Probability-coordinate robust likelihood head

Integrate this as the final scalar probabilistic head of a small MLP or transformer regression model. The backbone produces a scalar z, and use eta=eta_max*tanh(z), with eta_max initially set to 8 to keep quadrature and gradients numerically safe. Use the Cauchy chart u=0.5+atan(y)/pi and statistic t=u-0.5. Choose h(u)=1 on (0,1), so A(eta)=log integral of exp(eta*(u-0.5)) du and A'(eta) is the corresponding mean statistic. For an MVP, precompute A and A' on a grid of 4096 eta values over [-8,8] using Gauss-Legendre quadrature with 64 or 128 nodes, then linearly interpolate during training. For each batch, execute: z=backbone(x); eta=eta_max*tanh(z); u=0.5+atan(y)/pi; t=u-0.5; A=interp_A(eta); nll=-eta*t+A-log(1/(pi*(1+y*y))); loss=mean(nll). The Jacobian term is necessary for calibrated value-space likelihoods but has no gradient with respect to eta. The formulas provide the chart, score, Fisher information, and barycenter; estimate quadrature tables numerically and measure gradient bounds empirically. First compare against Gaussian NLL, Huber regression, and Student-t NLL on y=x+epsilon with Cauchy noise and with 5 percent extreme outliers. Use a two-layer MLP and equal training budgets. Record per-example gradient magnitude versus |y|, optimization divergence rate, clean-test MAE, test NLL, and 90/95 percent interval coverage. Success is a bounded or saturating score for extreme targets, fewer unstable updates, improved contaminated-data accuracy, and better calibrated intervals at comparable compute.