周期为2的山谷分配在2×2k地图上的精确标记层序分类
Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps
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- Alferov University（阿尔费罗夫大学）
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中文总结 AI 辅助
本文分类了周期为2的山谷分配下2×2k地图的标记层序，给出可实现枢轴对的充要条件及状态计数，并单独处理k=1的特殊情形。
中文摘要 AI 辅助
我们分类了普通正交2×2k地图的标记最终层序，这些地图的山谷分配在每个折痕族中具有周期为2的周期性。对于k≥2，局部可平折性在64个形式周期词中保留了16个。八个褶词具有唯一的标记状态。其余八个词在标记对称性下归结为两个由行枢轴描述的常数行族。对于规范词MMMMVV，有序枢轴对(a,b)可实现当且仅当a=1、a=m或a=b，其中m=2k-1。对于MVMMMM，可实现性等价于BR(a)=BR(b)。在这两个族中，每个兼容的有序枢轴对确定唯一的标记状态。所得状态计数为0、1、6k-5和4k-3。k=1的情况单独处理，因为两个形式周期位没有物理折痕载体。
英文摘要
We classify the labelled final layer orders of ordinary orthogonal 2 x 2k maps whose mountain-valley assignments are periodic with period two in each crease family. For k >= 2, local flat-foldability leaves 16 of the 64 formal period words. Eight pleat words have a unique labelled state. The remaining eight words reduce, under labelled symmetries, to two constant-row families described by row pivots. For the canonical word MMMMVV, an ordered pivot pair (a,b) is realizable exactly when a = 1, a = m, or a = b, where m = 2k - 1. For MVMMMM, realizability is equivalent to BR(a) = BR(b). In both families each compatible ordered pivot pair determines a unique labelled state. The resulting state counts are 0, 1, 6k - 5, and 4k - 3. The case k = 1 is treated separately because two formal period bits have no physical crease carriers.
