OpenAI reports the ℓ¹-Bass conjecture for every discrete group

For every discrete group G, the Hattori–Stallings trace of every idempotent matrix over the complex Banach algebra ℓ¹(G) has finite support, contained in conjugacy classes of finite-order elements.

Sources for this record

What was already known

Bass-type support results were known for various restricted group classes and for algebraic group rings; the ℓ¹ version additionally requires finite support for completed-algebra idempotents.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Major

Resolves the ℓ¹-Bass conjecture for arbitrary discrete groups.

Limitations

The linked Lean scope checks Bass trace vanishing for the algebraic complex group ring C[G], not ℓ¹(G); it does not establish finite support for Banach-algebra idempotents.

Technical verification Partial

OpenAI reports a Lean check of a narrower theorem, case, or component; the selected claim extends beyond that scope.

Formal checking

The pinned OpenAI Lean scope note verifies only this narrower statement, not the full selected claim.

Checker
OpenAI-reported Lean 4 / Comparator
Scope
For every group G, the Hattori–Stallings trace on finitely generated projective C[G]-modules vanishes on conjugacy classes of infinite-order elements.
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and precise partial or mismatched Lean-scope review for non-notifying local inclusion.

Read this published revision