Fractional Mahalanobis radial head

Integrate this module at the binary classifier's final logit, after an existing encoder f_theta(x), rather than adding another MLP layer. Use an embedding width of 16-128. Maintain two learnable centers mu_0 and mu_1 and lower-triangular factors L_0 and L_1; compute Sigma_c=L_c L_c^T+epsilon I with epsilon=1e-4 to 1e-3, and compute each radius by triangular solves instead of explicitly inverting Sigma_c. Choose exponents alpha=[0.25,0.5,1.0,1.5,2.0] and learn coefficients b. Initialize mu_c and covariance factors from one pass of encoder embeddings, initialize the affine coefficient near its QDA value, and initialize all non-affine coefficients near zero. For each minibatch, compute z=f_theta(x), then r_c=(z-mu_c)^T Sigma_c^{-1}(z-mu_c), evaluate h(r_c)=b_0+sum_k b_k*r_c**alpha_k, add determinant and prior terms, and train with binary cross-entropy. Freeze the encoder for 1-3 warmup epochs so centers and covariances stabilize, then unfreeze it with a learning rate 5-10 times smaller than the head. Use Cholesky solves, softplus diagonal entries of L_c, gradient clipping, and optionally clip radii at the 99.9th percentile for numerical safety. Compare against the same encoder with a linear head, a two-layer MLP head, and affine QDA on frozen embeddings. Start with UCI tabular data and synthetic Gaussian versus Student-t mixtures. Success means lower test log-loss and higher accuracy on heavy-tailed data at equal parameter count, no degradation on Gaussian data, improved calibration, and stable covariance condition numbers.