Veronese Projective Feature Layer

Integrate this as the input embedding for vectors whose labels or geometry are invariant under sign reversal: normalized directions, eigenvectors, phase-free measurements, unoriented lines, or projective data. Given a batch Q in R^(B by (m+1)), first compute q_i = Q_i/(||Q_i||_2 + epsilon). For each sample form M_i = sqrt((m+1)/m) times (q_i q_i^T - I/(m+1)), then extract the upper-triangular entries, including the diagonal, and feed them to a small MLP, transformer token, or learned linear projection. Pseudocode: q = normalize(x); M = sqrt((m+1)/m) * (einsum('bi,bj->bij', q, q) - eye(m+1)/(m+1)); z = symmetric_upper_triangle(M); h = encoder(z). The formula supplies the exact invariant feature map and its normalization; no learned parameters are needed in this layer. Check numerically that trace(M) is approximately zero and that phi(q) equals phi(-q) to floating-point tolerance. Estimate conditioning from the batch covariance and compare against raw quadratic features q q^T. For the first cheap experiment, use m = 2 or m = 3 and a synthetic classification task in which q and -q have identical labels. Compare a standard MLP on q with sign augmentation against the fixed Veronese layer at equal parameter count. Evaluate clean accuracy, accuracy after random sign flips, data efficiency, and training speed. Success means exact sign invariance, higher accuracy with fewer augmentation samples, and better test accuracy at small training-set sizes. Add an ablation using raw q q^T without trace removal or normalization to test whether the paper's traceless spherical construction improves optimization.