SIGNIFICANCE

Checked 2026-08-01 · source current · v8

Claim from OpenAI

The unit group L_F2(1,2)× of the binary Leavitt algebra is not sofic.

Quoted from the source · OpenAI · 2026-08-01, p.78

Author involvement

Compiled from OpenAI's public release, manuscript, and repository. OpenAI did not participate in or confirm this Significance record.

Record note · Significance · 2026-08-11

Summary · for readers

In one sentence
OpenAI says one specific infinite group — the unit group of a small algebra built from two symbols — cannot be approximated by finite shuffles. That property is called soficity, and the manuscript states this group does not have it.

Reader summary · Significance · 2026-08-07

Claim
OpenAI affirme qu'un groupe infini précis — le groupe des éléments inversibles d'une petite algèbre construite à partir de deux symboles — ne peut pas être approché par des permutations finies. Cette propriété s'appelle la soficité, et le manuscrit soutient que ce groupe ne la possède pas.
Checked
Significance a reconstruit le code Lean publié à un commit fixé et a enregistré que la compilation et la vérification des axiomes sont allées à leur terme, les deux journaux étant empreintés. Le fichier du manuscrit a été téléchargé et empreinté. Ce sont des vérifications portant sur des artefacts, pas sur les mathématiques.
Not checked
Personne n'a établi le lien entre le théorème 1.1 du manuscrit et l'énoncé que contient réellement le code Lean ; la cible formelle du dépôt est un énoncé d'existence plus large. Aucune évaluation par un mathématicien n'est consignée ici. Les trois invitations ouvertes désignent ce travail.

Plain-language restatement · Significance editor · 2026-08-07 Translation into fr, attributed to its translator.

Claim
تؤكد OpenAI أن زمرة لا نهائية محددة — زمرة العناصر القابلة للعكس في جبر صغير مبني من رمزين — لا يمكن تقريبها بتباديل منتهية. تسمّى هذه الخاصية الصوفية، ويذهب المخطوط إلى أن هذه الزمرة لا تتمتع بها.
Checked
أعادت Significance بناء شيفرة Lean المنشورة عند إصدار مثبّت، وسجّلت أن عملية البناء وفحص البديهيات اكتملتا، مع بصمة رقمية لكلا السجلّين. كما جرى تنزيل ملف المخطوط وحساب بصمته. هذه فحوص على المصنوعات الرقمية لا على الرياضيات نفسها.
Not checked
لم يتتبع أحد الصلة بين المبرهنة 1.1 في المخطوط والعبارة التي تتضمنها شيفرة Lean فعلياً؛ فالهدف الصوري في المستودع عبارة وجود أوسع. ولا يتضمن هذا السجل أي تقييم من رياضي. والدعوات المفتوحة الثلاث تشير إلى هذا العمل.

Plain-language restatement · Significance editor · 2026-08-07 Translation into ar, attributed to its translator.

Start here · for reviewers

Main deduction

Start with the manuscript's Theorem 1.1 and the companion Comparator target, then trace whether they state the same mathematical result.

Manuscript Theorem 1.1, pp. 78–80; Comparator challenge · Read the statement correspondence before attempting the longer proof. · Significance’s interpretation · Significance editor · 2026-08-17

Risk points

  • The central risk is the gap between the manuscript's concrete unit-group theorem and the formal target's broader finitely-presented existence statement.

    Theorem 1.1 and Comparator/Challenge.lean · Significance’s interpretation · Significance editor · 2026-08-17

  • The proof's use of property (T), Kun's theorem, and the expander decomposition is the main mathematical reading target after correspondence.

    Sections 5–8 · Significance’s interpretation · Significance editor · 2026-08-17

Useful background

  • Basic soficity, property (T), expander graphs, and the role of a finitely presented group presentation.

    Manuscript Sections 2–4 and the cited Kun theorem · Significance’s interpretation · Significance editor · 2026-08-17

Needs checking

  • Compare the theorem represented by the Lean/Comparator target with the concrete group named in the manuscript. — The existing receipt establishes execution and axioms, not informal/formal correspondence.

    Challenge.lean, Solution.lean, and manuscript Theorem 1.1 · Significance’s interpretation · Significance editor · 2026-08-17

  • Suggest another focused check

Formalization handoff

A compact map for someone who wants to formalize this claim. It records preparation and open work, not a mathematical verdict.

Target
The manuscript's Theorem 1.1 and the Comparator target for a finitely presented non-sofic group. Significance’s interpretation · Significance editor · 2026-08-20
System
Lean 4 with the repository's Comparator challenge Significance’s interpretation · Significance editor · 2026-08-20
Work state
Artifact reproduced
Code
formalization repository · commit c510a55434c0… · Lean 4.32.0; toolchain pinned by digest in the receipt

Definitions

  • Soficity, the concrete unit group named in the manuscript, and finitely presented group must be compared before treating the paper and code as the same target.

    Significance’s interpretation · Significance editor · 2026-08-20

Prerequisites

  • Basic group presentations, sofic approximations, property (T), and the repository's Comparator workflow.

    Significance’s interpretation · Significance editor · 2026-08-20

Open questions

  • Trace the declaration chain and record exactly where the concrete Leavitt-algebra statement does or does not meet the broader formal target.

    Significance’s interpretation · Significance editor · 2026-08-20

  • If extending the formalization, identify the smallest missing declaration rather than attempting the whole manuscript at once.

    Significance’s interpretation · Significance editor · 2026-08-20

Paper/code correspondence: The pinned receipt confirms that the formal artifact builds and its declared axiom policy is respected; correspondence with the manuscript remains a separate human reading task. Significance’s interpretation · Significance editor · 2026-08-20

Significance’s interpretation · Significance editor · 2026-08-20

Review activity

Independent reruns
1
Math assessments
0
Written reviews
0
Open checks
2

What Significance checked: Significance rebuilt the published Lean code at a pinned commit and recorded that the build and the axiom check ran to completion, with both logs hashed. The manuscript file was downloaded and hashed. These are checks on artifacts, not on the mathematics.

Still open: Nobody has traced the manuscript's Theorem 1.1 to the statement the Lean code actually contains; the repository's formal target is a broader existence statement. No mathematician's assessment of the argument is recorded here. The three open invitations name this work.

Careful wording

OpenAI has published a claimed proof with a Lean formalisation checked by its own pipeline; the link between the paper's theorem and the formal statement has not been traced by anyone else, and no independent mathematical assessment is on record.

Wording · Significance · 2026-08-07

Copy for sharing

Copy and paste this summary.

OpenAI has published a claimed proof with a Lean formalisation checked by
its own pipeline; the link between the paper's theorem and the formal
statement has not been traced by anyone else, and no independent
mathematical assessment is on record.

Checked: Significance rebuilt the published Lean code at a pinned commit and
recorded that the build and the axiom check ran to completion, with both
logs hashed. The manuscript file was downloaded and hashed. These are checks
on artifacts, not on the mathematics.

Not checked: Nobody has traced the manuscript's Theorem 1.1 to the statement
the Lean code actually contains; the repository's formal target is a broader
existence statement. No mathematician's assessment of the argument is
recorded here. The three open invitations name this work.

As of 2026-08-01T14:09:51Z (freshness: current)
Full record: https://hjyuh.github.io/significance/2026-openai-nonsofic-groups/
Significance records evidence. It does not judge the mathematics.
Scope
The manuscript identifies a concrete countable unit group; its public Comparator target separately states existence of a finitely presented non-sofic group. Significance’s interpretation · Significance editor · 2026-08-01
Excluded
The manuscript says this result does not determine whether the unit group is hyperlinear. Quoted from the source · OpenAI · 2026-08-01, p.80
Source
OpenAI, Ten Advances in Mathematics and Theoretical Computer Science · official-release-2026-08-01 · retrieved 2026-08-01T14:09:51Z
sha256 f318c6508c9d…

Evidence — 2 entries

  1. Formal code published by the source

    ev-openai-lean-artifact

    Published by the source. Any independent rerun appears as a separate entry.

    OpenAI published NonSoficGroup.lean and a Comparator challenge targeting existence of a finitely presented non-sofic group. At the time of this source-publication entry, Significance had not yet attached an independent execution receipt; the later formal-artifact entry records that reproduction.

    Open link @ c510a55434c0… · Quoted from the source · OpenAI · 2026-08-01 · View source

  2. Independent formal-code run

    ev-lean-c510a55434c0… · execution record attached

    Statement in the paper

    The unit group L_F2(1,2)× of the binary Leavitt algebra is not sofic.

    Statement represented in lean4

    Open link @ c510a55434c0…

    At the pinned commit, ComparatorChallenges/D_NonSoficGroup.json maps solution module NonSoficGroup to SoficGroups.SourceTopLevelCompressionFinal.exists_finitelyPresented_nonsofic_group. This identifies the declaration the repository asks Comparator to check; it does not independently establish correspondence with the manuscript's concrete Leavitt-algebra theorem.

    Significance’s interpretation · Significance editor · 2026-08-02 · View source · The match between statements is a human judgement, not an automated check.

    Allowed assumptionsStandard classical Lean profile — propext, Classical.choice, Quot.sound
    Code buildCompleted · 2026-08-02 · log fingerprint d0eb06613908…
    Assumption checkCompleted · 2026-08-02 · log fingerprint d0eb06613908…
    Technical execution details
    Environmentsignificance-lean 0.1.0 · image fingerprint sha256:e18db1519e04… · Significance automated check

Plain-language explanation

A group here is a set of reversible operations you can undo and combine. Soficity asks whether an infinite group can be imitated, to any accuracy you like, by shuffling a finite deck of cards: every product you can form in the group is matched by a shuffle that agrees almost everywhere. The manuscript names the reversible elements of a small algebra built from two symbols. Its claim is that no finite shuffle imitates this group well enough. What Significance holds is the artifacts around that claim -- the manuscript file, the formal code, the logs of rebuilding it -- not a reading of the argument itself.

Explanation · Significance

Mathematical context

The manuscript proposes an explicit boundary for finite-permutation approximation: a concrete group built from the binary Leavitt algebra that is claimed not to admit sofic approximations. The public Lean target records a related existential finite-presentation statement, so correspondence between the concrete and formal claims remains a useful, separate audit task.

Importance

The announcement describes the result as addressing a central open question in group theory. Quoted from the source · OpenAI

Limits

  • This record does not issue a whole-paper correctness judgement.
  • Correspondence between formal and informal statements is attested, not machine-checked; a formalization may omit informal hypotheses.
  • The formal adapter performs narrow artifact checks. It is not a general mathematical verifier.
  • Open invitations identify work not yet represented by a completed evidence entry.