The CKLW strong-locality conjecture for strongly rational unitary vertex operator algebras
A simple unitary vertex operator algebra is CKLW-strongly local if its fields satisfy polynomial energy bounds and the von Neumann algebras generated by the closed smeared fields with commute for disjoint intervals of the circle, so that generates a conformal net. Carpi, Kawahigashi, Longo and Weiner proved this for unitary affine, Virasoro and moonshine VOAs and conjectured it for every simple unitary VOA (Conjecture 8.18). Commuting unbounded fields need not give commuting von Neumann algebras, so the point is analytic. Is every simple unitary vertex operator algebra strongly local?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Chiral conformal field theory: vertex operator algebras and conformal nets
- Posed by
- Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo and Mihaly Weiner (From vertex operator algebras to conformal nets and back, Conjecture 8.18)
- Year posed
- 2015
- Years open
- 11y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 27 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Theorem 1.1: every simple unitary strongly rational VOA has polynomial energy bounds and is CKLW-strongly local, generating an irreducible conformal net ; moreover is completely unitary (all simple modules unitarizable, fusion forms positive), every unitary module is strongly integrable, is completely rational, and the Carpi-Weiner-Xu functor is a braided unitary tensor equivalence . It does not treat non-rational or non-C2-cofinite unitary VOAs, so the general conjecture remains open.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.
Verification
No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against CKLW Conjecture 8.18 as cited. The proof was not refereed. The result covers only strongly rational VOAs (CFT type, self-contragredient, rational, C2-cofinite); the general conjecture stays open. Lean-checked on the release's own Comparator challenge VertexAlgebraNet together with its solution module OAI.RepresentationTheory.VertexAlgebra.MinimalComparator, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.MinimalVertex.main) assumes a simple CFT-type VOA with a unitary structure that is self-contragredient, rational and C2-cofinite, and concludes polynomial energy bounds, CKLW strong locality (interval algebras contained in the commutant of the complementary interval's algebra) and an irreducible conformal net. That is the headline claim of this partial entry. The 47 kB statement file was read for its main theorem and key definitions only. Not formalized: complete rationality, complete unitarity, the braided equivalence and the extension correspondence (parts 2 to 5).
Sources
- Lean proofLean Comparator challenge VertexAlgebraNet (not in formalization.yaml)Lean solution module OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean
- CodeOpenAI math release: Strongly rational unitary vertex operator algebras and conformal nets
- Problem recordCarpi, Kawahigashi, Longo and Weiner, Mem. Amer. Math. Soc. 254 (2018), Conjecture 8.18