# Can every group be approximated by finite permutations?

- ID: non-sofic-groups
- Provider: OpenAI
- Published: 2026-08-01
- Editorial check: 2026-10-29
- Status: formalized-provider-reported
- Evidence: Provider announcement; formal certificates reported, not rerun here
- Source: https://openai.com/index/ten-advances-in-mathematics/
- Public article: https://ai-tracking.sevensa.nl/en/discoveries/non-sofic-groups/

## Plain-language explanation

A group describes reversible operations and how they combine. One way to understand an infinite group is to imitate small portions using permutations of a finite set. Soficity asks whether increasingly accurate imitations of this kind are always possible.

## Research contribution

OpenAI reports a construction of a non-sofic group. If accepted under the stated definition, this supplies a counterexample to the universal approximation conjecture.

## Academic interpretation

The central issue is approximation of multiplication relations with small normalized error, not whether the group has a finite presentation or a simple drawing. An audit must check exactly which relations are preserved, how approximation error is measured, and why every sufficiently accurate finite model is ruled out. A readable exposition and a machine-checked statement answer complementary questions.

## Implications

The result could clarify the limits of a broad approximation principle used across group theory and related operator-algebra questions.

## Caveat

This is a source-reported construction with reported formalization. We have not independently checked the group construction or its certificate.

This is an editorial summary, NOT an independent mathematical proof check.