Entropy Collapse: A Universal Failure Mode of Intelligent Systems

Abstract

A foundational assumption in complex-system collapse studies is that critical transitions are second-order, preceded by early-warning signals like rising autocorrelation, variance, and critical slowing down (Scheffer, 2009). We show this fails for feedback-amplified adaptive systems. We prove entropy collapse - the irreversible contraction of effective state space when feedback amplification alpha exceeds novelty regeneration beta - is a first-order (discontinuous) phase transition. Four exact results: (1) Threshold alpha_c(beta) = 1/(1-beta), from Jacobian spectrum of Multiplicative-Weights operator. (2) Discontinuity: entropy order parameter m = 1 - H_ss/H_max jumps Delta m_0 = 0.698 at alpha_c, with hysteresis Delta H_hyst approx 2.73 nats (lower bound; up to 3.9 nats in simulations); no pre-transition warnings as autocorrelation and variance stay finite. (3) Relaxation exponent nu = 1, from transcritical bifurcation (R^2 = 0.9997 vs. simulation); universality across update mechanisms. (4) Two classes: feedback curvature kappa = f''(1/N) determines order - Class 1 (kappa > 0, convex, e.g., power-law) irreversible with nu = 1; Class 2 (kappa = 0, linear) reversible with nu = 1/2. Theorems validated in neural experiments on two-layer autoregressive transformer (SmallGPT, N=50 vocab, 92 conditions, 8 seeds/condition): Delta H_hyst^NN = 2.92 nats > 2.73 (Theorem 2); nu^NN = 1.14 +/- 0.13, R^2 = 0.977 (Theorem 3). This unifies AI model collapse (Shumailov et al., 2023), economic institutional sclerosis, and evolutionary genetic bottlenecks as first-order entropy-driven processes, evading standard early-warning monitoring.

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