# Figure Captions

## Fig. 1: `figures/Fig1_lambda1_success_failure_HRM_TRM.png`

First finite-time Lyapunov exponent distribution for successful versus failed inference trajectories in HRM and TRM checkpoints. Failures are shifted toward larger/more positive exponents, motivating chaos as a failure detector.

## Fig. 2: `figures/Fig2_full_spectrum_success_failure_shift.png`

Top Lyapunov spectrum for successful versus failed examples. The separation is not only a top-exponent effect; many leading modes shift toward expansion on failed examples.

## Fig. 3: `figures/Fig3_trajectory_perturbation_improves_peak_accuracy.png`

Peak exact accuracy for baseline training versus trajectory perturbation training. The perturbation method keeps the same supervised input/target pair but trains additional recurrent rollouts with small latent-state perturbations to reach the same answer. This is a ceiling/peak result; HRM later shows final-checkpoint collapse.

## Fig. 4: `figures/Fig4_optional_PTRM_Q_head_vs_stability.png`

Optional context. In PTRM-style stochastic multi-rollout inference, the learned Q-head score is correlated with a finite-difference stability proxy on mixed-success problems, suggesting that learned rollout selection may partly approximate a low-dimensional stability score.

# Caveats

- Lyapunov measurements are finite-time diagnostic estimates on sampled subsets, not full asymptotic exponents.
- Fig. 3 reports best-checkpoint/peak accuracy, not necessarily final-checkpoint accuracy.
- The gradient flossing analogue is still preliminary; the main question for Rainer is conceptual, not a claim that flossing does or does not work in RRMs.