Cubic-Group Averaged 3D Convolution

Integrate this at the first 3D convolution of a dense CNN, U-Net, neural operator, or lattice world model. For a scalar-density baseline, store one learnable kernel K with offsets in a cubic neighborhood and construct K_eq on every forward pass by applying the 48 integer coordinate permutations and sign flips, summing, and dividing by 48; use ordinary pointwise nonlinearities afterward. For a stronger version, use representation-valued channels: classify channels as scalar, vector, or tensor types, apply the projector P_R to the tensor formed from channel indices and spatial offsets, and use only equivariant linear maps between types. Pseudocode is: sample x; for each offset u set K_eq[u]=mean(K[g(u)] for g in G); y=conv(x,K_eq); y=activation(y). The group action, orbit list, and averaging rule are taken directly from the paper; the only empirical quantities are the learned kernel weights and validation error. First run a cheap synthetic test on 32^3 voxel fields generated from randomly rotated anisotropic Gaussian blobs or cubic obstacle patterns. Compare an unconstrained CNN, a six-orientation augmentation baseline, and the averaged CNN at equal parameter count and training examples. Measure equivariance error E_eq=mean_g ||f(gx)-g f(x)||_2/||f(x)||_2 and prediction error. The quantitative prediction is E_eq at numerical precision, typically below 1e-6 in float32 apart from boundary effects, while the unconstrained model has nonzero error; with only one-sixth to one-twelfth of the orientation coverage, the equivariant model should retain the augmented baseline's accuracy within 10 percent. On lattice-fluid data, compare wall profiles and anisotropic pair correlations separately by crystallographic direction.