Develop a criterion for useful enlargement of the test-function family

Identify a criterion that determines in advance when enlarging the family of Barankin test functions—for example, by adding site-potential or other residence-time perturbations—will improve the finite-perturbation variance bound for a nonlinear observable.

Background

The paper enlarges the perturbation family beyond edge-rate tilts by adding site potentials, which substantially improves the bound for the three-state motor but provides almost no improvement in the independent two-channel model. The authors investigate whether the coupling of candidate directions to the observable provides a cheap diagnostic, but this diagnostic fails: stronger coupling does not reliably predict a larger gain.

Consequently, the paper does not provide an a priori method for deciding whether additional test functions will contribute new useful span to the Barankin projection. Establishing such a criterion would reduce the need to enlarge candidate families empirically and measure the resulting bound.