Connes' rigidity conjecture for lattices

Connes' rigidity conjecture for lattices concerns whether the group von Neumann algebra of a rigid lattice, such as SL₃(Z), remembers the group. Two manuscripts in OpenAI's public mathematics collection, both dated 23 September 2026, claim a classification of all finite-index bimodules between scalar-twisted group factors of groups commensurable with property (T) lattices over characteristic-zero local fields and twisted factors of arbitrary ICC groups, and deduce that such factors recover the group, the cocycle and the amplification scale. The collection has no Lean formalization of these results.

Background

As recounted in the manuscripts' introductions:
- Connes' problems. Connes proved that the fundamental group of a property (T) group factor is countable. His later problems ask whether nonisomorphic ICC property (T) groups have nonisomorphic group factors and what the fundamental groups of these factors are. Popa's survey states a stronger recovery question that also controls the given isomorphism and its amplification scale.
- Counterexamples among property (T) groups. Independent constructions by OpenAI and by Zhou, cited in the manuscript, give nonisomorphic ICC property (T) groups with isomorphic group factors, so property (T) alone does not ensure reconstruction.
- Earlier rigidity theorems. Deformation and rigidity methods gave Ioana's recovery of Bernoulli actions, the W*-superrigid groups of Ioana, Popa and Vaes and of Chifan, Ioana, Osin and Sun, Vaes's classification of finite-index bimodules for generalized Bernoulli factors, and results of Donvil and Vaes and of Chifan, Fernández Quero, Osin and Tan for twisted wreath-like products.
- The lattice side. Bekka classified the characters of SL_n(Z), n ≥ 3, and Peterson established a finite-factor representation dichotomy for higher-rank lattices. Boutonnet and Houdayer developed stationary and noncommutative boundary methods for such lattices.

The original results

Arithmetic exhaustion, 23 September 2026. The main theorem of Arithmeticity of twisted finite correspondences for lattices over local fields concerns groups commensurable with lattices in finite products of noncompact absolutely simple groups with property (T) over R, C or finite extensions of Q_p, including quaternionic and Cayley rank-one groups. It states that every bifinite correspondence from a scalar-twisted group factor of such a group to a scalar-twisted factor of any countably infinite ICC group is a closed summand of a finite direct sum of models built from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio.
Stable recovery, 23 September 2026. The arithmetic category and stable recovery of lattice factors uses that theorem to compute the arithmetic correspondence category and states stable canonical recovery: a stable isomorphism to a twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain and equal amplification scales.
Scope, as stated: the class consists of groups commensurable with property (T) lattices over characteristic-zero local fields; nonzero correspondences force the groups to be abstractly commensurable. Lattices without property (T) and fields of positive characteristic are outside the class.
Method, as described: the proof constructs compatible boundary measurements inside the correspondence, using symmetric spaces and Bruhat–Tits buildings, whose simultaneous law produces finite-rank subgroup-graph fibres; the companion paper adds the algebraic calculus of summands, intertwiners and fusion.
Sources. Commit adc7f12 of the openai/math repository: Arithmeticity of twisted finite correspondences for lattices over local fields and The arithmetic category and stable recovery of lattice factors.
Formalization. The collection lists no Lean scope note or Comparator challenge for this family.
Within the collection, a related manuscript on type III factors is described on Connes' bicentralizer problem.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-07	Lattice factors: finite correspondences and stable recovery, an article of the Latent Seminar blog at lowrank.github.io	Reading, exposition	Explains family 286: twisted factors and module dimension; the lattice class and the recovery statements; explicit models and the cocycle ratio; boundary measurements and finite-dimensional fibers; why summands require a separate argument; recovering stable scales and the specified map; recovering an unknown factor from multiplication.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. Read on 31 October 2026.
2026-10-08	C. Houdayer, reaction in 100+ reactions to 100+ solutions, a collected post on the Proofs and Prompts blog	Comment	Houdayer, who worked on the conjecture with Boutonnet through noncommutative boundary theory, writes that he is genuinely thrilled by the announced solution, wants to understand the proof and plans to study it with his PhD students and postdocs during an IHP programme on operator algebras. He does not report a completed verification.	Blog post dated 8 October 2026, with reactions added through 19 October 2026. Read on 31 October 2026.
2026-10-16	J. Jardine, Lattice24, referee reviews rank_003 of the exhaustion manuscript and rank_450 of the companion in JJardine919/openai-math-audit	Review, AI-assisted audit	Both reviews report no gap. The exhaustion review notes that the claim is stronger than the abstract suggests, giving trivial fundamental group and stable rigidity for the lattice class, and calls for extra scrutiny of the sections it did not read line by line; it did not read the finite-panel and ray-estimate sections, most of the three sections that follow them, or parts of others. The companion review read the whole companion, notes that all its geometric content comes from the exhaustion theorem, and did not read that manuscript.	Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
2026-10-18	AI Proved What?, episode 286 A perfect fingerprint, proved at last?, remade version of 23 October 2026	Reading, exposition	Video explainer of the two preprints, 190 pages: SL₃ over Z and related lattices would be W*-superrigid and their factors would have trivial fundamental group. It recalls the August 2026 counterexamples among general property T groups, one from OpenAI and one by Zhou with AI help, and states that the preprints do not claim lattices without Kazhdan's property T, positive characteristic or general groups with property T.	Episode status, last checked 30 October 2026: posted as preprints, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, no, with about 2,300 unlinked lines covering some algebra of the model bimodules. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.

Notes

The background summarizes the manuscripts' own accounts of earlier work.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.