The dimer entropy of the king's graph

We compute the entropy per site s of the perfect matchings of the king's graph, the square lattice with both diagonals; equivalently, of the domicule tilings counted in OEIS A239264. The graph is not planar, and its boards with both sides at least 3 admit no Pfaffian orientation, so Kasteleyn's method does not apply. Row transfer matrices on cylinders of circumference up to 25, split by a conserved parity of the dimers crossing a loop, converge exponentially; a Lüscher-type extrapolation with predictive backtests and an error budget gives s = 0.5655119110413253 ± 6·10⁻¹⁶, a non-rigorous estimate. Computer-assisted Collatz-Wielandt certificates, combined with a bound for the matchings across a cut that depends on the runs of crossing cells, prove 0.545343749727 ≤ s ≤ 0.582767477484. We also estimate the surface free energy of a free edge, σ = 0.242017935766251 ± 9·10⁻¹⁵, the corner free energy, which gives the asymptotics of A239273, and the decay length ξ = 1.235 ± 0.004 of the splitting between the two parity sectors. Two sign properties of the strip growth rates, convexity in the even width and a one-sided bound on the odd increments, hold numerically and would give 0.565511911035 ≤ s ≤ 0.565511911049.