Universal Beta Angular Calibration Loss

1. Integration point: attach the loss to a representation tensor h in a contrastive learner, classifier penultimate layer, or transformer residual stream. Use a small fixed probe dictionary A with k=8 or 16 unit vectors and uniform weights w_j=1/k; keep the probe out of the task prediction path. 2. Pseudocode: u_b=normalize(h_b); t_bj=A_j dot u_b; t_bj=sign(t_bj)*clamp(abs(t_bj),eps); g_b=sum_j w_j*A_j/t_bj; q_b=clamp(1/(norm(g_b)^2+delta),0,1); qbar=mean(q_b); compute L_mom from the target mean 1/p and variance 2(p-1)/(p^2(p+2)); optimize L_task+lambda*L_mom, with lambda ramped from zero during the first 10 percent of training. A stronger version uses a soft histogram and cross-entropy against the beta density, but the moment loss is cheaper and less sensitive to poles. 3. The target moments are analytic; estimate q moments, clipping rate, and validation correlations empirically. Sweep lambda over {1e-4,1e-3,1e-2}, use fixed random probes for a clean diagnostic, and compare against learned probes. 4. First experiment: train a 2-layer 512-wide MLP on CIFAR-10 or MNIST with and without the regularizer, using identical optimizer, seeds, and parameter budget. Compare cross-entropy, validation accuracy, expected calibration error, feature covariance spectrum, and gradient spikes. Success requires at least 0.5 percentage-point test improvement or lower calibration error without instability, while q mean and variance move toward the analytic targets. If the regularizer changes q but not task metrics, reject the method.