Sharp Sumset Support Regularizer

1. Integration point: use this in a sparse additive layer, sparse convolution, or discrete latent composition module with n branches. Each branch emits a nonnegative integer coordinate vector or a distribution over a finite integer grid {0,...,m}^d. For each minibatch example, define A_j as the set of grid locations whose branch activation exceeds threshold tau, and define A_out as the set of locations attainable after adding one selected support element from every branch. Do not apply the theorem directly to dense real-valued hidden states without first introducing quantization or a sparse-grid representation. 2. Pseudocode: for each example: A[j] = active_support(branch[j], tau); S = A[0]; for j in 1..n-1: S = {a+s for a in S for s in A[j]}; p = n*log(m+1)/log(n*m+1); target = sum_j log(len(A[j])+eps)/p; loss_sumset = relu(target - log(len(S)+eps)); loss = task_loss + lambda_sumset*loss_sumset. Use bitsets and integer shifts for exact S when d and m are small. For larger supports, sample K tuples from the branch supports, hash their sums, and estimate distinct-sum cardinality; for differentiable training, use soft occupancy r[s]=1-exp(-sum_t product_j q_j[a_j]), where q_j is the branch probability at coordinate a_j and t ranges over sampled tuples, then approximate |A_out| by sum_s r[s]. 3. Mathematics versus estimates: p and the lower-bound target are exact functions of n and m. Active cardinalities and output cardinality are empirical quantities; tau, K, and the hash estimator are engineering choices. Use a straight-through gradient through thresholding, or train with soft occupancy and evaluate the exact hard-support inequality periodically. 4. First experiment: construct a three-branch sparse MLP or 2-D sparse convolutional block on MNIST or CIFAR-10, with every branch producing integer-grid offsets in a 16x16 grid and an additive merge. Compare the baseline, entropy-only support regularization, and this bound with lambda chosen so its initial contribution is 1% of task loss. Record validation accuracy, active output support, unique summed coordinates, collision rate, and wall-clock/FLOPs at equal accuracy. Success means lower branch-to-output collision and improved accuracy or robustness at the same sparsity, without uncontrolled support growth; failure is increased computation with no validation improvement.