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The Kadison-Kaplansky Projection Conjecture

There is a finitely generated torsion-free group G=G0∗FNG=G_0*\mathbb{F}_N and a projection e∈Cr∗(G)e\in C_r^*(G) with 0<τ(e)<1/20<\tau(e)<1/2, so e≠0,1e\ne 0,1. It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. 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 Kadison-Kaplansky Projection Conjecture · Kadison-Kaplansky conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Lineage and corrections

Same family: An irrational-trace counterexample to reduced Baum-Connes evidence for this event

The Kadison-Kaplansky Projection Conjecture parent of this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: The Kadison-Kaplansky Projection Conjecture evidence for this event

The Kadison-Kaplansky Projection Conjecture evidence for this event

Same family: A torsion-free counterexample to reduced Baum-Connes injectivity evidence for this event

There is a finitely generated torsion-free group G=G0∗FNG=G_0*\mathbb{F}_N and a projection e∈Cr∗(G)e\in C_r^*(G) with 0<τ(e)<1/20<\tau(e)<1/2, so e≠0,1e\ne 0,1. It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group. parent of this event

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