The statement that applying the annihilation operator to the vacuum state yields the zero vector is a foundational definition in quantum mechanics and quantum field theory, making it common knowledge that does not require citation.
The claim is a fundamental definition in quantum mechanics (specifically $a|0\rangle = 0$, where $a$ is the annihilation operator and $|0\rangle$ is the vacuum state). None of the retrieved papers specifically test or focus on proving this elementary definition, as it is standard textbook knowledge in quantum theory. Therefore, it is classified as COMMON_KNOWLEDGE.