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Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark

Yuda Bi, Chenyu Zhang, Vince D Calhoun

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 $\Dloc(λ)=Cλ^4+O(λ^6)$, even though the observed spectrum itself is perturbed at $O(λ^2)$; detectability is therefore quartic, not quadratic, in coupling. The coefficient $C$ is obtained in closed form and vanishes as $(a-b)^2$ 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 $\lcpop(N)\propto(\log N/N)^{1/4}$, 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.

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-by-AR benchmark. The local Whittle/Kullback--Leibler distance from the true spectrum to the best nearby one-pole surrogate obeys , even though the observed spectrum itself is perturbed at ; detectability is therefore quartic, not quadratic, in coupling. The coefficient is obtained in closed form and vanishes as 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 , 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.
Paper Structure (50 sections, 2 theorems, 68 equations, 8 figures)

This paper contains 50 sections, 2 theorems, 68 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Solvable Model and Geometric Mechanism
  3. Quartic Detectability Law
  4. Numerical and Operational Tests
  5. Beyond the Strict One-Pole Null
  6. Discussion
  7. Appendix A: Supplementary Overview and Literature Positioning
  8. Appendix B: Exact Spectrum and Relative Perturbation
  9. Appendix C: Local Whittle Geometry of the One-Pole Manifold
  10. Relative Coordinates Near the Null Point
  11. Population Whittle Geometry
  12. Orthogonality of the tangent basis
  13. Appendix D: Projection Coefficients and Pseudo-True Shifts
  14. Basic integrals
  15. Projection coefficients
  16. ...and 35 more sections

Key Result

Theorem 1

For each fixed $(a,b)$ with $|a|<1$ and $|b|<1$, there exists $\lambda_0(a,b)>0$ such that for $|\lambda|<\lambda_0(a,b)$ the local Whittle minimizer branch within the one-pole projection class exists uniquely and the associated local minimum satisfies with $C\ge 0$ and

Figures (8)

  • Figure 1: Pole coalescence hides the driver. For this mechanism panel, $a=0.75$, $\lambda=0.12$, and $\sigma_\epsilon^2=\sigma_\eta^2=1$, while the hidden persistence is $b=0.50$, $0.70$, and $0.75=a$ from top to bottom. The exact driven spectrum is compared with the best-fit one-pole null spectrum for separated, near-coalescent, and merged timescales; the merged case corresponds to exact coalescence, $b=a$. The residual spectral shape is pronounced away from coalescence and is strongly suppressed as the hidden and intrinsic poles merge.
  • Figure 2: Detectability is quartic in coupling. Exact numerical minimization of the one-pole projection problem for $a=0.75$, $\sigma_\epsilon^2=\sigma_\eta^2=1$, and $b=0.50$, $0.70$, $0.75=a$. Away from coalescence the quartic residual is visible; timescale matching suppresses it through a vanishing prefactor $C$ (inset).
  • Figure 3: Model selection turns over near the predicted scaled boundary. Whittle-BIC detection probability is plotted against $\lambda/\lambda_c^{\mathrm{pop}}(N)$ for the baseline parameters $(a,b,\sigma_\epsilon^2,\sigma_\eta^2)=(0.95,0.8,1,1)$ and $N=256$--$2048$ ($200$ repetitions per point). The common crossover near unity is consistent with the asymptotic boundary but not with exact finite-size collapse.
  • Figure S1: Auxiliary pseudo-true checks. Left: the leading-order prediction for the best-fit AR pole shift. Right: the corresponding prediction for the best-fit innovation variance shift. Both formulas track the numerical pseudo-true motion of the null model, supporting the projection picture without being part of the principal PRL claim.
  • Figure S2: Supplementary numerical validation. Left: exact numerical Whittle minimization confirms the quartic coupling law. Right: the same population solver confirms the threshold scaling predicted by the closed-form coefficient.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Proposition 1