Arithmetic torus packets and equidistribution

Source and verification

This article explains family 015 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Lattices and diagonal motion

A full lattice in ${\mathbb{R}}^n$ is the integer span of a real basis. Its covolume is the volume of a fundamental parallelepiped. The space of covolume-one lattices is

$$X_n={\mathrm{SL}}_n(\mathbb{Z})\backslash {\mathrm{SL}}_n(\mathbb{R}).$$

Changing the integer basis leaves the lattice unchanged. The space has a natural invariant probability measure $m_n$, but it is not compact.

Positive diagonal matrices

$$\begin{array}{c}{A}_n\\ =\{\mathrm{diag}(e^{u_1},\dots ,e^{u_n}):{\sum }_j{u}_j=0\}\end{array}$$

stretch some coordinates while contracting others, preserving volume. Certain arithmetic lattices have compact $A_n$-orbits. A packet is a specified finite collection of these orbits, assembled using arithmetic equivalence conditions.

For a compact orbit $O$, let ${\nu }_O$ be its invariant probability and $\mathrm{vol}(O)$ its volume in fixed logarithmic coordinates. The packet measure is

$${\mu }_{\mathcal{P}}=\frac{{\sum }_{O\in \mathcal{P}}\mathrm{vol}(O){\nu }_O}{{\sum }_{O\in \mathcal{P}}\mathrm{vol}(O)}.$$

Each distinct orbit is counted once. Equidistribution concerns this complete, volume-weighted collection, not a chosen single orbit.

🏷️ Arithmetic data needed for the statements

A totally real number field of degree $n$ has $n$ real embeddings. Applying these embeddings to a full integer lattice in the field gives a lattice in ${\mathbb{R}}^n$, which can be normalized to covolume one.

An order is a full-rank subring containing one. For a lattice $M$ in the field, its multiplier order consists of the elements $\alpha$ satisfying $\alpha M\subseteq M$. Its discriminant is

$$D(M)=|\mathrm{Disc}(K)|[{\mathcal{O}}_K:\mathcal{O}(M){]}^2.$$

Thus the arithmetic size can grow through either the field discriminant or the order index. A field is primitive if it has no intermediate field strictly between itself and $\mathbb{Q}$.

The packet definition also includes local data: two lattices are locally homothetic if, at every prime, their completed lattices differ by multiplication by an invertible field element. This can distinguish packets with the same multiplier order. These definitions specify the input objects; a course in their algebraic classification is not needed for the dynamical argument below.

🏷️ Three distinct packet theorems

Degree	Arithmetic input	Discriminant tending to infinity
Fixed prime $n\ge 5$	Every full-lattice local homothety type	Multiplier-order discriminant
Four	Primitive totally real field; all proper invertible ideals of an arbitrary order	Order discriminant
Six	Primitive totally real field; full maximal-order ideal packet	Field discriminant

Each manuscript claims convergence of its packet measures to $m_n$, with no escape of mass. (OpenAI, 2026) (OpenAI, 2026b) (OpenAI, 2026c)

The quartic result permits unbounded order index but does not state the prime-degree theorem’s arbitrary-module generality. The sextic result uses maximal orders. The sources also include the prescribed sign translates in their packets; suppressing those signs can change the collection.

No convergence rate or individual-orbit theorem is asserted.

🏷️ Tightness before entropy

Weak convergence against compactly supported functions can lose mass on a noncompact space. The elementary model is a sequence of point masses ${\delta }_j$ on $\mathbb{R}$: every compactly supported test eventually integrates to zero, although each measure has mass one.

In lattice space, escape occurs through the cusp, where a lattice has a very short nonzero vector. For example, in dimension two,

$${\Lambda }_t=\mathbb{Z}(e^{-t},0)+\mathbb{Z}(0,e^t)$$

has covolume one while its shortest vector tends to zero. A useful packet estimate must control this behavior before applying rigidity to a probability limit.

The sources unfold packet averages of vector counts into ideal sums with local weights. They retain the residue of a Dedekind zeta function in the arithmetic count, so that it cancels the corresponding packet normalization. Bounds for local containment probabilities control conductor growth.

Short-vector estimates give tightness. Counts in small boxes away from coordinate hyperplanes give an $O(r^n)$ bound for small neighborhoods in a limiting measure.

🏷️ From local mass bounds to rigidity

A diagonal orbit has dimension $n-1$. A two-sided dynamical tube can be covered by approximately $r^{-(n-1)}$ ordinary neighborhoods at a radius $r$ decreasing exponentially with the observation time. If each neighborhood has mass $O(r^n)$, the tube has mass $O(r)$. Exponential decay of these tube masses is the mechanism producing positive entropy.

It is necessary to establish this for almost every ergodic component, not merely positive entropy of the average. The source does so for invariant probabilities dominated by a finite multiple of the limit. A positive family of zero-entropy components would otherwise survive.

In prime degree, the cited measure-classification theorem identifies the positive-entropy components with Haar measure. Composite degree allows proper homogeneous components, so the other two papers require further arguments.

For degree four, the obstruction has two blocks of dimension two. Exterior-square vectors detect it through the signed products

$$v_{12}{v}_{34},-v_{13}{v}_{24},v_{14}{v}_{23}.$$

Arithmetic estimates for these products, uniform in the order index, bound thin rectangles and exclude the proper components. The underlying local Fourier calculation accounts for ramification and conductor depth.

For degree six, the source estimates thin tubes using ordinary and dual lattice vectors. It then constructs conditional probabilities at intermediate resolutions before passing to a limit. These probabilities retain information that exact conditional measures on each finite packet would miss. Product rules and entropy estimates force activity in all required off-diagonal directions, excluding proper components. Simply disintegrating the weak limit would not supply these properties.

🏷️ Key steps and proof outlines

Uniform vector counts produce probability limits

Let ${\lambda }_1(\Lambda )$ be the length of the shortest nonzero vector of a covolume-one lattice. Mahler’s compactness criterion says that imposing ${\lambda }_1\ge \epsilon$ gives a compact subset of lattice space. Thus the first quantitative task is to make the probability of ${\lambda }_1<\epsilon$ small uniformly over packets of large discriminant.

Proof outline. Count vectors in a small neighborhood of the origin, and integrate that count over the volume-weighted packet. Unfolding the orbit integral turns it into an arithmetic sum over vectors in the corresponding ideals, with the local membership probabilities retained. Since the event of a short vector is bounded by the number of such vectors, an upper bound on this expected count controls escape to the cusp. The residue factor from the ideal-counting formula also occurs in the packet-volume normalization; keeping it until division prevents an uncontrolled field-dependent constant. For arbitrary orders or local module types, the local containment factors must be bounded uniformly in their depth. Only after these estimates does compactness supply a subsequence converging to a measure of total mass one. (OpenAI, 2026a)

Ordinary balls bound dynamical tubes

The prime-degree paper proves $\mu (xB(r))\le C_{\Omega }{r}^n$ for small ordinary neighborhoods, uniformly when $x$ lies in a fixed compact set $\Omega$. Let

$$a(t)=\mathrm{diag}(e^{{\lambda }_{1}t},\dots ,e^{{\lambda }_{n}t}),\sum _i{\lambda }_i=0,\lambda =\underset{i\ne j}{\min }|{\lambda }_i-{\lambda }_j|>0.$$

Here the ${\lambda }_i$ are distinct weights, unrelated to the shortest-vector notation above. A two-sided tube consists of matrices that remain near the identity after conjugation both by $a(t)$ and by $a(-t)$.

Proof outline. Conjugation multiplies the $(i,j)$ entry by $e^{({\lambda }_i-{\lambda }_j)t}$. Membership in both conjugated neighborhoods therefore forces every off-diagonal entry to be $O(e^{-\lambda t})$. Put $r=e^{-\lambda t}$. In the determinant expansion every non-diagonal permutation contributes at least two off-diagonal factors, so the product of the diagonal entries is $1+O(r^2)$. Correcting the final diagonal entry puts the matrix within $O(r)$ of a bounded set of diagonal matrices. That set has dimension $n-1$ and needs at most $C{r}^{-(n-1)}$ balls of radius $Cr$. Summing the ordinary-ball bound gives tube mass

$$C{r}^{-(n-1)}{r}^n=Cr=C{e}^{-\lambda t}.$$

The source applies an entropy criterion for two-sided tubes to get a positive lower bound, for example $\lambda /3$, on the entropy of $a(1)$. The extra power between $n$ and $n-1$ is exactly what survives the covering argument.

Positive entropy must survive restriction to components

An invariant measure can have positive entropy while assigning positive mass to zero-entropy components. The classification step therefore needs more than positivity for the full average.

Proof outline. The ball and tube estimates also hold for every invariant probability $\xi \le c\mu$, with a larger constant. Suppose a collection of zero-entropy ergodic components had total weight $w>0$. Normalize their average to a probability $\xi$. Then $\xi \le w^{-1}\mu$, so the same argument gives $h_{\xi }(a(1))\ge \lambda /3$. But entropy is the integral of the component entropies, making $h_{\xi }(a(1))=0$. This contradiction proves positivity in almost every component. In prime degree the cited rigidity theorem identifies each such component with Haar probability. Tightness then upgrades uniqueness of every subsequential limit to convergence of the whole packet sequence.

Quartic packets need an exterior-coordinate obstruction estimate

In degree four, positive entropy still permits a proper homogeneous component with two blocks of size two. The source associates such a component with exterior vectors having two bounded coordinates and four arbitrarily small coordinates. An exterior vector is a linear combination of oriented coordinate-area elements; a wedge of two vectors is one example, but sums of wedges are allowed. Its six coordinates are indexed by $ij$ with $1\le i<j\le 4$. A single wedge satisfies a vanishing quadratic relation, whereas the obstruction estimate below concerns a nonzero quadratic invariant.

Proof outline. Use the three signed opposite-coordinate products $v_{12}{v}_{34}$, $-v_{13}{v}_{24}$, and $v_{14}{v}_{23}$. Four coordinates bounded by $r$ make two of those products bounded by $r^2$. A fixed nonzero quadratic invariant imposes a trace relation on the three products. Thus the two small products occupy a rectangle of area $O(r^4)$ in that trace plane. The exterior counting theorem bounds their arithmetic multiplicity uniformly in the order index. Integrating along diagonal orbits introduces logarithmic factors: when two coordinates have fixed product $z$ and both are bounded by $h$, the available logarithmic interval has length ${\log }_{+}(h^2/|z|)$. The source sums these factors and still obtains an expected exterior-vector count $O_{m,L}(r^4)$ for fixed invariant $m\ne 0$ and bound $L$. Letting $r\downarrow 0$, and then taking the countable union over the possible invariant data, excludes the positive mass required by a proper component. (OpenAI, 2026b)

Sextic packets retain conditional information at intermediate scales

The sextic argument uses local charts with transverse coordinate $q$ and an off-diagonal plaque coordinate $h$. A plaque is a small piece obtained by moving along one specified matrix-entry direction. Partition the transverse coordinates into dyadic cells and let $p_l$ be the conditional probability vector of a fixed finite set of plaque bins at depth $l$.

Proof outline of the common-plateau lemma. Conditional probabilities are bounded martingales under refinement, so

$$\mathbb{E}\Vert p_v-p_u{\Vert }_2^2=\mathbb{E}\Vert p_v{\Vert }_2^2-\mathbb{E}\Vert p_u{\Vert }_2^2(u\le v).$$

The sum of these increments over disjoint intervals is at most one. Divide a long available range of logarithmic depths into many intervals. One interval has very small total increment; a weighted diagonal selection makes this true for every fixed chart and bin partition. Cauchy–Schwarz converts the estimate to small expected ${\ell }^1$ change throughout that interval. Randomly shifting the grids controls boundary mass, so neighboring coordinate descriptions give compatible conditionals.

The source keeps these probabilities as additional data when taking limits. Its product rule and tube estimates force nontrivial activity in every off-diagonal direction. A proper homogeneous component, however, lies in a countable union of pieces fixed by a nonidentity diagonal matrix. A direction between two unequal diagonal entries meets such a piece in at most countably many points, and the source’s conditional-measure lemma makes that direction inactive there. The contradiction excludes proper components. The plateau construction is essential: exact conditionals on individual finite packets, or disintegration of the weak limit alone, need not retain this intermediate-scale information. (OpenAI, 2026c)

🏷️ Connections

Multiple mixing discusses invariant measures and decorrelation in a different dynamical setting. Zero-free regions supplies background on analytic zeros, but the packet manuscripts state their own arithmetic inputs; their conclusions should not be inferred from that family alone.

🏷️ Manuscripts and verification scope

• Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type — pinned manuscript and companion source.
• Equidistribution of primitive quartic torus packets for arbitrary orders — pinned manuscript and companion source.
• Equidistribution of Primitive Sextic Torus Packets — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page describes the prime-degree theorem only; it does not document the quartic and sextic companions. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type.

🐻  OpenAI 2026b. Equidistribution of primitive quartic torus packets for arbitrary orders.

🐻  OpenAI 2026c. Equidistribution of Primitive Sextic Torus Packets.