Removal of the finite time horizon under a logarithmic gap

Determine whether the finite time horizon of order 1/|\lambda| required for general interacting spectator-sector perturbations can be removed when the spectator energies satisfy the logarithmic-gap condition.

Background

The paper proves that sufficiently weak spectator-sector interactions preserve the target annular roots only over a finite time horizon, with a certified horizon scaling as Θβ(|λ|{-1}) as the perturbation strength λ tends to zero. A two-mode interacting example demonstrates order-one root motion on this time scale in a weak-gap regime.

The authors state that this example does not resolve whether the finite horizon is necessary when the gap is logarithmic in the system size. Establishing or disproving removal of the horizon in that regime remains open.

References

Whether the horizon can be lifted entirely under the logarithmic-gap condition is not settled by that example, and we leave it open.

— Single-Particle Spectral Estimation  (2610.02183 - Chapman et al., 1 Oct 2026) in Section 7.2, “Recovery under spectator-sector perturbations,” immediately after Corollary “Stability in the logarithmic-gap regime”