Tameness criterion for primes in the two-dimensional case

Determine whether the second summand in the criterion for \(\nu_p(a_2(p)-1)=1\), namely \(\sum_{r\ne\pm1}j_r\) modulo \(p\), can cancel the Wall–Sun–Sun contribution \(2\ell\), thereby deciding tameness of primes \(p\) for the two-dimensional hard-square entropy problem when \(\nu_p(a_2(p)-1)\ge2\).

Background

For $d=2$, the paper expresses the number of translation orbits of independent sets of size $p$ in terms of Lucas numbers and independence numbers of circulant graphs with connection sets $\{\pm1,\pm r\}$. Writing $L_p=1+p\ell$ and the relevant circulant independence numbers as $1+pj_r$, the valuation $\nu_p(a_2(p)-1)$ equals one precisely when $2\ell+\sum_{r\ne\pm1}j_r\not\equiv0\pmod p$.

The Wall–Sun–Sun-related quantity $\ell$ is only one contribution. The paper states that the circulant contribution is structurally independent and may cancel it, so knowledge of whether $p$ is or is not Wall–Sun–Sun does not by itself resolve tameness in dimension two.