View source for The Baum–Connes and Kadison–Kaplansky conjectures

'''The Baum–Connes conjecture''' predicts that, for a countable discrete group, the assembly map from the equivariant ''K''-homology of its classifying space for proper actions to the ''K''-theory of its reduced group ''C''*-algebra is an isomorphism. '''The Kadison–Kaplansky conjecture''' predicts that the reduced group ''C''*-algebra of a torsion-free group contains no projections other than 0 and 1. Three manuscripts dated 23 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claim counterexamples to the coefficient-free reduced Baum–Connes conjecture, through a failure of rational injectivity and a separate failure of surjectivity, and to the projection form of the Kadison–Kaplansky conjecture.

== Background ==
* As recounted in the introductions of the manuscripts:
** Baum and Connes formulated the assembly conjecture in a manuscript circulated in 1982 and published in 2000; the formulation with the classifying space for proper actions is due to Baum, Connes and Higson (1994). The coefficient-free form uses the complex numbers with the trivial action; the stronger form allows a coefficient ''C''*-algebra.
** The conjecture is known for large classes of groups. Higson and Kasparov proved it with coefficients for groups acting properly by affine isometries on Hilbert space, including countable amenable groups. Mineyev and Yu proved the coefficient-free form for hyperbolic groups, and Lafforgue later proved the form with coefficients for hyperbolic groups. Guentner, Higson and Weinberger proved split injectivity with coefficients for countable linear groups.
** Higson, Lafforgue and Skandalis used expanders to construct counterexamples with suitable commutative coefficients; those constructions do not give a counterexample to the coefficient-free statement for groups.
** For a torsion-free group the canonical trace is integral on the image of coefficient-free assembly, and for every group Lück's trace theorem makes it rational there. Surjectivity therefore implies the Kadison–Kaplansky projection conjecture for torsion-free groups, and a projection of irrational trace obstructs surjectivity. Pimsner and Voiculescu proved the projection conjecture for free groups, and Mineyev and Yu deduced it for torsion-free subgroups of hyperbolic groups.

== The original results ==
* '''Rational injectivity (23 September 2026).''' Theorem 1.1 of the manuscript states that there are a finitely generated torsion-free group ''G'', an injective homomorphism from '''Z'''<sup>2</sup> to ''G'' and a class in ''H''<sup>2</sup>(''BG''; '''Z''') pairing to 1 with the embedded torus, such that the image in ''K''<sub>0</sub> of the reduced group ''C''*-algebra of a rank-zero Bott generator vanishes. The Dirac class of the torus then gives a class of infinite order in ''K''<sub>0</sub>(''BG'') in the kernel of reduced assembly, which disproves rational injectivity. Finite presentation is not part of the conclusion. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026 A Torsion-Free Counterexample to Reduced Baum–Connes Injectivity].
* '''Surjectivity, through an irrational trace (23 September 2026).''' Theorem 1.1 of the manuscript states that there are a finitely generated group and a projection in its reduced group ''C''*-algebra with irrational canonical trace; its ''K''<sub>0</sub>-class lies outside the image of coefficient-free reduced assembly. The group is not asserted to be torsion-free. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026 An Irrational Trace Counterexample to Reduced Baum–Connes].
* '''The Kadison–Kaplansky projection conjecture (23 September 2026).''' Theorem 1.1 of the manuscript states that there are a finitely generated torsion-free group and a projection ''e'' in its reduced group ''C''*-algebra with canonical trace strictly between 0 and 1/2, so ''e'' is neither 0 nor 1. The group is given by an infinite graphical presentation followed by a free product with a finitely generated free group. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026 A Torsion-Free Counterexample to the Kadison–Kaplansky Projection Conjecture].
* '''Formalization.''' The repository has no Lean formalization of these manuscripts.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 27 October 2026)
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_325.json referee review] of the injectivity manuscript in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || Verdict: no gap found. The review reads the introduction and the assembly section in detail and parts of the geometric and analytic sections, and lists no issue. || Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_444.json referee review] of the Kadison–Kaplansky manuscript in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || Verdict: no gap found. The review checks the scalar compression step and the spectral section line by line and lists no issue. || Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_503.json referee review] of the irrational-trace manuscript in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || Verdict: no gap found. The review checks the diagram, avoidance and overlap lemmas and the order of constants, the hyperbolicity argument at the level of its structure, and lists no issue. || Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-27 || G. Yu, [https://arxiv.org/abs/2610.35674v1 arXiv:2610.35674v1], ''Projections in reduced group C*-algebras with prescribed trace values in (0,1)'' || Extension || Building on the irrational-trace manuscript, constructs for every real number ''r'' strictly between 0 and 1 a finitely generated group ''G''<sub>''r''</sub> and a projection in its reduced group ''C''*-algebra with canonical trace ''r''. For irrational ''r'' its ''K''-theory class is not in the image of the reduced Baum–Connes map. The groups have torsion and satisfy the maximal strong Novikov conjecture. The acknowledgement states that the geometric and spectral techniques are developed from arguments in the OpenAI preprint and that all proofs were developed by the author. || arXiv v1.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
* The Lattice24 reviews are AI-assisted audits; a finding of no gap is not an independent verification of the complete proofs.

[[Category:OpenAI results]]