Structural Thresholds, Distributed Networks, and Irreversible Regimes From Physical Dissipation to Urban Drainage and Biological Recovery

Many complex systems continue to operate while progressively entering irreversible regimes. This persistence often creates a false sense of safety, as local performance indicators remain within acceptable ranges while global state accessibility collapses. This mismatch between functional continuity and structural safety constitutes a central blind spot in the analysis and governance of dissipative systems. In this article, we introduce a normalized irreversibility index grounded in physical principles of dissipation, recovery, and threshold exceedance. The framework formalizes irreversibility as a loss of accessible states occurring when the frequency of perturbations exceeds the system’s intrinsic recovery time. An explicit alphabet of regimes (O–C–K–Ø) is proposed to characterize structural positions independently of short-term functionality. The index is first established in a general physical setting, then applied to urban drainage networks using the Rainfall Failure Threshold (RFT) as an operational measure. Empirical data from Biarritz show that recurrent non-exceptional rainfall events drive the system into a structurally locked regime (K), despite continued operation and increasing compensatory expenditures. A comparative analysis with distributed urban systems illustrates how network topology alters irreversibility trajectories without requiring greater peak capacity. The framework is subsequently transferred to biological systems through a canonical application to bone tissue under cyclic mechanical stress. Using the notion of a Tissue Failure Threshold (TFT), we show that functional persistence and structural regeneration are not equivalent, and that compensatory remodeling can mask irreversible loss of accessible regenerative states, consistent with established results in systems biology. By unifying physical thresholds, network structure, recovery times, and compensation mechanisms, this work provides a falsifiable and trans-domain framework for diagnosing irreversible regimes. The results suggest that irreversibility is not primarily a consequence of extreme events or insufficient resources, but of indicators that fail to capture structural limits on future accessibility. The framework provides an explicit, step-by-step diagnostic procedure to determine whether recovery remains structurally accessible under repeated forcing.