Z-Irreversibility: Under Finite Correction Constraint

Most complex systems measure output variables. Few measure remaining correction capacity. We formalize structural irreversibility within bounded dynamical systems using Lyapunov framing and functional analysis. A system evolving in a Banach space with finite correction rate admits a critical threshold beyond which restoration to equilibrium becomes asymptotically impossible. The result is independent of domain semantics and follows directly from bounded control constraints under cumulative deviation growth.We then construct a diagnostic framework that operationalizes this theorem through measurable quantities: risk ratio R*, stability margin M, and uncertainty penalty U. The framework reveals that opacity directly accelerates irreversibility through structural mechanisms, not moral arguments.