The Ultimate Fate of Life Is Not Shared
Abstract
The limit set of Conway's Game of Life collects the configurations that can still appear at arbitrarily late times: those admitting predecessors of every finite depth. Answering a question of Salo and Törmä (ICALP 2022), we prove that two fixed finite patterns can each occur in the limit set, yet can never be found together in a single configuration of it, at any relative position; equivalently, the spatial translation action on the limit set is not topologically transitive. Our method is to make a persistent marker force a periodic lane to grow in sufficiently deep predecessors, until two perpendicular lanes are forced to intersect and prescribe incompatible values at a common cell. The argument uses two computer-verified local implications.
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BibTeX
@misc{gan2026ultimate,
title = {{The Ultimate Fate of Life Is Not Shared}},
author = {Ziyue Gan and Ziran Li and Jiadong Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34122},
eprint = {2609.34122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34122},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Ultimate Fate of Life Is Not Shared
AU - Ziyue Gan
AU - Ziran Li
AU - Jiadong Zhu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34122
UR - https://arxiv.org/abs/2609.34122
AB - The limit set of Conway's Game of Life collects the configurations that can still appear at arbitrarily late times: those admitting predecessors of every finite depth. Answering a question of Salo and Törmä (ICALP 2022), we prove that two fixed finite patterns can each occur in the limit set, yet can never be found together in a single configuration of it, at any relative position; equivalently, the spatial translation action on the limit set is not topologically transitive. Our method is to make a persistent marker force a periodic lane to grow in sufficiently deep predecessors, until two perpendicular lanes are forced to intersect and prescribe incompatible values at a common cell. The argument uses two computer-verified local implications.
ER -