Finite-Excitation Latent Replay

1. Integration point: use a latent ODE, state-space model, or recurrent model with a dynamics head that is linear in an adapter parameter vector \(\theta\); a neural feature map \(\Phi_\psi(z,u)\) can generate the regressors. Maintain a buffer of windows \((z(t_i),z(t_i+T),u_{i:i+T})\), where \(z\) is the encoder or recurrent hidden state. 2. Pseudocode: for every stored window, numerically integrate the feature matrix using \(\widehat{\Omega}_i=\sum_k\Phi_\psi(z_k,u_k)\Delta t\); form \(\widehat G=\sum_i\widehat\Omega_i^{\mathsf T}\widehat\Omega_i\); estimate latent uncertainty \(\varepsilon_i\) from an ensemble variance, dropout variance, or observer residual; compute \(q=\lambda_{\min}(\widehat G)-\sum_i(2\|\widehat\Omega_i\|\varepsilon_i+\varepsilon_i^2)\). If \(q>\gamma\), update \(\theta\) using the integral-regression residual; otherwise freeze or downweight the dynamics-adapter update. Select history windows greedily by maximizing the increase in \(\lambda_{\min}(\widehat G)\). 3. The integral regression and perturbation inequality are taken from the paper; the true matrix and exact state error are unavailable, so \(\varepsilon_i\) must be estimated empirically and inflated by a safety factor. 4. First cheap experiment: train a two- or four-dimensional latent state-space model on Lorenz-63 or a damped nonlinear oscillator, comparing Adam, ungated integral regression, and excitation-gated replay. Sweep observation noise and window length. The quantitative prediction is that parameter error decreases rapidly after \(q\) becomes positive, with the empirical transition from ill-conditioned to stable identification near \(q=0\). When true states are available for evaluation, measured \(\lambda_{\min}(G)\) should exceed the conservative lower bound \(q\), up to numerical integration error.