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Kaplansky's zero-divisor conjecture

Claims a counterexample: a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space and nonzero $\alpha,\beta\in\mathbb F_2[G]$ with $\alpha\beta=0$ . The group is the fundamental group of a cone complex over two random labelled graphs, with cancellation forced by odd label intersections; torsion-freeness comes from asphericity. It works in characteristic two only and does not disprove the zero-divisor conjecture over $\mathbb Q$ or $\mathbb C$ , nor the Atiyah conjecture.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

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: Kaplansky's zero-divisor conjecture · Kaplansky zero-divisor conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Kaplansky's zero-divisor conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source against the posed problem, and the Lean statement. formalization.yaml lists OAI.TorsionFreeZeroDivisors.main (file lean/OAI/Algebra/GroupRing/Main.lean, comparator ComparatorChallenges/TorsionFreeZeroDivisors.json, axioms propext, Quot.sound, Classical.choice). The comparator statement asserts a group $G$ that is finitely presented, torsion-free in the ordinary sense, has a finite connected 2-dimensional CW complex with contractible covering as $K(G,1)$ , and has nonzero $\alpha,\beta$ in MonoidAlgebra (ZMod 2) G with $\alpha\beta=0$ . That is the full headline claim. Not rebuilt here; the comparator run was not reproduced.