Research shows large language models continue to believe false statements even after being explicitly told they are incorrect. This finding directly threatens clinical AI applications where accuracy is non-negotiable. Healthcare systems deploying LLMs for documentation, clinical decision support, or EHR integration must now confront a fundamental reliability problem: these models cannot be corrected through standard instruction methods. The implications are severe for value-based care models that depend on accurate data capture and for healthcare organizations seeking AI vendors. Until this persistence-of-false-belief issue is solved, LLMs cannot be safely deployed for clinical workflows where patient safety depends on factual accuracy.