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A Counterexample to the Group-Ring Determinant Conjecture

OpenAI · September 23, 2026

Lean

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Abstract

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over $\mathbb Z[G]$ that is invertible over $\mathbb Q[G]$ and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI, Lean scope note for family 197 (lean/docs/197.md) (openai/math release, 2026-10-06). States that the formalization gives a finitely generated $G$ and a matrix over $\mathbb Z[G]$, invertible over $\mathbb Q[G]$, with determinant strictly between 0 and 1, defined via the trace of $\log(T^*T)$. The paper's spectral-measure integral conclusion is not included.

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Cite

@misc{OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A Counterexample to the Group-Ring Determinant Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/197.md. The Lean statements have not been checked by people against the paper's claims.

Family 197: A torsion-free group algebra that is not directly finite

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

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