收缩与递归：确定性动力学基于观测预测中的指数分离
Contraction versus Recurrence: An Exponential Separation in Observation-Based Prediction of Deterministic Dynamics
AI总结：
研究基于观测预测确定性动力学中收缩与递归方法的差异，通过形式化和实验验证两者指数分离，给出成本缩放规律及数值验证结果，还用准入协议补充定理，解释分形维数，在实际数据上验证了该方法。
AI中文摘要：
给定一个具有低维吸引子的遍历动力系统的标量可观测量，有两类方法用于重构和预测基础状态：基于递归的方法（类似物方法及其衍生方法），等待轨迹回到先前观测状态的 $\varepsilon$ 邻域；基于观测器的方法，在延迟重构上拟合收敛状态估计器。我们形式化并通过实验验证了两者之间的指数分离：递归的预期成本按 $\varepsilon^{-d}$ 缩放，其中 $d$ 是不变测度的逐点维度（这是Kac引理和定量庞加莱递归的结果），而可检测的线性观测器在 $\Theta(\log(1/\varepsilon)/(1-\rho(A_{cl})^2))$ 步内收敛，其中 $\rho(A_{cl})$ 是里卡蒂不动点的闭环谱半径。通过数值验证了这两个规律（洛伦兹吸引子上的返回时间指数为 -1.8，理论值为 -2.05；观测器成本在 $\log(1/\varepsilon)$ 上线性，$R^2 = 1.000$，在 $(1-\rho^2)^{-1}$ 上线性，$R^2 = 0.985$），在 $\varepsilon = 10^{-6}$ 和 $d\approx 2$ 时产生约 $10^{9}$ 的实测成本差距。我们用一个准入协议（Kac - 里卡蒂门）补充该定理，通过替代数据预测门控来确定信号是否属于该定理的类别；它还将“通用”分形维数的民间传说解释为由 $2\log_{10}N$ 界定的数据集大小伪像。在实际数据上，该门接纳圣达菲激光基准（$\hat D_2 = 2.0$）并拒绝月度太阳黑子序列，重现了历史低维性主张的既定分辨率。所有结果都可以从单个验证脚本（17/17检查）中重现。
英文摘要：
Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state: recurrence-based methods (the method of analogues and its descendants), which wait for the trajectory to return to an $\varepsilon$-neighborhood of a previously observed state, and observer-based methods, which fit a converging state estimator on the delay reconstruction. We formalize and empirically verify an exponential separation between the two: the expected cost of recurrence scales as $\varepsilon^{-d}$, where $d$ is the pointwise dimension of the invariant measure (a consequence of the Kac lemma and quantitative Poincare recurrence), whereas a detectable linear observer converges in $Θ(\log(1/\varepsilon)/(1-ρ(A_{cl})^2))$ steps, where $ρ(A_{cl})$ is the closed-loop spectral radius of the Riccati fixed point. Both laws are verified numerically (return-time exponent $-1.8$ on the Lorenz attractor against the theoretical $-2.05$; observer cost linear in $\log(1/\varepsilon)$ with $R^2=1.000$ and in $(1-ρ^2)^{-1}$ with $R^2=0.985$), yielding a measured cost gap of $\sim 10^{9}$ at $\varepsilon=10^{-6}$ for $d\approx 2$. We complement the theorem with an admission protocol (the Kac-Riccati gate) deciding whether a signal lies inside the theorem's class, via surrogate-data prediction gating; it also explains the folklore of "universal" fractal dimensions as a dataset-size artifact bounded by $2\log_{10}N$. On real data the gate admits the Santa Fe laser benchmark ($\hat D_2=2.0$) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims. All results reproduce from a single verification script (17/17 checks).
