Lexicographic spectral activation

1. Integration point: implement a PyTorch or JAX LexicographicSpectralFunction layer for batches of real symmetric or complex Hermitian matrices A in C^(d x d). The forward pass returns Y=f(A), where f may be log1p, sqrt, exp, soft thresholding, or a learned scalar spline applied to eigenvalues. Insert the layer after covariance pooling, on graph Laplacians, on attention Gram matrices, or on matrix-valued token embeddings. 2. Pseudocode: compute (lambda,Q)=eigh(A); group eigenvalues whose gaps are below a tolerance; construct each projector P_r=Q_r Q_r^T; return Y=sum_r f(lambda_r)P_r. For a Jacobian-vector product with perturbation B, construct G[r,s]=f^[1](lambda_r,lambda_s) and return J=sum_{r,s} G[r,s] P_r B P_s. For a Hessian-vector product with perturbations B and C, return H=sum_{r,s,t} f^[2](lambda_r,lambda_s,lambda_t)(P_r B P_s C P_t + P_r C P_s B P_t). Register these as custom JVP and HVP rules so the autodiff system does not differentiate eigenvectors. 3. Compute eigenvalues and eigenspaces numerically; compute divided differences using the recurrence above, switching to analytic derivatives such as f'(x) and f''(x) when all relevant arguments are within 1e-5 max(1,|x|,|y|). The paper's mathematics determines the ordered projector products and coefficients; the engineer must choose the tolerance, matrix dimension, and scalar function. 4. First cheap experiment: train a d=16 covariance-feature classifier on a synthetic SPD classification problem or CIFAR-10, comparing this layer against ordinary Q diag(f(lambda)) Q^T with framework autodiff and against finite-difference JVPs. Measure JVP and HVP error against high-precision finite differences, NaN frequency, gradient norms near repeated eigenvalues, validation accuracy, and backward wall-clock time. Success means at least 10x lower derivative error, no gradient spikes for eigenvalue gaps below 1e-4, and equal or better accuracy at comparable FLOPs; a secondary target is faster second-order or meta-learning training because exact HVPs avoid repeated forward evaluations.