Microcanonical Krylov Stability Monitor

Integrate the mechanism at the optimizer or recurrent-cell level. For optimizer monitoring, take $z=\theta$ and $F(\theta)=-\widehat{\nabla}_\theta L$; for an RNN or SSM, take $z=h$ and $F$ equal to the continuous-time hidden-state drift. Choose an observable $O_0(z)$ such as a random projection of the gradient, hidden activation, or perturbation energy $\|\delta h\|^2$. Every $q$ updates, collect a replay window of states and assign samples to shells using loss, gradient norm, or activation norm. Estimate $\mathcal{L}O_n$ by finite differences $(O_n(z_{t+1})-O_n(z_t))/\Delta t$, or by Jacobian-vector products when available. Run $m=8$--$32$ Lanczos steps per shell with reorthogonalization, storing $a_n$ and $b_n$, then solve the resulting small tridiagonal ODE to obtain $K_e(t)$. Do not backpropagate through the monitor initially. Estimate $\lambda_{K,e}$ from a linear fit of $\log(K_e+\epsilon)$ over the first few time points, smooth it with an exponential moving average, and set the shell-specific learning-rate or recurrent-step multiplier to $\min(1,c/(\lambda_{K,e}\eta+\epsilon))$. The paper supplies the phase-space generator, Lanczos recursion, and microcanonical decomposition; the neural implementation must estimate the generator, shell measure, and calibration constant $c$. First test a 2-layer MLP on MNIST and a small chaotic-copy RNN task, comparing SGD or Adam with and without monitoring across a learning-rate sweep. The falsifiable prediction is that shells with larger measured $\lambda_{K,e}$ reach loss or hidden-state divergence first, and that the instability boundary follows $\eta_c\approx c/\lambda_{K,e}$ within 20 percent. Shell-resolved control should reduce failures more effectively than a global average Krylov complexity.