Result 015, Number theory

Torus-packet equidistribution in prime, quartic, and sextic degrees

Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.

Lean formalization Proof

The bigger picture

Why it matters

Lattices are repeating grids of points. The manuscripts claim that certain complete families built from totally real number fields, called torus packets, eventually sample the space of grids uniformly, without any probability disappearing toward infinity.

What changes?

With each orbit weighted by its volume, the claimed limit is Haar probability, the lattice space's natural uniform measure. The cases are fixed prime degrees at least five with arbitrary full lattices and prescribed local types; primitive degree-four fields with invertible ideal classes of arbitrary orders; and primitive degree-six fields with maximal-order ideal classes, including all coordinate-sign translates. Primitive means no proper intermediate field. The discriminant tends to infinity: that of the multiplier order, arbitrary order, or field, respectively.

What does that help mathematicians do?

An order is a subring forming a full lattice in its number field; diagonal orbits track lattices under coordinatewise scaling. The reported convergence lets researchers replace volume-weighted averages of bounded continuous measurements over these complete packets by averages over the whole lattice space. No escape of mass rules out a hidden fraction drifting into increasingly degenerate lattices. The conclusion concerns complete packets, not individual orbits.

Are there practical applications?

The immediate value is foundational: these claims connect the arithmetic of ideal classes and orders with uniform geometric statistics of lattices. The quartic abstract specifically reports control at primes dividing the order index, supporting arbitrary orders rather than only maximal ones. No computational or technological application is established here.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

September 24, 2026 16 pages

We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,
  author = {{OpenAI}},
  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},
  year = {2026}
}

Equidistribution of primitive quartic torus packets for arbitrary orders

October 5, 2026 70 pages

Let K range over totally real quartic fields with no proper intermediate field, and let O\mathcal O be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible O\mathcal O-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as ∣Disc(O)∣|\mathop{\mathrm{Disc}}\nolimits (\mathcal O)| tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026,
  author = {{OpenAI}},
  title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}},
  year = {2026}
}

Equidistribution of Primitive Sextic Torus Packets

October 5, 2026 53 pages

We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026,
  author = {{OpenAI}},
  title = {{Equidistribution of Primitive Sextic Torus Packets}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/015.md.

Torus-packet equidistribution in prime, quartic, and sextic degrees

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.

Comparator links

Result Comparator statement
Equidistribution of prime-degree torus packets DukePrimeDegree.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.