Hidden Cost, Visible Calm: A Fold-Bifurcation Theory of Performance Facades and Abrupt Systemic Collapse with Multi-Reserve, Network and Passive-Identification Extensions

Many human, organisational and algorithmic systems can maintain near-target output while a hidden reserve is depleted by compensatory effort. We formulate a minimal dynamical theory in which reserve Φ follows a one-dimensional saddle-node system governed by a dimensionless facade-load index Λ, combining demand, baseline capacity, recovery and coherence. We derive a closed-form critical coherence, an exact finite-time collapse formula, a universal inverse-square-root bottleneck law, a strict Lyapunov potential, and a visibility result showing why effort cost can warn before output changes. The −1/2 lag exponent persists for a broad class of smooth divergent effort-cost functions. Extensions treat depletion-dependent coherence and a coherence-controlled Hopf loop, multiple reserves with heterogeneous recovery timescales, symmetric network coupling and absorbing sinks, and passive identification from quasi-static records. Numerical experiments verify the analytical results and test early-warning robustness under cost-law misspecification, heavy-tailed noise, missing observations, coarse sampling and smoothed reporting. In the baseline stochastic ensemble, cost variance increased while output remained statistically unchanged, and a cost-level alarm preceded simulated collapse substantially earlier than an output-level alarm. The results constitute a theoretical and computational framework; no claim of empirical validation for real systems is made.