Spanning Trees in Four Dimensions and a Modular Form of Level Six: A reduction of the hypercubic entropy to one value at a CM point (P15)

Summary: The number of spanning trees of a large lattice grows like e^(λN), and the entropy λ is a Mahler measure. In dimension two the known values are L-values of weight two, in dimension three they are L-values of weight three. We ask whether the ladder continues in dimension four. For the hyper-body-centred lattice it does, by a theorem of Papanikolas, Rogers and Samart, which we restate as λ_bcc(4) = 8 L′(f₈, 0) − 28 ζ′(−2). For the hypercubic lattice ℤ⁴ we find numerically, to more than 100 digits, λ_sc(4) = m(8 − Σᵢ (xᵢ + xᵢ⁻¹)) = 12 L′(g₆, 0) − (56/3) ζ′(−2) = (162/π⁴) L(g₆, 4) + (14/3) ζ(3)/π², where g₆ = (η(τ)η(2τ)η(3τ)η(6τ))² is the unique newform of weight 4 and level 6. This relation was not found in the literature we checked. We do not prove it, but we prove a chain of exact statements that reduces it to a single number. Along the line Re τ = 1/2 in the upper half-plane the entropy splits as λ_sc(4) = 36 L′(g₆, 0) − 2K, where K is an integral from the CM point τ* = 1/2 + i/(2√3) to the cusp. We evaluate the depth-one part of K by a Ramanujan-type identity at the fixed point of the Atkin–Lehner involution W₃, split the depth-two part by parity under W₃, and prove the even half, π I_E = ζ′(−2) + 9 L′(g₆, 0). The main relation is then equivalent to one statement about the odd half, K = 12 L′(g₆, 0) + (28/3) ζ′(−2). We show that no symmetry of Γ₀(6)⁺ can settle this, so the missing step is arithmetic. Two primers make the paper readable for physicists and for number theorists; labels separate proved, cited, classical, numerical and open statements. The companion code (Python scripts with mpmath, PARI/GP via cypari2 and python-flint; the eight pre-registered tests with their registrations and SHA-256 files; a runner that checks every step against its reference log; reference logs including the failed pre-registered items; the figure script and a reference checker) is included as P15_companion_code.zip. The figures used by the LaTeX source are in P15_figures.zip. spanning trees - lattice entropy - hypercubic lattice - Mahler measure - modular forms of weight four - newform of level six - L-functions of modular forms - Rogers–Zudilin method - Atkin–Lehner involution - CM point - iterated integrals - Eichler integrals - Domb numbers - lattice Green functions - Bessel moments - PSLQ - pre-registration - 11F11 (holomorphic modular forms of integral weight) - 11F67 (special values of automorphic L-series) - 11R06 (Mahler measure) - 05C30 (enumeration in graph theory) - 82B20 (lattice systems in equilibrium statistical mechanics) - 11Y60 (evaluation of number-theoretic constants)