Derangetropy Rank Warp

Integrate the method as a per-channel scalar normalization layer after an MLP activation or before a transformer projection; initially test it in a small MLP rather than inside attention. For a batch tensor A in R^(B times d), maintain one running CDF approximation per feature using K=128 fixed quantile knots q[j,k], where k ranges from 0 to K-1. Estimate each activation rank by piecewise-linear interpolation between adjacent running quantiles. Update the quantile estimates with an exponential moving average during training, but detach the knot values in the forward pass to avoid unstable gradients through sorting. Compute r_prime = r - sin(2 pi r)/(2 pi), clip it to [1e-4,1-1e-4], and either use r_prime directly or map it through the inverse standard-normal CDF followed by learned scale and bias. A differentiable batch-only variant can use r_i = (1/B) sum_j sigmoid((a_i-a_j)/tau), with tau annealed from 0.2 to 0.02, but this costs O(B squared times d), so the running-quantile version should be the main implementation. The kernel and antiderivative are computed exactly; CDFs and quantiles are estimated empirically. First test on CIFAR-10 with a three-layer width-512 MLP and on a small transformer language model, comparing against LayerNorm, RMSNorm, BatchNorm, and identity at equal parameter count. Record validation accuracy or perplexity, activation kurtosis, gradient-norm variance, and robustness to monotone rescaling and heavy-tailed perturbations. Success means lower gradient variance and equal-or-better clean validation performance, with improved accuracy under feature rescaling or outlier corruption and no more than 10 percent training-time overhead.