AI 中文总结
该研究为不可压缩欧拉方程的欧拉-雷诺次解构建微分拓扑框架，通过提升型公式构造极限空间，建立一阶理论，引入有限混合模型实现切方向并得到相计数界与极小性结果。
AI 中文摘要
我们为不可压缩欧拉方程的欧拉-雷诺次解几何构建了一套微分拓扑框架。通过转向提升型公式，其中速度、二次通量和无迹雷诺应力被视为独立变量，我们构造了一个由光滑严格次解的弱闭包得到的微分拓扑极限空间。尽管不存在任何基础流形结构，该空间的内部切空间仍提供了无穷小变形的内在概念。我们证明，内部切向量的环境实现满足分布意义下的线性化欧拉-雷诺方程，从而为提升型弱极限空间建立了自然的一阶理论。我们进一步描述了与欧拉轨迹兼容的切方向，并确定了自然速度、通量和全态观测的核。这些结果将可观测扰动与编码雷诺应力无穷小变化的隐藏应力-规范方向区分开来。最后，我们为提升型欧拉-雷诺状态引入了有限混合模型，其微分实现了显式切方向，并将雷诺应力与无穷小相分裂关联起来。在一般性假设下，每个偏应力张量都可由此类一阶混合缺陷实现，得到相计数界和观测层级的极小性结果。
英文摘要
We develop a diffeological framework for the geometry of Euler--Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler--Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler--Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.
