[#P2918] Weak vanishing of a horizontal line in Conway's Life

A flat mathematical diagram showing a horizontal line evolving in Conway's Game of Life. — A schematic view of a horizontal line evolving in Conway's Game of Life.

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?

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.

2Problem setup

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.

Remark 1. 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.

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 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] Relevance to this problem Defines the dated frontier and the exact remainder that research on this problem must resolve. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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. A complete resolution must satisfy this condition: 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. Open full record →
	Trace	Reported
attempt · Route 1 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] Relevance to this problem For Weak vanishing of a horizontal line in Conway's Life, record life-horizontal-line-weak-vanishing-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemWeak vanishing of a horizontal line in Conway's Life

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Positive cycle entropy for finite cyclic Rule 30cellular automata
A half-edge transient bound for majority dynamics on the square toruscellular automata
Strong block universality of Conway's Game of Lifecellular automata

How to cite

TheoremDB contributors, “Weak vanishing of a horizontal line in Conway's Life,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/life-horizontal-line-weak-vanishing

This page as plain text: life-horizontal-line-weak-vanishing.md

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.

[#P2918] Weak vanishing of a horizontal line in Conway's Life

1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance