Fast Lyapunov Indicator (λF)

Metric directly localizing saddle-like boundaries between solution routes by measuring maximal local trajectory separation.

Neighborhood — ranked by edge-count

paper

thinker

Claude Froeschlé
introduces
Originally developed the fast Lyapunov indicator (used here to measure asteroid orbit stability, repurposed to localize reasoning saddles).

cosine ≥ 0.65 · no typed edge

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

Fast Lyapunov indicator λF strongly correlates with the number of distinct solutions a reasoning trajectory passes through before convergingfinding0.872
Links saddle-crossing intensity to the number of candidate-answer switches during reasoning.
Finite-Time Lyapunov Exponents (FTLE)method0.774
Spectrum quantifying amplification/suppression of perturbations along independent latent directions, used to detect transient chaos onset during training.
Lyapunov Functionconcept0.739
Variational free energy acts as Lyapunov function for neuronal dynamics, ensuring convergence.
F-statistics and Linear Probes for Feature Selectionmethod0.714
Method to select d_steer top-activated SAE features for constructing control vectors
Fast Fourier Transform-Based Methodmethod0.687
Algorithm mentioned alongside Monte Carlo for computing pi, illustrating solution diversity.
Pre-bifurcation, all finite-time Lyapunov exponents are negative, so no transient chaos occursfinding0.686
Establishes the baseline (non-chaotic) regime before the bifurcation.
For simple factual tasks F0-F3, probe directions show a sharp geometric transition in middle layers, with late-layer probes converging to high cosine similarity; A3 and F4-F5 show no clear transition.finding0.680
Geometric evidence for convergence to stable truth directions only for simpler tasks.
Logistic Regression Probemethod0.669
Standard linear probing technique; compared to mass-mean probing for classification accuracy and causal implication