投影、记忆与标量早期预警信号的有效性：Schur--Volterra理论及其在AMOC中的应用
Projection, Memory, and the Validity of Scalar Early-Warning Signals: A Schur--Volterra Theory with an AMOC Application
AI总结：
本研究提出Schur-Volterra算子理论，统一分析投影导致的标量早期预警信号歧义，区分三种记忆机制，并通过AMOC实验验证EOF1-EOF2可揭示崩溃前的互补预警信息。
AI中文摘要：
早期预警信号通常由一个或少数几个可观测变量计算得出，尽管底层动力学是高维的。投影造成了结构性歧义：相同的标量记录可能反映谱稳定性丧失、隐藏状态记忆、非正态放大、物理平衡的转折点，或这些因素的组合。我们发展了一个统一的算子理论，使这些替代方案变得明确。对于已解析-隐藏块线性化，我们推导了精确的Volterra约化、有限和扩散马尔可夫提升，以及稳定性问题的Schur-预解式分解，区分了三种记忆概念：实现路径式、结构Schur返回和有限时域投影式。一个协方差定理给出了标量从接近单位圆的实乘子继承临界减速的充分条件：谱分离、有界特征向量条件数、临界模式的可观测性、该模式的随机激发以及准平稳强迫。然后我们证明了失效机制：临界模式可能不可见或未被激发；振荡穿越不一定提高滞后一阶相关；非正态族可以在固定稳定谱下产生发散的标量方差；基于物理的转折点不一定与谱边界重合；有限时域数据无法唯一识别长记忆。该理论通过一个4400年的CESM AMOC滞回实验加以说明。仅EOF1能检测到改变后的敏感性，但EOF1-EOF2是揭示恢复后改变的Schur返回和投影记忆的最小解析空间；F_ovS在崩溃前26年达到最小值，而模态裕度并未向零减小：这两个诊断探测互补机制。该应用是机制性的且基于单轨迹；未声称有经过验证的早期预警信号。
英文摘要：
Early-warning signals are usually computed from one or a few observables, although the underlying dynamics are high dimensional. Projection creates a structural ambiguity: the same scalar record may reflect spectral loss of stability, hidden-state memory, nonnormal amplification, a turning point of a physical balance, or a combination of these. We develop a unified operator theory that makes these alternatives explicit. For a resolved-hidden block linearization we derive exact Volterra reductions, finite and diffusive Markovian liftings, and a Schur-resolvent factorization of the stability problem, distinguishing three notions of memory: realized pathwise, structural Schur return, and finite-horizon projected. A covariance theorem gives sufficient conditions for a scalar to inherit critical slowing down from a real multiplier approaching the unit circle: spectral separation, bounded eigenvector conditioning, observability of the critical mode, stochastic excitation of that mode, and quasi-stationary forcing. We then prove failure mechanisms: a critical mode can be invisible or unexcited; an oscillatory crossing need not raise lag-one correlation; a nonnormal family can produce diverging scalar variance with a fixed stable spectrum; a physics-based turning point need not coincide with a spectral boundary; and finite-horizon data cannot identify long memory uniquely. The theory is illustrated with a 4400-year CESM AMOC hysteresis experiment. EOF1 alone detects altered susceptibility, but EOF1-EOF2 is the smallest resolved space revealing changed Schur return and projected memory after recovery; F_ovS reaches its minimum 26 years before collapse while the modal margin does not decrease toward zero: the two diagnostics probe complementary mechanisms. The application is mechanistic and single-trajectory; no validated early-warning signal is claimed.
