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

What OpenAI claims

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)

Papers (3)

1. Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. Equidistribution of primitive quartic torus packets for arbitrary orders

   5 October 2026 · Preprint, not peer reviewed · PDF

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

3. Equidistribution of Primitive Sextic Torus Packets

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.