Integrated-Growth Hopf Delay Scheduler

Integrate this at the optimizer or inference-controller level for a model whose effective update Jacobian changes slowly. A practical first target is a small RNN, state-space model, or MLP trained while the learning rate is ramped. Let the control variable be $\mu=\eta$ or a normalized training-time ramp, and estimate the Jacobian of one update map on a fixed probe batch. Every $K$ steps, use Jacobian-vector products and Arnoldi iteration on a low-dimensional Krylov subspace, or finite differences for a small model, to estimate the leading conjugate pair and set $\hat\alpha_t=\operatorname{Re}\hat\lambda_t$. Maintain $B_{t+1}=B_t+\Delta\mu\hat\alpha_t$. Before the crossing, reset $B$ whenever the monitored eigenmode changes. After $\hat\alpha_t$ becomes positive, continue the ramp only while $B_t<\epsilon\log(r_{\mathrm{max}}/r_0)$. Estimate $r_0$ from minibatch-gradient noise or an injected probe perturbation, estimate $r_{\mathrm{max}}$ from the largest perturbation that remains bounded over a calibration window, and estimate $\epsilon$ from the slope of the parameter ramp per update. If the budget is exceeded, freeze the ramp, halve the learning rate, or roll back the last update. The paper mechanism is the slow-passage amplitude law and exponentially small attracting-repelling manifold splitting; Jacobians, noise levels, and thresholds are estimated empirically. First run equal-compute CIFAR-10 training of a small ResNet or sequence benchmark training of an RNN with five linear ramp rates. Compare standard instantaneous spectral clipping, this integrated-growth scheduler, and an ablation with $B=0$ that disallows delay. Pre-register the signature: measured exit should satisfy $B_{\mathrm{exit}}\approx\epsilon\log(r_{\mathrm{max}}/r_0)$ and collapse across ramp rates when plotted against $B/\epsilon$. Reject the transfer if the collapse has more than 20% relative error, divergence regularly occurs before the predicted budget, or the proposed scheduler provides no larger stable post-crossing area than instantaneous clipping.