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Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

OpenAI constructs infinitely many pairwise nonisomorphic finitely generated ICC property-(T) groups whose tracial group von Neumann factors are mutually isomorphic. The family refutes Connes’s rigidity conjecture and a finite-to-one strengthening.

Sources for this record

What was already known

Connes proposed rigidity for ICC property-(T) groups; prior rigidity theorems covered special classes. Zhou later gave a different pairwise counterexample, without this infinite-family result.

AI involvement AI-led

OpenAI attributes the mathematical argument and Lean formalization to its Astra model. Humans selected problems and prepared the manuscript; chapter reviewers examined parts of the argument.

OpenAI’s original disclosure assigns the argument and subsequent Lean formalization to Astra; human manuscript preparation is separately credited.

Importance Major

The infinite family challenges a longstanding classification principle in operator algebras.

Limitations

A Lean certificate is reported for the formal theorem. Human reviewers checked the core binary-carry construction only in part; a complete independent review of the infinite-family argument is not documented.

Technical verification Formal (Lean 4.32.0 with Mathlib and Comparator + human)

The project reports a complete Lean certificate for the stated formal theorem. Human assessments have narrower scope; semantic matching and any manuscript gaps remain separately noted.

Independent checking

chapter-specific mathematical review of the published argument, with the limited coverage summarized in the audit

Checker
Chapter 4 human reviewers in the Sienicki et al. review corpus
Scope
principal binary-carry counterexample; not every step of infinite-family proof
Basis
Reported by the linked checker
Evidence source

Formal checking

Lean 4.32.0 formal certificate in ConnesRigidity.lean; comparator target for ConnesRigidity.exists_infinite_pairwise_nonisomorphic_propertyT_icc_groups_with_isomorphic_factors; manifest reports zero sorry and only standard axioms

Checker
OpenAI ten-proofs project, Lean kernel and Comparator configuration
Scope
infinite pairwise nonisomorphic property-(T) ICC family with isomorphic factors
Basis
Reported by the linked checker
Evidence source

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

Human understanding Full

The manuscript and Chapter 4 review explain the binary-carry construction, factor isomorphism and group nonisomorphism; Zhou explains a separate construction, not the OpenAI mechanism.

Correction and publication history