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The group-ring Determinant Conjecture (Luck)

Theorem 1.1: a finitely generated group $G_D$ , $n\ge1$ and $A\in M_n(\mathbb Z[G_D])$ , invertible over $\mathbb Q[G_D]$ with bounded inverse, whose Fuglede-Kadison determinant lies strictly between $0$ and $1$ ; the log integral is finite and negative. The group is nonsofic by Elek-Szabo. It disproves the unrestricted conjecture only; it says nothing about torsion-free groups, groups satisfying the Atiyah conjecture, or the determinant approximation conjecture, and it relies on the characteristic-two direct-finiteness counterexample.

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: The group-ring Determinant Conjecture (Luck) · Determinant Conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The group-ring Determinant Conjecture (Luck)

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Conjecture 13.2 of Luck as the paper quotes it; it claims a finitely generated group $G_D$ and $A\in M_n(\mathbb Z[G_D])$ invertible over $\mathbb Q[G_D]$ with $0<\det_{\mathcal N(G_D)}(T_A)<1$ . The proof was not refereed; it depends on the characteristic-two companion (also Lean-checked). lean/formalization.yaml lists a main result for this manuscript (comparator GroupRingDeterminant, declaration OAI.GroupRingDeterminant.main). ComparatorChallenges/GroupRingDeterminant.lean was read here: it asserts a finitely generated group, $n\ge1$ , an integral matrix invertible over $\mathbb Q[G]$ , an invertible bounded operator $T$ matching the left-regular action coordinatewise, and $0<\exp(\tfrac12\mathrm{Re}\,\tau(\log T^*T))<1$ . This states the headline for invertible matrices, the case that disproves the conjecture; the spectral-integral reformulation is not included. Not rebuilt here.