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+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
+
+---
+[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.]