Cross Spectra Break the Single-Channel Impossibility

Abstract

Lucente et al. proved that no time-irreversibility measure can detect departure from equilibrium in a scalar Gaussian time series from a linear system. We show that a second observed channel sharing the same hidden driver overcomes this impossibility: the cross-spectral block, structurally inaccessible to any single-channel measure, provides qualitatively new detectability. Under the diagonal null hypothesis, the cross-spectral detectability coefficient $\Scross$ (the leading quartic-order cross contribution) is \emph{exactly} independent of the observed timescales -- a cancellation governed solely by hidden-mode parameters -- and remains strictly positive at exact timescale coalescence, where all single-channel measures vanish. The mechanism is geometric: the cross spectrum occupies the off-diagonal subspace of the spectral matrix, orthogonal to any diagonal null and therefore invisible in any single-channel reduction. For the one-way coupled Ornstein--Uhlenbeck counterpart, the entropy production rate (EPR) satisfies $\EPRtot=α_2λ^2$ exactly; under this coupling geometry, $\Scross>0$ certifies $\EPRtot>0$, linking observable cross-spectral structure to full-system dissipation via $\EPRtot^{\,2}\propto\Scross$. Finite-sample simulations predict a quantitative detection-threshold split testable with dual colloidal probes and multisite climate stations.

Figures (10)

• Figure 1: Population mechanism and boundary characterization. Panel A: fractional cross-term dominance $C_{\mathrm{cross}}/C_{\mathrm{total}}$ across the $(a_1,a_2)\in[-0.9,0.9]^2$ parameter plane, evaluated from the closed-form coefficients on a dense mesh. Near the coalescence point $a_1=a_2=b$ (dashed lines, $b=0.7$), the auto contributions vanish and the cross term accounts for nearly all detectability---a direct visualization of the singularity removal. Panel B: along the coalescence path $a_1=a_2=b+\delta$, the inherited auto contribution collapses while the cross term remains finite and dominant. Panel C: only the aligned enriched branch ($c=b$) fully absorbs the coalescent cross residual; curves shown for $b=0.5,0.7,0.85$ on a refined $c$ grid.
• Figure 2: Detection threshold split under the diagonal null ($b=0.7$, $u_1=u_2=1/\sqrt{2}$, $\sigma_{\epsilon_1}^2=\sigma_{\epsilon_2}^2=\sigma_\eta^2=1$). Left: single-channel reduction. Right: two-channel diagonal reduction. Curves correspond to $N=512,1024,2048$ (line style and marker). The single-channel threshold $\lambda_{50}(\delta)$ rises strongly toward coalescence, while the two-channel threshold remains bounded and much flatter---the finite-sample signature of the coalescence singularity removal proved in Corollary \ref{['cor:coalescence']}.
• Figure S1: Targeted finite-sample controls. Panels A and B compare the single-channel plug-in curve, the two-channel plug-in curve, and a fixed-nuisance semi-oracle two-channel curve at $N=1024$ and $N=2048$, respectively. Panels C and D show the extended asymptotic trend of $r_{50}(N)$ at $\delta=0.10$ and $\delta=0.02$. The supplement therefore distinguishes population-level boundedness from finite-sample extraction cost without changing the main-text theorem statements.
• Figure S2: Hypothesis-class semantics. The two panels report, at $N=1024$ and $N=4096$, how often a one-parameter aligned cross-shape family is preferred over the diagonal family across three representative data-generating processes. The no-cross case stays at zero aligned-family preference, the instantaneous-common-input case is strongly absorbed by the aligned family, and the persistent-hidden-driver case shows intermediate-to-strong preference for the aligned family. This confirms that the two null classes answer different physical questions rather than stronger and weaker versions of the same one.
• Figure S3: Matched-information control. The single-channel and two-channel reductions are compared both with plug-in nuisance estimation and with fixed nuisance values set to the true autoregressive poles and innovation scales. This diagnostic equalizes nuisance knowledge across the two reductions, separating signal geometry from extraction cost: the single-channel fixed-nuisance curves still exhibit coalescence blow-up, whereas the two-channel fixed-nuisance curves remain markedly flatter.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1: Cancellation identity
• Theorem 2
• Corollary 1: Coalescence singularity removal
• Theorem 3: Exact EPR for one-way coupled OU
• Corollary 2: EPR--detectability relationship
• Remark 1: Identification hierarchy
• Proposition 1: Projection upper bound
• Proposition 2: Exact-coalescence benchmark