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arxiv: 2603.20917 · v2 · submitted 2026-03-21 · ❄️ cond-mat.stat-mech

Timescale Coalescence Makes Hidden Persistent Forcing Spectrally Dark

Yuda Bi , Chenyu Zhang , Vince D Calhoun This is my paper

Pith reviewed 2026-05-15 07:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords timescale coalescencespectral inferencehidden forcingAR(1) processWhittle distancemodel manifolddark regimepower spectrum
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The pith

When a hidden AR(1) driver shares the observed process timescale, the mismatch to the best one-pole spectral model scales as lambda to the fourth power rather than lambda squared.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a driven AR(1) process observed at coarse resolution and shows that the local Kullback-Leibler distance to the closest one-pole surrogate grows as C lambda^4 plus higher-order terms. This quartic onset occurs because the leading quadratic perturbation in the true spectrum can be absorbed by a small reparametrization along the tangent space of the reduced model manifold. The coefficient C is obtained in closed form and drops to zero proportionally to the square of the pole difference precisely at timescale coalescence, creating a spectrally dark regime in which the persistent forcing remains dynamically active yet locally invisible to one-pole inference. For drivers with richer AR(2) structure the coefficient stays positive for all parameter values, including coalescence, showing that the darkness is not automatic but requires exact pole-structure matching with the null family.

Core claim

In a solvable driven AR(1) benchmark the local Whittle distance from the true spectrum to the best nearby one-pole surrogate obeys D_loc(lambda) = C lambda^4 + O(lambda^6) even though the observed spectrum is perturbed at O(lambda^2). The quartic coefficient C vanishes as (a-b)^2 at timescale coalescence between hidden driver and observed process. For a non-degenerate AR(2) hidden driver the coefficient remains strictly positive for all parameters because the additional spectral structure cannot be absorbed by the two-dimensional tangent space of the one-pole family. The quartic law and the associated detection boundary lambda_pop(N) proportional to (log N / N)^{1/4} are therefore universal,

What carries the argument

Local Whittle/Kullback-Leibler distance on the one-pole model manifold, whose tangent space absorbs O(lambda^2) spectral perturbations only when the hidden driver shares the null pole structure.

If this is right

  • The dark regime boundary scales as lambda_pop(N) proportional to (log N / N)^{1/4}.
  • For AR(1) hidden drivers the quartic coefficient vanishes exactly at coalescence, rendering the forcing locally undetectable.
  • For AR(2) hidden drivers with nonzero second root the coefficient stays positive even at coalescence.
  • The quartic law holds universally inside the one-pole projection class whenever the hidden spectrum matches the null pole structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Standard one-pole spectral methods may systematically miss persistent forcing in systems whose internal timescales naturally coalesce.
  • Replacing the null model with a two-pole family would lift the dark regime for AR(2)-like drivers and restore quadratic sensitivity.
  • The same geometric absorption argument could be tested in higher-order autoregressive or state-space models to map the parameter region where hidden drivers remain invisible.

Load-bearing premise

The hidden driver must be an AR(1) process whose spectrum matches the null family's pole structure so that the O(lambda^2) perturbation can be absorbed by tangent-space reparametrization of the one-pole surrogate.

What would settle it

Compute the local KL distance for a driven AR(1) process with coalesced poles across a range of small lambda and verify whether the leading term is quartic; a clearly superior quadratic fit persisting to arbitrarily small lambda would falsify the claimed onset.

Figures

Figures reproduced from arXiv: 2603.20917 by Chenyu Zhang, Vince D Calhoun, Yuda Bi.

Figure 1
Figure 1. Figure 1: FIG. 1. Pole coalescence hides the driver. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Detectability is quartic in coupling. Population-level [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Model selection turns over near the predicted [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. No coalescence suppression for the AR(2) driver. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes spectral inference under coarse observation for driven AR(1) and AR(2) processes. For a solvable AR(1) hidden driver, it claims that the local Whittle/KL distance D_loc(λ) from the true spectrum to the best nearby one-pole surrogate scales as Cλ^4 + O(λ^6), even though the observed spectrum is perturbed at O(λ^2). This quartic onset arises geometrically because the O(λ^2) perturbation is absorbed by tangent-space reparametrization on the one-pole projection manifold, leaving only a normal residual; the coefficient C is obtained in closed form and vanishes as (a-b)^2 at timescale coalescence, defining a spectrally dark regime. For non-degenerate AR(2) drivers (z2 ≠ 0), C > 0 for all parameters, with smooth interpolation C ~ z2^4 as z2 → 0. The results classify the quartic law and boundary lcpop(N) ∝ (log N/N)^{1/4} as universal within the one-pole projection class, while the dark regime requires spectral matching to the null family's pole structure.

Significance. If the geometric derivations hold, the work supplies a precise, closed-form account of when hidden persistent forcing becomes locally invisible to reduced spectral models. The explicit AR(1) versus AR(2) contrast demonstrates that the dark regime is not an inevitable feature of the manifold but depends on pole-structure matching, providing a falsifiable classification. The parameter-free character of the leading-order coefficient C and the explicit scaling predictions strengthen the result's utility for time-series model selection and inference diagnostics.

major comments (2)
  1. [Derivation of C (AR(1) case)] The central derivation of the closed-form C for the AR(1) case (abstract and §3) rests on the local KL expansion and the decomposition into tangent and normal components of the one-pole manifold. The explicit algebraic steps showing how the O(λ²) spectral perturbation is absorbed by reparametrization, leaving a normal residual beginning at O(λ⁴), are not reproduced in the provided text; these steps are load-bearing for the claim that D_loc(λ) = Cλ^4 + O(λ^6) and for the vanishing of C as (a-b)².
  2. [AR(2) case and interpolation] §4, AR(2) comparison: the statement that the richer spectral structure cannot be absorbed by the two-dimensional tangent space, yielding C > 0 even at coalescence, requires the explicit normal-residual calculation for z2 ≠ 0 and the interpolation C ∼ z2^4. Without these steps, the claim that the dark regime is not geometrically inevitable remains unverified.
minor comments (2)
  1. [Abstract and §2] Notation: Dloc(λ) should be rendered consistently as D_loc(λ) or D_{loc}(λ) throughout; the current mixed form reduces readability.
  2. [§2] The definition of the local Whittle/KL distance and the precise normalization of the one-pole surrogate family should be restated once in §2 before the geometric argument begins.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments on our manuscript. We address each major comment below and will revise the text accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Derivation of C (AR(1) case)] The central derivation of the closed-form C for the AR(1) case (abstract and §3) rests on the local KL expansion and the decomposition into tangent and normal components of the one-pole manifold. The explicit algebraic steps showing how the O(λ²) spectral perturbation is absorbed by reparametrization, leaving a normal residual beginning at O(λ⁴), are not reproduced in the provided text; these steps are load-bearing for the claim that D_loc(λ) = Cλ^4 + O(λ^6) and for the vanishing of C as (a-b)².

    Authors: We agree that the explicit algebraic steps for the local KL expansion, tangent-space reparametrization, and normal-residual computation were summarized at a high level in §3 rather than expanded in full. In the revised manuscript we will insert the complete derivation: begin with the spectral perturbation S(ω;λ) = S₀(ω) + λ² δS(ω) + O(λ⁴), project onto the one-pole manifold, solve for the tangent correction that absorbs the O(λ²) term, and compute the leading O(λ⁴) normal residual that yields the closed-form C together with its (a-b)² prefactor at coalescence. This addition will make the quartic onset and the dark-regime condition fully self-contained. revision: yes

  2. Referee: [AR(2) case and interpolation] §4, AR(2) comparison: the statement that the richer spectral structure cannot be absorbed by the two-dimensional tangent space, yielding C > 0 even at coalescence, requires the explicit normal-residual calculation for z2 ≠ 0 and the interpolation C ∼ z2^4. Without these steps, the claim that the dark regime is not geometrically inevitable remains unverified.

    Authors: We acknowledge that the AR(2) normal-residual calculation and the z₂⁴ interpolation were stated without the intermediate algebra in §4. In revision we will add the explicit expansion: for z₂ ≠ 0 the additional spectral features generated by the second characteristic root lie outside the two-dimensional tangent space of the one-pole family, producing a nonzero normal component whose leading coefficient is strictly positive; we then expand this coefficient in powers of z₂ and recover the smooth interpolation C ∼ z₂⁴ as z₂ → 0. These steps will confirm that the dark regime is not an automatic geometric feature but requires pole-structure matching. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from model equations and manifold geometry

full rationale

The paper presents the quartic law D_loc(λ)=Cλ^4+O(λ^6) as a direct geometric consequence of the one-pole projection manifold: an O(λ²) perturbation from the AR(1) hidden driver is absorbed by tangent-space reparametrization, leaving a normal residual whose leading coefficient C is obtained in closed form and shown to vanish as (a-b)² at coalescence. The result is immediately contrasted with the AR(2) case (C>0 for all parameters, including coalescence) to establish that the dark regime is not generic. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation is independent of the target result and follows from the stated benchmark equations and manifold geometry.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on the analytical solvability of the driven AR(1) benchmark and the geometric structure of the one-pole model manifold; these are standard domain assumptions in time-series spectral analysis but are load-bearing for the quartic law and dark-regime identification.

free parameters (1)
  • AR(1) characteristic roots a and b
    The coefficient C is expressed in terms of (a-b)^2; these are model parameters whose difference controls the dark regime.
axioms (3)
  • domain assumption The driven AR(1) benchmark is solvable analytically for its spectrum.
    Invoked to obtain the true spectrum perturbation at O(λ^2) and the closed-form local distance.
  • domain assumption The reduced inference class is the two-dimensional manifold of one-pole surrogates.
    Defines the tangent-space reparametrization that absorbs the O(λ^2) term, leaving the normal residual at O(λ^4).
  • standard math Local distance is measured by Whittle or Kullback-Leibler divergence.
    Standard choice for spectral approximation quality; used to define D_loc(λ).

pith-pipeline@v0.9.0 · 5615 in / 1794 out tokens · 70698 ms · 2026-05-15T07:05:01.595726+00:00 · methodology

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    Projection coefficients The basic projection integrals are ⟨h,˜e1⟩= σ2 η σ2ϵ 1 1−b 2 ,⟨h,˜e 2⟩= σ2 η σ2ϵ 2b (1−ab)(1−b 2) .(S3) The second coefficient comes from a contour integral on the unit circle whose poles lie atz=aandz=b. The exact residue sum collapses to the rational form in Eq. (S3)

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  59. [59]

    Local pseudo-true shifts They imply the local pseudo-true shifts ˜σ2∗(λ) =σ 2 ϵ +λ 2 σ2 η 1−b 2 +O(λ 4),(S4) and ˜a∗(λ) =a+λ 2 σ2 η σ2ϵ b(1−a 2) (1−ab)(1−b 2) +O(λ 4).(S5) These shifts are auxiliary to the main theorem but provide a direct check that the best null model moves along the one-pole manifold exactly as predicted by the projection geometry. 12 ...

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  60. [60]

    Statement proved here In this appendix we prove Theorem 1. Equivalently, for|a|<1 and|b|<1, there existsλ 0(a, b)>0 such that for |λ|< λ 0(a, b) the local minimizer branch exists uniquely and Dmin KL,loc(λ) =C(a, b, σ ϵ, ση)λ4 +O(λ 6),(S1) with C(a, b, σϵ, ση) = σ4 η 2σ4ϵ b2(a−b) 2 (1−b 2)3(1−ab) 2 .(S2) Remark 1.The theorem is pointwise in parameter spac...

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  61. [61]

    (S5), together with the positive Gram determinant from Eq

    Existence and uniqueness of the local branch The local quadratic reduction in Eq. (S5), together with the positive Gram determinant from Eq. (S9), implies that the Hessian of the reduced objective is strictly positive definite at the null point. The implicit-function theorem therefore gives a unique local minimizer branch (˜a∗(λ),˜σ2∗(λ)) for sufficiently...

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  62. [62]

    (11), this proves Eq

    Projection formula and residual norm Let R=h−Π T h.(S3) Because the tangent directions are orthogonal, the residual norm is ∥R∥2 =∥h∥ 2 − ⟨h,˜e1⟩2 ∥˜e1∥2 − ⟨h,˜e2⟩2 ∥˜e2∥2 .(S4) 13 Substituting the exact projection integrals yields ∥R∥2 = σ4 η σ4ϵ 2b2(a−b) 2 (1−b 2)3(1−ab) 2 .(S5) Together with Eq. (11), this proves Eq. (S1). Positivity is immediate becau...

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  63. [63]

    It does not prove that the same expansion controls the global infimum over the full stationary one-pole class

    What this appendix does and does not prove The theorem above identifies the nearby pseudo-true branch and the asymptotic law for its local minimum. It does not prove that the same expansion controls the global infimum over the full stationary one-pole class. That stronger statement is instead checked numerically in Appendix I by exact global population mi...

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  64. [64]

    Coalescence law The quartic coefficient factorizes as C(a, b, σϵ, ση) = σ4 η 2σ4ϵ b2 (1−b 2)3(1−ab) 2 (a−b) 2,(S1) so C∝(a−b) 2 asb→a.(S2)

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  65. [65]

    (18) for the effective two-pole alternative with ∆k= 2

    Population boundary The corresponding leading-order population boundary is λpop c (N)= ∆k σ4 ϵ (1−b 2)3(1−ab) 2 σ4ηb2(a−b) 2 logN N 1/4 (1 +o(1)),(S3) which reduces to Eq. (18) for the effective two-pole alternative with ∆k= 2

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  66. [66]

    LetMbe a smooth spectral manifold throughS 0

    Abstract geometric criterion The solvable model above isolates a broader local criterion. LetMbe a smooth spectral manifold throughS 0. If the leading perturbation enters as Strue(λ) =S 0(1 +λ 2h+O(λ 4)),(S4) and if the local Whittle Hessian on the tangent spaceTofMis positive definite, then the same quadratic reduction gives a unique local minimizer bran...

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  67. [67]

    It does not prove the first nonzero post-quartic coefficient at strict coalescence, and it does not claim a universal law for arbitrary hidden-variable architectures

    What is not proved here The theorem package establishes the quartic law and its exact prefactor within the one-pole projection class. It does not prove the first nonzero post-quartic coefficient at strict coalescence, and it does not claim a universal law for arbitrary hidden-variable architectures. Appendix G: Enriched Null Families and the ARMA(1,1)Corollary

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  68. [68]

    Thus the tangent space is enlarged to T(1,1) = span{˜e1,˜e2,˜e3}.(S4)

    ARMA(1,1)one-pole null family The simplest enriched null class keeps a single relaxation pole but adds one moving-average zero: S(1,1) null (ω; ˜a, θ,˜σ2) = ˜σ2 Pθ(ω) P˜a(ω) .(S1) At the AR(1) null point (˜a, θ,˜σ2) = (a,0, σ 2 ϵ ), the relative expansion becomes S(1,1) null (ω; ˜a, θ,˜σ2) S0(ω) = 1 +u˜e1 +v˜e2 +w˜e3 +O(u 2 +v 2 +w 2 +uv+uw+vw),(S2) with ...

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  69. [69]

    The new projection coefficient is also explicit: ⟨h,˜e3⟩=− 2b σ2 η σ2ϵ (1−b 2) .(S7) Together with Eq

    Gram matrix and projection data The extra tangent direction has the exact inner products ⟨˜e1,˜e3⟩= 0,⟨˜e 2,˜e3⟩=−2,∥˜e 3∥2 = 2.(S5) Hence, for generica̸= 0, the three-dimensional Gram matrix is G(1,1) =   1 0 0 0 2 1−a 2 −2 0−2 2   ,detG (1,1) = 4a2 1−a 2 >0.(S6) The nongeneric pointa= 0 is degenerate because ˜e 2 and ˜e3 are collinear there (˜e2...

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  70. [70]

    Closed quartic coefficient Proposition 1.For generica̸= 0with|a|<1and|b|<1, local Whittle minimization over the ARMA(1,1)one-pole null family gives Dmin KL,loc,(1,1)(λ) =C (1,1)(a, b, σϵ, ση)λ4 +O(λ 6),(S8) with the exact coefficient C(1,1) = σ4 η 2σ4ϵ b4(a−b) 2 (1−b 2)3(1−ab) 2 =b 2C.(S9) 15 The proof is the same projection argument as in Appendix E, but...

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  71. [71]

    Symbolic identities checked exactly A separate symbolic pipeline verifies, without numerical substitution:

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  72. [72]

    the contour-integral identity that gives Eq. (S3)

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  73. [73]

    the exact residual norm in Eq. (S5)

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  74. [74]

    the factorization of the coalescence term (a−b) 2

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  75. [75]

    the positivity of the Gram determinant from Eq. (S9). These checks close the main algebraic loophole in the theorem package: the quartic coefficient does not rely on a single hand derivation

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  76. [76]

    The coefficient no longer rests on a single contour-integral route; it is fixed both by direct analysis and by an independent algebraic verification chain

    Why this matters The symbolic layer makes the theorem harder to attack. The coefficient no longer rests on a single contour-integral route; it is fixed both by direct analysis and by an independent algebraic verification chain. Appendix I: Exact Numerical KL Validation

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  77. [77]

    Baseline parameter set Unless otherwise stated, the numerical tests use the baseline parameters (a, b, σ2 ϵ , σ2 η) = (0.95,0.8,1,1).(S1) The exact-population validation never inserts the local asymptotic formula into the objective. Instead, for each coupling value we minimize the full Whittle/KL divergence over the stationary one-pole parameter domain an...

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  78. [78]

    For the baseline parameters in Eq

    Quartic and threshold laws Exact population minimization of the Whittle objective over the one-pole class gives a quartic slope of 2.67918 for λ≤0.15. For the baseline parameters in Eq. (S1), the quartic approximation remains accurate to within about 10% forλ≲0.15. SolvingN D min,num KL (λc) = logNoverN∈ {200,400,800,1600,3200,6400,12800}gives a threshold...

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  79. [79]

    At strict coalescence,D min,num KL /λ4 keeps decreasing asλ→0, confirming that the quartic coefficient truly vanishes

    Coalescence law Extracting the quartic coefficient on the windowλ∈ {0.005,0.0075,0.01,0.015}and sweepingbtowardagives a normalized coalescence slope of 1.922526. At strict coalescence,D min,num KL /λ4 keeps decreasing asλ→0, confirming that the quartic coefficient truly vanishes. 16 10□1 Coupling strength, λ 10□6 10□5 10□4 10□3 10□2 D min KL slope 4 solid...

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  80. [80]

    Their motion agrees with Eqs

    Pseudo-true shifts The same exact numerical optimization returns the best-fit one-pole parameters. Their motion agrees with Eqs. (S4) and (S5), providing an additional numerical check of the projection geometry. Appendix J: Operational Monte Carlo Validation

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