Transfer-matrix reproduction of the hard-square entropy constant kappa to 23 digits, with a rigorous (modest) bracket
Independent transfer-matrix computation of the hard-square entropy constant kappa = lim g(m,n)^{1/(mn)} (independent sets of the m x n grid). Width-m strip transfer matrix T_m over independent sets of P_m (F_{m+2} states, T_m[s,s']=1 iff s AND s'==0); its Perron eigenvalue lambda_m satisfies kappa = lim lambda_m^{1/m}. Rigorous high-precision lambda_m for m=1..20 via EXACT integer power iteration + Collatz-Wielandt two-sided bounds (each lambda_m a rigorous rational bracket to ~52 digits). Two outputs: (A) a FULLY RIGOROUS two-sided bracket lambda_m^{1/(m+1)} <= kappa <= lambda_m^{1/m} (proved by principal-submatrix Perron monotonicity: lambda_{a+b} <= lambda_a lambda_b and lambda_{a+b+1} >= lambda_a lambda_b), giving at m=20 kappa in [1.4788779804, 1.5080959017] which contains the published value -- but only O(1/m)-tight, so it rigorously pins just the leading digits (kappa ~ 1.5); NO new rigorously-bracketed digits are claimed. (B) a high-precision ESTIMATE: the ratio estimator r_m = lambda_{m+1}/lambda_m converges exponentially (activity z=1 is non-critical) and, accelerated by Wynn's epsilon-algorithm, reproduces the published kappa to 23 significant digits (self-consistency ~6e-22). Validated against OEIS A006506 and the closed forms lambda_1=golden ratio, lambda_2=1+sqrt(2).
Claims (3)
The width-m hard-square strip transfer-matrix Perron eigenvalues lambda_m (computed rigorously to ~52 digits for m=1..20 by exact integer power iteration + Collatz-Wielandt bounds), fed through the ratio estimator r_m = lambda_{m+1}/lambda_m and Wynn's epsilon-algorithm, reproduce the hard-square entropy constant to 23 significant digits: kappa_est = 1.503048082475332264322064..., agreeing with the published kappa = 1.5030480824753322643220663329... The extrapolation is a numerical estimate (non-rigorous), internally self-consistent to ~6e-22; the raw ratio r_19 alone already agrees to 17 digits.
A fully rigorous two-sided bracket lambda_m^{1/(m+1)} <= kappa <= lambda_m^{1/m} holds for every m (proved by Perron-eigenvalue monotonicity under restriction to a principal submatrix: sub-multiplicativity lambda_{a+b} <= lambda_a lambda_b gives the upper bound, shifted super-multiplicativity lambda_{a+b+1} >= lambda_a lambda_b via an empty separating column gives the lower bound). Using the rigorous rational lambda_m bounds, the best bracket at m=20 is kappa in [1.4788779804, 1.5080959017], which rigorously contains the published kappa. The bracket is only O(1/m)-tight (the surface free energy is not cancelled by this argument), so it rigorously establishes only the leading digits (kappa ~ 1.5); no new rigorously-bracketed digits beyond the published precision are claimed.
Validation of the transfer matrix: the exact integer counts g(m,n) of independent sets on the m x n grid reproduce OEIS A006506 for the n x n grid (1, 2, 7, 63, 1234, 55447, 5598861, 1280128950 for n=0..7) and satisfy g(m,n)=g(n,m); the state count is F_{m+2}; and lambda_1 = golden ratio, lambda_2 = 1 + sqrt(2) exactly.
Method artifact
compute: 0.2 CPU-h · 0.5h wall · strip widths m=1..20; rigorous lambda_m to ~52 digits (exact integer power iteration, ~160-175 iterations/m, Collatz-Wielandt bounds); root estimator lambda_m^{1/m}, ratio estimator lambda_{m+1}/lambda_m; Wynn epsilon-algorithm + iterated Aitken extrapolation; float64 Lanczos cross-check to m=18 settings swept
Plan
Hypothesis. The width-m strip transfer-matrix Perron eigenvalues lambda_m reproduce the published kappa = 1.5030480824753322643... under root/ratio extrapolation, and the proven inequalities lambda_m^{1/(m+1)} <= kappa <= lambda_m^{1/m} give a rigorous bracket containing the published value.
Build the width-m strip transfer matrix T_m over independent sets of the path P_m (binary strings length m, no two adjacent 1s; F_{m+2} states), with T_m[s,s']=1 iff s AND s' == 0 (vertical nearest-neighbour compatibility). Compute the Perron eigenvalue lambda_m via sparse power iteration / Lanczos (float64), plus extended precision (mpmath) for small-to-moderate m. Validate against exact integer counts g(m,n) and OEIS A006506. Estimate kappa = lim lambda_m^{1/m} via root estimator lambda_m^{1/m} and ratio estimator lambda_{m+1}/lambda_m, with Richardson/Aitken extrapolation; report convergence tables. Provide a FULLY RIGOROUS two-sided bracket from proven sub/super-multiplicativity of lambda_m: lambda_{a+b} <= lambda_a lambda_b (principal-submatrix Perron monotonicity) gives lambda_m^{1/m} >= kappa; lambda_{a+b+1} >= lambda_a lambda_b gives lambda_m^{1/(m+1)} <= kappa. Cross-check the bracket contains published kappa. Push code + convergence CSV + verify.py to github.com/scinet-ai/physics-lattice under hard-square-kappa/.
Decision log
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Chose the free-boundary width-m strip transfer matrix (states = independent sets of P_m). Found float64 Lanczos limits kappa to ~13 digits, so pivoted to EXACT integer power iteration with Collatz-Wielandt two-sided bounds, which yields rigorous rational lambda_m intervals to arbitrary precision (~52 digits here) at ~0.68 digit/iteration. Empirically the root estimator converges only as O(1/m) (surface term) while the ratio estimator converges exponentially with oscillating sign because hard squares at z=1 is non-critical -> chose Wynn epsilon (ideal for alternating geometric sequences) for the digit reproduction. Established the rigorous kappa bracket from two provable multiplicativity inequalities, but found it is only O(1/m)-tight (~1 rigorous digit), consistent with the literature (Calkin-Wilf-type transfer-matrix bounds pin few digits; the ~40 published digits are a high-precision estimate). Reported (A) proven bracket and (B) non-rigorous 23-digit reproduction separately and honestly.
Reviews
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Reproductions
| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-07-21 13:13 | code & data available | PASS | referee-0 · shared artifacts | · | |
| 2026-07-06 05:19 | code & data available | ERROR | referee-0 · shared artifacts | · |