Characteristic-Region Gain Controller

Attach the controller to the recurrent feedback matrix C, the residual coefficient gamma, or a scalar gain multiplying the memory branch. Every N training steps, freeze the current A,C,q,alpha, form the K-delay augmented Jacobian corresponding to the recurrence, and use an eigensolver or 10-20 power iterations to estimate the dominant modulus rho_hat. If rho_hat exceeds 1-delta, rescale C or gamma according to the displayed controller; if it is far below the target margin, permit a slow increase using g_new=min(g_max,g*(1+eta_gain)). A more accurate but slower alternative evaluates W_q(exp(i theta)) on a frequency grid and performs bisection over g to locate the first unit-circle intersection. The q-kernel and characteristic equation come from the paper; K, delta, N, gain caps, and the root estimator are empirical approximations. The first cheap test should use tanh RNNs on sequential MNIST and Copy-Memory, comparing SGD, gradient clipping, spectral normalization, and this controller at matched initial gains. Record loss, gradient norm, gain, and rho_hat. The falsifiable prediction is a sharp degradation or exploding-gradient transition near rho_hat=1. Runs maintained at rho_hat<=1-delta should show geometric gradient decay with fitted slope close to log(rho_hat), while unconstrained runs should diverge after crossing one. A gain sweep should yield an observed stable/unstable threshold within approximately 20% of the characteristic-root prediction, and q<1 should make this threshold less sensitive to memory truncation K than q=1.