Routh-Hurwitz Gain-Capped Optimizer

1. Integration point: implement this as a wrapper around SGD with momentum, Adam without its elementwise preconditioner, or a recurrent state-space update. Maintain a scalar effective feedback gain g representing learning rate multiplied by local curvature or feedback amplification; for a layerwise version, compute one cap per parameter block. Use the previous two or three parameter and gradient states to estimate local oscillatory dynamics. 2. Pseudocode: every K optimization steps, fit a least-squares local model z(t+1) = B₁z(t) + B₂z(t−1) + B₃z(t−2), or estimate the dominant complex eigenpair using Jacobian-vector-product probes. Convert the dominant pair into r̂, L̂, and ω̂₀; alternatively fit cubic coefficients directly and obtain â₁ and â₂ from the fitted polynomial. Compute g_max = ρâ₁â₂/κ with ρ = 0.8, then set g to the minimum of its proposed value and g_max before applying the optimizer update. Log χ = â₃/(â₁â₂). 3. Computed versus estimated: the cubic Routh-Hurwitz inequality is the paper-derived condition; all neural quantities are empirical local estimates and should be smoothed with an exponential moving average. Reject estimates with negative damping or poor fit residuals and fall back to a conservative fixed learning rate. 4. First cheap experiment: train a two-layer MLP on MNIST with SGD momentum while sweeping learning rate and momentum. Compare vanilla SGD, gradient clipping, and this cap. For each run measure loss variance, oscillation amplitude, fitted cubic roots, and χ. The falsifiable prediction is a sharp rise in oscillation amplitude and loss variance as χ approaches one, with the divergence boundary predicted by χ = 1 within approximately 20 percent. The capped method should keep χ below 0.8 and eliminate runs that diverge at the same nominal learning rate.