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The finitely presented Burnside problem: is every finitely presented periodic group finite?

Claims Theorem 1.1: there is a finitely presented graded $\mathbb F_2$ -algebra $R$ such that $\mathrm{St}_{12}(R)$ is infinite, ordinarily finitely presented and periodic; Corollary 1.2 adds Kazhdan's property (T), so it is nonamenable. The companion claims $\mathrm{St}_{12}(R)$ is residually finite and that the kernel $G$ of the map to $\mathrm{St}_{12}(\mathbb F_2)$ is an infinite finitely presented residually finite 2-group, necessarily of unbounded exponent. It does NOT give a finitely presented infinite group of bounded exponent (the bounded Burnside question for finite presentations), and element orders depend on the element.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The finitely presented Burnside problem: is every finitely presented periodic group finite? · Finitely presented Burnside problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The finitely presented Burnside problem: is every finitely presented periodic group finite?

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No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: an explicit unital $\mathbb F_2$ -algebra $R$ for which the Steinberg group $\mathrm{St}_{12}(R)$ is infinite, ordinarily finitely presented and periodic. The Lean challenge ComparatorChallenges/PeriodicGroup.json (theorem OAI.SourceBurnside.thm_main, solution module OAI.GroupTheory.PeriodicGroups.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/247.md. Its statement was read here: a ring $R$ that is an $\mathbb F_2$ -algebra with the Steinberg group on 12 indices (presented by the addition and commutator relations) infinite, finitely presented and periodic, together with the bare statement that some infinite finitely presented periodic group exists. That is the headline claim; no uniform exponent is asserted. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The residual-finiteness companion and the nil-algebra conclusions are not formalised.