Subject: Re: Question on gradient flossing vs forward trajectory stability in recursive reasoning models

Hi Rainer,

A short follow-up to my email of June 5 — we have since measured the things I was speculating
about, and two results seem worth sharing because they sharpen the question I asked you.

First, conditioning per-example finite-time Lyapunov spectra on both outcome and terminal
settling (n = 2048–8192, two architectures) shows that failure is almost exclusively
non-settling: in an official-recipe TRM at 87.6% accuracy, none of 254 failed trajectories
ever enters the low-velocity band that all successes occupy, and they remain locally expansive
to the end (median λ₁ +0.10 vs +0.01). "Converged to the wrong attractor" failures exist in
HRM but make up only ~0.5% of failures. The chaotic signature also survives two controls: it
persists after matching trajectories on displacement level (so it is not just re-measuring
non-convergence), and after binning by puzzle difficulty.

Second — and this is the part that genuinely surprised us — the signature is strictly
concurrent. Among puzzles still unsolved after a quarter of the inference budget, neither the
early-window exponents nor early state velocity predict which trajectories will eventually
succeed (AUC ≈ 0.5); in HRM the association even inverts, with eventually-successful
trajectories moving more in the early phase. So the failed trajectories are not "born chaotic":
chaos at the end and failure appear together.

This makes me think the right framing for my earlier question is reachability of the settled
region (escape from a long chaotic transient) rather than per-example landscape quality, which
would be consistent with your view of flossing as a learning-time tool rather than an
inference-time one. If you know of work that conditions finite-time exponents on trajectory
fate in this way — in transient-chaos settings or elsewhere — I would be grateful for a
pointer; we have not found a precedent.

Best,
Yuren

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[Notes, not part of the email: numbers from analysis_2x2/OBSERVATIONS.md addenda 1-2. Send only
if/after Rainer replies to the June 5 email, or as a gentle bump after ~2 weeks (June 19+).
The "born chaotic" phrasing mirrors his literature's transient-chaos vocabulary deliberately.]