Zoomed and Pole-Safe Rational Activation

Replace GELU in a small MLP or transformer feed-forward block with a K-node barycentric rational activation. Maintain support nodes z[k], values f[k], and weights w[k], and evaluate the displayed barycentric formula directly using float32 or float64 compensated summation. At each batch, compute detached preactivation quantiles a and b, using the 1st and 99th percentiles, then update exponential moving averages a=.99*a+.01*q01 and b=.99*b+.01*q99. Every 500 optimizer steps, relocate the nodes by applying u=(2s-a-b)/(b-a) to the old nodes and mapping the normalized coordinates into the new interval. Keep two support nodes outside [a,b] by a margin m=.1*(b-a). Add a pole-safety penalty L_pole=lambda*sum_k softplus(m-|p_k-c|)^2, where c=(a+b)/2 and p_k are generalized eigenvalues of the barycentric pencil A-lambda B. Compute p_k with a QZ or generalized eigensolver once per relocation, not during every forward pass. Refit f[k] after relocation by evaluating the previous activation on 128 points and solving a small least-squares interpolation problem; keep w[k] fixed initially, then unfreeze them in a second experiment. Clip the denominator magnitude with epsilon=1e-6. The paper supplies the pencil and separation criterion; activation quantiles, interval margins, and penalty weights are empirical. First test on CIFAR-10 with a four-layer MLP or tiny ViT, comparing GELU, a fixed rational activation, and the zoomed pole-safe version at equal parameter count. Record validation accuracy, loss spikes, gradient-norm outliers, and denominator minima. Success is equal-or-better accuracy with fewer outlier batches and no denominator blow-ups.