Bifurcation-Calibrated Stale-Gradient Controller

Integration point: train a small MLP or ResNet with an optimizer having two modes. Mode L can use Adam with momentum beta_L, while mode R uses a larger-step SGD update or a different diagonal preconditioner. Define a scalar gate s_t = u^T(theta_t - theta_ref), where u is a fixed random unit vector or the normalized recent parameter displacement and theta_ref is an exponential moving average of parameters. Apply the gate using a delayed value s_(t-d). Alternatively, use hysteresis: switch to mode R when s_t exceeds h and switch back to mode L only when s_t falls below -h. Set mu equal to d times the base learning rate for delay experiments, or use the normalized hysteresis width h as the perturbation parameter. Pseudocode: initialize a ring buffer of scalar sections; at step t read s_gate = buffer[t-d]; choose mode R if s_gate > 0 and mode L otherwise; compute the minibatch gradient g_t; apply theta_(t+1) = theta_t - eta P_mode g_t; append s_(t+1). Every K steps, collect local section-return pairs (y_i, y_next_i) and fit the displacement D_i = y_next_i - y_i to c1 y_i^(2n) + c2 mu / (abs(y_i)^(2q) + epsilon), where q = |k_R-k_L| and epsilon only prevents numerical overflow. Set V_hat = c1 and kappa_hat = abs(c2). Predict y_hat_star = (kappa_hat mu / abs(V_hat))^(1/(2(n+q))) when V_hat is negative. If y_hat_star exceeds a prescribed radius r, reduce delay d or hysteresis h; if exploration is desired, permit the cycle only while y_hat_star is below r and then anneal mu toward zero. Directly computed quantities are gradients, parameters, section coordinates, delay, and return pairs. V_2n, kappa_td, n, and effective tangency orders are estimated from short trajectories; n can be selected by comparing regressions for n = 1, 2, and 3. The first cheap experiment is a two-layer MLP on two-moons or MNIST, comparing Adam, switched updates with delays d in {1, 2, 4, 8, 16}, and the same switched updates with no delay. Measure return-map displacement, cycle radius, validation loss, and gradient norm. The falsifiable prediction is a log-log slope d log(abs(y_star)) divided by d log(mu) near 1/M, with measured amplitude within 20 percent of the fitted prediction. Increasing delay should produce a larger cycle according to this power law, and the displacement should change sign at y_star. The predicted safety boundary is mu_max = r^M abs(V_hat) / abs(kappa_hat); runs below this boundary should remain inside radius r, while runs above it should show either a larger cycle or instability.