State-compression ladder · Note M14 · Part VII · stress test
A chaotic system is where long-horizon differentiable training is worst — so it is the sharpest test of the space↔time idea. Here the factorization is not a speedup. It is the line between learning the flow and learning nothing.
Part VI found the multiple-shooting knee on a tame forced pendulum. The obvious objection: a non-chaotic system is easy, and long integration only mildly hurts. So we ran the opposite extreme — the Lorenz attractor, where nearby trajectories separate exponentially and one long differentiable shoot should be maximally ill-behaved.
Under chaos, pointwise trajectory error explodes even for a perfect field — two identical flows started a hair apart diverge regardless. So long-horizon MSE is the wrong scorecard, and we froze two chaos-appropriate metrics before running: the vector-field error (did we learn the flow itself, measured on attractor samples — immune to trajectory divergence), and the attractor statistics (free-run the learned field for a long time and check it reproduces the invariant measure, not any particular path).
On the tame pendulum, one long shoot was merely slower and slightly worse. On Lorenz it fails outright. The reason is the same mechanism the whole arc has been tracking, taken to its limit: exponential sensitivity makes the gradient through a long chaotic integration catastrophically ill-conditioned, and no usable field comes out the other end. Short shoots stay inside the horizon over which the flow is still predictable, so each segment poses a well-conditioned learning problem.
On a chaotic system the space↔time factorization is not a speedup — it is the difference between learning the dynamics and not learning them.
The most satisfying part is the second metric. A free run of the short-shoot field does not follow any particular ground-truth trajectory — it can't, chaos forbids it — yet it lands on the same butterfly and spends the right fraction of time in each wing: attractor error under 0.015. It learned the invariant measure, which is the correct notion of "got the dynamics right" for a chaotic system. The long shoot did not even do that (0.86).
So the same principle that started as "parallelize an RNN" now reads, at the hard end of dynamical learning:
Keep only the amount of contiguous time over which the system is actually predictable; spatialize the rest. Beyond that horizon, more sequential depth doesn't add information — under chaos it actively destroys the gradient.
Single-seed, one chaotic system, one integrator. The chaos-metric choices (vector-field error and attractor statistics) were frozen before results, and long-horizon pointwise MSE was deliberately excluded because it measures chaos, not the model. There is a faint hint of an interior optimum in the attractor metric — its minimum is at a 16-step block, with the one-step extreme slightly worse — which would echo Part VI's composition boundary, but it is too shallow at one seed to lean on. The categorical result — long shoot fails, short shoots succeed — is not shallow at all.
Part of the whitebox / MorphoHDL program. Isolated stress test extending the multiple-shooting arc (Part VI). Lorenz σ=10, ρ=28, β=8/3; shared 2-layer vector field; RK4 ground truth, Euler-integrated learned field; multiple shooting with a continuity penalty; metrics chosen for chaos (vector-field error and attractor statistics), frozen before the run.