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

For a discrete group GG and an idempotent matrix ee over the group ring CG\mathbb CG, the Hattori-Stallings trace assigns to each conjugacy class CC the coefficient ∑g∈C∑ieii(g)\sum_{g\in C}\sum_i e_{ii}(g), giving a map K0(CG)→⨁CC CK_0(\mathbb CG)\to\bigoplus_{C}\mathbb C\,C. Bass (1976) conjectured that for the integral group ring the trace is supported at the identity; the complex form asks that it vanish on every conjugacy class of infinite-order elements. Berrick, Chatterji and Mislin (2004) also stated an ℓ1\ell^1 form (finite support on finite-order classes for idempotents over ℓ1(G)\ell^1(G)), proved when the Bost assembly map is rationally surjective. Known for many classes of groups but open in general. Does the Hattori-Stallings trace of every finitely generated projective CG\mathbb CG-module vanish on infinite-order conjugacy classes, for every group GG?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
K-theory of group rings; Hattori-Stallings traces
Posed by
H. Bass, Euler characteristics and characters of discrete groups (Inventiones 1976); complex and l1 forms: Berrick, Chatterji and Mislin, Math. Annalen (2004), Conjectures 2.1 and 2.2
Year posed
1976
Years open
50y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 (24 September): for every discrete group GG, every x∈K0(CG)x\in K_0(\mathbb CG) and every infinite-order gg, HSG(x)([g])=0\mathrm{HS}_G(x)([g])=0; the integral (identity-support) form for ZG\mathbb ZG is deduced in Section 9. The 5 October companion proves the ℓ1\ell^1-Bass conjecture (BCM 2004, Conjecture 2.2): the trace of every idempotent over ℓ1(G)\ell^1(G) is supported on finitely many finite-order classes, with no assembly hypothesis, which implies the complex form. Neither manuscript addresses the reduced C∗C^*-algebra (the Kadison-Kaplansky question), where this release separately claims a counterexample.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization of the complex group-ring theorem accompanies it (Comparator challenge BassTrace, listed in lean/formalization.yaml). The l1 strengthening is a separate 5 October manuscript without Lean.

Verification

No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts (TeX source), read against the Bass conjecture as cited (BCM 2004 Conjecture 2.1; Chatterji-Mislin 2011 Conjecture 2). Lean: Comparator challenge BassTrace, declaration OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support (listed in lean/formalization.yaml). Its statement was read here: for every group GG and every class xx in a module-level K0(C[G])K_0(\mathbb C[G]), the Hattori-Stallings trace vanishes at every infinite-order conjugacy class and its support lies in finite-order classes. That is the complex headline. Not rebuilt here. The ℓ1\ell^1-Bass theorem of the 5 October companion is stronger and is NOT formalized; the manuscript says its proof does not use the complex theorem as input.

Sources

Changelog1 change

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