One integrand, two cycles: banana Feynman integrals at threshold and spanning trees of root lattices (Paper P19)

Feynman integrals of banana graphs and spanning-tree constants of crystal lattices are built from the same Laurent polynomial P_n = (1 + x_1 + … + x_n)(1 + 1/x_1 + … + 1/x_n). Integrated over the positive orthant it gives the two-dimensional equal-mass banana integral with n+1 propagators; integrated over the torus it gives the lattice Green function of the n-dimensional diamond lattice. We collect the classical facts that connect the two sides and add elementary observations that we have not found stated in this form. (A) At the threshold t = (n+1)² the torus side is exactly the spanning-tree constant of the root lattice A_n, λ(A_n) = m((n+1)² − P_n), and half of it is that of the diamond lattice. (C) The orthant side is finite at the threshold if and only if n ≥ 3. (D) Every λ(A_n) is a single Bessel moment, 2 ln(n+1) − 2(n+1) ∫₀^∞ K₁((n+1)x) (I₀(x)^(n+1) − 1) dx. For n = 2, 3 this places the triangular and face-centred cubic constants, (15√3/4π) L(χ₋₃, 2) and (48√3/π³) L(g, 3) with g = η(2τ)³η(6τ)³, at the banana threshold. For n = 4 the threshold fibre carries the weight-4 form of level 30 found by Hulek and Verrill, but λ(A₄), computed to 228 digits, satisfies no integer relation with the L-values of the rational newforms of weight 2 to 4 and level dividing 30 that we tested. For n = 3 both cycles carry the same form g at the threshold: Samart's Feynman value contains L(g, 2), the lattice constant L(g, 3). A pre-registered search at 240 digits finds no relation between the remaining factor of the Feynman value and L(g, 3), ζ(3) or π³. At the threshold the two cycles share a modular form, not a number. All statements are classical or elementary; every numbered claim is checked by companion code. Reading guide and labels: every result carries a status label (proved, classical, cited, numerical, negative in tested bases, open); a table in the introduction lists the results with their status. Each part opens with a short plain-language summary, a dictionary table translates between the Feynman and the spanning-tree side, and a glossary explains the terms. The companion archive contains the check program with two reference runs (30 and 60 digits, 28 checks each, all passed), the three pre-registered integer-relation searches with two runs each at different precision, the list of newforms used in the enlarged bases, and the two figure scripts (CC BY 4.0).