Chemotactic Instability Spectral Controller

Attach the controller to the chemotactic feature block or to any neural PDE or state-space layer containing learned cross-spatial transport. Every K training steps, freeze the current weights, evaluate the feature mean state, and obtain reaction Jacobian entries with automatic differentiation. For a small grid of spatial frequencies q, construct M(q; beta), calculate its largest real eigenvalue, and set s(beta) to the maximum across frequencies and samples. Update beta with the displayed feedback rule, clipping it to a nonnegative interval. Use gamma greater than zero for a contractive block, or target a small positive spectral abscissa for a short pattern-forming stage. If feature statistics vary strongly between samples, use the 90th percentile of s across the minibatch. The matrix formula comes from the paper; the Jacobian, state, and frequency set are estimated in the implementation. First run CIFAR-10 with a six-block CNN and compare fixed gains against the controller at equal compute. Inject sinusoidal perturbations into intermediate features and measure their growth or decay. The prediction is exponential perturbation decay when s is below zero, with a transition kink at s(beta) equal to zero. The measured gain threshold should be within 20 percent of the spectral estimate.