Prediction Accuracy Can Select the Scaling Law of Required Precision

Abstract

How much precision must be allocated, and to which uncertainty channels, to meet a prediction requirement? We formulate a minimum one-shot specification cost over initial state variables and persistent forcing parameters. The central result is that prediction accuracy can select the horizon-scaling law of this resource by activating different physical uncertainty channels. Within any common uniformly scale-separable regime with the same active channels, changing a fixed tolerance changes the required precision by only $\cO(1)$ bits. An exactly solvable multiscale toral system shows that one fixed dynamics and target can nevertheless switch from $B_N(\eps_c)=\Th(\log N)$ to $B_N(\eps_f)=\Th(N)$ when finer accuracy activates an expanding channel. A forced Lorenz--84 calculation then realizes the same channel-switching mechanism numerically over finite horizons in a single two-scale diagnostic field: a sampled post-transient eddy envelope supports a coarse branch controlled by the phase rate of a large-scale forcing, whereas finer accuracy activates exponentially sensitive atmospheric degrees of freedom and produces an approximately linear increase of the linearized one-shot cost over the resolved window. Thus accuracy can determine not only how much predictive precision is required, but which physical channel controls its growth with horizon.