Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one. (OpenAI’s Lean note)
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In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
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Let P be a finitely generated virtually polycyclic group, and let H be any finitely generated group. If H and P are quasi-isometric, then H is virtually polycyclic.
No Lean statement in the release is paired with this theorem.
In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
If a finitely generated group H is quasi-isometric to a lattice in a connected simply connected solvable Lie group, then a finite-index subgroup of H is a uniform lattice in some connected simply connected solvable Lie group.
No Lean statement in the release is paired with this theorem.
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PolycyclicRecognitionThe paper states
In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
Let G be a connected simply connected real-triangulable unimodular Lie group, with a fixed left invariant Riemannian metric, exponential height map \pi:G\to W, and a fixed norm on W. There is a finite subgroup \A_G\leq\GL(W) such that, for every K\geq1 and C\geq0, there is a constant B=B(G,K,C) with the following property. Every (K,C) self quasi-isometry F of G has an A_F\in\A_G satisfying
The words they share: bounded, connected, exponential, finite, fixed, group, height, invariant, isometry, left, metric, quasi, real, self, simply, subgroup, uniform, unimodular.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PolycyclicRecognition.lean, the release lists it as a supporting statement.
theorem group_recognition {P J : Type} [Group P] [Group J] [Group.FG P] [Group.FG J] (e : GroupCoarseEquivalence P J) (hP : IsVirtuallyPolycyclic P) : IsVirtuallyPolycyclic J ∧ ∃ Γ : Subgroup J, Γ.FiniteIndex ∧ ∃ (S : SimplyConnectedSolvableLieModel) (Λ : Subgroup S.carrier), IsUniformLattice Λ ∧ Nonempty (Γ ≃* Λ) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
The file states two more theorems: main_, lattice_.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: group_recognition (e : GroupCoarseEquivalence P J) (hP : IsVirtuallyPolycyclic P) [P, J f.g.] : IsVirtuallyPolycyclic J ∧ ∃ Γ finite-index, ∃ S Lie model, Λ, IsUniformLattice Λ ∧ Nonempty (Γ ≃* Λ) and lattice_recognition (a finite-covolume lattice in a simply connected solvable Lie group, quasi-isometric to a f.g. J, makes J virtually a uniform lattice in possibly another such group). ‘Quasi-isometric’ is encoded as GroupCoarseEquivalence (bornologous maps with finite-set errors), which for f.g. groups is equivalent to a quasi-isometry; polycyclic = explicit cyclic subnormal series. Both summary claims are stated.
Definitions the group asks readers to compare with the paper: Hand-built coarse geometry: GroupBornologous f := ∀ S finite, ∃ T finite, ∀ x y, x⁻¹ y ∈ S → (f x)⁻¹ (f y) ∈ T, GroupCoarseEquivalence (maps in both directions plus finite error sets) used instead of word-metric quasi-isometry; IsUniformLattice Λ := DiscreteTopology Λ ∧ ∃ K compact, ∀ g, ∃ γ k, k ∈ K ∧ γ k = g; SimplyConnectedSolvableLieModel packages Mathlib LieGroup/SimplyConnectedSpace/Group.IsSolvable. Plausible and standard but non-Mathlib. External package Gromov is in the import cone.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group.
The linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.
Quasi-isometric recognition of virtually polycyclic groups and solvable lattices PolycyclicRecognition
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main recognition theorem, its lattice corollary and the bounded-height theorem for self quasi-isometries of unimodular solvable Lie groups.
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PolycyclicRecognitionTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 1804 s; peak memory 8.5 GiB). | 10 October 2026 | Original report |
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Why this passage is shown
Theorem 1.1
In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
Why this passage is shown
Corollary 1.2
In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
Why this passage is shown
Theorem 1.4 (Uniform bounded height)
In Quasi-isometric recognition of virtually polycyclic groups, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, PolycyclicRecognition
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, PolycyclicRecognition
Lean statement
PolycyclicRecognition