The higher-dimensional Duke theorem for torus packets of totally real fields of prime degree at least five (with primitive quartic and sextic cases)
Ideal classes of a totally real field of degree give compact orbits of the diagonal group on , grouped into packets by local homothety type. Duke proved equidistribution for ; Einsiedler, Lindenstrauss, Michel and Venkatesh proved the cubic case and described the extension to higher prime degree as conditional on subconvexity; Lemke Oliver, Thorner and Zaman handled prime degree outside a sparse exceptional set of fields. For a fixed degree and discriminants tending to infinity, do the volume-weighted packet measures on converge to Haar probability measure, with no escape of mass?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Homogeneous dynamics; equidistribution of periodic torus orbits
- Posed by
- Einsiedler, Lindenstrauss, Michel and Venkatesh (Distribution of periodic torus orbits and Duke's theorem for cubic fields, Annals of Math. 2011, Section 1.6.3), following Linnik and Duke
- Year posed
- 2011
- Years open
- 15y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Principal Theorem 1.1: for each fixed prime and any totally real fields of degree with full lattices whose multiplier-order discriminant , the packet measures converge weakly to Haar measure on with no escape of mass, for every local homothety type, nonmaximal orders of unbounded index and every field (no exceptional set). Companions: the same for Picard packets of arbitrary orders in primitive totally real quartic fields, and for ideal-class packets of maximal orders in primitive totally real sextic fields. Not shown: individual orbits, rates, composite degrees with proper intermediate fields, nonmaximal sextic orders.
What the AI did
Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.
Verification
No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of all three manuscripts, read against the higher-dimensional Duke problem as they state it; proofs not refereed. Lean-checked on the release's Comparator challenge DukePrimeDegree with solution module OAI.NumberTheory.DukePrimeDegree.MainUnconditional, both fetched at the pinned commit; the challenge is not listed in lean/formalization.yaml. The statement was read here and not rebuilt: it defines full lattices, multiplier discriminants, local homothety, packet orbits and volume-weighted packet measures from scratch, and asserts weak convergence to a Haar probability plus tightness for every prime degree , which is the principal theorem. The quartic and sextic companions are not formalized. Composite degrees with intermediate fields remain open, hence partial.
Sources
- PaperCompanion: Equidistribution of primitive quartic torus packets for arbitrary ordersCompanion: Equidistribution of primitive sextic torus packets
- Lean proofLean Comparator challenge DukePrimeDegree (not in formalization.yaml)Lean solution module OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean
- CodeOpenAI math release: Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type
- Problem recordEinsiedler, Lindenstrauss, Michel, Venkatesh, Duke's theorem for cubic fields, Annals 173 (2011)