# P2822: Least uniform modulus of an abelian-square-free morphism on four letters

- ID: `P2822`
- Reference: `abelian-square-free-uniform-morphism-minimum`
- Page: https://theoremdb.org/statements/P2822
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the least integer \(L\ge2\) for which there is a morphism \(h:\{0,1,2,3\}^*\to\{0,1,2,3\}^*\) such that \(|h(a)|=L\) for every letter \(a\), and \(h(w)\) is abelian-square-free whenever \(w\) is abelian-square-free.

### Definitions

- **Definition.** A morphism satisfies \(h(uv)=h(u)h(v)\) and is determined by the four letter images.
- **Definition.** A word is abelian-square-free when it has no factor \(uv\) with \(|u|=|v|>0\) and equal Parikh vectors.
- **Definition.** The common image length \(L\) is called the uniform modulus.

### What counts as a solution

- Give an \(L\)-uniform morphism with a complete finite-test or direct proof that it preserves abelian-square-freeness, and prove that no such morphism exists for any smaller modulus.
- A negative certificate for a fixed modulus must cover unrestricted morphisms up to explicitly stated alphabet and reversal symmetries.

## Status

UNKNOWN as of 2026-07-31. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus. Give an \(L\)-uniform morphism with a complete finite-test or direct proof that it preserves abelian-square-freeness, and prove that no such morphism exists for any smaller modulus. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus. Give an \(L\)-uniform morphism with a complete finite-test or direct proof that it preserves abelian-square-freeness, and prove that no such morphism exists for any smaller modulus.

UNKNOWN as of 2026-07-31. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus.

A complete resolution must satisfy this condition: Give an \(L\)-uniform morphism with a complete finite-test or direct proof that it preserves abelian-square-freeness, and prove that no such morphism exists for any smaller modulus.

### Background and intake notes

The 85-uniform construction proves existence with a large structured certificate. Removing its symmetry restriction creates a finite sequence of reusable SAT instances, forbidden-factor lemmas, and verified morphism candidates.

- Original intake status: UNKNOWN as of 2026-07-27. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus.
- 2026-07-27 prior-art search checked Keränen's 1992 construction, the 2009 powerful substitution, later abelian-square-free morphism searches, and claims of minimality. The located minimality statement applies to the cyclic-permutation template used by the 85-uniform morphism.
- The known construction gives \(L\le85\). A lower-bound search must range over all four image words, rather than assume that the images are letter permutations of one seed word.
- Finite-test criteria for power-free morphisms can turn a candidate into a bounded verification problem, but the exact test set and its proof must be recorded.

- Recorded example: Keränen's construction supplies an admissible morphism at modulus 85, so the requested least value is at most 85.

### Open directions

- **Route 1** (reported): Give an \(L\)-uniform morphism with a complete finite-test or direct proof that it preserves abelian-square-freeness, and prove that no such morphism exists for any smaller modulus. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `abelian-square-free-uniform-morphism-minimum`, 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>Veikko Keränen, “Abelian squares are avoidable on 4 letters”. Lecture Notes in Computer Science (1992), 41-52. DOI 10.1007/3-540-55719-9_62. Keränen's 85-uniform construction and its symmetry-restricted minimality motivate the unrestricted minimum; this CC0 record was written on 2026-07-27. https://doi.org/10.1007/3-540-55719-9_62
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Least uniform modulus of an abelian-square-free morphism on four letters: UNKNOWN as of 2026-07-27. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus.
   - Source named by the research packet.
2. <a id="reference-2"></a>Veikko Keränen, “A powerful abelian square-free substitution over 4 letters”. Theoretical Computer Science 410(38-40) (2009), 3893-3900. DOI 10.1016/j.tcs.2009.05.027. Full journal article relevant to Least uniform modulus of an abelian-square-free morphism on four letters. https://doi.org/10.1016/j.tcs.2009.05.027
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Least uniform modulus of an abelian-square-free morphism on four letters: UNKNOWN as of 2026-07-27. An 85-uniform abelian-square-free endomorphism is known and was shown minimal inside its cyclic-symmetry form; the checked sources do not identify the unrestricted least modulus.