The Threshold Only Computers Know

Mathematics keeps two ledgers for percolation thresholds: thresholds it has derived and thresholds it has only weighed. The derived ledger holds the square bond threshold (exactly 1/2, self-duality, proved by Kesten in 1980), the triangular site threshold (exactly 1/2), the triangular bond threshold (exactly 2 sin(π/18)), and the honeycomb bond threshold (exactly 1 − 2 sin(π/18), Sykes–Essam duality). The weighed ledger holds the most famous threshold of all: the square-lattice site threshold, 0.59274605079210(2) (Jacobsen 2015), with a second weighing at 0.592746050786(3) (Mertens 2022) — fourteen digits, no closed form, and an exact expression remains an open problem. The paper prints both ledgers side by side, evaluates the exact forms to fifteen digits, and corrects one filing error found on inspection: the kagome site threshold is not a numerical near-collision with the honeycomb bond value but is exactly equal to it, by the site–bond correspondence, and belongs in the derived ledger. Grade: TWO LEDGERS — DERIVED WHERE SYMMETRY SPEAKS, WEIGHED WHERE IT DOESN'T, AND THE FAMOUS ONE IS WEIGHED. This paper is part of the breadth program of the author's corpus; it touches no claim of the physics core.