Minimum-entropy symmetry noise

Integrate this as a stochastic augmentation layer on a categorical latent, class index, token cluster, or graph-node label. Represent each latent state by an element of Z_k. For the exact two-point path, identify the two dominant states, map them to -1 and +1, estimate p from the minibatch histogram, choose an odd k, compute l = (k-1)/2 and delta = (2p-1)/(kp-l), then construct g with g(0) = 1-l delta and the sparse residue pattern from the paper. Sample y from g and replace latent state x by (x+y) modulo k before the downstream network. For a general categorical distribution f, form a linear constraint matrix A whose row for x contains the coefficients of sum_y f(y)g(x-y) minus sum_y f(y)g(-x-y). Optimize g on the simplex using projected gradient or Frank-Wolfe, minimizing entropy subject to A g = 0; use a small positive floor during optimization, then threshold tiny entries and renormalize. Compute f from an exponential moving average of minibatch histograms, while the symmetry constraints and entropy are exact. Start with MNIST or CIFAR-10, a k = 7 categorical bottleneck, and compare no augmentation, uniform Z_7 noise, and minimum-entropy noise at equal FLOPs. Record validation accuracy, calibration, latent reconstruction error, entropy H(g), and residual max_x absolute value of (f*g)(x) minus (f*g)(-x). Success means matching or exceeding uniform-noise accuracy with lower entropy and fewer distinct transformations per example.