Quartic anisotropy of the lattice Laplacian on simple, body-centred and face-centred cubic meshes: amplitude, sign and coupling dependence

A bounded scalar toy-model lattice calculation continuing paper 1. From the lattice symbol it computes the amplitude and sign of the quartic anisotropy kappa of the discrete Laplacian on simple, body-centred and face-centred cubic meshes: for nearest-neighbour coupling kappa/d_nn^2 = +1/12, -1/18, -1/48, and for a next-nearest shell of relative weight rho in closed form, with one sign change at rho = 1/2, 1/2, 1/8; a real-space lattice Green function fit confirms amplitudes and signs. It compares two parameter-free coupling rules and shows that a gap-only bond energy gives a correct gradient term only with an unstable bond (U''<0) and an exactly tuned on-site stiffness. The simple cubic part is related to known isotropic stencils (D3Q19, Thampi et al.); the bcc and fcc closed forms and the Voronoi case are the new parts, and the mapping to the short-range framework is paper 1's. It gives the argument for the withdrawal, in paper 1 version 4, of the earlier statement of a limit on the mesh spacing. Paper 1's kappa upper limits are used as inputs only; values of the mesh spacing at which the leading-order amplitude equals those limits lie outside the range of validity of the expansion and are not limits on the spacing. No data fit, forecast, tensor generalisation, derivation of the coupling from the balloon dynamics or preferred-frame analysis is included. The scripts used are available from the author on request. A one-page diagram of the results (paper2_diagram.html) is included.