State-compression ladder · Note M14 · Part IX · generalization
Breaking one long trajectory into short blocks worked on a pendulum, and again on the Lorenz attractor. So does the cut itself generalize? Across four dynamical systems it does — but three honest edges say it is not a free, uniform law.
Parts VI–VIII established the mechanism on a tame forced pendulum and stress-tested it on Lorenz. The open worry was over-fitting to two friendly systems. So we ran the same block-length sweep on two more: Van der Pol (a relaxation oscillator, dialled from mild to stiff) and Duffing (a driven cubic oscillator — strongly nonlinear but not stiff). The headline holds. The interesting part is exactly where it frays.
| system | character | best vs single shoot | optimum |
|---|---|---|---|
| Pendulum | mild | interior | T_block ~ 4 |
| Van der Pol (μ=1) | moderate | 9.9× | T_block 4 (interior) |
| Duffing | strong, non-stiff | 51× | T_block 1 (extreme) |
| Lorenz | chaotic | categorical | short |
| Van der Pol (μ=4) | stiff | 1.5× — all poor | — |
Wherever the surrogate integrator is adequate, short blocks beat the single long shoot — and on the most strongly nonlinear non-stiff system, Duffing, by the largest margin of all, 51×. The cut travels. Now the edges.
On the pendulum and Van der Pol the best block is in the middle — short enough for a well-conditioned gradient, long enough to still compose dynamics (the two-jobs tension of Part VI). On Duffing, shorter is simply better all the way down. Same cut, different sweet spot.
Look again at the Van der Pol curve where it pokes above its own single-shoot line. Splitting into just four long segments (T_block=128) trains worse than not splitting at all — and this reproduces in every one of three seeds (mean 0.606 vs 0.181 for the single shoot). A few still-long blocks, loosely stitched by a fixed continuity penalty, settle into an inconsistent solution that the fully-coupled single shoot avoids.
Decomposition is not monotone. Below some block length it wins big; just above it, a coarse split can be worse than the monolith it replaced. You have to cut past the coarse regime, not merely into it.
This is not a failure of the idea — the win at T_block=4 is ten-fold and robust — but it kills any naive "more splitting is always safer" reading. The likely lever is the coupling strength: a stronger or adaptive continuity weight for coarse splits should erase the valley, and that is its own testable claim.
Turn Van der Pol up to μ=4 and the clean story collapses: every block length is poor, the best beats the single shoot by only 1.5×, and the gradient-survival signature inverts. But this is not the decomposition failing. It is the integrator failing: a stiff relaxation oscillator is exactly where an explicit Euler surrogate is unstable, so no factorization of a broken integrator can help. Stiffness is orthogonal to the credit-path question — a separate axis that wants an implicit or adaptive solver, not a different cut.
Duffing is the control that proves this reading: it is more strongly nonlinear than mild Van der Pol, yet non-stiff — and it gives the largest win in the whole suite. Nonlinearity is not the problem; stiffness of the chosen integrator is.
Short-block multiple shooting robustly beats the single long shoot across strongly nonlinear systems — provided the surrogate integrator is stable on the system, and the blocks are short enough to clear the coarse-decomposition regime. The optimal block length is system-dependent, not a constant.
That is a smaller claim than "breaking trajectories always helps," and a truer one. The mechanism generalizes; the tuning does not come for free. Both facts matter, and a note that reported only the first would be the less useful one.
Duffing and stiff Van der Pol are single-seed; mild Van der Pol is three-seed and it is the one carrying the load-bearing "coarse valley" claim. All use the same explicit-Euler surrogate as the rest of the arc, which is precisely why the μ=4 result is read as an integrator limit rather than a credit result. The per-arm gradient-survival numbers are noisy (each is a property of a separately-trained field) and are not leaned on; free-running rollout is the metric throughout.
Part of the whitebox / MorphoHDL program. Generalization sweep extending the multiple-shooting arc (Parts VI–VIII). Van der Pol x' = y, y' = μ(1−x²)y − x (μ = 1 and 4); Duffing x' = y, y' = −0.3y + x − x³ + 0.2cos(1.2t); shared 2-layer vector field, explicit Euler surrogate, multiple shooting with a continuity penalty; free-run rollout the scoring metric.
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