FDT-Calibrated Rotational Optimizer

Integrate the method between minibatch gradient computation and the parameter update for one or more parameter blocks. Maintain the current gradient g_t and the previous gradient g_{t-1}. Normalize them to u_t and v_t, then construct a rank-two skew operator A_t = alpha times (u_t v_t-transpose minus v_t u_t-transpose). Use the update direction d_t = -g_t + A_t g_t and apply theta <- theta + eta d_t. Estimate a local stability margin every K steps using finite-difference probes: perturb theta in two random directions, measure the resulting gradient changes, and run a few Arnoldi iterations on the resulting Jacobian-vector product. If the estimated spectral radius of I plus eta J exceeds 0.98, reduce alpha and eta; otherwise increase alpha slowly up to a preset maximum. The paper supplies the symmetric/skew decomposition, covariance Lyapunov equation, and complex-eigenvalue mechanism. The implementation estimates J, C, and response empirically. For a response estimate, inject a small gradient perturbation at time t and record the parameter or loss response at later lags. First test on a two-layer MLP and MNIST against SGD, Adam, and momentum SGD, using alpha in {0, 0.1, 0.3, 0.6, 1.0}. Log loss, gradient autocorrelation, estimated spectral radius, and perturbation response. The prediction is that stronger alpha produces damped oscillations with envelope approximately exp(-s t) and period approximately 2 pi divided by absolute value of a. Instability should begin near rho(I + eta J) = 1, with the measured boundary within 20 percent of the estimated boundary. The fluctuation-response residual should increase with alpha even when the loss remains exponentially relaxing.