Fractional Boolean Sensitivity Regularization

Integrate the method at the model-output interface for a classifier or encoder whose input is x in {-1,1}^n, including binary feature vectors, token-presence indicators, or hard binary routing gates. Do not apply it directly to continuous hidden activations unless those activations are first discretized. For a first implementation with n <= 14, enumerate all cube points or use a fixed random subset, evaluate the scalar output f_theta(x), apply a fast Walsh-Hadamard transform to obtain the coefficients f_hat(S), multiply coefficient S by |S|^(1/p) for every nonempty S and set the constant coefficient to zero, then apply the inverse transform. Use the following loop: y = model(theta, x_cube); coeff = FWHT(y); coeff[0] = 0; for each spectral index S, set coeff[S] = coeff[S] * popcount(S)**(1/p); z = IFWHT(coeff); loss = task_loss + lambda * mean(abs(z)**p); backpropagate and update theta. For larger n, retain only low-order Walsh subsets or estimate coefficients by Monte Carlo sampling of chi_S(x); validate the approximation against exact transforms on small n. The paper supplies the sensitivity-to-fractional-energy inequality, while the Walsh diagonalization and truncation are implementation choices. Begin with a two-layer MLP on a synthetic parity-plus-noise task or binarized MNIST using n = 12 or 16 input bits. Compare ordinary cross-entropy, explicit one-bit-flip augmentation, and this penalty at matched FLOPs. Measure clean accuracy, accuracy under one-, two-, and four-bit flips, empirical sensitivity E[(sum_j |D_j f|^2)^(p/2)], and training loss at equal step count. Success means lower flip sensitivity and improved corrupted accuracy at comparable clean accuracy, or matching robustness with fewer explicit perturbation evaluations.