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Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero

Corollary 1.2(ii): for every torsion-free group $G$ and every commutative unital domain $R$ of characteristic zero, the only idempotents of $RG$ are $0$ and $1$ . Part (i): for torsion-free $G$ the canonical trace of an idempotent matrix over $\mathbb CG$ is the rank of its augmentation, so $\tau_*(K_0(\mathbb CG))=\mathbb Z$ . Both are deduced from the Bass trace theorem. Nothing is claimed in positive characteristic or about projections in $C^*_r(G)$ ; the release separately claims a torsion-free counterexample to the Kadison-Kaplansky projection conjecture there, which is consistent with this result.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 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 idempotent conjecture for torsion-free groups in characteristic zero · Kaplansky idempotent conjecture (char 0)

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction, Corollary 1.2 and the torsion-free section of the TeX source, read against the idempotent conjecture as cited. Lean: Comparator challenge BassTorsionFree, declaration OAI.TorsionFreeBass.torsion_free_corollary. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: for torsion-free $G$ , the trace of every idempotent matrix over $\mathbb C[G]$ equals the rank of its augmentation, the $K_0$ trace equals the augmentation rank with range $\mathbb Z$ , and for every commutative domain $R$ of characteristic zero every idempotent of $R[G]$ is $0$ or $1$ . The last clause is exactly the headline. Not rebuilt here. The paper notes its argument does not cover the reduced $C^*$ -algebra.

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The Bass trace conjecture for complex group rings of all discrete groups

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean Comparator challenge BassTorsionFree (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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The l1-Bass Conjecture for Discrete Groups (companion, stronger l1 form)

paper · pending

paper link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge BassTorsionFree (OpenAI math release)

lean artifact · pending

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Lean Comparator challenge BassTorsionFree (OpenAI math release) evidence for this event

Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero parent of this event

This event attributed to Unreleased internal OpenAI model

The l1-Bass Conjecture for Discrete Groups (companion, stronger l1 form) evidence for this event

VibeMathed record: Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero evidence for this event

The Bass trace conjecture for complex group rings of all discrete groups evidence for this event

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