Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gan, Ziyue, Li, Ziran, Zhu, Jiadong
Institutions: Xi'an Jiaotong University, University of Connecticut
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214728707