Curvature-weighted Finsler Mixup

Integrate this at the embedding output of a classifier, before the final prediction head. Add a small positive anisotropic metric network g_phi(h)=softplus(W h+b)+epsilon, producing per-coordinate inverse scales; define theta(h0,h1)=sqrt(sum_j g_phi((h0+h1)/2)_j (h0_j-h1_j)^2). Start with a straight-line geodesic proxy h_t=(1-t)h0+t h1, since exact Finsler geodesics are unnecessary for the MVP. For each minibatch, randomly pair examples, sample t from Beta(0.4,0.4), compute theta, K, and fixed N such as the embedding dimension, evaluate tau with numerically stable sinc and sinhc branches, and construct w0,w1 using the normalized tau values. Run the prediction head on h_t and add the weighted-label cross-entropy. Optionally estimate p_i as the model confidence on endpoint samples and add the hinge penalty in the final formula; stop gradients through p_i initially to avoid collapse. K should first be a fixed sweep over {-1,0,1}; later learn K as a clipped scalar or estimate it from local triangle distortion. Train the metric network jointly but regularize log g_phi toward zero. The first experiment is CIFAR-10 with a small ResNet-18, comparing ordinary Mixup, manifold Mixup, and this method at identical augmentation counts and FLOPs. Measure validation accuracy, calibration, robustness to anisotropic input corruptions, and training-loss variance. Success means improved accuracy or corruption robustness at equal compute without unstable confidence or metric-scale collapse.