G2 structured latent code

Modify the latent bottleneck of a VAE. Let the encoder output logits of shape [batch, latent_dimensions, K], representing posterior probabilities for K binary digits per latent coordinate, instead of Gaussian means and variances. Apply sigmoid logits and sample with Binary Concrete or a straight-through Bernoulli estimator. For each coordinate, compute x by setting x=0 and prod=1, then for k from 1 through K set g equal to g0 when the hard digit is zero and g0-1 when it is one, add digit*g_(1-digit)*prod to x, and multiply prod by g. Feed the resulting x vector to the existing decoder. Add the structured-prior penalty: estimate negative log prior as -log pi[a1] minus the sum of log P[a[k],a[k+1]], and subtract the posterior Bernoulli entropy or use a categorical KL approximation. Learn pi and each transition row with softmax logits, or share one matrix across coordinates. Compute the coordinate recurrence exactly; estimate reconstruction and KL terms by Monte Carlo. Start with g0=0.5, K=8 or 16, and compare iid digits with p00=p11=0.9. On MNIST or CIFAR-10, compare against a Gaussian VAE with the same decoder and latent bit budget. Measure reconstruction error, bits per dimension, active latent usage, and robustness to masked latent bits. Success means equal reconstruction with fewer latent storage values or improved sampling quality at the same code budget.