Block-exchange descent for insertion: commuting projections and minimum masks
Abstract
For fixed nonempty words u,v, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree r, the minimum over masks with commuting source projections is exactly r−1; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.
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BibTeX
@misc{ma2026blockexchange,
title = {{Block-exchange descent for insertion: commuting projections and minimum masks}},
author = {Haobo Ma and Wenlin Zhang and Charles E. Hughes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39941},
eprint = {2610.39941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39941},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Block-exchange descent for insertion: commuting projections and minimum masks
AU - Haobo Ma
AU - Wenlin Zhang
AU - Charles E. Hughes
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39941
UR - https://arxiv.org/abs/2610.39941
AB - For fixed nonempty words u,v, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree r, the minimum over masks with commuting source projections is exactly r−1; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.
ER -