Intrinsic Instantaneous Coarse-to-Fine Recoverability in the Lorenz-96 System

In multiscale chaotic systems, a basic closure question is how much of the unresolved fine scales is instantaneously determined by the resolved coarse scales on the attractor. In a Fourier description, we formalize this by asking, given a target mode $k$ and a lower-mode cutoff $k_{\rm cut}<k$, how much of mode $k$ is determined by the retained modes $0,\ldots,k_{\rm cut}$. We quantify this relation by the correlation-ratio functional $R(k\mid k_{\rm cut})$, interpreted as conditional-mean explained variance, and use it to build a scale-resolved recoverability map $(k,k_{\rm cut})\mapsto R(k\mid k_{\rm cut})$, whose structure is sharply organized by the nonlinear dynamics. Applying the diagnostic to the Lorenz-96 system for forcings $F=8,16,32,64$, we find that the recoverability maps are strongly nonuniform: low modes remain weakly constrained by still coarser observations, while high modes exhibit finite-band partial slaving once the retained cutoff reaches the energetic intermediate modes. The growth of substantial recoverability is organized around the quadratic triad-access scale $k_{\rm cut}\approx\lceil k/2\rceil$, consistent with the Fourier coupling rule $p+q\equiv k\pmod N$, while remaining shifted by regime-dependent statistics. Increasing $F$ preserves this geometric organization but reduces its amplitude, indicating greater conditional freedom of the unresolved modes in more strongly driven regimes. The maps show that instantaneous deterministic closure varies systematically across scales as a property of the invariant measure: retained modes provide nontrivial deterministic information in some regions, while other regions are dominated by conditional residual variance.