四体凸中心构型对所有质量的唯一性
Uniqueness of four-body convex central configurations for all masses
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中文总结 AI 辅助
本文证明了任意四个正质量在任意循环排序下存在唯一凸平面中心构型，通过Hessian下界与区间算术验证，并用Lean证明助手形式化验证。该方法解决了Simó-Yoccoz猜想，并推出退化构型为凹及对称定理。
中文摘要 AI 辅助
我们证明，对于任意四个正质量的选择以及物体的任意循环排序，存在且仅存在一个与该排序对应的严格凸平面中心构型（在相似意义下）。这回答了Albouy、Cabral和Santos列表中的问题10，Santoprete称之为Simó-Yoccoz猜想。主要步骤是凸中心构型处Hessian矩阵的一致下界：它至少是其径向部分的四分之一，而径向部分仅在平移和旋转时消失。Dziobek关系将Hessian矩阵的不定部分转化为一个平方的负倍数，而Cauchy-Schwarz论证将该项界定为径向部分乘以一个显式$2\ imes 2$矩阵（其中不出现质量）的迹。我们通过区间算术在归一化凸中心构型的三维集合上（使用Corbera、Cors和Roberts的坐标）证明该迹小于3/4。唯一性随后通过从四个相等质量情形的覆盖论证得出。该计算已用第二个独立编写的程序重复进行。Hessian界和唯一性定理（包括计算）已在Lean证明助手中形式化，并由其内核仅使用标准公理进行检查。作为推论，每个退化的四体中心构型都是凹的，凸中心构型对质量解析依赖，并且风筝形、等腰梯形和菱形的已知对称定理在几行内即可得出。解析依赖性和对称定理也已形式化。
英文摘要
We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a square, and a Cauchy-Schwarz argument bounds this term by the radial part times the trace of an explicit $2\times 2$ matrix in which the masses do not appear. We prove that this trace is less than 3/4 by interval arithmetic on the three-dimensional set of normalized convex central configurations, in the coordinates of Corbera, Cors and Roberts. Uniqueness then follows by a covering argument from the case of four equal masses. The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms. As consequences, every degenerate four-body central configuration is concave, the convex central configuration depends analytically on the masses, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines. The analytic dependence and the symmetry theorems are also formalized.
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