OpenAI claim summary for Math-429

Lück's Determinant Conjecture for Discrete Groups

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. An integral group-ring matrix is claimed to be rationally invertible while its Fuglede--Kadison determinant lies strictly between $0$ and $1$.

Mathematical statement of the claimed result

Result 1.

For some finitely generated discrete group $G_D$, integer $n\ge1$, and matrix \[ A\in M_n(\mathbb Z[G_D]),\qquad A\text{ invertible over }\mathbb Q[G_D], \] the operator induced by the left regular action $T_A$ on $\ell^2(G_D)^n$ has Fuglede--Kadison determinant \[ 0<\det_{\mathcal N(G_D)}(T_A)<1. \] Here $\mathcal N(G_D)$ is the group von Neumann algebra and the logarithmic spectral integral defining the determinant is finite and negative. The characteristic-two direct-finiteness result quoted in the proof is an input, not an additional determinant hypothesis. The convention is $(L_g\xi)(h)=\xi(g^{-1}h)$, and the matrix trace $\tau_n$ is unnormalized. For the normalized spectral law $\mu_A$ of $T_A^*T_A$, \[ \log\det_{\mathcal N(G_D)}(T_A)=\tfrac12\tau_n(\log(T_A^*T_A))=\tfrac n2\int\log t\,d\mu_A(t). \]

Outline of the argument

A characteristic-two inverse defect is lifted using finite-coordinate projections in a wreath-product setting. These projections detect a selected coefficient, and a tracial logarithmic derivative applied to block shears calculates a determinant below the conjectured integral lower bound.

Formal verification

For family 197, the documented Lean entries include the scalar inverse defects in the specified characteristics and an operator-invertible determinant counterexample. The characteristic-two search recipe is included, but the nonsofic consequence, the extra spectral-integral conclusion, and the later torsion-free construction are outside the selected statements. The listed Comparator files are GroupRingDeterminant.lean, KaplanskyDirectFiniteness.lean, KaplanskyFinitelyPresented.lean, OddKaplansky.lean. Named declarations include OAI.GroupRingDeterminant.main, OAI.KaplanskyCounterexample.finitelyPresented_counterexample, OAI.OddKaplansky.main_theorem. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, A Counterexample to the Group-Ring Determinant Conjecture, 2026-09-23; citation key OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: thm:input.
• Family 197: Lean scope.
• Comparator GroupRingDeterminant.json.
• Comparator GroupRingDeterminant.lean.
• Comparator KaplanskyDirectFiniteness.json.
• Comparator KaplanskyDirectFiniteness.lean.
• Comparator KaplanskyFinitelyPresented.json.
• Comparator KaplanskyFinitelyPresented.lean.
• Comparator OddKaplansky.json.
• Comparator OddKaplansky.lean.
• Family 197: Reasoning summary.
• OpenAI catalogue entry.