Counterdiabatic spectral transport

Use this as a correction inside a residual SSM or as a representation-transport term coupled to an optimizer. Choose a small learned matrix H_t acting on hidden width D. Every 10 optimizer steps, compute an Arnoldi frame U_t from a fixed probe batch of hidden states and store U_(t-1). Align each new column by multiplying it by the phase or sign of u_(t-1,j)^* u_(t,j), preventing arbitrary basis flips. Estimate dot(U)_t = (U_t - U_(t-1)) / Delta t. Form Omega_t = one-half times [U_t^* dot(U)_t - (U_t^* dot(U)_t)^*], which explicitly enforces anti-Hermiticity, then set G_t = i Omega_t. Retain only entries with absolute index difference at most one to obtain G_tri,t. In Arnoldi coordinates c_t = U_t^* h_t, add -i alpha G_tri,t c_t to the residual update and transform back using h_t = U_t c_t. Initially stop gradients through U_t and tune alpha in {0.01, 0.1, 1}; later test full differentiation. The connection and tridiagonal locality come from the paper, while finite differences and alpha are engineering choices. Start with a width-64 residual SSM on sequential MNIST or a small language-model fine-tuning task. Compare AdamW with and without the correction under a ten-times larger learning rate or abrupt interpolation between two operators. Measure loss spikes, hidden-state cosine drift, gradient norms, and final accuracy. Success means fewer transition-induced spikes and better accuracy at the same update count.