Algebraically Scrambled Augmentation Batches

Integrate this at the data-loader or augmentation-policy sampler, before the augmentation operators are applied. Let d be the number of continuous controls, for example crop scale, aspect ratio, color strength, blur, and rotation; map each digital-net coordinate u in [0,1]^d to a legal parameter through an inverse CDF or affine range map. Build a binary digital net of N points using a Sobol-style generator, then apply a random invertible binary matrix to the direction-number bits for matrix scrambling, or a linear digital scramble; do not use independent random coordinates and do not use full Owen scrambling in the first comparison. For every batch window, use a fresh scramble or a nested prefix of one scrambled sequence. Pseudocode: generate bits B[n,j,k]; sample invertible lower-triangular binary matrices A_j; set B'[n,j,:]=A_j B[n,j,:] mod 2; convert bits to u[n,j]=sum_k B'[n,j,k]2^{-k}; transform u to augmentation parameters; apply one vector per example. Compute R_infty exactly on each batch and estimate h_infty by evaluating distances from a 4,096-point uniform probe grid to P. The theorem supplies the qualitative target rho=O_P(log N); h and q are measured empirically. First test on CIFAR-10 with a ResNet-18 and continuous RandAugment-like policies, comparing IID uniforms, Latin hypercube, Owen-scrambled Sobol, and matrix-scrambled Sobol at identical augmentation FLOPs. Success means higher validation accuracy or faster loss reduction at equal optimizer steps, together with larger R_infty and smaller estimated rho_infty.