模Krinkle镶嵌中的Farey结构：中项拼接与单边生成原砖
Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge
AI总结：
本文研究模Krinkle镶嵌的Farey结构，证明相邻参数的Farey和可通过原砖的扇形扭转拼接实现，给出原砖分解规则及变体原砖的分离定理，还关联了螺旋臂数量与参数的关系。
AI中文摘要：
Imura提出的模Krinkle镶嵌（arXiv:2506.07638）是由既约分数m/k和整数t≥2参数化的非周期螺旋单形镶嵌族。本文证明，两个Farey相邻参数的Farey和（中项）(m₁+m₂)/(k₁+k₂)可通过对原砖的精确几何操作实现：(m₁+m₂,k₁+k₂)型原砖的下边界路径，是在对其边进行保边长的渐进旋转（扇形扭转）后，拼接两个父砖的下路径得到的。反之，每个原砖恰好存在一种扇形扭转拼接分解——非相邻参数永不拼接——其切割位置为k₁=m⁻¹ mod k，递归沿Stern-Brocot树下降至单条单位边。该操作的组合核心是Christoffel字的经典标准分解；本文的贡献在于其在圆方向系统上的精确等边实现，以及由此得到的模Krinkle族（包括新近引入的变体）的结构理论：本文证明了一个分离定理，即每个变体原砖都是公共递归生成的核心加上有限条方向不变的装饰边。作为Imura螺旋臂计数公式的推论，两个Farey父砖在无偏移镶嵌中可见：逆时针和顺时针螺旋臂的数量分别为tk₁和tk₂。
英文摘要：
The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction $m/k$ and an integer $t\ge 2$. We show that the Farey sum (mediant) $(m_1+m_2)/(k_1+k_2)$ of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the $(m_1+m_2,k_1+k_2)$-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition -- no non-adjacent parameters ever splice -- the cut position being $k_1=m^{-1}\bmod k$, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are $tk_1$ and $tk_2$.
