Pfaffian activation budget

(1) Exact integration point: use a one-hidden-layer tanh classifier, f(x)=b_0+sum_{j=1}^s v_j tanh(w_j^T x+b_j), and apply the certificate to restrictions x=x_0+t u along random lines through the training and validation data. Treat the hidden width s as the Pfaffian-chain length. Add group sparsity over input coordinates, with g_r=||W_{:,r}||_2, so that k is the number of coordinates whose group norm exceeds a threshold tau. (2) Pseudocode: for each candidate width s, initialize the MLP; on every batch compute logits and cross-entropy; compute g_r for every input coordinate; set k=sum_r 1[g_r>tau]; compute B=2^{s(s-1)/2} prod_i beta_i (sum_i beta_i-n+min(k,s) alpha+1)^s with alpha=2, n=1, and beta_1=1; add lambda*log1p(B) for architecture scoring. Because the exact k and B are discrete and astronomically large, do not backpropagate through the hard expression in the first implementation. Instead, use it after each epoch for pruning/model selection, or use k_soft=sum_r sigmoid((g_r-tau)/epsilon) during a differentiable architecture-search phase. Prune input groups and retrain after selection. (3) The paper supplies the analytic zero-count certificate. The implementation estimates active-variable count from trained weights and independently measures actual oscillation by evaluating f(x_0+t u)-c on a dense grid, removing values with absolute magnitude below a tolerance and counting sign changes. Also record the number of near-tangent stationary events using finite differences, since those are not regular roots covered by the theorem. (4) First cheap experiment: compare tanh MLP widths 4, 8, 16, and 32 on two-moons and a noisy one-dimensional sinusoid classification dataset against ordinary tanh training and tanh plus L2 weight decay, using matched optimization budgets. Evaluate test accuracy, calibration, adversarial perturbation stability, training-seed variance, and sign changes along 100 random lines. The expected signal is fewer boundary oscillations and lower seed variance at equal training accuracy, or equal test accuracy with a smaller selected width; the theoretical B itself need not be numerically tight.