An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
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Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The other possible matches are Theorem 1.1 in An infinite finitely presented residually finite 2-group () and Corollary 1.2(A periodic Kazhdan group) in the same paper ().
There is an infinite periodic group with an ordinary finite presentation. More precisely, the construction below gives a unital associative \F_2-algebra R for which \St_{12}(R) is infinite, finitely presented, and periodic.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We prove that the infinite, ordinarily finitely presented periodic Steinberg group \(\Gamma=\mathop{\mathrm{St}}\nolimits _{12}(R)\) of a companion paper is residually finite. Its finite-index subgroup \(G=\ker(\Gamma\to\mathop{\mathrm{St}}\nolimits _{12}(\mathbb F_2))\) is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion’s graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh’s algebraic finiteness question negatively.
One Lean statement in this family
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There is an infinite periodic group with an ordinary finite presentation. More precisely, the construction below gives a unital associative \F_2-algebra R for which \St_{12}(R) is infinite, finitely presented, and periodic.
The words they share: algebra, finite, finitely, gives, group, infinite, periodic, presented.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This 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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:thm_main : MainStatement ∧ BurnsideStatement only gives an infinite finitely presented PERIODIC group (Periodic G := ∀ g, ∃ m > 0, g^m = 1, namely Steinberg 12 R for some F₂-algebra R, and a general ∃ G version). The summary’s group is a residually finite 2-group (every element of 2-power order) with accompanying nil-algebra consequences; neither ‘residually finite’ nor ‘2-power order’ nor the algebra statements is stated (docs: 'no common exponent is asserted... separate nil-algebra and radical-algebra conclusions are outside these selected statements'). The negative answer to the finitely presented Burnside problem itself is covered.
Definitions the group asks readers to compare with the paper:Steinberg n R := PresentedGroup {r | SteinbergRel n R r} with relations x_{ij}(a+b) = x_{ij}(a) x_{ij}(b), [x_{ij}(a), x_{kl}(b)] = 1 for p.col ≠ q.row ∧ p.row ≠ q.col, and [x_{ij}(a), x_{jk}(b)] = x_{ik}(ab): the standard Steinberg presentation, hand-written (the carrier ring R is existentially chosen).
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:Residual finiteness and the 2-power order of every element (the October 5 manuscript’s contribution; undeclared), plus the nil algebra and algebraic-unitization clause (declared outside).
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The finitely presented Burnside question asks whether a finitely presented group in which every element has finite order must be finite. The formalization gives a negative answer by constructing an infinite finitely presented periodic group, including a witness realized as a Steinberg group over an algebra of characteristic two. Periodicity means that each element has some finite order; no common exponent is asserted. The paper’s separate nil-algebra and radical-algebra conclusions are outside these selected statements.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an infinite finitely presented periodic group, realised as St₁₂(R) for an F₂-algebra R. The nil-algebra conclusions and the Kazhdan corollary are not part of the statement.
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Family 247The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.