One Character per Laplace Symbol: Catalan's Constant, L(2, χ₋₃) and the Entropy of Planar Lattices

Summary: Three classical models of planar lattice statistics are spanning trees, close-packed dimers and the Ising model at its critical point. On the square lattice their entropies and free energies contain Catalan's constant G = L(2, χ₋₄), and on the triangular and honeycomb lattices they contain L(2, χ₋₃). All individual formulas have been known for decades. We arrange them in one dictionary. With Boyd's constants d_f = L′(χ₋f, −1) = f^(3/2) (4π)^(−1) L(2, χ₋f), every known case of the triangular family has the form c + q d₃ with c a rational multiple of the logarithm of a rational number and q rational. For spanning trees q = 5/ν, where ν is the number of sites per unit cell (triangular, honeycomb, kagome, 3.12.12 and net 27); for the critical Ising model q is half as large; for honeycomb dimers and for the residual entropy of the triangular antiferromagnet q = 1. The factor 5 has an arithmetic origin: 5/4 = 1 − χ₋₃(2) 2^(−2) is the inverse of the Euler factor at 2, and the spanning tree constant of the triangular lattice is (3√3/π) Σ(n,6)=1 χ₋₃(n) n^(−2). For the square lattice no such factor occurs because χ₋₄(2) = 0. At criticality the Ising kernel is a constant multiple λ of the Laplace symbol, which gives −βf_c = c + z/2 and fixes c. By the functional equation every entropy in the dictionary is a rational multiple of a derivative at the trivial zero s = −1, plus a logarithm. The rule has a clear limit. The 4·8·8 lattice has four-fold symmetry, and its constant contains G/π, but only together with an inverse tangent integral Ti₂(3 − 2√2) for which we find no relation to L-values. The character is chosen by the Laplace symbol, not by the symmetry. As an outlook we record that spanning trees on the four cubic lattices follow the same pattern one level up: their constants are rational multiples of L′(f, 0) for CM newforms f of weight 3, over ℚ(i), ℚ(√−3) and ℚ(√−6), and for the diamond and fcc lattices we correct two printed prefactors. No new physics is claimed. The paper is written for number theorists; Section 2 explains the physics from the beginning. The companion code (93 checks, Python with mpmath, sympy and numpy; PARI/GP via cypari2 for the cubic lattices) reproduces every displayed constant and every numerical statement. - Dirichlet L-function - Catalan's constant - Dirichlet character - Euler factor - functional equation - trivial zero - Boyd's constants - Mahler measure - Bloch–Wigner dilogarithm - Clausen function - inverse tangent integral - CM modular forms - spanning trees - matrix-tree theorem - dimer model - Kasteleyn's theorem - Ising model - Kramers–Wannier duality - residual entropy - triangular lattice - cubic lattices