✗ Mechanism failed 2026

Digital-Recurrence Lyapunov Monitor

Implementation & benchmark of arXiv:2608.16120 — When More Data Become Less Informative: Finite-Precision Periodicization and Collapse of Forecast-Error Lyapunov Estimates

Usefulness7/10

Difficulty5/10

Novelty8/10

Regularization Stability Accuracy Rnn Ssm World model Training dynamics Scheduler Dynamical systems Numerical analysis Statistics Random matrix

Source paper: When More Data Become Less Informative: Finite-Precision Periodicization and Collapse of Forecast-Error Lyapunov Estimates arXiv:2608.16120 ⓘ · analyzed Sep 1, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Add a numerical-health monitor that distinguishes genuine contraction or chaos from finite-precision periodicization. It tracks hidden-state recurrence, effective cycle length, and the divergence between single-rollout and independent-restart Lyapunov estimates, then triggers precision escalation, rollout truncation, perturbation, or training early stopping when the diagnostic enters the recurrence-collapse regime.

Formulas

$$\tau=\min\{t>s\geq0:\ z_t=z_s\},\qquad P=t-s,$$

$$\widehat\lambda_1(N)=\operatorname{slope}\left(\log\|\delta_k\|\right)_{k=1}^{N},\qquad \widehat\lambda_R(K)=\frac{1}{R}\sum_{r=1}^{R}\operatorname{slope}\left(\log\|\delta^{(r)}_k\|\right)_{k=1}^{K},$$

$$C(N)=\frac{|\widehat\lambda_1(N)-\widehat\lambda_R(K)|}{|\widehat\lambda_R(K)|+\epsilon},\qquad \text{alarm if }N\gtrsim\tau\text{ and }C(N)>c_0,$$

Mathematical statement

For a deployed transition $z_{t+1}=Q(F_\theta(z_t))$, define the first recurrence time $\tau=\min\{t>s\geq0:z_t=z_s\}$ and the observed period $P=t-s$. For continuous floating-point states, equality can be replaced by a hash collision or $\|z_t-z_s\|_\infty\leq\varepsilon$. Let $\widehat\lambda_1(N)$ be the forecast-error Lyapunov estimate from one length-$N$ trajectory and $\widehat\lambda_R(N)$ the mean estimate from $R$ independently restarted trajectories of short horizon $K$. Define the collapse ratio $C(N)=|\widehat\lambda_1(N)-\widehat\lambda_R(N)|/(|\widehat\lambda_R(N)|+\epsilon)$, where $\epsilon>0$ prevents division by zero. The paper predicts that $C(N)$ rises when $N$ exceeds the transient-plus-period recurrence scale, while independent restarts delay the effect. This creates an actionable alarm rather than treating a near-zero exponent as a reliable property of the learned system.

Implementation notes

1. Integration point: place the monitor around validation rollouts and, optionally, inside an adaptive training controller for RNNs, SSMs, neural simulators, or diffusion samplers with deterministic low-precision state updates. Keep a shadow fp32/fp64 run only for calibration when affordable. 2. Pseudocode: maintain a bounded hash table of recent quantized hidden states; record the first repeated hash, estimated period, and transient. Every validation interval, calculate one long-rollout exponent $\widehat\lambda_1(N)$ and a restart estimate from $R$ independent segments of length $K$. Compute $C(N)$. If recurrence is detected and $C(N)>c_0$, shorten rollout windows, increase precision, inject a tiny controlled state perturbation, or use restart averaging; log the intervention rather than silently reporting the collapsed exponent. 3. Exact quantities are state hashes and finite-difference forecast errors; the threshold $c_0$, tolerance $\varepsilon$, and minimum repeat count should be selected on a held-out precision calibration run. 4. First cheap experiment: use a 2-layer tanh RNN on Mackey-Glass or Lorenz-63, compare fp16, bf16, fp32, and fp64, and sweep rollout length logarithmically. Predict a kink: $C(N)$ should remain low before the measured recurrence scale and rise sharply afterward, while the single-rollout exponent decreases toward zero. Reproducing the paper's logistic-map control requires the collapse near $N=20,000$--$30,000$ in float32 and no comparable collapse in float64.

Verification

Mechanism failed

Stage 1 · Toy mechanism gate: Failed ✗

Stage 2 · Mechanism transferred to benchmark: Not tested

Stage 2 · Practical benchmark result: Not run

Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a bounded state-recurrence monitor with exact/grid-hash detection, finite-difference Lyapunov slopes, restart averaging, collapse ratio, and alarm logic. The logistic-map sanity check found float32 recurrence at tau=4686 with period 4344, while float64 had no recurrence through 20,000 steps. However, C(N) rose to about 0.91 by N=512 in both float32 and float64, well before float32 recurrence, indicating separation saturation rather than recurrence collapse; therefore the claimed effect was not validated.

Agent confidence: 9/10

Baseline: float64: no exact recurrence through 20,000 steps; C(512)=0.913 and C(2048)=0.977
Idea: float32: first recurrence tau=4686, period=4344; C(512)=0.914 and C(8192)=0.980; float16 recurrence tau=37, period=1 but finite-difference perturbation was unrepresentable

Limitations: Only a deterministic logistic-map toy system was tested, not an RNN, neural simulator, or training controller. The Lyapunov estimator uses raw finite-difference separation slopes without renormalization, so chaotic-distance saturation confounds the collapse ratio. No intervention effectiveness, wall-clock overhead, approximate hashing, or held-out threshold calibration was evaluated.

How to run: python3 digital_recurrence.py > results.json

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Artifacts

Implementation overview ⬇ Download all as ZIP 3 files · code, reports and structured results

🐍 digital_recurrence.py 4.1 KB View Raw
📄 report.md 1.5 KB View
📊 results.json 4.4 KB View Raw