Show HN: I applied Lyapunov stability theory to detect when LLM agents spiral

Lyapunov-stability monitor for multi-turn LLM agents. Detects token spirals, classifies failure patterns, and tells you why a task failed — no extra LLM calls.

from state_harness import GrowthRatioGuard, FailureReport

guard = GrowthRatioGuard(token_budget=50_000)

with guard:
    for turn in agent_loop:
        result = llm.invoke(turn.prompt)
        guard.record_step(tokens_used=result.usage.total_tokens)

report = FailureReport.from_guard(guard)
print(report)
⚠️  STABILITY TRIPPED at turn 12

Pattern: Context Accumulation Spiral (confidence: 92%)
  • Last 5 turns all exceeded 1.5× baseline (4/4 were accelerating).
  • Peak growth ratio: 5.2× baseline.
  • Without intervention, projected cost was $0.0396 (actual: $0.0039).

Energy: ▁▁▁▁▁▂▂▃▄▆█
  Baseline: 1050 tokens/turn
  Peak ratio: 5.2× baseline

Cost: $0.0039 (saved ~$0.0357 by tripping early)

Suggested actions:
  🔴 1. Enable RG history compression in your agent loop.
     → Compressing older messages reduces prompt tokens by 40-60%.
  🟡 2. Lower the growth ratio threshold to 1.8×.
     → A lower threshold would have caught it earlier.
  🟢 3. Add a sliding-window context strategy.
     → Send only the last N messages plus a summary of earlier ones.

Production multi-agent systems fail at rates of 41–87% (Kore.ai 2026). When an agent spirals — replaying full context, retrying a broken tool, drifting off-task — a budget cap will kill it, but tells you nothing about why.

State-harness monitors token consumption relative to a warmup baseline via a Lyapunov energy function. When the growth ratio exceeds a threshold for W consecutive steps, it trips and classifies the failure pattern (context spiral, retry storm, policy drift) with fix suggestions — from the energy trajectory alone, no LLM calls.

pip install state-harness

and wrap your agent loop.

Pattern	Signal	Example
Context Spiral
Token growth accelerating beyond baseline	Agent replaying full history each turn
Retry Storm
Low-variance repeated calls	Tool failing, agent retrying identically
Policy Drift
VSA similarity score dropping	Agent going off-topic mid-conversation
Early Explosion
Token spike in first 3 turns	Oversized system prompt or tool response
Budget Exhaustion
Cumulative spend hits ceiling	Complex task, not necessarily broken

State-harness does not improve resolve rates — a naive budget cap achieves comparable task success (multi-trial results below). The value is:

Failure diagnostics— classified failure patterns with actionable fixes, not just "budget exceeded." No extra LLM calls.** Compute efficiency on long loops**— 38.6% fewer search nodes and 30% less wall time on SWE-bench by terminating dead-end branches early.

Validated across 3,175 runs (4 benchmarks, 5-condition ablation, multi-trial with bootstrap CIs). Zero false positives across 7 models incl. 4 local via Ollama. Details in Benchmarks.

Search-tree agents(MCTS, beam search) — per-branch caps look fine in isolation; tree-level cost explosion is silent.** Platform teams at scale**— failure classification at the edge, exported as OpenTelemetry attributes.** Benchmarking**— the ~4–5% nondeterminism floor means single-run deltas <8% are noise.

Not needed for chatbots, RAG, single-turn apps, or ReAct loops with <10 turns — max_iterations

• budget cap suffice.

pip install state-harness

Python ≥ 3.10. Pre-built wheels for Linux, macOS, Windows (x86_64 + ARM64). No Rust toolchain needed.

git clone https://github.com/vishal-dehurdle/state-harness.git
cd state-harness

python -m venv .venv && source .venv/bin/activate

curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh

pip install maturin
maturin develop --release

pip install pytest
pytest tests/

GrowthRatioGuard

normalizes token usage against a baseline — trips only on disproportionate growth, not natural context-window accumulation.

from state_harness import GrowthRatioGuard, StabilityViolation

guard = GrowthRatioGuard(
    token_budget=100_000,     # hard ceiling
    ratio_threshold=2.0,      # trip when turn is 2× the baseline
    window=3,                 # 3 consecutive escalating turns to trip
    budget_gate=8_000,        # don't trip until 8K tokens spent
)

with guard:
    for turn in agent_loop:
        try:
            result = llm.invoke(turn.prompt)
            guard.record_step(
                tokens_used=result.usage.total_tokens,
                errors=0,
            )
        except StabilityViolation as e:
            print(f"Agent killed: {e}")
            break

print(f"Total cost: {guard.total_tokens} tokens")
print(f"Baseline: {guard.baseline} tokens/turn")
print(f"Peak ratio: {guard.current_ratio}×")

After any execution (tripped or not):

from state_harness import FailureReport

report = FailureReport.from_guard(guard, model="gemini-2.5-flash")

print(report)

import json
print(json.dumps(report.to_dict(), indent=2))

Classifies the failure pattern, provides evidence, estimates cost, and suggests fixes — no LLM calls.

For lower-level control using raw token counts (no normalization):

from state_harness import BoundaryGuard

with BoundaryGuard(token_budget=100_000, lambda_=1.0, window=5) as guard:
    for turn in agent_loop:
        result = llm.invoke(turn.prompt)
        guard.record_step(
            tokens_used=result.usage.total_tokens,
            errors=0,
            tool_name="search",
        )
python
from state_harness import boundary_guard

@boundary_guard(
    token_budget=50_000,
    token_counter=lambda r: r.usage.total_tokens,
)
def agent_step(prompt: str):
    return llm.invoke(prompt)
python
from langgraph.prebuilt import create_react_agent
from state_harness.adapters import monitor_graph

agent = create_react_agent(model, tools=[search, calculate])
safe = monitor_graph(agent, token_budget=100_000)

result = safe.invoke({"messages": [("user", "Fix the login bug")]})

print(safe.total_tokens)  # cumulative usage
print(safe.tripped)       # did stability trip?
print(safe.report)        # full FailureReport with pattern + suggestions

For streaming:

for chunk in safe.stream({"messages": [("user", "Refactor this module")]}):
    print(chunk)

With a trip callback (e.g., for Slack alerts):

safe = monitor_graph(
    agent,
    token_budget=100_000,
    on_trip=lambda report: slack.send(f"Agent tripped: {report.pattern}"),
)

Advanced: per-tool wrapping with LangGraphMiddleware #

from state_harness import BoundaryGuard
from state_harness.adapters import LangGraphMiddleware

guard = BoundaryGuard(token_budget=150_000)
middleware = LangGraphMiddleware(guard)

@middleware.wrap_tool
def search_database(query: str):
    return db.search(query)

with guard:
    result = agent.invoke({"messages": [...]})
python
from crewai import Agent, Task, Crew
from state_harness.adapters import CrewAICallback

callback = CrewAICallback(token_budget=200_000)

crew = Crew(
    agents=[researcher, writer],
    tasks=[research_task, write_task],
    step_callback=callback.step_callback,
    task_callback=callback.task_callback,
)

result = crew.kickoff()
print(callback.report)  # FailureReport
callback.close()
python
from state_harness import BoundaryGuard
from state_harness.adapters import VanillaHook

guard = BoundaryGuard(token_budget=50_000)
hook = VanillaHook(guard)

with guard:
    for step in agent_loop:
        hook.before_call(tool_name="search")
        result = execute_tool(step)
        hook.after_call(tokens_used=result.tokens)
state-harness simulate 1000 1200 1500 2000 3000 5000 8000 --budget 50000

state-harness analyze report.json
state-harness analyze report.json --json    # JSON output
state-harness analyze report.json --otel    # OpenTelemetry attributes

state-harness batch --dir ./reports/ --output results.csv

FailureReport

supports multiple output formats:

report = FailureReport.from_guard(guard)

report.to_json()            # pretty-printed
report.to_json(indent=None) # compact, single line

with open("results.csv", "w") as f:
    f.write(FailureReport.csv_header() + "\n")
    for r in reports:
        f.write(r.to_csv_row() + "\n")

from opentelemetry import trace
span = trace.get_current_span()
span.set_attributes(report.to_otel_attributes())

Three mechanisms, implemented in Rust (via PyO3):

graph TD
    A["Agent Loop"] --> B["GrowthRatioGuard\n(Python SDK)"]
    B --> |"Normalizes tokens → growth ratio\nWarmup baseline · Budget gate"| C{" "}
    C --> D["Lyapunov Monitor\nV(k) = S + λθ\nΔV ≥ 0?"]
    C --> E["RG Decimator\nTF-IDF\nCompression"]
    C --> F["Holographic Engine\n(VSA)\nDrift Detection"]
    
    style D fill:#1a1a1a,stroke:#555,color:#e8e8e8
    style E fill:#1a1a1a,stroke:#555,color:#e8e8e8
    style F fill:#1a1a1a,stroke:#555,color:#e8e8e8
    style B fill:#0d1117,stroke:#30363d,color:#e6edf3

Component	Purpose	Speed
Lyapunov Monitor
Tracks energy derivative ΔV(k). Trips when ΔV ≥ 0 for W consecutive steps.	~1μs/step
RG Decimator
RG-inspired decimation of conversation history (TF-IDF scoring). Retains structurally important messages.	~100µs/compress
Holographic Engine
VSA-based policy drift detection. Binds domain invariants to high-dimensional vectors.	~10μs/check

5-condition ablation across 4 benchmarks (3,175 total runs). Full methodology in the research paper.

Condition	Lyapunov	RG Decimation	VSA Dual-Gate	Description
A. Baseline
—	—	—	Unmonitored agent
B. Lyapunov-only
✅	—	—	Energy monitoring, no intervention
C. Lyapunov+RG
✅	✅	—	+ history compression on violation
D. Full-stack
✅	✅	✅	+ policy drift gating
E. Naive Cap
—	—	—	Hard budget cap (control)

Benchmark	Runs	Stability Trips	Cost Savings (D vs A)	Resolve-Rate Δ	Diagnostics
MINT (reasoning + coding)
1,136	0	~0%	−0.7pp (noise)	N/A (no trips)
τ³-bench (customer service)
750	0	8.1%	within ±12pp nondeterminism	N/A (no trips)
SWE-bench Verified (coding)
333 + 148	~38%	38.6% (nodes)	−3.6pp (within ±4–5% noise)	Pattern classification
Custom Local (4 models)
240	3 (true pos.)	15.2%	0pp	Pattern classification
MINT Local (Qwen3:4B)
568	0	~0%	+1.8pp	N/A (no trips)

Resolve-rate deltas fall within LLM nondeterminism (~4–5% stdev). No trips on short/medium loops (1,886 runs). Savings concentrate on long-loop search trees.

37 Django instances, SWE-bench Verified. Agent: moatless-tools SearchTree, 50-node budget. Model: Gemini 2.5 Flash.

Condition	Resolved	Rate	Total Nodes	Wall Time	Nodes/Resolve
A. Baseline
15 / 37	40.5%	945	80 min	63.0
B. Lyapunov
16 / 37	43.2%	620	69 min	38.8
D. Full-stack
14 / 37	37.8%	580
56 min
41.4
E. Naive Cap
21 / 37	56.8%	876	77 min	41.7

Note:Single-trial resolve rates have ~±8pp standard error. E's apparent 56.8% is not statistically significant vs A's 40.5%. Multi-trial results below confirm this.

Full-stack monitoring: 38.6% fewer nodes (945 → 580), 30% less wall time (80 → 56 min). Baseline had 7 tasks burning the full 50-node budget (all failed); with monitoring, zero hit ceiling. Lyapunov alone (Condition B, ~5 lines of code) delivers ~90% of the savings.

Ablation — each mechanism contributes independently:

Layer Added	Compute (nodes)	Δ vs Baseline	Cumulative Reduction
A. No monitoring	945	—	—
B. + Lyapunov	620	−325	34.4%
D. + RG + VSA	580	−40	38.6%

Lyapunov alone delivers ~90% of the benefit. RG and VSA add incremental value.

3 trials per condition (A, D, E) across all 37 instances — 333 total runs. 12 runs stuck in Docker (28+ min), counted as failures:

Condition	Trial 1	Trial 2	Trial 3	Mean ± σ
A. Baseline
18/37 (48.6%)	16/37 (43.2%)	15/37 (40.5%)	44.1% ± 4.1%
D. Full-stack
15/37 (40.5%)	16/37 (43.2%)	14/37 (37.8%)	40.5% ± 2.7%
E. Naive Cap
19/37 (51.4%)	15/37 (40.5%)	17/37 (45.9%)	45.9% ± 5.4%

Cross-condition variance (2.9%) ≤ within-condition nondeterminism (4.1%). All differences fall within the noise band.

The ~4% within-condition stdev converges with τ³-bench (±4.6%), establishing a ~4–5% nondeterminism floor for Gemini 2.5 Flash on code tasks. Single-run deltas <8% are unreliable.

Bootstrap CIs (10,000 resamples) and Welch's t-tests: A−D = +3.6pp [−0.9, +8.1], p ≈ 0.17; A−E = −1.8pp [−8.1, +4.5], p ≈ 0.68; D−E = −5.4pp [−10.8, 0.0], p ≈ 0.09. Full analysis in paper §7.3.1.

50 tasks × 3 trials × 5 conditions = 750 total runs. Agent handles airline reservations via tool calls. Model: Gemini 2.5 Flash. Concurrency=1.

Condition	Trial Pass	Rate	Task Pass (maj)	Rate	Cost	Cost Δ
A. Baseline
99/150	66.0%	35/50	70.0%	$2.47	—
B. Lyapunov-only
83/150	55.3%	28/50	56.0%	$2.42	−2.0%
C. Lyapunov+RG
79/150	52.7%	26/50	52.0%	$1.69	−31.8%
D. Full-stack
86/150	57.3%	30/50	60.0%	$2.28	−8.1%
E. Naive Cap
81/150	54.0%	26/50	52.0%	$2.33	−5.7%

Key findings:

Zero stability trips across 750 runs. All airline tasks classified as stable; no interventions.Pass-rate variance is nondeterminism. Naive cap (E, zero monitoring) drops −16pp from baseline —worsethan full-stack (D, −10pp). The ~10–16pp spread is intrinsic variance.25% of tasks flip pass/fail within the same condition across trials (~±12pp nondeterminism floor).8.1% cost savings from passive monitoring (zero interventions).

284 tasks × 4 conditions = 1,136 total runs across GSM8K (48), MATH (100), HumanEval (45), MBPP (91). Agent uses up to 5 turns per task.

Condition	GSM8K	MATH	Total	Tokens
A. Baseline
91.7%	39.0%	29.2%
1,909,582
B. Lyapunov
91.7%	41.0%	29.9%
1,904,421
C. Lyapunov+RG
89.6%	37.0%	28.2%
1,910,926
D. Full-stack
87.5%	39.0%	28.5%
1,949,708

Zero stability violations across 1,136 runs. Token usage invariant (<2% overhead).

Failed tasks cost disproportionately more:

Task	Success Avg	Failure Avg	Ratio
GSM8K	2,613 tok	8,857 tok	3.4×
MATH	5,154 tok	8,188 tok	1.6×

HumanEval and MBPP show 0% across all conditions — a MINT framework limitation in code execution evaluation, consistent across conditions (harness does not introduce new failure modes).

20 custom tasks (5 easy, 10 medium, 5 hard) × 4 models × 3 conditions = 240 runs. Hardware: Apple M4 MacBook Pro, 16 GB RAM, Ollama local inference.

Model	Size	Baseline	Harness	Naive Cap	Token Savings	FP
Llama 3.2:3B
2.0 GB	45%	45%	60%	1.2%	0
Phi-4-Mini
2.5 GB	30%	30%	40%	20.7%	0
Qwen3:4B
2.5 GB	30%	30%	40%	0.9%	0
Gemma4:E4B
9.6 GB	35%	35%	70%	37.9%	0

Key findings:

Zero false positives across 80 harness runs— 4 model families, 3 difficulty tiers. Growth-ratio generalizes without threshold retuning.** Small-model self-sabotage:*Naive cap beats baseline by +17.5pp avg (+12.5pp median). Small models solve early turns correctly, then destroy solutions in later turns. Strongest on Gemma4:E4B (+35pp).Model-family behavioral signatures: Llama 3.2:3B:Classic spirals (ratios: 2.3×, 5.9×, 7.6×) — 3 true-positive tripsPhi-4-Mini:Spike-and-recover — 20.7% passive savingsQwen3:4B:255K tokens but flat ratios (≤1.06×) — stable despite 3× volumeGemma4:E4B:*Decreasing ratios — 37.9% passive savings, zero trips

Deploying ≤4B models via Ollama? State-harness works out of the box (zero false positives). The self-sabotage finding suggests adding a turn limit (2–3 turns) for open-ended code generation.

Task	Harness (max=5)	Naive Cap (max=2)	Δ
GSM8K	37.5%	27.1%	+10.4pp
MATH	0.0%	0.0%	—
HumanEval	11.1%	11.1%	—
MBPP	14.3%	14.3%	—
Total
12.7%
10.9%
+1.8pp

Zero interventions across 284 tasks. With max 5 turns and W=3, the monitor cannot trigger within available post-warmup turns — a structural guarantee.

Full reproduction steps (all three benchmarks) #

git clone https://github.com/vishal-dehurdle/state-harness.git
git clone https://github.com/sierra-research/tau-bench.git tau3-bench

cd state-harness
python -m venv .venv && source .venv/bin/activate
pip install maturin && maturin develop --release

cd ../tau3-bench
uv sync
cp ../state-harness/tau3_integration/harness_agent.py src/tau2/agent/
cp ../state-harness/tau3_integration/naive_cap_agent.py src/tau2/agent/

export GOOGLE_CLOUD_PROJECT=your-project-id
export VERTEXAI_LOCATION=asia-south1

bash benchmarks/tau3/run_5phase_airline.sh

bash benchmarks/swe_bench/run_benchmark.sh
bash benchmarks/swe_bench/run_benchmark_dbe.sh

bash benchmarks/mint/run_mint_fullstack.sh

Ablation conditions are controlled via environment variables:

Variable	Values	Effect
HARNESS_RG
on / off
Enable/disable RG history compression
HARNESS_VSA
on / off
Enable/disable VSA policy drift detection
HARNESS_RATIO_THRESHOLD
float (e.g., 2.0 )
Override growth ratio threshold
HARNESS_BUDGET_GATE
int (e.g., 8000 )
Override minimum spend before trip

See benchmarks/ for setup, configs, and reproduction instructions.

Multi-trial SWE-bench— 333 runs (3 trials × 3 conditions × 37 instances) confirming non-invasiveness within ±4% noise band - Local model validation— 240 runs across 4 open-weight models (Llama, Phi, Qwen, Gemma) + 568 MINT runs on Qwen3:4B - Terminal-Bench— Terminal-based agent tasks; command-line tool loops where spirals manifest as repeated failed commands - SWE-bench Pro— Harder, contamination-resistant variant of SWE-bench - Cross-model validation— 7 models total: GPT-4o-mini, Claude Haiku 4.5, Gemini 2.5 Flash + Llama 3.2:3B, Phi-4-Mini, Qwen3:4B, Gemma4:E4B

37 SWE-bench instances— A larger sample would improve statistical power (n=3 trials gives limited degrees of freedom for t-tests).** No causal intervention**— The harness currently kills spiraling tasks. Redirect/repair is on the roadmap.** Physics-inspired, not physics-equivalent**— Terms like "Renormalization Group" and "Lyapunov stability" are used as structural inspirations. The mathematical mapping is analogical, not isomorphic.Custom benchmark scale— The 20-task local battery is smaller than standard benchmarks. The self-sabotage finding (mean +17.5pp, median +12.5pp) is consistent across 4 models but requires larger-scale replication.

Parameter	Default	Description
token_budget
100,000	Hard ceiling on cumulative tokens
ratio_threshold
2.0	Growth ratio above which a turn counts as "escalating" (domain-tuned: airline=2.0, retail=2.5, telecom=2.0)
window
3	Consecutive escalating turns before circuit breaker trips
warmup_turns
3	Turns used to establish baseline (no monitoring during warmup)
budget_gate
8,000	Minimum cumulative tokens before the monitor can trip (retail: 12,000)
lambda_
1.0	Error weighting in the Lyapunov energy function

Environment variable overrides (highest precedence, for threshold sweeps):

Env Var	Description
HARNESS_RATIO_THRESHOLD
Override ratio_threshold (e.g., 2.5 )
HARNESS_BUDGET_GATE
Override budget_gate (e.g., 12000 )

Tuning tips:

More aggressive(catch spirals earlier):ratio_threshold=1.8, window=2

More conservative(fewer false positives):ratio_threshold=2.5, window=3

High-value tasks: Increasebudget_gate

to 20K+ to let expensive tasks run longerComplex domains(retail, multi-tool): Start withratio_threshold=2.5

Lyapunov stability: V(k) = S(k) + λθ(k) models token consumption as a dynamical system. ΔV ≥ 0 for W consecutive steps → unstable.** Renormalization Group (RG): Message compression via coarse-graining — eliminates high-frequency noise, preserves scale-invariant task objectives. Vector Symbolic Architecture (VSA)**: Domain policies bound to high-dimensional bipolar vectors (10,000-d, i8), enabling constant-time drift detection outside the LLM context window.

Implements the framework from:

Empirical Lyapunov Stability: Growth-Ratio Energy Functions as Leading Indicators of Agent Task FailureVishal Verma, 2026[Read the full paper →]

Full ablation, multi-trial validation, local-model results, and failure taxonomy. Key results reproduced in Benchmarks above.

If you use this library or refer to these findings in your research, please cite the preprint:

@misc{verma2026empirical,
  author       = {Verma, Vishal},
  title        = {Empirical Lyapunov Stability: Growth-Ratio Energy Functions as Leading Indicators of Agent Task Failure},
  month        = jun,
  year         = 2026,
  publisher    = {Zenodo},
  version      = {1.0.0},
  doi          = {10.5281/zenodo.20722987},
  url          = {https://doi.org/10.5281/zenodo.20722987}
}

Based on the theoretical framework from:

The Fluid Dynamics of Multi-Agent AI: Resolving d'Alembert's Paradox of Generative WorkflowsVishal Verma, 2026[Read →]

See CONTRIBUTING.md for dev setup, code style, and PR guidelines.

Adaptive threshold— Auto-tune τ based on task complexity signal from early turns - Causal intervention— Instead of killing spiraling tasks, redirect them (prompt injection, tool restriction) - Streaming support— Token-level monitoring for streaming LLM responses - Multi-model validation— 7 models validated: GPT-4o-mini, Claude Haiku 4.5, Gemini 2.5 Flash + 4 local models via Ollama - Dashboard / observability— Optional lightweight UI for monitoring energy trajectories in real-time

See SECURITY.md. Do not open public issues for security reports.

Split-core licensing:

Component	License	Notes
Rust Core (src/ )
BSL 1.1	Free for non-commercial + ARR < $1M. Converts to Apache 2.0 on May 26, 2030.
Python SDK (python/ )
Apache 2.0	Fully permissive.

See LICENSE.md for full details.

source & further reading

github.com — original article