着色平铺定理：分形分割的缩放定律与分离维度
The Chromatic Tiling Theorem: Scaling Laws and the Separation Dimension of Fractal Partitions
AI总结：
本文提出分离维度以量化分形分割边界的Hausdorff维度，定义了几何正则分割并证明其存在性，通过着色平铺定理给出了图色数的精确上界，揭示了着色复杂度由分离集维度决定。
摘要：
本文在分形分割的组合复杂度与其界面的内蕴几何之间建立了严格的定量联系。我们引入了分离维度（Separation Dimension, $\sepdim$）的概念，这是一种量化瓦片之间边界Hausdorff维度的新颖特征。一种自然但有缺陷的方法是将着色复杂度与分形的环境拓扑边界相关联。我们证明，这种外在观点对于一大类自相似集是不可行的。相反，我们以分离维度为核心的内蕴框架提供了正确的表述。我们定义了一类新的性质良好的分割，称为几何正则分割（Geometrically Regular Partitions, GRPs），并证明了它们在Sierpinski Carpet等典型分形上的存在性。我们的主要结果——着色平铺定理——为与此类分割相关联的图的色数（$χ$）提供了精确的上界，证明其被界定为 $χ(\tiling) \leq \Cconst (r_{\max}/r_{\min})^{\sepdim}$。这一结果表明，着色复杂度的缩放并非由分形本身的维度决定，而是由分割诱导的分离集的维度决定。最后，我们提出将所有容许分割上的分离维度最小化作为分形几何中的一个新变分问题。
英文摘要：
This paper establishes a rigorous, quantitative link between the combinatorial complexity of a fractal partition and the intrinsic geometry of its interfaces. We introduce the concept of the \emph{Separation Dimension} ($\sepdim$), a novel characteristic that quantifies the Hausdorff dimension of the boundaries between tiles. A natural but flawed approach would be to relate coloring complexity to the fractal's ambient topological boundary. We demonstrate that this extrinsic view is untenable for a vast class of self-similar sets. Instead, our intrinsic framework, centered on the Separation Dimension, provides the correct formulation. We define a new class of well-behaved partitions, termed \emph{Geometrically Regular Partitions (GRPs)}, and prove their existence on canonical fractals such as the Sierpinski Carpet. Our main result, the Chromatic Tiling Theorem, provides a sharp upper bound for the chromatic number ($χ$) of the graph associated with such a partition, proving that it is bounded by $χ(\tiling) \leq \Cconst (r_{\max}/r_{\min})^{\sepdim}$. This result demonstrates that the scaling of coloring complexity is governed not by the dimension of the fractal itself, but by the dimension of the separation set induced by the partition. We conclude by proposing the minimization of the separation dimension over all admissible partitions as a new variational problem in fractal geometry.
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