发现智能系统崩溃的全新机制：熵坍缩导致突变式失效，无法提前预警。
Entropy Collapse: A Universal Failure Mode of Intelligent Systems
- 提出熵坍缩理论，揭示反馈放大超过新信息生成时系统会突变崩溃。
- 实验验证熵滞回量达2.92纳特，无早期预警信号，与理论预测一致。
- 适用于AI模型、经济制度、基因演化等场景，适合关注系统风险的研究者。
复杂系统崩溃研究常假设临界转变是二阶的，伴有自相关性、方差上升等预警信号（Scheffer, 2009）。本文发现，对于反馈放大型自适应系统，该假设不成立。我们证明：当反馈放大系数α超过新信息再生系数β时，有效状态空间不可逆收缩，形成一阶（不连续）相变——熵坍缩。四个严格结果：(1) 阈值α_c(β) = 1/(1−β)，由乘法权重算子的雅可比谱导出；(2) 熵序参量m = 1 − H_ss/H_max在α_c处跳跃Δm₀ = 0.698，滞回量ΔH_hyst ≥ 2.73纳特（模拟中可达3.9纳特），自相关与方差始终有限，无预兆；(3) 松弛指数ν = 1，来自叉积分岔（模拟拟合R² = 0.9997），跨更新机制具有普适性；(4) 分两类：曲率κ = f''(1/N) > 0（凸型，如幂律）为不可逆，ν = 1；κ = 0（线性）为可逆，ν = 1/2。定理在两层自回归Transformer（SmallGPT，N=50词表，92条件，每条件8种子）神经实验中验证：ΔH_hyst^NN = 2.92纳特 > 2.73；ν^NN = 1.14 ± 0.13，R² = 0.977。该理论统一解释了AI模型崩溃（Shumailov et al., 2023）、经济制度僵化、进化遗传瓶颈等一阶熵驱动过程，规避传统预警监测。
原文摘要 · Abstract (English)
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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