Abstract: Robots engaged in fast physical interactions often need to act before the intent of another agent is fully known. Anticipatory goalkeeping illustrates this challenge. Waiting provides more reliable information about the target but reduces the physical opportunity for interception, whereas acting early preserves reachability but requires initiating motion under uncertainty. Given a fixed closed-loop save controller, we formulate the decision of when to initiate motion as a policy-conditional finite-horizon optimal stopping problem. Building on this formulation, we propose monotone optimal stopping (MOS), a structured release-timing method for dynamic robotic interception. The quadruped save policy is trained with reinforcement learning, while MOS determines when the policy should be activated from the evolving robot state and target belief. Rather than predicting a release time or relying on confidence alone, MOS learns the return advantage of acting now over waiting for one more observation. We derive a direct Bellman recursion for this act-versus-wait margin and impose monotonicity only with respect to physical urgency, reflecting the irreversible loss of interception opportunity as time elapses. This structure enables early activation for dynamically demanding saves while preserving closed-loop adaptation when later observations change the predicted target. Under a single-crossing condition, MOS admits a threshold release boundary with a bounded approximation error. Extensive simulation studies show that MOS improves the mean save rate from 67.7% to 74.4% over a parameter-matched learned gate and increases reversal saves from 52.1% to 66.5%. Real-robot experiments further demonstrate rapid interception and post-release direction correction under human shot-direction feints.