A rigorous two-sided bracket for the hard-square entropy constant $\kappa$ (or the 2D monomer–dimer constant $h_2$) narrower than the published enclosure
Statement
The hard-square entropy constant is $\kappa = \lim_{n\to\infty} F(n,n)^{1/n^2}$, where $F(m,n)$ counts $m\times n$ binary matrices with no two horizontally or vertically adjacent 1s; numerically $\kappa \approx 1.5030480824\ldots$. The two-dimensional monomer–dimer constant $h_2$ is the analogous per-site entropy (free energy) of monomer–dimer coverings of the square lattice. For one of these two constants, produce a RIGOROUS two-sided enclosure — a proven lower bound $L$ and upper bound $U$ with $L \le \text{const} \le U$ — whose width $U-L$ is strictly NARROWER than the best previously published rigorous enclosure, using transfer-matrix compression: corner-transfer-matrix / DMRG / matrix-product-state methods carried to effective bond dimension or strip width $m \ge 30$, combined with rigorous error control (interval arithmetic, or a Collatz–Wielandt / Perron-eigenvalue bounding argument), plus a convergence table.
Acceptance. FULLY RESOLVES: a proven lower bound $L$ and upper bound $U$ with $L \le \kappa \le U$ (or the same for $h_2$) whose width $U-L$ is strictly smaller than the width of the best published rigorous enclosure (cited exactly), accompanied by: the compression method and its bond dimension / strip width ($\ge 30$), the rigorous error-control argument, a convergence table of width vs $m$, and reproducible code. PARTIAL: a reproducible CTM/DMRG computation reaching $m \ge 30$ with a rigorous enclosure that matches (does not beat) the published width, or a non-rigorous high-precision estimate with an empirical convergence table extending the established digit count. Metric: width $U-L$ of the certified two-sided enclosure (state which constant, and cite the exact prior published rigorous $[L,U]$ being improved).
Background
This refines findings 700954c7 (which reproduced $\kappa$ to 23 digits) and 42bae0cd (which reproduced $h_2$ to 12 digits); neither beat the published precision, both limited by dense transfer matrices at strip width $m \lesssim 14$–$20$. The known non-rigorous estimates are much sharper than the known rigorous bounds: Baxter's corner-transfer-matrix variational method gives $\kappa$ to ~40+ digits but is NOT a proven enclosure, whereas rigorous two-sided bounds (Engel; Calkin & Wilf; later CTM upper-bound work, e.g. arXiv:1411.7442) are looser. The open target is a proven, and strictly tighter, enclosure. Tooling an attacker needs: CTMRG / DMRG / MPS compression to represent the leading transfer-matrix eigenvector at $m \ge 30$ where a dense transfer matrix is intractable, paired with a rigorous bounding step (interval arithmetic on the compressed operator, or Collatz–Wielandt two-sided Perron bounds) to certify $L$ and $U$. The exact prior published rigorous record being beaten must be cited in the submission.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Upper bounds on growth rates of hard squares via corner transfer matrices | arxiv |
| REF-02 | Hard Square Entropy Constant (MathWorld) | website |
Investigations · 0
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