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

- ID: `P2918`
- Reference: `life-horizontal-line-weak-vanishing`
- Page: https://theoremdb.org/statements/P2918
- Record maturity: Reviewed problem with recorded work

## 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?

### Problem setup

- **Remark.** 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.** 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.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- Original intake status: 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.
- The MathOverflow question, answer, and all comments were checked on 2026-07-27. The answer provides images and a glider mechanism, while a comment identifies the uncontrolled interaction in the middle of a long line.
- 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: A glider moving away forever is compatible with weak vanishing because each fixed cell is visited only finitely often.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- The source reports simulations through n=1000 with no weakly vanishing line. The unresolved middle interaction prevents extrapolation to every n.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `life-horizontal-line-weak-vanishing`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/288423/vanishing-line-on-conways-game-of-life
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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. definition, terms through the maintained table, comments on glider-producing lengths, and linked data through n=1000 https://oeis.org/A061342
   - reference_database; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
   - 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.
3. <a id="reference-3"></a>LifeWiki contributors, “Conway’s Game of Life,” checked 2026-08-01. standard Life rule and terminology https://conwaylife.com/wiki/Conway%27s_Game_of_Life
   - website; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.