The bridge breaks, proved at last?
Three AI preprints claim counterexamples to Baum–Connes and Kadison–Kaplansky
Covers result 285 in OpenAI’s catalog: all 3 papers. See its connections to other results and the work it cites.
4:07. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Every group has an algebra of operators built from it, the reduced group C*-algebra. If the group has an element of finite order, averaging over that element’s powers gives a projection, an on-off switch in the algebra; in the cyclic group of order 6, projections have traces that are multiples of 1/6. The Kadison–Kaplansky conjecture says that a torsion-free group, with no such elements, has no projections except 0 and 1. It would follow from the Baum–Connes conjecture of 1982, which says that a bridge, the assembly map, from the geometry of a group to the K-theory of its operator algebra, is a perfect match. In 2002, Higson, Lafforgue and Skandalis showed that, given Gromov’s then unpublished monster groups, whose Cayley graphs carry expanders, a version with coefficients fails. The version without coefficients stayed open: on October 1, 2026, Valette wrote that as of September 2026, no counterexample is known.
Result 285 is three OpenAI preprints, 184 pages, dated September 23 and released October 6, 2026. The first claims a finitely generated torsion-free group where assembly is not even rationally injective: a torus class of infinite order becomes zero in the K-theory of the operator algebra. The second claims a finitely generated torsion-free group whose reduced group C*-algebra contains a projection with trace strictly between 0 and 1/2, a counterexample to Kadison–Kaplansky, and with it a failure of surjectivity. The third gives another failure of surjectivity, through a projection with irrational trace, in a group that does have torsion.
What it doesn’t claim
A counterexample to the Novikov conjecture, which the paper says it does not disprove; groups with finitely many defining relations, or explicit ones. The group ring itself still has no such idempotents: the projection lives only in the completed algebra.
Where it stands
- Posted as preprints
- OpenAI, three papers, 184 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No (only a small, unlinked Heisenberg cohomology lemma)
Last checked October 19, 2026. The preprints have not been revised. In Lean, only a small Heisenberg cohomology lemma is formalized, not linked to the item. OSVerify has not reviewed them, we found no assessment by specialists, and one survey notes that the order of parameter choices needs close checking.
How the proof goes
- Expanders planted in a group. All three plant expanders in Cayley graphs by graphical small cancellation, following Gromov and Osajda. A spectral gap puts the projection onto an operator’s kernel inside the C*-algebra itself. In the third paper, that projection’s trace is a sum of powers of 1/p with ever longer gaps, an irrational number, while by Lück’s theorem every class coming from geometry through the assembly map has rational trace.
- Towers, squeezes and a Hilbert hotel. The torsion-free examples lift towers of expanding graphs to covers over nilpotent groups. For the projection, a free product squeezes a small matrix projection into a one-by-one projection with the same trace. For injectivity, a torus inside a Heisenberg group is detected by cohomology, yet its image in the operator algebra is cancelled by shifting an infinite tower up a floor, twice.
The numbers in the video
- Ours: projection traces in the cyclic group of order 6 are 0, 1/6, …, 5/6, 1.
- Ours: a cycle’s spectral gap shrinks toward zero, while Cayley graphs of SL2 mod p level off near 0.2, up to 103,776 vertices.
- Ours: a toy trace with digits 0.1010010001…, never repeating.
Read the source
- A Torsion-Free Counterexample to Reduced Baum–Connes Injectivity (PDF), 85 pages.
- A Torsion-Free Counterexample to the Kadison–Kaplansky Projection Conjecture (PDF), 76 pages.
- An Irrational-Trace Counterexample to Reduced Baum–Connes (PDF), 23 pages.
- The collection’s catalog, where this is result 285.
The 3 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Every group has an algebra of operators built from it, the reduced group C*-algebra. If the group has an element of finite order, averaging over that element’s powers gives a projection, an on-off switch in the algebra. Our computer checks the cyclic group of order 6: its projections have traces that are multiples of 1/6. The Kadison–Kaplansky conjecture says that a torsion-free group, with no such elements, has no projections except zero and one. It would follow from the Baum–Connes conjecture of 1982, which says that a bridge, the assembly map, from the geometry of a group to the K-theory of its operator algebra, is a perfect match. It has been proved for large classes of groups. In 2002, Higson, Lafforgue and Skandalis showed that, given Gromov’s then unpublished monster groups, whose Cayley graphs carry expanders, a version with coefficients fails. The version without coefficients stayed open. Five days before this release, Valette wrote that as of September 2026, no counterexample is known. Expanders are graphs that stay well connected at every size. Our computer finds that a cycle’s spectral gap shrinks toward zero as it grows, while Cayley graphs of SL₂ over the integers mod p level off near 0.2, even with 100,000 vertices.
1:23The claims
Result 285 is three OpenAI preprints, 184 pages, dated September 23 and released October 6, 2026. The first claims a finitely generated torsion-free group where assembly is not even rationally injective: a torus class of infinite order becomes zero in the K-theory of the operator algebra. The second claims a finitely generated torsion-free group whose reduced group C*-algebra contains a projection with trace strictly between 0 and 1/2, a counterexample to Kadison–Kaplansky, and with it a failure of surjectivity. The third gives another failure of surjectivity, through a projection with irrational trace, in a group that does have torsion. Not claimed: a counterexample to the Novikov conjecture, which the paper says it does not disprove; groups with finitely many defining relations, or explicit ones; and the group ring itself still has no such idempotents.
2:25How the proof goes
All three plant expanders in Cayley graphs by graphical small cancellation, following Gromov and Osajda. A spectral gap puts the projection onto an operator’s kernel inside the C*-algebra itself. In the third paper, that projection’s trace is a sum of powers of 1/p with ever longer gaps. Our toy version reads 0.1010010001…, never repeating, so the number is irrational, while by Lück’s theorem every class coming from geometry through the assembly map has rational trace. The torsion-free examples lift towers of expanding graphs to covers over nilpotent groups. For the projection, a free product squeezes a small matrix projection into a one-by-one projection with the same trace. For injectivity, a torus inside a Heisenberg group is detected by cohomology, yet its image in the operator algebra is cancelled by shifting an infinite tower up a floor, twice, a Hilbert hotel argument inside the operator algebra.
3:30Where it stands
The preprints are unrefereed and unchanged since release. In Lean, only a small Heisenberg cohomology lemma is formalized, not linked to the item. OSVerify has not reviewed them, we found no assessment by specialists, and one survey notes that the order of parameter choices requires close checking. If they hold, the geometry of a group and the algebra of its operators can tell different stories, even without torsion, though so far only for monstrous groups built from infinitely many defining relations.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.