Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913006600847360 |
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| author | Bi, Yuda Zhang, Chenyu Calhoun, Vince D |
| author_facet | Bi, Yuda Zhang, Chenyu Calhoun, Vince D |
| contents | 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 $\Dloc(λ)=Cλ^4+O(λ^6)$, even though the observed spectrum itself is perturbed at $O(λ^2)$. The quartic onset is a geometric consequence of the reduced model manifold: the $O(λ^2)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark Bi, Yuda Zhang, Chenyu Calhoun, Vince D Statistical Mechanics 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 $\Dloc(λ)=Cλ^4+O(λ^6)$, even though the observed spectrum itself is perturbed at $O(λ^2)$. The quartic onset is a geometric consequence of the reduced model manifold: the $O(λ^2)$ 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. |
| title | Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2603.20917 |