Persistent language-model agents increasingly combine tool use, tiered memory, reflective prompting, and runtime adaptation. In such systems, behavior is shaped not only by current prompts but by mutable internal conditions that influence future action. This paper introduces layered mutability, a framework for reasoning about that process across five layers: pretraining, post-training alignment, self-narrative, memory, and weight-level adaptation. The central claim is that governance difficulty rises when mutation is rapid, downstream coupling is strong, reversibility is weak, and observability is low, creating a systematic mismatch between the layers that most affect behavior and the layers humans can most easily inspect. I formalize this intuition with simple drift, governance-load, and hysteresis quantities, connect the framework to recent work on temporal identity in language-model agents, and report a preliminary ratchet experiment in which reverting an agent's visible self-description after memory accumulation fails to restore baseline behavior. In that experiment, the estimated identity hysteresis ratio is 0.68. The main implication is that the salient failure mode for persistent self-modifying agents is not abrupt misalignment but compositional drift: locally reasonable updates that accumulate into a behavioral trajectory that was never explicitly authorized.