$\lambda$-Reachability: Geometric-Horizon Safety Bellman Equations for Humanoid Safety
arXiv:2606.16022v1 Announce Type: new Abstract: We introduce $\lambda$-Reachability, a scalable approach to Hamilton--Jacobi safety analysis for high-dimensional robotic systems. Unlike prior discounted formulations that rely on fixed one-step Bellman updates, $\lambda$-Reachability employs a stochastic multi-step estimator of the safety value, using a geometrically distributed rollout horizon together with a randomly absorbed terminal. Conceptually analogous to TD($\lambda$), $\lambda$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($\lambda$), where the terminal value is always incorporated in learning targets, the terminal safety value in $\lambda$-Reachability is only used at a probability controlled by parameter $\delta$. We formally show that for $\delta<1$, the update induces a contraction mapping that allows temporal-difference learning; as $\lambda \to

