Cycle-breaking BB optimizer

1. Integration point: replace the scalar learning-rate calculation in a full-batch or low-noise optimizer for a flattened parameter vector theta. For large models, apply the rule independently to parameter blocks or layers. Store theta_{t-1}, g_{t-1}, the last seven normalized gradients, and the last seven scalar step sizes. 2. Pseudocode: compute g_t. If t>0, set s=theta_t-theta_{t-1}, y=g_t-g_{t-1}, a1=(s dot s)/(s dot y+delta), and a2=(s dot y)/(y dot y+delta); clip both to [alpha_min,alpha_max]. Set q_t=g_t/(||g_t||+delta). If t>=7, compute D=1-(q_t dot q_{t-7})^2 and c=abs(log(a1/(a_{t-7}+delta))). If D<epsilon_q and c<epsilon_alpha for m consecutive checks, enter fallback mode for r updates and choose a2 or 0.5*a2; otherwise choose a1. Update theta_{t+1}=theta_t-alpha_t g_t, then append q_t and alpha_t. 3. The BB formulas and projective-dynamics interpretation come from the paper. The implementation estimates recurrence empirically from normalized gradients, rather than estimating the Hessian eigenvalues. Use delta=1e-12, epsilon_q=1e-3, epsilon_alpha=0.05, m=2, r=7, and set alpha_min and alpha_max from a warmup, such as [1e-6,10 times the median warmup step]. 4. First experiment: optimize a 4D diagonal quadratic with eigenvalues [1,3,10,30] from many random initializations, comparing BB1, BB2, clipped BB1, and the cycle-breaking rule. Then train a 2-layer MLP on MNIST with full batches or batch size 4096. Record objective gap, gradient norm, trigger frequency, loss oscillation, time to a fixed loss, and final accuracy. Success means fewer long plateaus or lower loss at equal gradient evaluations specifically on runs where BB1 exhibits seven-step recurrence, with no substantial degradation on non-triggered runs.