Longitudinal healthcare agents require persistent state tracking under temporal uncertainty. In domains like chronic disease management, patient states—medications, symptoms, and vital signs—evolve continuously over months. Existing memory architectures for Large Language Models (LLMs) are inherently retrieval-centric: they treat memory as a static repository of past interactions, failing to resolve conflicting or superseded information when queried for the current patient state. We propose a shift to state-centric memory. Our framework introduces (1) a bi-temporal state representation that decouples event time from ingestion time and tracks temporal validity windows, (2) an incremental state arbitration mechanism using four operators—SUPPORT, REFINE, SUPERSEDE, and BRANCH-CONFLICT—to handle evolving medical facts without destructive overwriting, and (3) a confidence-thresholded evidence escalation layer for robust, efficient memory access. Evaluated on a longitudinal diabetes management suite as a representative biomedical state tracking task, our method achieves a Unique-F1 of 0.85 and Conflict-F1 of 0.98, substantially improves upon long-context LLMs (0.38 / 0.89) and standard vector memory (0.30 / 0.60), demonstrating that agentic AI in longitudinal biomedical settings requires continuous, evidence-grounded arbitration rather than simple retrieval.