Sampling-Invariant Disturbance Budget

Integrate this at the update rule of an SGD-like optimizer or at the recurrent transition of an RNN or SSM. First choose a compact region B that empirically contains the attractor, using late-training checkpoints and a robust coordinatewise or Mahalanobis envelope. At each update, inject a bounded disturbance δ_k representing minibatch replacement, gradient compression, clipped Gaussian noise, or an explicit adversarial vector. Estimate L using power iteration on the update Jacobian, automatic differentiation on sampled states, or the maximum observed local Jacobian norm. Run M replicas at a fixed disturbance radius r; declare escape when the state leaves B for q consecutive steps; binary-search the largest nonescaping radius μ̂_h. Report μ̂_h/h and optionally cap the next learning rate so the desired continuous-time disturbance rate stays below this threshold. The theorem supplies the normalization and bounds; L, B, and μ̂_h are estimated empirically. The first cheap test is a two-layer MLP on MNIST with SGD, comparing several step sizes h and explicit bounded gradient perturbations. The falsifiable prediction is that μ̂_h/h approaches a common limit as h tends to zero, lies between the theoretical bounds using the estimated L within roughly 20 percent, and usually has linear residual error in h.