Conditioned Sylvester Neural Layer

Integrate this into an implicit recurrent or state-space layer. Store trainable A, B, and C, solve AX-XB=C with a differentiable Sylvester solver, and use X as the transition in h_{t+1}=phi(X h_t+W u_t+b). A minimal training loop is: (1) construct A, B, and C, optionally restricting A and B to diagonal or small block-diagonal form; (2) compute X with a batched Sylvester solve; (3) estimate sigma_max(X) by power iteration on X transpose times X; (4) estimate sigma_min(X) by power iteration on the inverse or by a differentiable small-matrix SVD; (5) add lambda_cond times the smoothed log condition number to the task loss; (6) for diagonal or block-diagonal coefficients, compute eigenvalue gaps and add the resonance term in the displayed formula; (7) backpropagate through the solve. The mathematical quantities supplied by the paper are the Sylvester structure and the small-denominator risk. Singular values, eigenvalues, and coefficients are estimated numerically from the current batch-independent parameters. First test on sequential MNIST or a delayed-copy task with a small state size, comparing an unconstrained recurrent matrix, an unregularized Sylvester layer, and the conditioned layer at matched parameter counts. Record long-horizon accuracy, gradient-norm variance, hidden-state norms, solver residuals, and loss spikes. Success means fewer exploding trajectories, smoother gradients, and higher delayed-copy accuracy at equal parameter count without materially increasing per-step solve cost.