Conceptual replacement for the finite rooted-region certificate

Develop a conceptual inequality that replaces the finite computational certificate used to bound the local free energy of rooted regions in the proof of the case (r,d)=(4,2).

Background

To prove the Balogh–Bollobás–Narayanan conjecture for (r,d)=(4,2), the paper handles certain rooted regions by an exhaustive finite computation. The rooted-region certificate verifies an inequality for all configurations satisfying specified degree and multiplicity constraints, using exact integer arithmetic.

The authors explicitly leave open the task of replacing this computational verification with a conceptual inequality, which would provide a more structural proof of the local free-energy bound.