Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark

Abstract

Under coarse observation, unresolved slow forcing can remain dynamically active yet locally invisible to reduced spectral inference. For a solvable driven AR$(1)$ benchmark, the local Whittle/Kullback--Leibler distance from the true spectrum to the best nearby one-pole surrogate obeys $\Dloc(\lambda)=C\lambda^4+O(\lambda^6)$, even though the observed spectrum itself is perturbed at $O(\lambda^2)$. The quartic onset is a geometric consequence of the reduced model manifold: the $O(\lambda^2)$ perturbation is partially absorbed by tangent-space reparametrization, and only the normal residual survives. We obtain $C$ in closed form for an AR$(1)$ hidden driver and show that $C$ vanishes as $(a-b)^2$ at timescale coalescence, identifying a spectrally \emph{dark} regime. We then show that this dark regime is not geometrically inevitable: for a non-degenerate AR$(2)$ hidden driver (second characteristic root $z_2\neq 0$), $C>0$ for all parameter values, including single-root coalescence, because the richer spectral structure cannot be absorbed by the two-dimensional tangent space. The quartic coefficient interpolates smoothly between the two cases as $C\sim z_2^4$ when the second characteristic root vanishes. Together, the AR$(1)$ and AR$(2)$ results yield a classification within the one-pole projection class: the quartic law and the boundary $\lcpop(N)\propto(\log N/N)^{1/4}$ are universal features of the projection geometry within this class, while the dark regime requires the hidden driver's spectrum to match the null family's pole structure.