Projection-Induced Determinism: An Epistemic Constraint on Early Warning Signals with a Mathematical Characterization of Sign Inversion

This preprint introduces Projection-Induced Determinism (PID) as a formal epistemic constraint on collapse detection in complex systems. Classical early warning signal (EWS) frameworks—particularly those based on Critical Slowing Down—implicitly assume that the observation process preserves the qualitative behavior of system dynamics. We show that this assumption can fail under dimensionality reduction.   PID arises when a high-dimensional stochastic system is observed through a lower-dimensional projection, leading to a distortion of covariance structure and, under specific conditions, a reversal of the sign of dynamical indicators. We provide a mathematical characterization of this effect based on the covariance derivative D(\lambda) = \partial_\lambda \Sigma_x and its spectral decomposition.   The core result establishes that sign inversion of trace-based indicators under linear projection is only possible when D(\lambda) is indefinite. In particular, if D(\lambda) is positive semidefinite—corresponding to classical critical slowing down—projection can attenuate but cannot invert early warning signals. This identifies a sharp boundary condition for the validity of EWS methods.   We further derive necessary and sufficient conditions for sign inversion in terms of the projection-weighted eigenstructure of D(\lambda), and provide a geometric interpretation based on the alignment between the observation subspace and the collapse-relevant eigenspaces. Two complementary indices are introduced: a subspace suppression index quantifying the invisibility of collapse-relevant directions, and a trend alignment index capturing the directional consistency between observed and true system dynamics.   Rather than replacing existing approaches, PID defines the conditions under which early warning signals remain diagnostically valid. This reframes collapse detection as a joint problem of system dynamics and observability, with implications for ecological monitoring, high-dimensional data analysis, and model-based inference.     Projection-Induced Determinism Early Warning Signals Complex Systems Collapse Detection Critical Transitions     Covariance Structure Spectral Decomposition Linear Projection Eigenvalue Analysis High-Dimensional Systems     Critical Slowing Down Multivariate Early Warning Signals Fisher Information Spectral Entropy Effective Rank     Observability Epistemic Constraints Dimensionality Reduction System Collapse Complexity Science