Convergence of the Modular Data of a Solvable Gaussian Network: Conjectures, a Band-Edge Obstruction, and a Controlled Numerical Window

A solvable Gaussian network, the ground state of a scalar field on a weighted lattice with a truncated exponential interaction kernel, admits exact computation of the entropies and modular Hamiltonians of arbitrary regions. In the companion paper this model was used to test the matter sector of Jacobson's entanglement-equilibrium argument, with a first law whose contact term is consistent with the conformal value 2π ξ_c. That consistency is a statement about finite lattices extrapolated to the continuum; the present work isolates the step that was extrapolated: the convergence of the lattice modular data of a ball to the continuum modular Hamiltonian computed by Longo and Morsella. We formulate convergence conjectures (C1, C1a, C1a', C1b) and prove none of the deep ones. What is proved is elementary and stated with its proof status: the exact first law with the modular-Hamiltonian matrices of a restricted Gaussian state; a bulk identity for lattice balls, Σ_u κ_u = K(ℓ_A)(1 + O(ℓ^(-2))); sharp cell estimates for smooth lattice sums; a purely geometric test of the bulk identity over 4 ≤ ℓ ≤ 32; a non-Lipschitz obstruction at the band edge ν = 1/2, illustrated exactly by a two-oscillator model; and perturbation bounds with an accumulated-weight estimate that organises one proof strategy for C1a around a counting statement, provided a mode-wise condition holds—which fails at the extreme edge. For the counting we state a hypothesis E1' at fixed threshold. The numerical section reports every value computed here, over 4 ≤ ℓ ≤ 12 with five dilute band probes per radius. The two-parameter decomposition R_c - 1 = a + b / x_eff^2 gives b = 0.959, 0.985, 0.997 and a = 0.039, 0.027, 0.021 at ℓ = 8, 10, 12, both moving towards the conformal values 1 and 0; the rates are not resolved, the direction of the trend is not robust against removing the smallest-band probe, and the dominant systematic of the pipeline (the k = 0 infrared prescription) shifts b by up to 0.23 at ℓ = 8, an order of magnitude above the ℓ-trend itself. The ℓ ≤ 32 field data of the companion paper remain cited, not recomputed. All numbers generated in this work are reproduced by distributed programs with a test suite.