连接连续板块模型与板块边界网络拓扑的极限定理
A limit theorem linking continuous plate models to the topology of the plate boundary network
- Chungnam National University（忠南国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过极限定理统一了板块构造的离散与连续描述，证明形状函数收敛于板块指示函数，并建立了加权欧拉示性数恒等于2的拓扑不变量，为板块重建的拓扑审计提供了数学基础。
AI中文摘要：
板块构造有两种描述方式：一种是离散描述，其中刚性板块铺满球面并在三价连接处相遇；另一种是连续描述，其中Bercovici和Wessel（1994）用有限边界半宽度$\delta^*$的光滑形状函数替代硬板块轮廓。我们证明这两种描述通过一个极限定理相联系。将形状函数推广到带符号测地距离构造，我们证明它们几乎处处以$1/\delta^*$的指数速率收敛到板块指示函数；在显式结构假设下，尖锐板块马赛克是球面的有限正则三价CW分解；并且由形状函数的超水平集构建的、计入分量数的加权欧拉示性数$\chi_w^{\delta^*}(\tau)$，对所有低于显式阈值$\delta_0(\tau)$的$\delta^*$都等于$V-E+F=2$，其面、边和顶点项分别收敛到板块数、边界弧数和三连接点数。该定理在正到达、分离和有界扇形假设下对带符号距离偏移覆盖无条件证明，并通过一个比较引理转移到归一化单位分割场，对阈值$\tau<1/3$成立，该引理的连接球步骤（一个平面单交叉性质）通过数值验证。残差$|\chi_w-2|$定义了扩散板块边界的拓扑扩散指数。对于现今的PB2002网络，这些假设被测量为非空的，其中$\delta_0(0.15)\approx2.8$公里。该定理是配套论文（Kim，2026，提交至《Geoscience Frontiers》）中板块重建拓扑审计的数学基础。
英文摘要：
Plate tectonics is described in two registers: a discrete one, in which rigid plates tile the sphere and meet at trivalent junctions, and a continuous one, in which Bercovici and Wessel (1994) replace hard plate outlines by smooth shape functions of finite boundary half-width $δ^*$. We prove that the two registers are connected by a limit theorem. Generalizing the shape functions to a signed-geodesic-distance construction, we show that they converge almost everywhere, exponentially in $1/δ^*$, to the plate indicator functions; that the sharp plate mosaic is a finite regular trivalent CW decomposition of the sphere under an explicit structural hypothesis; and that a component-counted weighted Euler characteristic $χ_w^{δ^*}(τ)$, built from the super-level sets of the shape functions, equals $V-E+F=2$ for all $δ^*$ below an explicit threshold $δ_0(τ)$, with its face, edge and vertex terms converging separately to the numbers of plates, boundary arcs and triple junctions. The theorem is proved unconditionally on the signed-distance offset cover under positive-reach, separation and bounded-sector hypotheses, and transferred to the normalized partition-of-unity field for thresholds $τ<1/3$ by a comparison lemma whose junction-ball step, a planar single-crossing property, is certified numerically. The residual $|χ_w-2|$ defines a topological diffuseness index for diffuse plate boundaries. For the present-day PB2002 network the hypotheses are measured to be non-vacuous, with $δ_0(0.15)\approx2.8$ km. The theorem is the mathematical foundation of the topological audit of plate reconstructions presented in a companion paper (Kim, 2026, submitted to Geoscience Frontiers).
