Sector-Parametrized Neural Operator

(1) Integration point: use this parameterization for the recurrent transition matrix in a small RNN, the state matrix of a linear/nonlinear SSM, or the linear map inside a residual block. Let the feature width be m. Start with fixed diagonal T=diag(exp(q_i)) and fixed t<0 satisfying T^{-1}+tT>0; do not apply the transform to the nonlinear activation or bias. (2) Pseudocode: sample unconstrained B from trainable parameters; compute C=B^T B+epsilon I; compute R=C^{-1/2} by eigendecomposition or 5--10 Newton--Schulz iterations; set sigma=B R; set A=(T^{-1}+tT)^{1/2}; set W=T^{-1}+A sigma T^{-1/2}; run h_next=phi(W h+x U+b), or use W as the SSM transition. The singular values of sigma are s/sqrt(1+s^2)<1, so the contraction condition is enforced without clipping. (3) The paper supplies the affine map and the exact admissibility condition. The contraction construction, inverse-square-root approximation, and optional real-valued transpose implementation are engineering adaptations. Estimate the largest singular value of W empirically with 5-step power iteration, and monitor the minimum eigenvalue of W+W^T when positive-real behavior is desired. (4) First experiment: compare vanilla RNN, spectral-normalized RNN, and this layer on permuted-MNIST or sequential CIFAR-10 using equal hidden width and FLOPs. Train at learning rates where the vanilla model becomes unstable, and record gradient norms, exploding-state frequency, validation accuracy, and loss versus update count. A successful result is no divergence at larger learning rates, lower Jacobian products over long sequences, and equal-or-better accuracy without per-step projection.