AI 中文总结
该研究针对保测动力学下纤维到截面提升的结构障碍，证明了希尔伯特空间势的互易性、Koopman记忆通道的可识别性、变分记忆的反转刚性等结论，给出分支粘贴的相关条件及强紧性准则。
AI 中文摘要
许多非线性和变分问题是在概率空间上逐纤维提出的，而与动力学相关的对象是通过保测变换耦合的可测截面。我们研究这种纤维到截面提升产生的结构障碍。首先，对于实希尔伯特空间上的全局连续可微势，闭子空间之间无影响是互易的；对于有限正交分解，这会在依赖分量上产生加性分解。其次，有限Koopman记忆通道在严格轨道分离条件下可精确识别。逐通道解决经典希尔伯特空间势准则后，会在正向和反向记忆之间产生移位伴随平衡。在全局有限差分层面，若具有任意有限动力学记忆的截面映射是全局连续可微势的梯度，其真正活跃的滞后在时间反演下不变。因此全局变分记忆是反转完备的，单侧记忆坍缩为当前状态局域性。连续邻近算子通过其凸势表示继承相同的刚性；不连续邻近选择表明连续性通常无法去除。最后，在两个相干分支逐点非退化条件下，消失的分支粘贴残差等价于分支标签的渐近不变性。对于有限生成保测遍历作用，强遍历性是所有分支粘贴近似零序列相对紧性的精确阈值。我们还记录了完整可测选择提升的强紧性准则以及非凸Wasserstein最小步的示例。
英文摘要
Many nonlinear and variational problems are posed fibrewise over a probability space, whereas the dynamically relevant objects are measurable sections coupled by measure-preserving transformations. We study structural obstructions created by this fibre-to-section lifting. First, for a global continuously differentiable potential on a real Hilbert space, absence of influence between closed subspaces is reciprocal; for finite orthogonal decompositions this yields an additive decomposition over dependency components. Second, finite Koopman memory channels are identifiable exactly under a sharp orbit-separation condition. Resolving the classical Hilbert-space potentiality criterion channel by channel then gives a shifted-adjoint balance between forward and reverse memories. At the global finite-difference level, if a section map with arbitrary finite dynamical memory is the gradient of a global continuously differentiable potential, its genuinely active lags are invariant under time reversal. Thus globally variational memory is reversal-complete, and one-sided memory collapses to present-state locality. Continuous proximity operators inherit the same rigidity through their convex-potential representation; a discontinuous proximal selection shows that continuity cannot generally be removed. Finally, under pointwise nondegeneracy of two coherent branches, vanishing branchpasting residual is equivalent to asymptotic invariance of the branch labels. For finitely generated ergodic probability-preserving actions, strong ergodicity is therefore the exact threshold for relative compactness of all branch-pasting approximate-zero sequences. We also record a strong-compactness criterion for full measurable-selection lifts and a nonconvex Wasserstein minimizing-step illustration.
Comments32 pages, no figures
