Bernstein resolvent activation

(1) Integration point: implement this as a drop-in replacement for GELU or SiLU in an MLP block, but apply it to nonnegative features after RMSNorm plus ReLU, or use it as a positive gate multiplying a residual branch. For signed activations, use the odd extension sign(x)φ(|x|), and compare it separately because the Bernstein guarantee itself is only on x≥0. (2) Pseudocode: initialize fixed log-spaced poles λ_k=exp(linspace(log λ_min, log λ_max, K)); store unconstrained parameters u_a,u_c,u_λ and map them with softplus: a=softplus(u_a), c_k=softplus(u_c[k]), λ_k=softplus(u_λ[k])+λ_floor. Set y=epsilon+relu(h), s=y**tau with tau either fixed to 0.5 or learned through tau=0.5*sigmoid(u_tau). Return b+a*s+sum(c_k*s/(s+λ_k)). Backpropagate normally. Optionally normalize the output by dividing by the value of φ at a calibration quantile of the input so its scale matches the baseline activation. (3) Computed from the paper's mathematics: positivity of coefficients, positive poles, and the certified τ≤1/2 range. Estimated empirically: K, pole range, epsilon, output normalization, and whether the monotone-concave prior helps the task. Numerically verify monotonicity on a dense grid and estimate derivative signs with automatic differentiation. (4) First experiment: train a 4-layer CIFAR-10 MLP or small ViT MLP block with identical parameter count using GELU, Softplus, and the resolvent activation for K∈{4,8,16}. Compare training stability, gradient-norm variance, calibration, and test accuracy at equal FLOPs. A successful result is fewer exploding or vanishing-gradient events and equal-or-better validation accuracy, or faster loss descent at equal optimizer steps. Then ablate τ=0.75 versus τ=0.5 to test whether violating the paper's sharp range causes worse derivative behavior or optimization instability.