State-compression ladder · Note M14 · Part VI
The space↔time principle leaves neural networks behind. In a differentiable dynamical system it shows both failure modes at once — and three different optima that don't line up.
Parts III–V showed space↔time factorization helping neural sequence models — serialized pixels, then real audio. Part VI asks whether the same principle survives when the "sequence" is the evolution of a physical dynamical system itself. It does, and it comes back sharper.
Train a small shared vector field ẋ = fθ(x,u) to reproduce a forced damped pendulum over 1024 steps. The baseline is one long differentiable integration — the Neural-ODE analogue of the full-recurrence GRU. The alternative has an exact numerical-analysis name, multiple shooting: break the trajectory into many short segments, each integrating the same field from its own shooting state, all in parallel, with a continuity penalty tying the end of each segment to the start of the next. The shooting states are free variables, not frozen teacher points — otherwise it collapses to local derivative fitting.
The load-bearing test is the free-running rollout: after training, integrate the learned field from x₀ for the whole horizon with no shooting states, and compare to ground truth. Short segments can fit fragments perfectly while learning a field that fails to compose — the lesson M13 taught the hard way.
The single long shoot is worst on every axis: badly conditioned gradients (ρG ≈ 320), slow to train (it never reaches the target error in budget), and a worse free rollout. But the fully-fragmented extreme is not the answer either — and that is the new part.
Temporal depth provides the learning signal and constrains composition. Long horizons wreck the first; extremely short horizons wreck the second. The optimum keeps just enough contiguous time to identify dynamics that compose correctly.
Gradient conditioning improves monotonically as the block shrinks — so ρG explains why the long end is bad, but it cannot explain the whole optimum. If conditioning alone governed performance, the one-step block would win. It doesn't. Something the gradient story misses turns the curve back up.
At a block of a single step the fragment loss is essentially zero — the field fits every one-step transition almost perfectly — yet the free-running rollout is worse than at four steps. That is M13's lesson in its cleanest possible form:
Local correctness is not global dynamical correctness.
It is an almost absurdly explicit argument against judging a learned dynamical system by one-step or derivative losses alone: here the one-step error is near-perfect and the composed field is still not the best. And the gradient pathology generalizes across the whole programme. In the recurrent nets, long depth made gradients vanish (ρG → 0). In the ODE, long depth makes them explode (ρG ≫ 1). Opposite failure, identical remedy: reduce unnecessary causal depth.
The central sentence of the whole arc is now this:
Shortening causal depth improved optimization — until the temporal fragments became too short to constrain globally composable dynamics.
Which turns space↔time from a trick into a design rule that has held across recurrent nets, raw audio, and now a differentiable physical system:
Do not maximize temporal depth, and do not eliminate it. Keep only the amount of time required to make the local computation globally meaningful. Equivalently: factor a long causal problem into the shortest local temporal units that still constrain the global solution.
The next questions follow naturally: whether cheap local or direct credit between segments can replace exact backprop (the path to on-chip and physical learning), and how the boundary moves for a chaotic system — Lorenz — where shortening the gradient horizon should help most, and where the right metrics are statistical rather than pointwise.
Part of the whitebox / MorphoHDL program. Part VI of a series with M13.5 (I), M14 (II), and Parts III–V. Forced damped pendulum, non-chaotic; shared 2-layer vector field; multiple shooting with free shooting states and a continuity penalty; free-running rollout as the load-bearing metric; corrected target alignment; the interior-minimum shape confirmed across three seeds.