Necessity of logarithmic inverse-temperature scaling

Determine whether the logarithmic inverse-temperature scaling used by SPICES to guarantee its stability criteria is ultimately necessary, particularly when only coarse-grained features of the single-particle energy density are required.

Background

The SPICES recovery guarantees use an inverse temperature with logarithmic dependence on the system size to ensure contour separation and a nonvanishing normalized signal. The discussion notes that practical applications may seek only coarse-grained spectral information, such as bandgaps, rather than resolving individual energies.

The unresolved issue is whether the logarithmic scaling is an artifact of the present certification argument or remains necessary for the SPICES task, especially in this relaxed coarse-grained setting.

References

We use logarithmic inverse temperature dependence to guarantee these, but it is not clear a priori that this requirement is ultimately necessary.

— Single-Particle Spectral Estimation  (2610.02183 - Chapman et al., 1 Oct 2026) in Section 2, Discussion

Hence 2\le L/\Delta_\Omega<c/2, so the window is wider than the gap by at most a constant factor. This comes from certifying radial separation and the signal floor through the logarithmic-temperature condition, not from \cref{thm:spices-recovery}, which takes q and b as inputs; we do not know whether the condition is necessary.

— Single-Particle Spectral Estimation  (2610.02183 - Chapman et al., 1 Oct 2026) in Remark “Width of the window,” Section 7.2, “Recovery under spectator-sector perturbations”