A measured neighbourhood of the Andrews-Curtis conjecture: the census to total length 14, the box around AK(3), exact peaks on AC-19, and a Lean 4 archive
Year: 2026
Type: preprint
Abstract: We report a computational study of balanced two-generator presentations of the trivial group under Andrews-Curtis moves. The study does not prove or disprove the Andrews-Curtis conjecture, and the status of AK(3) is unchanged. It measures the neighbourhood of the conjecture and attaches a replayable certificate or an exhausted finite search to every statement. We extend the canonical enumeration of Havas and Ramsay from total relator length 13 to 14 and classify every presentation with unimodular abelianisation up to that length. At length 14 the unimodular stratum has 57,765 presentations: 57,727 present the trivial group, 34 present SL(2,5) and 4 present infinite groups. All but 50 of the presentations of the trivial group are decided with explicit certificates, and the 50 lie in at most nine AC-classes, so there are at most 20 AC-classes at length 14. We enumerate the set of presentations reachable from AK(3) without exceeding total length 25 (2.9 billion orbits under a symmetry quotient) and find that it does not contain the standard presentation. A second hard class at length 13 is absent from every such box up to cap 24, and it is AC-equivalent to AK(3) exactly when AK(3) is AC-trivial. For the rows of total length at most 18 in the AC-19 dataset of Fagan et al. we determine the exact minimal peak of an AC-trivialisation for 51,766 of 57,115 rows. We also describe a Lean 4 archive of 829 files whose axiom reports name only propext, Classical.choice and Quot.sound. It contains a proof that AK(n) presents the trivial group for every n, soundness theorems for two certificate formats, kernel-checked non-triviality witnesses and bounded negatives, and a classification of the Metzler presentations at k = 4. We list fifteen corrections to published statements, all minor. Code, certificates, ledgers and Lean files: https://github.com/pragyaangaur/Andrews-Curtis-Conjecture. Use of AI: the computations, certificates, Lean proofs and reports were produced by a multi-agent system on Open Math OS, with GLM 5.3 as coordinator and DeepSeek V4.1 Flash agents as members and referees. The paper was drafted with Claude (Anthropic) from the campaign's reports. The author designed and directed the campaign, checked the paper, and is its sole author.
Author: Gaur, Pragyaan
Institution: National Institute of Technology Agartala
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225455877