Lipschitz-Certified Cache Refresh

1. Integration point: apply this controller to transformer KV caching, recurrent hidden-state reuse, or a latent state in a learned world model. Define a certificate vector $c_t$ from quantities whose drift can be bounded, such as logit margin, normalized hidden-state displacement, predictive entropy, constraint residual, or deviation from a freshly recomputed state. Define an admissible set, for example $\mathcal{A}=\{c:m(c)\ge m_{\min},\ e(c)\le e_{\max}\}$, where $m$ is classification margin and $e$ is a prediction-error proxy. 2. Pseudocode: at each cheap sampling point, compute the certificate from the stale state; estimate $\widehat L_t$ using the maximum certificate slope over the last $k$ samples multiplied by a safety factor $q>1$; compute distance to the constraint boundary; if that distance is below $\widehat L_t(\Delta t+\ell)$, regenerate the KV cache or latent state, otherwise reuse it. In pseudocode: c=certificate(stale_state,x); Lhat=q*max_recent_slope(c); margin=distance_to_invalid_boundary(c); if margin <= Lhat*(dt+latency): state=fresh_forward(x); else: state=stale_state. 3. Computed from the paper: robust set contraction by the drift budget $L\tau$, the horizon rule, and the memoryless regeneration policy. Estimated empirically: $L$ or $\widehat L_t$, latency $\ell$, and the certificate-to-safety boundary; calibrate $q$ on a held-out stream so that violations are below a chosen target. 4. First cheap experiment: use a small GPT-2-style transformer on WikiText-2 or a synthetic long-context copy task, compare fixed cache refresh intervals against this policy at matched average FLOPs. Record the first-refresh horizon and safety violations, defining violation as a margin or perplexity threshold crossing before refresh. The quantitative prediction is that, for held-out sequences whose certificate drift satisfies the measured Lipschitz bound, no violation occurs during the next $\tau$ steps; empirically the violation rate should be near zero below the boundary $\operatorname{dist}(c,\mathcal{A}^{\mathsf c})=\widehat L_t\tau$, and refresh frequency should increase sharply when latency or the drift multiplier crosses this boundary.