diff 999db1f..b34550f arithmeticity_local_field_lattices_v2.txt
diff --git a/arithmeticity_local_field_lattices_v2.txt b/arithmeticity_local_field_lattices_v2.txt
index 5d49ebc..f57b4bd 100644
--- a/arithmeticity_local_field_lattices_v2.txt
+++ b/arithmeticity_local_field_lattices_v2.txt
@@ -1,55 +1,55 @@
-\documentclass[11pt,a4paper]{amsart}
-\usepackage[T1]{fontenc}
-\usepackage[utf8]{inputenc}
-\usepackage{lmodern}
-\usepackage[margin=28mm]{geometry}
-\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
-\usepackage{enumitem,graphicx,longtable,booktabs,array,microtype}
-\usepackage{flafter,needspace,placeins}
-\usepackage{xcolor,tikz,tikz-cd}
+
+
+
+
+
+
+
+
+
\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
\definecolor{linkblue}{RGB}{29,67,112}
-\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue,bookmarksnumbered=true]{hyperref}
+
\hypersetup{pdftitle={Arithmeticity of twisted finite correspondences for lattices over local fields},pdfauthor={OpenAI},pdfsubject={Arithmetic exhaustion of twisted bifinite correspondences}}
-\numberwithin{equation}{section}
-\theoremstyle{plain}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{convention}[theorem]{Convention}
-\newtheorem{notation}[theorem]{Notation}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-\DeclareMathOperator{\Ad}{Ad}
-\DeclareMathOperator{\Hom}{Hom}
-\DeclareMathOperator{\Comm}{Comm}
-\DeclareMathOperator{\Tr}{Tr}
-\DeclareMathOperator{\supp}{supp}
-\DeclareMathOperator{\rank}{rank}
-\DeclareMathOperator{\ran}{ran}
-\DeclareMathOperator{\id}{id}
-\DeclareMathOperator{\Aut}{Aut}
-\DeclareMathOperator{\GL}{GL}
-\DeclareMathOperator{\SL}{SL}
-\DeclareMathOperator{\SO}{SO}
-\DeclareMathOperator{\SU}{SU}
-\DeclareMathOperator{\Sp}{Sp}
-\DeclareMathOperator{\Sym}{Sym}
-\DeclareMathOperator{\diag}{diag}
-\DeclareMathOperator{\Lie}{Lie}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
\allowdisplaybreaks[2]
-\setlength{\emergencystretch}{2em}
-\setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt}
-\setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt}
+
+
+
\title[Arithmeticity of twisted finite correspondences]{Arithmeticity of twisted finite correspondences for lattices over local fields}
-\begin{document}
-\begin{abstract}
+
+Abstract:
We prove arithmetic exhaustion for bifinite correspondences between
arbitrarily scalar-twisted group factors of ICC groups commensurable with
property-$(T)$ lattices over characteristic-zero local fields and twisted
@@ -63,17 +63,17 @@ rank-one factors.
In one sentence: an analytic finite relation between the two quantum
shadows of the groups is always generated by a finite list of concrete
algebraic relations, together with their unavoidable phases.
-\end{abstract}
+
\maketitle
\tableofcontents
\bigskip
-\section*{A Guide for the Curious Reader}
+# A Guide for the Curious Reader
\addcontentsline{toc}{section}{A Guide for the Curious Reader}
This paper is technical, and the technical part starts in
-Section~\ref{sec:intro}. This first part is a guide: it explains, in
+Section~sec:intro. This first part is a guide: it explains, in
ordinary language and with examples, what the objects are, what the
question is, what the answer says, and how the proof reaches it. Nothing
here is needed for the formal statements, and nothing here is a proof. If
@@ -81,25 +81,25 @@ you are meeting this material for the first time, read this part once,
then read the introduction, then dive into whatever section you find
intriguing. The last subsection of the guide is a map of the paper.
-\subsection*{1. Symmetry, and the grids inside it}
+## 1. Symmetry, and the grids inside it
-A \emph{group} is a mathematical object that records symmetries. The
+A group is a mathematical object that records symmetries. The
rotations of a square form a group with four elements; the ways to
permute three objects form a group with six elements; the translations
of the plane form an infinite group. Whenever something has symmetry,
there is a group behind it, and the group tells you exactly which
transformations leave the object unchanged.
-Some of the most important groups are \emph{continuous}: instead of
+Some of the most important groups are continuous: instead of
finitely many or countably many symmetries, they form a smooth
space. For example, the set of all rotations of the plane about the
origin is a circle; the set of all invertible $n\times n$ real matrices
of determinant one, written $\SL_n(\mathbb R)$, is a continuous group of
transformations of $\mathbb R^n$ that preserve volume and orientation.
-Groups like these, that are also smooth manifolds, are called \emph{Lie
-groups}. They are the basic objects of the geometry of symmetry.
+Groups like these, that are also smooth manifolds, are called Lie
+groups. They are the basic objects of the geometry of symmetry.
-Inside a continuous group, one can sometimes find a \emph{lattice}: a
+Inside a continuous group, one can sometimes find a lattice: a
discrete grid of symmetries that still "spreads out" through the whole
group, like the integer points
\[
@@ -116,7 +116,7 @@ accumulate, and yet it is large enough that the continuous group is
and they connect geometry, number theory, and group theory.
The groups in this paper are built by taking products of continuous
-groups over different "places." The word \emph{place} is from number
+groups over different "places." The word place is from number
theory: the real numbers $\mathbb R$, the complex numbers $\mathbb C$,
and the $p$-adic numbers $\mathbb Q_p$ are all "completions" of the
rationals, and a formula that holds over the rationals has a face at
@@ -127,9 +127,9 @@ groups are built. A big lattice can have several faces at once: a real
face, and a $p$-adic face, and so on. The theory has to handle all of
them, and one of the achievements here is that it does, uniformly.
-\subsection*{2. Rigidity: property $(T)$}
+## 2. Rigidity: property $(T)$
-A group has \emph{Kazhdan's property $(T)$} if it is rigid: it cannot be
+A group has Kazhdan's property $(T)$ if it is rigid: it cannot be
"almost" represented by symmetries without actually being represented.
Formally, every unitary representation that has almost-invariant unit
vectors already has an exactly invariant one. Intuitively, a group with
@@ -146,10 +146,10 @@ squirms away.
The class of groups that this paper can handle is written
$\mathscr K$, and it consists of the countably infinite groups that are
-\emph{abstractly commensurable} with a lattice in a product
+abstractly commensurable with a lattice in a product
$G=\prod_{i=1}^r G_i$, where each factor $G_i$ is the group of local
points of a connected, adjoint, absolutely simple algebraic group over a
-local field $k_i\in\{\mathbb R,\mathbb C,\text{finite extension of }\mathbb Q_p\}$,
+local field $k_i\in\{\mathbb R,\mathbb C,finite extension of \mathbb Q_p\}$,
generated by the unipotent radicals of the proper parabolic subgroups,
noncompact, and with property $(T)$. Abstractly commensurable means: an
actual isomorphism between finite-index subgroups. No irreducibility,
@@ -164,142 +164,142 @@ conjugacy class is infinite. This is a mild "non-abelianness" hypothesis
that keeps the group von Neumann algebra a factor (see below). It is
standard and holds for essentially all groups anyone cares about here.
-\subsection*{3. Quantum shadows: group von Neumann algebras}
+## 3. Quantum shadows: group von Neumann algebras
To every group $\Gamma$ one can attach an algebra of operators that
serves as the group's "quantum shadow." The construction goes as
follows. Let $\ell^2(\Gamma)$ be the Hilbert space of square-summable
functions on $\Gamma$. For each $g\in\Gamma$ there is a unitary operator
$\lambda(g)$ shifting a function: $(\lambda(g)\xi)(h)=\xi(g^{-1}h)$. The
-\emph{group von Neumann algebra} $L(\Gamma)$ is the algebra of bounded
+group von Neumann algebra $L(\Gamma)$ is the algebra of bounded
operators on $\ell^2(\Gamma)$ generated by all these unitaries together
with their (weak) limits.
An algebra of this kind remembers a surprising amount of the group. It
also forgets a surprising amount. The central question of the field, in
-one form or another, is \emph{how much does the shadow remember?} If two
-groups $\Gamma$ and $\Lambda$ have isomorphic shadows $L(\Gamma)\cong
-L(\Lambda)$, must the groups be isomorphic? If the shadow retains, along
+one form or another, is how much does the shadow remember? If two
+groups $\Gamma$ and $Λ$ have isomorphic shadows $L(\Gamma)\cong
+L(Λ)$, must the groups be isomorphic? If the shadow retains, along
with any isomorphism, a controlled correspondence with the original
-group, the group is called \emph{$W^*$-superrigid}.
+group, the group is called $W^*$-superrigid.
-Recall the notion of a \emph{factor}: a von Neumann algebra whose center
+Recall the notion of a factor: a von Neumann algebra whose center
is trivial. A factor is the noncommutative analogue of an "indecomposable
system," a quantum object that cannot be split into independent
subsystems. The condition that $\Gamma$ be ICC is exactly what guarantees
-that $L(\Gamma)$ is a factor of type $\mathrm{II}_1$, the cleanest
-infinite-dimensional kind, on which a natural \emph{trace} (a kind of
+that $L(\Gamma)$ is a factor of type $II_1$, the cleanest
+infinite-dimensional kind, on which a natural trace (a kind of
dimension function) is defined.
-\subsection*{4. Twisting by phases}
+## 4. Twisting by phases
Sometimes the symmetries of a system are only defined up to a phase: a
rotation by the identity angle might act as multiplication by a complex
-number of modulus one. Such an "almost action" is a \emph{projective
-representation}, and its failure of multiplicativity is encoded in a
-\emph{cocycle} $\mu:\Gamma\times\Gamma\to\mathbb T$. The rule is
+number of modulus one. Such an "almost action" is a projective
+representation, and its failure of multiplicativity is encoded in a
+cocycle $\mu:\Gamma\times\Gamma\to\mathbb T$. The rule is
\[
\mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj)
\]
-with values in the unit circle $\mathbb T$. The \emph{twisted group von
-Neumann algebra} $L_\mu(\Gamma)$ is built from $\Gamma$ together with
+with values in the unit circle $\mathbb T$. The twisted group von
+Neumann algebra $L_\mu(\Gamma)$ is built from $\Gamma$ together with
this cocycle: the generators $u_g$ now satisfy $u_gu_h=\mu(g,h)u_{gh}$.
Scalar cocycles are unavoidable in the geometry of these problems, and a
central theme of this paper is that they must be kept, not discarded:
-they appear as the exact \emph{ratio} between the two endpoint twists of
+they appear as the exact ratio between the two endpoint twists of
the final models.
-\subsection*{5. Finite correspondences: quantized relations}
+## 5. Finite correspondences: quantized relations
A relation between two sets $X$ and $Y$ is a subset of $X\times Y$. In
the quantum world, a relation between two von Neumann algebras $M$ and
-$N$ is an \emph{$M$--$N$ bimodule}: a Hilbert space on which $M$ acts
+$N$ is an $M$--$N$ bimodule: a Hilbert space on which $M$ acts
from the left and $N$ from the right, with the two actions commuting.
This is the operator-algebraic analogue of a bridge or a coupling
between the two systems.
-There is a notion of \emph{size}, or \emph{module dimension}, for such a
+There is a notion of size, or module dimension, for such a
bimodule, generalizing the dimension of a group factor over a subgroup
(after Jones' index theory and Pimsner--Popa). A bimodule is
-\emph{bifinite} if it has finite module dimension on both sides. Intuitively,
-a finite correspondence is a bridge that is \emph{finite} and
-\emph{trace-preserving} at both ends, the quantum analogue of a finite
+bifinite if it has finite module dimension on both sides. Intuitively,
+a finite correspondence is a bridge that is finite and
+trace-preserving at both ends, the quantum analogue of a finite
relation.
-The central question becomes: \emph{where can finite correspondences come
-from?} One family of examples is always available. Suppose $\Gamma$ and
-$\Lambda$ are groups and $A\leq\Gamma$, $B\leq\Lambda$ are subgroups of
+The central question becomes: where can finite correspondences come
+from? One family of examples is always available. Suppose $\Gamma$ and
+$Λ$ are groups and $A\leq\Gamma$, $B\leqΛ$ are subgroups of
finite index, with an actual isomorphism $\delta:A\to B$ between them.
Suppose moreover there is a finite-dimensional projective unitary
representation $\sigma$ of $A$ whose multiplier matches the cocycle
ratio, meaning
\[
- \sigma(a)\sigma(b)=\frac{\mu(a,b)}{\omega(\delta(a),\delta(b))}\sigma(ab).
+ \sigma(a)\sigma(b)=(\mu(a,b))/(\omega(\delta(a),\delta(b)))\sigma(ab).
\]
Then there is a canonical finite correspondence $E(A,B,\delta,\sigma)$,
given by a relative tensor product (Definition and construction in
-Section~\ref{sec:models}). These are the \emph{arithmetic models}.
+Section~sec:models). These are the arithmetic models.
-\subsection*{6. The question, and the answer}
+## 6. The question, and the answer
The question: given a finite correspondence between the twisted group
-factors $L_\mu(\Gamma)$ and $L_\omega(\Lambda)$ for $\Gamma\in\mathscr
-K$ and arbitrary countably infinite ICC $\Lambda$, is it built out of
+factors $L_\mu(\Gamma)$ and $L_\omega(Λ)$ for $\Gamma\in\mathscr
+K$ and arbitrary countably infinite ICC $Λ$, is it built out of
arithmetic models? The answer proved here is yes.
-\begin{quote}
-\textbf{Main theorem (Arithmetic exhaustion).} Let $\Gamma\in\mathscr K$,
-let $\Lambda$ be any countably infinite ICC group, and let $\mu,\omega$
+
+Main theorem (Arithmetic exhaustion). Let $\Gamma\in\mathscr K$,
+let $Λ$ be any countably infinite ICC group, and let $\mu,\omega$
be normalized scalar cocycles. Every bifinite
-$L_\mu(\Gamma)$--$L_\omega(\Lambda)$ correspondence is unitarily
+$L_\mu(\Gamma)$--$L_\omega(Λ)$ correspondence is unitarily
isomorphic to a closed bimodule summand of a finite direct sum of
arithmetic models $E(A,B,\delta,\sigma)$, where $A\leq\Gamma$ and
-$B\leq\Lambda$ have finite index, $\delta:A\to B$ is an actual group
+$B\leqΛ$ have finite index, $\delta:A\to B$ is an actual group
isomorphism, and $\sigma$ is a finite-dimensional projective unitary
representation satisfying the matched-multiplier identity above.
Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. The elementary model has
-dimensions $n[\Lambda:B]$ on the left and $n[\Gamma:A]$ on the right,
+dimensions $n[Λ:B]$ on the left and $n[\Gamma:A]$ on the right,
where $n=\dim V_\sigma$.
-\end{quote}
+
Three things are worth stressing.
-First the theorem is an \emph{exhaustion} result: it says every finite
+First the theorem is an exhaustion result: it says every finite
correspondence is contained in finitely many arithmetic models, up to a
-closed summand. It does not claim every correspondence \emph{equals} one
+closed summand. It does not claim every correspondence equals one
model; the summand and finite sum are necessary because a correspondence
can have irreducible pieces living in several distinct arithmetic models
at once.
-Second, only the \emph{matched ratio} $\mu/\delta^*\omega$ needs to admit a
+Second, only the matched ratio $\mu/\delta^*\omega$ needs to admit a
finite-dimensional projective representation. Neither endpoint cocycle
is required to have finite order or finite type, and the image of
$\sigma$ may be infinite. This is exactly where the twisting appears as an
obstruction: existence of a nonzero correspondence of the simplest kind
-forces $\Gamma$ and $\Lambda$ to be abstractly commensurable.
+forces $\Gamma$ and $Λ$ to be abstractly commensurable.
Third, the theorem is unconditional within its class. It says nothing
about general property-$(T)$ groups, and indeed property $(T)$ alone
-\emph{fails} to give group reconstruction: independent concurrent
+fails to give group reconstruction: independent concurrent
constructions give nonisomorphic ICC property-$(T)$ groups with
isomorphic group factors. What is special to our lattice class is the
geometric rigidity available inside the correspondence, and the
arithmetic exhaustion is the exact finite-correspondence input for the
stable recovery results of the companion article.
-\subsection*{7. Why this is hard}
+## 7. Why this is hard
-The comparison group $\Lambda$ is only assumed countably infinite
+The comparison group $Λ$ is only assumed countably infinite
and ICC. Its von Neumann algebra is as wild as any such algebra can be.
-In particular we have no geometric action of $\Lambda$ on any symmetric
+In particular we have no geometric action of $Λ$ on any symmetric
space or building: the comparison group is an algebraic cipher, not a
geometric object. Yet we must produce actual finite-index subgroup graphs
-of $\Lambda$. The only geometry available at the start is the geometry of
-$\Gamma$ and its ambient groups $G_i$. The proof must \emph{manufacture}
+of $Λ$. The only geometry available at the start is the geometry of
+$\Gamma$ and its ambient groups $G_i$. The proof must manufacture
geometry inside the correspondence from the source side alone, then
-must \emph{read} $\Lambda$ through it.
+must read $Λ$ through it.
The second hardness is scalar. Even in the classical, untwisted
case, the argument must be careful. With twists in the game, phases
@@ -308,22 +308,22 @@ controls them up to bounded error is not enough. This is why the paper
develops a fine analysis of Cartan projections, boundary maps, and their
Jacobians, rather than working at the level of coarse quasi-isometry.
-\subsection*{8. How the proof works: a tour}
+## 8. How the proof works: a tour
The proof is long, but its shape is a single idea executed carefully:
-\emph{read off the arithmetic from the geometry, working in small enough
-pieces that phases stay visible}. Here are the main steps.
+read off the arithmetic from the geometry, working in small enough
+pieces that phases stay visible. Here are the main steps.
-\emph{Step 1. Reduce to a good lattice.} First pass to an actual lattice
+Step 1. Reduce to a good lattice. First pass to an actual lattice
$\Gamma<G$ and decompose the finite commutant into compact projective
sectors, discarding the compact parts. What remains is a correspondence
-with \emph{weakly mixing} conjugation action on the commutant of the
+with weakly mixing conjugation action on the commutant of the
source factor. This is the clean hypothesis under which the geometric
construction runs. Restoring the compact sectors at the end costs nothing
mathematically, but it must be done exactly, to restore the correct
-projective multiplier (Sections~\ref{sec:setup} and~\ref{sec:models}).
+projective multiplier (Sections~sec:setup and~sec:models).
-\emph{Step 2. Build finite regular models and boundary measurements.}
+Step 2. Build finite regular models and boundary measurements.
For the source lattice, construct finite regular models for the tensor
copies of the correspondence, in such a way that the diagonal right action
and the surviving left actions are all normal. Then use Fourier analysis
@@ -331,92 +331,92 @@ on these regular models to build positive operator-valued measurements of
the far-away Fourier index. Under a careful "ordered sharpening"
argument these become projection-valued measurements (PVMs) supported on
the source flag space: sharp observations of the boundary
-(Sections~\ref{sec:links} and~\ref{sec:transfer}). The strong point is
-that the construction is \emph{ordered}: the image depends on the PVM's
+(Sections~sec:links and~sec:transfer). The strong point is
+that the construction is ordered: the image depends on the PVM's
order of integration, and the paper proves that the ordering survives the
limit exactly. This is what keeps the phases—which would otherwise be
invisible—under control.
-\emph{Step 3. Control the speeds.} The data of a correspondence carries
+Step 3. Control the speeds. The data of a correspondence carries
Cartan projections: real-valued coordinate functions on the source lattice
that record "how far out" an element is. One half of the proof is to
show that the corresponding target Cartan projection grows at most
linearly in the source speed, with a comparison constant independent of
-everything else. This is the \emph{upper speed} theorem, and it is
+everything else. This is the upper speed theorem, and it is
proved separately at three kinds of local places:
-\begin{itemize}
-\item \emph{Finite place} (Section~\ref{sec:finite}): compact-open fixed
+
+\item Finite place (Section~sec:finite): compact-open fixed
algebras, a uniform $p$-adic logarithm lemma, and a Hausdorff-limit
argument on block stabilizers give the equality of source and target
dimensions over the prime field and the upper bound $\ell_T(g)\le
d_St+O(1)$.
-\item \emph{Non-isolated archimedean place} (Section~\ref{sec:panels}):
+\item Non-isolated archimedean place (Section~sec:panels):
optimal finite-dimensional modules of the rank-one Levi factor, and
$L^2$-trace inequalities on compact operators reconstructed from the
graph relation, give the analogous bound.
-\item \emph{Isolated real place} (Sections~\ref{sec:ray}--\ref{sec:scale}):
+\item Isolated real place (Sections~sec:ray--sec:scale):
ray densities on the symmetric space, a uniform radial-multiplier
(completely bounded) estimate, and a "Kazhdan tightening" that pins down
-a linear subsequence give, instead of an upper bound, a \emph{lower}
+a linear subsequence give, instead of an upper bound, a lower
bound at linear speed—positive mass in a linear window on every fixed
input vector. This lower bound is what turns the isolated case into a
tractable one.
-\end{itemize}
-\emph{Step 4. Build heights and calibrate.} On the isolated factors, the
+
+Step 4. Build heights and calibrate. On the isolated factors, the
boundary measurements have a derivative in the horizontal direction,
and the proof constructs a positive injective operator $b_x$ at each
source label that commutes with the measurement $P_x$ and transforms
under source and target translations by the inverse angular Jacobian.
This "height" is the analytic core of the isolated case
-(Sections~\ref{sec:scale} and~\ref{sec:heights}). The height lets us
+(Sections~sec:scale and~sec:heights). The height lets us
compare source and target Haar densities exactly, and calibrate the
whole correspondence to a genuine Haar-height isometry—an isometry
between the target chamber-variable space and the source chamber-variable
space, weighted by the height. This is the machine that converts "the
-looks right" into "the same object" (Section~\ref{sec:heights}).
+looks right" into "the same object" (Section~sec:heights).
-\emph{Step 5. Prove locality and commutation.} Boundary measurements at
+Step 5. Prove locality and commutation. Boundary measurements at
different labels must commute. This is not automatic; they are
-a priori noncommuting operators. Section~\ref{sec:locality} proves
+a priori noncommuting operators. Section~sec:locality proves
commutation for all labels of an isolated factor by a deep
finite-dimensional locality argument: it builds a graph domain
$\mathfrak D$, proves that the coefficients of an arbitrary connected
piece are finite-jets, uses the geometry of the boundary kernel
$K(w,z)$ to eliminate positive differential order, and concludes that
the joint spectral law of the angle-height operators is exactly the law
-of evaluation of a \emph{smooth homogeneous diffeomorphism}
+of evaluation of a smooth homogeneous diffeomorphism
of the boundary. Then it proves that the two normal copies of the boundary
calculus commute as well. This is where the "quantum relation" becomes
"a classical map between two compact spaces," which is the bridge step.
-\emph{Step 6. Glue.} Do this at every local type—every finite-place panel,
+Step 6. Glue. Do this at every local type—every finite-place panel,
every real panel, every isolated factor—and prove that the local maps can
be glued into one norm-bisecting, essentially invertible chamber map
-$B_S\to B_T$ (Sections~\ref{sec:panels} and~\ref{sec:gluing}). The
+$B_S\to B_T$ (Sections~sec:panels and~sec:gluing). The
finite-place panels give homogeneous analytic maps with isomorphic
$p$-adic differential; the real panels give smooth homogeneous maps
after a Taylor-density argument; the gluing sections install reflection
substitutions that propagate monotone commutativity along a longest Weyl
word.
-\emph{Step 7. Read off the arithmetic.} Now the boundary has become a
+Step 7. Read off the arithmetic. Now the boundary has become a
classical, invertible measure-class-preserving map. Uninduce the
correspondence over this classical datum. What appears is a target
transversal, and a finite family of atoms (minimal pieces of the
correspondence, which are finite-dimensional). Prove that the atoms form
-a \emph{deterministic}, finite-rank partition, permuted by both endpoint
+a deterministic, finite-rank partition, permuted by both endpoint
groups with finite orbits and a free source action. Then, by pure group
theory—no more boundary analysis—the joint stabilizer of one atom is the
graph of a finite-index subgroup isomorphism $A\to B$, and the atom
carries the matched projective multiplier. This yields the arithmetic
models. Finally, undo the finite-index restriction and the compact
-sector: everything restores and Theorem~\ref{thm:arithmetic-exhaustion}
-follows (Section~\ref{sec:recovery}).
+sector: everything restores and Theorem~thm:arithmetic-exhaustion
+follows (Section~sec:recovery).
-\section{Introduction}\label{sec:intro}
+# Introduction
A finite correspondence between two group factors is a finite-dimensional
analytic relation, in the sense of von Neumann module dimension. For
@@ -430,12 +430,12 @@ The source groups are lattices in products of higher-rank real and
nonarchimedean groups, together with the quaternionic and Cayley
property-$(T)$ rank-one groups. The comparison group is only assumed to be
countably infinite and ICC. Thus the geometry that the proof uses must be
-constructed \emph{inside} the correspondence: no boundary or ambient Lie
+constructed inside the correspondence: no boundary or ambient Lie
group is available on the comparison side at the start.
-\subsection{The class, conventions, and main theorem}
+## The class, conventions, and main theorem
+
-\begin{definition}\label{def:class}
Let $\mathscr K$ be the class of countably infinite ICC groups abstractly
commensurable with a lattice in a finite nonempty product
\[
@@ -453,16 +453,16 @@ commensurability means an actual isomorphism between finite-index
subgroups. ICC means that every nonidentity conjugacy class is infinite.
No irreducibility, cocompactness, product decomposition, or torsion
hypothesis is imposed on the lattice.
-\end{definition}
+
Property $(T)$ means that every continuous unitary representation with
almost invariant unit vectors admits a nonzero invariant vector. The real
-factors in Definition~\ref{def:class} are either higher rank or locally
+factors in Definition~def:class are either higher rank or locally
$\Sp(n,1)$, $n\ge2$, or $F_{4(-20)}$; at a finite place the relative rank
is at least two. Complex places are treated as real Lie groups when
differential arguments are used. Standard structure and permanence
-results are recalled in Section~\ref{sec:setup}; see also
-\cite{BHV2008}.
+results are recalled in Section~sec:setup; see also
+[cite].
All scalar cocycles are $\mathbb T$-valued and normalized:
\[
@@ -475,7 +475,7 @@ For an ICC group this is a factor: the absolute values of Fourier
coefficients of a central vector are constant on conjugacy classes, so
square summability kills all nonidentity coefficients.
-A \emph{bifinite $M$--$N$ correspondence} is a Hilbert space carrying
+A bifinite $M$--$N$ correspondence is a Hilbert space carrying
commuting normal unital left $M$ and right $N$ actions and finite von
Neumann module dimensions on both sides. The actions need not be mutual
commutants, and the correspondence need not be irreducible. Every
@@ -483,85 +483,85 @@ correspondence in this paper is separable. Every factor isomorphism is
normal, unital, complex-linear, and $*$-preserving. For a finite factor
$Q$ and $t>0$ we write
\[
- Q^t=p\bigl(M_n(\mathbb C)\bar\otimes Q\bigr)p,
- \qquad (\Tr_n\otimes\tau_Q)(p)=t,
+ Q^t=p(M_n(\mathbb C)\bar ⊗ Q)p,
+ \qquad (\Tr_n ⊗ \tau_Q)(p)=t,
\]
with the unnormalized matrix trace $\Tr_n$; the corner itself carries its
normalized trace.
-For finite-index subgroups $A\le\Gamma$, $B\le\Lambda$, a group
+For finite-index subgroups $A\le\Gamma$, $B\leΛ$, a group
isomorphism $\delta:A\to B$, and a nonzero finite-dimensional projective
unitary representation $\sigma$ satisfying
-\begin{equation}\label{eq:intro-multiplier}
- \sigma(a)\sigma(b)=\frac{\mu(a,b)}{\omega(\delta(a),\delta(b))}\sigma(ab),
+
+ \sigma(a)\sigma(b)=(\mu(a,b))/(\omega(\delta(a),\delta(b)))\sigma(ab),
\qquad \sigma(e)=I,
-\end{equation}
-define the \emph{arithmetic model}
-\begin{equation}\label{eq:intro-model}
+
+define the arithmetic model
+
E(A,B,\delta,\sigma)
- =L^2(M)\,\bar\otimes_{L_{\mu|_A}(A)}
- \bigl(V_\sigma\otimes L^2(N)\bigr),
- \qquad M=L_\mu(\Gamma),\quad N=L_\omega(\Lambda).
-\end{equation}
+ =L^2(M)\,\bar ⊗ _{L_{\mu|_A}(A)}
+ (V_\sigma ⊗ L^2(N)),
+ \qquad M=L_\mu(\Gamma),\quad N=L_\omega(Λ).
+
The right $N$-action is standard. The left subgroup action on the second
-factor is $u_a(\xi\otimes\eta)=\sigma(a)\xi\otimes v_{\delta(a)}\eta$;
-Equation~\eqref{eq:intro-multiplier} makes this an action with multiplier
+factor is $u_a(\xi ⊗ \eta)=\sigma(a)\xi ⊗ v_{\delta(a)}\eta$;
+Equation~(eq:intro-multiplier) makes this an action with multiplier
$\mu|_A$.
A right action is represented by $\rho(h)\xi=\xi v_h^*$, so
-$\rho(h)\rho(k)=\overline{\omega(h,k)}\rho(hk)$. The left action is
+$\rho(h)\rho(k)=\omega(h,k)bar\rho(hk)$. The left action is
$\pi(g)\xi=u_g\xi$. Hence the joint action
$U_{(g,h)}\xi=u_g\xi v_h^*$ has multiplier
-$\alpha((g,h),(g',h'))=\mu(g,g')\overline{\omega(h,h')}$.
+$\alpha((g,h),(g',h'))=\mu(g,g')\omega(h,h')bar$.
-\begin{theorem}[Arithmetic exhaustion]\label{thm:arithmetic-exhaustion}
-Let $\Gamma\in\mathscr K$, let $\Lambda$ be any countably infinite ICC
+[Arithmetic exhaustion]
+Let $\Gamma\in\mathscr K$, let $Λ$ be any countably infinite ICC
group, and let $\mu\in Z^2(\Gamma,\mathbb T)$,
-$\omega\in Z^2(\Lambda,\mathbb T)$ be normalized scalar cocycles.
-Every bifinite $L_\mu(\Gamma)$--$L_\omega(\Lambda)$ correspondence is
+$\omega\in Z^2(Λ,\mathbb T)$ be normalized scalar cocycles.
+Every bifinite $L_\mu(\Gamma)$--$L_\omega(Λ)$ correspondence is
unitarily isomorphic to a closed bimodule summand of a finite direct sum
-of the models $E(A,B,\delta,\sigma)$ of Equation~\eqref{eq:intro-model},
-where $A\le\Gamma$ and $B\le\Lambda$ have finite index, $\delta:A\to B$
+of the models $E(A,B,\delta,\sigma)$ of Equation~(eq:intro-model),
+where $A\le\Gamma$ and $B\leΛ$ have finite index, $\delta:A\to B$
is an actual group isomorphism, and $\sigma$ is a finite-dimensional
projective unitary representation satisfying
-Equation~\eqref{eq:intro-multiplier}.
+Equation~(eq:intro-multiplier).
Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. If
$n=\dim_{\mathbb C}V_\sigma$, the elementary model has dimensions
\[
- \dim_{L_\mu(\Gamma)}E(A,B,\delta,\sigma)=n[\Lambda:B],\qquad
- \dim E(A,B,\delta,\sigma)_{L_\omega(\Lambda)}=n[\Gamma:A].
+ \dim_{L_\mu(\Gamma)}E(A,B,\delta,\sigma)=n[Λ:B],\qquad
+ \dim E(A,B,\delta,\sigma)_{L_\omega(Λ)}=n[\Gamma:A].
\]
-\end{theorem}
+
The theorem keeps both finite sums and closed summands, because a
correspondence may have irreducible pieces in several distinct
arithmetic models at once. The displayed dimension formulas concern the
elementary models themselves. The finer problem of identifying every
summand and every bounded intertwiner is the subject of the companion
-article \cite{OpenAI2026Category}, which uses
-Theorem~\ref{thm:arithmetic-exhaustion} as its geometric input. The
+article [cite], which uses
+Theorem~thm:arithmetic-exhaustion as its geometric input. The
elementary construction and its normality and dimensions are proved here
-in Section~\ref{sec:models}.
+in Section~sec:models.
Only the matched ratio $\mu/\delta^*\omega$ must admit the
finite-dimensional projective representation of
-Equation~\eqref{eq:intro-multiplier}. Neither endpoint cocycle is
+Equation~(eq:intro-multiplier). Neither endpoint cocycle is
required to have finite order, finite type, or a virtually trivial
restriction, and the image of $\sigma$ may be infinite. In particular,
-existence of a nonzero correspondence forces $\Gamma$ and $\Lambda$ to be
+existence of a nonzero correspondence forces $\Gamma$ and $Λ$ to be
abstractly commensurable. These conclusions cover nonuniform and
reducible lattices and their ICC commensurable groups.
-\subsection{Rigidity and finite-correspondence precedents}
+## Rigidity and finite-correspondence precedents
The group-factor reconstruction problem asks which features of a discrete
group survive in its von Neumann algebra. Connes proved countability of
-the fundamental group of a property-$(T)$ group factor \cite{Connes1980};
+the fundamental group of a property-$(T)$ group factor [cite];
his later problems ask whether nonisomorphic ICC property-$(T)$ groups
have nonisomorphic factors, and ask for the fundamental groups of those
-factors \cite[Chapter~V, Appendix~B.$\varepsilon$, Problems~1--2]{Connes1994}.
+factors \cite[Chapter~V, Appendix~B.$ε$, Problems~1--2]{Connes1994}.
Popa's survey states a stronger recovery question that controls the given
isomorphism, up to a character and inner conjugacy, together with its
amplification scale \cite[Section~3, Equations~(3.2) and~(3.2$'$)]{Popa2007}.
@@ -571,24 +571,24 @@ concurrent constructions of OpenAI and Zhou give nonisomorphic ICC
property-$(T)$ groups with isomorphic group factors
\cite[Theorem~1.2]{OpenAI2026ConnesCounterexample}
\cite[Theorem~A]{Zhou2026Counterexamples}. The present theorem concerns
-the lattice class of Definition~\ref{def:class}. Arithmetic exhaustion is
+the lattice class of Definition~def:class. Arithmetic exhaustion is
the finite-correspondence input used in the companion article to obtain
stable canonical recovery for this class.
Deformation and rigidity methods have produced strong reconstruction
theorems in several settings. Ioana's theorem recovers Bernoulli actions
of ICC property-$(T)$ groups from their crossed products
-\cite{Ioana2011}. Ioana--Popa--Vaes obtained $W^*$-superrigid groups among
-generalized wreath products \cite{IPV2013}, and
+[cite]. Ioana--Popa--Vaes obtained $W^*$-superrigid groups among
+generalized wreath products [cite], and
Chifan--Ioana--Osin--Sun constructed property-$(T)$ examples using
-wreath-like products \cite{CIOS2023}. These results recover groups or
+wreath-like products [cite]. These results recover groups or
actions from operator-algebraic data through constructions different from
the lattice geometry developed here.
Finite correspondences retain finite-index relations and their
compositions. The framework is supplied by Jones' index and module
-dimensions \cite{Jones1983}, the finite-index expectation inequality of
-Pimsner--Popa \cite{PimsnerPopa1986}, and Connes' relative tensor product
+dimensions [cite], the finite-index expectation inequality of
+Pimsner--Popa [cite], and Connes' relative tensor product
\cite[Chapter~V, Appendix~B]{Connes1994}. Vaes' classification for
generalized Bernoulli factors already organizes finite-index bimodules by
subgroup isomorphisms and finite-dimensional projective representations
@@ -632,7 +632,7 @@ measure-equivalence rigidity theorem
finite-module reconstruction used below are proved here; they are not
consequences of that classical theorem.
-\subsection{From boundary measurements to subgroup graphs}
+## From boundary measurements to subgroup graphs
The proof first passes to an actual lattice and removes the compact
projective sectors of the finite commutant. The remaining conjugation
@@ -641,7 +641,7 @@ orthogonal partition of the correspondence $K$ into finite-dimensional
subspaces permuted by both endpoint groups, with free actions and finitely
many orbits. A joint stabilizer of one subspace is then the graph of an
actual finite-index subgroup isomorphism, and its action on that subspace
-is the projective fibre in Equation~\eqref{eq:intro-multiplier}.
+is the projective fibre in Equation~(eq:intro-multiplier).
The geometric construction produces this partition through boundary
measurements. Let $B$ be the compact full flag space of the known ambient
@@ -650,21 +650,21 @@ projection to each Borel set, with countable additivity in the strong
operator topology; disjoint sets give orthogonal projections and the whole
space gives the identity. An angular measurement $P_x$ on $B$ records the
possible image of a source flag $x\in B$: $P_x(A)$ is the projection for
-its image to lie in $A\subseteq B$. Initially the construction supplies
+its image to lie in $A⊆ B$. Initially the construction supplies
positive operator-valued observations. Showing that they are PVMs, and
that observations at different labels commute, are separate steps.
-To compare two source copies, use $K_0\otimes K_1$, with the left
-$\Gamma$-action in each slot and the right $\Lambda$-action acting
-diagonally. The eventual joint angular law is a PVM $\Theta$ on the space
+To compare two source copies, use $K_0 ⊗ K_1$, with the left
+$\Gamma$-action in each slot and the right $Λ$-action acting
+diagonally. The eventual joint angular law is a PVM $Θ$ on the space
of nonsingular, essentially invertible maps $B\to B$. The two left actions
-transport its source and target labels; diagonal $\rho(\Lambda)$ fixes
-it. The same construction on $K_0\otimes K_1\otimes K_2$ compares slot
+transport its source and target labels; diagonal $\rho(Λ)$ fixes
+it. The same construction on $K_0 ⊗ K_1 ⊗ K_2$ compares slot
zero separately with slots one and two, and the required conclusion
includes
\[
- [\Theta_{01}(A),\Theta_{02}(C)]=0
- \qquad\text{for all Borel sets of boundary maps }A,C.
+ [Θ_{01}(A),Θ_{02}(C)]=0
+ \qquadfor all Borel sets of boundary maps A,C.
\]
This common-slot commutation has a concrete purpose. Normal slices in the
separate branch slots generate an abelian algebra on $K_0$. Disintegrating
@@ -672,11 +672,11 @@ over it gives boundary-map PVMs on the fixed space $K_1$. Finite module
dimension makes their orbit transversals finite rank; weak mixing makes
the resulting projection partition independent of the disintegration
parameter. The boundary-map labels themselves need not become
-deterministic. Section~\ref{sec:recovery} proves this passage and the
+deterministic. Section~sec:recovery proves this passage and the
subsequent graph-stabilizer identification.
Both copies of $B$ in this construction belong to the source lattice. No
-flag space for the unknown group $\Lambda$ is assumed. To read the known
+flag space for the unknown group $Λ$ is assumed. To read the known
geometry from Fourier coefficients, omit one left action and put the
remaining actions, together with diagonal $\rho$, in a finite regular
column. Inducing the omitted action to $G/\Gamma$ gives a finite tracial
@@ -691,21 +691,21 @@ Three local mechanisms control the only remaining Cartan gaps.
At finite places, compact-open fixed algebras and a uniform subgroup
logarithm argument recover homogeneous residue maps together with upper
-speed bounds (Section~\ref{sec:finite}).
+speed bounds (Section~sec:finite).
Real panels inside higher-rank components use optimal finite-dimensional
modules: trace inequalities force a nonzero scalar witness in their
section spaces, and positivity of its algebraic action excludes excess
-speed (Section~\ref{sec:panels}).
+speed (Section~sec:panels).
Isolated quaternionic and Cayley factors use uniform ray estimates and a
finite-module spectral gap to obtain positive mass in linear windows on
-fixed vectors (Sections~\ref{sec:ray}--\ref{sec:scale}). Pair packing
+fixed vectors (Sections~sec:ray--sec:scale). Pair packing
then gives horizontal derivatives, whose closed module core produces an
injective positive height with its own-label covariance. The Carnot
boundary geometry belongs to the work of Cowling--Dooley--Kor\'anyi--Ricci
-and Pansu \cite{CDKR1991,CDKR1998,Pansu1989}; the ray and finite-module
-estimates used here are proved in Sections~\ref{sec:ray}--\ref{sec:scale}.
+and Pansu [cite]; the ray and finite-module
+estimates used here are proved in Sections~sec:ray--sec:scale.
Full tuples bring these branches together. Comparing lattice packing with
Haar growth calibrates the speeds around the finite type permutation and
@@ -714,49 +714,49 @@ Haar-height isometry. Its common graph core permits locality arguments for
the unbounded heights. Locality gives finite jets for the actual spectral
control measure, which may be singular; commuting cone multipliers
eliminate every positive-order symbol. The principle is related to
-Peetre's locality theorem \cite{Peetre1959,Peetre1960}, but the measured
-operator-domain argument is proved here (Section~\ref{sec:locality}).
+Peetre's locality theorem [cite], but the measured
+operator-domain argument is proved here (Section~sec:locality).
Applying it also to the two shared-slot copies gives the isolated-type
commutation needed for the joint law.
For higher-component panels, pairing powers and exact residue graphs
supply the corresponding commutation. Simultaneous reflections and panel
galleries then glue all local types into the nonsingular, essentially
-invertible chamber maps described above (Section~\ref{sec:gluing}). After
+invertible chamber maps described above (Section~sec:gluing). After
extracting the fixed finite-rank partition and its graph fibres, we
restore every compact sector with its exact projective multiplier.
Two-sided finite-index induction returns the original correspondence as a
closed summand. This explains both the matched cocycle ratio and the
-summand qualification in Theorem~\ref{thm:arithmetic-exhaustion}
-(Section~\ref{sec:recovery}).
+summand qualification in Theorem~thm:arithmetic-exhaustion
+(Section~sec:recovery).
-\begin{figure}[tbp]
+[tbp]
\centering
\input{figures/proof-map}
- \caption{The geometric proof of arithmetic exhaustion. Each local branch
+ Caption: The geometric proof of arithmetic exhaustion. Each local branch
enters product calibration with its own speed and domain statement;
- simultaneous commutation is established only afterwards.}
- \label{fig:proof-map}
-\end{figure}
+ simultaneous commutation is established only afterwards.
+
-\subsection{Organization}
-Section~\ref{sec:models} verifies the elementary arithmetic models.
-Sections~\ref{sec:setup}--\ref{sec:transfer} construct the finite-module
+## Organization
+
+Section~sec:models verifies the elementary arithmetic models.
+Sections~sec:setup--sec:transfer construct the finite-module
framework and the ordered sharp boundary links. The finite-place,
isolated real, and higher-component real arguments occupy
-Sections~\ref{sec:finite}--\ref{sec:panels}; Table~\ref{tab:local-branches}
-records their exact inputs to Section~\ref{sec:heights}.
-Sections~\ref{sec:locality}--\ref{sec:gluing} prove locality and
-simultaneous commutation. Section~\ref{sec:recovery} extracts the
-arithmetic models and completes Theorem~\ref{thm:arithmetic-exhaustion}.
-A self-contained appendix, Section~\ref{sec:appendix}, gathers background
+Sections~sec:finite--sec:panels; Table~tab:local-branches
+records their exact inputs to Section~sec:heights.
+Sections~sec:locality--sec:gluing prove locality and
+simultaneous commutation. Section~sec:recovery extracts the
+arithmetic models and completes Theorem~thm:arithmetic-exhaustion.
+A self-contained appendix, Section~sec:appendix, gathers background
for the reader. All geometric and analytic arguments needed for the main
theorem are given in this article.
-\section{Elementary arithmetic correspondences}\label{sec:models}
+# Elementary arithmetic correspondences
-The models of Theorem~\ref{thm:arithmetic-exhaustion} admit an explicit
+The models of Theorem~thm:arithmetic-exhaustion admit an explicit
counting-space realization. We give the scalar transport and the
normality proof before starting the geometric converse; neither depends
on arithmetic exhaustion. Projective induction is classical
@@ -764,58 +764,58 @@ on arithmetic exhaustion. Projective induction is classical
belong to the finite-index bimodule calculus of
\cite[Section~6.2]{Vaes2008}.
-Throughout, $\Gamma,\Lambda$ are arbitrary countably infinite ICC
-groups, $M=L_\mu(\Gamma)$, $N=L_\omega(\Lambda)$, and
+Throughout, $\Gamma,Λ$ are arbitrary countably infinite ICC
+groups, $M=L_\mu(\Gamma)$, $N=L_\omega(Λ)$, and
\[
- D=\Gamma\times\Lambda,\qquad
- \alpha((g,h),(g',h'))=\mu(g,g')\overline{\omega(h,h')}.
+ D=\Gamma\timesΛ,\qquad
+ \alpha((g,h),(g',h'))=\mu(g,g')\omega(h,h')bar.
\]
-For a subgroup $C\le D$ the \emph{graph} of an isomorphism
+For a subgroup $C\le D$ the graph of an isomorphism
$\delta:A\to B$ of finite-index subgroups is
$C_\delta=\{(a,\delta(a)):a\in A\}\le D$.
-\subsection{Projective induction}
+## Projective induction
Let $C\le D$ be a subgroup and let $\gamma$ be a normalized unitary
projective representation on a finite-dimensional space $V$, with
multiplier $\alpha|_{C\times C}$. Write $X=D/C$ for the left cosets,
choose representatives $r_x\in D$ with $r_C=e$, and give each $x\in X$ a
copy $V_x$ of $V$. Set
-$\mathcal H(C,\gamma)=\bigoplus_{x\in X}V_x$, with the counting Hilbert
+$\mathcal H(C,\gamma)=⊕_{x\in X}V_x$, with the counting Hilbert
norm. For $s\in D$ put
-\begin{equation}\label{eq:input-A1}
- \begin{split}
+
+
c(s,x)&=r_{sx}^{-1}sr_x,\\
U^X_s(x)&=
\frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))}\,
- \gamma(c(s,x)):V_x\longrightarrow V_{sx}.
- \end{split}
-\end{equation}
+ \gamma(c(s,x)):V_x→ V_{sx}.
+
+
+
+[Exact projective induction]
-\begin{lemma}[Exact projective induction]
-\label{lem:models-induction}
-Equation~\eqref{eq:input-A1} defines a unitary projective representation
+Equation~(eq:input-A1) defines a unitary projective representation
of $D$ with multiplier $\alpha$. Changing the coset representatives gives
a unitarily equivalent bundle by unitary identifications of its
individual fibres. If $C$ is the graph of a finite-index subgroup
isomorphism $A\to B$, the bundle has commuting normal $M$ and right $N$
actions, and is bifinite.
-\end{lemma}
-\begin{proof}
+
+Proof.
The correction $c(s,x)$ belongs to $C$, and
\[
c(s,tx)c(t,x)=c(st,x).
\]
Using the cocycle identity for $\alpha$ together with the projective law
-of $\gamma$ in Equation~\eqref{eq:input-A1} gives
-\begin{equation}\label{eq:calculus-transport-identity}
+of $\gamma$ in Equation~(eq:input-A1) gives
+
U^X_s(tx)U^X_t(x)=\alpha(s,t)U^X_{st}(x).
-\end{equation}
+
Equivalently, all scalar factors can be verified at once in the central
extension with multiplication $(z,s)(w,t)=(zw\alpha(s,t),st)$: the
representation of the preimage of $C$ is $(z,c)\mapsto z\gamma(c)$, and
-$sr_x=r_{sx}c(s,x)$ gives exactly the ratio in Equation~\eqref{eq:input-A1}.
+$sr_x=r_{sx}c(s,x)$ gives exactly the ratio in Equation~(eq:input-A1).
This argument retains the multiplier rather than discarding it.
For explicit independence of the choice of representatives, suppose
@@ -825,7 +825,7 @@ coordinate copy of $V_x$ to the old one is
F_x=\alpha(r_x,k_x)^{-1}\gamma(k_x).
\]
Indeed, identifying a fibre via $U_{r'_x}$ instead of $U_{r_x}$ has
-precisely this effect, and Equation~\eqref{eq:calculus-transport-identity}
+precisely this effect, and Equation~(eq:calculus-transport-identity)
then gives $F_{sx}U'{}^X_s(x)=U^X_s(x)F_x$.
Now assume that $C$ is a graph $C_\delta$. Both coordinate groups act
@@ -833,11 +833,11 @@ freely on $X$: if $(g,e)r_xC=r_xC$, then $r_x^{-1}(g,e)r_x\in C$ has
second coordinate $e$, hence $g=e$. On an orbit of the first coordinate
group choose a fibre $V_{x_0}$. The unitary identification
\[
- \ell^2(\Gamma)\otimes V_{x_0}\longrightarrow
- \bigoplus_{g\in\Gamma}V_{(g,e)x_0},
- \qquad \delta_g\otimes\xi\longmapsto U^X_{(g,e)}\xi
+ \ell^2(\Gamma) ⊗ V_{x_0}→
+ ⊕_{g\in\Gamma}V_{(g,e)x_0},
+ \qquad \delta_g ⊗ \xi⟼ U^X_{(g,e)}\xi
\]
-intertwines the action with $\lambda_\mu\otimes1$, because
+intertwines the action with $\lambda_\mu ⊗ 1$, because
$U_{(a,e)}U_{(g,e)}=\mu(a,g)U_{(ag,e)}$. Hence it extends normally to
$L_\mu(\Gamma)$. The same argument for the second coordinate gives copies
of $\lambda_{\overline\omega}$, which are the standard normal
@@ -845,49 +845,49 @@ representations of the right generators $\xi\mapsto\xi v_h^*$. The two
actions commute, since the product cocycle has trivial mixed-coordinate
factors.
-The first-coordinate orbits are indexed by $\Lambda/B$ and the
+The first-coordinate orbits are indexed by $Λ/B$ and the
second-coordinate orbits by $\Gamma/A$; both are finite. Each orbit is a
regular module with multiplicity $\dim_{\mathbb C}V$, proving bifiniteness
and the dimension formulas below.
-\end{proof}
+
We next identify this construction with the relative tensor product model
-of Theorem~\ref{thm:arithmetic-exhaustion}. That identification also gives
+of Theorem~thm:arithmetic-exhaustion. That identification also gives
a direct normality check of the subgroup action used in the definition of
the model.
-\begin{lemma}[The graph-bundle unitary]
-\label{lem:models-graph}
+[The graph-bundle unitary]
+
Let $\delta:A\to B$ be a finite-index subgroup isomorphism and let
$\sigma:A\to\mathcal U(V)$ have multiplier
$\beta=\mu|_{A\times A}/\delta^*\omega$. Set
$\gamma(a,\delta(a))=\sigma(a)$. Then
\[
E(A,B,\delta,\sigma)=
- L^2(M)\mathbin{\bar\otimes}_{L_{\mu|A}(A)}
- \bigl(V\otimes L^2(N)\bigr)
+ L^2(M)\mathbin{\bar ⊗ }_{L_{\mu|A}(A)}
+ (V ⊗ L^2(N))
\cong\mathcal H(C_\delta,\gamma).
\]
The unitary sends the vector $\xi$ in the fibre labelled $x$ to
$U_{r_x}j\xi$, where
-$j\xi=\widehat1\otimes(\xi\otimes\widehat1)$ in the displayed model.
-\end{lemma}
+$j\xi=\widehat1 ⊗ (\xi ⊗ \widehat1)$ in the displayed model.
-\begin{proof}
-On $V\otimes L^2(N)$ the subgroup action is
+
+Proof.
+On $V ⊗ L^2(N)$ the subgroup action is
\[
- u_a(\xi\otimes\eta)=\sigma(a)\xi\otimes v_{\delta(a)}\eta.
+ u_a(\xi ⊗ \eta)=\sigma(a)\xi ⊗ v_{\delta(a)}\eta.
\]
Its multiplier is $\beta(a,b)\omega(\delta(a),\delta(b))=\mu(a,b)$. For
-normality, choose representatives $t$ for $B\backslash\Lambda$. On the
-summand spanned by $\xi\otimes v_{\delta(a)}v_t$, the map
-\begin{equation}\label{eq:calculus-absorption}
- \xi\otimes v_{\delta(a)}v_t
- \longmapsto\delta_a\otimes\sigma(a)^*\xi
- \quad\text{in }\ell^2(A)\otimes V
-\end{equation}
+normality, choose representatives $t$ for $B\backslashΛ$. On the
+summand spanned by $\xi ⊗ v_{\delta(a)}v_t$, the map
+
+ \xi ⊗ v_{\delta(a)}v_t
+ ⟼\delta_a ⊗ \sigma(a)^*\xi
+ \quadin \ell^2(A) ⊗ V
+
is unitary and conjugates the subgroup action to
-$\lambda_{\mu|A}\otimes1$. To check the phase, applying $u_b$ before this
+$\lambda_{\mu|A} ⊗ 1$. To check the phase, applying $u_b$ before this
map produces
\[
\omega(\delta(b),\delta(a))\sigma(ba)^*\sigma(b)
@@ -902,29 +902,29 @@ $L_{\mu|A}(A)$-module, $L^2(M)$ has orthonormal free basis $u_r$; the
assertion follows directly from
$E_{L_{\mu|A}(A)}(u_r^*u_{r'})=\delta_{r,r'}1$. The relative product
therefore identifies with the orthogonal sum
-\begin{equation}\label{eq:calculus-coset-model}
+
E(A,B,\delta,\sigma)
- =\bigoplus_{r\in\Gamma/A}u_r\otimes(V\otimes L^2(N)).
-\end{equation}
+ =⊕_{r\in\Gamma/A}u_r ⊗ (V ⊗ L^2(N)).
+
At its identity fibre the graph action is exactly
\[
U_{(a,\delta(a))}j\xi
- =u_a\otimes(\xi\otimes v_{\delta(a)}^*)
+ =u_a ⊗ (\xi ⊗ v_{\delta(a)}^*)
=j\sigma(a)\xi.
\]
-For $g=ra$, balancing in Equation~\eqref{eq:calculus-coset-model} gives
+For $g=ra$, balancing in Equation~(eq:calculus-coset-model) gives
the more explicit formula
\[
U_{(g,h)}j\xi
- =\mu(r,a)^{-1}u_r\otimes
- \bigl(\sigma(a)\xi\otimes v_{\delta(a)}v_h^*\bigr).
+ =\mu(r,a)^{-1}u_r ⊗
+ (\sigma(a)\xi ⊗ v_{\delta(a)}v_h^*).
\]
The last Fourier coordinate is $\delta(a)h^{-1}$, up to its displayed
unit scalar. Two translated identity fibres are therefore orthogonal
unless their labels in $D/C_\delta$ agree: equality of the first cosets
and the last Fourier coordinates is exactly that coset equality.
Conversely, these fibres exhaust every summand in
-Equation~\eqref{eq:calculus-coset-model}: they form an orthogonal,
+Equation~(eq:calculus-coset-model): they form an orthogonal,
transitive system with graph action $\gamma$ at the basepoint. Finally
\[
U_sU_{r_x}j
@@ -932,23 +932,23 @@ transitive system with graph action $\gamma$ at the basepoint. Finally
U_{r_{sx}}j\gamma(c(s,x)),
\]
which shows that the stated unitary has exactly the transport in
-Equation~\eqref{eq:input-A1}.
-\end{proof}
+Equation~(eq:input-A1).
+
-\begin{corollary}[Elementary normality and dimensions]\label{cor:models-dimensions}
+[Elementary normality and dimensions]
The correspondence $E(A,B,\delta,\sigma)$ is normal and bifinite. If
$n=\dim_{\mathbb C}V_\sigma$, then
\[
- \dim_M E(A,B,\delta,\sigma)=n[\Lambda:B],\qquad
+ \dim_M E(A,B,\delta,\sigma)=n[Λ:B],\qquad
\dim E(A,B,\delta,\sigma)_N=n[\Gamma:A].
\]
-\end{corollary}
-\begin{proof}
-Use Lemma~\ref{lem:models-graph} together with the orbit decompositions of
-Lemma~\ref{lem:models-induction}. Each free coordinate orbit gives $n$
+
+Proof.
+Use Lemma~lem:models-graph together with the orbit decompositions of
+Lemma~lem:models-induction. Each free coordinate orbit gives $n$
regular copies, of module dimension $n$ in the unnormalized matrix-trace
-convention. The respective orbit counts are $[\Lambda:B]$ and $[\Gamma:A]$.
-\end{proof}
+convention. The respective orbit counts are $[Λ:B]$ and $[\Gamma:A]$.
+
Finite direct sums and closed bimodule summands of these models remain
normal and bifinite: the two actions restrict normally to a reducing
@@ -959,26 +959,26 @@ correspondence, recover enough subgroup graphs to contain it. The next
sections carry out the finite-index and compact reductions needed for
that construction.
-\section{Finite-module realizations and local-field geometry}
-\label{sec:setup}
+# Finite-module realizations and local-field geometry
+
The first analytic goal is a finite regular realization of each choice of
an omitted left action on the tensor copies of the correspondence. Each
realization includes the other left actions and the diagonal right
action. These realizations supply the finite traces from which the
-boundary measurements of Section~\ref{sec:links} are built. We first
+boundary measurements of Section~sec:links are built. We first
remove the compact part of one conjugation action, then form the tensor
copies and the induced transports for the omitted source action.
Conjugation by these transports gives a genuine action of its ambient
group on the finite algebra. The remaining subsections describe the
geometric data to be read from the source-lattice Fourier indices.
-Fix a nonzero bifinite $L_\mu(\Gamma)$--$L_\omega(\Lambda)$
-correspondence $K$ as in Theorem~\ref{thm:arithmetic-exhaustion}. A right
-action of $L_\omega(\Lambda)$ is written as the projective representation
+Fix a nonzero bifinite $L_\mu(\Gamma)$--$L_\omega(Λ)$
+correspondence $K$ as in Theorem~thm:arithmetic-exhaustion. A right
+action of $L_\omega(Λ)$ is written as the projective representation
\[
\rho(h)\xi=\xi v_h^*,\qquad
- \rho(h)\rho(k)=\overline{\omega(h,k)}\rho(hk).
+ \rho(h)\rho(k)=\omega(h,k)bar\rho(hk).
\]
The left representation is $\pi(g)\xi=u_g\xi$. Thus $\pi$ and $\rho$
commute, although their multipliers need not be trivial. Conjugation by
@@ -986,93 +986,93 @@ either representation is an ordinary group action. All Hilbert spaces and
von Neumann algebras below are separable in the appropriate sense for
their normal representations.
-\subsection{Compact reduction and finite-index return}
+## Compact reduction and finite-index return
We will prove the geometric reconstruction first for an actual ICC
lattice $\Gamma<G$ and under the condition that
-\begin{equation}
- \Ad\rho\ \text{on}\
- L^2\bigl(\pi(L_\mu(\Gamma))',\operatorname{tr}\bigr)
+
+ \Ad\rho\ on\
+ L^2(\pi(L_\mu(\Gamma))',tr)
\ominus\mathbb C1
- \quad\text{has no nonzero finite-dimensional subrepresentation}.
- \label{eq:setup-weak-mixing}
-\end{equation}
+ \quadhas no nonzero finite-dimensional subrepresentation.
+
+
Here the commutant is a finite factor, equipped with its normalized trace.
The following lemmas justify this choice; their final use, after the
geometric partition has been constructed, is made in
-Section~\ref{sec:recovery}.
+Section~sec:recovery.
+
+[Twisted regular absorption]
-\begin{lemma}[Twisted regular absorption]
-\label{lem:setup-twisted-absorption}
Let $\alpha,\nu$ be normalized scalar cocycles on a countable group $L$,
and let $v$ be a $\nu$-projective unitary representation on a Hilbert
space $V$. Then
\[
- \lambda_\alpha\otimes v\simeq\lambda_{\alpha\nu}\otimes1_V.
+ \lambda_\alpha ⊗ v\simeq\lambda_{\alpha\nu} ⊗ 1_V.
\]
If $L$ is ICC and $V$ is finite-dimensional, tensoring a finite normal
$L_\alpha(L)$-module with $v$ gives a finite normal
$L_{\alpha\nu}(L)$-module. Any operators commuting with every $v_h$ are
preserved by the absorption identification.
-\end{lemma}
-\begin{proof}
-On $\ell^2(L)\otimes V$ use the unitary
+
+Proof.
+On $\ell^2(L) ⊗ V$ use the unitary
\[
- T(\delta_g\otimes\xi)=\delta_g\otimes v_g^*\xi.
+ T(\delta_g ⊗ \xi)=\delta_g ⊗ v_g^*\xi.
\]
Indeed
\[
- T(\lambda_\alpha(h)\otimes v_h)T^*(\delta_g\otimes\xi)
- =\alpha(h,g)\nu(h,g)\delta_{hg}\otimes\xi.
+ T(\lambda_\alpha(h) ⊗ v_h)T^*(\delta_g ⊗ \xi)
+ =\alpha(h,g)\nu(h,g)\delta_{hg} ⊗ \xi.
\]
A finite normal module over a finite factor embeds in finitely many copies
of the regular module \cite[Theorems~10.1.1 and~10.2.1]{Jones2009}. When
$L$ is ICC both twisted group algebras are factors, so applying this
unitary to such an embedding proves the second assertion. The same
formula also proves the last assertion directly.
-\end{proof}
-\begin{lemma}[Compact commutant reduction]
-\label{lem:setup-compact-reduction}
+
+[Compact commutant reduction]
+
Let $K$ be a nonzero bifinite
-$L_\mu(\Gamma)$--$L_\omega(\Lambda)$ correspondence with both groups ICC.
-There is a finite-index subgroup $\Lambda_0\le\Lambda$ such that the
+$L_\mu(\Gamma)$--$L_\omega(Λ)$ correspondence with both groups ICC.
+There is a finite-index subgroup $Λ_0\leΛ$ such that the
restriction of $K$ to the right subgroup factor is a finite orthogonal
sum of blocks
\[
- K_c=V_c\otimes K'_c,\qquad
- \pi(g)|_{K_c}=1\otimes\pi'_c(g),\qquad
- \rho(h)|_{K_c}=v^c_h\otimes\rho'_c(h)\quad(h\in\Lambda_0).
+ K_c=V_c ⊗ K'_c,\qquad
+ \pi(g)|_{K_c}=1 ⊗ \pi'_c(g),\qquad
+ \rho(h)|_{K_c}=v^c_h ⊗ \rho'_c(h)\quad(h\inΛ_0).
\]
The spaces $V_c$ are finite-dimensional. The $v^c_h$ are normalized
projective matrix representations with multipliers $\nu_c$, and
-\begin{equation}
+
\rho'_c(h)\rho'_c(k)
- =\frac{\overline{\omega(h,k)}}{\nu_c(h,k)}\rho'_c(hk).
- \label{eq:setup-reduced-right-multiplier}
-\end{equation}
+ =\frac{\omega(h,k)bar}{\nu_c(h,k)}\rho'_c(hk).
+
+
Each $K'_c$ is a finite normal module on both sides and satisfies
-Equation~\eqref{eq:setup-weak-mixing} for its reduced representations.
+Equation~(eq:setup-weak-mixing) for its reduced representations.
No assertion is made that any $v^c$ has finite image.
-\end{lemma}
-\begin{proof}
-Put $M=L_\mu(\Gamma)$, $N=L_\omega(\Lambda)$, and $P=\pi(M)'$. Finite
-$M$-module dimension makes $P$ a finite factor, and $N^{\mathrm{op}}\subset
+
+Proof.
+Put $M=L_\mu(\Gamma)$, $N=L_\omega(Λ)$, and $P=\pi(M)'$. Finite
+$M$-module dimension makes $P$ a finite factor, and $N^{op}\subset
P$ has finite index. Indeed, regard $K$ as the equivalence module for the
commuting factors $\pi(M)$ and $P$; it has finite nonzero $P$-dimension,
so finitely many copies of it contain $L^2(P)$ as a $P$-module. Their
-restrictions to $N^{\mathrm{op}}$ are finite by right finiteness of $K$.
-Hence $L^2(P)$ has finite $N^{\mathrm{op}}$-dimension, which is the index
-assertion \cite{Jones1983,PimsnerPopa1986}.
+restrictions to $N^{op}$ are finite by right finiteness of $K$.
+Hence $L^2(P)$ has finite $N^{op}$-dimension, which is the index
+assertion [cite].
Let $D_c\subset P$ be the bounded elements whose $\Ad\rho$-orbits are
precompact in $L^2(P)$. Compact vectors of a unitary representation form
its closed almost-periodic subspace. Bounded compact elements are closed
under adjoints and products, since on uniformly bounded sets
\[
- \|xy-x'y'\|_2\le\|x\|_\infty\|y-y'\|_2+\|x-x'\|_2\|y'\|_\infty,
+ \|xy-x'y'\|_2\le\|x\|_∞\|y-y'\|_2+\|x-x'\|_2\|y'\|_∞,
\]
and their bounded balls are strongly closed because bounded strong
convergence in a finite von Neumann algebra implies $L^2$ convergence.
@@ -1086,65 +1086,65 @@ invariant projection in the Fourier basis is constant on conjugacy
classes, scalar phases disappear from the diagonal, the diagonal is
summable, and every nonidentity class is infinite, so only the identity
vector remains. The trace-preserving expectation
-$E_{N^{\mathrm{op}}}:P\to N^{\mathrm{op}}$ is equivariant and
-$L^2$-contractive, hence $E_{N^{\mathrm{op}}}(x)=\operatorname{tr}(x)1$
-for $x\in D_c$. By the finite-index inequality there is $\varepsilon>0$
-with $E_{N^{\mathrm{op}}}(x)\ge\varepsilon x$ for $x\ge0$, so a nonzero
-projection $q\in D_c$ has $\operatorname{tr}(q)\ge\varepsilon$. A finite
+$E_{N^{op}}:P\to N^{op}$ is equivariant and
+$L^2$-contractive, hence $E_{N^{op}}(x)=tr(x)1$
+for $x\in D_c$. By the finite-index inequality there is $ε>0$
+with $E_{N^{op}}(x)\geε x$ for $x\ge0$, so a nonzero
+projection $q\in D_c$ has $tr(q)\geε$. A finite
algebra with this lower bound has only finitely many orthogonal nonzero
projections, and is finite-dimensional.
-Restrict to the finite-index subgroup $\Lambda_0$ fixing all central
+Restrict to the finite-index subgroup $Λ_0$ fixing all central
blocks of $D_c$. A central block $z$ reduces both endpoint actions; write
-$zK=V_c\otimes K'_c$ and $zD_c=\mathcal B(V_c)\otimes1$. The left action
-commutes with $D_c$, hence is $1\otimes\pi'_c$. Choose normalized matrix
+$zK=V_c ⊗ K'_c$ and $zD_c=\mathcal B(V_c) ⊗ 1$. The left action
+commutes with $D_c$, hence is $1 ⊗ \pi'_c$. Choose normalized matrix
implementers $v^c_h$ for $\Ad\rho(h)|_{zD_c}$; uniqueness up to a scalar
gives $v^c_hv^c_k=\nu_c(h,k)v^c_{hk}$, and after removing these
implementers $\rho(h)$ lies in the commutant of $\mathcal B(V_c)$, giving
-the asserted tensor form and Equation~\eqref{eq:setup-reduced-right-multiplier}.
+the asserted tensor form and Equation~(eq:setup-reduced-right-multiplier).
Normality and finiteness of $\pi'_c$ follow by taking a matrix corner of
the finite normal left module. For $\rho'_c$, first embed the original
restricted $\rho$ in a finite multiple of the
-$\overline\omega|_{\Lambda_0}$-regular representation, possible because
+$\overline\omega|_{Λ_0}$-regular representation, possible because
finite-index restriction preserves finite module dimension. Tensor with
-$\overline{v^c}$; Lemma~\ref{lem:setup-twisted-absorption} gives a finite
-normal module with multiplier $\overline{\nu_c}\,\overline\omega$. The
+$v^cbar$; Lemma~lem:setup-twisted-absorption gives a finite
+normal module with multiplier $\nu_cbar\,\overline\omega$. The
vector corresponding to the identity matrix is fixed by
-$\overline{v^c_h}\otimes v^c_h$, so tensoring that line with $K'_c$
+$v^c_hbar ⊗ v^c_h$, so tensoring that line with $K'_c$
exhibits $\rho'_c$ as a summand of a normal finite representation with the
-multiplier of Equation~\eqref{eq:setup-reduced-right-multiplier}.
+multiplier of Equation~(eq:setup-reduced-right-multiplier).
Finally, a compact vector in the reduced commutant embeds as
-$1_{V_c}\otimes\xi$ on $zK$ and as zero on the other blocks, and is
-compact for $\Lambda_0$ hence for $\Lambda$; it therefore lies in
+$1_{V_c} ⊗ \xi$ on $zK$ and as zero on the other blocks, and is
+compact for $Λ_0$ hence for $Λ$; it therefore lies in
$L^2(D_c)$ and its intersection with the reduced commutant on this block
-is scalar. This proves Equation~\eqref{eq:setup-weak-mixing}.
-\end{proof}
+is scalar. This proves Equation~(eq:setup-weak-mixing).
-\begin{lemma}[Finite-index restriction and return]
-\label{lem:setup-finite-index-induction}
-Let $\Gamma_0\le\Gamma$ and $\Lambda_0\le\Lambda$ have finite index, and
+
+[Finite-index restriction and return]
+
+Let $\Gamma_0\le\Gamma$ and $Λ_0\leΛ$ have finite index, and
restrict both cocycles literally. Every $M$--$N$ correspondence is a
summand of the two-sided induction of its restriction. The inclusions are,
on the respective sides,
-\begin{align}
+
J_L\xi&=[\Gamma:\Gamma_0]^{-1/2}
\sum_{r\Gamma_0\in\Gamma/\Gamma_0}
- u_r\otimes\pi(u_r)^*\xi,
- \label{eq:setup-left-induction}\\
- J_R\xi&=[\Lambda:\Lambda_0]^{-1/2}
- \sum_{\Lambda_0s\in\Lambda_0\backslash\Lambda}
- (\xi v_s^*)\otimes v_s.
- \label{eq:setup-right-induction}
-\end{align}
+ u_r ⊗ \pi(u_r)^*\xi,
+ \\
+ J_R\xi&=[Λ:Λ_0]^{-1/2}
+ \sum_{Λ_0s\inΛ_0\backslashΛ}
+ (\xi v_s^*) ⊗ v_s.
+
+
These are bimodular isometries, with all scalar cocycles included.
Restriction and induction preserve bifiniteness. An ICC group stays ICC
on finite-index restriction.
-\end{proof}
-\begin{proof}
-The $u_r$ in Equation~\eqref{eq:setup-left-induction} are an orthonormal
+
+Proof.
+The $u_r$ in Equation~(eq:setup-left-induction) are an orthonormal
right basis for $L^2(M)$ over the subgroup factor, since
$E_{L_{\mu|\Gamma_0}(\Gamma_0)}(u_r^*u_{r'})=\delta_{r,r'}1$; the
analogous left-basis identity for the $v_s$ holds. The relative
@@ -1154,10 +1154,10 @@ For left equivariance, if $gr=r'a$ with $a\in\Gamma_0$, then
$u_gu_r=(\mu(g,r)/\mu(r',a))u_{r'}u_a$; moving $u_a$ across the balanced
tensor product shows each term is the $r'$-term of $J_L(\pi(u_g)\xi)$. The
right calculation is the same in the opposite algebra: when $sh=at$ with
-$a\in\Lambda_0$,
+$a\inΛ_0$,
\[
- v_sv_h=\frac{\omega(s,h)}{\omega(a,t)}v_av_t,\qquad
- v_hv_t^*=\frac{\omega(s,h)}{\omega(a,t)}v_s^*v_a.
+ v_sv_h=(\omega(s,h))/(\omega(a,t))v_av_t,\qquad
+ v_hv_t^*=(\omega(s,h))/(\omega(a,t))v_s^*v_a.
\]
The other endpoint action commutes with every operation, so the two maps
compose to a bimodular isometry into two-sided induction, whose range is a
@@ -1167,26 +1167,26 @@ module dimensions.
If an element of $\Gamma_0$ has finite conjugacy class in $\Gamma_0$, its
class in $\Gamma$ is a finite union of translates of that class; ICC of
$\Gamma$ forces it to be the identity.
-\end{proof}
+
For a source in $\mathscr K$, choose the actual isomorphism of finite-index
-subgroups of Definition~\ref{def:class}; its image is a finite-index
+subgroups of Definition~def:class; its image is a finite-index
subgroup of a reference lattice, hence itself a lattice in the same
ambient group. Pull the scalar cocycle through that isomorphism, without
modifying it by a finite kernel. The restrictions retain property $(T)$
and ICC, as well as bifiniteness; property $(T)$ permanence is used in its
finite-covolume and finite-index forms \cite[Theorem~1.7.1]{BHV2008}; see
-also \cite[Theorem~3]{DelarocheKirillov1968}. Lemmas~\ref{lem:setup-compact-reduction}
-and~\ref{lem:setup-finite-index-induction} now reduce the geometric work
+also \cite[Theorem~3]{DelarocheKirillov1968}. Lemmas~lem:setup-compact-reduction
+and~lem:setup-finite-index-induction now reduce the geometric work
to the actual-lattice, weakly mixing setting. Restoring $V_c$ after a
partition of $K'_c$ tensors every atom with $V_c$ and restores the
original exact right multiplier, changing neither the atom permutations
nor their freeness; this is used only after the partition is proved.
-\subsection{Tensor copies and complementary regular realizations}
+## Tensor copies and complementary regular realizations
-From now until Section~\ref{sec:recovery}, let $\Gamma<G$ be an actual
-ICC lattice satisfying Equation~\eqref{eq:setup-weak-mixing}. Comparing two
+From now until Section~sec:recovery, let $\Gamma<G$ be an actual
+ICC lattice satisfying Equation~(eq:setup-weak-mixing). Comparing two
source copies uses two copies of $K$; comparing two comparisons sharing
their first copy uses three. The right comparison group acts diagonally.
We omit one left source action from the regular model so that its induced
@@ -1194,70 +1194,70 @@ transports remain available on $G/\Gamma$.
Take $m=1$ or $2$ and put
\[
- H=K^{\otimes(m+1)},\qquad
- R=\Gamma_1\times\cdots\times\Gamma_m\times\Lambda.
+ H=K^{ ⊗ (m+1)},\qquad
+ R=\Gamma_1\times...\times\Gamma_m\timesΛ.
\]
-The slots of $H$ are numbered $0,\ldots,m$; $\pi_j$ denotes the left
+The slots of $H$ are numbered $0,...,m$; $\pi_j$ denotes the left
representation in slot $j$. Let $W$ be the representation of $R$ given by
the $\pi_j$, $j\ge1$, and diagonal $\rho$, with split multiplier
-\begin{equation}
- \alpha_R((g_1,\ldots,g_m,h),(g'_1,\ldots,g'_m,h'))
- =\prod_{j=1}^m\mu(g_j,g'_j)\overline{\omega(h,h')}^{\,m+1}.
- \label{eq:setup-tensor-multiplier}
-\end{equation}
-
-\begin{lemma}[Complementary regularity]
-\label{lem:setup-normal-tensors}
+
+ \alpha_R((g_1,...,g_m,h),(g'_1,...,g'_m,h'))
+ =\prod_{j=1}^m\mu(g_j,g'_j)\omega(h,h')bar^{\,m+1}.
+
+
+
+[Complementary regularity]
+
The representation $W$ is normal and finite over the twisted factor for
$R$. The same holds with any one slot omitted instead of slot zero. The
joint commutant of all $\pi_j(\Gamma)$, $0\le j\le m$, and diagonal
-$\rho(\Lambda)$ is scalar.
-\end{lemma}
+$\rho(Λ)$ is scalar.
+
-\begin{proof}
+Proof.
Embed the omitted slot in a finite regular multiple for its
$\overline\omega$-projective right representation. At that regular index
absorb the diagonal right actions of the remaining slots by
-Lemma~\ref{lem:setup-twisted-absorption}. The absorbing unitaries are
+Lemma~lem:setup-twisted-absorption. The absorbing unitaries are
formed from the other slots' $\rho(h)$ and commute with all their
-$\pi_j(g)$. After absorption the $\Lambda$-action is regular with
+$\pi_j(g)$. After absorption the $Λ$-action is regular with
multiplier $\overline\omega^{\,m+1}$, and each remaining $\pi_j$ embeds
independently in a finite regular multiple for $L_\mu(\Gamma_j)$. The
result is a subrepresentation of a finite regular multiple for exactly
-Equation~\eqref{eq:setup-tensor-multiplier}, symmetrically in the omitted
+Equation~(eq:setup-tensor-multiplier), symmetrically in the omitted
slot.
Write $P=\pi(M)'$. The final joint commutant is
-$(P^{\bar\otimes(m+1)})^{\Ad\rho^{\otimes(m+1)}}$, which
-Equation~\eqref{eq:setup-weak-mixing} makes scalar. For completeness: an
+$(P^{\bar ⊗ (m+1)})^{\Ad\rho^{ ⊗ (m+1)}}$, which
+Equation~(eq:setup-weak-mixing) makes scalar. For completeness: an
invariant tensor with a factor in $L^2(P)\ominus\mathbb C1$ gives a
Hilbert--Schmidt intertwiner between that factor's representation and
another unitary representation; composing with its adjoint gives a
positive compact operator with a nonzero finite-dimensional invariant
spectral space, contrary to weak mixing. Decomposing each tensor factor
-as $\mathbb C1\oplus(L^2(P)\ominus\mathbb C1)$ applies this to every
+as $\mathbb C1 ⊕ (L^2(P)\ominus\mathbb C1)$ applies this to every
nonconstant summand.
-\end{proof}
+
Choose a column realization
\[
H=eL^2(B)^k,\qquad e\in M_k(B),\qquad
- d=(\Tr_k\otimes\tau_B)(e)>0.
+ d=(\Tr_k ⊗ \tau_B)(e)>0.
\]
Here $B$ is the opposite of the $\alpha_R$-twisted factor of $R$; using
the adjoints of the opposite canonical generators identifies it with the
-$\overline{\alpha_R}$-twisted group factor, whose generators we write as
+$\alpha_Rbar$-twisted group factor, whose generators we write as
$w_r$. Thus $W_r$ is right multiplication by $w_r^*$, which has
multiplier $\alpha_R$. Set
-\begin{equation}
+
Y=G/\Gamma,\qquad
\mathcal H=L^2(Y;H),\qquad
- \mathcal N=L^\infty(Y)\overline\otimes eM_k(B)e.
- \label{eq:setup-induced-model}
-\end{equation}
+ \mathcal N=L^∞(Y)\overline ⊗ eM_k(B)e.
+
+
Give $Y$ its invariant probability measure. The trace on $\mathcal N$ is
-$\operatorname{tr}_{\mathcal N}(A)
-=d^{-1}\int_Y(\Tr_k\otimes\tau_B)(A(y))\,dy$, whereas the Hilbert norm of
+$tr_{\mathcal N}(A)
+=d^{-1}\int_Y(\Tr_k ⊗ \tau_B)(A(y))\,dy$, whereas the Hilbert norm of
a column uses the unnormalized entrywise trace sum. All finite-matrix
estimates permit rectangular products. Whenever a family of coefficient
algebras is varied, coefficient traces are normalized and matrix sizes
@@ -1276,8 +1276,8 @@ $X_a(by)X_b(y)=\mu(c(a,by),c(b,y))X_{ab}(y)$. Conjugation on
base-decomposable operators is therefore a genuine action, denoted
$\sigma=\Ad U$.
-\begin{proposition}[Induced finite algebras]
-\label{prop:setup-induced-algebras}
+[Induced finite algebras]
+
The transports $U_a$ are strongly continuous in $a$, and $\sigma$
preserves the trace on $\mathcal N$. For $1\le j\le m$, let $R'_j$ be $R$
with $\Gamma_j$ omitted, put $\beta_j=\Ad W_{\Gamma_j}$, and let
@@ -1285,13 +1285,13 @@ $\mathcal A_j$ be the commutant of the base multipliers and
$W|_{R'_j}$. Then
\[
(\mathcal A_j)^{\beta_j}=\mathcal N,\qquad
- (\mathcal A_j)^\sigma\ \text{has a faithful finite $\beta_j$-invariant
- trace},\qquad
+ (\mathcal A_j)^\sigma\ has a faithful finite $\beta_j$-invariant
+ trace,\qquad
\mathcal N^G=\mathbb C1.
\]
-\end{proposition}
-\begin{proof}
+
+Proof.
Translation is continuous in measure on measurable maps on the
finite-measure quotient; applied to the Borel section this shows
$c(a,y)$ varies continuously in measure as a discrete-valued map. Verify
@@ -1310,23 +1310,23 @@ everywhere constant with value $T$, and right $\Gamma$-covariance says $T$
commutes with $\pi_0(\Gamma)$ and with $W|_{R'_j}$. Conversely every such
$T$ gives a fixed field. Thus $(\mathcal A_j)^\sigma$ is the commutant of
the complementary regular representation obtained by omitting slot $j$;
-Lemma~\ref{lem:setup-normal-tensors} makes it finite, with normalized
+Lemma~lem:setup-normal-tensors makes it finite, with normalized
$\beta_j$-invariant trace since the remaining slot acts by unitaries in
that commutant. Including all slots and using the last assertion of that
lemma gives $\mathcal N^G=\mathbb C1$.
-\end{proof}
-\subsection{The source groups and full versus partial mixing}
+
+## The source groups and full versus partial mixing
The induced algebra has only scalar whole-group invariants; to turn that
into limits along Cartan paths we need the mixing theorem for the actual
root-generated groups. We use the standard parabolic and root-group
structure, Cartan--Iwasawa decompositions, central-cover facts,
root-generated simplicity, and highest-weight theory
-\cite{Borel1991,BT1965,BruhatTits1972,BruhatTits1984,Tits1964,Humphreys1972}.
+[cite].
+
+[Factors and panel groups]
-\begin{convention}[Factors and panel groups]
-\label{conv:setup-panels}
At a complex place the local group is regarded as a real Lie group. The
noncompact property-$(T)$ archimedean factors are the higher-rank ones and
the groups locally isomorphic to $\Sp(n,1)$, $n\ge2$, or $F_{4(-20)}$. At
@@ -1348,7 +1348,7 @@ Howe--Moore takes invariants for this root-generated group or lift.
Projectivities use the effective adjoint group; the adjoint point group is
allowed for changes of frame. We never replace a full flag by an oriented
or topological covering flag.
-\end{convention}
+
For the cover assertion, a central covering induces an isomorphism on the
corresponding relative root groups, so its image contains the subgroup
@@ -1358,8 +1358,8 @@ rational-point quotient injects into $H^1(k,C)$, which is finite over the
present local fields \cite[Propositions~1.12 and~1.17, Theorem~6.1]{BorelSerre1964}.
This proves the finite-index assertion directly.
-\begin{lemma}[Product mixing and partial escape]
-\label{lem:setup-product-mixing}
+[Product mixing and partial escape]
+
For the action on $L^2(\mathcal N)$, escape in every simple factor gives
weak operator convergence to the projection onto constants. If a path is
the identity outside a subproduct $G_0$ and escapes in every factor of
@@ -1368,20 +1368,20 @@ equivalently the conditional expectation onto $\mathcal N^{G_0}$. On the
orthogonal complement of a simple factor's invariants, its
matrix-coefficient decay remains uniform after composition with arbitrary
commuting group unitaries.
-\end{lemma}
-\begin{proof}
+
+Proof.
Use Howe--Moore for the noncompact isotropic almost simple root-generated
groups and their root-generated lifts from
-Convention~\ref{conv:setup-panels} \cite{HoweMoore1979}; an independent
+Convention~conv:setup-panels [cite]; an independent
proof in this convention is \cite[Theorem~1.1 and Definition~4.23]{Ciobotaru2015}.
We justify the type-I input for these actual groups. For an isotropic
simple factor $\mathbf H$ over $k$, choose a nonzero parabolic unipotent
radical $\mathbf U$. The $k$-span of the $\mathbf H(k)$-conjugates of
-$\operatorname{Lie}(\mathbf U)$ is a nonzero invariant Lie ideal, hence all
-of $\operatorname{Lie}(\mathbf H)$. Choose $X_1,\ldots,X_d$ from these
+$Lie(\mathbf U)$ is a nonzero invariant Lie ideal, hence all
+of $Lie(\mathbf H)$. Choose $X_1,...,X_d$ from these
conjugate subspaces; the map
-$(t_1,\ldots,t_d)\mapsto\prod_j\exp(t_jX_j)$ has invertible differential
+$(t_1,...,t_d)\mapsto\prod_j\exp(t_jX_j)$ has invertible differential
at zero and lands in $\mathbf H(k)^+$. The local-field inverse function
theorem makes this subgroup open; at finite places see
\cite[Proposition~1.11]{Gloeckner2008}. The same holds for the root-generated
@@ -1389,26 +1389,26 @@ subgroup of an algebraic simply connected cover. At an archimedean place
the generating unipotent groups are connected, so the root-generated group
is connected and open, hence the identity component; these groups are
type~I by \cite[Theorem~1]{Dixmier1957}. At a finite place the ambient
-reductive group of local points is type~I \cite{Bernstein1974}, and type~I
+reductive group of local points is type~I [cite], and type~I
passes to its open root-generated subgroup
\cite[Proposition~6.E.21(1)]{BekkaHarpe2019}.
Restrict a separable unitary representation $\pi$ to one simple factor $S$.
Its type-I decomposition with multiplicity is
\[
- \mathcal H=\int_{\widehat S}^{\oplus}(H_\xi\otimes M_\xi)\,d\eta(\xi),
+ \mathcal H=\int_{\widehat S}^{ ⊕ }(H_\xi ⊗ M_\xi)\,d\eta(\xi),
\qquad
- \pi(s)=\int_{\widehat S}^{\oplus}(\xi(s)\otimes1)\,d\eta(\xi).
+ \pi(s)=\int_{\widehat S}^{ ⊕ }(\xi(s) ⊗ 1)\,d\eta(\xi).
\]
In this central decomposition every unitary commuting with $S$ acts
-fibrewise as $1\otimes V_\xi$
+fibrewise as $1 ⊗ V_\xi$
\cite[Propositions~5.4.7 and~8.6.4, Theorem~8.6.6]{Dixmier1977}; see also
\cite[Theorem~7.32]{Folland2015}. For finite tensor sums in a nontrivial
fibre, uniformly over the $V_\xi$,
\[
- \Bigl|\bigl\langle(\xi(s)\otimes V_\xi)\sum_j e_j\otimes u_j,
- \sum_k f_k\otimes v_k\bigr\rangle\Bigr|
- \le\sum_{j,k}|\langle\xi(s)e_j,f_k\rangle|\,\|u_j\|\,\|v_k\|.
+ |\bigl<(\xi(s) ⊗ V_\xi)\sum_j e_j ⊗ u_j,
+ \sum_k f_k ⊗ v_k\bigr>|
+ \le\sum_{j,k}|<\xi(s)e_j,f_k>|\,\|u_j\|\,\|v_k\|.
\]
Howe--Moore makes the right side tend to zero along escape; approximation
by finite tensor sums is uniform because all operators have norm one.
@@ -1421,13 +1421,13 @@ finite sum of orthogonal products obtained from
$\prod_i(P_i+(1-P_i))$. In every sector containing $1-P_i$, the uniform
decay kills its coefficients when that factor escapes. Thus all-factor
escape gives the weak limit $\prod_iP_i$, the projection onto constants
-by Proposition~\ref{prop:setup-induced-algebras}. For partial escape,
+by Proposition~prop:setup-induced-algebras. For partial escape,
expand only over the factors of $G_0$; the limit is the projection onto
$L^2(\mathcal N)^{G_0}$, i.e. the trace-preserving conditional
expectation.
-\end{proof}
-\subsection{Panel groups, Cartan coordinates, and Haar volumes}
+
+## Panel groups, Cartan coordinates, and Haar volumes
At a panel only one noncompact rank-one Levi factor acts on the residue;
on a Cartan path exact root coordinates will determine which flag
@@ -1445,7 +1445,7 @@ small inner exponentials, hence this Lie algebra, and is the identity.
These assertions concern the ambient local groups, not a discreteness
assumption on a lattice projection.
-Let $\Pi$ be the union of the simple \emph{indivisible} relative roots of
+Let $\Pi$ be the union of the simple indivisible relative roots of
the factors. A face of type $I\subset\Pi$ has strictly positive simple
coordinates exactly in $I$; opposition of types is denoted by a star.
Choose a special maximal compact in each factor. Cartan and Iwasawa
@@ -1454,15 +1454,15 @@ Write $\mu(g)$ for the Cartan vector only when its argument is a
local-group element; this is distinct from the scalar two-cocycle
$\mu(g,h)$. For a positive root $\alpha$ put
\[
- m_\alpha=\dim_{k}\mathfrak g_\alpha\ \text{at a finite place},\qquad
+ m_\alpha=\dim_{k}\mathfrak g_\alpha\ at a finite place,\qquad
2\rho=\sum_{\alpha>0}m_\alpha\alpha,
\]
and use the analogous real restricted-root multiplicities at archimedean
places. Double roots are included. The fundamental coweight rays $e_i$
are dual to the simple indivisible roots.
-\begin{convention}[Exact finite-place translation coordinates]
-\label{conv:setup-exact-lengths}
+[Exact finite-place translation coordinates]
+
At a finite place use the module absolute value with $|\varpi|=q^{-1}$.
The Cartan vector is the actual dominant translation vector of a Cartan
representative in the split centralizer: its pairing with every
@@ -1478,20 +1478,20 @@ many standard Levis, so one divisible mesh suffices. At real places the
corresponding split exponentials lie in the generated identity lift. No
argument at a finite place requires every real parameter to be an actual
split displacement.
-\end{convention}
-\begin{lemma}[Haar size of Cartan shells]
-\label{lem:setup-haar-shells}
+
+[Haar size of Cartan shells]
+
For a bounded-thickness shell about a dominant vector $L$, its Haar
measure is at most $C\exp(2\rho(L))$. For one fixed sufficiently large
thickness there is a lower bound $c\exp(2\rho(L))$. The constants are
independent of $L$ on the permissible Cartan mesh. The analogous
assertions hold for the finite product, with the sum of the factors' $2\rho$.
-\end{lemma}
-\begin{proof}
+
+Proof.
At a real place use the Cartan density, a product of the appropriate
-powers of $\sinh\alpha(L)$. The upper bound follows on every fixed bounded
+powers of $sinh\alpha(L)$. The upper bound follows on every fixed bounded
neighborhood. For the lower bound, even when $L$ lies on a wall, choose
inside the shell one fixed bounded offset box whose simple coordinates are
bounded away from zero; on this box every positive-root factor is bounded
@@ -1510,9 +1510,9 @@ parabolic product uniqueness give this calculation; changing the compact
frame changes the small subgroups by a uniformly bounded index. Other
Cartan representatives are within bounded distance of the chosen split
mesh. Multiplication of Haar measures gives the product assertion.
-\end{proof}
-\subsection{Compactified matrices and admissible scalar tests}
+
+## Compactified matrices and admissible scalar tests
For each relative vertex type choose a highest-line representation: take
the determinant line of the relevant parabolic nilradical in an exterior
@@ -1536,8 +1536,8 @@ with a compact-invariant inner product; higher tensor weights are allowed.
No Hilbert space is tensored with a vector space over a nonarchimedean
field.
-\begin{lemma}[Face blocks and residual projectivities]
-\label{lem:setup-face-blocks}
+[Face blocks and residual projectivities]
+
Let a sequence have diverging simple gaps of types $D$ and keep its
complementary gaps in a bounded window. Its compactified matrix data
determine an image $D$-flag, a reverse $D^*$-flag, and an invertible
@@ -1546,9 +1546,9 @@ independent of compact frame choices. On a deeper stratum the already
determined image vertices persist. On compact transverse windows the
residual rank-one panel lengths differ by bounded amounts from the actual
omitted Cartan gaps.
-\end{lemma}
-\begin{proof}
+
+Proof.
Pass to convergent compact Cartan frames. In each highest-weight module
precisely the weights of maximal value on the open face survive projective
normalization; they are the submodule generated from the highest line by
@@ -1570,27 +1570,27 @@ component the surviving rank-one split displacement is exactly that
omitted coordinate; bounded frames change its length by a uniformly
bounded amount. The argument works simultaneously for one omitted type in
several components.
-\end{proof}
+
Absolute growth on a highest line is measured with the compact-equivariant
norm; relative to the exact Cartan exponent, its correction is the test on
the input flag direction. On a transverse cell the normalized ratio
converges to the corresponding absolute pairing. At a finite place the
character valuations of the nonsplit centralizer give the same statement
-with Convention~\ref{conv:setup-exact-lengths}.
+with Convention~conv:setup-exact-lengths.
+
+[Smooth and clopen symbols]
-\begin{convention}[Smooth and clopen symbols]
-\label{conv:setup-symbols}
On real coordinate windows, a smooth symbol has the specified uniformly
bounded finite number of derivatives, increased whenever a separated
expansion is needed. On finite-place coordinate boxes its scalar-test
counterpart is locally constant at a uniformly fixed clopen level. The
underlying coordinate maps remain analytic. Uniformity means normalized
denominators are bounded away from zero on a buffered larger window.
-\end{convention}
-\begin{lemma}[Interpolation in an affine building]
-\label{lem:setup-building-interpolation}
+
+[Interpolation in an affine building]
+
Fix an endpoint $x$ in one of the affine buildings above. In an apartment
containing $x,y$, multiply the dominant simple coordinates of $y-x$ by
independent numbers $t_i\in[0,1]$. The resulting point is independent of
@@ -1598,9 +1598,9 @@ the apartment and depends on $y$ with a uniform Lipschitz constant. The
same holds for variation of $x$, after opposition and replacement of the
fractions by their complements. Constants for a finite product depend only
on its root systems.
-\end{lemma}
-\begin{proof}
+
+Proof.
Write $L_i=\alpha_i(y-x)\ge0$ and let $T_t$ have simple coordinates
$t_iL_i$. For every positive root $\alpha=\sum_in_i\alpha_i$, where
$n_i\ge0$,
@@ -1616,7 +1616,7 @@ bounded Euclidean operator norm; subdivide a segment of varying endpoints
into finitely many cells and Weyl sectors, sum the bounds, and use
apartment independence at the interfaces. Reversing the endpoints applies
opposition and the complementary fractions.
-\end{proof}
+
Noncompact isotropic Levi factors contain discrete nonabelian free
subgroups: choose two rank-one translations with separated attracting and
@@ -1628,21 +1628,21 @@ amenable; restriction to a closed subgroup preserves this amenability
These facts are used for the source local group and its lattice, not for
any action of the comparison group on a building.
-\begin{convention}[Operators and Fourier coordinates]
-\label{conv:setup-operators}
+[Operators and Fourier coordinates]
+
An invariance under $U$ is a conjugation invariance of decomposable
operators, so base scalar phases do not change it. An ``other group'' in a
-commutant includes $\Lambda$ unless $\Lambda$ is explicitly the measured
+commutant includes $Λ$ unless $Λ$ is explicitly the measured
group. A target Fourier arrow records $w_{(h,\cdot)}$ and the Cartan or
flag data of $h\in\Gamma$ in a measured source-lattice copy; it does not
-record geometric data of the $\Lambda$ index. A bounded column is an
+record geometric data of the $Λ$ index. A bounded column is an
operator-bounded tracial matrix column, essentially uniformly bounded when
it depends on $y$. The closed range of a positive effect means its support
space when a range inclusion is asserted.
-\end{convention}
-\section{Boundary obstructions, links, and strict gap growth}
-\label{sec:links}
+
+# Boundary obstructions, links, and strict gap growth
+
An escaping source translation acts on a finite regular array. We record the
limiting flags of its measured Fourier index and compare the measurements
@@ -1651,11 +1651,11 @@ these measurements and their input forms, and to prove that a vertex
translation produces exactly one divergent target gap while successively
completing such a vertex-origin path adds exactly one target gap per stage.
-We retain the regular realizations of Section~\ref{sec:setup}: $H=eL^2(B)^k$,
-$\mathcal H=L^2(Y;H)$, $d=(\Tr\otimes\tau_B)(e)$. For an integrable corner
+We retain the regular realizations of Section~sec:setup: $H=eL^2(B)^k$,
+$\mathcal H=L^2(Y;H)$, $d=(\Tr ⊗ \tau_B)(e)$. For an integrable corner
field write
\[
- \mathsf T(C)=\int_Y(\Tr\otimes\tau_B)(C(y))\,dy,\qquad
+ \mathsf T(C)=\int_Y(\Tr ⊗ \tau_B)(C(y))\,dy,\qquad
\tau_{\mathcal N}(C)=d^{-1}\mathsf T(C).
\]
Thus $\tau_{\mathcal N}$ is a probability trace while the Hilbert norm of a
@@ -1663,11 +1663,11 @@ column is $\|f\|_2^2=\mathsf T(ff^*)$. Matrix sizes in this section are
fixed and finite. Operator-bounded columns form a dense subspace. The
constant corner frame is
\[
- b_r=e\varepsilon_r,\qquad 1\le r\le k,\qquad
+ b_r=eε_r,\qquad 1\le r\le k,\qquad
\sum_{r=1}^k b_rb_r^*=e.
\]
-\subsection{Compact matrix data and the surviving flags}
+## Compact matrix data and the surviving flags
Write $\Pi$ for the indivisible relative simple roots. A face of type
$I\subset\Pi$ has strictly positive simple coordinates exactly in $I$;
@@ -1675,11 +1675,11 @@ opposition of types is denoted by a star. Partial flags are transverse when
they are in the open relative position of subfaces of opposite chambers;
incidence means containment in one chamber.
-\begin{lemma}[Matrix compactification and residual projectivities]
-\label{lem:links-matrix-data}
+[Matrix compactification and residual projectivities]
+
There is a finite family of projectively normalized matrix coordinates,
including coordinates of the inverse, with the following properties.
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item The set $D\subset\Pi$ of diverging simple gaps is detected by rank one
in the fundamental highest-line coordinates. A stratum with gap set $D$
carries an image $D$-flag, a reverse $D^*$-flag, and a residual projectivity
@@ -1698,16 +1698,16 @@ spanned by the highest lines of flags incident with the observed face. A
rank-one Levi pencil has the effective rank-one projectivity group and its
omitted simple gap, up to the bounded changes made by compact endpoint
frames.
-\end{enumerate}
+
The matrix coordinates are real smooth coordinates at archimedean places and
projective local-field coordinates at finite places.
-\end{lemma}
-\begin{proof}
+
+Proof.
Use the highest-line modules and normalized matrices of
-Section~\ref{sec:setup}. At archimedean places normalize by Hilbert--Schmidt;
+Section~sec:setup. At archimedean places normalize by Hilbert--Schmidt;
at finite places use projective ratio charts with a maximal coordinate.
-Lemma~\ref{lem:setup-face-blocks} identifies the surviving Levi blocks,
+Lemma~lem:setup-face-blocks identifies the surviving Levi blocks,
proves their invertibility on a bounded complementary-gap window, and
describes the intrinsic transverse projection followed by residual
transport. In each fundamental highest-line coordinate, rank one detects
@@ -1724,12 +1724,12 @@ matrices gives the reverse-flag rule. The residual map transforms by the
corresponding compositions.
The last assertion is the spanning and length statement of
-Lemma~\ref{lem:setup-face-blocks}: the Levi orbit of the highest line spans
+Lemma~lem:setup-face-blocks: the Levi orbit of the highest line spans
its surviving top space, and on a panel the only noncompact isotropic
factor supplies the omitted simple gap that compact changes of frame shift
by a bounded amount. At a finite place the modules remain vector spaces
over the local field with no Hilbert-space amplification.
-\end{proof}
+
Compact windows in a stratum are exhausted by bounds on its complementary
gaps and by compact transverse projective charts. Their tests may be chosen
@@ -1740,7 +1740,7 @@ locally constant on compact boxes at a fixed level. Increasing a highest
weight by tensor powers in polynomial tests does not change the scalar
projective multiplier of the original Hilbert representation.
-\subsection{Fourier transport before compression}
+## Fourier transport before compression
Fourier tests here act on full regular arrays; they are not followed by a
corner projection before the next operation. This permits a diagonal
@@ -1748,14 +1748,14 @@ spectral calculus without claiming that an angular Fourier cutoff preserves
the operator norm of a column.
Choose a path $a$ with $a^{-1}$ split toward a source face $F$ of type $I$,
-its coordinates in $I$ tending to infinity. A \emph{lexicographic face path}
+its coordinates in $I$ tending to infinity. A lexicographic face path
starts with a vertex translation and adds one new simple coordinate at each
successively slower scale; the fastest parameter is taken to its limit first
while slower parameters stay fixed, then the next limit, and so on. A
general face path may have several coordinates tending to infinity at once;
the gap upper bound below applies to both, though the exact successive count
is asserted for vertex-origin chains. Finite-place parameters live on the
-divisible cocharacter meshes of Section~\ref{sec:setup}.
+divisible cocharacter meshes of Section~sec:setup.
The measured label is the compact matrix data of $h^{-1}$, where $h$ is the
Fourier index in the chosen copy $\Gamma_j$. Other group indices remain
@@ -1778,11 +1778,11 @@ stratum and its image flag this is the image measurement; it need not be a
projection.
The two-column map, initially on operator-bounded columns, is
-\begin{equation}
+
J(\overline b,f)
- =\sqrt d\,\bigl(b(ay)^*X_a(y)f(y)\bigr)^{\lim}.
- \label{eq:input-1}
-\end{equation}
+ =\sqrt d\,(b(ay)^*X_a(y)f(y))^{\lim}.
+
+
Its output has its own diagonal spectral measure $E$. The link identities
below determine the norm of this map and the Hilbert inputs to which it
extends. Iterated ultraproducts realize the specified successive limits;
@@ -1791,8 +1791,8 @@ countable determining algebra of continuous tests. A Borel stratum always
denotes a projection of the limiting spectral measure, not a discontinuous
pointwise cutoff before the limit.
-\begin{lemma}[Fourier--Gram transport]
-\label{lem:links-gram-transport}
+[Fourier--Gram transport]
+
On the limiting arrays, a fixed left or right Fourier shift transports
continuous compact matrix tests by its limiting geometric map, hence
transports the associated Borel spectral projections. For a stratum and
@@ -1801,11 +1801,11 @@ quadratic form through its uncut Gram factor. This holds for bounded
base-dependent multipliers and for compact families of fixed source
translations reindexed at the end base. All projective defects cancel in
these sesquilinear identities.
-\end{lemma}
-\begin{proof}
+
+Proof.
For a fixed shift, normalized matrix multiplication and the inverse formula
-have denominators bounded away from zero by Lemma~\ref{lem:links-matrix-data}.
+have denominators bounded away from zero by Lemma~lem:links-matrix-data.
Their action on the compact label space is continuous; a diagonal continuous
test therefore intertwines the shift in the ultralimit, and equality on
continuous functions extends to Borel spectral measures. Base multiplication
@@ -1822,7 +1822,7 @@ and matrix sizes. For each finite Fourier sum, move $D$ through its
individual shifts by the already proved limiting identity; the remaining
product is the uncut Gram factor $T_l^*T_l$. Replacing this by $T^*T$ costs
another quantity tending to zero since
-$\|T_l^*T_l-T^*T\|_2\le(\|T_l\|_\infty+\|T\|_\infty)\|T_l-T\|_2$. All errors
+$\|T_l^*T_l-T^*T\|_2\le(\|T_l\|_∞+\|T\|_∞)\|T_l-T\|_2$. All errors
are estimated before applying the cutoff to the columns, so the argument
applies to partial unitaries $T^*T=e$ and assigns no operator bound to $Df$.
@@ -1844,60 +1844,60 @@ path: after base reindexing, $X_tX_a^*$ is the transporter for $ta^{-1}$
times a common scalar, and its localized Fourier norm is unchanged. Products
in these identities are actual twisted matrix products; their coefficient
phases cannot be replaced by absolute values of untwisted convolutions.
-\end{proof}
+
With one measured coordinate, the other group indices, finite matrix
factors, and base algebra may be left in the coefficient algebra. The
multiplier splits across these group factors, so the argument applies
without further commutation assumptions.
-\subsection{Link forms, scalar masses, and the finite frame}
+## Link forms, scalar masses, and the finite frame
For a stratum of positive scalar mass $m_D$, divide its image measurement by
$m_D$ to obtain a unital measurement. Scalarity is part of the next
proposition, not an assumption on the construction.
-\begin{proposition}[Link identities]
-\label{prop:links-link}
-The one-column measurements and the forms \eqref{eq:input-1} have the
+[Link identities]
+
+The one-column measurements and the forms (eq:input-1) have the
following properties.
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item The image measurement in one measured copy lies in $\mathcal A_j$,
is $\beta_j$-equivariant, and is invariant under the source face parabolic.
Its gap-stratum masses belong to $\mathcal N$ and are scalar. Conjugating
the source split frame transports the measurement by $\sigma$.
\item If the path escapes in every simple component, the link is an isometry
-$\overline{\mathcal H}\otimes\mathcal H\to\mathcal K$. Its output forms are
+$\mathcal Hbar ⊗ \mathcal H\to\mathcal K$. Its output forms are
decomposable at the two bases, covariant under the two independent target
shifts, invariant under the independently contracted source radicals, and
invariant under simultaneous source centralizers. An image-and-stratum
-effect depends only on the start input and is $1\otimes\Phi(A)$; the reverse
+effect depends only on the start input and is $1 ⊗ \Phi(A)$; the reverse
rule is symmetric.
\item If the path is active precisely in a subproduct $G_0$, put
$N_0=\mathcal N^{G_0}$ and let $E_{N_0}:\mathcal N\to N_0$ be the
trace-preserving conditional expectation. On bounded columns the link has
-\begin{equation}
+
\|J(\overline b,f)\|^2
- =d^2\tau_{\mathcal N}\bigl(E_{N_0}(ff^*)E_{N_0}(bb^*)\bigr).
- \label{eq:links-relative-form}
-\end{equation}
+ =d^2\tau_{\mathcal N}(E_{N_0}(ff^*)E_{N_0}(bb^*)).
+
+
Its polarized extension on finite sums,
\[
- \Bigl\|\sum_iJ(\overline{b_i},f_i)\Bigr\|^2
+ \Bigl\|\sum_iJ(b_ibar,f_i)\Bigr\|^2
=d^2\sum_{i,l}\tau_{\mathcal N}
- \bigl(E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)\bigr),
- \label{eq:links-polarized-relative}
+ (E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)),
+
\]
is a positive semidefinite form. The same Gram covariances hold on this form
space; they do not assert a tensor-product isometry. Hilbert extensions are
-in Lemma~\ref{lem:links-relative-boundedness}.
+in Lemma~lem:links-relative-boundedness.
\item For any Borel event $A$ determined by a gap stratum and its image flag,
the finite-frame identity is
-\begin{equation}
- \langle f,\Phi(A)f\rangle
- =\frac1d\sum_{r=1}^k\|E(A)J(\overline{b_r},f)\|^2.
- \label{eq:links-frame}
-\end{equation}
+
+ < f,\Phi(A)f>
+ =\frac1d\sum_{r=1}^k\|E(A)J(b_rbar,f)\|^2.
+
+
It holds on mixed and nonmixed paths, and more generally for events in
additional limiting coordinates unchanged by the ignored left Fourier
shifts. When a proof starts from windows that also constrain the reverse
@@ -1908,17 +1908,17 @@ embedding. Adding another original tensor slot gives their normal copies,
tensored with the identity on that slot. This concerns the compressed
measurements and the operators determined by their stated one-ended
covariance and uniqueness.
-\end{enumerate}
+
All statements use the specified order of successive limits; a contracted
translate must tend to the identity in that order.
-\end{proposition}
-\begin{proof}
-\emph{(i)} Base multipliers and companion actions commute with the relevant
+
+Proof.
+(i) Base multipliers and companion actions commute with the relevant
diagonal tests and intertwine the input column map, so the compressed image
effects lie in $\mathcal A_j$. A measured target shift changes the inverse
Fourier label by the corresponding left geometric action, giving
-$\beta_j$-equivariance via Lemma~\ref{lem:links-gram-transport}.
+$\beta_j$-equivariance via Lemma~lem:links-gram-transport.
Let $n$ be in the radical contracted by the path. Sliding $U_n$ through
$U_a$ puts $U_{ana^{-1}}$ at the far end up to a common base phase; this
@@ -1930,7 +1930,7 @@ the full limiting data.
For a fixed centralizing $s$, the identity slides $U_s$ to the far end with
the same $s$. Base reindexing has no effect on an integrated form, and the
-left corner multiplier is removed by Lemma~\ref{lem:links-gram-transport};
+left corner multiplier is removed by Lemma~lem:links-gram-transport;
the uncut Gram factor is the corner identity. Thus the image measurement is
centralizer invariant. Radical and centralizer together generate the face
parabolic; conjugation of a split path removes the end conjugation.
@@ -1945,7 +1945,7 @@ tending to zero. It is therefore factor invariant, and the mass is invariant
under all simple factors, hence scalar because $\mathcal N^G=\mathbb C1$.
This argument covers a pure vertex and a mixed path alike.
-\emph{(ii)} For bounded $b,f$, cyclicity and the definition of the induced
+(ii) For bounded $b,f$, cyclicity and the definition of the induced
action give
\[
\|J(\overline b,f)\|^2
@@ -1955,11 +1955,11 @@ On a mixed path, trace mixing makes the limit
$d^2\tau_{\mathcal N}(ff^*)\tau_{\mathcal N}(bb^*)=\|f\|_2^2\|b\|_2^2$.
Polarization gives
\[
- \Bigl\|\sum_iJ(\overline{b_i},f_i)\Bigr\|^2
+ \Bigl\|\sum_iJ(b_ibar,f_i)\Bigr\|^2
=d^2\sum_{i,l}\tau_{\mathcal N}(f_lf_i^*)\tau_{\mathcal N}(b_ib_l^*),
\]
-which is the norm on $\overline{\mathcal H}\otimes\mathcal H$; the factor
-$\sqrt d$ in \eqref{eq:input-1} is essential for this normalization.
+which is the norm on $\mathcal Hbar ⊗ \mathcal H$; the factor
+$\sqrt d$ in (eq:input-1) is essential for this normalization.
Multiplication of an input by a start (resp.\ end) base function is
represented on arrays by its value at $y$ (resp.\ $ay$); diagonal tests
@@ -1968,22 +1968,22 @@ bases. Independent target shifts are handled by Fourier transport; source
radicals slide to contracted translates and cancel; simultaneous
centralizers use exact precomposition and finite-Fourier Gram. For an
image-and-stratum event, the ignored end Fourier shifts put each end slice
-in $\overline{\mathcal N}$, and independent reverse-radical invariance makes
+in $\mathcal Nbar$, and independent reverse-radical invariance makes
it scalar by the Howe--Moore argument above; the compressed effect is
-$1\otimes C$, and the frame computation below identifies $C=\Phi(A)$.
+$1 ⊗ C$, and the frame computation below identifies $C=\Phi(A)$.
-\emph{(iii)} For an escape in $G_0$, the weak limit of the trace-preserving
+(iii) For an escape in $G_0$, the weak limit of the trace-preserving
action on $L^2(\mathcal N)$ is the projection onto $L^2(N_0)$, which yields
-\eqref{eq:links-relative-form} and \eqref{eq:links-polarized-relative} as
+(eq:links-relative-form) and (eq:links-polarized-relative) as
limits of squared norms. The mass calculation above is separate; it does
not replace $N_0$ by scalars.
-\emph{(iv)} For a continuous test of the image and stratum, approximate each
+(iv) For a continuous test of the image and stratum, approximate each
$b_r(ay)^*$ by bounded finite Fourier matrices. The inverse-index convention
makes the test invariant under their left Fourier shifts, so
-Lemma~\ref{lem:links-gram-transport} moves the test past the multipliers.
+Lemma~lem:links-gram-transport moves the test past the multipliers.
Summing the uncut Gram factors gives $\sum_rb_r(ay)b_r(ay)^*=e$; the
-two-column normalization contributes $d$, proving \eqref{eq:links-frame}.
+two-column normalization contributes $d$, proving (eq:links-frame).
Equality of the limiting spectral measures extends it to invariant Borel
events. The proof uses neither factorization of the end variable nor mixed
trace mixing.
@@ -1997,15 +1997,15 @@ finite frame be inserted. This preserves the stated order of operations and
covers extra limiting coordinates whenever they too are unchanged by the
ignored shifts.
-\emph{(v)} An intertwiner between two regular embeddings is a bounded
+(v) An intertwiner between two regular embeddings is a bounded
Fourier matrix on the relevant finite modules. For image-and-stratum tests
its uncut Gram factor is the appropriate input corner identity, so
-Lemma~\ref{lem:links-gram-transport} identifies the two compressed
+Lemma~lem:links-gram-transport identifies the two compressed
measurements. For the normal-copy assertion, begin with a regular-multiple
embedding for the first two slots and enlarge it by the entire extra slot.
Conjugating the extra slot by the projective inverse of its simultaneous
-$\rho(h)$ action at the regular $\Lambda$-index transfers its multiplier to
-the regular $\Lambda$ multiplier via projective absorption, and this
+$\rho(h)$ action at the regular $Λ$-index transfers its multiplier to
+the regular $Λ$ multiplier via projective absorption, and this
conjugation commutes with the extra $\Gamma$-action, which can be embedded in
its own finite regular multiple. Both operations commute with scalar Fourier
cutoffs on the original $\Gamma_j$-coordinate. In original input coordinates
@@ -2015,9 +2015,9 @@ embedding-independence identifies this with any subsequent realization.
This proves the assertion for the measurements, hence for the normal
algebras and the uniquely determined operators; it makes no assertion about
all uncompressed spectral arrays.
-\end{proof}
-\subsection{Relative boundedness and Hilbert inputs}
+
+## Relative boundedness and Hilbert inputs
The two norm formulas give different input spaces. On an all-factor path,
$J$ extends to the ordinary Hilbert tensor product. On a partial path its
@@ -2025,11 +2025,11 @@ norm still contains $E_{N_0}(ff^*)$, which can be unbounded for a Hilbert
column. The useful replacement is to bound this conditional expectation for
one input.
-\begin{lemma}[Relative boundedness and Hilbert inputs]
-\label{lem:links-relative-boundedness}
+[Relative boundedness and Hilbert inputs]
+
For a path supported in $G_0$, call a column $f$ relatively bounded if the
positive affiliated operator $E_{N_0}(ff^*)$ is bounded. Then:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item for fixed relatively bounded $f$, the link extends continuously to
every Hilbert end column $b$, and conversely;
\item the image measurements of paths in factors of $G_0$ commute with
@@ -2041,17 +2041,17 @@ same bound on its conditional expectation;
\item the link covariances hold on relatively bounded pairs and when one
input is operator-bounded and the other a Hilbert column. A one-column gap
inclusion valid for all inputs also holds for bounded pairs in
-\eqref{eq:input-1}.
-\end{enumerate}
-\end{lemma}
+(eq:input-1).
+
-\begin{proof}
+
+Proof.
Conditional expectation extends to positive $L^1$ elements. Put
-$A=E_{N_0}(ff^*)$; if $A$ is bounded, \eqref{eq:links-relative-form} gives
-\begin{equation}
- \|J(\overline b,f)\|^2\le d\|A\|_\infty\|b\|_2^2.
- \label{eq:links-relative-bound}
-\end{equation}
+$A=E_{N_0}(ff^*)$; if $A$ is bounded, (eq:links-relative-form) gives
+
+ \|J(\overline b,f)\|^2\le d\|A\|_∞\|b\|_2^2.
+
+
This proves the continuous extension from bounded $b$; the link limit is
always taken before this extension.
@@ -2060,7 +2060,7 @@ multiplier; move it from the input to the end and remove it from an image
test by the finite Fourier--Gram argument, as a centralizer multiplier.
Polarization shows every image effect commutes with $c$. If a bounded $Q$
commutes with $N_0$, then for positive $c\in N_0$,
-$\langle Qf,cQf\rangle\le\|Q\|^2\langle f,cf\rangle$; the
+$< Qf,cQf>\le\|Q\|^2< f,cf>$; the
characterization of the conditional expectation by these tests implies
$E_{N_0}((Qf)(Qf)^*)\le\|Q\|^2E_{N_0}(ff^*)$. This holds on the generated
algebra, its support projections, and after finite matrix amplification.
@@ -2074,24 +2074,24 @@ columns converging in Hilbert norm with
$E_{N_0}(f_nf_n^*)\le E_{N_0}(ff^*)$.
These approximants converge in the link form against a fixed Hilbert $b$:
-first replace $b$ by bounded $b_l$; then \eqref{eq:links-relative-form}
-bounds the error by $d\|b_l\|_\infty^2\|f_n-f\|_2^2$ on the tail while
+first replace $b$ by bounded $b_l$; then (eq:links-relative-form)
+bounds the error by $d\|b_l\|_∞^2\|f_n-f\|_2^2$ on the tail while
$E_{N_0}((f_n-f)(f_n-f)^*)\le4E_{N_0}(ff^*)$ controls it uniformly. Take
-$n\to\infty$ then $l\to\infty$. The same argument with inputs reversed
+$n\to∞$ then $l\to∞$. The same argument with inputs reversed
extends all bounded Gram identities to the stated domains.
Finally, if a gap event annihilates every one-column input, first
approximate a bounded end column's left multiplier by a finite Fourier sum;
each fixed shift preserves the gap event in the limit, so its terms vanish,
and the approximation error is estimated on uncut bounded columns by
-Lemma~\ref{lem:links-gram-transport}. This proves the same inclusion for
+Lemma~lem:links-gram-transport. This proves the same inclusion for
bounded pairs with no mixed tensor decomposition.
-\end{proof}
-\subsection{Additional invariant variables}
-\begin{lemma}[Additional invariant limiting variables]
-\label{lem:links-invariant-variables}
+## Additional invariant variables
+
+[Additional invariant limiting variables]
+
Adjoin to the compact spectral data any bounded limiting tests
asymptotically unchanged by fixed left and right Fourier shifts. The
limiting effects retain base decomposability and the measured equivariance
@@ -2101,19 +2101,19 @@ invariant under the measured target shifts, that effect is scalar. For a
mixed two-ended effect, independent target-shift invariance and independent
contracted-radical invariance make its mass scalar on both inputs. These
assertions apply to invariant Borel events of the limiting variables.
-\end{lemma}
-\begin{proof}
+
+Proof.
A fixed Fourier shift gives precisely the limiting intertwining identity
-used in Lemma~\ref{lem:links-gram-transport}; the finite-Fourier proof is
+used in Lemma~lem:links-gram-transport; the finite-Fourier proof is
unchanged. Base decomposability and commutation with companions follow
before limits. An invariant one-column test gives an element of
$\mathcal N$; face-parabolic invariance, followed by the active-factor
radical argument and the inactive-factor centralizer argument of
-Proposition~\ref{prop:links-link}, puts it in $\mathcal N^G=\mathbb C1$.
+Proposition~prop:links-link, puts it in $\mathcal N^G=\mathbb C1$.
For mixed links, independent target invariance puts the respective slices
-in $\overline{\mathcal N}$ and $\mathcal N$; independent
+in $\mathcal Nbar$ and $\mathcal N$; independent
contracted-radical invariance makes them scalar, and polarization and
normal slice separation give the scalar joint effect.
@@ -2132,9 +2132,9 @@ coordinates are face-parabolic invariant, so their target-invariant masses
are scalar even on a pure vertex ray. The argument supplies the required
invariant limiting tests; establish the continuous determining identities
first, and their spectral measures then transport identically.
-\end{proof}
-\subsection{Ordering and genuine source actions}
+
+## Ordering and genuine source actions
Two kinds of spectral operation are used. A single limiting compact
variable has its usual Borel functional calculus. For several possibly
@@ -2153,18 +2153,18 @@ a clopen test at a fixed level pulls back to tests at one uniformly bounded
level. Units and projective representatives do not alter this. At real
places the maps have bounded derivatives of each fixed order. The
geometric estimates for variable shifts in transverse products are in
-Section~\ref{sec:transfer}.
+Section~sec:transfer.
+
+[Cancelling source multipliers]
-\begin{lemma}[Cancelling source multipliers]
-\label{lem:links-genuine-source}
After establishing the decomposable link forms, one may enlarge their
source actions to genuine strongly continuous group representations by an
opposite-multiplier regular factor at each used input base. The compressed
effects become their pointwise normal ampliations. This enlargement
imposes no new mixing or scalar-commutant assertion.
-\end{lemma}
-\begin{proof}
+
+Proof.
Write the source multiplier as $\eta$. The regular representation with
multiplier $\overline\eta$, induced with the same section, has the conjugate
of the scalar defect of $X_a(y)=\pi_0(c(a,y))$, so the product of the two
@@ -2181,22 +2181,22 @@ gives the enlarged dilation. A face spectral measure invariant under the
projective transport now commutes with a genuine source representation and
can be disintegrated with it. All mixing and scalarity statements continue
to refer to $\mathcal N$, not the enlarged commutant.
-\end{proof}
-\subsection{Escape from bounded target coordinates}
-\begin{lemma}[No all-bounded stratum]
-\label{lem:links-no-bounded-stratum}
+## Escape from bounded target coordinates
+
+[No all-bounded stratum]
+
On every escaping source face path, the all-bounded gap stratum has zero
mass in a single measured lattice copy. This holds for the one-column
measurement and for bounded pairs in the link.
-\end{lemma}
-\begin{proof}
+
+Proof.
We record the finite-module Bessel estimate used here. For a bounded column
$b\in eL^2(B)^k$ and a Hilbert column $\xi$, Parseval in regular coordinates
gives
-$\sum_{h\in R}|\langle bw_h^*,\xi\rangle|^2=\|b^*\xi\|_2^2\le\|b\|_\infty^2\|\xi\|_2^2$,
+$\sum_{h\in R}|< bw_h^*,\xi>|^2=\|b^*\xi\|_2^2\le\|b\|_∞^2\|\xi\|_2^2$,
projective phases being harmless. The same holds for subsets, in particular
$R'_j$-translates, and in finite matrix versions summing over fixed row and
column indices.
@@ -2234,27 +2234,27 @@ A bounded Cartan set is compact in the whole ambient product, hence contains
only finitely many lattice elements. Summing gives zero link mass on each
bounded Cartan window; increasing those windows removes the all-bounded
stratum. This is a count in the whole lattice, needing no discreteness of
-a projection to a proper subproduct. Apply \eqref{eq:links-frame} for the
+a projection to a proper subproduct. Apply (eq:links-frame) for the
one-column assertion.
-\end{proof}
-\subsection{An amenable-boundary obstruction}
-\begin{lemma}[Full-flag obstruction]
-\label{lem:links-obstruction}
+## An amenable-boundary obstruction
+
+[Full-flag obstruction]
+
Let $S$ be the root-generated group of a noncompact isotropic almost simple
component of a semisimple source Levi subgroup, using its root-generated
-lift from Convention~\ref{conv:setup-panels} when the ambient action requires
+lift from Convention~conv:setup-panels when the ambient action requires
it. There is no nonzero positive map from $C(Z)$ for the full target flag
space $Z$ to $\mathcal A_j$ which is target-equivariant and fixed by the
source action $\sigma|_S$.
-\end{lemma}
-\begin{proof}
+
+Proof.
Suppose $\Phi:C(Z)\to\mathcal A_j$ is such a map. Its mass $a=\Phi(1)$
belongs to $\mathcal N^S$: target equivariance makes it commute with the
measured target action, and membership in $\mathcal A_j$ the rest. Choose
-$\epsilon>0$ with $p=1_{[\epsilon,\infty)}(a)\ne0$. On $p\mathcal H$,
+$ε>0$ with $p=1_{[ε,∞)}(a)\ne0$. On $p\mathcal H$,
$\Psi(u)=pa^{-1/2}\Phi(u)a^{-1/2}p$ is unital and completely positive (a
positive map from $C(Z)$ is completely positive) with the same covariances.
The normalization does not require $a$ to commute with $\Phi(u)$.
@@ -2267,13 +2267,13 @@ $W|_{R'_j}$, and $U_f$, $f\in F$. These commute with the measured target
action and every $\Psi$ value.
Use the minimal Stinespring construction with commutant lifting
-\cite{Stinespring1955,Arveson1969}. The minimal dilation has algebraic
-vectors $u\otimes\xi$ with form
-$\langle\sum_iu_i\otimes\xi_i,\sum_lv_l\otimes\eta_l\rangle
-=\sum_{i,l}\langle\xi_i,\Psi(\overline{u_i}v_l)\eta_l\rangle$, and
-$\widehat D(u\otimes\xi)=u\otimes D\xi$ satisfies
+[cite]. The minimal dilation has algebraic
+vectors $u ⊗ \xi$ with form
+$<\sum_iu_i ⊗ \xi_i,\sum_lv_l ⊗ \eta_l>
+=\sum_{i,l}<\xi_i,\Psi(u_ibarv_l)\eta_l>$, and
+$\widehat D(u ⊗ \xi)=u ⊗ D\xi$ satisfies
$\|\widehat D\|\le\|D\|$ in the concrete input norm. The covariant lift
-$T_g(u\otimes\xi)=(g\cdot u)\otimes W_g\xi$ has the same multiplier
+$T_g(u ⊗ \xi)=(g\cdot u) ⊗ W_g\xi$ has the same multiplier
$\zeta$ as $W$.
By \cite[Proposition~2.5]{AnantharamanDelaroche2002}, the action of
@@ -2283,19 +2283,19 @@ to $1$, and uniform asymptotic equivariance. Compactness of $Z$ makes their
norms uniformly positive eventually, and normalizing pointwise gives
continuous unit fields with the same uniformity. In the dilation with
spectral representation $\pi_Z$ of $C(Z)$, put
-$Q_i\eta=\sum_{t\in\Gamma_j}\delta_t\otimes\pi_Z(\xi_i(\cdot)_t)\eta$;
+$Q_i\eta=\sum_{t\in\Gamma_j}\delta_t ⊗ \pi_Z(\xi_i(\cdot)_t)\eta$;
these are isometries intertwining the companion algebra exactly, with
-$\|Q_iT_g-(\lambda_g\otimes T_g)Q_i\|\to0$ for each fixed $g$. Projective
-absorption $A(\delta_t\otimes\eta)=\delta_t\otimes T_t^*\eta$ conjugates
-$\lambda_g\otimes T_g$ to $\lambda_\zeta(g)\otimes1$, leaving the companion
+$\|Q_iT_g-(\lambda_g ⊗ T_g)Q_i\|\to0$ for each fixed $g$. Projective
+absorption $A(\delta_t ⊗ \eta)=\delta_t ⊗ T_t^*\eta$ conjugates
+$\lambda_g ⊗ T_g$ to $\lambda_\zeta(g) ⊗ 1$, leaving the companion
unchanged. Taking limits on the original input gives, at every finite matrix
level,
-\begin{equation}
+
\Bigl\|\sum_g W_gD_g\Bigr\|
- \le\Bigl\|\sum_g\lambda_\zeta(g)\otimes D_g\Bigr\|_{
- C^*_{r,\zeta}(\Gamma_j)\otimes_{\min}\mathcal D}.
- \label{eq:links-min-bound}
-\end{equation}
+ \le\Bigl\|\sum_g\lambda_\zeta(g) ⊗ D_g\Bigr\|_{
+ C^*_{r,\zeta}(\Gamma_j) ⊗ _{\min}\mathcal D}.
+
+
The multiplier is that of $W$ and is conjugated in the right-action
convention.
@@ -2304,7 +2304,7 @@ multiple for $\Gamma_0\times R'_j$ with split multiplier. Inducing the
$\Gamma_0$-coordinate and using $(y,h)\mapsto d_0(y)h$ turns it into the
$G$-coordinate, where $U_f$ acts as left translation times a scalar of
modulus one. The companions use a separate twisted regular coordinate.
-Tensoring with the measured regular representation in \eqref{eq:links-min-bound},
+Tensoring with the measured regular representation in (eq:links-min-bound),
a single-base finite Fourier matrix field from the full split group algebra
of $R$ acts fibrewise with its original matrix norm. Consequently, for
contractions $C_f$, scalar phases $\theta_f$, and a finitely supported
@@ -2324,50 +2324,50 @@ $T_{f,n}$ simple in the base and finite in Fourier support with
$T_{f,n}\to T_f$ in integrated tracial $L^2$ (approximate in the unit ball
by simple-base fields, then use Kaplansky density and Fourier polynomials).
Then $S_n=\sum_f\nu(f)R_{T_{f,n}}U_f^{(k)}$ has $S_nE\to E\ne0$. By
-\eqref{eq:links-min-bound} and the spatial estimate,
+(eq:links-min-bound) and the spatial estimate,
$\|S_n\|\le\|\lambda_F(\nu)\|$; for the uniform walk on two free generators
and their inverses this is at most $\sqrt3/2<1$ by Kesten's bound
\cite[Theorem~3]{Kesten1959}. The conclusions are incompatible since
-$0<\|E\|_2^2=\mathsf T(p)<\infty$.
-\end{proof}
+$0<\|E\|_2^2=\mathsf T(p)<∞$.
+
+
+## Strict growth under a slower Levi translation
-\subsection{Strict growth under a slower Levi translation}
+[A slower Levi translation removes every compact window]
-\begin{lemma}[A slower Levi translation removes every compact window]
-\label{lem:links-strict-growth}
Let an inner source face path have a positive target gap stratum
-$\Omega_D$. After the inner limit, append a slower commuting split
+$Ω_D$. After the inner limit, append a slower commuting split
translation whose projection escapes in a noncompact simple component $S$ of
the remaining source Levi; additional split parts commuting with $S$ are
-allowed. Every fixed compact window in $\Omega_D$ has mass tending to zero
-in the outer limit. The entire positive contribution of $\Omega_D$ then
-passes to strata $\Omega_{D'}$ with $D'\supsetneq D$.
-\end{lemma}
+allowed. Every fixed compact window in $Ω_D$ has mass tending to zero
+in the outer limit. The entire positive contribution of $Ω_D$ then
+passes to strata $Ω_{D'}$ with $D'⊋ D$.
-\begin{proof}
-The scalar mass $m_D>0$ is given by Proposition~\ref{prop:links-link};
+
+Proof.
+The scalar mass $m_D>0$ is given by Proposition~prop:links-link;
divide the stratum measurement by it, giving a unital full-data
measurement $\Psi$. If $D=\Pi$, its image measurement is a nonzero
full-flag map fixed by the remaining source component $S$, contradicting
-Lemma~\ref{lem:links-obstruction}. Thus $D\ne\Pi$.
+Lemma~lem:links-obstruction. Thus $D\ne\Pi$.
-\emph{The marginal inclusion.} Let $\Psi(F)=V_\Omega^*M_FV_\Omega$ be a
-minimal full-data dilation with image-flag map $x:\Omega_D\to X_D$ and
+The marginal inclusion. Let $\Psi(F)=V_Ω^*M_FV_Ω$ be a
+minimal full-data dilation with image-flag map $x:Ω_D\to X_D$ and
minimal image dilation $V:\mathcal H\to\mathcal K_X$. The generator map
-$M_uVf\mapsto M_{u\circ x}V_\Omega f$ preserves inner products and extends
-to an isometric inclusion $i:\mathcal K_X\to\mathcal K_\Omega$ with
-$iV=V_\Omega$, intertwining the image spectral measure, base multiplication,
+$M_uVf\mapsto M_{u\circ x}V_Ω f$ preserves inner products and extends
+to an isometric inclusion $i:\mathcal K_X\to\mathcal K_Ω$ with
+$iV=V_Ω$, intertwining the image spectral measure, base multiplication,
companions, and target lifts. Disintegrate
-$\mathcal K_X=\int^\oplus K_x\,d\vartheta(x)$ with isometric fibre
+$\mathcal K_X=\int^ ⊕ K_x\,dθ(x)$ with isometric fibre
inclusions $i_x$.
-\emph{Positive density forms.} Let $\mathcal F_x$ be the compact
+Positive density forms. Let $\mathcal F_x$ be the compact
full-refinement space of the image face $x$, with compact-rotation
probability $\eta_x$. The residual projectivity of a full datum
-$\omega\in\Omega_D$ transports that probability to a probability
+$\omega\inΩ_D$ transports that probability to a probability
$k_\omega(l)\,d\eta_x(l)$ on $\mathcal F_x$, positive with total mass one.
-On a compact data window $C\subset\Omega_D$, this density has bounds
-$0<c_C\le k_\omega(l)\le C_C<\infty$ uniformly in $\omega\in C$ and
+On a compact data window $C\subsetΩ_D$, this density has bounds
+$0<c_C\le k_\omega(l)\le C_C<∞$ uniformly in $\omega\in C$ and
$l\in\mathcal F_x$, because on such a window the residual transformations
and inverses range in a compact set, and their Jacobians against
$\eta_x$ are positive and bounded; at a finite place use analytic Haar
@@ -2375,26 +2375,26 @@ densities in finitely many compact-open charts.
For fixed $(x,l)$, define bounded increasing
$D_{n,x}(l)=i_x^*M_{\min(k_\omega(l),n)}i_x$. The limiting quadratic form
-$q_{x,l}(v)=\|M_{\sqrt{k_\omega(l)}}i_xv\|^2$ is closed. Its domain is dense
+$q_{x,l}(v)=\|M_{√(k_\omega(l))}i_xv\|^2$ is closed. Its domain is dense
for almost every $(x,l)$: choose $f_r$ with $Vf_r(x)$ total in $K_x$
almost everywhere (the measurable projection onto the fibrewise orthogonal
complement kills angular multiples of $V\mathcal H$, so vanishes by
minimality), then Tonelli and $\int k\,d\eta=1$ give
\[
- \int_{X_D}\int_{\mathcal F_x}q_{x,l}(Vf_r(x))\,d\eta_x(l)d\vartheta(x)
+ \int_{X_D}\int_{\mathcal F_x}q_{x,l}(Vf_r(x))\,d\eta_x(l)dθ(x)
=\|f_r\|^2.
\]
For the associated positive $D_x(l)$, measurability follows from the
bounded approximants, and $\int_{\mathcal F_x}q_{x,l}(v)\,d\eta_x(l)=\|v\|^2$
for $v\in K_x$. Consequently
-\begin{equation}
+
(Tf)(x,l)=D_x(l)^{1/2}(Vf)(x)
- \label{eq:links-density-isometry}
-\end{equation}
+
+
is a bounded isometry from $\mathcal H$ to
-$\mathcal R=\int^\oplus_{X_D}L^2(\mathcal F_x,\eta_x;K_x)d\vartheta(x)$.
+$\mathcal R=\int^ ⊕ _{X_D}L^2(\mathcal F_x,\eta_x;K_x)dθ(x)$.
-\emph{Equivariance.} For a target $\gamma$, set
+Equivariance. For a target $\gamma$, set
$r_{\gamma,x}(l')=d(\gamma_*\eta_x)/d\eta_{\gamma x}(l')$. Then
$k_{\gamma\omega}(\gamma l)=r_{\gamma,x}(\gamma l)k_\omega(l)$, and since
$i$ intertwines target lifts, the compressed form is
@@ -2411,17 +2411,17 @@ $U_s^X(M_uVf)=M_uVU_sf$; extend it to $\widetilde U_s$ on $\mathcal R$ by
its action on $K_x$ with induced base reindexing and identity on $l$. These
lifts use only the image marginal.
-\emph{The finite-trace intertwiner space.} Let $\mathscr E_0$ be the bounded
+The finite-trace intertwiner space. Let $\mathscr E_0$ be the bounded
intertwiners $\mathcal H\to\mathcal R$ for the base and target actions;
$B^*C\in\mathcal N$ for $B,C\in\mathscr E_0$, and
-$\langle B,C\rangle_{\mathscr E}=\tau_{\mathcal N}(B^*C)$ completes to
+$< B,C>_{\mathscr E}=\tau_{\mathcal N}(B^*C)$ completes to
$\mathscr E$ with
-\begin{equation}
+
\|B\|_{\mathscr E}^2=\frac1d\sum_{r=1}^k\|Bb_r\|^2,
\qquad
- \|Bf\|^2\le d\|f\|_\infty^2\|B\|_{\mathscr E}^2,
- \label{eq:links-intertwiner-frame}
-\end{equation}
+ \|Bf\|^2\le d\|f\|_∞^2\|B\|_{\mathscr E}^2,
+
+
the second by testing $B^*B$ against the positive matrix $ff^*$. Source
conjugation $\mathscr U_sB=\widetilde U_sBU_s^*$ is isometric because
$\sigma_s$ preserves $\tau_{\mathcal N}$, and base defects cancel since $B$
@@ -2430,12 +2430,12 @@ equality and the two source lifts. The representation has no $S$-invariant
vector: if it had, a bounded intertwiner whose projection onto the
invariant subspace is nonzero has orbit convex hull containing that
projection, and finite convex combinations $B_n$ with
-$\|B_n\|_{\mathrm{op}}\le\|B\|_{\mathrm{op}}$ converge in $\mathscr E$; the
+$\|B_n\|_{op}\le\|B\|_{op}$ converge in $\mathscr E$; the
bounded-column estimate gives strong convergence on bounded columns, and
-$B_n\to B_\infty$ with $B_\infty$ intertwining base and target and $S$-fixed.
-Then $u\mapsto B_\infty^*M_{u(l)}B_\infty$ is a nonzero positive full-flag
+$B_n\to B_∞$ with $B_∞$ intertwining base and target and $S$-fixed.
+Then $u\mapsto B_∞^*M_{u(l)}B_∞$ is a nonzero positive full-flag
map into $\mathcal A_j$, target-equivariant and $S$-invariant, contradicting
-Lemma~\ref{lem:links-obstruction}.
+Lemma~lem:links-obstruction.
Howe--Moore now gives weak decay of translates of $T$ along escape in $S$,
also when multiplied by commuting translations. Disintegrating into
@@ -2445,44 +2445,44 @@ matrix coefficients of the irreducible $S$-representations, tending to
zero uniformly over the multiplicity unitaries; dominated convergence
finishes.
-\emph{The positive compact-window comparison.} For a compact window
-$C\subset\Omega_D$, let $L_C=i^*M_{1_C}i$; it is decomposable and positive,
+The positive compact-window comparison. For a compact window
+$C\subsetΩ_D$, let $L_C=i^*M_{1_C}i$; it is decomposable and positive,
$V^*L_CV=\Psi(C)$. The density lower bound gives $D_x(l)\ge c_CL_C(x)$ in
form order, and operator monotonicity of square roots yields
-\begin{equation}
- D_x(l)^{1/2}\ge\sqrt{c_C}\,L_C(x).
- \label{eq:links-positive-window}
-\end{equation}
+
+ D_x(l)^{1/2}\ge√(c_C)\,L_C(x).
+
+
Write the extended path $a_tb_s$, inner $t$-limit first. The exact cocycle
formula and change of base $z=b_sy$ show the inner full-data law is
precomposed by $U_{b_s}$: $\Psi_s(F)=U_{b_s}^*\Psi(F)U_{b_s}$, with
unchanged image marginal. Set $T^{(s)}=\widetilde U_{b_s}^*TU_{b_s}$. The
marginal lift fixes $(x,l)$, so its fibre expression is
$(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X$ applied to $Vf$; inequality
-\eqref{eq:links-positive-window} persists. With $J_0f(x,l)=Vf(x)$,
-$B\mapsto\langle Bf,J_0f\rangle$ is continuous on $\mathscr E$ by
-\eqref{eq:links-intertwiner-frame}. Weak decay and the positive inequality
+(eq:links-positive-window) persists. With $J_0f(x,l)=Vf(x)$,
+$B\mapsto< Bf,J_0f>$ is continuous on $\mathscr E$ by
+(eq:links-intertwiner-frame). Weak decay and the positive inequality
give
-$0\le\sqrt{c_C}\langle f,\Psi_s(C)f\rangle\le\langle T^{(s)}f,J_0f\rangle\to0$;
+$0\le√(c_C)< f,\Psi_s(C)f>\le< T^{(s)}f,J_0f>\to0$;
density of bounded columns and $0\le\Psi_s(C)\le1$ extend the vanishing to
every input. This proves disappearance of the actual translated window
mass.
-\emph{The entire stratum contribution.} Each $\Psi_s$ has mass one on
-$\Omega_D$. Take its outer limit on $\overline{\Omega_D}$. Every continuous
-test compactly supported in $\Omega_D$ has zero limiting effect by the
+The entire stratum contribution. Each $\Psi_s$ has mass one on
+$Ω_D$. Take its outer limit on $Ω_Dbar$. Every continuous
+test compactly supported in $Ω_D$ has zero limiting effect by the
window estimate, so a countable exhaustion shows the outer law gives no
-mass to $\Omega_D$. Lemma~\ref{lem:links-matrix-data} places the remaining
-support in $\overline{\Omega_D}\setminus\Omega_D
-\subseteq\bigcup_{D'\supsetneq D}\Omega_{D'}$. Restoring $m_D$ proves the
+mass to $Ω_D$. Lemma~lem:links-matrix-data places the remaining
+support in $Ω_Dbar∖Ω_D
+⊆∪_{D'⊋ D}Ω_{D'}$. Restoring $m_D$ proves the
assertion for the entire positive contribution; further subpieces are
dominated by the positive total measurements.
-\end{proof}
-\subsection{Gap counts and vertex quotients}
-\begin{proposition}[Gap count and vertex quotients]
-\label{prop:links-gap-count}
+## Gap counts and vertex quotients
+
+[Gap count and vertex quotients]
+
In one measured copy, every positive stratum of a source path toward a face
of type $I$ has at most $|I|$ infinite gaps. A vertex path has exactly one
infinite gap. In a successive single-gap lex chain starting at a vertex,
@@ -2493,14 +2493,14 @@ For a quotient of a faster and a slower vertex translation in a common
apartment, there is exactly one target gap if the two source vertices are
coaxial (equal or opposite on the same translation axis), and at least two
if they are not.
-\end{proposition}
-\begin{proof}
+
+Proof.
Put $r=|\Pi|$ and start with a positive target stratum $D$ on a face of
type $I$. Append the $r-|I|$ missing source coordinates at successively
slower scales. At each stage the fundamental split vector for a missing
root has a nonzero projection to its remaining-Levi component, whose other
-split parts commute; so Lemma~\ref{lem:links-strict-growth} applies. If a
+split parts commute; so Lemma~lem:links-strict-growth applies. If a
full target stratum were reached before the source face were complete, the
obstruction in that lemma would already contradict. Otherwise the tracked
contribution passes through a strictly increasing chain of length
@@ -2508,7 +2508,7 @@ $r-|I|$; since $D$ has at most $r$ types, $|D|+r-|I|\le r$, proving
$|D|\le|I|$. If $I=\Pi$, this is the automatic ceiling and no
remaining-Levi obstruction is invoked.
-For a vertex, Lemma~\ref{lem:links-no-bounded-stratum} excludes zero gaps
+For a vertex, Lemma~lem:links-no-bounded-stratum excludes zero gaps
and the upper bound excludes two or more. The same strict-growth argument
applied successively from the vertex gives at least $k$ gaps at stage $k$,
while the upper bound gives at most $k$.
@@ -2518,32 +2518,32 @@ fundamental ray $v$ of type $j$. If the slower vertex is coaxial, the
quotient is again a vertex translation at the faster scale (possibly with
opposed orientation), and the vertex conclusion applies. If its vector $w$
is not coaxial, the projection to a simple factor of the remaining Levi is
-nonzero, since the common kernel of $\Pi\setminus\{j\}$ is $\mathbb Rv$.
+nonzero, since the common kernel of $\Pi∖\{j\}$ is $\mathbb Rv$.
Choosing a dominant chamber inside the remaining Levi, the slower quotient
component escapes in a noncompact simple factor with other split parts
commuting; strict growth from the one-gap faster limit forces at least one
additional target gap. All statements retain the faster-first order of
limits.
-\end{proof}
-\section{Transverse Fourier products and geometric transfer}
-\label{sec:transfer}
+
+# Transverse Fourier products and geometric transfer
+
We continue with the induced finite-module realization of
-Section~\ref{sec:links}. All target indices here are indices of a measured
-copy of $\Gamma$; the comparison group $\Lambda$ remains in the coefficient
-algebra, and no flag space carries an action of $\Lambda$. Our objective is
+Section~sec:links. All target indices here are indices of a measured
+copy of $\Gamma$; the comparison group $Λ$ remains in the coefficient
+algebra, and no flag space carries an action of $Λ$. Our objective is
a sharp flag measurement at each source flag, with an exact equation for
the residual projectivity on a missing panel.
-\subsection{A transverse product estimate}
+## A transverse product estimate
For a scalar function $f$ of the measured Fourier index, let $D_f$ denote
diagonal multiplication of the Fourier array; it is an $L^2$ operator, and
we do not assume it preserves operator-bounded columns. We estimate
-\begin{equation}
- (D_g B_0)(D_f A_0),\qquad\text{with index product }qp,
- \label{eq:input-2}
-\end{equation}
+
+ (D_g B_0)(D_f A_0),\qquadwith index product qp,
+
+
where $A_0,B_0$ are operator-bounded with Fourier expansions
$A_0=\sum_pa_pw_p$, $B_0=\sum_qb_qw_q$, operator-valued coefficients in the
unmeasured algebra, $p$ an index of $A_0$ and $q$ an index of $B_0$.
@@ -2553,17 +2553,17 @@ below uses the actual Hilbert norms. Rectangular matrices are included by
embedding in one fixed square matrix algebra and using the corresponding
corners.
-\begin{proposition}[Transverse products]
-\label{prop:transfer-transverse}
+[Transverse products]
+
Fix face types, a thickness bound, and sufficiently small compact
transverse face patches at the basepoint $o$. Suppose the supports of $f$
and $g$ lie within that thickness of the respective positive cones, with
all specified positive gaps sufficiently large, and that the ideal
directions of $po$ and $q^{-1}o$ belong to the fixed transverse patches.
Then $\mu(qp)=\mu(q)+\mu(p)+O(1)$, uniformly on the supports. The product
-in \eqref{eq:input-2} has a uniform tracial $L^2$ bound in either of the
+in (eq:input-2) has a uniform tracial $L^2$ bound in either of the
following situations:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item For each coordinate contributed by both cones, one of its two
supports is restricted to a moving interval of a fixed bounded width. The
separate cutoff factors may be arbitrary uniformly bounded functions.
@@ -2571,16 +2571,16 @@ separate cutoff factors may be arbitrary uniformly bounded functions.
smooth with uniformly bounded sufficiently high seminorms in fixed compact
normalized matrix charts; at finite places this means locally constant with
a uniformly bounded level.
-\end{enumerate}
+
Write $f=h_Af_1$, $g=h_Bg_1$. In case (i), all four factors may be
arbitrary uniformly bounded scalar functions; in case (ii), each factor
has the smooth seminorm bounds or fixed clopen level specified. Then
-\begin{equation}
+
\|(D_gB_0)(D_fA_0)\|_2^2
- \le C\|A_0\|_\infty\|B_0\|_\infty
+ \le C\|A_0\|_∞\|B_0\|_∞
\|D_{h_A}A_0\|_2\|D_{h_B}B_0\|_2.
- \label{eq:transfer-homogeneous}
-\end{equation}
+
+
The constant includes the bounds on the scalar factors. The same estimate
holds for a joint symbol $\sum_{q,p}h_B(q)h_A(p)m(q,p)b_qa_pw_qw_p$, where
$w_qw_p$ is the actual twisted monomial product and $m$ is a smooth joint
@@ -2589,11 +2589,11 @@ derivatives. Consequently an additional smooth cutoff factor whose
one-column Hilbert norm tends to zero makes the product tend to zero,
provided the other bounds and the other cut Hilbert norm remain bounded;
for a fixed finite trace of mass $t$,
-$\|(D_gB_0)(D_fA_0)\|_2\le C\sqrt t\|A_0\|_\infty\|B_0\|_\infty$.
-\end{proposition}
+$\|(D_gB_0)(D_fA_0)\|_2\le C\sqrt t\|A_0\|_∞\|B_0\|_∞$.
-\begin{proof}
-\emph{Common apartments and addition.} On compact transverse patches, the
+
+Proof.
+Common apartments and addition. On compact transverse patches, the
two ideal faces admit common apartment or flat sections with origins in a
bounded set. After erasing the bounded thickness, $po=z+A+O(1)$ and
$q^{-1}o=z-B+O(1)$ with $A,B$ positive and $z$ bounded, so the vector
@@ -2604,24 +2604,24 @@ ambient product and no discreteness of a proper projection.
We need Lipschitz stability of interpolation: from a fixed endpoint,
multiply the simple coordinates $L_i$ of a dominant vector by $t_i\in[0,1]$.
In an archimedean polar chart, the angular differential in a positive-root
-direction is multiplied by $\sinh(\alpha(t\cdot L))/\sinh(\alpha(L))\le1$
+direction is multiplied by $sinh(\alpha(t\cdot L))/sinh(\alpha(L))\le1$
and the radial differential by the bounded linear map $L\mapsto t\cdot L$;
a wall stabilizer fixes the interpolated vector since zero simple
coordinates stay zero. Approximation of paths by regular paths gives
continuity across walls; reversing endpoints uses opposition and the
complementary fractions. For finite-place buildings, the same statement is
-Lemma~\ref{lem:setup-building-interpolation}. The product of real and
+Lemma~lem:setup-building-interpolation. The product of real and
finite-place bounds covers the whole ambient space.
-\emph{The common shift.} Suppose $qp=q'p'$ and $k=p(p')^{-1}$. With
+The common shift. Suppose $qp=q'p'$ and $k=p(p')^{-1}$. With
primed coordinates $z'+A',z'-B'$, Cartan addition gives $A+B=A'+B'+O(1)$.
Interpolate from the negative endpoint with $t_i=B_i'/(A_i'+B_i')$ (any
choice in $[0,1]$ if the denominator vanishes); these return $z'$ in the
primed model, and Lipschitz equivariance gives
-\begin{equation}
+
ko=z+B'-B+O(1).
- \label{eq:transfer-common-shift}
-\end{equation}
+
+
The fractions are bounded and held fixed while applying the estimate.
In case (i), every overlapping coordinate has bounded variation on at
@@ -2635,7 +2635,7 @@ In case (ii), an unbounded difference can occur only in the single overlap
coordinate $i$; otherwise we are in the bounded-shift case. Put
$\delta=B_i'-B_i$. Then $A_i-A_i'=\delta+O(1)$ and
$p=kp'$, $q'=qk$, $(q')^{-1}=k^{-1}q^{-1}$. Applying
-\eqref{eq:transfer-common-shift} to both orientations gives
+(eq:transfer-common-shift) to both orientations gives
$ko=z+\delta e_i+O(1)$ and $k^{-1}o=z'-\delta e_i+O(1)$. With $U$ the
patch of type-$i$ vertices incident to the positive endpoint patch and $V$
the patch of type-$i^*$ vertices incident to the negative endpoint patch,
@@ -2661,8 +2661,8 @@ multiplication has uniformly bounded smooth derivatives there. At a
finite place, maximal-coordinate ratio charts give uniformly nonzero
denominators.
-\emph{Gram coefficients and twists.} Expand the squared norm in
-\eqref{eq:input-2} and group pairs with $qp=q'p'$ by $k=p(p')^{-1}$. After
+Gram coefficients and twists. Expand the squared norm in
+(eq:input-2) and group pairs with $qp=q'p'$ by $k=p(p')^{-1}$. After
trace cyclicity the summand is the pairing of two Gram coefficients, one
made from $a_{p'}a_p^*$ and the other from $b_q^*b_{q'}$ (adjoints changed
in the opposite orientation). The cocycle factors are exactly the factors
@@ -2671,72 +2671,72 @@ convolution by an absolute-value convolution. For bounded shifts, row and
column Cauchy--Schwarz bound each weighted Gram coefficient in operator
norm by the appropriate uncut column norm squared; its trace norm is
bounded by the relevant cut Hilbert norm squared, and tracial
-Cauchy--Schwarz gives \eqref{eq:transfer-homogeneous}.
+Cauchy--Schwarz gives (eq:transfer-homogeneous).
For unbounded shifts, if $p=kp'$ is the length-increased side, retain
-$\overline{h_A(p')}$ in $f(kp')\overline{f(p')}=
-\overline{h_A(p')}f(kp')\overline{f_1(p')}$ and expand the rest in the
+$h_A(p')bar$ in $f(kp')f(p')bar=
+h_A(p')barf(kp')f_1(p')bar$ and expand the rest in the
shorter argument $p'$. On the other side, retain $h_B(q)$ in
-$g(q)\overline{g(qk)}=h_B(q)g_1(q)\overline{g(qk)}$; its shorter geometric
+$g(q)g(qk)bar=h_B(q)g_1(q)g(qk)bar$; its shorter geometric
argument is $q^{-1}$. Conjugating Gram coefficients may conjugate these
formulas, with no effect on the cut Hilbert norm; the opposite orientation
interchanges primes, retaining $h_A(p)$ and $h_B(q')$. Extend with an
initial-patch cutoff with slack. For the first Gram sequence, choose
conjugated basis functions so its full weight has the form
-\begin{equation}
- w_k(r)=\sum_nc_n(k)\overline{h(r)u_n(r)},\qquad
- \|u_n\|_\infty\le1,\qquad
- \sum_n\sup_k|c_n(k)|<\infty.
- \label{eq:transfer-unary-expansion}
-\end{equation}
+
+ w_k(r)=\sum_nc_n(k)h(r)u_n(r)bar,\qquad
+ \|u_n\|_∞\le1,\qquad
+ \sum_n\sup_k|c_n(k)|<∞.
+
+
The strong order of sum and supremum follows from Fourier expansion on a
slightly larger real box with uniform derivative bounds; at finite places
a finite clopen mask basis replaces it. Minkowski, Plancherel, and the
Hilbert ideal inequality give
$\|(\sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k\|_{\ell^2(L^2)}
-\le(\sum_n\sup_k|c_n(k)|)\|A_0\|_\infty\|D_hA_0\|_2$.
+\le(\sum_n\sup_k|c_n(k)|)\|A_0\|_∞\|D_hA_0\|_2$.
The other side is symmetric. Cauchy--Schwarz in the shift variable then
-proves \eqref{eq:transfer-homogeneous}, and the small-factor assertion.
+proves (eq:transfer-homogeneous), and the small-factor assertion.
Smooth joint symbols follow from summable separated expansions retaining
the restrictions with slack; approximating bounded columns by uniformly
bounded Fourier polynomials with unary $L^1$ convergence gives the asserted
$L^2$ bound. No step requires operator-norm control of an angularly cut
column.
-\end{proof}
-\begin{remark}[Uniformity used below]
-\label{rem:transfer-uniformity}
+
+[Uniformity used below]
+
In case (i) the constant is insensitive to the location of the
bounded-width intervals and to arbitrary bounded separate masks. In case
(ii) all smooth seminorms and finite-place clopen levels are held fixed
during one use of the estimate. A gap threshold may be fixed first and
exhausted only after the relevant inner limit.
-\end{remark}
-\subsection{Fullness and the permutation of types}
+
+## Fullness and the permutation of types
For an archimedean target vertex module $V$, let $n(v)$ be a unit
representative of its highest line; $n(v)n(v)^*$ is independent of the
phase. We also allow the trivial one-dimensional module; at a finite target
place only this trivial-module version is used.
-\begin{lemma}[Fullness of open patches]
-\label{lem:transfer-fullness}
+[Fullness of open patches]
+
Let $\Phi_z$ be a nonzero family of image measurements as in
-Proposition~\ref{prop:links-link}, transported over its source label space
+Proposition~prop:links-link, transported over its source label space
and normalized by its positive scalar stratum mass to be unital. For every
nonempty open target patch $O$, the closed sum, over conull source labels
$z$, of the ranges of
-\begin{equation}
- \int_O n(v)n(v)^*\otimes\Phi_z(dv)
- \label{eq:transfer-line-effect}
-\end{equation}
+
+ \int_O n(v)n(v)^* ⊗ \Phi_z(dv)
+
+
is the full amplified input space. For the trivial module this is the
ordinary patch-fullness assertion; the assertion passes to normal copies
in the three-slot system.
-\end{lemma}
-\begin{proof}
+
+Proof.
Omit $O$ first. Source covariance makes the uncut range join a projection
over $(\mathcal A_1)^\sigma$; target covariance transports its range by
finite-dimensional linear target matrices, unchanged by positive
@@ -2748,8 +2748,8 @@ Let a positive matrix $A$ have weight spaces with strictly increasing
eigenvalues and let $P$ be a projection over a finite algebra. Filtering
$\ran P$ by its intersections with the weight filtration and taking the
successive leading projections onto the weight spaces gives, by polar
-decomposition, an orthogonal sum $Q$ with $\operatorname{tr}(Q)=
-\operatorname{tr}(P)$ and $\ran Q$ strongly approximated by $A^n\ran P$.
+decomposition, an orthogonal sum $Q$ with $tr(Q)=
+tr(P)$ and $\ran Q$ strongly approximated by $A^n\ran P$.
Since the range projections have constant finite trace, they converge in
trace $L^2$ to $Q$. Apply pointwise to the induced field; a convergent
orbit of a trace-isometric action is fixed (the distance between
@@ -2767,26 +2767,26 @@ the chosen frames along increasingly long translations, so suitable
translates of $O$ exhaust the measure. Applied to a faithful
scalarization of a countable total family, the complements of the
translated patch joins have trace tending to zero, giving patch fullness.
-Normal copies follow from normal ampliation in Proposition~\ref{prop:links-link}.
-\end{proof}
+Normal copies follow from normal ampliation in Proposition~prop:links-link.
+
-A normalized image branch of a pure vertex path is a \emph{candidate}. Its
+A normalized image branch of a pure vertex path is a candidate. Its
label is a source vertex, its values are target vertices. Different
candidates may use different paths, limits, and strata. Face-parabolic
invariance makes their transport well defined on the vertex space, and
strong continuity is used only for these constructed families.
-\begin{proposition}[Type comparison]
-\label{prop:transfer-types}
+[Type comparison]
+
There is a unique permutation $p$ of the full relative type list $\Pi$
such that every candidate at source type $i$ takes values in target type
$p(i)$; it preserves opposition, $p(i^*)=p(i)^*$. For noncoaxial source
vertices of dual types, all ordered products of candidates vanish on
transverse target patches. The permutation preserves the collection of
isolated types.
-\end{proposition}
-\begin{proof}
+
+Proof.
Choose paths $t,a$ whose inverses tend to the two source vertices in a
common apartment. Take successive limits, $t$ faster. Pairing the ordered
cut effects between bounded columns and using trace cyclicity gives a
@@ -2795,21 +2795,21 @@ after base reindexing; the latter is the transporter of $ta^{-1}$ up to a
common scalar, with index product $pq^{-1}$.
If coaxial, the quotient path has exactly one target gap
-(Proposition~\ref{prop:links-gap-count}). Nondual transverse target types
+(Proposition~prop:links-gap-count). Nondual transverse target types
would force two distinct large gaps. The bounded-overlap version of
-Proposition~\ref{prop:transfer-transverse} bounds the cut-product norm
+Proposition~prop:transfer-transverse bounds the cut-product norm
uniformly at fixed angular patches and thickness, and the localized
quotient matrix norm tends to zero on the forbidden strata by the finite
frame columns; Cauchy--Schwarz makes the pairing zero.
If noncoaxial, the quotient has at least two gaps. Dual transverse target
vertex cones put the cut product in a fixed thickened single-vertex cone;
-the smooth version of Proposition~\ref{prop:transfer-transverse} with
+the smooth version of Proposition~prop:transfer-transverse with
fixed smooth cuts again compares with a forbidden localized quotient norm
and gives zero.
Fullness gives nonzero ordered cut product for generic source pairs
-(apply Lemma~\ref{lem:transfer-fullness} successively). Generic dual
+(apply Lemma~lem:transfer-fullness successively). Generic dual
source vertices are opposite, hence coaxial; generic nondual source types
are noncoaxial. Thus candidates at dual source types have dual target
types, and at nondual source types nondual target types. If $T_i$ is the
@@ -2822,34 +2822,34 @@ continuity: if the image type were isolated, nonopposition would mean
equality; diagonal support of two unital effects would make them equal,
and gallery connectivity would make the candidate family constant, giving
a source-invariant target-equivariant flag probability against contraction
-(Lemma~\ref{lem:transfer-fullness}). Thus self-opposite higher-rank types
+(Lemma~lem:transfer-fullness). Thus self-opposite higher-rank types
do not map to isolated types; non-self-opposite ones do not either by
opposition preservation. Since $p$ is a permutation, it permutes the
isolated types.
-\end{proof}
-\subsection{Equality at isolated types}
+
+## Equality at isolated types
Fix an isolated archimedean source type and its matched target boundary
$Z$.
-\begin{lemma}[Disjoint angular forms]
-\label{lem:transfer-disjointness}
+[Disjoint angular forms]
+
For almost every independent pair of source labels, possibly from two
different candidate families at the fixed isolated type, there is no
common nonzero target-equivariant positive map dominated by the two
candidate maps (complete-positive domination). After induction, there is
no common nonzero linear or conjugate-linear functional on
-$C^\infty(Z)\odot\mathcal H$ bounded by both angular form norms
-\begin{equation}
- b\longmapsto\Bigl(\int_Z\langle b(v),\Phi_x(dv)b(v)\rangle\Bigr)^{1/2}.
- \label{eq:input-3}
-\end{equation}
+$C^∞(Z)\odot\mathcal H$ bounded by both angular form norms
+
+ b⟼(\int_Z< b(v),\Phi_x(dv)b(v)>)^{1/2}.
+
+
The generic conclusion holds throughout the transverse source orbit by
source covariance.
-\end{lemma}
-\begin{proof}
+
+Proof.
Undo source induction: simultaneous covariance yields measurable unital
CP maps $F_x:C(Z)\to B(H)$ with
$F_{\gamma x}=\pi_0(\gamma)F_x\pi_0(\gamma)^*$, equivariant for $R$; the
@@ -2874,9 +2874,9 @@ Fix the central support $z$ of one functional of the second family.
$z\mathfrak C^{**}$ has separable predual, and the components $\phi_x^z$
form a measurable norm-separable family: test domination on a countable
norm-dense positive family, let
-$n(x)=\sup\{\|\theta\|:0\le\theta\le\phi_x,\ \theta\text{ is }z\text{-normal}\}$
+$n(x)=\sup\{\|\theta\|:0\le\theta\le\phi_x,\ \theta is z-normal\}$
be upper semianalytic, and apply Jankov--von Neumann
-\cite{Kechris1995} to obtain measurable dominated $\theta_q(x)$ with
+[cite] to obtain measurable dominated $\theta_q(x)$ with
norm $>q$ on $\{n>q\}$; rational $q_m(x)<n(x)$ with $n(x)-q_m(x)<1/m$
gives $\|\phi_x^z-\theta_{q_m(x)}(x)\|<1/m$. No simultaneous measurable
choice of second-family central supports is needed.
@@ -2893,7 +2893,7 @@ For each fixed label of the second family, the two functionals are
disjoint a.e. Orthogonality is Borel (distance equals sum of norms), and
Fubini gives the first assertion for independent pairs.
-Reinduce and consider \eqref{eq:input-3}. A nonzero scalar functional
+Reinduce and consider (eq:input-3). A nonzero scalar functional
bounded by both gives the parallel sum of the two squared forms. Build it
as the quotient of the direct sum of the two completions by the span of
pairs $(h,-h)$ from the common algebraic domain. This subspace is reducing
@@ -2903,25 +2903,25 @@ gives a nonzero decomposable equivariant CP map dominated by both
original maps. Disintegration gives forbidden subordinate maps on a
positive-measure set of base points, contradiction. Source covariance
transports the generic conclusion over the single transverse orbit.
-\end{proof}
-\begin{remark}[A consequence for sharp measurements]
-\label{rem:transfer-disjoint-ranges}
-If the two measurements in Lemma~\ref{lem:transfer-disjointness} are
+
+[A consequence for sharp measurements]
+
+If the two measurements in Lemma~lem:transfer-disjointness are
sharp, the ranges $P_x(U)P_y(V)\mathcal H$ for disjoint angular patches
$U,V$ have dense sum. A vector orthogonal to all these ranges would,
by regular approximation, have the same spectral vector for the two
PVMs; pairing gives a common functional bounded by the two forms,
contrary to the lemma.
-\end{remark}
-\begin{proposition}[Sharpness at isolated types]
-\label{prop:transfer-isolated-sharpness}
+
+[Sharpness at isolated types]
+
At an isolated source vertex, every candidate is sharp, and all candidate
choices agree.
-\end{proposition}
-\begin{proof}
+
+Proof.
Take fast and slow paths to the same source vertex $x$; suppose two
angular effects on disjoint matched target patches have nonzero ordered
product. Choose smooth buffered angular cutoffs with bounded complementary
@@ -2931,11 +2931,11 @@ patch disjoint from that of $A$. The finite corner-frame identity and
exhaustion of fixed windows give bounded columns $f,f',b'$ for which the
conjugate-linear functional, on $b(v)=\sum_ih_i(v)b_i$ with smooth $h_i$
and bounded columns $b_i$, is the slow limit of
-\begin{equation}
- \sum_i\Bigl\langle D_{h_i(q^+)B_1}[b'(ay)^*X_a(y)f'(y)],\,
- b_i(ay)^*X_a(y)(Lf)(y)\Bigr\rangle.
- \label{eq:input-4}
-\end{equation}
+
+ \sum_i\Bigl< D_{h_i(q^+)B_1}[b'(ay)^*X_a(y)f'(y)],\,
+ b_i(ay)^*X_a(y)(Lf)(y)\Bigr>.
+
+
Here $q^+$ is the image of the ordinary index $q$ (the reverse angular
variable in the matched component), and smooth boundary tests on the
large-gap windows extend through the simple top singular line in the
@@ -2944,7 +2944,7 @@ normalized proximal matrix chart.
Bound it by the fast uncut form at $x$. At fixed slow time replace $Lf$
by its defining fast weak limit along $t$; the result pairs
$X_tX_a^*b_i(ay)$ with the transverse product of $D_AX_tf$ and the adjoint
-of the first column in \eqref{eq:input-4}, with indices $p,q^{-1}$.
+of the first column in (eq:input-4), with indices $p,q^{-1}$.
Transfer $h_i$ by evaluation at the image of $qp^{-1}\to q^+$. The
transfer error is uniform: normalized products are smooth with bounded
derivatives on the buffered patches, so replacing $q$'s normalized matrix
@@ -2952,7 +2952,7 @@ by its rank-one top truncation writes the difference of the two angular
tests as a finite sum of entries of the rank-one defect times symbols with
uniformly bounded seminorms. The slow cut column multiplied by this defect
tends to zero in Hilbert norm; the small-factor assertion of
-Proposition~\ref{prop:transfer-transverse} kills the product error. The
+Proposition~prop:transfer-transverse kills the product error. The
remaining cut product has a uniform Hilbert norm. The squared norm of the
summed pairing column has fast limit equal to the fast uncut angular form
on $\sum_ih_iU_a^*b_i$, which by parabolic invariance at $a$ equals the
@@ -2962,85 +2962,85 @@ Move the end columns by the reverse radical $n$ (invariance unchanged),
giving a bound by the fast form on $\sum_ih_iU_nb_i$, hence the form on $b$
at $n^{-1}x$. Opposite radical moves cover an open Bruhat cell. Choose
two generic moved labels there; the same nonzero functional is bounded by
-both forms, contradicting Lemma~\ref{lem:transfer-disjointness}.
+both forms, contradicting Lemma~lem:transfer-disjointness.
Thus candidate products at the same source vertex are supported on
equality of target values. For unital POVMs $E,F$ with this property,
regular approximation gives $E(S)F(T)=0$ for disjoint Borel $S,T$;
unitality gives $E(S)=E(S)F(S)=F(S)$, so $E(S)^2=E(S)$ and the candidates
are equal PVMs.
-\end{proof}
-\subsection{Incidence range containments}
+
+## Incidence range containments
Let $z\sim F$ mean $z$ is incident to $F$. Fix a positive stratum of a
mixed source face path or a pure vertex path $F$; denote its normalized
image-face marginal by $\Psi_F$ and the scalar mass of the unnormalized
stratum by $c_F>0$. For an open patch $O$ of target type $p(i)$, put
-$\mathcal I_i(O)=\{x:\text{the target face }x\text{ admits incident
-type-}p(i)\text{ vertex in }O\}$. For an archimedean highest-line module
+$\mathcal I_i(O)=\{x:the target face x admits incident
+type-p(i) vertex in O\}$. For an archimedean highest-line module
$V$, let $R_x$ be the orthogonal projector onto
-$\operatorname{span}\{n(v):v\sim x,\ \operatorname{type}(v)=p(i)\}$, and
+$span\{n(v):v\sim x,\ type(v)=p(i)\}$, and
define
-\begin{equation}
- L_z=\overline{\ran}\int n(v)n(v)^*\otimes\Phi_z(dv).
- \label{eq:transfer-Lz}
-\end{equation}
-\begin{proposition}[Incidence containment]
-\label{prop:transfer-incidence}
+ L_z=\ranbar\int n(v)n(v)^* ⊗ \Phi_z(dv).
+
+
+
+[Incidence containment]
+
The following inclusions hold:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item The support space of $\Psi_F(\mathcal I_i(O))$ is contained in
$\overline{\sum_{z\sim F}\ran\Phi_z(O)}$.
\item If the path is mixed and $p(i)$ is archimedean, in every even highest
-power the support of $\int R_x\otimes\Psi_F(dx)$ is contained in
+power the support of $\int R_x ⊗ \Psi_F(dx)$ is contained in
$\overline{\sum_{z\sim F}L_z}$.
-\end{enumerate}
+
For (i), a stronger individual-pair statement holds: on a compact
full-data window that sends a buffered transverse end patch $V_1$ into
-$O$, if $f\in\ker\Phi_z(O)$ and $b\in\overline{\ran}\Phi_z(V_1)$, then
-$E(\mathrm{window})J(\bar b,f)=0$. On a mixed path $f$ may be any Hilbert
+$O$, if $f\in\ker\Phi_z(O)$ and $b\in\ranbar\Phi_z(V_1)$, then
+$E(window)J(\bar b,f)=0$. On a mixed path $f$ may be any Hilbert
column; on a nonmixed path it is relatively bounded in the sense of
-Lemma~\ref{lem:links-relative-boundedness}. In both cases $b$ may be any
+Lemma~lem:links-relative-boundedness. In both cases $b$ may be any
vector in the stated closed end range.
-\end{proposition}
-\begin{proof}
-We prove (i) by annihilating the common kernel $\mathcal C=\bigcap_{z\sim
+
+Proof.
+We prove (i) by annihilating the common kernel $\mathcal C=∩_{z\sim
F}\ker\Phi_z(O)$. For mixed paths take any $f\in\mathcal C$ and
operator-bounded $f_l\to f$. For nonmixed paths supported in $G_0$ with
$N_0=\mathcal N^{G_0}$: if $i$ is in another component, ordinary fullness
gives $\mathcal C=0$ and the inclusion is immediate; otherwise
-Lemma~\ref{lem:links-relative-boundedness}(ii) makes $\mathcal C$
+Lemma~lem:links-relative-boundedness(ii) makes $\mathcal C$
$N_0$-invariant and part (iii) gives $f_l\to f$ with
$E_{N_0}(f_lf_l^*)\le E_{N_0}(ff^*)$; in either case
-$\langle f_l,\Phi_z(O)f_l\rangle\to0$ for each $z\sim F$.
+$< f_l,\Phi_z(O)f_l>\to0$ for each $z\sim F$.
Fix a compact window sending a buffered transverse $V_1$ into $O$,
$z\sim F$, and an operator-bounded $b_0$. A vertex path $s^{-1}$ toward
$z$ with a smooth angular cut in $V_1$ and fixed complementary thickness
realizes its cut end effect by an auxiliary weak limit; exhausting cuts
and varying $b_0$ gives a total set in
-$\overline{\ran}\Phi_z(V_1)$. For fixed main time $a$ and index $l$,
+$\ranbar\Phi_z(V_1)$. For fixed main time $a$ and index $l$,
insert the auxiliary $s$-limit: cyclicity pairs
$X_s(ay)X_a(y)f_l(y)$ with the product of cut $X_s(ay)b_0(ay)$ and
$b''(ay)^*X_a(y)f''(y)$. If $q,h$ are the two product indices, the
observed inverse is $h^{-1}q^{-1}$; transversality and the buffer put its
image in $O$ once the auxiliary gap is large, so the pairing column has
-norm at most $\langle f_l,\Phi_z(O)f_l\rangle^{1/2}$. The main split
+norm at most $< f_l,\Phi_z(O)f_l>^{1/2}$. The main split
translation fixes $z$ (trivially on Levi chamber refinements incident to
-$F$), so the bound is independent of main time. Proposition~\ref{prop:transfer-transverse}
+$F$), so the bound is independent of main time. Proposition~prop:transfer-transverse
gives a uniform Hilbert bound on the other cut product depending only on
-the fixed data, not on $\|f_l\|_\infty$. Take the main-link limits with $l$
-fixed, then $l\to\infty$; on mixed paths the tensor isometry, on nonmixed
+the fixed data, not on $\|f_l\|_∞$. Take the main-link limits with $l$
+fixed, then $l\to∞$; on mixed paths the tensor isometry, on nonmixed
paths the common conditional expectation bound, justifies the passage.
Keep $f$ fixed, exhaust auxiliary cuts, and take the closed span of the
resulting end ranges. Continuity in the entire Hilbert end variable
-(mixed: tensor isometry; nonmixed: Lemma~\ref{lem:links-relative-boundedness})
-gives $E(\mathrm{window})J(\bar b,f)=0$ for all
-$b\in\overline{\ran}\Phi_z(V_1)$.
+(mixed: tensor isometry; nonmixed: Lemma~lem:links-relative-boundedness)
+gives $E(window)J(\bar b,f)=0$ for all
+$b\in\ranbar\Phi_z(V_1)$.
Move the end occurrence also by the independent reverse radical. As
$z\sim F$ and these moves vary, the resulting labels cover a conull
@@ -3051,8 +3051,8 @@ For the frame step: exhaust the compact windows to the event $A$ of the
chosen gap stratum and image condition $\mathcal I_i(O)$; monotone
continuity gives $E(A)J(\bar b,f)=0$. Then $A$ is invariant under ignored
left Fourier shifts, and the frame identity
-$c_F\langle f,\Psi_F(\mathcal I_i(O))f\rangle=\frac1d\sum_r
-\|E(A)J(\overline{b_r},f)\|^2=0$ (with $b_r$ the constant frame of $e$)
+$c_F< f,\Psi_F(\mathcal I_i(O))f>=\frac1d\sum_r
+\|E(A)J(b_rbar,f)\|^2=0$ (with $b_r$ the constant frame of $e$)
combined with positivity and density gives (i). Arbitrary Borel output
subprojections are imposed after buffered tests.
@@ -3060,10 +3060,10 @@ For (ii), write $S(h)$ for the Hilbert-normalized real module matrix at
$h^{-1}$. If $(f_i)\perp L_z$ and $(b_j)$ lies in the amplified line
range on a patch transverse to the output reverse window, the same
computation gives
-\begin{equation}
- E(\mathrm{window})\sum_{ij}E(S_{ij}(h))J(\overline{b_j},f_i)=0.
- \label{eq:input-5}
-\end{equation}
+
+ E(window)\sum_{ij}E(S_{ij}(h))J(b_jbar,f_i)=0.
+
+
The auxiliary fast symbol is $S(q)S(q)^*$ with cutoffs, and
$S(h)S(q)S(q)^*=\rho(h,q)S(qh)S(q)^*$ with $\rho$ and derivatives bounded
on transverse supports. Move $S(qh)^*$ to the full pairing column; its
@@ -3072,16 +3072,16 @@ defining $L_z$, with the same stabilizer translation, so it vanishes on
the approximating kernel input. The reverse radical may again move the
end; amplified fullness makes the resulting end columns span the whole
end space. The surviving top block of $S(h)$ has range containing every
-incident highest line (Lemma~\ref{lem:links-matrix-data}); bounded local
+incident highest line (Lemma~lem:links-matrix-data); bounded local
right inverses or finite local spanning columns transfer the annihilation
-in \eqref{eq:input-5} to $R_x$. Compressing with mixed input independence
-from Proposition~\ref{prop:links-link} proves (ii).
-\end{proof}
+in (eq:input-5) to $R_x$. Compressing with mixed input independence
+from Proposition~prop:links-link proves (ii).
+
+
+## Sharp flags and Coxeter compatibility
-\subsection{Sharp flags and Coxeter compatibility}
+[Propagation of incidence words]
-\begin{lemma}[Propagation of incidence words]
-\label{lem:transfer-incidence-words}
Consider a finite word of operations, each replacing either one of a pair
of vertices by an incident vertex of a prescribed type, and ending with
dual types. The word is successful if some sequence of choices ends at
@@ -3090,14 +3090,14 @@ If a word fails at a source pair, the ordered candidate product at that
pair is supported on target pairs for which the permuted word fails. The
sets of words successful at the least position are unchanged by the type
permutation $p$.
-\end{lemma}
-\begin{proof}
+
+Proof.
The base failure rule is nonopposition of dual types
-(Proposition~\ref{prop:transfer-types}); at equal self-opposite vertices
+(Proposition~prop:transfer-types); at equal self-opposite vertices
it follows by continuity. Suppose all incident replacements of one or both
source vertices satisfy a previously proved failure rule. If an incident
-target pair violates it, proposition~\ref{prop:transfer-incidence} puts
+target pair violates it, proposition~prop:transfer-incidence puts
the corresponding output ranges inside sums of the incident source-label
ranges, which the prior rule makes orthogonal; local homogeneous sections
ensure nearby faces still admit incident values in chosen neighborhoods.
@@ -3108,7 +3108,7 @@ maximal parabolic Weyl subgroup; an incidence move permits the successive
maximal-parabolic blocks. Maximal attainable Bruhat positions are the
Demazure maxima, increasing in the initial position by the Coxeter subword
rule. Ending opposite at dual types means reaching the longest chamber
-position; parabolic incidence \cite{Borel1991,Tits1964} gives monotonicity
+position; parabolic incidence [cite] gives monotonicity
of success.
If a word is successful at the least target position it is successful at
@@ -3117,15 +3117,15 @@ source pair, failure propagation would force the unital ordered source
product to have empty support, impossible. This gives inclusion of
success languages in one direction; iterating $p$ (finite order) closes
the cyclic chain.
-\end{proof}
-\begin{lemma}[An inequality-detecting word]
-\label{lem:transfer-inequality-word}
+
+[An inequality-detecting word]
+
Every higher-rank component has a vertex type $j$ and an incidence word
that fails exactly when two initial type-$j$ vertices are equal.
-\end{lemma}
-\begin{proof}
+
+Proof.
For type $A$, take an end node and move the second vertex to an incident
vertex of the opposite end type; success is equivalent to a hyperplane in
the second point's position avoiding the first. For other crystallographic
@@ -3136,11 +3136,11 @@ $s_j$ factor in a reduced longest word, $r-1$ round trips cannot end
opposite from equality, and $r$ can from the least nontrivial double
coset $W_{\widehat j}s_jW_{\widehat j}$. Every unequal position is above
that least nontrivial double coset, so success at all unequal pairs
-follows from Lemma~\ref{lem:transfer-incidence-words}.
-\end{proof}
+follows from Lemma~lem:transfer-incidence-words.
+
+
+[Sharp flag system]
-\begin{proposition}[Sharp flag system]
-\label{prop:transfer-sharp-flags}
All candidates are sharp and independent of candidate choice. For each
source vertex $z$, face $F$, and chamber $C$ write the resulting PVMs as
$P_z,P_F,P_C$. Incident source vertex measurements commute and are
@@ -3148,26 +3148,26 @@ supported on incidence of their target values. Every lexicographic or
mixed face path to $F$ observes exactly the vertices of $p(F)$ with these
measurements. Incident source labels commute across the two normal
measured copies.
-\end{proposition}
-\begin{proof}
-Lemma~\ref{lem:transfer-inequality-word} and its permuted inverse give
+
+Proof.
+Lemma~lem:transfer-inequality-word and its permuted inverse give
failure of the corresponding word at equality of source
$p^{-1}(j)$, so every ordered product of two candidates at that source
label is supported on the target diagonal. The unital-POVM argument
-(Proposition~\ref{prop:transfer-isolated-sharpness}) makes them equal
+(Proposition~prop:transfer-isolated-sharpness) makes them equal
projections, supplying a sharp type in every target component.
Extending a vertex path lexicographically to a full chamber preserves the
marginal. If a sharp vertex label is incident to that chamber,
-Proposition~\ref{prop:transfer-incidence} puts the marginal
+Proposition~prop:transfer-incidence puts the marginal
ranges inside the sharp PVM's ranges; unitality forces equality. The
tested candidate commutes with the sharp incident measurement, and their
ordered product is supported on target incidence.
For remaining target types in the sharp component, take two candidates at
the same source label with different incidence families of the sharp
-type; Proposition~\ref{prop:transfer-incidence} puts the ranges near one
+type; Proposition~prop:transfer-incidence puts the ranges near one
vertex in the sum of sharp incident ranges, orthogonal to the ranges near
the other. Their self-products vanish where families disagree. Equal
families must then determine a unique vertex: the stabilizer of the
@@ -3176,13 +3176,13 @@ exactly the vertex parabolic. Diagonal support gives sharpness and
uniqueness, component by component.
For a general mixed face path of type $I$: the type-$i$ source vertex $z_i$
-incident to $F$ is unique for $i\in I$. Proposition~\ref{prop:transfer-incidence}(i)
+incident to $F$ is unique for $i\in I$. Proposition~prop:transfer-incidence(i)
puts $\Psi_F(\mathcal I_i(O))$ inside $P_{z_i}(O)$. A target face omitting
$p(i)$ admits two distinct incident vertices of that type, so disjoint
$O,O'$ make $\Psi_F(\mathcal I_i(O)\cap\mathcal I_i(O'))$ supported in
both $P_{z_i}(O)$ and $P_{z_i}(O')$, hence zero. Countably many such
pairs exhaust the omitted-type event, so every $p(i)$ for $i\in I$
-occurs. The upper bound of Proposition~\ref{prop:links-gap-count} forces
+occurs. The upper bound of Proposition~prop:links-gap-count forces
the stratum type to equal $p(I)$, with marginals $P_{z_i}$; sharp
marginals put the joint measurement in their common multiplicative domain,
giving $P_F$. Lexicographic path marginals are uniquely identified and
@@ -3191,19 +3191,19 @@ marginal invariant under both the source face parabolic of $F$ and its
own parabolic would by normalization give a source-invariant boundary
measurement, contradicting contraction). In the three-slot system, the
two normal copies are marginals of one common dilation and commute.
-\end{proof}
-\begin{proposition}[Coxeter compatibility and ordered galleries]
-\label{prop:transfer-galleries}
+
+[Coxeter compatibility and ordered galleries]
+
The permutation $p$ preserves the Coxeter matrix. Along a source chamber
gallery, the ordered product of the chamber PVMs is supported on target
galleries of the permuted type, allowing stutters. Smooth functions
vanishing on that target gallery manifold annihilate the ordered product,
with arbitrary smooth multipliers; at finite places the same holds for
locally constant tests by compact clopen partitions.
-\end{proposition}
-\begin{proof}
+
+Proof.
Adjacent source chambers share their sharp common panel; inserting its
projection between consecutive spectral slots makes differences of the
two common-panel coordinates annihilate the product. The common-panel
@@ -3219,9 +3219,9 @@ Coxeter length, hence the permuted word is reduced longest and no stutter
is possible. An alternating rank-two word of length $m_{ij}$ is reduced
and extends to a reduced longest word; its permuted segment is reduced, so
$m_{ij}\le m_{p(i)p(j)}$; cycling the finite permutation gives equality.
-\end{proof}
-\subsection{Exact residue links}
+
+## Exact residue links
Take a mixed lex face path missing one type in a higher-rank component (or
one type in each of several). Fix one such component; its source rank-one
@@ -3236,39 +3236,39 @@ transports, identify the reference opposite panels by transverse
projection; the residual output coordinate $g$ is then a rank-one target
projectivity from the end pencil to the start pencil.
-\begin{proposition}[Residue graph equation]
-\label{prop:transfer-residue-graph}
+[Residue graph equation]
+
On compact output windows the mixed link satisfies
-\begin{equation}
- E(dg)J\bigl(\overline{Q_z}(dq)\otimes P_z(du)\bigr)
- \quad\text{is supported on}\quad u=gq.
- \label{eq:input-6}
-\end{equation}
+
+ E(dg)J(Q_zbar(dq) ⊗ P_z(du))
+ \quadis supported on\quad u=gq.
+
+
The equation means annihilation by smooth joint tests vanishing on the
graph, with locally constant tests in finite-place angular variables;
conjugate spectral measures conjugate projections. It holds with bounded
Borel output parameters after the angular identities are established. The
joint law is invariant under simultaneous source Levi transport, and
exact input translations may be made before any further limit.
-\end{proposition}
-\begin{proof}
+
+Proof.
Work in an archimedean component. On the ordinary-end Hilbert space use
the slots in order: reverse panel, end missing vertex at $z$, observation
at the start label (possibly moved by the reverse radical), with value
$v$ in the last slot. A source gallery shortened from the longest by the
missing reflection connects the matched chambers; its target gallery has
-the same shortened type (Proposition~\ref{prop:transfer-galleries}). The
+the same shortened type (Proposition~prop:transfer-galleries). The
shortened Weyl position $s_iw_0=w_0s_{i^*}$ has all other descents, so
parabolic saturation cannot reach $w_0$; the missing vertices are
nonopposite. A vertex transverse to a panel of dual missing type has
exactly one incident completion whose missing vertex is nonopposite (the
ordinary rank-one parabolic projection), so the ordered gallery equation
-says the end missing vertex is $\operatorname{pr}(v)$ on a transverse
+says the end missing vertex is $pr(v)$ on a transverse
window; integrating intermediate slots by identity projections preserves
the equation.
-Use \eqref{eq:input-5} at $z$, moving only its end occurrence by the
+Use (eq:input-5) at $z$, moving only its end occurrence by the
reverse radical. Sharpness permits the start column to be
$(1-n(P_z)n(P_z)^*)$ applied to an arbitrary column and the end column to
be the moved line projector; the link is annihilated by
@@ -3276,7 +3276,7 @@ $(1-n(u)n(u)^*)S(h)n(v)$. The two angular inputs commute (separate
tensor slots); separation extends to arbitrary smooth joint angular
multipliers. On the stratum the matrix action is the residual projectivity
after transverse projection, so the equations are the line graph
-$u=g\operatorname{pr}(v)$, which generates its smooth vanishing ideal on
+$u=gpr(v)$, which generates its smooth vanishing ideal on
compact transverse windows (the highest-line embedding is immersive).
Output coordinates may be Borel parameters with locally uniform angular
derivative bounds.
@@ -3285,20 +3285,20 @@ Combine the two steps in order. Preapply at the end a reverse-panel cut
followed by a moved-vertex cut on a buffered transverse rectangle; leave
the start vector arbitrary. Insert the $Q_z$ slit next to the reverse
panel (they commute); the shortened gallery replaces its value by
-$\operatorname{pr}(v)$. Integrate out that slot; pull the sharp
-reverse-panel projection through $J$ (Proposition~\ref{prop:links-link});
-the line equation gives $u=g\operatorname{pr}(v)=gq$. This proves
-\eqref{eq:input-6} on those ordered ranges; no discontinuous joint symbol
+$pr(v)$. Integrate out that slot; pull the sharp
+reverse-panel projection through $J$ (Proposition~prop:links-link);
+the line equation gives $u=gpr(v)=gq$. This proves
+(eq:input-6) on those ordered ranges; no discontinuous joint symbol
has been applied to noncommuting end measurements.
The ordered ranges are total: their closed join is target invariant, fixed
by a split source subgroup escaping in every ambient component, so scalar
-mass (Proposition~\ref{prop:links-link}). Varying $z$ gives nonzero ranges
+mass (Proposition~prop:links-link). Varying $z$ gives nonzero ranges
(visible start label, patch fullness); Levi transitivity gives nonzeroness
for every $z$. Hence the join is the whole end space.
For a finite target place, replace the Hilbert line argument by the
-individual-pair assertion of Proposition~\ref{prop:transfer-incidence} on
+individual-pair assertion of Proposition~prop:transfer-incidence on
scalar clopen patches. The shortened gallery substitution holds by
clopen partitions. The same total range argument applies; no complex
Hilbert input is tensored with a nonarchimedean representation.
@@ -3308,17 +3308,17 @@ summable smooth separated expansions in real slots. Finally, exact cocycle
composition (made before limiting) shows simultaneous source Levi action
changes the array only by base reindexing and a common scalar phase, so
tested forms are invariant. If a genuine action is needed, use the
-conjugate-multiplier regular amplification (Section~\ref{sec:links}); it
+conjugate-multiplier regular amplification (Section~sec:links); it
tensors by identity and cancels the phase defect without altering any
finite-trace mixing assertion.
-\end{proof}
-\begin{corollary}[Fibrewise links and polynomial fullness]
-\label{cor:transfer-fibre-links}
+
+[Fibrewise links and polynomial fullness]
+
Disintegrate over the two sharp endpoint faces with genuine source
amplifications if necessary. For product-almost every pair of endpoint
-fibres, the mixed isometry \eqref{eq:input-1}, the graph equation
-\eqref{eq:input-6}, simultaneous source Levi invariance, and exact input
+fibres, the mixed isometry (eq:input-1), the graph equation
+(eq:input-6), simultaneous source Levi invariance, and exact input
translations hold. On almost every individual fibre, for every even
highest power in an archimedean target module, the line ranges over $z$
span at least the Hilbert fibre tensored with the span of all ordinary
@@ -3327,9 +3327,9 @@ with linear Hilbert-vector coefficients that holds spectrally at every
residue label holds at every ordinary residue value; the assertion
includes algebraic subspace membership expressed by Pluecker equations
and chartwise relations after the denominator procedure.
-\end{corollary}
-\begin{proof}
+
+Proof.
Sharp face marginals intertwine the link dilation, so its spectral span
disintegrates over the product of the two endpoint spectral measures; the
strongly continuous source Levi representations commute with those PVMs
@@ -3349,21 +3349,21 @@ directly; no countable norm-dense class of all bounded Borel functions is
used.
Disintegrate the amplified incidence containment of
-Proposition~\ref{prop:transfer-incidence} on the same family; on almost
+Proposition~prop:transfer-incidence on the same family; on almost
every fibre the span contains the Hilbert fibre tensored with the span of
all ordinary incident highest lines. A polynomial vanishing on spectral
line ranges is orthogonal to their span, hence to each ordinary incident
highest line; norm powers homogenize different even degrees.
For a relation known only on a chart $\{\Delta\ne0\}$, clear a common
-homogeneous denominator and call the numerator $F_\mathrm{num}$; global
-polynomial fullness applied to $\Delta F_\mathrm{num}$ vanishes spectrally
+homogeneous denominator and call the numerator $F_num$; global
+polynomial fullness applied to $\Delta F_num$ vanishes spectrally
on the chart by the relation and on $\{\Delta=0\}$ by the factor; divide
by $\Delta$ on the chart. Pluecker equations and common local
denominators place algebraic membership tests in this form.
-\end{proof}
-\section{Finite-place panels: fixed algebras and uniform upper speed}
-\label{sec:finite}
+
+# Finite-place panels: fixed algebras and uniform upper speed
+
Take a mixed lexicographic face path missing one type in a non-isolated
component, or one type in each of several. Write $B_S,B_T$ for the source
@@ -3383,14 +3383,14 @@ goal is a fixed homogeneous analytic homeomorphism $\tau:B_S\to B_T$ and
measurable target projectivity observations $a,b$ so that the endpoint
families are evaluations of $z\mapsto a\tau(z)$ and $z\mapsto b\tau(z)$.
-\subsection{The paired endpoint algebra}
+## The paired endpoint algebra
For a locally constant scalar $f$ on $B_T$, define the paired
-decomposable operator on $L^2(T;H')\oplus L^2(T;I)$ by
-\begin{equation}
- F_{z,f}(t)=\bigl(f(tP_z),f(tQ_z)\bigr),\qquad t\in T,
- \label{eq:finite-generators}
-\end{equation}
+decomposable operator on $L^2(T;H') ⊕ L^2(T;I)$ by
+
+ F_{z,f}(t)=(f(tP_z),f(tQ_z)),\qquad t\in T,
+
+
with spectral evaluation $f(tP_z)=\int f(tu)P_z(du)$ and similarly at
the end. Let $D_*$ be the von Neumann algebra generated by these over a
countable dense set of labels and clopen tests; $D_*$ has separable
@@ -3398,30 +3398,30 @@ predual.
In the intended model $f$ would be $f(ta\tau(z))$, with frame/projectivity
entering only via $x=ta$. The result below says the actual paired
-algebra is exactly $L^\infty(T)$ with left translation, establishing
+algebra is exactly $L^∞(T)$ with left translation, establishing
commutation of the endpoint families and the homogeneous form.
-\begin{proposition}[The fixed algebra]
-\label{prop:finite-frame}
+[The fixed algebra]
+
For a sufficiently small compact-open $U<T$, the fixed algebra
$D_*^U$ is atomic (a countable product of finite matrix algebras), and
$D_*^T=\mathbb C1$.
-\end{proposition}
-\begin{proof}
+
+Proof.
A $U$-fixed decomposable operator is essentially constant on left
cosets, so there are normal evaluations $\pi_t^+(d)=d_t^+$,
$\pi_r^-(d)=d_r^-$ for $Ut,Ur\in U\backslash T$, jointly faithful on the
countable set $U\backslash T$. Identify
-$\overline I\otimes H'\cong\operatorname{HS}(I,H')$ via
-$\overline\xi\otimes\eta\mapsto|\eta\rangle\langle\xi|$. For fixed
-cosets $Ut,Ur$, put $\Omega_{t,r}=\{g:Utg=Ur\}$ and
-$K=J^*E(\Omega_{t,r})\zeta\in\operatorname{HS}(I,H')$. Then for every
+$\overline I ⊗ H'\congHS(I,H')$ via
+$\overline\xi ⊗ \eta\mapsto|\eta><\xi|$. For fixed
+cosets $Ut,Ur$, put $Ω_{t,r}=\{g:Utg=Ur\}$ and
+$K=J^*E(Ω_{t,r})\zeta\inHS(I,H')$. Then for every
$d\in D_*^U$,
-\begin{equation}
+
d_t^+K=Kd_r^-.
- \label{eq:finite-hs-intertwining}
-\end{equation}
+
+
Varying $\zeta$, these intertwiners force the fibres $d_t^+$ to be graded
by the normal representation types: each Schur subspace remains
finite-dimensional per coset, giving a countable direct sum of finite
@@ -3429,16 +3429,16 @@ matrix algebras. Ergodicity of $T$ acting transitively on $U\backslash
T$ forces $D_*^T=\mathbb C1$: any $T$-fixed normal functional on the
countable Cartesian product vanishes off the identity coset, and the
resulting character is scalar.
-\end{proof}
-\begin{corollary}[Separation of archimedean types]
-\label{cor:finite-arch-separation}
+
+[Separation of archimedean types]
+
A connected noncompact simple real source cannot be matched to a
finite-place target. Hence the type permutation preserves both the
archimedean and nonarchimedean subsets.
-\end{corollary}
-\begin{proof}
+
+Proof.
A continuous action of a connected group fixes every central atom of an
atomic von Neumann algebra (a normal state supported on one atom takes
values zero and one on its translates, so the atom has open stabilizer).
@@ -3450,12 +3450,12 @@ the source compact); a homomorphism with zero differential from a
connected Lie group is trivial. Thus the source would fix every $D_*^U$
pointwise. As compact open $U$ decreases to the identity, the normal
averages $\int_U\alpha_u(d)du$ converge ultraweakly to $d$, so the ranges
-generate $D_*$; contradiction to Proposition~\ref{prop:finite-frame}.
+generate $D_*$; contradiction to Proposition~prop:finite-frame.
Finiteness and bijectivity of the type permutation give the symmetric
assertion.
-\end{proof}
-\subsection{Corner reduction and finite blocks}
+
+## Corner reduction and finite blocks
Both panel groups are now nonarchimedean. Denote the residue
characteristics by $p_S,p_T$. Choose small torsion-free logarithmic
@@ -3466,30 +3466,30 @@ has open stabilizer by the normal-state test). The image of that compact
profinite stabilizer in the automorphism group of its finite matrix block
is profinite and Chinese-Lie-Lie closed, hence finite; take a smaller
$V$ inside its open kernel. Put
-\begin{equation}
+
D_0=eD_*e,\qquad D_0^U=\mathbb Ce.
- \label{eq:finite-corner}
-\end{equation}
-Choose decreasing compact open normal subgroups $U_n\triangleleft U$ with
+
+
+Choose decreasing compact open normal subgroups $U_n△left U$ with
intersection $\{1\}$; then $D_n=D_0^{U_n}$ is finite-dimensional, since
an orthogonal family of nonzero projections has at most $[U:U_n]$ members
(their average mass forces $\lambda_a\ge[U:U_n]^{-1}$).
-\begin{lemma}[Block stabilizers]
-\label{lem:finite-blocks}
+[Block stabilizers]
+
The simple blocks of $D_n$ form a transitive $U$-set; the stabilizer
$L_n<U$ of a chosen block acts on it by an irreducible projective unitary
representation. If $G_n<V\times U$ is the stabilizer of that block for
the commuting source and target actions, then $G_n$ is closed, its
projection onto $V$ is surjective, and
-\begin{equation}
- [K_n,K_n]\ \text{acts trivially on}\ D_n,\qquad
+
+ [K_n,K_n]\ acts trivially on\ D_n,\qquad
K_n=\{v\in V:(v,1)\in G_n\}.
- \label{eq:finite-block-commutators}
-\end{equation}
-\end{lemma}
+
+
-\begin{proof}
+
+Proof.
Ergodicity of $U/U_n$ on $D_n$ makes its center $U$-transitive. The
source $V$ commutes with $D_*$ and acts by $\sigma$-conjugation, so it
normalizes $G_n$. For each $v\in V$, surjectivity onto $V$ follows by
@@ -3497,25 +3497,25 @@ choosing a block $b$, then a conjugate $\sigma_v(b)$ at the vacuous level.
Commutators $[K_n,K_n]$ commute both with the action and with $v$; the
source and target normal subgroups meet in the identity because the
target $U$ is torsion-free, so their nonparallel action kills the block.
-\end{proof}
+
Let $H$ be any closed subgroup of the fixed product $V\times U$ with
fixed faithful multiplicative coordinates. The following uniform lemma
controls its logarithms.
-\begin{lemma}[Uniform logarithm saturation]
-\label{lem:finite-uniform-log}
+[Uniform logarithm saturation]
+
There is a constant $c$, depending only on $V\times U$, such that for
every closed subgroup $H\subset V\times U$,
-\begin{equation}
+
\exp(p^c(\Lie H\cap M_H))\subset H,
- \label{eq:finite-uniform-log}
-\end{equation}
+
+
where $M_H=\log H$ in the chosen coordinates and the image is a fixed
small ball.
-\end{lemma}
-\begin{proof}
+
+Proof.
Use the Jordan--Holder filtration of the prime-field Lie algebra in the
given coordinates, and write $\exp(x)$ for the resulting truncated series
on $p^c$. By Cayley--Hamilton on the integer action lattice, the relevant
@@ -3523,16 +3523,16 @@ partial matrices stay integral, so every element of $p^c(\Lie H\cap M_H)$
arises from a closed-subgroup element up to a bounded shift, saturated by
the standard BCH/Dynkin estimates at the fine $p$-adic scale. Details are
as in the original.
-\end{proof}
-\begin{lemma}[Limiting source kernel]
-\label{lem:finite-limit-kernel}
+
+[Limiting source kernel]
+
The source $V$-action on $D_0$ has zero Lie-algebra kernel: for every
open subgroup $V_1<V$ acting trivially on $D_0$, we have $V_1=\{1\}$.
-\end{lemma}
-\begin{proof}
-As $n\to\infty$, $D_n$ exhausts $D_0$ ultraweakly (normal averaging and
+
+Proof.
+As $n\to∞$, $D_n$ exhausts $D_0$ ultraweakly (normal averaging and
strong continuity). Since $V$ is pro-$p$, any nontrivial open kernel
contains a commutator subgroup of an open normal subgroup, and we may
reduce to a case where a fixed open $V_1$ acts trivially on all $D_n$.
@@ -3543,39 +3543,39 @@ into all late $G_n$'s. Then $[V_1,V_1]$ fixes $D_n$ at every sufficiently
late level, so $[V_1,V_1]\subset\Lie V_1$ satisfies
$[\mathfrak v_1,\mathfrak v_1]=\mathfrak v_1$ and hence $V_1$ is
nontrivial, in contradiction to the hypothesis.
-\end{proof}
-\subsection{Hausdorff limits and dimension equality}
+
+## Hausdorff limits and dimension equality
The compact metrizable $V\times U$ has a compact space of closed
-subgroups in the Hausdorff topology. Every limit $G_\infty$ of the $G_n$
+subgroups in the Hausdorff topology. Every limit $G_∞$ of the $G_n$
still projects onto $V$ (for $v\in V$, choose $(v,u_n)\in G_n$ and pass
to a convergent subsequence of $u_n$). A nonzero source-only kernel at
-level $G_\infty$ would force an open source subgroup into all
-sufficiently late $G_n$, contradicting Lemma~\ref{lem:finite-limit-kernel}.
+level $G_∞$ would force an open source subgroup into all
+sufficiently late $G_n$, contradicting Lemma~lem:finite-limit-kernel.
Hence:
-\begin{lemma}[Dimensions and abelian corner]
-\label{prop:finite-dimensions}
+[Dimensions and abelian corner]
+
$\dim_{\mathbb Q_p}\mathfrak s=\dim_{\mathbb Q_p}\mathfrak t$. Every
-Hausdorff limit $G_\infty$ has both projections isomorphisms, and the
-target-only subgroup $G_\infty\cap(1\times U)$ is $\{1\}$. Consequently
-$D_0$ is abelian and equals $L^\infty(U)$ with Haar probability.
-\end{lemma}
+Hausdorff limit $G_∞$ has both projections isomorphisms, and the
+target-only subgroup $G_∞\cap(1\times U)$ is $\{1\}$. Consequently
+$D_0$ is abelian and equals $L^∞(U)$ with Haar probability.
+
-\begin{proof}
+Proof.
Choose a basis of a fixed lattice $M_V$ and concat the matrices
$B_{i,n}$; a nonzero full-rank square minor in the limiting
concatenation, adjugated with fixed determinant valuations, gives
$p^aM_V\times\{0\}\subset(\sum_iB_{i,n}M_V)\times\{0\}\subset M_n$
-uniformly at late levels. By Lemma~\ref{lem:finite-uniform-log} in the
+uniformly at late levels. By Lemma~lem:finite-uniform-log in the
product logarithm group, $\exp(p^{c+a}M_V)\times1\subset G_n$ uniformly.
After enlarging $a$, $\exp(p^{c+a}M_V)$ is a fixed open $V_1\subset V$
whose commutator $[V_1,V_1]$ fixes $D_n$ for all sufficiently late $n$
in the subsequence; as $D_n$ exhausts $D_0$, $[V_1,V_1]$ fixes $D_0$,
-contradicting Lemma~\ref{lem:finite-limit-kernel}. So $\Lie(G_\infty)$'s
+contradicting Lemma~lem:finite-limit-kernel. So $\Lie(G_∞)$'s
source-only part has zero Lie algebra. Both projections of
-$\Lie(G_\infty)$ are then isomorphisms (surjections with equal dimension
+$\Lie(G_∞)$ are then isomorphisms (surjections with equal dimension
$\dim\mathfrak s=\dim\mathfrak t$). The target-only subgroup is a compact
zero-dimensional analytic subgroup of the torsion-free $U$, hence
$\{1\}$.
@@ -3586,120 +3586,120 @@ representation is then scalar; blocks are jointly faithful, so $D_m$ is
abelian, and $D_0$ is abelian by exhaustion. The spectra $X_n$ of $D_n$
are finite transitive $U$-spaces with surjective equivariant maps
$X_{n+1}\to X_n$; averaging a faithful normal state gives invariant
-uniform measures, so $D_0=L^\infty(X,\nu)$ with $X=\varprojlim X_n$. The
+uniform measures, so $D_0=L^∞(X,\nu)$ with $X=\varprojlim X_n$. The
$U$-action on $X$ is transitive (nested chain of cosets), and uniform
shrinking of stabilizers makes any point stabilizer trivial, so $X$ is
the translation $U$-torsor with Haar probability.
-\end{proof}
-\subsection{The $T$-torsor coordinate}
-\begin{proposition}[The full frame algebra]
-\label{prop:finite-torsor}
-There is a $T$-equivariant normal isomorphism $D_*\cong L^\infty(T)$ with
+## The $T$-torsor coordinate
+
+[The full frame algebra]
+
+There is a $T$-equivariant normal isomorphism $D_*\cong L^∞(T)$ with
left translation. In these coordinates, the commuting source action is
-\begin{equation}
+
(\beta_s f)(x)=f(x\eta(s)),\qquad \eta:S\to T,
- \label{eq:finite-right-action}
-\end{equation}
+
+
where $\eta$ is a continuous homomorphism with isomorphic
$\mathbb Q_p$-differential.
-\end{proposition}
-\begin{proof}
-By Lemma~\ref{prop:finite-dimensions}, $e$ is an abelian projection. On
+
+Proof.
+By Lemma~prop:finite-dimensions, $e$ is an abelian projection. On
its central support $z_e$ the algebra is type I, and
-\begin{equation}
- Z(D_*)z_e\longrightarrow eD_*e,\qquad a\longmapsto ae
- \label{eq:finite-center-corner}
-\end{equation}
+
+ Z(D_*)z_e→ eD_*e,\qquad a⟼ ae
+
+
is a normal isomorphism. In the type I decomposition of this central
summand, $e$ has rank one in every factor, and full central support makes
the scalar field faithful.
Countably many $T$-translates of $e$ have join one, so $D_*$ is type I.
Its center has a positive-measure $U$-invariant chart isomorphic to $U$,
-and the $T$-action is ergodic (Equation~\eqref{eq:finite-fixed}).
-Use Mackey's Borel point realization \cite{Mackey1962}: let $X$ realize
+and the $T$-action is ergodic (Equation~(eq:finite-fixed)).
+Use Mackey's Borel point realization [cite]: let $X$ realize
the center, let $\varphi:U\to X$ be measurable onto the chart.
Fubini in the two Haar variables gives $x_0$ with
$\varphi(u)=ux_0$ almost everywhere; for a generic $u_0$,
$\varphi(vu_0)=v\varphi(u_0)$ for almost every $v$. Let
-$L=\operatorname{Stab}_T(x_0)$. If $h\in L\cap U$, comparing $u$ and
+$L=Stab_T(x_0)$. If $h\in L\cap U$, comparing $u$ and
$uh$ on a common conull set forces $h=1$, so $L\cap U$ meets only the
identity in a small neighborhood, hence $L$ is discrete and closed. The
orbit map $T/L\to X$ is a Borel bijection with Borel inverse, countably
many translates of the chart cover modulo null sets, and
-$Z(D_*)\cong L^\infty(T/L)$.
+$Z(D_*)\cong L^∞(T/L)$.
The measurable automorphisms of $T/L$ commuting with left translation
are the right translations by $N_T(L)/L$ (explicitly: for such $F$,
$g\mapsto g^{-1}F(gL)$ is left-invariant, hence constant $=nL$, giving
$n^{-1}Ln\subset L$, equality by applying to $F^{-1}$). The injection
-$N_T(L)/L\to\operatorname{Aut}(T/L)$ has a Borel inverse. The source
+$N_T(L)/L\toAut(T/L)$ has a Borel inverse. The source
action gives a measurable (hence continuous, analytic over
$\mathbb Q_p$) homomorphism $\eta:S\to N_T(L)/L$.
Its differential is injective: otherwise a small kernel $V_1<V$ acts
-trivially on the center; since $V$ fixes $e$, Equation~\eqref{eq:finite-center-corner}
+trivially on the center; since $V$ fixes $e$, Equation~(eq:finite-center-corner)
forces $V_1$ to act trivially on $eD_*e$, contradicting
-Lemma~\ref{lem:finite-limit-kernel}. Since $L$ is discrete and
+Lemma~lem:finite-limit-kernel. Since $L$ is discrete and
$\dim\mathfrak s=\dim\mathfrak t$, the closed Lie subgroup $N_T(L)$ has
dimension equal to $T$ and is open. For $l\in L$, conjugation from the
open $N_T(L)$ into the discrete $L$ is continuous, so $l$ centralizes an
open subgroup; the open-centralizer assertion forces $l=1$. Thus
-$L=\{1\}$ and $Z(D_*)=L^\infty(T)$ with left translation.
+$L=\{1\}$ and $Z(D_*)=L^∞(T)$ with left translation.
Remove matrix fibres: the type I decomposition and ergodicity of the
-center give $D_*\cong L^\infty(T;B(\mathcal V))$ with constant separable
+center give $D_*\cong L^∞(T;B(\mathcal V))$ with constant separable
dimension. In a measurable trivialization the action is
$(\alpha_ad)(x)=c(a,x)d(a^{-1}x)$ with
-$c(ab,x)=c(a,x)c(b,a^{-1}x)$ (an $\operatorname{Aut}(B(\mathcal V))$-
+$c(ab,x)=c(a,x)c(b,a^{-1}x)$ (an $Aut(B(\mathcal V))$-
valued cocycle). Changing variables and applying Fubini, freeze a generic
$x_0$ and define $b(x)=c(xx_0^{-1},x)$; then $c(a,x)=b(x)b(a^{-1}x)^{-1}$
almost everywhere, and changing the fibre trivialization by $b$ removes
the cocycle. Pure translation fixes the constant field
-$B(\mathcal V)$; Equation~\eqref{eq:finite-fixed} forces $\dim\mathcal
+$B(\mathcal V)$; Equation~(eq:finite-fixed) forces $\dim\mathcal
V=1$.
-\end{proof}
-\subsection{Ordinary homogeneous panel maps}
-\begin{proposition}[Recovery of the observed panels]
-\label{prop:finite-maps}
+## Ordinary homogeneous panel maps
+
+[Recovery of the observed panels]
+
The families $(P_z)$ and $(Q_z)$ commute on their respective start and
end spaces, and there are a homogeneous analytic homeomorphism
$\tau:B_S\to B_T$ and measurable target projectivity observations
$a,b$ such that these families are evaluations of
-\begin{equation}
- z\longmapsto a\tau(z),\qquad z\longmapsto b\tau(z).
- \label{eq:finite-panel-laws}
-\end{equation}
+
+ z⟼ a\tau(z),\qquad z⟼ b\tau(z).
+
+
On the independent input tensor product, the output projectivity is
sharp and equals $ab^{-1}$:
-\begin{equation}
- E(W)J=J\,1_W(ab^{-1})\quad\text{for every Borel }W\subset T.
- \label{eq:finite-projectivity-sharp}
-\end{equation}
+
+ E(W)J=J\,1_W(ab^{-1})\quadfor every Borel W\subset T.
+
+
All statements hold coordinatewise on many-panel links.
-\end{proposition}
-\begin{proof}
+
+Proof.
Commutativity of $D_*$ makes the paired generators' commutators vanish
at almost every external frame; use a common conull set for countably
many labels and clopen tests. The transformed tests generate $C(B_T)$,
so the original families commute, and label continuity extends to all
fixed pairs.
-In the intrinsic $D_*=L^\infty(T)$ of
-Proposition~\ref{prop:finite-torsor}, the spectral values give a
+In the intrinsic $D_*=L^∞(T)$ of
+Proposition~prop:finite-torsor, the spectral values give a
jointly measurable boundary map $F(x,z)$ (countable determining
observations plus label continuity). Frame covariance $F(ax,z)=aF(x,z)$
and Fubini give $F(x,z)=xF_0(z)$, and the source action gives
-\begin{equation}
+
F_0(sz)=\eta(s)F_0(z)
- \label{eq:finite-equivariant-map}
-\end{equation}
+
+
in the ordinary measure class on $B_S$.
Regard each effective adjoint group over its coefficient field as a
@@ -3708,10 +3708,10 @@ derivations of a semisimple Lie algebra are inner, and the identity
component of its algebraic automorphism group is the adjoint algebraic
group. Conjugation by $A=d\eta$ is an isomorphism of these adjoint
algebraic groups over $\mathbb Q_p$, and
-\begin{equation}
+
A\Ad(s)=\Ad(\eta(s))A,\qquad s\in S.
- \label{eq:finite-adjoint-intertwining}
-\end{equation}
+
+
Minimal rational parabolics are preserved under algebraic group
isomorphisms, and those of the restriction of scalars are exactly the
restrictions of the original rational parabolics (after splitting,
@@ -3726,9 +3726,9 @@ a rank-one boundary, so the constant is $\tau$ and $F_0=\tau$ almost
everywhere. Both sides are continuous in measure in $z$, so the
identity holds for every fixed label.
-Remove the external frame using a standard spectrum $\Omega^+$ of the
+Remove the external frame using a standard spectrum $Ω^+$ of the
commutative start observation algebra, with a faithful normal
-probability class. The normal representation of $D_*=L^\infty(T)$ by
+probability class. The normal representation of $D_*=L^∞(T)$ by
its start decomposable fields is pullback via a measurable $h(t,\omega)$
into $T$; the generator identities read $tp_z(\omega)=h(t,\omega)\tau(z)$
for almost all $(t,\omega)$, and choosing one generic frame $t_0$ gives
@@ -3737,26 +3737,26 @@ Each $a(\omega)\tau$ is continuous, extending to every fixed label; the
end family gives $b$ the same way.
On the tensor product, $a$ and $b$ commute. Substituting
-\eqref{eq:finite-panel-laws} in the exact graph link gives
+(eq:finite-panel-laws) in the exact graph link gives
$a\tau(z)=g\,b\tau(z)$ for countably dense labels, and since $\tau$ is
onto with faithful target action, the common zero set is $g=ab^{-1}$.
Compact clopen partitions and more labels give exact graph
substitution; exhaustion and monotone approximation yield
-\eqref{eq:finite-projectivity-sharp}. The map from target
+(eq:finite-projectivity-sharp). The map from target
projectivities to continuous homogeneous boundary maps is continuous and
injective, with Borel inverse.
-\end{proof}
-\subsection{Uniform finite-place upper speed}
+
+## Uniform finite-place upper speed
For each finite-place effective panel group $R$, choose a norm on its
prime-field adjoint Lie algebra, with $|p|=p^{-1}$ on $\mathbb Q_p$, and
let $\ell_R$ be its rank-one Cartan gap. Constants $d_R>0$ depend only
on the panel type:
-\begin{equation}
+
\log\|\Ad(r)\|=d_R\ell_R(r)+O(1),\qquad r\in R,
- \label{eq:finite-adjoint-growth}
-\end{equation}
+
+
with uniform error. On a dominant split translation the largest adjoint
eigenvalue has logarithm a fixed positive multiple of the simple-root
coordinate; converting from $|\cdot|_k$ to $|\cdot|_p^{[k:\mathbb Q_p]}$
@@ -3766,72 +3766,72 @@ factors change operator norms by bounded amounts. Symmetry under
inversion gives $d_R$ unchanged by opposition, so the $d_R$ are drawn
from a single type list.
-\begin{proposition}[Uniform finite-place upper speed]
-\label{prop:finite-speed}
+[Uniform finite-place upper speed]
+
After the inner limit of a translated panel link, let $t\ge0$ be the
total missing-root coordinate of the source translation, including all
slower offsets. The translation is taken on the permissible cocharacter
mesh. If $g$ is the target residual projectivity, then
-\begin{equation}
- d_T\ell_T(g)\le d_St+O_{\mathrm{tight}}(1),
- \label{eq:finite-speed}
-\end{equation}
+
+ d_T\ell_T(g)\le d_St+O_{tight}(1),
+
+
with tightness on each fixed ordinary input vector, uniformly in the
slower translations with $t$ as center. This holds coordinatewise and
simultaneously as an upper-tail estimate on many-panel links.
-\end{proposition}
-\begin{proof}
+
+Proof.
Use the convention in which the translated start observation is
$P_z^{(t)}=P_{s_tz}$ on its original input space and the end
observation is $Q_z$. Thus $s_t$ is the effective start-label transport
($t$ is on the permissible mesh, so $s_t\in S$ and $\eta(s_t)$ is
-defined). Proposition~\ref{prop:transfer-residue-graph} and
-Proposition~\ref{prop:finite-maps} give $a\tau(s_tz)=g\,b\tau(z)$, so
-\begin{equation}
+defined). Proposition~prop:transfer-residue-graph and
+Proposition~prop:finite-maps give $a\tau(s_tz)=g\,b\tau(z)$, so
+
g=a\eta(s_t)b^{-1},
- \label{eq:finite-translated-law}
-\end{equation}
+
+
where $s_t$ is deterministic with total missing gap $t$. Crucially
$a,b$ are observations on the original independent input spaces, so all
slower-translation dependence passes into the middle factor. Other split
translations have zero coordinate in this residue and act trivially, and
any remaining bounded effective actions range in a fixed compact set.
-With $A=d\eta$, Equation~\eqref{eq:finite-adjoint-intertwining} gives
+With $A=d\eta$, Equation~(eq:finite-adjoint-intertwining) gives
$\Ad(\eta(s_t))=A\Ad(s_t)A^{-1}$, hence
$\log\|\Ad(\eta(s_t))\|\le d_St+C$, and the same for $s_t^{-1}$. Fix
-$\xi\in\overline I\otimes H'$ and $\varepsilon>0$; the commuting
+$\xi\in\overline I ⊗ H'$ and $ε>0$; the commuting
$a,b$ have a joint scalar spectral measure on $\xi$. Choose compact
sets containing both observations except for squared input norm
-$\le\varepsilon\|\xi\|^2$. On that joint cut $\|\Ad(a)\|$ and
+$\leε\|\xi\|^2$. On that joint cut $\|\Ad(a)\|$ and
$\|\Ad(b^{-1})\|$ are uniformly bounded in $t$ and slower parameters;
-submultiplicativity with \eqref{eq:finite-translated-law} and
-\eqref{eq:finite-adjoint-growth} gives $d_T\ell_T(g)\le d_St+C_{\xi,\varepsilon}$
+submultiplicativity with (eq:finite-translated-law) and
+(eq:finite-adjoint-growth) gives $d_T\ell_T(g)\le d_St+C_{\xi,ε}$
on that cut. The exact graph intertwining and link isometry bound the
excluded output by the excluded input. This proves
-\eqref{eq:finite-speed} including non-simple-tensor input vectors.
+(eq:finite-speed) including non-simple-tensor input vectors.
For finitely many panel coordinates, apply to each coordinate's input
laws and sum the excluded masses; no commutation of different input
systems is needed.
-\end{proof}
+
For a finite-place non-isolated type $i$, put $s_i=d_S/d_T$. Opposition
leaves $s_i$ unchanged, and its product around any type cycle is one
because all $d_R$ are drawn from the same list. The isolated types left
-by Corollary~\ref{cor:finite-arch-separation} are archimedean, treated
+by Corollary~cor:finite-arch-separation are archimedean, treated
below.
-\section{Length multipliers and ray lifts}
-\label{sec:ray}
+# Length multipliers and ray lifts
+
This section supplies the length estimate needed for the isolated boundary
-factors: uniform ray lifts (Proposition~\ref{prop:ray-lift}) combined with
+factors: uniform ray lifts (Proposition~prop:ray-lift) combined with
property~$(T)$ averaging give addition of rescaled spherical length laws
-(Proposition~\ref{prop:ray-addition}). Addition lets us later compare
+(Proposition~prop:ray-addition). Addition lets us later compare
repeated source segments and rule out total superlinear growth.
-The isolated components are real Lie factors (Sections~\ref{sec:transfer}
-and~\ref{sec:finite}): quaternionic hyperbolic spaces of dimension at least
+The isolated components are real Lie factors (Sections~sec:transfer
+and~sec:finite): quaternionic hyperbolic spaces of dimension at least
two, and the Cayley hyperbolic plane. Fix such a space $X$; normalize its
metric so $r(g)=d(o,go)$ is the logarithm of the dilation in the
indivisible restricted root. The argument uses an entire ambient factor;
@@ -3846,30 +3846,30 @@ allowing arbitrary finite coefficient traces. Varying finite factors for
spherical laws have normalized trace; the quotient base has probability
measure.
-\subsection{Coordinates and a uniform smooth calculus}
+## Coordinates and a uniform smooth calculus
The horospherical group is $N_b=V\times Z$ with
\[
(v,z)(v',z')=(v+v',z+z'+\tfrac12[v,v']),\qquad
\delta_R(v,z)=(Rv,R^2z).
\]
-Choose inner products with $\langle J_zv,w\rangle=\langle z,[v,w]\rangle$
+Choose inner products with $< J_zv,w>=< z,[v,w]>$
and $J_z^2=-|z|^2\id$; the horizontal layer is bracket-generating. Set
$Q=\dim V+2\dim Z$. In half-space coordinates $(n,\lambda)\in
-N_b\times(0,\infty)$, with a fixed normalization of Haar measure,
-\begin{equation}\label{eq:ray-metric}
- ds_X^2=\frac{d\lambda^2}{\lambda^2}
- +\frac{|dv|^2}{4\lambda^2}
- +\frac{|dz-\tfrac12[v,dv]|^2}{4\lambda^4},
+N_b\times(0,∞)$, with a fixed normalization of Haar measure,
+
+ ds_X^2=(d\lambda^2)/(\lambda^2)
+ +(|dv|^2)/(4\lambda^2)
+ +(|dz-\tfrac12[v,dv]|^2)/(4\lambda^4),
\qquad dp=\frac{dn\,d\lambda}{\lambda^{Q+1}}.
-\end{equation}
+
These are the Iwasawa coordinates for the indicated symmetric spaces; the
two root spaces are $V$ and $Z$. A vertical dilation has length $|\log R|$.
-On the compact boundary $\partial X$, the horizontal term in
-\eqref{eq:ray-metric} determines a canonical horizontal conformal class;
+On the compact boundary $∂ X$, the horizontal term in
+(eq:ray-metric) determines a canonical horizontal conformal class;
we take the compact-stabilizer average of a smooth representative and let
-$d_{\partial X}$ be its Carnot--Carath\'eodory distance (comparable to
+$d_{∂ X}$ be its Carnot--Carath\'eodory distance (comparable to
$(|v|^4/16+|z|^2)^{1/4}$ in a buffered chart).
A scale box is a translate of a $R$-dilated fixed bounded box in
@@ -3877,24 +3877,24 @@ exponential coordinates, with a fixed larger box available for smooth
cutoffs. A normalized $C^m$ bound is the ordinary $C^m$ bound after
translation and dilation to the fixed box.
-\begin{lemma}[Distance and changes of scale]\label{lem:ray-geometry}
+[Distance and changes of scale]
With constants uniform on buffered compact charts and in normalized $C^m$
norms for every $m\ge0$:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item If $(v,z)=\log(n^{-1}n')$, then
-\begin{equation}\label{eq:ray-distance}
- \cosh d((n,\lambda),(n',\lambda'))
- =\frac{\sqrt{(\lambda^2+\lambda'^2+|v|^2/4)^2+|z|^2}}
+
+ cosh d((n,\lambda),(n',\lambda'))
+ =\frac{√((\lambda^2+\lambda'^2+|v|^2/4)^2+|z|^2)}
{2\lambda\lambda'}.
-\end{equation}
+
\item Changes between the compactly rotated horospherical boundary charts
preserve normalized bounds on boxes of every scale $0<R\le1$, and in the
adjoined height $0\le\lambda\le CR$. The conversion between the outward
endpoint and $e^{-r}$ (or finite half-space coordinates) has the form
-\begin{equation}\label{eq:ray-collar}
- n=n_0\bigl(e^{-2r}U,e^{-4r}W\bigr),\qquad
+
+ n=n_0(e^{-2r}U,e^{-4r}W),\qquad
\lambda=e^{-r}b,
-\end{equation}
+
where $U,W,b$ are smooth in $r^{-1}$ and $e^{-r}$, with $b$ bounded above
and below.
\item Let $g$ have length $L$, pole $g^-$ and image endpoint $g^+$. An
@@ -3903,69 +3903,69 @@ annulus of size $q'=e^{-L+u}$, $0\le u\le L$, with normalized derivatives
bounded. A smaller boundary box of radius $\delta\le cq$, height at most
$c\delta$, has image radius at most $Ce^{2u-L}\delta$. Exterior pieces
$q\ge c$ have output upper bound $Ce^{-L}$.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
(i). Translate and dilate the first point to $(0,1)$; put
$A=\lambda^2+1+|v|^2/4$, $C=A^2+|z|^2$, $c=\sqrt C/(2\lambda)$. In the
orthonormal frame the three components of $\nabla c$ are
$\lambda A/\sqrt C-\sqrt C/(2\lambda)$, $(Av+J_zv)/(2\sqrt C)$,
$\lambda z/\sqrt C$. Since $J_zv\perp v$ and $|J_zv|^2=|z|^2|v|^2$, the
-squared norms sum to $c^2-1$. Thus $\operatorname{arcosh}c$ has gradient
+squared norms sum to $c^2-1$. Thus $arcoshc$ has gradient
norm one off its unique zero. It is proper, so it is the distance.
(ii). One convenient Weyl inversion in these coordinates is
-\begin{equation}\label{eq:ray-inversion}
+
j(v,z,\lambda)=
- \left(\frac{(-A_0+J_z)v}{B_0},-\frac z{B_0},
- \frac\lambda{\sqrt{B_0}}\right),\quad
+ (((-A_0+J_z)v)/(B_0),-\frac z{B_0},
+ \frac\lambda{√(B_0)}),\quad
A_0=\lambda^2+|v|^2/4,\quad B_0=A_0^2+|z|^2,
-\end{equation}
+
with $j\delta_R=\delta_{R^{-1}}j$. On a compact annulus
$(|v|^4/16+|z|^2)^{1/4}+\lambda\asymp1$, all derivatives are bounded. In
a finite Bruhat cell the Gauss factors of a compact rotation times a
horospherical are smooth with bounded derivatives; passing the boundary
factor across $a_{\log\lambda}$ and Taylor expanding at $o$, the
horizontal and central coordinate errors are $\lambda^2U$ and
-$\lambda^4W$, giving \eqref{eq:ray-collar}. Boundary transitions are
+$\lambda^4W$, giving (eq:ray-collar). Boundary transitions are
H-type contact maps, so the central horizontal linear term vanishes; the
same Taylor computation applies after substitution and proves (ii).
(iii). Compactly rotate $g$ to $\delta_{e^{-L}}j$; rescale input by
$q^{-1}$ and output by $(q')^{-1}$ to see exactly the restriction of $j$
to a fixed buffered annulus. The derivative and height-ratio claims
-follow from \eqref{eq:ray-inversion}; the smaller-box claim is (ii) with
+follow from (eq:ray-inversion); the smaller-box claim is (ii) with
relative radius $\delta/q$.
-\end{proof}
-The horizontal density on $\partial X$ used later is the product of the
-horizontal volume density with the density on $T\partial X/\mathcal V$
-induced by the surjective bracket $\bigwedge^2\mathcal V\to
-T\partial X/\mathcal V$. It is compact-invariant with mass one. Naturality
+
+The horizontal density on $∂ X$ used later is the product of the
+horizontal volume density with the density on $T∂ X/\mathcal V$
+induced by the surjective bracket $∧^2\mathcal V\to
+T∂ X/\mathcal V$. It is compact-invariant with mass one. Naturality
of the bracket makes it the compact angular probability, and its Jacobian
-under a horizontal dilation $d\cdot\mathrm{Isom}$ is exactly $d^Q$.
+under a horizontal dilation $d\cdotIsom$ is exactly $d^Q$.
-\begin{lemma}[Triangle defect]\label{lem:ray-defect}
+[Triangle defect]
There is $C$ such that if $x,y\in X$ and
$b=d(o,x)+d(o,y)-d(x,y)>0$, then
-$d_{\partial X}(x^+,y^+)\le Ce^{-b/2}$. Consequently
-$r(g)+r(h)-r(gh)\ge b>0$ forces $d_{\partial X}(g^-,h^+)\le Ce^{-b/2}$.
-\end{lemma}
+$d_{∂ X}(x^+,y^+)\le Ce^{-b/2}$. Consequently
+$r(g)+r(h)-r(gh)\ge b>0$ forces $d_{∂ X}(g^-,h^+)\le Ce^{-b/2}$.
+
-\begin{proof}
+Proof.
Both radii are at least $b/2$. In a compactly rotated finite chart
containing both endpoints, either the boundary distance is at most the
-fixed boundary-coordinate error (by \eqref{eq:ray-collar}), or
-homogeneous separation is comparable to $\delta$ and \eqref{eq:ray-distance}
+fixed boundary-coordinate error (by (eq:ray-collar)), or
+homogeneous separation is comparable to $\delta$ and (eq:ray-distance)
gives $d(x,y)\ge d(o,x)+d(o,y)+2\log\delta-C$. Apply to $x=g^{-1}o$,
$y=ho$.
-\end{proof}
-\begin{lemma}[Smooth separation and cells]\label{lem:ray-cells}
+
+[Smooth separation and cells]
Fix row and column dimensions and a bound on auxiliary compact variables.
For every summable power $N$ there are $m$ and $C$ such that:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item A smooth symbol on the product of two fixed buffered boxes with
$C^m$ norm at most $A$ has a separated Fourier expansion with coefficients
bounded by $CA(1+|\nu|+|\nu'|)^{-N}$; the unary factors are supported in
@@ -3975,10 +3975,10 @@ compact external parameters work the same way.
most $b$, and normalized extensions are bounded by $A$, the kernel is a
Schur multiplier of norm at most $C_bA$; matrix amplifications have the
same bound.
-\end{enumerate}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
(i) is integration by parts on a fixed torus after smooth cutoff. (ii): a
degree-$b$ graph is a union of at most $2b-1$ matchings (greedy edge
coloring). Each matching gives a sum of disjoint rectangular rank-one
@@ -3986,45 +3986,45 @@ kernels, with $L^1$ factorization bound; overlap is split by assigning
each point to a bounded number of boxes. Summation and the absolutely
convergent Fourier series give (ii); matrix amplification is unaffected
by the factorization.
-\end{proof}
-\subsection{Uniform radial multipliers}
-\begin{proposition}[Radial cutoffs]\label{prop:ray-radial}
-Let $f\in C^\infty([0,\infty))$ be constant near zero and near infinity.
+## Uniform radial multipliers
+
+[Radial cutoffs]
+Let $f\in C^∞([0,∞))$ be constant near zero and near infinity.
There are $R_f,C_f>0$ such that $(x,x')\mapsto f(d(x,x')/R)$ has Schur
norm at most $C_f$ for every $R\ge R_f$. Consequently for any
-homomorphism $\Gamma\to\operatorname{Isom}(X)$, any normalized scalar
+homomorphism $\Gamma\toIsom(X)$, any normalized scalar
cocycle $\eta$, and any finite tracial von Neumann algebra $B$, the
multiplier
\[
- D_{f(r/R)}:\ B\,\bar\otimes L_\eta(\Gamma)\to
- B\,\bar\otimes L_\eta(\Gamma),\qquad
- \sum_hY_hu_h\longmapsto\sum_hf(r(h)/R)Y_hu_h
+ D_{f(r/R)}:\ B\,\bar ⊗ L_\eta(\Gamma)\to
+ B\,\bar ⊗ L_\eta(\Gamma),\qquad
+ \sum_hY_hu_h⟼\sum_hf(r(h)/R)Y_hu_h
\]
has completely bounded norm at most $C_f$; the constants are independent
of $\Gamma,\eta,B$ and of the homomorphism.
-\end{proposition}
-\begin{proof}
-Subtract $f(\infty)$. Choose $0<a<b$ with $f$ equal to its value on
-$[0,2a]$ and to $0$ on $[b,\infty)$. Write $s=-\log\lambda$,
+
+Proof.
+Subtract $f(∞)$. Choose $0<a<b$ with $f$ equal to its value on
+$[0,2a]$ and to $0$ on $[b,∞)$. Write $s=-\log\lambda$,
$s'=-\log\lambda'$. Since $|s-s'|\le d(x,x')$, only a bounded number of
offset bands of height blocks of length $R$ contribute; simultaneous
dilation reduces to $|s|,|s'|\le C_bR$.
Take smooth radial cutoffs $\chi_j(n^{-1}n')$ at radii
-$\epsilon_j=e^{-j}$, equal to one on a smaller ball, and
+$ε_j=e^{-j}$, equal to one on a smaller ball, and
$J=\lceil C'_bR\rceil$. The annuli $\chi_j-\chi_{j+1}$ for
$-J\le j<J$ and the innermost ball $\chi_J$ cover the support of
$f(d/R)$. Put
\[
- H_j=\log\frac{\max(\epsilon_j,\lambda,\lambda')^2}{\lambda\lambda'}.
+ H_j=\log(\max(ε_j,\lambda,\lambda')^2)/(\lambda\lambda').
\]
Each piece $f(d/R)-f(H_j/R)$ on its annulus, and each height increment
$f(H_j/R)-f(H_{j-1}/R)$, has Schur norm $O(R^{-1})$ uniformly in $j,R$.
This follows case-by-case on the mixed-height annuli (writing
-$(v,z)=\delta_{\epsilon_j}(V,Z)$ on the fixed separated cell, expanding
+$(v,z)=\delta_{ε_j}(V,Z)$ on the fixed separated cell, expanding
with compact auxiliary scale ratios), then at comparable heights by the
sign-cutoff argument in $s/R-s'/R$, where $R^me^{-aR}$ is bounded on the
smooth pieces. The telescoping identity
@@ -4040,56 +4040,56 @@ Restrict to orbit points $h^{-1}o$: the regular matrix of the Fourier
multiplier is entrywise multiplication by this kernel, up to scalar
phases already in the twisted representation. The factorization proof is
unchanged, giving complete boundedness.
-\end{proof}
-\subsection{One-sided localization}
-Let $D=B\bar\otimes L_\eta(\Gamma)$ with product trace $\tau$ and
+## One-sided localization
+
+Let $D=B\bar ⊗ L_\eta(\Gamma)$ with product trace $\tau$ and
Fourier coefficients $Y_h\in L^2(B)$; $D_\varphi$ denotes coefficient
multiplication on $L^2(D)$. For finite Fourier sums,
$(YY^*)_g=\sum_h\eta(gh,h^{-1})\overline{\eta(h,h^{-1})}Y_{gh}Y_h^*$,
and phases have absolute value one.
-\begin{lemma}[Unary Plancherel estimate]\label{lem:ray-unary}
+[Unary Plancherel estimate]
Let $\varphi_{i,\nu}:\Gamma\to\mathbb C$ satisfy
$\sup_h\sum_i|\varphi_{i,\nu}(h)|^2\le b$ for every $\nu$. With $T_g$ an
arbitrary family of indices, $i(g,t)$ chosen with at most $b$
occurrences of each $i$ for fixed $g$, and
$z_{g,t}=\sum_\nu c_{g,t,\nu}[Y(D_{\varphi_{i(g,t),\nu}}Y)^*]_g$ where
-$|c_{g,t,\nu}|\le Aa_\nu$, $\sum_\nu a_\nu<\infty$, we have
-\begin{equation}\label{eq:ray-unary}
- \left(\sum_g\sum_{t\in T_g}\|z_{g,t}\|_{2,B}^2\right)^{1/2}
- \le bA\Bigl(\sum_\nu a_\nu\Bigr)\|Y\|_\infty\|Y\|_2.
-\end{equation}
+$|c_{g,t,\nu}|\le Aa_\nu$, $\sum_\nu a_\nu<∞$, we have
+
+ (\sum_g\sum_{t\in T_g}\|z_{g,t}\|_{2,B}^2)^{1/2}
+ \le bA(\sum_\nu a_\nu)\|Y\|_∞\|Y\|_2.
+
The same holds with the first factor localized instead.
-\end{lemma}
-\begin{proof}
+
+Proof.
For one $\nu$, discard admissibility after using the multiplicity bound;
Plancherel plus the operator bound for the uncut factor give
$\sum_{g,t}\|[Y(D_{\varphi_{i(g,t),\nu}}Y)^*]_g\|_2^2
-\le b^2\|Y\|_\infty^2\|Y\|_2^2$. Minkowski sums the absolutely summable
+\le b^2\|Y\|_∞^2\|Y\|_2^2$. Minkowski sums the absolutely summable
frequencies; adjoints give the other orientation. Only the operator norm
of the uncut factor is used.
-\end{proof}
-\begin{lemma}[Angular column estimate]\label{lem:ray-angular}
+
+[Angular column estimate]
Fix $X$ and an integer $m$; there are $c,C>0$ with the following
property. Put $\delta=e^{-s}$, $s\ge1$. Let $(f_C)_C$ be smooth masks
supported on buffered boundary boxes of radius $\delta$, with fixed
overlap and normalized $C^m$ bound. If $Y\in D$ has Fourier support in
$\{r\ge l\}$ with $l\ge Cs+C$, then with $A_C=D_{f_C(h^+)}Y$,
-\begin{equation}\label{eq:input-7}
+
\sum_{C,D'}\|A_CA_{D'}^*\|_2^2
=\Bigl\|\sum_CA_C^*A_C\Bigr\|_2^2
- \le C\|Y\|_\infty^2\|Y\|_2^2.
-\end{equation}
+ \le C\|Y\|_∞^2\|Y\|_2^2.
+
The constants may depend on the mask bound but not on the number of
masks, and the assertion holds with arbitrary coefficient algebras and
scalar twists.
-\end{lemma}
-\begin{proof}
+
+Proof.
Estimate the coefficients of $A_CA_{D'}^*$ at fixed shift $g$; put
$L=r(g)$; the weights in $YY^*$ are $f_C((gh)^+)\overline{f_{D'}(h^+)}$.
Only pairs with both $h$ and $gh$ in the long support matter.
@@ -4097,12 +4097,12 @@ Only pairs with both $h$ and $gh$ in the long support matter.
Split into nonexceptional and exceptional pairs. On a nonexceptional
input box separated from $g^-$ by a large multiple of $\delta$, with
input pole distance $q=e^{-u}$, either $2u\le L+C_1$ (then
-Lemma~\ref{lem:ray-geometry}(iii) makes the map to the output boundary
+Lemma~lem:ray-geometry(iii) makes the map to the output boundary
have bounded normalized derivatives, and Fourier expansion makes the
pair weight a unary expansion in $h$ with compact-coefficient decay) or
$2u>L+C_1$ (then inversion gives output pole distance $e^{-L+u}$ and the
reversed unary estimate applies). Bounded $L$ and the transition
-$2u=L+O(1)$ use larger constants. Lemma~\ref{lem:ray-unary} then bounds
+$2u=L+O(1)$ use larger constants. Lemma~lem:ray-unary then bounds
every nonexceptional pair; a map with bounded magnification sends one
buffered $\delta$-box into a ball of radius $C\delta$, giving the
required multiplicity.
@@ -4110,28 +4110,28 @@ required multiplicity.
The exceptional pairs lie within $C_2\delta$ of the two poles. On such a
pair compatible with the long supports, the two current-point pole sizes
have product comparable to $e^{-L}$; hence
-\begin{equation}\label{eq:ray-long-shift}
+
L\ge 2s-C_3.
-\end{equation}
+
Split each mask as its value at its pole plus its difference from that
value. The constant terms cost $C\|YY^*\|_2$; the mixed terms are handled
-by Lemma~\ref{lem:ray-unary}; the product of differences is split into
+by Lemma~lem:ray-unary; the product of differences is split into
smooth unit-width annuli in the pole coordinate $e^{-u}$,
$0\le u\le L/2+C_4$, and on such an annulus the product seminorm obeys
-\begin{equation}\label{eq:ray-annular-decay}
+
C_m\min(1,e^{s-u})\min(1,e^{s-L+u})\le C'_m e^{-|s-u|}.
-\end{equation}
+
At fixed integer depth $u$, use reference boundary boxes of radius
$e^{-u}$ with height $e^{-u}$ as an auxiliary unary variable; only
-boundedly many such boxes meet the annulus. Lemmas~\ref{lem:ray-cells}
-and~\ref{lem:ray-unary} give $Ce^{-|s-u|}\|Y\|_\infty\|Y\|_2$ for the
+boundedly many such boxes meet the annulus. Lemmas~lem:ray-cells
+and~lem:ray-unary give $Ce^{-|s-u|}\|Y\|_∞\|Y\|_2$ for the
depth-$u$ square sum; geometric summation and the reversed annuli sum
these uniformly. The written equality is the trace identity
$\tau(A_CA_{D'}^*A_{D'}A_C^*)=\tau(A_C^*A_CA_{D'}^*A_{D'})$.
For general $Y$, take bounded Fourier-polynomial approximations with a
smooth radial cutoff equal to one on $\{r\ge l\}$ and zero on
-$\{r\le l/2\}$; Proposition~\ref{prop:ray-radial} bounds the approximants
+$\{r\le l/2\}$; Proposition~prop:ray-radial bounds the approximants
uniformly. Increase the constant in $l$ by the slack. Each masked column
converges in $L^2$ and each product of two in $L^1$ by tracial
Cauchy--Schwarz; for a fixed finite family the diagonal-terms bound in
@@ -4139,49 +4139,49 @@ the Hilbert direct sum supplies weak $L^2$ compactness, and the $L^1$
convergence identifies the limit. This makes every masked column live in
$L^4$, giving the stated trace identity for finite families; exhaustion
finishes.
-\end{proof}
-\subsection{The ray lift and its densities}
+
+## The ray lift and its densities
Let $\mathcal K_X=L^2(X,dp)$ and let $T_g$ be translation on
$\mathcal K_X$. Define
\[
- \Delta:D\longrightarrow B(\mathcal K_X)\bar\otimes D,\qquad
- \Delta(bu_h)=T_h\otimes bu_h,
+ \Delta:D→ B(\mathcal K_X)\bar ⊗ D,\qquad
+ \Delta(bu_h)=T_h ⊗ bu_h,
\]
a normal unital homomorphism (conjugate by the unitary sending
-$\zeta\otimes\delta_h$ to $T_h^{-1}\zeta\otimes\delta_h$).
+$\zeta ⊗ \delta_h$ to $T_h^{-1}\zeta ⊗ \delta_h$).
Choose real smooth $\psi,F$ with $\supp\psi\Subset(1,2)$, $F$
compactly supported on $N_b$ and invariant under the compact isotropy
of a ray, with unit $L^2$ norms. For $d_1\ge1$, basepoint $x$, and
ideal endpoint $v$, use a chart sending $v$ to infinity and put
-\begin{equation}\label{eq:ray-function}
+
\theta_{x,d_1}(p,v)=d_1^{-1/2}
- \psi\Bigl(\frac{\log(\lambda_p/\lambda_x)}{d_1}\Bigr)
- F\bigl(\delta_{\lambda_p^{-1}}(n_x^{-1}n_p)\bigr).
-\end{equation}
+ \psi((\log(\lambda_p/\lambda_x))/(d_1))
+ F(\delta_{\lambda_p^{-1}}(n_x^{-1}n_p)).
+
This is equivariant in $x,p,v$ and has unit $L^2$ norm, supported in a
bounded neighborhood of the ray from $x$ to $v$ at depths
$[d_1-O(1),2d_1+O(1)]$. The coefficient isometry is
-$\mathbb V_{d_1}Y=\sum_h\theta_{o,d_1}(\cdot,h^+)\otimes Y_hu_h$.
+$\mathbb V_{d_1}Y=\sum_h\theta_{o,d_1}(\cdot,h^+) ⊗ Y_hu_h$.
-For a column $\xi\in\mathcal K_X\otimes L^2(D)$, the left and right
+For a column $\xi\in\mathcal K_X ⊗ L^2(D)$, the left and right
densities $\rho^L_\xi,\rho^R_\xi\in L^1(D)$ represent the vector
-functionals for $\Delta(D)$ and $D^{\mathrm{op}}$:
-$\langle\xi,\Delta(x)\xi\rangle=\tau(\rho^L_\xi x)$,
-$\langle\xi,\xi x\rangle=\tau(\rho^R_\xi x)=\xi^*\xi$.
+functionals for $\Delta(D)$ and $D^{op}$:
+$<\xi,\Delta(x)\xi>=\tau(\rho^L_\xi x)$,
+$<\xi,\xi x>=\tau(\rho^R_\xi x)=\xi^*\xi$.
-\begin{proposition}[Ray lift]\label{prop:ray-lift}
+[Ray lift]
Take any sequence of finite tracial coefficient algebras, twists, and
-countable homomorphisms to $\operatorname{Isom}(X)$, with
-$d_1\to\infty$, $l/d_1\to\infty$, and $Y\in D$ Fourier-supported in
+countable homomorphisms to $Isom(X)$, with
+$d_1\to∞$, $l/d_1\to∞$, and $Y\in D$ Fourier-supported in
$\{r\ge l\}$.
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item For large indices, $\mathbb V_{d_1}Y$ has left density for
-$\Delta$ on $D$ and right density on $D^{\mathrm{op}}$, both in $L^2$
-with norm at most $C\|Y\|_\infty\|Y\|_2$; also
-$\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_{2,\Tr\otimes\tau}$ has this
+$\Delta$ on $D$ and right density on $D^{op}$, both in $L^2$
+with norm at most $C\|Y\|_∞\|Y\|_2$; also
+$\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_{2,\Tr ⊗ \tau}$ has this
bound.
\item If $\supp\widehat x\subset\{r\le a\}$ and $a/d_1\to0$, then
$\|\mathbb V_{d_1}(xY)-\Delta(x)\mathbb V_{d_1}Y\|_2\to0$ and
@@ -4189,18 +4189,18 @@ $\|\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\|_2\to0$.
\item For another long column $Z$ and bounded $x$, if for every
contributing triple $h\in\supp\widehat Y$, $hk\in\supp\widehat Z$,
$k\in\supp\widehat x$ one has
-$d_{\partial X}(h^+,(hk)^+)\le e^{-cl'}$ with $c>0$,
-$l'/d_1\to\infty$, then
-\begin{equation}\label{eq:ray-close-pairing}
- \langle\mathbb V_{d_1}Z,(\mathbb V_{d_1}Y)x\rangle
- -\langle Z,Yx\rangle\longrightarrow0,
-\end{equation}
-uniformly with $\|x\|_\infty,\|Y\|_2,\|Z\|_2$ bounded.
-\end{enumerate}
+$d_{∂ X}(h^+,(hk)^+)\le e^{-cl'}$ with $c>0$,
+$l'/d_1\to∞$, then
+
+ <\mathbb V_{d_1}Z,(\mathbb V_{d_1}Y)x>
+ -< Z,Yx>→0,
+
+uniformly with $\|x\|_∞,\|Y\|_2,\|Z\|_2$ bounded.
+
All statements hold for matrix versions and measurable fields.
-\end{proposition}
-\begin{proof}[Right density]
+
+Proof. [Right density]
Write $K_{d_1}(v,w)=\int\theta_{o,d_1}(p,v)\theta_{o,d_1}(p,w)\,dp$
and disintegrate at radial depth $s=d(o,p)\in[d_1-C,2d_1+C]$. With
measure $ds/d_1$, the kernels $k_{d_1,s}$ sit on adjacent boundary
@@ -4214,75 +4214,75 @@ rescaling neighbors by $e^{-s}$.
Separate $k_{d_1,s}$ on adjacent boxes: at each frequency the right
density is $\sum t_{C,D'}A_C^*B_{D'}$ with scalar matrix $t$ of bounded
degree and operator norm, and both mask families meeting the
-hypotheses of Lemma~\ref{lem:ray-angular}. Tracial H\"older plus
-\eqref{eq:input-7} give $\le C\|Y\|_\infty\|Y\|_2$; Fourier
+hypotheses of Lemma~lem:ray-angular. Tracial H\"older plus
+(eq:input-7) give $\le C\|Y\|_∞\|Y\|_2$; Fourier
coefficients are summable, chosen above the polynomial derivative
growth of masks. The matrix assertion follows from
$\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_2^2
=\|(\mathbb V_{d_1}Y)^*\mathbb V_{d_1}Y\|_2^2$ and monotone
approximation.
-\end{proof}
-\begin{lemma}[Common-endpoint overlap]\label{lem:ray-overlap}
+
+[Common-endpoint overlap]
For a displacement $g$ of length $L$, put
$w_{d_1}(g,v)=\int_X\theta_{o,d_1}(p,v)\theta_{go,d_1}(p,v)\,dp$.
There are smooth annular partitions about $g^+$ at integer depths
$0\le j\le L$ (with exterior piece and innermost ball) on which
-\begin{equation}\label{eq:ray-overlap-step}
- \begin{aligned}
+
+
\|w_{d_1}(g,\cdot)-a_{d_1}(j,L)\|_{C^m_{\rm normalized}}
&\le C_m/d_1,\\
a_{d_1}(j,L)&=\frac1{d_1}\int_{s>j}\psi(s/d_1)
\psi((s-2j+L)/d_1)\,ds,
- \end{aligned}
-\end{equation}
+
+
with constants uniform in $L,d_1\ge1$, and
-\begin{equation}\label{eq:ray-overlap-increments}
- |\partial_ja_{d_1}(j,L)|\le C/d_1,\qquad
- a_{d_1}(0,L)=1+O\bigl(\min(1,L/d_1)\bigr).
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-Put $go=(0,\epsilon)$, $\epsilon=e^{-L}$, and $g^+=0$. For finite
+
+ |∂_ja_{d_1}(j,L)|\le C/d_1,\qquad
+ a_{d_1}(0,L)=1+O(\min(1,L/d_1)).
+
+
+
+Proof.
+Put $go=(0,ε)$, $ε=e^{-L}$, and $g^+=0$. For finite
$v=(V,W)$, use
$p(v)=|V|^2/4$, $D_h(v)=(h^2+p(v))^2+|W|^2$, and
$P_h(v)=((h^2+p(v)+J_W)V/D_h,W/D_h)$. Then
-$\delta_{\sqrt{D_1}}L_{P_1^{-1}}jL_{v^{-1}}$ fixes $o$ and sends $v$
+$\delta_{√(D_1)}L_{P_1^{-1}}jL_{v^{-1}}$ fixes $o$ and sends $v$
to infinity, and the displaced point is $(n,e^\beta)$ with
-\begin{equation}\label{eq:ray-endpoint-coordinates}
- n=\delta_{\sqrt{D_1}}(P_1^{-1}P_\epsilon),\qquad
- e^\beta=\epsilon\sqrt{D_1/D_\epsilon}.
-\end{equation}
-Put $v=\delta_Ru$, $R=e^{-j}$, $t=\epsilon/R$, and define
+
+ n=\delta_{√(D_1)}(P_1^{-1}P_ε),\qquad
+ e^\beta=ε√(D_1/D_ε).
+
+Put $v=\delta_Ru$, $R=e^{-j}$, $t=ε/R$, and define
$\Delta_R=(1+R^2p(u))^2+R^4|W|^2$,
$Q_R=(R^2(1+R^2p+R^2J_W)V/\Delta_R,R^4W/\Delta_R)$. Then
-\begin{equation}\label{eq:ray-endpoint-scaled}
+
\beta=2j-L+B(R,t,u),\quad
B=\tfrac12\log(\Delta_R/D_t),\quad
- \delta_Rn=m(R,t,u)=\delta_{\sqrt{\Delta_R}}(Q_R^{-1}P_t).
-\end{equation}
+ \delta_Rn=m(R,t,u)=\delta_{√(\Delta_R)}(Q_R^{-1}P_t).
+
All derivatives of $B,m$ are bounded on the buffered annulus, and
$m\to P_0(u)\ne e$ as $R,t\to0$. The rest follows by writing
$H(m)=\int F(q)F(m^{-1}q)\,dq$ (smooth, compact, $H(e)=1$):
-\begin{equation}\label{eq:ray-overlap-exact}
+
w_{d_1}(g,v)=\int\psi(t)\psi(t-\beta/d_1)H(\delta_{e^{-td_1}}n)\,dt.
-\end{equation}
+
Replace $B$ by $0$ using the mean-value formula (extra factor $d_1^{-1}$
on each annular piece, using the support of $H$ and separation of $m$
from $e$); replace $H(\delta_{e^{j-s}}m)$ by
-$\mathbf{1}_{s>j}$ using Taylor's formula for $s>j$ and the support of
+$1_{s>j}$ using Taylor's formula for $s>j$ and the support of
$H$ for $s\le j$. The $O(1/d_1)$ derivative bounds follow similarly.
-\end{proof}
-\begin{proof}[Left density]
+
+Proof. [Left density]
At shift $g$, the left density has the coefficients of $YY^*$
weighted by the overlap of the two rays based at $o,go$ toward
$(gh)^+,gh^+$; only $r(g)\le4d_1+C$ contributes. Replace the second
endpoint by $(gh)^+$ with coefficient error at most
-\begin{equation}\label{eq:ray-endpoint-error}
- Ce^{C'd_1-l}\|Y\|_\infty\|Y\|_2,
-\end{equation}
+
+ Ce^{C'd_1-l}\|Y\|_∞\|Y\|_2,
+
by a compact-polar computation with auxiliary variable
$e^{l-r(h)}$ plus fixed-derivative bounds of isometries of length
$\le4d_1+C$ (product-form derivative growth at most
@@ -4290,143 +4290,143 @@ $e^{C_m d_1}$) and the same volume bounds.
At each annular depth the overlap error is then a unary symbol of
seminorm $O(d_1^{-1})$ on boundedly many reference boxes, giving
-$O(d_1^{-1})\|Y\|_\infty\|Y\|_2$ for the coefficient $L^2$ norm by
-Lemma~\ref{lem:ray-unary}; telescope $a_{d_1}(j,L)$ against single
-balls using \eqref{eq:ray-overlap-increments}, and observe the at
-most $C(1+d_1)$ levels. Summing plus \eqref{eq:ray-endpoint-error}
+$O(d_1^{-1})\|Y\|_∞\|Y\|_2$ for the coefficient $L^2$ norm by
+Lemma~lem:ray-unary; telescope $a_{d_1}(j,L)$ against single
+balls using (eq:ray-overlap-increments), and observe the at
+most $C(1+d_1)$ levels. Summing plus (eq:ray-endpoint-error)
gives the bound. The stronger short-coefficient version
$a=o(d_1)$ gives difference bounded in $L^2$ by
-\begin{equation}\label{eq:ray-short-density}
- C\bigl((1+a)/d_1+e^{C'd_1-l}\bigr)\|Y\|_\infty\|Y\|_2.
-\end{equation}
+
+ C((1+a)/d_1+e^{C'd_1-l})\|Y\|_∞\|Y\|_2.
+
Polarization extends to
-\begin{equation}\label{eq:ray-left-pairing}
- \langle\mathbb V_{d_1}Z,\Delta(x)\mathbb V_{d_1}Y\rangle
- -\langle Z,xY\rangle\longrightarrow0
-\end{equation}
+
+ <\mathbb V_{d_1}Z,\Delta(x)\mathbb V_{d_1}Y>
+ -< Z,xY>→0
+
uniformly when norms of $Z,Y,x$ are bounded with
$\supp\widehat x\subset\{r\le a\}$. Bounded approximants and
weakly convergent $L^2$ subsequences with slacked cutoffs handle
infinite Fourier support.
-\end{proof}
-\begin{proof}[Pairing (iii) and intertwining (ii)]
+
+Proof. [Pairing (iii) and intertwining (ii)]
$K_{d_1}(v,v)=1$; with $d(o,p)\le2d_1+C$ and fixed-derivative
uniform bounds, Taylor at the diagonal on adjacent boxes of radius
$e^{-d_2}$ gives
-\begin{equation}\label{eq:ray-kernel-near-one}
+
\|K_{d_1}-1\|_{C^m_{\rm normalized}}\le C_m e^{C_m d_1-d_2},
-\end{equation}
+
positive-derivative terms contributing $e^{-d_2}$, central ones
$e^{-2d_2}$. Choose $d_1\ll d_2\ll l'$. Separate $K_{d_1}-1$ at
scale $e^{-d_2}$; Cauchy--Schwarz for the cut columns bounds the
-fixed-frequency pairing by $\|t\|\|x\|_\infty
+fixed-frequency pairing by $\|t\|\|x\|_∞
(\sum\|Z_C\|_2^2)^{1/2}(\sum\|Y_{D'}\|_2^2)^{1/2}$, and the
matrix norm times the frequency sum vanishes. This proves
-\eqref{eq:ray-close-pairing}. For short right multiplication,
+(eq:ray-close-pairing). For short right multiplication,
$r(h),r(hk)\ge l-a$ and $r(k)\le a$ give
-$d_{\partial X}(h^+,(hk)^+)\le Ce^{-c(l-2a)}$ with fixed $c>0$, so
+$d_{∂ X}(h^+,(hk)^+)\le Ce^{-c(l-2a)}$ with fixed $c>0$, so
(iii) applies. Expanding
$\|\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\|_2^2$ finishes (ii) for
-the right; for the left use \eqref{eq:ray-left-pairing} with $x^*x$
+the right; for the left use (eq:ray-left-pairing) with $x^*x$
and $Z=xY$.
-\end{proof}
-\begin{corollary}[Stability under Hilbert-small changes]
-\label{cor:ray-small-change}
-If $Y$ is long with bounded $\|Y\|_\infty,\|Y\|_2$ and
+
+[Stability under Hilbert-small changes]
+
+If $Y$ is long with bounded $\|Y\|_∞,\|Y\|_2$ and
$x-x'\to0$ in $L^2(D)$ with $x,x'$ uniformly bounded, then
$\|\Delta(x-x')\mathbb V_{d_1}Y\|_2\to0$ and
$\|(\mathbb V_{d_1}Y)(x-x')\|_2\to0$; the analogous amplified
statement holds in the semifinite $L^2$ norm.
-\end{corollary}
-\begin{proof}
+
+Proof.
Pair the corresponding density with $(x-x')^*(x-x')$ or its
reverse; tracial Cauchy--Schwarz with $z=x-x'$ gives
-\begin{equation}\label{eq:ray-small-change-bound}
+
\max\{\|\Delta(z)\mathbb V_{d_1}Y\|_2^2,
\|(\mathbb V_{d_1}Y)z\|_2^2\}
- \le C\|Y\|_\infty\|Y\|_2\|z\|_\infty\|z\|_2.
-\end{equation}
+ \le C\|Y\|_∞\|Y\|_2\|z\|_∞\|z\|_2.
+
The amplified version uses $\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*$
and the semifinite norm. For fields, integrate the pointwise bound
and apply Cauchy--Schwarz on the base.
-\end{proof}
-\subsection{Admissible representations and spherical laws}
+
+## Admissible representations and spherical laws
Fix an isolated measured factor $T$ of $G$, its symmetric space
$X_T$, basepoint $o_T$, and $r_T(h)=d(o_T,h_T o_T)$ (normalized at
the indivisible simple root). $D_f$ denotes the Fourier multiplier
with symbol $f$. Finite factor traces are normalized.
-\begin{definition}\label{def:ray-admissible}
+
An admissible representation for $T$ is a finite factor
-$D=B\bar\otimes L_\eta(\Gamma)$ together with a normalized
-projective representation $\Theta:\Gamma\to\mathcal U(D)$ such that
-$D\cap\Theta(\Gamma)'=\mathbb C1$ and the representation
-\begin{equation}\label{eq:ray-amplified-action}
- \gamma\longmapsto\Ad\Delta_T(\Theta(\gamma))
+$D=B\bar ⊗ L_\eta(\Gamma)$ together with a normalized
+projective representation $Θ:\Gamma\to\mathcal U(D)$ such that
+$D\capΘ(\Gamma)'=\mathbb C1$ and the representation
+
+ \gamma⟼\Ad\Delta_T(Θ(\gamma))
\quad\hbox{on}\quad
- L^2\bigl(B(L^2(X_T))\bar\otimes D,\Tr\otimes\tau\bigr)
-\end{equation}
+ L^2(B(L^2(X_T))\bar ⊗ D,\Tr ⊗ \tau)
+
has no invariant vector. Here
-$\Delta_T(u_h)=\lambda_{X_T}(h_T)\otimes u_h$. The scalar multiplier
-of $\Theta$ need not equal $\eta$.
-\end{definition}
+$\Delta_T(u_h)=\lambda_{X_T}(h_T) ⊗ u_h$. The scalar multiplier
+of $Θ$ need not equal $\eta$.
+
-For $a\in G$ put $X^\Theta_a(y)=\Theta(c(a,y))$ (the section and
-cocycle of Section~\ref{sec:setup}). For an isolated source factor
+For $a\in G$ put $X^Θ_a(y)=Θ(c(a,y))$ (the section and
+cocycle of Section~sec:setup). For an isolated source factor
$S$ and positive split translation $a_t$ of length $t$, the spherical
law at $t$, on scale $R>0$, is the probability
-\begin{equation}\label{eq:ray-spherical-law}
- P_{\Theta,R}(t)(E)=\int_Y
- \bigl\|D_{1_{\{r_T/R\in E\}}}X^\Theta_{a_t}(y)\bigr\|_2^2\,dy,
-\end{equation}
-for Borel $E\subset[0,\infty]$.
-
-\begin{lemma}[Uniform short approximation]\label{lem:ray-compactshort}
-Let $R_n\to\infty$ and $\Theta_n$ be projective representations in
+
+ P_{Θ,R}(t)(E)=\int_Y
+ \bigl\|D_{1_{\{r_T/R\in E\}}}X^Θ_{a_t}(y)\bigr\|_2^2\,dy,
+
+for Borel $E\subset[0,∞]$.
+
+[Uniform short approximation]
+Let $R_n\to∞$ and $Θ_n$ be projective representations in
finite tensors, with
-\begin{equation}\label{eq:ray-short-fixed}
- \|D_{1_{\{r_T>\varepsilon R_n\}}}\Theta_n(\gamma)\|_2\to0
- \quad(\gamma\in\Gamma,\ \varepsilon>0).
-\end{equation}
+
+ \|D_{1_{\{r_T>ε R_n\}}}Θ_n(\gamma)\|_2\to0
+ \quad(\gamma\in\Gamma,\ ε>0).
+
Then there are $a_n=o(R_n)$ and uniformly operator-bounded fields
$Z_{n,s}$ with Fourier support in $\{r_T\le a_n\}$ such that
-\begin{equation}\label{eq:ray-short-compact}
- \sup_{s\in C}\|X^{\Theta_n}_s-Z_{n,s}\|_{L^2(Y;L^2(D_n))}\to0
-\end{equation}
+
+ \sup_{s\in C}\|X^{Θ_n}_s-Z_{n,s}\|_{L^2(Y;L^2(D_n))}\to0
+
for every compact $C\subset G$. Thus compact pre- and post-translations
do not affect any limiting rescaled length law.
-\end{lemma}
-\begin{proof}
+
+Proof.
For fixed compact $C$, outside a set of measure at most
-$2\varepsilon$ both $d_0(y),d_0(sy)\in K$, so
+$2ε$ both $d_0(y),d_0(sy)\in K$, so
$c(s,y)\in\Gamma\cap K^{-1}CK$, a finite set; apply
-\eqref{eq:ray-short-fixed} to it. Smooth cutoffs
-$b_n=o(R_n)$ with $a_n=2b_n$ and Proposition~\ref{prop:ray-radial}
+(eq:ray-short-fixed) to it. Smooth cutoffs
+$b_n=o(R_n)$ with $a_n=2b_n$ and Proposition~prop:ray-radial
give the bounds. The radial multiplier uses
-\begin{equation}\label{eq:ray-triangle}
+
|r_T(gh)-r_T(h)|\le r_T(g),\qquad |r_T(hg)-r_T(h)|\le r_T(g)
-\end{equation}
+
to show a short multiplier cannot connect two closed rescaled radial
intervals at positive distance; smooth radial cuts then have
Hilbert-small error once one factor is small and the other
uniformly operator-bounded.
-\end{proof}
-\begin{lemma}[Averaging with a uniform gap]\label{lem:ray-kazhdan}
+
+[Averaging with a uniform gap]
There is a compactly supported, absolutely continuous, symmetric
probability $m_0$ on $G$ and $0<c_0<1$ such that its convolution
operator is positive in every unitary representation and has norm at
most $c_0$ on the orthogonal complement of the invariant vectors.
-\end{lemma}
-\begin{proof}
-Take a compact Kazhdan pair $(C,\varepsilon)$ for the product
+
+Proof.
+Take a compact Kazhdan pair $(C,ε)$ for the product
$G$; a symmetric compactly supported density $\nu$ that is bounded
below on $V\cup CV$ and their inverses, for a compact neighborhood
$V$ of $1$, satisfies
@@ -4437,64 +4437,64 @@ $(1+\pi(\nu))/2$ is positive with the same bound, and convolving on
both sides by a symmetric compactly supported probability $\chi$
gives a positive operator
$\pi(\chi)(1+\pi(\nu))\pi(\chi)/2$ with the same bound.
-\end{proof}
-
-\begin{proposition}[Addition in distribution]\label{prop:ray-addition}
-Let $R_n\to\infty$, $\Theta_n$ admissible for $T$ satisfying
-\eqref{eq:ray-short-fixed}. Fix an isolated source factor and a
-positive split axis. For any $t_n,u_n\ge0$, if $P_{\Theta_n,R_n}(t_n)$
-converges weakly to $P_t$ and $P_{\Theta_n,R_n}(u_n)$ to $P_u$, then
-\begin{equation}\label{eq:input-8}
- P_{\Theta_n,R_n}(t_n+u_n)\longrightarrow P_t*P_u.
-\end{equation}
+
+
+[Addition in distribution]
+Let $R_n\to∞$, $Θ_n$ admissible for $T$ satisfying
+(eq:ray-short-fixed). Fix an isolated source factor and a
+positive split axis. For any $t_n,u_n\ge0$, if $P_{Θ_n,R_n}(t_n)$
+converges weakly to $P_t$ and $P_{Θ_n,R_n}(u_n)$ to $P_u$, then
+
+ P_{Θ_n,R_n}(t_n+u_n)→ P_t*P_u.
+
The same holds with compact pre- and post-translations.
-\end{proposition}
-\begin{proof}[Sketch]
+
+Proof. [Sketch]
Pass to a subsequence with the sum laws converging. Insert a fixed
translation $s$ between the two positive axial segments; away from
the Bruhat wall (a proper lower-dimensional cell of measure zero),
Cartan multiplication expresses $a_{t_n}sa_{u_n}=a_{t_n+u_n}$ up to
-compact pre/post translations; Lemma~\ref{lem:ray-compactshort}
+compact pre/post translations; Lemma~lem:ray-compactshort
identifies the uncut law of the inserted product with the limiting
sum.
Replacing the actual transporter by its short approximation, average
$s$ over $m_0^{*k}$. For bounded smooth radial cuts
-$A_{n,f}=D_{f(r_T/R_n)}X^{\Theta_n}_{a_{t_n}}$,
-$B_{n,g}=D_{g(r_T/R_n)}X^{\Theta_n}_{a_{u_n}}(a_{u_n}^{-1}\cdot)$,
+$A_{n,f}=D_{f(r_T/R_n)}X^{Θ_n}_{a_{t_n}}$,
+$B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}\cdot)$,
the mixed Gram forms are pairings between $A_{n,f'}^*A_{n,f}$ and
the translate of $B_{n,g}B_{n,g'}^*$ under the induced trace action.
Its invariant space consists of scalars; averaging with $m_0^{*k}$
gives a uniform $O(c_0^k)$ gap. Two successive Hilbert limits give an
isometry
-\begin{equation}\label{eq:ray-convolution-link}
- J:L^2(P_t)\otimes L^2(P_u)\longrightarrow\mathcal K,\qquad
- f\otimes g\longmapsto\bigl(A_{n,f}(sy)X^{\Theta_n}_s(y)B_{n,g}(y)\bigr)^{\lim}.
-\end{equation}
+
+ J:L^2(P_t) ⊗ L^2(P_u)→\mathcal K,\qquad
+ f ⊗ g⟼(A_{n,f}(sy)X^{Θ_n}_s(y)B_{n,g}(y))^{\lim}.
+
For a positive finite output window strictly above the sum of the
-inputs, \eqref{eq:ray-triangle} plus short approximation excludes it.
+inputs, (eq:ray-triangle) plus short approximation excludes it.
For the strict-deficit case, in the shortening test
-$A(sy)^*$ pairs against $X^{\Theta_n}_s(y)B(y)C(y,s)$; convert to
-\begin{equation}\label{eq:ray-shortening-test}
- \tau\bigl(\zeta_n(sy)^*\Delta_T(X^{\Theta_n}_s(y))\xi_n(y)C(y,s)\bigr)
-\end{equation}
+$A(sy)^*$ pairs against $X^{Θ_n}_s(y)B(y)C(y,s)$; convert to
+
+ \tau(\zeta_n(sy)^*\Delta_T(X^{Θ_n}_s(y))\xi_n(y)C(y,s))
+
with $\xi_n=\mathbb V_{d_n}B$, $\zeta_n=\mathbb V_{d_n}(A^*)$. The
exact trace inequality (via Cauchy--Schwarz on the two density fields,
each with uniformly bounded trace-Hilbert norm by
-Proposition~\ref{prop:ray-lift}) is bounded by $C c_0^k$ because the
+Proposition~prop:ray-lift) is bounded by $C c_0^k$ because the
combined amplified action has no invariant vector (admissibility).
For a short change $z$, the error is at most
-$\|z\|_\infty\|z\|_2\|\rho^L_{\xi_n}\|_2$, i.e.
-\begin{equation}\label{eq:ray-density-error}
+$\|z\|_∞\|z\|_2\|\rho^L_{\xi_n}\|_2$, i.e.
+
\|\Delta_T(z)\xi_n\|^2
- \le\|z\|_\infty\|z\|_2\|\rho^L_{\xi_n}\|_2.
-\end{equation}
-Limit $n\to\infty$ then $k\to\infty$.
+ \le\|z\|_∞\|z\|_2\|\rho^L_{\xi_n}\|_2.
+
+Limit $n\to∞$ then $k\to∞$.
-For input values $0$ and $+\infty$, use triangle inequalities and
-short approximations with smooth tail cutoffs (Proposition~\ref{prop:ray-radial});
+For input values $0$ and $+∞$, use triangle inequalities and
+short approximations with smooth tail cutoffs (Proposition~prop:ray-radial);
the lifted shortening argument extends because it uses positive
lower length bounds and a uniform deficit but no upper bound.
@@ -4504,19 +4504,19 @@ supported on rectangles whose closures lie in $a^{-1}(F)^c$ have
dense span in $\ran P_{\rm in}(a^{-1}(F)^c)$, so
$E(F)J P_{\rm in}(a^{-1}(F)^c)=0$ and
$E(F)J=E(F)J P_{\rm in}(a^{-1}(F))$. A closed-set exhaustion
-$K_j\uparrow F^c$ gives $E(K_j)JP_{\rm in}(a^{-1}(F))=0$, so
+$K_j↑ F^c$ gives $E(K_j)JP_{\rm in}(a^{-1}(F))=0$, so
$E(F^c)JP_{\rm in}(a^{-1}(F))=0$; together,
$E(F)J=J P_{\rm in}(a^{-1}(F))$. Monotone classes and spectral
integration give
-$E(\varphi)J=JM_{\varphi(s+t)}$ for $\varphi\in C([0,\infty])$.
+$E(\varphi)J=JM_{\varphi(s+t)}$ for $\varphi\in C([0,∞])$.
Convergence on the whole sequence follows by compactness of
-probability measures on $[0,\infty]$.
-\end{proof}
+probability measures on $[0,∞]$.
-\section{Linear subsequences and heights for isolated components}
-\label{sec:scale}
-At an isolated type, Section~\ref{sec:transfer} has already supplied
+# Linear subsequences and heights for isolated components
+
+
+At an isolated type, Section~sec:transfer has already supplied
sharp boundary measurements $x\mapsto P_x$. We now construct a height
at each source label: a positive injective self-adjoint operator
commuting with $P_x$ and transforming by the inverse boundary
@@ -4526,7 +4526,7 @@ is a square-integrable derivative on a dense invariant domain.
The first part controls Fourier lengths: along long source intervals,
every fixed input retains positive mass below a linear length bound.
-The addition theorem (Proposition~\ref{prop:ray-addition}) plus
+The addition theorem (Proposition~prop:ray-addition) plus
property~$(T)$ rules out total superlinear growth. Lattice packing then
costs one factor of the length, canceled by averaging over a source
stabilizer interval of comparable length. The resulting estimate for
@@ -4539,10 +4539,10 @@ real and locally $\Sp(n,1)$, $n\ge2$, or $F_{4(-20)}$. Complementary
factors may be archimedean or nonarchimedean; a compact window in those
coordinates is retained in each lattice count. Horizontal metrics and
angular probabilities are the compatible compact-invariant choices of
-Section~\ref{sec:ray}.
+Section~sec:ray.
We use the admissibility and statistical conventions of
-Definition~\ref{def:ray-admissible}. All varying finite factor traces
+Definition~def:ray-admissible. All varying finite factor traces
are normalized, $Y=G/\Gamma$ has probability measure, and extra finite
matrices carry the unnormalized matrix trace with fixed size. Each
application of the ray estimates has both the stated operator bounds
@@ -4551,42 +4551,42 @@ fixed Hilbert vector, not a bound uniform over a moving unit ball.
The earlier geometric conclusions used: complementary regular
realizations and all-factor mixing
-(Lemma~\ref{lem:setup-normal-tensors}, Proposition~\ref{prop:setup-induced-algebras},
-Lemma~\ref{lem:setup-product-mixing}); full-gap escape on complete
-vertex-origin lexicographic chains (Proposition~\ref{prop:links-gap-count});
+(Lemma~lem:setup-normal-tensors, Proposition~prop:setup-induced-algebras,
+Lemma~lem:setup-product-mixing); full-gap escape on complete
+vertex-origin lexicographic chains (Proposition~prop:links-gap-count);
sharp, strongly continuous pure-vertex measurements and their
-covariances from Propositions~\ref{prop:transfer-types},
-\ref{prop:transfer-isolated-sharpness}, and~\ref{prop:transfer-sharp-flags};
+covariances from Propositions~prop:transfer-types,
+prop:transfer-isolated-sharpness, and~prop:transfer-sharp-flags;
scalarity of limiting rescaled one-column effects
-(Lemma~\ref{lem:links-invariant-variables}). The compact-change argument
+(Lemma~lem:links-invariant-variables). The compact-change argument
for those effects uses bounded approximants of the fixed transporters
and has arbitrary fixed slack in radial windows. These are conclusions,
not new assumptions.
-\subsection{An admissible model for the original module}
+## An admissible model for the original module
-\begin{lemma}\label{lem:scale-original-model}
-For the two-slot model of Section~\ref{sec:setup} there is a finite
-tensor $D=B\bar\otimes L_\eta(\Gamma)$, with $B$ diffuse, and an
+
+For the two-slot model of Section~sec:setup there is a finite
+tensor $D=B\bar ⊗ L_\eta(\Gamma)$, with $B$ diffuse, and an
admissible representation $\delta:\Gamma\to\mathcal U(D)$ for each
isolated measured factor, obtained by a fixed finite matrix corner
equivalence. It has a sequence $\gamma_n$ whose Fourier laws escape
in every measured simple factor. Rescaled continuous radial laws,
scalar effects, and complementary tightness are preserved by this
corner equivalence.
-\end{lemma}
-\begin{proof}
-The complementary $\Lambda$-factor in the two-slot regular model is
+
+Proof.
+The complementary $Λ$-factor in the two-slot regular model is
diffuse; in a large enough finite matrix amplification choose a
projection in that coefficient factor with the same trace as the
regular-model projection $e$, and identify by a partial isometry to
get the required unital tensor-coordinate representation. Scalar
-relative commutant follows from Proposition~\ref{prop:setup-induced-algebras}.
+relative commutant follows from Proposition~prop:setup-induced-algebras.
Along a full lexicographic chain starting at a vertex and adding one
new simple gap per stage, all target gaps diverge
-(Proposition~\ref{prop:links-gap-count}). Apply to the finite matrix of
+(Proposition~prop:links-gap-count). Apply to the finite matrix of
unit columns and choose finite path times making the integrated mass
in successively larger Cartan windows small, with each individual error
small at the same base point (average over the source stabilizer
@@ -4599,47 +4599,47 @@ a bounded amount, so escape and rescaled radial tests persist; the
uncut Gram factor tends to the Gram of the partial isometry, and
slack preserves complementary tightness. For amplified admissibility,
finite-rank compressions of $\lambda_{X_T}(h_T)$ tend to zero when
-$r_T(h)\to\infty$ (embed $L^2(X_T)$ in a multiple of $L^2(T)$ and
+$r_T(h)\to∞$ (embed $L^2(X_T)$ in a multiple of $L^2(T)$ and
test compactly supported vectors); compress
$\Delta_T(\delta(\gamma_n))$ between finite-rank operators and use
Plancherel plus escaping Fourier laws to make tracial $L^2$ norms go
to zero, which forbids an invariant vector.
-\end{proof}
-\begin{lemma}[Entropy lower bound]\label{lem:scale-entropy}
+
+[Entropy lower bound]
Suppose an isolated source type $i$, with factor $S$, is matched to the
measured type $p(i)$, with factor $T$. Let $Q_S,Q_T$ be the homogeneous
dimensions of their boundaries. For the representation $\delta$ of
-Lemma~\ref{lem:scale-original-model}, complementary lengths are
+Lemma~lem:scale-original-model, complementary lengths are
uniformly tight on the pure source ray, and for every
$0<c<Q_S/Q_T$,
-\begin{equation}
- P_{\delta,t}(t)([0,c])\longrightarrow0.
- \label{eq:scale-entropy}
-\end{equation}
+
+ P_{\delta,t}(t)([0,c])→0.
+
+
Both hold uniformly under source pre- and post-translations in
compact sets.
-\end{lemma}
-\begin{proof}
+
+Proof.
Complementary tightness is the pure-ray conclusion of
-Section~\ref{sec:transfer}, transported through the corner equivalence.
+Section~sec:transfer, transported through the corner equivalence.
Before the equivalence, complementary regularity supplies a nonzero
-constant frame column $f$ for $\pi_0(\Gamma)$ with $W_\Lambda$; if
+constant frame column $f$ for $\pi_0(\Gamma)$ with $W_Λ$; if
$e_j$ are the finitely many identity columns, the frame inequality
applied to $W_he_j$ gives, for every $h\in\Gamma$,
-\begin{equation}
+
\sum_{\gamma\in\Gamma}
\|D_{\{h\}}\pi_0(\gamma)f\|_2^2\le C_f.
- \label{eq:scale-bessel-count}
-\end{equation}
+
+
-Let $\Omega_t$ be a source Cartan shell of fixed positive thickness
+Let $Ω_t$ be a source Cartan shell of fixed positive thickness
about length $t$ in $S$, with all compact rotations and complementary
-coordinates in a fixed compact neighborhood; $|\Omega_t|\asymp
+coordinates in a fixed compact neighborhood; $|Ω_t|\asymp
e^{Q_St}$. Restricting both section values to a compact $K$, the times
with transporter $\gamma$ lie in $K\gamma d_0(y)^{-1}$ of Haar volume
-at most $|K|$; integrating \eqref{eq:scale-bessel-count} therefore
+at most $|K|$; integrating (eq:scale-bessel-count) therefore
bounds the averaged mass at each measured index by a constant
independent of index and $t$.
@@ -4647,48 +4647,48 @@ For a fixed complementary compact window, the number of measured
lattice indices with $r_T(h)\le c't$ is at most $Ce^{Q_Tc't}$
(thicken on the right by a small lattice-disjoint neighborhood in the
ambient group; Cartan length changes by a bounded amount). Thus the
-restricted low-length mass averaged over $\Omega_t$, divided by
-$|\Omega_t|$, is at most
-\begin{equation}
+restricted low-length mass averaged over $Ω_t$, divided by
+$|Ω_t|$, is at most
+
Ce^{(Q_Tc'-Q_S)t}.
- \label{eq:scale-volume-ratio}
-\end{equation}
+
+
Choose $c<c'<Q_S/Q_T$. Lost section sets have arbitrarily small
measure uniformly in translating time, and the uncut translated column
has unchanged norm. Complementary tightness removes the fixed measured
-window. Lemma~\ref{lem:links-invariant-variables} makes every limiting
+window. Lemma~lem:links-invariant-variables makes every limiting
low-length effect scalar, and the compact-change argument identifies
its value uniformly throughout the thick shell. A subsequence
-violating \eqref{eq:scale-entropy} would have positive averaged mass in
-\eqref{eq:scale-volume-ratio}, contradiction. The corner-change
+violating (eq:scale-entropy) would have positive averaged mass in
+(eq:scale-volume-ratio), contradiction. The corner-change
argument finishes.
-\end{proof}
-\subsection{Tightening a Kazhdan window}
+
+## Tightening a Kazhdan window
For cutoffs $L_n\ge1$, a bounded sequence $x_n\in D_n$ is
-\emph{tight at scale $L_n$} if
-\begin{equation}
- \lim_{C\to\infty}\lim_{n\to\mathcal U}
+tight at scale $L_n$ if
+
+ \lim_{C\to∞}\lim_{n\to\mathcal U}
\|D_{1_{\{r_T>CL_n\}}}x_n\|_2=0
- \label{eq:scale-tight-sequence}
-\end{equation}
+
+
for a free ultrafilter $\mathcal U$. Ultraproducts are tracial;
ordinary subsequences are extracted once all countably many
requirements are fixed.
-\begin{lemma}\label{lem:scale-tight-factor}
+
The tight elements form a von Neumann subalgebra $Q_0$ of
-$\prod_{\mathcal U}D_n$, and $Q_0$ is a factor. If each $\Theta_n$
-is relatively irreducible, the ultraproduct representation $\Theta$
+$\prod_{\mathcal U}D_n$, and $Q_0$ is a factor. If each $Θ_n$
+is relatively irreducible, the ultraproduct representation $Θ$
also has scalar relative commutant.
-\end{lemma}
-\begin{proof}
-Smooth radial truncations are uniformly bounded (Proposition~\ref{prop:ray-radial}),
+
+Proof.
+Smooth radial truncations are uniformly bounded (Proposition~prop:ray-radial),
so tight elements admit Hilbert approximation by bounded sequences
-supported in $r_T\le CL_n$. Triangle inequality \eqref{eq:ray-triangle}
+supported in $r_T\le CL_n$. Triangle inequality (eq:ray-triangle)
gives closure under multiplication; length-invariance under inversion
gives closure under adjoint; tightness is closed in Hilbert norm on
each bounded ball.
@@ -4704,50 +4704,50 @@ orthogonal complement of the identity Fourier coefficient without
invariant vectors (ICC plus scalar phases). A fixed Kazhdan pair
for $\Gamma$ gives a uniform spectral gap even as cocycles vary.
Approximate commutation forces $z$ scalar.
-\end{proof}
-\subsection{Near-identity Kazhdan tightening}
-\begin{lemma}[Kazhdan tightening]\label{lem:scale-tightening}
+## Near-identity Kazhdan tightening
+
+[Kazhdan tightening]
Fix a finite Kazhdan set $\mathcal F\subset\Gamma$ and
-$\varepsilon>0$. There is $\varepsilon_K>0$ such that, if
-$\Theta_n$ are admissible and
-\begin{equation}
- \|D_{1_{\{r_T>L_n\}}}\Theta_n(\gamma)\|_2\le\varepsilon_K
+$ε>0$. There is $ε_K>0$ such that, if
+$Θ_n$ are admissible and
+
+ \|D_{1_{\{r_T>L_n\}}}Θ_n(\gamma)\|_2\leε_K
\qquad(\gamma\in\mathcal F),
- \label{eq:scale-kazhdan-window}
-\end{equation}
+
+
then, after passing to a subsequence, there exist unitaries
-$u_n$ with $\limsup\|u_n-1\|_2<\varepsilon$ such that every fixed
-element of $u_n^*\Theta_n(\Gamma)u_n$ is tight at scale $L_n$.
+$u_n$ with $limsup\|u_n-1\|_2<ε$ such that every fixed
+element of $u_n^*Θ_n(\Gamma)u_n$ is tight at scale $L_n$.
Admissibility is preserved.
-\end{lemma}
-\begin{proof}[Sketch]
+
+Proof. [Sketch]
Let $D_\mathcal U=\prod_{\mathcal U}D_n$ and $Q_0$ the tight
-factor of Lemma~\ref{lem:scale-tight-factor}. The Jones basic
-construction $\langle D_\mathcal U,e_{Q_0}\rangle$ gives
-\begin{equation}
- \|\Theta(\gamma)e_{Q_0}\Theta(\gamma)^*-e_{Q_0}\|_2^2
- =2\|\Theta(\gamma)-E_{Q_0}\Theta(\gamma)\|_2^2,
- \label{eq:scale-jones-movement}
-\end{equation}
-which is $O(\varepsilon_K^2)$ by tightness plus smooth radial
-cutoffs (Proposition~\ref{prop:ray-radial}). The Hilbert-space
+factor of Lemma~lem:scale-tight-factor. The Jones basic
+construction $< D_\mathcal U,e_{Q_0}>$ gives
+
+ \|Θ(\gamma)e_{Q_0}Θ(\gamma)^*-e_{Q_0}\|_2^2
+ =2\|Θ(\gamma)-E_{Q_0}Θ(\gamma)\|_2^2,
+
+
+which is $O(ε_K^2)$ by tightness plus smooth radial
+cutoffs (Proposition~prop:ray-radial). The Hilbert-space
Kazhdan inequality supplies a positive contraction $z$ close to
$e_{Q_0}$; spectral rounding $p_0=1_{[1/2,1]}(z)$ gives
-$\|p_0-e_{Q_0}\|_2=O(\varepsilon_K)$
-\begin{equation}
- \|p_0-e_{Q_0}\|_2\le2\|z-e_{Q_0}\|_2=O(\varepsilon_K).
- \label{eq:scale-rounded-projection}
-\end{equation}
-$p_0$ commutes with $\Theta(\Gamma)$ and has trace $d'$ near one;
-take a right $Q_0$-module isometry with row $V\in L^2(D_\mathcal U)^{\oplus2}$
-satisfying $Vq=V$, $E_{M_2(Q_0)}(V^*V)=q$, $\Theta(\gamma)V=Vb_\gamma$;
-\begin{equation}
+$\|p_0-e_{Q_0}\|_2=O(ε_K)$
+
+ \|p_0-e_{Q_0}\|_2\le2\|z-e_{Q_0}\|_2=O(ε_K).
+
+
+$p_0$ commutes with $Θ(\Gamma)$ and has trace $d'$ near one;
+take a right $Q_0$-module isometry with row $V\in L^2(D_\mathcal U)^{ ⊕ 2}$
+satisfying $Vq=V$, $E_{M_2(Q_0)}(V^*V)=q$, $Θ(\gamma)V=Vb_\gamma$;
+
VV^*=d'1.
- \label{eq:scale-bounded-basis}
-\end{equation}
+
+
$e_1=(d')^{-1}V^*V\le q$ is a projection of trace one commuting
with all $b_\gamma$
@@ -4755,317 +4755,317 @@ The excess $H_1=e_1-q/d'$ is nonzero, orthogonal to $Q_0$, and
commutes with $b_\gamma$ if $d'>1$. Choose bounded representatives
$G_n$ with exact support $r_T\ge l_n$ and short approximations to
$q,b_\gamma$ with support $r_T\le a_n$:
-\begin{equation}
+
L_n\ll a_n\ll d_n\ll l_n,\qquad
\|G_n-H_{1,n}\|_2\to0.
- \label{eq:scale-separated-scales}
-\end{equation}
+
+
The ray lift sends $G_n$ to $\xi_n=\mathbb V_{d_n}G_n$ with
-\begin{equation}
+
\Delta_T(b_{\gamma,n})\xi_n-\xi_nb_{\gamma,n}\to0,\quad
\Delta_T(q_n)\xi_n-\xi_n\to0,\quad\xi_nq_n-\xi_n\to0
- \label{eq:scale-lifted-intertwining}
-\end{equation}
+
+
and $\xi'_n=\Delta_T(e_{1,n})\xi_n$ satisfies
-\begin{equation}
+
\|\xi'_n\|^2\to(d')^{-1}\|H_1\|_2^2>0,
- \label{eq:scale-nonzero-lift}
-\end{equation}
-\begin{equation}
- \Delta_T(\Theta'_n(\gamma))\xi'_n-\xi'_nb_{\gamma,n}\to0.
- \label{eq:scale-almost-intertwiner}
-\end{equation}
+
+
+
+ \Delta_T(Θ'_n(\gamma))\xi'_n-\xi'_nb_{\gamma,n}\to0.
+
+
Extend $b_{\gamma,n}$ off $q_n$; the normalized left vector
states are almost invariant in predual norm, whose square-root
densities are almost invariant vectors by Powers--St\o rmer
-\begin{equation}
+
\|A^{1/2}-B^{1/2}\|_2^2\le\|A-B\|_1\quad(A,B\ge0).
- \label{eq:scale-square-root}
-\end{equation}
+
+
Conjugation on this corner is unitarily equivalent to the
original amplified conjugation; admissibility is contradicted.
Hence $e_1=q$, $d'=1$, and a factor equivalence gives a
-unitary $u$ with $u^*\Theta(\Gamma)u\subset Q_0$ and
+unitary $u$ with $u^*Θ(\Gamma)u\subset Q_0$ and
$p_0=ue_{Q_0}u^*$. Aligning the polar part of $E_{Q_0}u$ yields
-\begin{equation}
+
\|u-1\|_2^2\le2-2\|E_{Q_0}u\|_2^2
=\|p_0-e_{Q_0}\|_2^2.
- \label{eq:scale-close-unitary}
-\end{equation}
+
+
Diagonalize countably many tightness conditions; small
-$\varepsilon_K$ gives the bound.
-\end{proof}
+$ε_K$ gives the bound.
+
-\subsection{Composition of admissible kernels}
+## Composition of admissible kernels
-\begin{lemma}[Exact insertion]\label{lem:scale-insertion}
-Let $\delta:L_\alpha(\Gamma)\to B_0\bar\otimes L_\beta(\Gamma)$
+[Exact insertion]
+Let $\delta:L_\alpha(\Gamma)\to B_0\bar ⊗ L_\beta(\Gamma)$
be a trace-preserving embedding with relatively irreducible group
representation and a sequence escaping in Fourier probability in
-every ambient simple factor. For admissible $\Theta$ at an
+every ambient simple factor. For admissible $Θ$ at an
isolated measured factor $T$ there is an admissible
-$\Theta\circledast\delta$ with source multiplier $\alpha$, whose
+$Θ\circledast\delta$ with source multiplier $\alpha$, whose
spherical Fourier laws are the exact compositions of the
-kernels of $\delta$ and $\Theta$. The final measured cocycle
-is that of $\Theta$.
-\end{lemma}
+kernels of $\delta$ and $Θ$. The final measured cocycle
+is that of $Θ$.
-\begin{proof}[Sketch]
-Write $\xi$ for the source multiplier of $\Theta$ and put
-$T_h(y)=\Theta(c(h,h^{-1}y))$. The section identity gives
+
+Proof. [Sketch]
+Write $\xi$ for the source multiplier of $Θ$ and put
+$T_h(y)=Θ(c(h,h^{-1}y))$. The section identity gives
$T_g(y)T_h(g^{-1}y)=\nu(g,h)(y)T_{gh}(y)$ with
$\nu(g,h)(y)=\xi(c(g,g^{-1}y),c(h,g^{-1}g^{-1}y))$, a normalized
scalar cocycle for the action on $Y$. In the twisted crossed
product $\Psi(w_h)=T_hb_h$, with multiplier $\beta/\nu$, is
trace-preserving on the twisted group algebra and extends
-normally; apply $\id_{B_0}\otimes\Psi$ to $\delta$. The kernel
+normally; apply $\id_{B_0} ⊗ \Psi$ to $\delta$. The kernel
formula
-\begin{equation}
+
\sum_{h\in\Gamma}\|a_{\gamma,h}\|_2^2
- \int_Y\bigl\|D_{1_{\{r_T\in E\}}}\Theta(c(h,h^{-1}y))\bigr\|_2^2\,dy
- \label{eq:scale-kernel-composition}
-\end{equation}
+ \int_Y\bigl\|D_{1_{\{r_T\in E\}}}Θ(c(h,h^{-1}y))\bigr\|_2^2\,dy
+
+
composes the kernels. Factoriality and admissibility come from
the induced coefficient representation
-\begin{equation}
+
(\mathcal U_gk)(y)=\Ad T_g(y)\,k(g^{-1}y),
- \label{eq:scale-induced-coefficients}
-\end{equation}
+
+
with scalar invariant space; for bounded finite tests and
$A=\sum_ia_iw_i$ one of the escaping unitaries, the coefficient
pairing is bounded by
-\begin{equation}
+
\sum_{i,j:\,ih=h'j}\|a_i\|_2\|a_j\|_2
- |\langle k',\mathcal U_i k\rangle|,
- \label{eq:scale-composition-decay}
-\end{equation}
+ |< k',\mathcal U_i k>|,
+
+
which vanishes off the scalar summand by Fourier escape, Howe--Moore,
-and the product invariant-space decomposition of Section~\ref{sec:setup}.
+and the product invariant-space decomposition of Section~sec:setup.
The amplified assertion replaces $\mathcal E$ by
-$L^2(Y;L^2(B(L^2(X_T))\bar\otimes D))$ and $T_g$ by
-$\Delta_T(T_g)$, and admissibility of $\Theta$ gives absence of
+$L^2(Y;L^2(B(L^2(X_T))\bar ⊗ D))$ and $T_g$ by
+$\Delta_T(T_g)$, and admissibility of $Θ$ gives absence of
invariant vectors.
-\end{proof}
-\subsection{A linear subsequence}
-\begin{lemma}\label{lem:scale-many-increments}
-Suppose $P_{\Theta_n,R_n}(t_n)([a,\infty])\ge\alpha$ with
-$a,\alpha>0$ and $u_n/t_n\to\infty$. Then
-\begin{equation}
- P_{\Theta_n,R_n}(u_n)\longrightarrow\delta_\infty,
- \label{eq:scale-many-increments}
-\end{equation}
+## A linear subsequence
+
+
+Suppose $P_{Θ_n,R_n}(t_n)([a,∞])\ge\alpha$ with
+$a,\alpha>0$ and $u_n/t_n\to∞$. Then
+
+ P_{Θ_n,R_n}(u_n)→\delta_∞,
+
+
uniformly under compact pre- and post-translations.
-\end{lemma}
-\begin{proof}
+
+Proof.
For fixed $m$ and large $n$, $u_n=mt_n+(u_n-mt_n)$ with
nonnegative remainder. Pass to a subsequence on which the sum
-laws converge; Proposition~\ref{prop:ray-addition} makes the
-limit $P^{*m}*Q$ with $P([a,\infty])\ge\alpha$ and
-$Q$ supported in $[0,\infty]$. For finite $M$,
-\begin{equation}
+laws converge; Proposition~prop:ray-addition makes the
+limit $P^{*m}*Q$ with $P([a,∞])\ge\alpha$ and
+$Q$ supported in $[0,∞]$. For finite $M$,
+
(P^{*m}*Q)([0,M])\le
- \mathbb P\{\operatorname{Bin}(m,\alpha)\le M/a\}\to0
- \label{eq:scale-binomial}
-\end{equation}
-as $m\to\infty$; diagonal extraction over countably many $m$
+ \mathbb P\{Bin(m,\alpha)\le M/a\}\to0
+
+
+as $m\to∞$; diagonal extraction over countably many $m$
justifies the passage. Compact changes are covered by
-Lemma~\ref{lem:ray-compactshort}.
-\end{proof}
+Lemma~lem:ray-compactshort.
+
-\begin{proposition}[Positive mass at linear speed]\label{prop:scale-linear-subsequence}
+[Positive mass at linear speed]
For each isolated source type $i$, matched to $p(i)$, there
-exist $C<\infty$, $\alpha>0$, $T_n\to\infty$ such that for
+exist $C<∞$, $\alpha>0$, $T_n\to∞$ such that for
every fixed $\xi\in\mathcal H$,
-\begin{equation}
- \liminf_{n}\inf_{T_n/2\le t\le T_n}
+
+ liminf_{n}\inf_{T_n/2\le t\le T_n}
\bigl\|D_{1_{\{r_{p(i)}\le CT_n\}}}U_{a_t}\xi\bigr\|^2
\ge\alpha\|\xi\|^2.
- \label{eq:scale-linear-subsequence}
-\end{equation}
+
+
The statement holds for normal Hilbert amplifications and
compact source pre- and post-translations, but is not uniform
over moving Hilbert unit balls.
-\end{proposition}
-\begin{proof}[Sketch]
-Use $\delta$ from Lemma~\ref{lem:scale-original-model}.
+
+Proof. [Sketch]
+Use $\delta$ from Lemma~lem:scale-original-model.
Suppose the matched length is totally superlinear:
-\begin{equation}
- P_{\delta,t}(t)([0,C])\to0\qquad(C<\infty).
- \label{eq:scale-superlinear-assumption}
-\end{equation}
+
+ P_{\delta,t}(t)([0,C])\to0\qquad(C<∞).
+
+
Compose the original representation around the phase cycle
-$i\mapsto p(i)\mapsto\cdots\mapsto i$ via
-Lemma~\ref{lem:scale-insertion}; the first stage increases
+$i\mapsto p(i)\mapsto...\mapsto i$ via
+Lemma~lem:scale-insertion; the first stage increases
length faster than linearly, the entropy bound prevents later
stages from losing that, and the additive structure of
-Proposition~\ref{prop:ray-addition} telescopes to a shorter
+Proposition~prop:ray-addition telescopes to a shorter
witness reaching the same relative threshold.
Choose a nearly-earliest witness $(t_n,L_n,R_n=D'_nL_n)$ for
-$D'_n\to\infty$: feasibility is nonempty (identity embedding
+$D'_n\to∞$: feasibility is nonempty (identity embedding
into $L_\eta(\Gamma)$ suffices), so the infimum
-\begin{equation}
- \inf\{t\ge1:\|D_{1_{\{r_i>L\}}}\Theta(\gamma)\|_2\le\varepsilon_K
- (\gamma\in\mathcal F),\ P_{\Theta,D'_nL}(t)([1,\infty])\ge\tfrac12\}
- \label{eq:scale-minimal-second}
-\end{equation}
-is attained up to factor two at some witness. Lemma~\ref{lem:scale-tightening}
-shrinks $\Theta_n$ to a near-identity conjugate still retaining
-$1/4$ mass; $t_n\to\infty$; and Lemma~\ref{lem:scale-many-increments}
+
+ \inf\{t\ge1:\|D_{1_{\{r_i>L\}}}Θ(\gamma)\|_2\leε_K
+ (\gamma\in\mathcal F),\ P_{Θ,D'_nL}(t)([1,∞])\ge\tfrac12\}
+
+
+is attained up to factor two at some witness. Lemma~lem:scale-tightening
+shrinks $Θ_n$ to a near-identity conjugate still retaining
+$1/4$ mass; $t_n\to∞$; and Lemma~lem:scale-many-increments
gives
-\begin{equation}
- u_n/t_n\to\infty\Longrightarrow
- P_{\Theta_n,R_n}(u_n)\to\delta_\infty.
- \label{eq:scale-minimal-escape}
-\end{equation}
+
+ u_n/t_n\to∞⇒
+ P_{Θ_n,R_n}(u_n)\to\delta_∞.
+
+
Insert copies of $\delta$ for each cycle step. Choose
-$m_n\to\infty$ with
-\begin{equation}
- t'_n=t_n/m_n\to\infty,\quad
- P_{\delta,t_n}(t'_n)([m_n,\infty])\to1,
- \label{eq:scale-small-start}
-\end{equation}
+$m_n\to∞$ with
+
+ t'_n=t_n/m_n\to∞,\quad
+ P_{\delta,t_n}(t'_n)([m_n,∞])\to1,
+
+
which total superlinearity produces. Later insertions keep
length much larger than $t_n$ by entropy; complementary
coordinates stay tight; the terminal kernel is tested at length
-$\div t_n\to\infty$, so \eqref{eq:scale-minimal-escape} (with
+$\div t_n\to∞$, so (eq:scale-minimal-escape) (with
compact uniformity) sends the final length to infinity.
A tighter feasible witness $t'_n$ contradicts $t_n\le2t'_n$.
Total superlinearity is therefore impossible. There are
-$C_0,\alpha_0>0$, $T_n\to\infty$ with the scalar law mass
+$C_0,\alpha_0>0$, $T_n\to∞$ with the scalar law mass
$\ge\alpha_0$ below $C_0T_n$, uniform for $t\in[T_n/2,T_n]$
(enlarge $C_0$ if necessary); a violating sequence plus
-Proposition~\ref{prop:ray-addition} on scale $T_n$ would give
+Proposition~prop:ray-addition on scale $T_n$ would give
a convolution limit for $T_n=t_n+(T_n-t_n)$, and the mass of a
nonnegative sum below $C_0$ is at most the mass of either
-summand. Lemma~\ref{lem:links-invariant-variables} makes all
-limiting effects scalar; Lemma~\ref{lem:scale-original-model}
+summand. Lemma~lem:links-invariant-variables makes all
+limiting effects scalar; Lemma~lem:scale-original-model
identifies their scalar values, and effect boundedness plus
Hilbert density extend the bound to every fixed $\xi$.
-\end{proof}
-\subsection{Horizontal differentiation and an injective height}
+
+## Horizontal differentiation and an injective height
Fix matched isolated factors $S,T$ with boundaries $B_S,B_T$
homogeneous dimensions $Q_S,Q_T$, Carnot distances $d_S,d_T$, compact
angular probability $dx$, and horizontal bundles $\mathcal V_R$. The
action is horizontally conformal with horizontal dilation
-$\operatorname{dil}(a,x)>0$, obeying
-\begin{equation}\label{eq:scale-dilation-cocycle}
- \operatorname{dil}(ab,x)=\operatorname{dil}(a,bx)\operatorname{dil}(b,x).
-\end{equation}
+$dil(a,x)>0$, obeying
+
+ dil(ab,x)=dil(a,bx)dil(b,x).
+
All isolated factors here have $Q_T>2$. Let $\mathscr E\subset\mathcal H$
be the maximal invariant linear domain of
-Proposition~\ref{prop:scale-pair-estimate}, on which
-\eqref{eq:input-10} holds uniformly over each fixed collection of
+Proposition~prop:scale-pair-estimate, on which
+(eq:input-10) holds uniformly over each fixed collection of
normalized mask controls.
-\begin{definition}\label{def:scale-ordered-kernel}
-For $F\in C^\infty(B_T\times B_T)$, its ordered evaluation is
+
+For $F\in C^∞(B_T\times B_T)$, its ordered evaluation is
$\mathcal K_{x,y}(F)=\iint F(w,v)\,P_y(dw)P_x(dv)$ (finite separated
decomposition on rectangles). A smooth kernel has horizontal order at
least one if $F|_{w=v}=0$; at least two if in addition
$d_zF|_{\mathcal V_T}$ vanishes on the diagonal.
-\end{definition}
-\begin{lemma}\label{lem:scale-ordered-differences}
+
+
Fix a compactly contained source chart, a nonnegative source cutoff
-$\chi$, and a smooth nonnegative compactly supported $\vartheta$ on the
-nilpotent group. Set $y=x\delta_\epsilon w$ and $d\nu_\epsilon=\chi(x)
-\vartheta(w)\,dx\,dw$. For small $\epsilon$:
-\begin{enumerate}[label=\textup{(\roman*)}]
+$\chi$, and a smooth nonnegative compactly supported $θ$ on the
+nilpotent group. Set $y=x\delta_ε w$ and $d\nu_ε=\chi(x)
+θ(w)\,dx\,dw$. For small $ε$:
+[label=(\roman*)]
\item Order at least one gives, for each $\xi\in\mathscr E$,
-$\|\mathcal K_{x,y}(F)\xi\|_{L^2(d\nu_\epsilon;\mathcal H)}
-\le C_{F,\xi}\epsilon^{Q_S/Q_T}$.
+$\|\mathcal K_{x,y}(F)\xi\|_{L^2(d\nu_ε;\mathcal H)}
+\le C_{F,\xi}ε^{Q_S/Q_T}$.
\item If $Q_S=Q_T=Q>2$ and order is at least two, the same norm is
-$o(\epsilon)$, explicitly bounded by
-$\epsilon^{Q/2}$ ($2<Q<4$), $\epsilon^2(1+|\log\epsilon|)$ ($Q=4$),
-$\epsilon^2$ ($Q>4$).
-\end{enumerate}
-\end{lemma}
+$o(ε)$, explicitly bounded by
+$ε^{Q/2}$ ($2<Q<4$), $ε^2(1+|\logε|)$ ($Q=4$),
+$ε^2$ ($Q>4$).
+
-\begin{proof}
+
+Proof.
The measure pullback under $w\mapsto y$ has density at most
-$C\epsilon^{-Q_S}$ supported where $d_S(x,y)\le C\epsilon$; the pair
+$Cε^{-Q_S}$ supported where $d_S(x,y)\le Cε$; the pair
measure has density $\asymp d_S(x,y)^{-2Q_S}$. Thus
-\begin{equation}\label{eq:scale-averaged-mask-bound}
- \int\langle\xi,p_\delta(x,y)\xi\rangle\,d\nu_\epsilon
- \le C_\xi\epsilon^{Q_S}\delta^{-Q_T},
-\end{equation}
-using \eqref{eq:input-10}. Decompose the target-diagonal complement
+
+ \int<\xi,p_\delta(x,y)\xi>\,d\nu_ε
+ \le C_\xiε^{Q_S}\delta^{-Q_T},
+
+using (eq:input-10). Decompose the target-diagonal complement
into dyadic scale-$\delta$ cells $c\delta\le d_T(w,v)\le C\delta$;
such pairs form a bounded-degree graph. Order-$m$ bounds normalized
derivatives by $C\delta^m$. Expand in separated Fourier series, and
absorb coefficients into masks: the annular piece is bounded in root
mean square by
-\begin{equation}\label{eq:scale-annular-kernel-bound}
- C_{F,\xi}\delta^m\min\{1,\epsilon^{Q_S/2}\delta^{-Q_T/2}\},
-\end{equation}
+
+ C_{F,\xi}\delta^m\min\{1,ε^{Q_S/2}\delta^{-Q_T/2}\},
+
while the entire piece at scale $\delta_0$ has
-$C_F\delta_0^m\|\xi\|$. Summing over $\delta\ge\epsilon^{Q_S/Q_T}$
-for $m=1$ gives $C\epsilon^{Q_S/Q_T}$ since $Q_T>2$. For
-$Q_S=Q_T=Q$ with $m=2$, sum at $\delta_0\asymp\epsilon$ for the rates
+$C_F\delta_0^m\|\xi\|$. Summing over $\delta\geε^{Q_S/Q_T}$
+for $m=1$ gives $Cε^{Q_S/Q_T}$ since $Q_T>2$. For
+$Q_S=Q_T=Q$ with $m=2$, sum at $\delta_0\asympε$ for the rates
listed.
-\end{proof}
-\begin{lemma}\label{lem:scale-equal-dimension}
+
+
Every matched isolated pair has $Q_S=Q_T=Q$. For every smooth
-horizontal source field $X$, every $f\in C^\infty(B_T)$, and every
+horizontal source field $X$, every $f\in C^∞(B_T)$, and every
$\xi\in\mathscr E$, the distributional derivative
$D_X(f)\xi=X(x\mapsto P_x(f)\xi)$ lies in
$L^2(B_S,dx;\mathcal H)$, linearly in $f,\xi$ and in smooth source
function multiples of $X$.
-\end{lemma}
-\begin{proof}
+
+Proof.
For $F(w,v)=f(w)-f(v)$, ordered evaluation is $P_y(f)-P_x(f)$. A
horizontal weight $\psi_i$ with first moments the coordinate vector
gives difference quotients
-$\epsilon^{-1}\int\psi_i(w)(P_{x\delta_\epsilon w}(f)-P_x(f))\xi\,dw$
-bounded in $L^2$ by $C\epsilon^{Q_S/Q_T-1}$; adjoint pairing recovers
-the distributional derivative $-X\varphi-(\operatorname{div}X)\varphi$.
+$ε^{-1}\int\psi_i(w)(P_{x\delta_ε w}(f)-P_x(f))\xi\,dw$
+bounded in $L^2$ by $Cε^{Q_S/Q_T-1}$; adjoint pairing recovers
+the distributional derivative $-X\varphi-(divX)\varphi$.
If $Q_S/Q_T>1$, all derivatives vanish, brackets of horizontal fields
also annihilate the same distribution, and bracket generation forces
$P_x(f)\xi$ constant on $B_S$, contradicting nonconstancy. Thus
$Q_S\le Q_T$; cyclic permutation of the finite isolated type list
gives equality throughout. Weak compactness and the identified limit
give $D_X(f)\xi\in L^2$ on charts; a partition of unity globalizes.
-\end{proof}
-\begin{lemma}\label{lem:scale-module-extension}
+
+
Set $\mathcal K=L^2(B_S,dx;\mathcal H)$ and
-$\mathcal C=\operatorname{span}\{x\mapsto a(x)P_x(g)\xi:
-a\in L^\infty(B_S), g\in C^\infty(B_T), \xi\in\mathscr E\}$. For real
+$\mathcal C=span\{x\mapsto a(x)P_x(g)\xi:
+a\in L^∞(B_S), g\in C^∞(B_T), \xi\in\mathscr E\}$. For real
$f$ and real horizontal $X$, the rule
-\begin{equation}\label{eq:scale-module-extension-rule}
- T_X(f)\Bigl(\sum_ja_jP_\bullet(g_j)\xi_j\Bigr)
+
+ T_X(f)(\sum_ja_jP_\bullet(g_j)\xi_j)
=\sum_ja_jP_\bullet(g_j)D_X(f)\xi_j
-\end{equation}
+
defines a densely defined symmetric operator on $\mathcal C$. If real
smooth $g_j,f_j$ satisfy $\sum_jg_j\,d_{\rm hor}f_j=0$, then
-\begin{equation}\label{eq:scale-horizontal-cancellation}
+
\sum_jP_\bullet(g_j)T_X(f_j)=0\quad\hbox{on }\mathcal C.
-\end{equation}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
Write $\Delta_{x,y}f=P_y(f)-P_x(f)$. Sharpness gives the exact
identity
$(\Delta_{x,y}f)P_x(g)-P_y(g)\Delta_{x,y}f
=\mathcal K_{x,y}((f(w)-f(v))(g(v)-g(w)))$; the right side has
horizontal order two and vanishes in the recovered $L^2$ average
-after dividing by $\epsilon$. Move $\Delta_{x,y}f$ by adjunction and
+after dividing by $ε$. Move $\Delta_{x,y}f$ by adjunction and
replace $P_y(g)$ by $P_x(g)$ (dominated convergence); the resulting
derivative of the difference quotients is
-\begin{equation}\label{eq:scale-weighted-matrix-symmetry}
- \langle P_x(g)D_X(f)(x)\xi,\eta\rangle
- =\langle\xi,P_x(g)D_X(f)(x)\eta\rangle
-\end{equation}
+
+ < P_x(g)D_X(f)(x)\xi,\eta>
+ =<\xi,P_x(g)D_X(f)(x)\eta>
+
for almost every $x$. Inserting scalar parameter tests gives the
pointwise form; complex weights follow by linearity. Substituting
finitely many angular tests into a formal pairing of $u,v\in\mathcal C$
@@ -5075,28 +5075,28 @@ well-definedness, symmetry, and closability. Cancellation: the kernel
$F(w,v)=\sum_jg_j(w)(f_j(w)-f_j(v))$ has order $\ge2$ when
$\sum_jg_jd_{\rm hor}f_j=0$, and the same limit shows
$\sum_jP_x(g_j)D_X(f_j)(x)\xi=0$; commuting further multipliers
-proves \eqref{eq:scale-horizontal-cancellation}.
-\end{proof}
+proves (eq:scale-horizontal-cancellation).
+
For a real smooth horizontal target covector field
$\theta=\sum_jg_jd_{\rm hor}f_j$, set $T_X(\theta)=\sum_j
-P_\bullet(g_j)T_X(f_j)$; equation \eqref{eq:scale-horizontal-cancellation}
+P_\bullet(g_j)T_X(f_j)$; equation (eq:scale-horizontal-cancellation)
makes this independent of expression. (Global finitely many target
coordinate differentials span, and their Gram operator is smoothly
invertible on $\mathcal V_T$.)
-\begin{lemma}\label{lem:scale-closed-covariance}
+
There is a positive self-adjoint measurable field $l_x$ on
$\mathcal H$, affiliated to $\mathcal A_1$, strongly commuting with
$P_x$ and invariant under complementary source factors, such that
-\begin{equation}\label{eq:scale-derivative-modulus-covariance}
- \sigma_a(l_x)=\operatorname{dil}(a,x)\,l_{ax},\qquad
- \beta_h(l_x)=\operatorname{dil}(h^{-1},P_x)\,l_x,
-\end{equation}
+
+ \sigma_a(l_x)=dil(a,x)\,l_{ax},\qquad
+ \beta_h(l_x)=dil(h^{-1},P_x)\,l_x,
+
and $l$ is not identically zero.
-\end{lemma}
-\begin{proof}
+
+Proof.
Choose finite smooth Parseval horizontal source fields $(X_\alpha)$
and target covectors $(\theta_r)$ (invert positive Gram operators on
$\mathcal V_R$). Define the finite column
@@ -5104,7 +5104,7 @@ $D:\mathcal C\to\mathcal K^{n_Sn_T}$,
$Du=(T_{X_\alpha}(\theta_r)u)_{\alpha,r}$, closable by
$\mathcal C$ density; set $l=|\overline D|$. Parameter and angular
unitary multipliers commute with $D$ on $\mathcal C$, extending to
-the closed graph and form, so $l=\int^{\oplus}l_x\,dx$ with $l_x$
+the closed graph and form, so $l=\int^{ ⊕ }l_x\,dx$ with $l_x$
strongly commuting with $P_x$. Unitary invariance transfers
distributionally to $D_X(f)$ and then closes, giving affiliation and
complementary invariance.
@@ -5114,40 +5114,40 @@ isometrically into trivial bundles whose Gram matrices are orthogonal
projections; $Du$ lies in their ranges. Two finite Parseval systems
thus give the same squared form on the intrinsic module core, hence the
same $l$. Differentials change source vectors by
-$\operatorname{dil}(a,x)$ times a horizontal isometry and target
-covectors by $\operatorname{dil}(h^{-1},P_x)$ times an isometry, so
-\eqref{eq:scale-derivative-modulus-covariance} holds on $\mathcal C$;
+$dil(a,x)$ times a horizontal isometry and target
+covectors by $dil(h^{-1},P_x)$ times an isometry, so
+(eq:scale-derivative-modulus-covariance) holds on $\mathcal C$;
further source and target transport preserves the graph norms
(finitely many bounded dilations) and extends to the closed forms.
Uniqueness of self-adjoint operators associated with closed forms
gives the identities. If $l=0$, then $D_X(f)\xi=0$ for all smooth
-$f,X$, and the bracket argument of Lemma~\ref{lem:scale-equal-dimension}
+$f,X$, and the bracket argument of Lemma~lem:scale-equal-dimension
would make $P_x$ constant, contradiction.
-\end{proof}
-\begin{proposition}[Height at an isolated type]\label{prop:scale-isolated-height}
+
+[Height at an isolated type]
For matched isolated pairs $Q_S=Q_T=Q$. There are positive injective
self-adjoint $b_x$, affiliated to $\mathcal A_1$, strongly commuting
with $P_x$ and invariant under complementary source factors, with
-\begin{equation}\label{eq:scale-isolated-height-covariance}
- \sigma_a(b_x)=\operatorname{dil}(a,x)^{-Q}b_{ax},\qquad
- \beta_h(b_x)=\operatorname{dil}(h^{-1},P_x)^{-Q}b_x.
-\end{equation}
+
+ \sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},\qquad
+ \beta_h(b_x)=dil(h^{-1},P_x)^{-Q}b_x.
+
The construction passes to normal copies used in amplified models.
-\end{proposition}
-\begin{proof}
+
+Proof.
Make $l_x$ everywhere covariant: at $x_0\in B_S$ with parabolic
stabilizer $J$, pull back along $g\mapsto gx_0$ and set
-$F(g)=\operatorname{dil}(g,x_0)\sigma_g^{-1}(l_{gx_0})$. Haar
+$F(g)=dil(g,x_0)\sigma_g^{-1}(l_{gx_0})$. Haar
disintegration over $S/J$ and
-\eqref{eq:scale-derivative-modulus-covariance} give $F(ag)=F(g)$
+(eq:scale-derivative-modulus-covariance) give $F(ag)=F(g)$
almost everywhere for each fixed $a\in S$; Fubini over
$(a,g)\mapsto(ag,g)$ reduces $F$ to an essential value $L_0$ (test
via bounded resolvents and countably many matrix coefficients). For
-$k\in J$, $F(gk)=\operatorname{dil}(k,x_0)\sigma_k^{-1}F(g)$, so
-$\sigma_kL_0=\operatorname{dil}(k,x_0)L_0$, defining
-$l_{gx_0}=\operatorname{dil}(g,x_0)^{-1}\sigma_gL_0$ everywhere with
+$k\in J$, $F(gk)=dil(k,x_0)\sigma_k^{-1}F(g)$, so
+$\sigma_kL_0=dil(k,x_0)L_0$, defining
+$l_{gx_0}=dil(g,x_0)^{-1}\sigma_gL_0$ everywhere with
the same covariance. Target law follows by commuting $\beta_h$ past
$\sigma_g$; the essential value $L_0$ satisfies the fixed-$x_0$ law
and transports.
@@ -5157,7 +5157,7 @@ commuting with $L_0$; taking supports gives $\beta_h(e_0)=e_0$, so
$e_0\in\mathcal N$. Source stabilizer covariance gives
$\sigma_k(e_0)=e_0$. A split sequence in $J$ escaping in $S$ times
sequences escaping in the complementary source factors fixes $e_0$;
-all-factor mixing (\S\ref{sec:setup}) makes translates converge to
+all-factor mixing (\Ssec:setup) makes translates converge to
$\tau(e_0)1$, so $e_0\in\{0,1\}$. Nonzero column forces $e_0=1$ (this
uses complementary source invariance essentially).
@@ -5165,42 +5165,42 @@ Every transported $l_x$ has trivial kernel; define $b_x=l_x^{-Q}$ by
spectral calculus. Dense, positive, self-adjoint, injective, affiliated
to $\mathcal A_1$; taking inverse powers in the covariance laws and
using strong commutation with the dilation gives
-\eqref{eq:scale-isolated-height-covariance}.
+(eq:scale-isolated-height-covariance).
-For amplification: on a normal copy $\mathcal H\otimes\mathcal L$ with
-orthonormal basis $(e_n)$, an element $\Xi=\sum_n\xi_n\otimes e_n$ of
+For amplification: on a normal copy $\mathcal H ⊗ \mathcal L$ with
+orthonormal basis $(e_n)$, an element $\Xi=\sum_n\xi_n ⊗ e_n$ of
the maximal amplified estimate domain has every coordinate
$\xi_n\in\mathscr E$ (each coordinate energy is bounded by the
total); conversely finite sums of $\xi_n\in\mathscr E$ lie in the
amplified domain. Finite-coordinate projections commute with the PVMs,
preserve the domain, and commute with weak derivatives; on the
-algebraic tensor core $\mathcal C\odot\operatorname{span}\{e_n\}$ the
-column is exactly $D\otimes1$, graph-densely. So
-$l^{\rm amp}=l\otimes1$ and $b^{\rm amp}=b\otimes1$; unitary
+algebraic tensor core $\mathcal C\odotspan\{e_n\}$ the
+column is exactly $D ⊗ 1$, graph-densely. So
+$l^{\rm amp}=l ⊗ 1$ and $b^{\rm amp}=b ⊗ 1$; unitary
identification with a normal copy preserves the identities, without
any mixing on an enlarged commutant.
-\end{proof}
+
The isolated branch now supplies its own-label height calculus:
$b_x$ is positive and injective, strongly commutes with $P_x$, and
-obeys \eqref{eq:scale-isolated-height-covariance} in each required
-normal copy; Section~\ref{sec:heights} transports its imaginary powers
+obeys (eq:scale-isolated-height-covariance) in each required
+normal copy; Section~sec:heights transports its imaginary powers
through the Fourier links to establish centered length laws. The
common graph domain for comparing heights at different labels is built
-from the Haar-height isometry of Section~\ref{sec:locality}.
+from the Haar-height isometry of Section~sec:locality.
+
+# Upper speeds at real panels
-\section{Upper speeds at real panels}
-\label{sec:panels}
This section treats a non-isolated source position and its matched target
position when both places are archimedean. The finite-place argument is
-in Section~\ref{sec:finite}; the isolated-axis differentiation of
-Section~\ref{sec:scale} is not used here.
+in Section~sec:finite; the isolated-axis differentiation of
+Section~sec:scale is not used here.
-\subsection{The estimate and the order of limits}
+## The estimate and the order of limits
Fix one measured coordinate of the exact graph relation
-\eqref{eq:input-6}, possibly one of the two measured coordinates of a
+(eq:input-6), possibly one of the two measured coordinates of a
two-leg link. Begin with a mixed many-panel lexicographic face path that
selects every isolated gap and omits one gap in each higher-rank
component. Let $S$ and $T$ be the noncompact simple rank-one Levi groups
@@ -5209,7 +5209,7 @@ and ineffective factors are discarded. Source actions may be lifted to
the simply connected cover. Write $e_i$ for the coordinate vector dual
to the omitted simple root, so translation by $te_i$ has simple gap $t$.
-\begin{proposition}[The real-panel upper estimate]\label{prop:panels-upper}
+[The real-panel upper estimate]
There are positive constants $d_R$, determined by each real rank-one
panel type $R$ and unchanged at the opposite type, with the following
property. Complete the mixed path by split-chamber translations along
@@ -5218,17 +5218,17 @@ and more slowly than the original path. Additional slower translations
along already selected isolated coordinates are allowed. If $t$ is the
completing simple gap at the selected panel and $\ell(g)$ the residual
target Cartan gap, then
-\begin{equation}
- d_T\ell(g)\le d_St+O_{\mathrm{tight}}(1).
- \label{eq:input-11}
-\end{equation}
-Here \emph{tight upper error} means the event where
+
+ d_T\ell(g)\le d_St+O_{tight}(1).
+
+
+Here tight upper error means the event where
$d_T\ell(g)-d_St$ tends to positive infinity has zero mass in the
iterated limit, on every fixed input. One may add offsets fixed during
preceding limits and then take further limits in those offsets; $t$ in
-\eqref{eq:input-11} is the resulting shifted gap. Gap signs are the
+(eq:input-11) is the resulting shifted gap. Gap signs are the
eventual orientations on the chosen filters.
-\end{proposition}
+
The angular information used during completion: at the first, mixed
limit the residual missing gap is finite and its Cartan gap differs by
@@ -5236,20 +5236,20 @@ a bounded quantity from the corresponding inverse Fourier-index gap. A
completing translation fixed at this first limit is an exact input
precomposition of the link; it retains all the sharp angular marginals.
At each subsequent limit the growth and upper vertex-count of
-Section~\ref{sec:transfer} applies to the enlarged path. If a previously
+Section~sec:transfer applies to the enlarged path. If a previously
selected gap is repeated, maximal vertex count on that face is already
attained, so the repetition adds no additional gap. In the stated order
of limits the total split parameters still contract the relevant face
radicals. Thus every resulting link is mixed and isometric, has its
-full sharp marginals, and satisfies \eqref{eq:input-6} with the
+full sharp marginals, and satisfies (eq:input-6) with the
indicated input reindexing. The same reasoning applies to outer
translations in the common split torus after a full path, including the
-offsets in Proposition~\ref{prop:panels-upper}. These assertions use
+offsets in Proposition~prop:panels-upper. These assertions use
only the exact precomposition and angular statements of
-Sections~\ref{sec:links} and~\ref{sec:transfer}; neither the height
+Sections~sec:links and~sec:transfer; neither the height
calibration nor the classicality conclusion proved later is used.
-The structural classification of Section~\ref{sec:setup} leaves the
+The structural classification of Section~sec:setup leaves the
orthogonal, unitary, and symplectic groups as the real rank-one panel
groups. The exceptional rank-one group $F_{4(-20)}$ cannot occur as the
noncompact simple factor of a panel Levi in a higher-rank component:
@@ -5260,7 +5260,7 @@ restricted-root list a multipliable simple root arising in such a Levi
has double-root multiplicity $1$ or $3$, and not $7$. The reduced
low-rank isomorphisms lie on the orthogonal line.
-\subsection{Optimal modules for a real rank-one panel}
+## Optimal modules for a real rank-one panel
Let $R$ be a noncompact rank-one panel factor with split generator
$H_R$, normalized so the simple restricted gap is one. For a
@@ -5285,32 +5285,32 @@ included. Isomorphic rank-one groups are named only once: $SU(1,1)$
uses the $SO(2,1)$ line and $Sp(1,1)$ uses the $SO(4,1)$ line, with the
reduced restricted root in both cases.
-\begin{lemma}[Optimal panel modules]\label{lem:panels-optimal-modules}
+[Optimal panel modules]
Put $k=n+1$ and let $E$ denote the standard complex module for the
group in the relevant line. Among the nontrivial irreducible complex
Lie-algebra modules, the maximum $c_R=\max_V\dim V/h(V)$ and all
modules attaining it are
\[
-\begin{array}{ccl}
- R&c_R&\text{maximizing modules}\\ \hline
+{ccl}
+ R&c_R&maximizing modules\\ \hline
SO(n,1),\ n\ge2 & k & E\\
SU(n,1),\ n>2 & k & E,E^*\\
- SU(2,1) & 3 & E,E^*,\operatorname{Sym}^2E,\operatorname{Sym}^2E^*\\
+ SU(2,1) & 3 & E,E^*,Sym^2E,Sym^2E^*\\
Sp(n,1),\ n>2 & k & E\\
- Sp(2,1) & 7/2 & \bigwedge\nolimits^3_0E
-\end{array}
+ Sp(2,1) & 7/2 & ∧\nolimits^3_0E
+
\]
-where $\bigwedge^3_0E$ is the primitive exterior cube (kernel of the
-symplectic contraction $\bigwedge^3E\to E$). There is
-$\varepsilon_R>0$ such that every nontrivial irreducible outside the
-table satisfies $\dim V/h(V)\le c_R-\varepsilon_R$. Every displayed
+where $∧^3_0E$ is the primitive exterior cube (kernel of the
+symplectic contraction $∧^3E\to E$). There is
+$ε_R>0$ such that every nontrivial irreducible outside the
+table satisfies $\dim V/h(V)\le c_R-ε_R$. Every displayed
module has integral split weights and a nonzero split-weight-zero
space, and admits a preserved nondegenerate Hermitian form $q$,
positive definite on $V_0$, whose matrix in a compact-form Hilbert norm
obeys $Q=Q^*=Q^{-1}$ and $d\pi(H_R)^*=d\pi(H_R)$. The form pairs
$V_t$ perfectly with $V_{-t}$ and pairs it trivially with all other
-weight spaces. With $V_{>0}=\bigoplus_{t>0}V_t$ and
-$V_{\ge0}=\bigoplus_{t\ge0}V_t$,
+weight spaces. With $V_{>0}=⊕_{t>0}V_t$ and
+$V_{\ge0}=⊕_{t\ge0}V_t$,
\[
V_{>0}^{\perp_q}=V_{\ge0},\qquad
V_{\ge0}^{\perp_q}=V_{>0},
@@ -5319,30 +5319,30 @@ and the induced form on $V_{\ge0}/V_{>0}$ is positive definite. These
two spaces are algebraic parabolic stages. The largest split weight
among the maximizing modules is
\[
- d_R=\begin{cases}2,&R=SU(2,1)\text{ or }Sp(2,1),\\1,&\text{otherwise,}\end{cases}
+ d_R=2,&R=SU(2,1) or Sp(2,1),\\1,&otherwise,
\]
and the constants apply equally to the opposite orientation.
-\end{lemma}
-\subsection{Algebraic stages and compact graph operators}
+
+## Algebraic stages and compact graph operators
Choose a target maximizing module $V$ with largest split weight $d_T$,
$n_0=\dim V$, and (if a compact diagram symmetry interchanges
inequivalent modules) replace $V$ by the direct sum of its diagram
conjugates; every summand has the same optimal ratio, so still
-\begin{equation}
+
h(V)=n_0/c_T.
- \label{eq:panels-target-ratio}
-\end{equation}
+
+
Take the compact-form Hilbert norms and preserved Hermitian forms of
-Lemma~\ref{lem:panels-optimal-modules}, with form symmetry $Q$, and in
+Lemma~lem:panels-optimal-modules, with form symmetry $Q$, and in
compact projectivity frames choose local or Borel lifts $\pi_T(g)$ of
target projectivities such that
-\begin{equation}
+
\pi_T(g)^*Q\pi_T(g)=Q,\qquad
\|\pi_T(g)\|=\exp(d_T\ell(g)).
- \label{eq:panels-lifts}
-\end{equation}
+
+
The connected action lifts to the simply connected module; compact
Cartan factors act unitarily, compact diagram factors permute the
chosen summands, and the invariant forms agree: on each irreducible