Real quadratic fields enter lattice spanning trees via the unit, not the L-value: A closed family, an angle rule and a parity rule (Paper P18)
Year: 2026
Type: preprint
Abstract: The spanning-tree constants of the classical planar lattices are values of Dirichlet L-functions of imaginary quadratic fields, such as Catalan's constant G = L(2, χ₋₄). Real quadratic fields appear as well, but in a different way: through the logarithm of a unit, as ln(2+√3) for the snub square lattice or ln(1+√2) for the 4·8·8 lattice. We explain when and how this happens for the one-parameter family of Laplace symbols D_{β,γ} = (2β+γ) − β(c₁+c₂) − γc₁c₂, which contains the square, king, snub square, 4·8·8 and Union Jack lattices and the Shastry–Sutherland net with arbitrary dimer conductance. (A) We prove a closed form for the Mahler measure of every member: for γ ≤ β it is ln(γ/4) + (1 − 2x/π) ln ε + π⁻¹[4G + 4Cl₂(x) − Cl₂(2x)] with sin x = γ/β and ε = cot(x/2); for γ ≥ β the unit term is absent. The real quadratic unit thus always enters with an angle as its prefactor. (B) By Niven's theorem the formula closes in standard constants only for γ/β ∈ {0, ½, 1}; this proves the snub square value (10/3)G/π + (1/3) ln(2+√3), gives the Shastry–Sutherland net for every dimer conductance by an explicit determinant, and explains why the 4·8·8 constant keeps an irreducible piece at the angle arcsin(1/3). (C) Whenever x/π is rational, the constant splits into derivatives at the first trivial zeros: L′(0, χ) for even characters (unit logarithms) and L′(−1, χ) for odd characters (values L(2, χ)). The split is the parity of the integrand in the angle. This is the classical parity rule of the gamma factor, stated in this series in an earlier paper as a parity dictionary. A short descriptive section records that in quasicrystals the same units appear as inflation factors, with algebraic gap labels. All statements are classical; every numbered claim is checked by companion code. Reading guide and labels: every result carries a status label (proved, proved computer-assisted, numerical, cited, descriptive); Table 1 lists the results with their status and the check labels. Each section opens with a short plain-language summary, and a glossary explains the terms. The companion archive contains the check program with two reference runs (40 and 80 digits, 120 checks each, 0 failed), the integer-relation search for the 4·8·8 remainder with its log, and the three figure scripts (CC BY 4.0).
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225750618