Existence of bi-cover certificates for the (3,7) and (7,3) tilings

Determine whether the regular tilings with parameter pairs (3,7) and (7,3) admit certificates with bi-covers, rather than merely certificates.

Background

The paper introduces finite (p,q)-certificates consisting of permutation data and a cocycle; such certificates yield monic cocycles on the corresponding regular hyperbolic tilings and consequently prove that the adjacency operator has no nonzero square-integrable eigenfunctions. A bi-cover is an additional combinatorial structure consisting of compatible vertical and horizontal covers, and it is required for the vertical, horizontal, and diagonal sewing operations used to generate certificates for larger parameter pairs.

The existence proof establishes certificates with bi-covers for most terminal parameter pairs. For the exceptional pairs (3,7) and (7,3), the paper supplies explicit certificates but does not establish that they admit bi-covers. The authors therefore cannot apply the same sewing framework to these cases in the general construction, leaving the existence of bi-cover certificates for these two tilings unresolved.