Lyapunov Stability Without Persistent Time

This work extends classical Lyapunov stability theory to systems in which time, admissibility, and objecthood are not guaranteed to persist. By introducing geometry-attached time, stability is formulated relative to the existence of an admissible regime rather than an assumed global clock. The framework clarifies phenomena such as stable chaos, stable irregular dynamics, and learning under nonstationary or ephemeral conditions, where Lyapunov inequalities may hold locally while asymptotic reasoning fails due to loss of regime validity. The notion of ephemeral objecthood is introduced to distinguish instability of dynamics from collapse of existence itself. Classical Lyapunov theory is preserved within its domain of validity; the contribution of this work is to make that domain explicit and to provide a minimal extension for systems governed by transient admissibility.