AI 中文总结
本文证明Sadovskii涡在固定水平平移后线性化层面局部刚性，通过自由边界框架、显式屏障近似及奇偶类分析，结合有限秩比较与区间验证实现非退化性证明。
AI 中文摘要
我们证明了归一化Sadovskii涡补丁在水平平移模下的非退化性。基于Huang和Tong发展的自由边界与根映射框架，我们首先获得精确Sadovskii剖面的精确定量近似，将其自由边界限制在围绕有理近似构造的两个显式屏障之间。然后利用这种定位来控制接触构型处的自然加权二次型。在奇类中，基态恒等式表明水平平移是唯一的零模。在偶类中，膨胀给出负方向，而边界紧致部分上的有限秩比较结合解析接触尾部估计，在余维一子空间上产生正间隙。因此，一旦平移被固定，Sadovskii偶极子在线性化层面是局部刚性的。计算机辅助部分仅限于对有限集合显式不等式的严格区间验证；连续统、接触和算子约简均通过解析方法证明。
英文摘要
We prove nondegeneracy, modulo horizontal translation, of the normalized Sadovskii vortex patch. Building on the free-boundary and root-map framework developed by Huang and Tong, we first obtain a precise quantitative approximation of an exact Sadovskii profile, trapping its free boundary between two explicit barriers constructed around a rational approximation. This localization is then used to control the natural weighted quadratic form at the touching configuration. In the odd class, a ground-state identity shows that horizontal translation is the only zero mode. In the even class, dilation gives a negative direction, while a finite-rank comparison on a compact part of the boundary together with analytic contact-tail estimates yields a positive gap on a codimension-one subspace. Consequently, the Sadovskii dipole is locally rigid at the linearized level once translation is fixed. The computer-assisted part is confined to rigorous interval verification of a finite collection of explicit inequalities; the continuum, contact and operator reductions are proved analytically.
