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Safe reinforcement learning (RL) aims to learn policies that optimize rewards while satisfying constraints. Predominant approaches rely on soft-constrained policy optimization, which has achieved empirical success but does not provide formal safety guarantees for the learned policy. In contrast, methods with strict guarantees typically rely on explicit certificate functions, whose construction requires the direct synthesis and verification of control-invariant sets, a process that scales poorly with state dimension and often yields overly conservative behavior. In this paper, we present the Provably Safe, yet Scalable RL (PS2-RL) framework, a novel two-phase architecture for learning provably safe policies in a scalable manner, designed to overcome the key bottlenecks of prior methods. Rather than explicitly computing invariant sets, PS2-RL leverages a learned backup policy to forward-integrate the system dynamics, generating an implicit control-invariant set online. In the first phase, the backup policy is trained with the safe-arrival value function to achieve rapid, safe recovery to the certified base set for invariant-set construction. In the second phase, an RL policy is trained end-to-end through a differentiable projection layer that strictly enforces the safety guarantees induced by the learned backup policy. By maximizing the volume of the implicit control-invariant set in the first phase, the resulting PS2 policy from the second phase is performant and scalable, while maintaining provable safety. Crucially, PS2-RL imposes no restrictions on the underlying RL algorithm and can be plugged into any existing training pipeline. We establish theoretical guarantees for the proposed framework and evaluate it on robotic control tasks with state dimensions up to 10, a regime in which prior provably safe RL methods struggle or become impractical.

Many everyday robot manipulation skills are affordance-dependent, with success determined by whether the robot contacts the functional object region required by the subsequent action. Current simulation data generators obtain contacts from generic grasp estimators or per-object manual contact annotations, but generic estimators rank stable grasps without task semantics and often select contacts that are misaligned with the downstream action, while manual contact annotations must be rewritten for each new object and task. To solve these challenges, we introduce AffordSim, a scalable data generator and benchmark that integrates open-vocabulary 3D affordance prediction into simulation-based trajectory generation. Given a natural-language task description, AffordSim synthesizes a task-relevant scene, emits affordance queries, grounds them on object surfaces, samples region-conditioned grasps, and selects executable candidates with motion planning. It further randomizes object pose, texture, lighting, image noise, and cross-viewpoint backgrounds for sim-to-real transfer. We instantiate AffordSim as a 50-task benchmark across diverse manipulation skills, five robot embodiments, and 500+ rigid and articulated objects. AffordSim achieves 93% of the trajectory collection success rate of manual contact annotations on affordance-critical tasks and 89% on hard composite tasks. Vision-language-action policies trained on AffordSim data transfer zero-shot to a real Franka FR3, reaching 24% average success.