The hard-square entropy constant

Find a closed form for the hard-square entropy constant κ ≈ 1.5030480825, the growth rate of independent sets in the square grid.

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State the problem, from HorizonMath

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	1	+# The hard-square entropy constant
	2	+
	3	+How many ways are there to place tokens on an $n \times n$ grid so that no two tokens are side by side? The count grows like $\kappa^{n^2}$ with $\kappa \approx 1.5030$. Its triangular-lattice cousin, the hard-hexagon model, was solved exactly by Baxter, but $\kappa$ itself has no known closed form.
	4	+
	5	+## Problem
	6	+
	7	+Let $F(m,n)$ be the number of $m \times n$ matrices with entries in $\{0,1\}$ in which no two 1s are horizontally or vertically adjacent. Equivalently, $F(m,n)$ counts the independent sets of the $m \times n$ grid graph. The hard-square entropy constant is
	8	+
	9	+$$
	10	+\kappa = \lim_{n \to \infty} F(n,n)^{1/n^2} = 1.50304808247533226432\ldots
	11	+$$
	12	+
	13	+The double limit $\lim_{m,n\to\infty} F(m,n)^{1/(mn)}$ exists and has the same value [1]. Find a closed-form expression for $\kappa$: a finite combination of algebraic numbers, standard constants and special functions such as $\Gamma$, hypergeometric functions, elliptic integrals and polylogarithms. Transfer-matrix eigenvalues, truncated series and infinite products without a closed evaluation do not count.
	14	+
	15	+## Background
	16	+
	17	+In statistical mechanics this is the hard-core lattice gas on $\mathbb{Z}^2$: particles on lattice sites, with nearest neighbours excluded. At unit activity every allowed configuration has weight 1, so $\kappa$ is the partition function per site and $\log \kappa \approx 0.40749$ is the entropy per site [3]. In symbolic dynamics, $\log\kappa$ is the topological entropy of the hard-square shift [4]. Baxter solved the hard-hexagon model, the same exclusion rule on the triangular lattice, exactly in 1980 [2]. No exact solution of the square-lattice model is known.
	18	+
	19	+## What is known
	20	+
	21	+- $\kappa = 1.5030480824753322643220663294755536893857810\ldots$. Baxter (1999) computed the partition function per site at unit activity to 43 decimal places with corner transfer matrices [3]. These digits are listed in [OEIS A085850](https://oeis.org/A085850), and HorizonMath uses them as its reference value.
	22	+- Calkin and Wilf (1998) proved that the double limit exists and that $1.503047782\ldots \le \kappa \le 1.5035148\ldots$ [1].
	23	+- Pavlov (2012) proved that differences of strip entropies converge to $\log\kappa$ at least exponentially fast. Hence $\log\kappa$ can be computed to within $1/n$ in time polynomial in $n$. He also noted that no closed form is known [4].
	24	+
	25	+## What would count as progress
	26	+
	27	+- A closed-form expression that matches $\kappa$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants. A candidate that also reproduces all 43 published decimals would be compelling evidence.
	28	+- A proof that some expression equals $\kappa$, for instance through an exact solution of the hard-square model at unit activity.
	29	+
	30	+## References
	31	+
	32	+1. N. J. Calkin and H. S. Wilf, "The number of independent sets in a grid graph", SIAM J. Discrete Math. 11 (1998), 54–60. [doi:10.1137/S089548019528993X](https://doi.org/10.1137/S089548019528993X)
	33	+2. R. J. Baxter, "Hard hexagons: exact solution", J. Phys. A 13 (1980), L61–L70. [doi:10.1088/0305-4470/13/3/007](https://doi.org/10.1088/0305-4470/13/3/007)
	34	+3. R. J. Baxter, "Planar lattice gases with nearest-neighbour exclusion", Ann. Comb. 3 (1999), 191–203. [doi:10.1007/BF01608783](https://doi.org/10.1007/BF01608783), [arXiv:cond-mat/9811264](https://arxiv.org/abs/cond-mat/9811264)
	35	+4. R. Pavlov, "Approximating the hard square entropy constant with probabilistic methods", Ann. Probab. 40 (2012), 2362–2399. [doi:10.1214/11-AOP681](https://doi.org/10.1214/11-AOP681)
	36	+
	37	+## Source
	38	+
	39	+Imported from [HorizonMath hard_square_entropy](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L704-L714), curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.