Combinatorial group theory · balanced presentations · free-group transformations

Andrews–Curtis Conjecture

Collaboration beta

Can every balanced presentation of the trivial group be simplified to the obvious free-basis presentation using four elementary relator moves? The source develops a sharply bounded rank-two frontier, but the unrestricted conjecture remains open.

F_{n} / ⟨ ⟨ r_{1}, \dots, r_{n} ⟩ ⟩ = 1 \Rightarrow (r_{1}, \dots, r_{n}) \sim_{AC} (x_{1}, \dots, x_{n}) .

Known results and sources

Andrews–Curtis asks whether every balanced trivial presentation can be simplified to the free basis without stabilization.

Research problem

Exact mathematical statement

Let $F_n=F(x_1,…,x_n)$ . An ordered tuple $(r_1,…,r_n)$ is a balanced trivial presentation if

F_n/⟪ r_1,…,r_n⟫=1.

The ordinary, non-stable Andrews–Curtis Conjecture asks whether every such tuple can be transformed to the free-basis tuple $(x_1,…,x_n)$ using relator inversion, multiplication by another relator, conjugation in the free group, and permutation of tuple positions. Stabilization is not part of this statement.

Problem infographic

Problem at a glance

Scientific explainer for the Andrews–Curtis Conjecture showing a balanced trivial presentation, the four allowed relator moves, the unresolved free-basis target, and a separate bounded length-seven research frontier with four obstructed hard cores. — The conjecture asks whether four elementary relator moves always reach the free basis; the lower band shows a bounded normalized frontier developed in the retained source, not a proof of the unrestricted statement.

Current mathematical picture

Where work on Andrews–Curtis Conjecture stands

Open conjecture

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureOne-sided primitive exposure

The retained Bass–Serre, amalgam, permutation, class-two, and Fox-orbit certificates block the literal one-sided directions at their stated scope. Analyze the penultimate-relator conjugacy condition for either three-block family, extend the bounded searches with proof-preserving pruning, and separately close the normalization/minimality seam.

Route status · Narrowed route

Main reductionCurrent reduction

In the studied normalized slice, adjacent movement across the isolated square requires at least three multiplication blocks and at least two target switches. The minimal three-block abelian skeletons fall into an identity family and a signed-swap family.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeSolve the penultimate-relator obstruction symbolically.Task status · Work already reported in progress

Work mapped so far

Andrews–Curtis Conjecture in numbers

932retained lines of mathematical investigation932 in the current working snapshot

Argument development: 824 · 88%
Explored or eliminated routes: 25 · 3%
Computational analysis: 19 · 2%
Open obligations: 29 · 3%
Definitions and setup: 35 · 4%

9selected mapped statements2routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJames J. Andrews, Morton L. CurtisProposed1965Status checkedAug 9, 2026

Source recordedAug 9, 2026

Recommended next task

Push the bounded signed-swap frontier beyond radius six.

Suggested move: Extend the signed-swap search to radius seven using centralizer normalization, symmetry reduction, exact transcripts, and proof-preserving pruning.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeOne-sided primitive exposure

Route status · Narrowed route

Narrowed routeNilpotent quotients as a complete invariant

The source gives a tuple mapping onto A5 to demonstrate the limitation of nilpotent information. Use deletion quotients, relation modules, Fox data, or global normalization control rather than another nilpotent truncation.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Push the bounded signed-swap frontier beyond radius six.Suggested move: Extend the signed-swap search to radius seven using centralizer normalization, symmetry reduction, exact transcripts, and proof-preserving pruning.

Ready to work on

Close the normalization-to-global seam.Suggested move: Prove controlled exponent normalization for a hypothetical rank-two counterexample without losing the complexity bound that makes the finite slice relevant.

Ready to work on

Solve the penultimate-relator obstruction symbolically.Suggested move: Derive a source-checkable necessary and sufficient condition for the penultimate relator in one of the two explicit three-block systems to be conjugate, up to inversion, to its target.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 9, 2026

Current statusOpen conjecture

As of 2026-08-09, the classical, unstabilized Andrews–Curtis conjecture remains open: no proof or accepted counterexample is known. It asks whether every normally generating n-tuple of the rank-n free group—equivalently, every balanced presentation of the trivial group—can be reduced to the standard basis or presentation using ordinary Nielsen moves on relators and arbitrary conjugation, without stabilization. Bounded-length theorems, explicit candidate families, and computational trivializations or equivalences concern subclasses or individual presentations. The stable or weak conjecture permits generator–relator stabilization and is a distinct weaker statement; a 2025 peer-reviewed correction says a 2024 claim did not settle the stable status of AK(3). Counterexamples to the stricter cancellative cyclic unstabilized variant do not refute the classical conjecture.

[7][8][12]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-07-26
PreprintCarreras released explicit machine-checkable AC-equivalence certificates among several length-14 frontier presentations and AK(3)-related classes. The preprint leaves AK(3) open and does not resolve the universal conjecture.[17][18]
2026-07-08
Peer reviewedFairbank, Lisitsa, and Vernitski published an automated-reasoning and classifier study with linked experiment data. It describes the Andrews–Curtis conjecture as unsolved and reports that current methods still struggle on open instances.[13][14]
2026-06-19
Peer reviewedThe ICML 2026 Two-Hump study reported stronger search methods, simplifications of many previously open Miller–Schupp instances, and released large datasets of known AC-trivial presentations. These are scoped instance results, not a proof of the conjecture.[15][16]
2025-12
Peer reviewedLisitsa's journal article retained machine-generated reductions but explicitly corrected the earlier conclusion: it does not establish stable AC-triviality of AK(3), whose stable status remains unresolved.[12]

Mathematical neighborhood

Related results and reusable starting points

Current focusAndrews–Curtis conjecture

Equivalent formulationNormally generating tuple formulation

For a free group of rank n, the conjecture says that every normally generating n-tuple lies in the AC orbit of a free basis; this is equivalent to the balanced trivial-group presentation formulation.

[5][8]

Equivalent formulationCyclic Andrews–Curtis conjecture

At the universal level, arbitrary conjugations may be replaced by cyclic operations on cyclically reduced relator tuples without changing the truth value of the classical conjecture.

[7]

Weaker or relaxed formStable or weak Andrews–Curtis conjecture

The stable or weak conjecture also permits adding or deleting a new generator with its matching relator. A classical trivialization is stable, but a stable sequence need not be an unstabilized one; both universal statements remain unresolved.

[2][7]

Equivalent formulationFinite contractible 2-complex 3-deformation conjecture

The stabilized presentation problem has the topological form that every finite contractible two-complex 3-deforms to a point. This correspondence belongs to the stable move system and should not erase the stabilization distinction.

[2][7]

Stronger or generalized formCancellative cyclic Andrews–Curtis variant

The cancellative cyclic formulation without stabilization allows fewer reductions and is false. Its counterexamples do not refute the classical conjecture; after stabilization the cancellative cyclic formulation is equivalent to the stable conjecture.

[7]

Solved special caseBounded rank-two relator-length frontier

All balanced two-generator trivial-group presentations of total relator length at most 12 are AC-trivial. At length 13, every case is trivial or AC-equivalent to AK(3), so unconditional verification stops at 12 while AK(3) remains open.

[5][6]

Related problemAkbulut–Kirby presentation family AK(n)

AK(n) is a family of balanced presentations of the trivial group. AK(2) is AC-trivial; AK(n) for n greater than 2 remain classical candidates, and the corrected stable status of AK(3) is also unresolved.

[3][8]

Related problemMiller–Schupp presentation families

Miller–Schupp families provide structured search benchmarks and hard candidate classes. Trivializations or equivalences for individual members narrow that benchmark frontier but do not decide every balanced presentation.

[13][15]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal statement · partial resource linkedFormal Conjectures pull request 4625: Andrews–Curtis conjecture
The open pull request proposes Lean definitions for relator tuples, ordinary AC moves, generated equivalence, normal generation, and a conjecture statement. As of 2026-08-09 it is unmerged, has no review, and contains no proof.
[19]
dataset · source linked; not reproduced by ProofAtlasFairbank–Lisitsa–Vernitski experiment data
The peer-reviewed automated-reasoning study links a Zenodo dataset containing experiment material, proofs, and outputs. ProofAtlas has not rerun it.
[13][14]
software · source linked; not reproduced by ProofAtlasAC-SolverX code and AC-trivial presentation datasets
The public companion repository provides search code, trained-model material, and the AC-19 and AC-1M datasets of known AC-trivial presentations. Availability does not imply independent reproduction or coverage of all balanced presentations.
[15][16]
certificate · not independently reproducedLength-14 Andrews–Curtis equivalence certificates
The July 2026 preprint and archived artifact provide a small verifier and explicit move certificates for several claimed equivalences. ProofAtlas did not replay them, and they do not trivialize AK(3) or resolve the conjecture.
[17][18]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA reviewed and merged formal representation of finite-rank free groups, indexed relator tuples, balanced presentations, and the equivalence between normal generation and presentation of the trivial group.
Formalization targetChecked definitions of each ordinary Andrews–Curtis move and their finite reflexive-symmetric-transitive closure, with permutation and left/right Nielsen conventions reconciled.
Formalization targetA reviewed exact formal statement quantifying over all finite ranks and normally generating rank-sized tuples, without accidentally adding stabilization or restricting to rank two.
Formalization targetSeparate checked definitions for stable moves, cyclic reductions, and cancellative variants so results cannot cross variant boundaries silently.
Formalization targetA small reviewed certificate checker for explicit elementary move sequences together with imported certificates for bounded cases or candidate equivalences.
Formalization targetA formal proof of the universal classical conjecture or a checked balanced trivial-group presentation with a proof that it is outside the ordinary AC orbit; neither was located in the bounded search.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma3 of 93
computational claim2 of 92
negative result1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementEvery balanced trivial presentation should be reducible to the free basis by inversion, multiplication, conjugation, and swapping—without stabilization.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementNilpotent quotients straighten uniformlyintermediate
retained route statementNormalized short relators are eliminatedintermediate
retained route statementLength-seven slice classifiedintermediate
retained route statementFour one-sided square edges are blockedintermediate
retained route statementAdjacent paths need two target switchesintermediate
retained route statementTwo bounded three-block searches return no solutionintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureOne-sided primitive exposurereported failure
Useful failureNilpotent quotients as a complete invariantreported failure
Research targetSolve the penultimate-relator obstruction symbolically.in progress reported
Research targetPush the bounded signed-swap frontier beyond radius six.open
Research targetClose the normalization-to-global seam.open
ComputationTwo embedded bounded searches explore minimal three-block systems through conjugator radii six and four.The source reports no signed-swap solution through radius six and no neutral-middle solution through radius four; the embedded code has not been rerun by ProofAtlas. · reported unreproduced
Narrowed routeOne-sided primitive exposureThe retained Bass–Serre, amalgam, permutation, class-two, and Fox-orbit certificates block the literal one-sided directions at their stated scope. Analyze the penultimate-relator conjugacy condition for either three-block family, extend the bounded searches with proof-preserving pruning, and separately close the normalization/minimality seam.
Narrowed routeNilpotent quotients as a complete invariantThe source gives a tuple mapping onto A5 to demonstrate the limitation of nilpotent information. Use deletion quotients, relation modules, Fox data, or global normalization control rather than another nilpotent truncation.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeSolve the penultimate-relator obstruction symbolically.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointPush the bounded signed-swap frontier beyond radius six.

Andrews–Curtis Conjecture · ready to start

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Research contextPrepared context for any AI agent

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Current routes and known obstacles
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Sources and references19 cited works · next context review by Nov 9, 2026

The mathematical context was checked on Aug 9, 2026. Status can be refreshed sooner after a material result or claim.

1
Free Groups and Handlebodiesoriginal source · J. J. Andrews, Morton L. Curtis · Proceedings of the American Mathematical Society · 1965 · DOI 10.2307/2033843 · MR MR0173241 · accessed Aug 9, 2026
2
Group Presentations and Formal Deformationspeer reviewed result · Perrin Wright · Transactions of the American Mathematical Society · 1975 · DOI 10.2307/1997282 · MR MR0380813 · accessed Aug 9, 2026
3
A Potential Smooth Counterexample in Dimension 4 to the Poincaré Conjecture, the Schoenflies Conjecture, and the Andrews–Curtis Conjecturepeer reviewed result · Selman Akbulut, Robion Kirby · Topology · 1985 · DOI 10.1016/0040-9383(85)90010-2 · MR MR0816520 · accessed Aug 9, 2026
4
Balanced Presentations of the Trivial Groupsurvey or monograph · R. G. Burns, Olga Macedońska · Bulletin of the London Mathematical Society · 1993-11-01 · DOI 10.1112/blms/25.6.513 · accessed Aug 9, 2026
5
Balanced Presentations of the Trivial Group on Two Generators and the Andrews–Curtis Conjecturepeer reviewed result · Alexei D. Miasnikov, Alexei G. Myasnikov · Groups and Computation III, De Gruyter · 2001 · ARXIV math/0304305 · DOI 10.1515/9783110872743.257 · accessed Aug 9, 2026
6
Breadth-First Search and the Andrews–Curtis Conjecturepeer reviewed result · George Havas, Colin Ramsay · International Journal of Algebra and Computation · 2003 · DOI 10.1142/S0218196703001353 · accessed Aug 9, 2026
7
On Conjectures of Andrews and Curtispeer reviewed result · Sergei V. Ivanov · Proceedings of the American Mathematical Society · 2018-03-09 · ARXIV 1606.08196 · DOI 10.1090/proc/13710 · accessed Aug 9, 2026
8
Andrews–Curtis Groupspeer reviewed result · Robert H. Gilman, Alexei G. Myasnikov · Groups, Complexity, Cryptology · 2025-07-02 · ARXIV 2506.23031 · DOI 10.46298/jgcc.2024.16.1.15972 · accessed Aug 9, 2026
9
Low-dimensional topology, problems inencyclopedia · Encyclopedia of Mathematics, EMS Press · 2001 [1994]; status notes dated 1999 · accessed Aug 9, 2026
10
Andrews–Curtis Conjectureencyclopedia · Wikipedia contributors · Wikipedia · accessed Aug 9, 2026
11
What Makes Math Problems Hard for Reinforcement Learning: A Case Studypreprint · Ali Shehper, Anibal M. Medina-Mardones, Lucas Fagan, Bartłomiej Lewandowski, Angus Gruen, Yang Qiu, Piotr Kucharski, Zhenghan Wang, Sergei Gukov · arXiv · 2024-08-27 · ARXIV 2408.15332 · accessed Aug 9, 2026
12
Stable Andrews–Curtis Trivialization of AK(3) Revisited: A Case Study Using Automated Deductionpeer reviewed result · Alexei Lisitsa · Journal of Computational Algebra · 2025-12 · ARXIV 2501.18601 · DOI 10.1016/j.jaca.2025.100041 · accessed Aug 9, 2026
13
Probabilistic Automaton Classifier Applied to Examples Related to the Andrews–Curtis Conjecturepeer reviewed result · Michael Fairbank, Alexei Lisitsa, Alexei Vernitski · Journal of Automated Reasoning · 2026-07-08 · DOI 10.1007/s10817-026-09759-8 · accessed Aug 9, 2026
14
Data for Probabilistic Automaton Classifier Applied to Examples Related to the Andrews–Curtis Conjecturesoftware or dataset · Michael Fairbank, Alexei Lisitsa, Alexei Vernitski · Zenodo · 2026 · DOI 10.5281/zenodo.16088111 · accessed Aug 9, 2026
15
The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learningpeer reviewed result · Lucas Fagan, Michele Tarquini, Ali Shehper, Maksymilian Manko, Angus Gruen, Coco Huang, Giorgi Butbaia, Davide Passaro, Sergei Gukov · Accepted at the 43rd International Conference on Machine Learning (ICML 2026) · 2026-06-19 · ARXIV 2606.21611 · accessed Aug 9, 2026
16
AC-SolverX: Code and Datasets for the Two-Hump Problemsoftware or dataset · Math-AI-Caltech contributors · GitHub · 2026 · accessed Aug 9, 2026
17
Machine-checkable Equivalence Certificates at the Length-14 Andrews–Curtis Frontierpreprint · Josep Carreras · arXiv · 2026-07-26 · ARXIV 2607.23611 · accessed Aug 9, 2026
18
Andrews–Curtis Certificate Campaign: Artifact Releasesoftware or dataset · Josep Carreras · GitHub and Zenodo · 2026-07 · DOI 10.5281/zenodo.21499081 · accessed Aug 9, 2026
19
Pull Request 4625: Formalize the Andrews–Curtis Conjectureformalization · Google DeepMind Formal Conjectures contributors · Google DeepMind formal-conjectures repository · 2026-07-25 · accessed Aug 9, 2026

Important qualifications

This record scopes its compact status to the classical, unstabilized Andrews–Curtis conjecture. Older topological literature sometimes uses the same name for a stabilized 3-deformation statement, so every variant claim here identifies its move set.
The stable or weak conjecture permits adding and deleting a generator with a matching relator and is weaker than the classical statement; results or failures for one are not silently transferred to the other.
A 2024 preprint claimed that AK(3) is stably AC-trivial, but a 2025 peer-reviewed correction reports that the required prior theorem was incorrect and that the stable status of AK(3) remains unresolved.
Ivanov's counterexample to the cancellative cyclic formulation without stabilization concerns a stricter move system and does not refute the classical conjecture.
The July 2026 Carreras certificate result is a preprint; ProofAtlas did not replay its certificates or independently check its exhaustive-search claims.
Linked computational code, datasets, proof outputs, and certificates are available external resources, not computations independently reproduced by ProofAtlas.
The Google DeepMind Formal Conjectures pull request was open and unreviewed on the collection date; it is not a merged canonical Lean statement and contains no proof.
The bounded public search found no other exact reviewed proof-assistant statement or proof; this does not establish global nonexistence.
Wikipedia remains in the current research map only as an ordinary recognition link and is not used as status, priority, or mathematical-result authority.
No unreviewed source material, packet-derived source, or private material was read during this external collection.
This administrative metadata grants no proof, counterexample, novelty, review, acceptance, publication, or mathematical-progress authority.

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