenously injected stress is varied before outcome measurement, using horizon 64 and dependency depth one. Stress predicts collapse with an AUROC of 0.979, and the first nonzero stress condition moves the model from a measurable zero-stress stability floor to near-deterministic collapse. Because stress is manipulated rather than inferred from the same error signal used to define collapse, this result provides more than a correlation between internal difficulty and failure.

Figure 4: Controlled stress causes a sharp transition from mostly stable behavior to near-deterministic collapse over horizon 64.

The implication is that small perturbations can have strongly nonlinear consequences when the agent is operating near a state-dependent failure boundary. However, the result is established in a controlled synthetic substrate; it does not by itself demonstrate that naturally occurring stress in benchmark environments has comparable causal leverage.

Benchmark traces provide a complementary result. In $\tau$-bench Airline, information-only tasks show little local-global mismatch, whereas more complex task classes produce gaps between approximately 0.62 and 0.68. GAIA intermediate-conclusion steps produce the largest reported gap, $\Delta^{LG}=0.857$. These actions remain locally coherent even after the evidence state has collapsed.

Figure 5: Local action validity can remain high after global evidence or task-state fidelity has substantially degraded.

This result directly challenges evaluations that use executable tool calls or immediate admissibility as proxies for reliable reasoning. A locally valid action is not evidence that the agent’s belief state, constraints, or evidence set remain correct. The measurement does, however, depend on the fidelity and granularity of the benchmark-specific gold-state extractor.

The surface-invariance experiments reinforce a structural interpretation. Paraphrases, renamings, distractors, order changes, and stylistic transformations preserve macro avalanche statistics when they preserve the underlying task graph. In StatefulPuzzle, six surface variants maintain comparable avalanche size, collapse rate, and collapse timing; reported KS distances remain below the corresponding same-distribution null threshold. Similar qualitative stability appears in HotpotQA and Retail.

Figure 6: Macro collapse statistics remain comparatively stable under prompt-surface changes that preserve task structure.

The result suggests that the measured dynamics are not reducible to superficial wording. It does not establish universality: the relevant invariance is conditional on preserving the task graph, and the paper later finds that intervention regimes and geometric signatures vary substantially across substrates.

Memory, bounded divergence, and metastability

Temporal analyses reject a white-noise interpretation of several error streams. StatefulPuzzle remains in a long-memory regime across tested horizons, with spectral error exponents in the range $\alpha_e \in [1.38,1.52]$ and valid-horizon DFA estimates of approximately 1.34–1.38. The authors explicitly discount short-series DFA artifacts by treating DFA as reliable only at sufficiently long horizons and using spectral estimates as the primary cross-horizon diagnostic.

HotpotQA supplies a mechanism-sensitive comparison. Full-context retrieval nearly decorrelates the error stream, whereas narrow top-two retrieval shifts errors toward flicker-like persistence. Thus, temporal dependence is not presented as an invariant property of the model alone; it changes with the information channel and retrieval regime.

Figure 7: Error persistence depends on horizon and information access, with narrow retrieval producing stronger temporal dependence in HotpotQA.

The dependency-depth experiment is framed as weak chaos rather than unbounded chaos. With a fixed initial discrepancy, exponential fits are preferred at depth one. At depth two, the mean AIC difference changes sign, and the fraction of paired trajectories favoring power-law fits increases from 0.133 at depth one to 0.667–0.700 at depths two through eight. Mean divergence rises from 0.120 at depth one to approximately 1.88–1.95 at larger depths, while saturation occurs within the finite state space.

Figure 8: Recursive dependency depth changes the shape of bounded divergence, with an AIC crossover at depth two rather than unbounded chaotic growth.

The authors correctly avoid interpreting these fits as estimates of a positive Lyapunov exponent. The power-law preference describes the shape of finite-system propagation before saturation. The result supports a depth-dependent propagation transition, but it does not establish deterministic chaos in the dynamical-systems sense.

The metastability analysis further limits the interpretation. ALFWorld agents escape wrong belief basins rapidly, but they also under-exploit success-associated basins. The measured escape probability from wrong basins is 0.97, compared with 0.82 from success-associated basins. This pattern is inconsistent with a simple account in which agents become rigidly trapped in incorrect metastable states. It is better characterized as shallow basin structure with excessive exploration or insufficient exploitation. Consequently, the paper’s title-level reference to metastable belief dynamics denotes partial, substrate-specific evidence rather than robust wrong-basin lock-in.

Geometry, finite-size effects, and capability boundaries

The geometry analyses test whether error clusters reflect the topology of evidence or task dependencies. HotpotQA, with fixed two-hop evidence topology, produces relatively stable fractal dimensions between 0.835 and 0.904, with a spread of 0.069. ALFWorld exhibits considerably larger topology-conditioned variation: reported values are 0.639 for long-chain tasks, 1.042 for container tasks, and 1.50 for multi-room tasks.

Figure 9: Error geometry is stable under fixed evidence topology but varies with embodied task-graph structure.

The implication is that fractal dimension should not be treated as a universal scalar characteristic of an agent. It is an estimand conditioned on the substrate’s graph structure, state representation, and error-cluster construction. The variation across ALFWorld task classes supports structural conditioning, but it also makes cross-benchmark numerical comparisons difficult without stronger normalization.

Finite-size scaling is most clearly demonstrated by varying trajectory horizon in StatefulPuzzle. The maximum avalanche size increases monotonically from 7 to 490, with every adjacent-horizon comparison remaining significant after false-discovery correction; the reported corrected tests are below $10^{-8}$. This establishes a horizon-dependent cutoff: longer trajectories permit larger cascades, while the finite task constrains the maximum event size.

The Game-of-Life experiment demonstrates why capacity controls are indispensable. For grid sizes from 4 through 64, the measured fractal dimension increases from 1.09 to 1.49. At grid sizes of at least 96, however, there are no valid generated trajectories. The absence of a signature at those sizes cannot be interpreted as evidence against critical-like dynamics, because the model has left the measurable operating regime.

Figure 10: Avalanche size grows with horizon in StatefulPuzzle, while Game of Life exposes a separate capacity boundary.

Figure 11: A missing diagnostic signature is interpretable only when the model can generate valid trajectories on the evaluated substrate.

This capability-matched-support condition is one of the paper’s most important methodological points. A benchmark can fail to reveal a dynamical signature either because the signature is absent or because the model cannot produce valid trajectories. These cases are observationally distinct only if validity is measured independently.

Limits on stronger SOC interpretations

The paper reports several results that qualify rather than strengthen the universal interpretation. In Retail, natural early stress predicts an independent reward-error label with only modest performance: AUROC 0.616. After accounting for task difficulty, the incremental cross-validated contribution of stress is small.

Figure 12: Natural stress is modestly predictive of independent Retail reward errors and adds little beyond task difficulty.

This result separates controlled stress sensitivity from observational precursor value. The controlled intervention demonstrates causal responsiveness in StatefulPuzzle; the Retail analysis does not show that naturally extracted stress is a strong or generally useful early-warning signal.

Prompt-regime clustering provides similarly limited evidence for discrete universality classes. Macro statistics remain stable across variants, with KS distances of approximately 0.03–0.17, but six interpretable clusters are not the statistically preferred partition. Silhouette analysis favors two clusters, with a reported silhouette score of 0.321, and bootstrap ARI indicates continuous rather than sharply separated regimes.

Figure 13: Stable macro statistics coexist with continuous prompt-regime structure rather than sharply discrete universality classes.

Intervention optima are also substrate-specific. Retail performs best under no intervention or high verification, with a reported best score of 0.50, whereas ALFWorld favors exploration-heavy or memory-heavy regimes, with a best score of 0.433.

Figure 14: Verification and exploration have different operating trade-offs in Retail and ALFWorld.

This contradicts the idea of a single near-critical intervention balance transferable across environments. The measured control policy must be conditioned on the substrate’s error ecology, task topology, and state observability.

Several open questions remain within the paper’s own scope. The validity of the conclusions depends on substrate-specific state maps and distance functions, whose construction may affect fidelity, stress, and avalanche estimates. Persistence models explain a substantial portion of Retail burstiness, so the incremental evidence for critical organization beyond correlated dependence is limited. Pure power laws are rejected in favor of truncated or lognormal tails, and no unique critical point is identified. Finally, the experiments use one model interface, fixed decoding settings, and a finite set of benchmarks; whether the same diagnostics preserve their interpretation across model families and independently designed state extractors remains unresolved.

Conclusion

The paper develops a finite, measurement-oriented account of long-horizon agent collapse. Its most consistent evidence concerns stress-sensitive failure, local-global state divergence, long-memory errors, dependency-conditioned propagation, topology-dependent error geometry, and horizon-dependent avalanche cutoffs. The strongest numerical findings are the controlled-stress AUROC of 0.979, the GAIA local-global gap of 0.857, the depth-two divergence transition, and avalanche growth from 7 to 490 with horizon.

The results support the use of SOC-inspired diagnostics for world-model reliability, but not the stronger claim that LLM agents possess universal critical dynamics. Natural stress has modest predictive value, persistence explains part of the observed clustering, belief basins are shallow rather than rigidly trapping, regime structure is continuous, and intervention optima are substrate-specific. The paper’s principal contribution is therefore methodological: long-horizon agent evaluation should measure intermediate world-state fidelity and trajectory dynamics rather than relying solely on terminal reward or stepwise action validity (2609.17419).