Knowledge acquisition from black-box large language models (LLMs) offers a practical approach to adapting smaller student models. However, many existing methods rely on predefined queries or student feedback to guide acquisition, leaving autonomous expansion of knowledge targets an important challenge. We propose \textbf{MapSteal}, a student-independent framework for structured recursive knowledge acquisition. Starting from a small set of initial roots, MapSteal represents acquisition targets using structured states and recursively expands them through teacher-proposed directions. Question realization, proposal admission, and deduplication constrain the acquisition process, while root-aware scheduling organizes exploration under a fixed QA budget. The resulting question--answer pairs are used to adapt student models without requiring student feedback during acquisition. Experiments with Qwen3.5 students at 2B, 4B, and 9B scales show competitive downstream performance across six medical and cybersecurity benchmarks. On the 4B medical evaluation, Full MapSteal improves average accuracy by 1.17 percentage points over a simplified recursive acquisition variant. Cross-domain evaluations show that gains depend on the domain and evaluation panel. These results demonstrate the feasibility of structured, student-independent recursive acquisition while highlighting the distinction between generating acquisition targets and obtaining useful training supervision. Our code is available at \href{https://github.com/LabRAI/Mapsteal} {https://github.com/LabRAI/Mapsteal}.

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