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Baseband processors (BPs) in cellular devices implement complex radio protocols, and memory corruption vulnerabilities in these implementations can lead to critical security breaches, including remote code execution. Traditional approaches to detecting such vulnerabilities rely on reverse engineering or emulation. However, these methods face significant scalability challenges due to proprietary firmware and architectural complexities. Over-the-air (OTA) testing offers broader applicability but poses challenges in managing UE state, detecting crashes, and ensuring protocol coverage. We present OTABase, an OTA testing framework that enables efficient detection of memory crashes in LTE base-bands by leveraging protocol specifications. OTABase combines three key techniques: a network-side state control mechanism for efficient management of UE states and connections, a specification-guided test case generation targeting memory crashes in NAS and RRC protocols, and a two-phase crash detection oracle utilizing protocol-based liveness checks and manufacturer debug features. Evaluating OTABase on six commercial BPs from three major manufacturers, unearthed seven previously unpatched memory crashes. Among these, three were assigned CVEs, including one out-of-bounds write vulnerability that allows remote code execution. Additionally, we extend OTABase to 5G basebands for PoC, demonstrating its generalizability and practical utility.