Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark

Abstract: Under coarse observation, detectability of unresolved slow forcing can be projection-controlled: only the component of the hidden-induced deformation normal to a reduced null manifold remains locally visible. We establish this exactly in a solvable driven AR$(1)$-by-AR$(1)$ benchmark. The local Whittle/Kullback--Leibler distance from the true spectrum to the best nearby one-pole surrogate obeys $\Dloc(λ)=Cλ<sup>4+O(λ<sup>6)$, even though the observed spectrum itself is perturbed at $O(λ<sup>2)$; detectability is therefore quartic, not quadratic, in coupling. The coefficient $C$ is obtained in closed form and vanishes as $(a-b)<sup>2$ when the hidden and intrinsic timescales coalesce, identifying a spectrally \emph{dark} regime in which the leading perturbation is tangent to the reduced manifold. This yields a population boundary $\lcpop(N)\propto(\log N/N)<sup>{1/4}$, with Whittle-BIC crossover near that scale. The benchmark exposes a broader geometric principle in reduced inference: tangent hidden effects are absorbed by reparametrization, whereas only surviving normal components control local distinguishability.