Lattice factors: finite correspondences and stable recovery

Source and verification

This article explains family 286 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.

Overview

A group von Neumann algebra remembers multiplication through operators, but an isomorphism need not preserve the original group basis. This family claims strong recovery for a specified class of lattices, including arbitrary scalar twists. The first manuscript classifies finite correspondences up to finite sums and summands; the companion identifies the summands and their maps. The resulting integer dimension formulas force stable isomorphisms to preserve scale and recover the given map.

🏷️ Twisted factors and module dimension

For a countable discrete group $\Gamma$, the group von Neumann algebra $L(\Gamma )$ is the weak operator closure of its left regular representation on ${\ell }^2(\Gamma )$. It carries the trace $\tau (u_g)=1_{\{e\}}(g)$. The vectors $u_g$ form an orthonormal basis of its trace Hilbert space $L^2(L(\Gamma ))$.

A normalized scalar 2-cocycle $\mu :{\Gamma }^2\to \mathbb{T}$, where $\mathbb{T}=\{z:|z|=1\}$, changes the multiplication to $u_g{u}_h=\mu (g,h)u_{gh}$. Associativity is exactly the identity

$$\begin{array}{c}\mu (g,h)\mu (gh,k)=\mu (h,k)\mu (g,hk),\\ \mu (e,g)=\mu (g,e)=1.\end{array}$$

The resulting algebra is $L_{\mu }(\Gamma )$. A group is ICC if each nonidentity conjugacy class is infinite. For an ICC group, $L_{\mu }(\Gamma )$ is a factor: its center consists of scalars. To see the mechanism, conjugation preserves the absolute values of Fourier coefficients of a central vector; square summability forces those coefficients to vanish on every infinite nonidentity class.

A bifinite correspondence from a finite factor $M$ to a finite factor $N$ is a Hilbert space $H$ with commuting normal unital left $M$ and right $N$ actions, finite in module dimension on both sides. Normality means the actions preserve increasing bounded suprema of positive operators. The actions need not be mutual commutants.

Module dimension differs from ordinary Hilbert-space dimension. A finite right $N$-module has the form $p{L}^2(N{)}^n$, with $p\in M_n(N)$ a projection, and

$$\dim H_N=({\mathrm{Tr}}_n\otimes {\tau }_N)(p).$$

The matrix trace is unnormalized. This dimension is a positive real number for a nonzero module and equals one for $L^2(N)$ itself. Left module dimension is defined analogously. The companion’s assertion that both dimensions are integers for these particular correspondences is therefore substantial.

An amplification $N^t$ is a corner $p{M}_n(N)p$ with unnormalized trace of $p$ equal to $t>0$, equipped with its normalized corner trace. Stable isomorphism permits comparison of such corners, not only of the original factors.

🏷️ The lattice class and the recovery statements

A lattice is a discrete subgroup of a locally compact group with finite Haar covolume. Groups are abstractly commensurable if they have isomorphic finite-index subgroups. Kazhdan’s property $(T)$ says that a unitary representation with almost invariant unit vectors has a nonzero invariant vector.

The manuscripts use the class $\mathcal{K}$ of countably infinite ICC groups abstractly commensurable with lattices in finite nonempty products of noncompact property-$(T)$ groups ${\mathbf{H}}_i(k_i{)}^{+}$. Here $k_i$ is $\mathbb{R}$, $\mathbb{C}$, or a finite extension of ${\mathbb{Q}}_p$; ${\mathbf{H}}_i$ is connected, adjoint, and absolutely simple; and $+$ denotes the subgroup generated by the unipotent radicals of proper parabolic subgroups. The real factors include higher-rank groups and the quaternionic and Cayley property-$(T)$ rank-one groups. The finite-place factors have relative rank at least two.

For a concrete example, ${\mathrm{SL}}_3(\mathbb{Z})$ and its finite-index subgroups belong to this class. The general theorem allows mixed local fields, reducible and nonuniform lattices, and torsion. Property $(T)$ alone is not the asserted hypothesis. (OpenAI, 2026) (OpenAI, 2026b)

Stable canonical recovery claimed by the companion

Let $\Gamma \in \mathcal{K}$, let $\Lambda$ be any countably infinite ICC group, and let $\mu ,\omega$ be arbitrary normalized scalar cocycles. Then

$$L_{\mu }(\Gamma {)}^t\cong L_{\omega }(\Lambda {)}^s$$

holds precisely when $t=s$ and an isomorphism $\delta :\Gamma \to \Lambda$ carries the cocycles to one another up to a normalized scalar cochain $b$. At unit scale, every specified isomorphism has the form

$$\theta (u_g)=wb(g)v_{\delta (g)}{w}^{\ast }$$

for a unitary $w$ in the target. All isomorphisms are normal, unital, complex-linear, and preserve the adjoint. (OpenAI, 2026b)

The cochain condition is the exact multiplication constraint

$$\mu (g,h)=\frac{b(g)b(h)}{b(gh)}\omega (\delta (g),\delta (h)).$$

If both cocycles are trivial, $b$ is a character. Taking the same factor at both endpoints gives trivial fundamental group: $M^t\cong M$ forces $t=1$.

The family also classifies arbitrary finite-index factor neighbors. A separable ${\mathrm{II}}_1$ factor $P$ admits a nonzero bifinite correspondence with $L_{\mu }(\Gamma )$ precisely when $P$ is an amplification of a twisted factor $L_{\omega }(\Lambda )$ with matched finite-index subgroup data described next. No group presentation for $P$ is assumed in advance. (OpenAI, 2026b)

🏷️ Explicit models and the cocycle ratio

Put $M=L_{\mu }(\Gamma )$ and $N=L_{\omega }(\Lambda )$. Choose finite-index subgroups $A\le \Gamma$, $B\le \Lambda$, and an actual isomorphism $\delta :A\to B$. A finite-dimensional projective unitary representation $\sigma$ supplies the remaining data:

$$\sigma (a)\sigma (b)=\frac{\mu (a,b)}{\omega (\delta (a),\delta (b))}\sigma (ab).$$

The ratio is forced by the subgroup action on $V_{\sigma }\otimes L^2(N)$: define $u_a(\xi \otimes \eta )=\sigma (a)\xi \otimes v_{\delta (a)}\eta$. Multiplying two such operators yields the required source multiplier $\mu$ precisely when the displayed equation holds.

Induce this action to $M$ using the relative tensor product

$$E=L^2(M){\overline{\otimes }}_{L_{\mu |A}(A)}(V_{\sigma }\otimes L^2(N)).$$

Relative tensoring identifies $(\xi a)\otimes \eta$ with $\xi \otimes (a\eta )$. Its Hilbert norm uses the operator-valued inner product of bounded module vectors, rather than the ordinary tensor-product norm.

There is a concrete counting-space model. Let $D=\Gamma \times \Lambda$ and let $C_{\delta }=\{(a,\delta (a)):a\in A\}$ be the graph subgroup. Place one copy of $V_{\sigma }$ at every left coset in $D/C_{\delta }$. The joint action $U_{(g,h)}\xi =u_g\xi v_h^{\ast }$ moves these fibers and has multiplier $\mu (g,g^{\prime })\overline{\omega (h,h^{\prime })}$. The right adjoint accounts for the conjugate.

Both coordinate groups act freely on this coset space. Their orbit counts are $[\Lambda :B]$ and $[\Gamma :A]$, respectively. Each orbit is a twisted regular module with multiplicity $n={\dim }_{\mathbb{C}}{V}_{\sigma }$. Thus

$$\begin{array}{rl}{\dim }_{M}E& =n[\Lambda :B],\\ \dim E_N& =n[\Gamma :A].\end{array}$$

The first manuscript’s arithmetic exhaustion theorem asserts that every bifinite correspondence is a closed bimodule summand of a finite sum of these models. Only the matched cocycle ratio must have a finite-dimensional projective representation. Neither individual cocycle must satisfy that condition, and the representation need not have finite image. (OpenAI, 2026a)

🏷️ Boundary measurements and finite-dimensional fibers

The main analytic work of the first manuscript starts with an arbitrary correspondence, without a known basis on the comparison side. After finite-index restrictions and removal of compact projective sectors, it works in a weakly mixing setting: the relevant conjugation representation on the mean-zero part of the finite commutant has no finite-dimensional invariant subspace.

The geometric input constructs observations on the flag boundary of the known source lattice. Both boundary copies come from the source; no ambient geometry for $\Lambda$ is assumed. Initially the observations are positive operator-valued measures: countably additive laws assigning positive operators to measurable sets, with total mass the identity. A projection-valued measure takes orthogonal projection values, so disjoint sets have orthogonal ranges. The proof must show that the observations are projection-valued and that distinct observations commute.

The required output is a simultaneous projection-valued law of nonsingular, essentially invertible boundary maps on $H_0\otimes H_1$. On three copies, the laws comparing slots $(0,1)$ and $(0,2)$ must commute even though they share slot zero. This shared-slot statement is stronger than constructing two separate boundary maps.

Here is its role in recovery. Take normal slices in the branch slots. Shared-slot commutation makes the resulting algebra on $H_0$ abelian, allowing disintegration into ordinary projection-valued laws on the fixed space $H_1$. A measurable transversal for the source-group action gives an orthogonal decomposition into translates of one projection. Finite module dimension makes that projection finite rank, so its spectral measure has finitely many atoms.

The remaining disintegration parameter could still change the atom partition. The source records each partition by a finite set of finite-rank projection orbits, using Hilbert–Schmidt distance. Group representations extending normally to the twisted group factor have matrix coefficients vanishing at infinity: approximate their $L^1$ coefficient densities by finite Fourier polynomials. This ensures the orbit distance separates distinct projection orbits.

Weak mixing makes the diagonal action on two copies of the partition distribution ergodic. Consequently the distance between two independent samples is almost surely constant. Separability forces that constant to be zero: some ball of every positive radius has positive probability, so two samples have positive probability of being arbitrarily close. The partition is therefore fixed almost everywhere. The boundary-map labels themselves need not be deterministic.

Both endpoint groups permute the resulting finite-dimensional atoms. The source action is free with finitely many orbits. Normal coefficient decay makes a comparison-group atom stabilizer finite; ICC and finite source-orbit count make it trivial. Finite right module dimension then gives finitely many comparison-group orbits. The joint stabilizer of an atom is the graph of a finite-index subgroup isomorphism, and its projective action has exactly the cocycle ratio above.

Restoring compact sectors restores the original scalar multiplier. Two-sided finite-index induction gives the original correspondence as a closed summand of a finite sum. The full geometric construction preceding this recovery—ordered boundary transfer, local scale estimates, domain control, locality, and gluing—is a substantive manuscript input, not independently verified in this exposition. (OpenAI, 2026a)

🏷️ Why summands require a separate argument

Integer dimensions of the containing models do not prove integer dimensions of their summands. The companion resolves this using germs of subgroup isomorphisms: two isomorphisms represent the same germ when they agree on a common finite-index subgroup.

For a germ $d$, the action of $D$ is $(g,h)d=\mathrm{Ad}(h)d\mathrm{Ad}(g^{-1})$. Its full stabilizer $C_d$ contains any chosen presenting graph $C_{\delta }$ at finite index. ICC makes $C_d$ itself a graph: an element centralizing a finite-index subgroup has a finite conjugacy class and must be the identity.

Several cosets of $C_{\delta }$ can describe the same germ. Their fibers must be grouped into one finite-dimensional representation of $C_d$ before classifying projections onto summands.

A projection that mixes presenting fibers

Present the identity germ using a subgroup $A<\Gamma$ of index $k>1$. The model has dimensions $(k,k)$. Its grouped fiber is the permutation representation on ${\ell }^2(\Gamma /A)$; the constant vectors form an invariant line selecting the standard correspondence of dimensions $(1,1)$. The selecting matrix has every entry $1/k$. Its individual diagonal entries are not projections. Inspecting the original one-dimensional fibers separately would miss the actual summand. (OpenAI, 2026b)

The crucial finite-orbit test says that the only finite $C_d$-orbit on the germ space is $\{d\}$. If a finite-index part of $C_d$ fixes another germ, the two subgroup maps induce equal conjugations on a common finite-index domain. Triviality of finite-index centralizers then makes the germs equal.

Now let $T$ intertwine two grouped-fiber models, with blocks $T_{yx}:V_x\to W_y$. A finite-dimensional source fiber gives

$$\sum _y\Vert T_{yx}{\Vert }_{\mathrm{HS}}^2\le \dim (V_x)\Vert T{\Vert }^2<\infty .$$

Equivariance makes each block’s Hilbert–Schmidt norm constant along the stabilizer orbit of its target label. A nonzero block must therefore lie in a finite orbit, forcing $y=x$. Every bounded bimodule map is fiberwise after grouping, and every summand is obtained from invariant subspaces of the finite-dimensional fiber representations.

Thus the actual correspondence, including any selected summand, has dimensions

$$\begin{array}{rl}{\dim }_{M}H& =\sum _d{n}_d[\Lambda :B_d],\\ \dim H_N& =\sum _d{n}_d[\Gamma :A_d],\end{array}$$

where the nonzero fiber ranks $n_d$ and subgroup indices are positive integers. This classification also describes adjoints and composition of maps. The companion computes conjugation and relative tensor composition of correspondences through projective induction and finite double-coset formulas, retaining all cocycle phases. (OpenAI, 2026b)

🏷️ Recovering stable scales and the specified map

An isomorphism $\theta :M^t\to N^s$ determines an equivalence correspondence between $M$ and $N$. The corner column $p{L}^2(M{)}^n$ has dimensions $(1/t,t)$ with normalized corner trace. Relative tensor products multiply left dimensions and right dimensions. Combining the two corner columns with $\theta$ therefore gives dimensions

$$\left(\frac{t}{s},\frac{s}{t}\right).$$

Both are positive integers by the actual-summand classification, and their product is one. Hence $t=s$. Both dimension sums then equal one, forcing a single orbit, a one-dimensional fiber, and subgroup indices one on both sides. The recovered subgroup isomorphism is consequently an isomorphism of the whole groups; its one-dimensional projective representation is the cochain $b$.

This identifies the correspondence of $\theta$. Recovering the specified map requires one more step. For actual corner projections $p\in M_n(M)$ and $q\in M_m(N)$, put $\phi (u_g)=b(g)v_{\delta (g)}$. Equal unnormalized traces give a rectangular partial isometry $z$ with

$$\begin{array}{rl}{z}^{\ast }z& ={\phi }^{(n)}(p),\\ z{z}^{\ast }& =q.\end{array}$$

The resulting corner map has the same correspondence as $\theta$. A right-linear unitary on the standard trace module is multiplication by a target unitary. Absorbing that unitary into $z$ yields the exact formula $\theta (x)=z{\phi }^{(n)}(x)z^{\ast }$. Thus the argument controls the given map in its given corners. (OpenAI, 2026b)

🏷️ Recovering an unknown factor from multiplication

For a bifinite $M$–$P$ correspondence with no presentation of $P$, the commutant of the right action is an amplification $Q=P^r$, containing $M$ as a finite-index subfactor. Arithmetic exhaustion grades $L^2(Q)$ by finite-dimensional spaces $K_d$ indexed by virtual automorphism germs of $\Gamma$.

These spaces initially contain $L^2$ vectors, which need not be bounded operators. For $\xi \in K_d$, orthogonality of distinct right translates implies $E_M({\xi }^{\ast }\xi )=\Vert \xi {\Vert }_2^{2}1$. Apply the finite-index expectation inequality to bounded spectral truncations of ${\xi }^{\ast }\xi$ to obtain

$$\Vert \xi {\Vert }_{\infty }\le [Q:M{]}^{1/2}\Vert \xi {\Vert }_2.$$

Products are now legitimate. On a common finite-index subgroup domain, the finite-dimensional orbit criterion gives $K_d{K}_f\subset K_{df}$ and $K_d^{\ast }=K_{d^{-1}}$.

The identity space $K_1$ is a finite-dimensional algebra. Compress by a minimal projection $p\in K_1$. If $0\ne x\in p{K}_{d}p$, both $x^{\ast }x$ and $x{x}^{\ast }$ are positive scalar multiples of $p$, with equal scalars by traciality. Normalizing $x$ makes it a corner unitary; any other vector of that degree is its scalar multiple.

The nonzero degrees form a countable group, since products of these unitaries cannot vanish. Their multiplication defines a scalar cocycle, and their trace-orthonormal basis identifies $pQp$ with a twisted group factor. The source further proves ICC, commensurability with $\Gamma$, and the exact cocycle-ratio condition by examining the identity matrix block. For an untwisted source, that ratio condition becomes finite type of the target cocycle. For arbitrary source twist, only the ratio is required. (OpenAI, 2026b)

🏷️ Verification scope

The review covers the elementary models, selected boundary-to-partition recovery, full-stabilizer and intertwiner arguments, integer-dimension deduction, specified corner maps, and bounded-product reconstruction. It does not independently verify the first manuscript’s complete boundary geometry, local-field estimates, or unbounded-operator gluing. The complete fusion formulas and external rigidity theorems are also outside the selected proof audit.

🏷️ Key steps and proof outlines

Convert a random partition into a deterministic one

The first manuscript’s geometric argument produces a simultaneous boundary law, and disintegration produces a distribution of finite-rank projection partitions. The final determinism argument uses a general probability principle that is useful beyond this setting.

Let $P$ be a random element of a separable metric space recording these partitions. Shared-slot commutation and weak mixing imply that the diagonal action on two independent samples is ergodic. Their bounded distance $d(P,P^{\prime })\wedge 1$ is invariant, hence is almost surely a constant $c$.

For every $\epsilon >0$, separability gives a countable cover by balls of radius $\epsilon$. At least one ball has positive probability. Two independent samples both belong to that ball with positive probability, so their distance is at most $2\epsilon$ with positive probability. Constancy forces $c\le 2\epsilon$ for every $\epsilon$, and hence $c=0$. Fubini then says almost every sample equals one fixed partition. (OpenAI, 2026a)

More precisely, the random variable records a finite set of projection-orbit classes. Decay of matrix coefficients makes the orbit metric separating, and its Hausdorff metric separates finite sets. A constant such set recovers an actual fixed partition by taking the union of all the represented projection orbits. The boundary labels and the chosen representatives of individual orbits are not required to become constant.

Square summability forbids off-label intertwiners

Once fibers are grouped by the full stabilizer of an isomorphism germ, let $T$ be a bounded intertwiner and fix a finite-dimensional source fiber $V_x$. For an orthonormal basis $e_1,\dots ,e_d$ of that fiber,

$$\sum _y\Vert T_{yx}{\Vert }_{\mathrm{HS}}^2=\sum _{j=1}^d\sum _y\Vert P_{y}T{e}_j{\Vert }^2=\sum _{j=1}^d\Vert T{e}_j{\Vert }^2\le d\Vert T{\Vert }^2.$$

Equivariance keeps each block norm constant along the source stabilizer’s orbit of $y$. An infinite orbit cannot carry a nonzero constant in this summable family. The finite-orbit lemma then leaves only $y=x$. Thus an orthogonal projection onto a correspondence summand is fiberwise in the grouped model. Its fiber ranks are integers, which explains why the dimension formulas survive passage to arbitrary closed summands. (OpenAI, 2026b)

This is stronger than knowing that the containing module has integer dimension. A subprojection in an arbitrary finite factor can have noninteger trace; here the orbit argument reduces every relevant subprojection to an ordinary finite-dimensional invariant subspace.

Establish boundedness before multiplying recovered vectors

For an unknown finite-index extension $M\subset Q$, the recovered spaces $K_d$ initially lie only in $L^2(Q)$. To multiply them, first take $\xi \in K_d$ and set $a={\xi }^{\ast }\xi \in L^1(Q{)}_{+}$. Distinct right translates of $\xi$ have distinct orthogonal degrees, so all nonidentity Fourier coefficients of $E_M(a)$ vanish. Its identity coefficient is $\Vert \xi {\Vert }_2^2$, hence

$$E_M(a)=\Vert \xi {\Vert }_2^{2}1.$$

Write $\lambda =[Q:M]$ and use bounded spectral truncations $a_R=\min (a,R)$. Positivity of expectation gives $E_M(a_R)\le E_M(a)$. The finite-index inequality $a_R\le \lambda E_M(a_R)$ therefore gives $a_R\le \lambda \Vert \xi {\Vert }_2^{2}1$ uniformly in $R$. Letting $R\to \infty$ proves $\xi \in Q$ and $\Vert \xi {\Vert }_{\infty }\le \sqrt{\lambda }\Vert \xi {\Vert }_2$.

Now orbit spans of products lie in the finite-dimensional span of products of basis elements. The intrinsic orbit criterion proves $K_d{K}_f\subseteq K_{df}$ and $K_d^{\ast }=K_{d^{-1}}$. This ordering avoids manipulating products of arbitrary unbounded $L^2$ vectors.

A minimal identity corner turns degrees into unitary lines

Choose a minimal projection $p$ of the finite-dimensional identity algebra $K_1$. For nonzero $x\in p{K}_{d}p$, the product rules give $x^{\ast }x=\alpha p$ and $x{x}^{\ast }=\beta p$. Faithful traciality forces $\alpha =\beta >0$, so $x/\sqrt{\alpha }$ is unitary in the corner. If $y$ has the same degree, $x^{\ast }y$ is scalar, making $y$ a scalar multiple of $x$.

Products of these unitaries cannot vanish, so the surviving degrees form a group. Choosing one unitary $v_d$ in each line gives $v_d{v}_f=\omega (d,f)v_{df}$ with $|\omega (d,f)|=1$; associativity is exactly the cocycle identity. Their orthonormal trace basis identifies the actual corner multiplication with a twisted regular representation. The later subgroup and cocycle-ratio analysis supplies the arithmetic restrictions. These analytic bridges occur in the recovery section of the first manuscript and the arbitrary-endpoint section of the companion.

See Also

• Reduced assembly, projections, and traces introduces group operator algebras and unnormalized trace conventions; this family works with von Neumann factors and finite module dimensions.
• Fixed points in reflexive spaces offers a different use of group actions and invariant structures; the present rigidity statement needs the specified lattice class and finite correspondences.

🏷️ Manuscripts and verification scope

• Arithmeticity of twisted finite correspondences for lattices over local fields — pinned manuscript and companion source.
• The arithmetic category and stable recovery of lattice factors — pinned manuscript and companion source.

No family scope page was present at lean/docs/286.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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. Arithmeticity of twisted finite correspondences for lattices over local fields.

🐻  OpenAI 2026b. The arithmetic category and stable recovery of lattice factors.