Exact simplex-lattice quantization

Integrate this module immediately before an embedding or MLP consumes a simplex-valued latent or histogram. Given a probability vector p of length N, select a quantization step Delta, center the input as y=(p-1/N)/Delta, and run the lattice quantizer. Compute a_i=floor(y_i), f_i=y_i-a_i, A=sum_i a_i, and S2=sum_i f_i^2. For a GPU-friendly MVP, assign each f_i to one of B uniform buckets and accumulate two arrays: C[b], the number of coordinates in bucket b, and S[b], the sum of their fractional parts. Scan buckets from largest to smallest, maintaining the prefix count m and prefix residual sum R. At each candidate m, evaluate F(m)=S2+m-2R-(A+m)^2/N. If a bucket contains multiple coordinates and exact selection is needed, sort only that boundary bucket or use a deterministic threshold plus a local scan. Set b_i=1 for the selected m* coordinates and zero elsewhere, then return q=Q(a+b) and p_hat=1/N+Delta*q. During training use the straight-through estimator p_st=p+stopgrad(p_hat-p), so the forward pass is quantized while gradients pass through p. The paper's mathematics determines the projector, candidate objective, and prefix scan; Delta and B are engineering choices estimated from calibration data. For rational inputs with common denominator D, encode residuals as integer numerators and evaluate a scaled objective with integer arithmetic. Start with a small CIFAR-10 classifier receiving normalized histogram features, or a Dirichlet-vector autoencoder. Compare scalar rounding, largest-remainder rounding, and A_n^* quantization at 4, 6, and 8 bits. Measure reconstruction MSE or KL, downstream accuracy, exact sum preservation, invalid-coordinate frequency, and latency. Success is lower distortion or higher accuracy at equal bit rate, exact constraint preservation, and throughput within 2x of scalar rounding; benchmark the bucket kernel against full sorting as N increases.