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 BN(epsc)=Th(log N) to BN(epsf)=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.
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BibTeX
@misc{kim2026prediction,
title = {{Prediction Accuracy Can Select the Scaling Law of Required Precision}},
author = {Song-Ju Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00352},
eprint = {2610.00352},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00352},
}
RIS
TY - UNPB
TI - Prediction Accuracy Can Select the Scaling Law of Required Precision
AU - Song-Ju Kim
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00352
UR - https://arxiv.org/abs/2610.00352
AB - 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.
ER -