The group-ring Determinant Conjecture (Luck)

For a discrete group $G$ and a matrix $A$ over $\mathbb Z[G]$ , let $T_A$ act on $\ell^2(G)^n$ by the regular representation and let $\mu_A$ be the spectral measure of $T_A^*T_A$ with respect to the group trace. A nonsingular integer matrix has $|\det|\ge1$ ; the Determinant Conjecture (Luck 2002, Conjecture 13.2) asks for the analogue $\int_{(0,\infty)}\log t\,d\mu_A(t)\ge0$ , which for invertible $A$ says the Fuglede-Kadison determinant is at least $1$ . It was known for amenable groups (Schick) and sofic groups (Elek-Szabo), and underlies approximation and torsion results for $L^2$ -invariants. Does the Determinant Conjecture hold for every group?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: L2-invariants; Fuglede-Kadison determinants over group von Neumann algebras
Posed by: Wolfgang Luck, L2-Invariants: Theory and Applications to Geometry and K-Theory (2002), Conjecture 13.2
Year posed: 2002
Years open: 24y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 35 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof takes the characteristic-two direct-finiteness counterexample of the companion manuscript as its only nonstandard input.

Verification

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.

The group-ring Determinant Conjecture (Luck)

What was actually shown

What the AI did

Verification

Sources

Discussion