feynman
Study pack · Pure mathematics

Connes’s rigidity conjecture

Operator algebras · as it stood on 31 July 2026 · run unseen-2026-11-07/connes-rigidity · Claude Opus 5.5

Not enough to judgeClaim 4 of 6

The isomorphism is asserted, not built.

1unsupported2unsupported3unsupported4breaks here5unsupported6unsupported
claims
6
holes
8
lenses
3
records cited
3 of 4
retracted precedents
0

What happened, 3 August 2026. An independent counterexample confirms the conjecture is false. A concurrent paper, posted on 3 August 2026 and made with AI assistance independently of the lab, constructs two non-isomorphic ICC property (T) groups with isomorphic group von Neumann algebras, so the conjecture is false whatever becomes of this manuscript's argument. A later human audit found no substantive error in the manuscript, though it rested on a single review of it, and the stronger infinite family has not been independently reproduced. Asked first, the model said it didn’t know how it ended.

01

The idea, as it stood

A manuscript produced by an AI model, released on 1 August 2026 with nine others, claims to disprove Connes's rigidity conjecture, which holds that an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. It constructs an infinite family of pairwise non-isomorphic ICC property (T) groups whose group von Neumann algebras are all isomorphic. A Lean certificate accompanies the principal theorem, and the manuscript was prepared by people with the model's help.

  1. Every group in the constructed family has Kazhdan's property (T).mathematical · unsupported
  2. Every group in the family is ICC, so its group von Neumann algebra is a II_1 factor.mathematical · unsupported
  3. The family is infinite and its groups are pairwise non-isomorphic as abstract groups.mathematical · unsupported
  4. The group von Neumann algebras of all members are isomorphic to one another as von Neumann algebras, by genuine isomorphisms rather than stable isomorphism, amplification, finite-index passage or a cocycle twist.mathematical · contested · it breaks here
  5. Any two members meet every hypothesis of the conjecture as posed and so refute it, which requires the family to be countable and to avoid every class of property (T) groups already proved determined by its von Neumann algebra.mathematical · unsupported
  6. The Lean certificate proves the principal theorem as stated, using the standard definitions of property (T), ICC, group von Neumann algebra and isomorphism, with no added axioms, unproved lemmas or extra hypotheses.mathematical · unsupported
02

The holes, and who found them

123456John von NeumannAlan TuringRichard Feynman

More than one lens found a hole in claim 6, each on its own. That is a stronger signal than one lens alone.

#HoleLensHow seriousClaimStatus
1The isomorphism is asserted, not builtRichard Feynmanfatal4Open
2A proof checker settles derivations, not meaningAlan Turingserious6Open
3Different kernels are not different groupsAlan Turingserious3Open
4Property (T) for infinitely many groups needs one uniform argumentAlan Turingserious1Open
5The family must miss every class already proved rigidRichard Feynmanserious5Open
6Only countably many members can share one algebraRichard Feynmanserious5Open
7Three checks that can fail as oneJohn von Neumannserious6Open
8ICC must be shown, and the factor step needs a non-trivial groupRichard Feynmanminor2Open
03

What it rests on

as it stood, 31 July 2026201320202026Claim 1reasoning onlyClaim 2reasoning onlyClaim 31 recordClaim 42 recordsClaim 52 recordsClaim 6reasoning only
  1. Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidityChifan et al. · 2023 · Annals of Mathematics · cited on claim 3, 4, 5 · doi:10.4007/annals.2023.198.3.6
  2. RIGIDITY FOR VON NEUMANN ALGEBRAS OF GRAPH PRODUCT GROUPS II. SUPERRIGIDITY RESULTSChifan et al. · 2024 · Journal of the Institute of Mathematics of Jussieu · cited on claim 5 · doi:10.1017/s147474802400015x
  3. A class of superrigid group von~Neumann algebrasIoana et al. · 2013 · Annals of Mathematics · cited on claim 4 · doi:10.4007/annals.2013.178.1.4
  4. Superrigidity of group von Neumann algebrasBerbec · 2014 · Lirias · retrieved, not cited · record
04

The tests that settle it

The deciding test · days of work

Audit the Lean certificate's principal theorem: print its axioms, replay it in a fresh checker against an unmodified library, confirm it is the whole refutation rather than a lemma, and unfold its statement to primitive definitions, comparing each with the standard one: countable discrete groups; property (T) through a Kazhdan pair over all unitary representations; ICC; the weak-operator closure of the left regular representation on ℓ²; a unital *-isomorphism onto the other algebra; non-isomorphism as the absence of any group isomorphism. Then prove negative controls with the same definitions: the integers lack property (T), the integers are not ICC, and the algebras of Z and F2 (the free group on two generators) are not isomorphic.

It fails if: The axiom list shows sorry or anything beyond propext, Classical.choice and Quot.sound; or the unfolded statement is weaker than two non-isomorphic ICC property (T) groups with isomorphic group von Neumann algebras, for instance distinct kernels in place of non-isomorphism, or isomorphic Hilbert spaces, amplifications, corners or twisted algebras in place of isomorphic algebras; or the same definitions prove that the integers have property (T) or that the algebras of Z and F2 are isomorphic.

For holeTestResult
1For two named members G1 and G2, write θ on the generators of G1 as explicit unitaries U_s in L(G2) and verify three things: the U_s satisfy every defining relation of G1 exactly; τ(U_g) = 0 for every non-identity g in G1, which makes θ a trace-preserving embedding; and the U_s generate all of L(G2), with θ(1) = 1. Against the idea: a relation that holds only up to a scalar (a cocycle twist), a non-zero trace at some non-identity element, or images that generate only a corner, an amplification or a proper subalgebra of L(G2).Proposed
2Print the axioms of the principal theorem, replay it in a fresh checker against an unmodified library, confirm it is the whole refutation rather than a lemma, and unfold its statement to primitive definitions. With the same definitions, prove three negative controls: the integers fail property (T), the integers are not ICC, and the algebras of Z and F2 are not isomorphic. Against the idea: sorry or any axiom beyond propext, Classical.choice and Quot.sound, a definition that departs from the standard one, or a negative control whose negation follows from the certificate's definitions.Proposed
3For each pair of members, compute a group invariant that differs, such as the abelianisation or the number of homomorphisms onto a fixed finite group, and state why the algebra does not determine it. Against the idea: non-isomorphism resting only on different kernels, presentations or parameter values, or an explicit isomorphism between two members.Proposed
4Name the single property (T) group, or other uniform mechanism, behind the family, and check that the proof derives property (T) for an arbitrary member with an explicit Kazhdan pair. Against the idea: property (T) checked for only finitely many members, or assumed rather than derived.Proposed
5Check every member, or the construction's parameters, against the exact hypotheses of the main superrigidity theorems in E1 and E3, and record which hypothesis fails for each member. Against the idea: any member that meets all of them.Proposed
6State the family's cardinality and whether every member is a quotient of one property (T) group; for pairs of members, compute the largest trace-norm distance over a Kazhdan set between the images of the generators under the claimed isomorphisms. Against the idea: an uncountable family of quotients of one property (T) group, or two members whose images lie within the Kazhdan constant of each other.Proposed
7Have a mathematician outside the project write the formal statement of the conjecture's negation from the standard definitions without seeing the certificate, then check that the certificate's principal theorem proves exactly that statement, or one proved equivalent to it. Against the idea: the two statements can be matched only by changing a definition or adding a hypothesis.Proposed
8For each member, show it is infinite and that every non-identity element has a centraliser of infinite index, equivalently that no non-identity element has finitely many conjugates. Against the idea: a member with a non-identity element that has finitely many conjugates.Proposed

Proposed tests that haven’t run are the agenda for what comes next. Download them as a draft pre-registration, ready to edit and register on OSF.

05

Two answers, matched blind

The same model answered twice: once plainly, as a standard AI, and once through Feynman. Both answers’ findings were matched blind, without the ending.

The lenses found, the standard AI missed

The isomorphism is asserted, not built Feynman's question, what distinguishes an explanation from a convincing story, led the lens to ask how the isomorphism is built, arguing relabelling and amenability cannot supply it. The cited 2013 and 2023 papers, postdating the lens, added cases where the algebra remembers the group. The standard AI instead warned a weaker sense of 'same algebra' could slip in.

Property (T) for infinitely many groups needs one uniform argument Turing's 'limits of any procedure' led the lens to hold that a finite calculation, however certified, settles only finitely many members, so an infinite family needs one uniform proof of property (T). It cites nothing. The standard AI instead said each member of an infinite family must be verified, and doubted the certificate covers the hard step.

Only countably many members can share one algebra Feynman's 'guess, compute, compare' led the lens to derive a consequence to check: if members are quotients of one property (T) group, only countably many can share an algebra. The cited 2023 paper, postdating the lens, supplied a construction yielding 2^ℵ0 groups, a continuum. The standard AI instead read the infinite family as needing each member verified.

ICC must be shown, and the factor step needs a non-trivial group Feynman's 'do not fool yourself' and 'a name is not an understanding' led the lens to test the ICC label, noting the trivial group is vacuously ICC yet has algebra C, and quotients can lose ICC. It cites nothing. The standard AI instead took ICC as the same as L(G) being a factor, to restate the conjecture.

The standard AI found, the lenses missed

A weaker sense of 'same algebra' is easy to slip in The standard AI drew on common near-misses (amplified, stable, virtual or twisted isomorphism) and the cited Connes and Jones result that, for ICC groups, L(G) has property (T) exactly when G does, reducing the conjecture to W*-superrigidity. Feynman’s lenses instead held that the isomorphism is asserted, not built, and a proof checker settles derivations, not meaning.

Extraordinary claim with a poor base rate The standard AI drew on the conjecture standing since about 1980, cited rigidity evidence from Cowling–Haagerup and Popa, the first W*-superrigid property (T) groups in 2023, and its view that AI claims on famous problems have mostly failed. Feynman’s lenses instead made prior rigidity a requirement: the family must miss every class already proved rigid.

06

The finding

The message reports a refutation without any of its steps: the isomorphism of algebras is not built, the Lean certificate is unaudited, and the packet holds only rigidity results in which the algebra remembers the group.

It held up: An independent counterexample confirms the conjecture is false. What would overturn the review’s reading: The axiom list shows sorry or anything beyond propext, Classical.choice and Quot.sound; or the unfolded statement is weaker than two non-isomorphic ICC property (T) groups with isomorphic group von Neumann algebras, for instance distinct kernels in place of non-isomorphism, or isomorphic Hilbert spaces, amplifications, corners or twisted algebras in place of isomorphic algebras; or the same definitions prove that the integers have property (T) or that the algebras of Z and F2 are isomorphic.

07

Cite it, disclose it, reproduce it

A paragraph for your methods section

We used Feynman (feynman.works), an AI research-review tool running on Claude Opus 5.5, to critique “Connes’s rigidity conjecture” on 2026-11-07 (run unseen-2026-11-07/connes-rigidity). It reconstructed 6 claims, retrieved 4 published records dated no later than 2026-07-31, and raised 8 objections through reasoning frameworks drawn from the published work of John von Neumann, Alan Turing and Richard Feynman. Each citation in its output was checked against the records it retrieved, and we read every source we relied on. Its objections informed our analysis; the conclusions, and any errors, are our own.

The lenses

  • John von NeumannMathematics & strategy · 1928–1957What are the incentives, available strategies and consequences of each move?
  • Alan TuringComputation & intelligence · 1936–1954What observable test would let us distinguish the competing claims?
  • Richard FeynmanPhysics & scientific method · 1948–1980What experiment would distinguish this explanation from a convincing story?

BibTeX

@misc{feynman-unseen-2026-11-07-connes-rigidity,
  title = {{Feynman review: Connes’s rigidity conjecture}},
  author = {{Feynman Works}},
  year = {2026},
  howpublished = {\url{https://www.feynman.works/pack/connes-rigidity}},
  note = {Run unseen-2026-11-07/connes-rigidity, Claude Opus 5.5, 2026-11-07}
}

@article{chifan2023wreath,
  title = {{Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidity}},
  author = {Chifan and others},
  year = {2023},
  journal = {Annals of Mathematics},
  doi = {10.4007/annals.2023.198.3.6}
}

@article{chifan2024rigidity,
  title = {{RIGIDITY FOR VON NEUMANN ALGEBRAS OF GRAPH PRODUCT GROUPS II. SUPERRIGIDITY RESULTS}},
  author = {Chifan and others},
  year = {2024},
  journal = {Journal of the Institute of Mathematics of Jussieu},
  doi = {10.1017/s147474802400015x}
}

@article{ioana2013class,
  title = {{A class of superrigid group von~Neumann algebras}},
  author = {Ioana and others},
  year = {2013},
  journal = {Annals of Mathematics},
  doi = {10.4007/annals.2013.178.1.4}
}

Run unseen-2026-11-07/connes-rigidity on Claude Opus 5.5. The review is reproducible from its record: the claims, the records retrieved before 31 July 2026, and each lens’s card. https://www.feynman.works/pack/connes-rigidity