Torus-packet equidistribution in prime, quartic, and sextic degrees
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
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. (OpenAI’s Lean note)
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The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let (K_i,\OO_i,\sigma_i) be any sequence in which K_i is a primitive totally real quartic field, \OO_i is an order in K_i, and \sigma_i is an ordering of its real embeddings. If |\Disc(\OO_i)|\to\infty, then
\int_{X_4} f\,d\mu_{\OO_i,\sigma_i}
\longrightarrow \int_{X_4} f\,dm_4
\qquad\text{for every }f\in C_c(X_4).
There is no escape of mass: for every \varepsilon>0 there is a compact \mathcal K\subset X_4 such that \mu_{\OO_i,\sigma_i}(\mathcal K)\ge1-\varepsilon for all sufficiently large i.
No Lean statement in the release is paired with this theorem.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let K_i be totally real primitive fields of degree six, with |\Disc(K_i)|\to\infty, and choose any ordering \sigma_i of the embeddings of each field. Then
There is no escape of mass: for every \epsilon>0 there is a compact C\subset X_6 such that \mu_{K_i,\sigma_i}(C)\ge1-\epsilon for all sufficiently large i.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
One more possible match
Theorem 1.1in Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type, 24 September 2026.
Reported proof verification
One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
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.
Let K range over totally real quartic fields with no proper intermediate field, and let \(\mathcal O\) be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible \(\mathcal O\)-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as \(|\mathop{\mathrm{Disc}}\nolimits (\mathcal O)|\) tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
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.
One Lean statement in this family
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For every fixed prime n\ge5 and every sequence (K_i,M_i,\sigma_i) as above with D(M_i)\to\infty,
\mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.
The convergence is weak convergence of probability measures. In particular, it holds against every continuous compactly supported function, and no mass escapes into the cusp.
The words they share: continuous, convergence, prime, function, weak, measures, probability, degree, equidistribution, local, packets, mass, escapes, sequence.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DukePrimeDegree.lean, the release lists it as a supporting statement.
theoremprime_degree_packet_measure_equidistribution_unconditional (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1) (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)] [∀ i, NumberField.IsTotallyReal (K i)] (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i)) (σ : ∀ i, OrderedEmbeddings (d+1) (K i)) (hn : ∀ i, Module.finrank ℚ (K i)=d+1) (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) : ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧ WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧ IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Prime degree >= 5 only; the primitive quartic (Picard packets) and primitive sextic (maximal-order ideal-class packets) results of the summary are not stated.
Definitions the group asks readers to compare with the paper:Whole packet-measure apparatus is custom (packetOrbits, orbitBasepoint := Classical.epsilon..., orbitalHaar := if h : HasAddFundamentalDomain... then... else 0, latticePoint := if h : ∃ g,... then ⟦h.choose⟧ else ⟦1⟧): junk else branches are unreachable for genuine torus packets, but the encoding of ‘complete volume-weighted packet’ is not checkable against a standard Mathlib notion
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:The primitive quartic (arbitrary orders) and primitive sextic (maximal orders) equidistribution theorems are not stated; only prime degree ≥ 5.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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.
AI-drafted comparisons, with no recorded human review
Equidistribution of prime-degree torus packetsDukePrimeDegree
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: weak convergence of prime-degree torus-packet measures to Haar probability measure, with tightness, for arbitrary local type.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Equidistribution of prime-degree torus packetsDukePrimeDegree
One report, at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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Papers elsewhere in the release that this family’s papers cite
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References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Identify the theorem the catalogue describes
Family 015No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1
In Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
fixed, mass, prime
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“partial”
What their term means
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…