Universal critical timescales in slow non-Hermitian dynamics

Non-Hermitian systems driven along slow parametric loops undergo non-adiabatic transitions whose outcome depends sensitively on the driving speed, yet no explicit formula has been available for the critical timescale $T_{\mathrm{c}\mathrm{r}}$ at which these transitions develop. Using a $2\times 2$ Hamiltonian with circular parameter trajectories, we derive $T_{\mathrm{c}\mathrm{r}}=\mathcal{G}\ln (1\mathrm{/}\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid })$ in closed form for non-encircling loops, phase-shifted loops, offset loops, and loops encircling exceptional points, where $\mathcal{G}$ is a geometry-dependent growth factor and $\mathrm{\Delta }$ is the instability seed. This formula sharply separates the regime where the system remains in the averagely dominant eigenstate ($T<T_{\mathrm{c}\mathrm{r}}$) from the superadiabatic regime where the instantaneous dominant eigenstate takes over ($T>T_{\mathrm{c}\mathrm{r}}$), resolving the apparent tension between the previous literature. We identify two competing seeds: a geometric Stokes multiplier and the finite-precision floor. When the geometric seed vanishes, precision alone governs the transition, yielding $T_{\mathrm{c}\mathrm{r}}\propto m\ln \beta$, linear in the number of precision bits $m$. This provides a purely forward-evolution manifestation of precision-induced irreversibility (PIR)~\cite{PIR}, demonstrating that the fundamental limit identified through echo protocols also controls the outcome of slow non-Hermitian dynamics without requiring time reversal. For PT-symmetric energy spectra, $T_{\mathrm{c}\mathrm{r}}$ additionally determines the onset of chirality: the dynamics is non-chiral for $T<T_{\mathrm{c}\mathrm{r}}$ and chiral for $T>T_{\mathrm{c}\mathrm{r}}$.