You observe a dynamical system through a measurement function — not the full state, but a projection of it. The question: can you predict the future of the observable from its past, without knowing the underlying state?
The paper on predictability of observables (arXiv: 2603.20641) shows that for linear systems, the observable satisfies a closed differential equation whose order is determined by the system and the observation operator. This minimal-order closure has an equivalent representation as a discrete delay equation: the current rate of change of the observable depends on finitely many past values.
For nonlinear systems, exact closure is generally impossible — the observable's dynamics depend on the full state, which is unobserved. But the paper introduces “diminishing ambiguity”: given sufficient output history, the instantaneous dynamics can be approximately determined. The approximation improves with more history, yielding a delay differential equation representation.
The through-claim: observability creates effective dimension reduction through delay. The full system may be high-dimensional, but the observable — one measurement — satisfies a lower-dimensional equation, at the cost of requiring memory (past observations). Delay trades space for time: instead of knowing the full state now, you use the partial observation over a window of past time. The information is the same; the representation is different.
2603.20641. Dynamical systems / observability / delay differential equations / dimension reduction / Takens embedding.