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A negative solution to the finite lattice representation problem

OpenAI · September 24, 2026

Lean

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Abstract

We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.

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Notes

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work in progress

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.

Comments and discussion

  • William DeMeo, OpenAI and the finite lattice representation problem (blog post, 2026-10-07). An AI-assisted reading of both manuscripts that finds Sections 2 and 8–10 correct given Schreier's theorem, says the argument rests on the unrefereed chain-rigidity Theorem 3.2 with an unbounded chain parameter B, and concludes "plausible and not yet refereed." [AI-generated per source] [family-level]

Formalization

  • OpenAI math release, lean/OAI/GroupTheory/FiniteLattice (Lean folder, release commit adc7f12). DeMeo's post attributes these Lean files to this paper and says they cover fences, the normal-subgroup test and four gluing lemmas, but not Theorem 3.2, the lattice construction, or Theorem 1.2; the post did not build the library.
  • OpenAI math release, lean/docs/206.md (mirror) (Lean scope note). The family's scope note covers only the companion paper's colored-graph criterion and states that the subgroup-interval conclusions are outside the formalization. [family-level]
  • franzhusch, Finite-Lattice-Representation Problem (google-deepmind/formal-conjectures issue #1795, opened 2026-01-20). An open issue proposing a Lean formalization of the affirmative conjecture that every finite lattice is Con(A) for a finite algebra A, which this paper claims to refute; it has no linked pull request or comments. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026,
  author = {{OpenAI}},
  title = {{A negative solution to the finite lattice representation problem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf}{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/206.md. The Lean statements have not been checked by people against the paper's claims.

Family 206: Finite lattice representation and undecidability

Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.

Other papers in this family: