Establish the tightness of the variational-ladder bounds on fiducial noise models

Determine the residual inefficiency of the variational ladder on the fiducial millimeter/submillimeter noise cells and establish whether the true source-confusion ceilings exceed the reported lower bounds of 3.8 for extended sources and 2.4 for compact sources.

Background

The variational ladder supplies certified lower bounds on the advantage ceiling, while exact ceilings are available only for covariance mixtures and certain calibration models. Its tightness is calibrated at approximately 0.9 using one exactly solvable beam-sharing companion, but that calibration does not establish the remaining gap for the fiducial noise models.

In particular, the source-confusion results are reported as lower bounds rather than exact ceilings. The paper explicitly acknowledges that the residual inefficiency of the ladder on the fiducial cells is unresolved and that the actual confusion ceilings could be higher. A solution would require either a tighter variational representation, an independent upper bound, or an estimator that closes the gap.

References

The ladder's values are lower bounds whose tightness is calibrated at $\sim 0.9$ on one exactly solvable model (\S\ref{sec:sensitivity}); on the fiducial cells the residual inefficiency is unknown and the confusion ceilings may sit higher than the quoted $3.8$ and $2.4$.

— Beyond the BLUE I: the advantage ceiling - how much can any estimator beat the matched filter in mm/submm survey data?  (2609.10475 - Basu, 9 Sep 2026) in Section 6, Discussion, subsection 6.3, “Limitations and scope”