Strongly monotone spectral residual block

1. Integration point: use the block as a replacement for a matrix-valued residual MLP in covariance-aware attention, graph neural networks with matrix-valued node states, or a small symmetric latent state in a world model. The input and output are S\in\operatorname{Sym}(n); enforce symmetry after every learned matrix operation with (Y+Y^T)/2. 2. Pseudocode: initialize S; at each layer symmetrize S; eigendecompose S=Q\operatorname{diag}(\lambda)Q^T; evaluate f(\lambda)=
abla\phi(\lambda) from a shared convex symmetric potential; form Y=Q\operatorname{diag}(f(\lambda))Q^T; set S_{new}=S+\alpha_tY. On a small subset of eigenvalue vectors in each minibatch, compute J_f by automatic differentiation, set L_t to the largest eigenvalue of (J_f+J_f^T)/2, and use \alpha_t=c/(L_t+10^{-6}) with c between 0.1 and 1. Initially stop gradients through L_t. 3. The paper supplies the global matrix monotonicity implication; the Lipschitz estimate and adaptive scale are empirical additions. Validate the guarantee by sampling random S,T and reporting the minimum ratio \langle F(S)-F(T),S-T\rangle/\|S-T\|_F^2. 4. First experiment: use 8 imes8 channel covariance states in a four-block graph classifier or tiny transformer. Compare a standard residual MLP, an unconstrained spectral residual, and this strongly monotone block at learning rates 1x, 3x, and 10x baseline. Evaluate divergence frequency, gradient norms, clean accuracy, and accuracy under Gaussian covariance perturbations. Success is no divergence at high learning rate, a measured monotonicity ratio near or above 1, smoother gradients, and improved noisy-input accuracy.