In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.

中文翻译：

---

平面$$4$$-体问题中凸中心构形的唯一性

在本文中，我们提供了严格的计算机辅助证明（CAP），证明了在平面四体问题中，对于属于闭域的给定顺序的任何四个固定正质量，存在唯一的凸中心配置。质量空间。该证明采用了 Krawczyk 算子和隐函数定理 (IFT)。值得注意的是，我们证明了隐函数定理可以与区间分析相结合，使我们能够估计隐函数存在的区域的大小，并将我们的发现从一个质点扩展到它的邻域。