[#P2918] Weak vanishing of a horizontal line in Conway's Life
Contents
Problem. In Conway's Game of Life on \(\mathbb Z^2\), start with exactly \(L_n=\{(0,0),(1,0),\ldots,(n-1,0)\}\) alive. Does there exist an integer \(n\ge25\) such that every fixed cell is dead at all sufficiently large times?
Agent access
Work in ChatGPTThe agent reads this problem's research brief and asks you before it saves anything to TheoremDB.
Definitions and notation
1Context
Exact evolution traces, glider decompositions, and periodic-core certificates can be reused for nearby line lengths. The cellwise definition separates escaping activity from recurrent activity near the origin.
2Definitions
Definition 1 (Life uses the eight neighboring cells: a live cell survives with two or three live neighbors, and a dead cell). Life uses the eight neighboring cells: a live cell survives with two or three live neighbors, and a dead cell is born with exactly three live neighbors.
Definition 2 (A configuration weakly vanishes if for every cell x there). A configuration weakly vanishes if for every cell x there is a time T_x after which x is always dead; live cells may continue moving outward forever.
3What counts as a solution
- Exhibit an integer n>=25 and prove that each fixed cell is eventually permanently dead, or prove that every n>=25 leaves some fixed cell alive at arbitrarily large times.
- A positive result may use a finite-time decomposition into escaping spaceships, but it must certify all debris and show that no component can return to a fixed bounded region.
1Status
Saved packet · July 31, 2026
Saved packet status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n. Exhibit an integer n>=25 and prove that each fixed cell is eventually permanently dead, or prove that every n>=25 leaves some fixed cell alive at arbitrarily large times.[1]
1Packet records
Recent contributions
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-31. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n.
- OEIS A061342 was checked for the evolution and lifespan data attached to finite horizontal rows. Its finite table does not decide weak vanishing for all n>=25.
- The ConwayLife wiki was checked for terminology and known line-pattern behavior. No theorem settling this quantified family was located.
- A local corpus search for Life line, horizontal row, weak vanishing, and A061342 found no duplicate.
Recorded example 1. A glider moving away forever is compatible with weak vanishing because each fixed cell is visited only finitely often.
Computational notes
- The source reports simulations through n=1000 with no weakly vanishing line. The unresolved middle interaction prevents extrapolation to every n.
How the 2 records connect
ProblemWeak vanishing of a horizontal line in Conway's Life
All 1 recorded relations between these records and the problem
2See also
- Strong block universality of Conway's Game of Lifecellular automata
- Doubling law for rotated XOR cycle lengthscellular automata
- Linear complexity of binary Sierpinski-triangle rowscellular automata
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Cite this problem statement
Cite the original sources separately.
“Weak vanishing of a horizontal line in Conway's Life.” TheoremDB. P2918. Problem statement; statement text SHA-256 5926bfb85361761e6feed75c568d1f25dc53f265731d5d7cda4f9abfa2e43910. https://theoremdb.org/statement/?ref=P2918
@misc{theoremdb-problem-5926bfb85361761e6feed75c568d1f25dc53f265731d5d7cda4f9abfa2e43910,
title = {{Weak vanishing of a horizontal line in Conway's Life}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 5926bfb85361761e6feed75c568d1f25dc53f265731d5d7cda4f9abfa2e43910},
url = {https://theoremdb.org/statement/?ref=P2918}
}Plain text: Built Markdown snapshot
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. MathOverflow: Vanishing line on Conway's Game of Life. Question 288423, its answer, and every visible comment were checked on 2026-07-27. Question 288423, its answer, and every visible comment were checked on 2026-07-27. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Weak vanishing of a horizontal line in Conway's Life: UNKNOWN as of 2026-07-27. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n.Source named by the research packet.
- Alex Fink, “A061342: Period of the stationary component produced by a row of n cells in Conway’s Game of Life,” On-Line Encyclopedia of Integer Sequences, submitted June 6, 2001; definition and sequence extended by Eric M. Schmidt, May 24, 2014; checked 2026-08-01. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.Reused material: definition, terms through the maintained table, comments on glider-producing lengths, and linked data through n=1000.Reuse basis: fair use reviewed · rights holder: The OEIS Foundation Inc. and the credited contributors · checked 2026-08-01 by Philip Weiss, TheoremDB staff.Required attribution: Alex Fink, “A061342: Period of the stationary component produced by a row of n cells in Conway’s Game of Life,” On-Line Encyclopedia of Integer Sequences, submitted June 6, 2001; definition and sequence extended by Eric M. Schmidt, May 24, 2014; checked 2026-08-01.UNKNOWN as of 2026-07-27. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n.Also cited at definition, terms through the maintained table, comments on glider-producing lengths, and linked data through n=1000.Source used to assess the problem's recorded status.For Weak vanishing of a horizontal line in Conway's Life, this source records computed behavior of one-cell-thick Life lines and leaves the universal weak-vanishing question unsettled.
- MathOverflow: Vanishing line on Conway's Game of Life, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source reports that line lengths through 1000 fail to weakly vanish and gives glider-based heuristics for large lengths. The answer does not prove persistence for every n or exhibit a weakly vanishing n.Also cited at standard Life rule and terminology.Source used to assess the problem's recorded status.For Weak vanishing of a horizontal line in Conway's Life, this source supplies background for the cellular-automaton convention; the line-length evidence is carried by OEIS A061342 and the MathOverflow thread.
An original CC0 reformulation motivated by the cited MathOverflow question, with weak vanishing stated cellwise; no page prose was copied.
Discussion
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