Interpolation of the odd part of the period

Determine whether the odd part of the period of the all-ones class, namely $L(n)/2^{\nu_2(L(n))}$, admits a continuous or analytic interpolation on suitable residue classes in the 2-adic integers.

Background

The main theorem gives the exact 2-adic valuation of the period L(n), but says nothing about the odd factor remaining after that power of 2 is removed. The authors compare this issue with known 2-adic interpolation phenomena for related combinatorial sequences.

They ask whether a suitable graph-tower framework could support such an interpolation while retaining a canonical class analogous to the all-ones class and allowing comparison between the order of that class and the order of the entire graph Jacobian.

References

Cohn's theorem suggests asking whether the odd part $L(n)/2{\nu_2(L(n))}$ has a continuous or analytic interpolation on suitable residue classes in $$. Graph analogues of Iwasawa theory describe $p$-primary growth of entire graph Jacobians in voltage $p$-towers and $_\elld$-towers. The wired squares here are not supplied with such a tower structure, so those results do not provide this interpolation. Is there a compatible family of graphs and maps which retains a canonical class analogous to the all-ones class, and for which its order can be studied alongside the order of the whole group?

— The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square  (2609.10625 - Akyar et al., 9 Sep 2026) in Section “Further directions,” paragraph “Interpolation and graph towers”