Extend the high-precision value of the hard-square entropy constant $\kappa$

The hard-square model is the lattice gas on $\mathbb{Z}^2$ in which occupied sites may not be nearest neighbours (nearest-neighbour exclusion). Let $g(m,n)$ be the number of valid configurations (independent sets) on an $m\times n$ grid; the hard-square entropy constant is $\kappa = \lim_{m,n\to\infty} g(m,n)^{1/(mn)}$ (equivalently the $z=1$ per-site partition function). Unlike the hard-HEXAGON model (solved exactly by Baxter), hard squares has no known closed form. GOAL: compute more rigorously-bracketed digits of $\kappa$ than currently known, or independently reproduce the known digits, via corner-transfer-matrix / tensor-network (CTMRG, VUMPS) or transfer-matrix numerics with an explicit error/interval estimate.

Acceptance. FULLY RESOLVES: report additional correct digits of $\kappa$ beyond the current published precision, bracketed by a rigorous lower and upper bound (e.g. from monotone CTM sequences), with (i) the code and (ii) a convergence table showing successive bracket widths. PARTIAL: independently reproduce the published digits with a distinct method and a stated interval / error estimate, or compute exact transfer-matrix strip counts $g(m,n)$ for larger widths $m$ (exact integers, matching OEIS A006506 on the overlap). State the method, bond dimension / strip width, and arithmetic precision used.