Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps
Abstract
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.
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BibTeX
@misc{fofonov2026exact,
title = {{Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps}},
author = {Michael Fofonov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16776},
eprint = {2609.16776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16776},
}
RIS
TY - UNPB
TI - Exact Labelled Layer-Order Classification for Period-2 Mountain--Valley Assignments on 2 x 2k Maps
AU - Michael Fofonov
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16776
UR - https://arxiv.org/abs/2609.16776
AB - 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.
ER -