Recent breakthroughs in generative simulation have harnessed Large Language Models (LLMs) to generate diverse robotic task curricula, yet these open-loop paradigms frequently produce linguistically coherent but physically infeasible goals, stemming from ungrounded task specifications or misaligned objective formulations. To address this critical limitation, we propose FATE (Feasibility-Aware Task gEneration), a closed-loop, self-correcting framework that reimagines task generation as an iterative validation-and-refinement process. Unlike conventional methods that decouple generation and verification into discrete stages, FATE embeds a generalist embodied agent directly into the generation loop to proactively guarantee the physical groundedness of the resulting curriculum. FATE instantiates a sequential auditing pipeline: it first validates static scene attributes (e.g., object affordances, layout compatibility) and subsequently verifies execution feasibility via simulated embodied interaction. Critical to its performance, upon detecting an infeasible task, FATE deploys an active repair module that autonomously adapts scene configurations or policy specifications, converting unworkable proposals into physically valid task instances. Extensive experiments validate that FATE generates semantically diverse, physically grounded task curricula while achieving a substantial reduction in execution failure rates relative to state-of-the-art generative baselines.

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