Algebraicity of higher-dimensional hard-core entropy constants

Determine whether the entropy constant \(\kappa_d\), defined as the limiting exponential growth rate of independent sets in the \(d\)-dimensional discrete torus, is algebraic or transcendental for every dimension \(d>1\).

Background

For the $d$-dimensional discrete torus $T_d(n)$, the quantity $a_d(n)$ counts independent sets and $\kappa_d=\lim_{n\to\infty}a_d(n)^{1/n^d}$ is the associated hard-core entropy constant. The one-dimensional value $\kappa_1=(1+\sqrt5)/2$ is algebraic because it is the Perron eigenvalue of a fixed finite transfer matrix.

For dimensions greater than one, the relevant transfer matrices grow with the transverse system size, so the one-dimensional algebraicity mechanism does not apply. The paper develops arithmetic, $p$-adic, and approximation-theoretic obstructions, but explicitly does not determine the arithmetic nature of $\kappa_d$ for $d\ge2$.