When an LLM agent runs a long task, it accumulates state: progress markers, beliefs, constraints, uncertainties, tool debt, and intermediate plans. Standard evaluation asks one question: did the final answer pass? This paper asks a different set of questions. Did stress build up before the failure? Did errors arrive as isolated mistakes or as cascading avalanches? Did the agent keep producing locally valid actions after its internal world model had already diverged from reality? Can you measure these dynamics with the same statistical vocabulary used to study sandpiles, neural cascades, and crackling noise in physical systems?
The answer across 22 experiments spanning puzzles, tool use, embodied navigation, multi-hop retrieval, general-assistant reasoning, and Game of Life is: yes, partially, and with important limits. Collapse in long-horizon agent trajectories is measurable, correlated, structured, and bounded. It is not universal, not critical in the strong physical sense, and not governed by a single shared mechanism across substrates.
What Gets Measured and How
The framework does not require access to the model's internal activations. It works entirely from logged interaction traces. For each task and each timestep, a substrate-specific extractor maps the history of observations, actions, and environment responses to a measured world state: progress, belief, constraint state, uncertainty, risk or tool debt, memory or retrieval context, and plan intention. A gold extractor provides the ground truth from simulator state, database state, supporting facts, or audit logs. The world-state fidelity is a weighted root-mean-square distance between measured and gold states, clipped to [0,1] and subtracted from 1. Local action validity measures syntax, admissibility, and one-step environment consistency.
The local-global gap is defined as local validity minus global fidelity. This is the paper's central diagnostic quantity. When local validity is high but global fidelity is low, the agent is executing syntactically correct actions against a world model that has already diverged from reality. In GAIA Level-1, this gap reaches 0.857 on intermediate-conclusion steps: the agent's reasoning is locally coherent while its evidence state has collapsed. In tau-bench Airline, information-only tasks show near-zero gaps, while nontrivial task classes show large positive gaps. The gap is not an artifact of evaluation design. It reflects a real property of long-horizon agent behavior: local validity is insufficient for identifying global state fidelity, a fact the paper proves constructively in Proposition 1.
Stress, Avalanches, and the Sandpile Analogy
Stress is a weighted combination of unresolved uncertainties, contradictions, retrieval conflicts, unverified assumptions, and tool-error debt. An avalanche is a contiguous sequence of timesteps where the world-state error exceeds a threshold. The paper tests whether these avalanches behave like the cascades in self-organized critical systems: whether they cluster in time, whether their sizes follow heavy-tailed distributions, whether stress predicts their onset, and whether their maximum size scales with the system's finite horizon.
In StatefulPuzzle-SOC, with horizon 64 and dependency depth one, injected stress predicts collapse with AUROC 0.979. The first nonzero stress level moves the agent from a visible zero-stress floor to near-deterministic collapse. This is a controlled intervention, not an observational correlation: stress is injected before outcome analysis, and the agent's response changes sharply. In tau-bench Retail, observed avalanches are too clustered for a matched independent Bernoulli error process. Truncated power laws, lognormal tails, and a Markov-persistence null each explain part of the pattern, but none alone captures the full clustering structure.
The paper is careful about what it claims here. It does not claim universal power laws. It does not claim a unique critical exponent. It claims that error sequences are correlated beyond independent per-step noise, that stress modulates collapse probability, and that the avalanche structure contains temporal information that a Bernoulli process misses. These are finite estimands tested against explicit null models, not asymptotic universality claims.
Error Memory and the Dependency-Depth Transition
Two results address how errors propagate through time and through task structure. First, error sequences exhibit long memory. In StatefulPuzzle, spectral exponents stay in the long-memory regime across horizons after DFA correction. In HotpotQA, retrieval width changes the temporal regime: full context nearly decorrelates the error stream, while narrow top-2 retrieval pushes the spectrum toward flicker-like persistence. This is a mechanism check that makes sense: when the agent retrieves more evidence, errors become more independent because each step has access to fresh information. When retrieval is narrow, errors persist because the agent is repeatedly making decisions from the same incomplete picture.
Second, dependency depth creates a bounded divergence transition. At depth one, exponential fits to divergence between trajectories with a fixed initial discrepancy are preferred. From depth two onward, power-law fits are preferred. But the state space is finite and divergence saturates. This is not unbounded chaos with a positive Lyapunov exponent. It is a finite-system transition where deeper recursive dependencies change the shape of divergence without producing runaway behavior. The distinction matters for reliability science: deeper tasks fail differently, but they do not fail without limit.
Error Geometry on Task Graphs
Error clusters follow the topology of the task's dependency graph. In HotpotQA, fixed two-hop evidence graphs yield stable fractal dimensions (D_f = 0.835 to 0.904, spread 0.069). In ALFWorld, fractal dimensions vary with task-graph structure: long-chain tasks at 0.639, container tasks at 1.042, multi-room tasks at 1.50. The fractal dimension is not a universal constant. It is a graph-conditioned diagnostic that reflects how errors distribute across the task's structural backbone. This is useful information for task design: tasks with higher fractal dimensions have more spatially distributed error patterns, which may require different mitigation strategies than tasks where errors cluster locally.
Finite-Size Scaling and the Capability Boundary
In StatefulPuzzle, the maximum avalanche cutoff grows monotonically from 7 to 490 as the horizon increases from 8 to 256. All adjacent-horizon tests remain statistically significant after false-discovery correction. This is finite-size scaling in the SOC sense: larger horizons permit larger avalanches, while the process remains constrained by the finite task. The result is expected but important to verify: it confirms that the measurement framework detects the system-size dependence that the theoretical framework predicts.
The Game of Life scan measures the model's capability boundary directly. For grid sizes 4 through 64, fractal dimension rises from 1.09 to 1.49. For grid sizes of 96 and above, there are zero valid trajectories because the model cannot emit valid grid states. This is a capacity-limited result, not a dynamics result. The paper separates these cleanly: a missing diagnostic signature is interpretable only when the model can produce valid trajectories on the substrate. When it cannot, the absence of a signature reflects model capacity, not measured dynamics.
What Does Not Hold: The Limits of the Analogy
The paper tests three stronger interpretations of the SOC analogy and reports mixed results. Natural stress in tau-bench Retail is only a modest precursor to collapse. Early stress is above chance, but its incremental value over task difficulty is small. This is a meaningful negative result: in real agent traces, natural stress accumulation is not as predictive as the controlled-injection experiments suggest. The controlled experiments prove the mechanism exists; the observational experiments show it is not the dominant signal in natural operating conditions.
Macro regimes are stable but not sharply discrete. Prompt variants show consistent macro statistics (KS distances 0.03 to 0.17), but clustering is moderate: two clusters are silhouette-optimal while six clusters are interpretable, and bootstrap adjusted Rand index indicates continuous regime structure. The universality-class interpretation from statistical physics does not transfer cleanly to agent dynamics.
Intervention optima are substrate-dependent. Retail is best with no intervention or high verification. ALFWorld is best with exploration-heavy or memory-heavy regimes. The same verification-exploration balance does not transfer across substrates. There is no single near-critical operating point that works everywhere. This is perhaps the most practically important negative result: it means there is no universal tuning recipe for agent reliability, and each task domain requires its own calibration.
What This Means for Agent Evaluation
The paper proposes four changes to how long-horizon agents are evaluated. First, benchmarks should log an agent-implied task state, not only final reward and executable actions. The world-state fidelity trajectory contains information that terminal success does not. Second, reports should include the local-global mismatch between action validity and world-state fidelity. This quantity identifies silent collapse: the agent is still doing things that look right while its internal model of the task has already diverged. Third, error sequences should be compared with both Bernoulli and persistence nulls. A Bernoulli process can match the mean error rate but miss burst size; a persistence model can match temporal correlation but miss structural propagation. Fourth, diagnostics should be repeated across horizon, dependency depth, and task topology, because the same model shows different propagation regimes under different graph structure.
For developers building long-horizon agents, the practical takeaway is that terminal success is an unreliable indicator of internal state health. An agent that passes the final check may have spent many steps operating against a stale world model, accumulating stress that happened not to trigger collapse within the observed horizon. The local-global gap provides a way to detect this condition without requiring ground-truth state access at every step: if local action validity is consistently high while a proxy for global fidelity is declining, the agent is in a silent-collapse regime.
The code and measurement framework are available at github.com/Hik289/agent-self-organized-criticality. The paper's 22 experiments span seven substrates with frozen measurement maps fixed before outcome analysis. All statistical tests use Benjamini-Hochberg false-discovery control at q=0.05, and every major signature is compared against an explicit null model. The results do not identify physical SOC in LLM agents, but they do establish that trajectory-level dynamical diagnostics add information that terminal reward and stepwise validity do not capture.