Diagonal-Balanced Hessenberg Dynamics

Integrate this into the transition operator of a delayed RNN, SSM, or companion-form polynomial layer. Form the augmented state z_t = [h_t; h_(t-1); ...; h_(t-r)], whose transition is a sparse block matrix T containing the learned A block, the delayed block -gamma I, and identity shift blocks. Introduce positive diagonal coordinates D = diag(exp(s_1), ..., exp(s_m)) over the augmented state and execute the transition in balanced coordinates as ztilde_(t+1) = D^(-1) T D ztilde_t; decode with z_t = D ztilde_t. In pseudocode: T = build_augmented_transition(A, gamma); D = diag(exp(s)); Tbal = solve_left(D, T @ D); rowbound = max(sum(abs(Tbal), dim=1)); loss += lambda_bal * softplus(rowbound - target)^2; z = Tbal @ z + input. For large widths, do not materialize T: compute the row sums from its sparse blocks and optimize s with 10 to 20 gradient steps every 100 training steps, or update s using a detached copy of T. The exact paper-derived quantity is the similarity-transformed infinity norm and its spectral-radius upper bound. The actual reduction in transient growth must be measured because non-normal powers can behave differently from a one-step bound. Begin with width 64, delay r = 2, and a synthetic linear system plus sequential MNIST. Compare no scaling, ordinary spectral normalization, and learned diagonal balancing at equal wall-clock budget. Track maximum hidden norm, gradient clipping frequency, Jacobian products ||T^k||, spectral radius, and task loss. Success means lower peak activation and gradient variance with unchanged validation accuracy, fewer training failures at the same learning rate, or faster convergence at equal FLOPs.