2026/03/28 by Bernd von Mallinckrodt · 1 voice
Earth and Planetary Sciences · Economics, Econometrics and Finance · Environmental Science · #Complex Systems and Time Series Analysis #Earthquake Detection and Analysis #Ecosystem dynamics and resilience
paper · doi:10.5281/zenodo.19285222
openalex publication_date 2026/03/28 · openalex created_date 2026/03/29 · openalex updated_date 2026/07/01
This preprint develops a unified theoretical framework for understanding collapse and its detectability in complex systems by integrating structural and epistemic perspectives. The Compression–Response–Transition Index (CRTI) formalism characterises collapse as a consequence of increasing structural compression of accessible state space (Φ) and declining adaptive capacity (R), combined into a viability ratio T = R/Φ. The paper introduces a complementary mechanism, Projection-Induced Determinism (PID), which formalises how low-dimensional observation of high-dimensional systems systematically distorts estimates of both structural compression and adaptive capacity. By modelling observation as a projection operator P: ℝⁿ → ℝᵏ, the framework shows that the observed viability ratio Tobs is not a direct reflection of the system’s intrinsic state, but a joint function of system dynamics and observation geometry, expressed as Tobs = g(Ttrue, P). A semi-formal characterisation of the distortion function g is provided, linking projection alignment with the eigenstructure of the system covariance matrix. This analysis yields explicit conditions under which projection produces monotonic distortion, as well as conditions under which sign inversion occurs, leading to potentially misleading early warning signals. The framework further establishes that collapse detection is observer-dependent as a structural consequence of dimensionality reduction, rather than a contingent artifact of noise or finite data. A mechanism-dependent taxonomy (bifurcation-driven, noise-amplified, and structural-compression-driven collapse) clarifies when classical early warning signals are expected to succeed or fail. The CRTI framework is shown to be specifically informative in systems where collapse is driven by progressive loss of adaptive dimensionality rather than by classical bifurcation dynamics. A refined collapse criterion Tc is introduced as a system-dependent threshold scaling with environmental complexity and perturbation dimensionality. The paper derives three falsifiable predictions regarding projection-induced distortion and outlines a simulation programme based on parameterised Ornstein–Uhlenbeck systems to test these predictions. Implications are discussed for early warning methodology, model reduction, and the epistemology of collapse detection. Core Framework structural compression adaptive capacity viability ratio CRTI Projection-Induced Determinism Theory & Methods early warning signals critical transitions complex systems dimensionality reduction covariance structure effective rank information theory collapse dynamics resilience model reduction observer effects Ashby’s Law of Requisite Variety Ornstein–Uhlenbeck process