State-Range Observer Gain Scheduler

1. Integration point: attach the scheduler to a neural ODE, continuous-time RNN, or learned world model receiving intermittent noisy measurements. Use a base gain $G_0$ and candidate multipliers $\mathcal{R}=\{1,1.25,1.5,2,3,4\}$. The scheduler changes only the residual-correction branch. 2. Pseudocode: before each observation update, compute Jacobian-vector products of $f_{\theta}$ at the current state; for each candidate $\rho$, form $H_t(\rho)=U_K^{\mathsf T}(J_t-\rho G_0C)U_K$; estimate the largest eigenvalue of $PH_t+H_t^{\mathsf T}P$ by power iteration; choose the smallest candidate satisfying the negative-margin condition; then update $z$ with $f_{\theta}(z)+\rho G_0(y-Cz)$. Add hysteresis by increasing $\rho$ immediately after a violation and decreasing it only after ten safe steps. 3. The Lyapunov inequality and gain-selection rule come from the observer mechanism; $J_t$, $P$, the state-range coverage, and the noise tradeoff are estimated. Solve for $P$ once using an SDP on stored Jacobians or learn it jointly. 4. First experiment: use a neural ODE on noisy partially observed Lorenz-63, comparing fixed low gain, fixed high gain, and scheduled gain. Sweep the estimated maximum projected eigenvalue. The predicted signature is negative reconstruction-error slope below the stability boundary and positive slope above it; the measured sign-change boundary should predict failure within 20 percent. During safe episodes, scheduling should reduce residual variance by at least 20 percent relative to fixed high gain while retaining its recovery rate after an unstable excursion.