Lipschitz Disagreement Coverage

Integrate this into learned transition-model verification, with an optional training regularizer. For each candidate model, maintain a reference transition function f, learned transition g, and a probe set of state-action points z=(s,a). At every probe compute e(z)=||f(z)-g(z)||. When e(z0)=eta exceeds epsilon, estimate local Lipschitz constants for both models using spectral-norm products of MLP layers or empirical slopes L_hat=max_j ||h(z_j)-h(z0)||/d(z_j,z0), then set L=max(L_f,L_g). Construct rho=(eta-epsilon)/(2L). Generate additional verification points inside D intersected with the metric ball B(z0,rho), clipping points to the legal state-action box and using kappa=2^{-(d+m)} at a corner. Pseudocode: find z0=max-error probe; compute eta and L; if eta>epsilon, rho=(eta-epsilon)/(2L); sample K directions u_k uniformly on the unit sphere; set z_k=clip(z0+rho*u_k,D); compare f(z_k) and g(z_k); update the gate distribution toward regions with high disagreement. For training, add lambda*max(0,e(z)-epsilon)^2 or penalize L_hat when eta is large. The theorem determines rho and the volume guarantee; actual gate mass p and local L estimates are empirical. Begin with a two-dimensional cart or pendulum simulator and an MLP transition model, comparing uniform verification against adaptive ball sampling at equal model evaluations. Success is higher detection probability and lower planner regret at fixed N, especially for boundary-localized errors.