p-adic convergence of strip-transfer approximants

Determine whether the quantities \(\beta(\widehat M_d(n))\), obtained as the \(p\)-adic Teichmüller-trace limits of the strip transfer matrices \(\widehat M_d(n)\), converge \(p\)-adically as \(n\to\infty\).

Background

The paper associates to each finite strip transfer matrix $\widehat M_d(n)$ a canonical $p$-adic quantity $\beta(\widehat M_d(n))$, defined from the eventual $p$-adic limit of traces of its $p^j$-th powers. These are $p$-adic analogues of finite-size spectral approximations to the hard-core entropy constant.

It is unresolved whether these finite-size $p$-adic invariants themselves converge as the strip width grows. Such convergence would provide a strip-based construction of a $p$-adic limit for $\kappa_d$ independent of the torus-based companion $\kappa_d^{(p)}$.