Weakly Normally Hyperbolic Cyclic Optimizer

1. Integration point: modify SGD with momentum or AdamW so the learning rate and momentum are deterministic periodic functions of a phase variable, for example $\eta(\phi)=\eta_0[1+a\sin\phi]$ and $\mu(\phi)=\mu_0+b\cos\phi$, with $\phi_{k+1}=\phi_k+2\pi/P$ modulo $2\pi$. Include optimizer state in $z$, not only weights. First construct the phase-averaged reference optimizer by replacing periodic coefficients with their cycle averages. 2. Pseudocode: initialize $(\theta,m,\phi)$; at each minibatch compute $g=\nabla_\theta L(\theta)$, update $m\leftarrow\mu(\phi)m+g$, update $\theta\leftarrow\theta-\eta(\phi)m$, then update $\phi\leftarrow(\phi+2\pi/P)\bmod 2\pi$. Every $K$ steps, record states at equal phase values and estimate the one-cycle Jacobian using finite differences, automatic Jacobian-vector products, or tangent perturbations. 3. Computed from the mechanism: phase lifting, cycle averaging, tangent/normal decomposition, and the normal-hyperbolicity condition. Estimated empirically: tangent and normal exponents, minibatch perturbation amplitude, and the Floquet multiplier $\mu_N$. Estimate local tangent spaces with PCA on phase-aligned cycle points and measure transverse decay from perturbed replicas. 4. Cheap experiment: train a two-layer MLP on MNIST using constant AdamW, cosine cyclic AdamW, and this phase-augmented optimizer at matched average learning rate and update count. Sweep forcing amplitude $a$ and period $P$. The falsifiable prediction is a stability boundary when the measured transverse multiplier crosses $\mu_N=1$; below the boundary, transverse perturbations decay exponentially with slope approximately $\log|\mu_N|/P$, while above it they fail to decay. The stable region should disappear when the measured gap $\gamma=\alpha-\beta$ becomes comparable to forcing or minibatch-noise amplitude.