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+\hypersetup{pdftitle={Arithmeticity of twisted finite correspondences for lattices over local fields},pdfauthor={OpenAI},pdfsubject={Arithmetic exhaustion of twisted bifinite correspondences}}
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+\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}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+
+\begin{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
+group factors of arbitrary countable ICC groups. Every such correspondence
+is a closed summand of a finite direct sum of models obtained from actual
+finite-index subgroup isomorphisms and finite-dimensional projective
+representations of the matched cocycle ratio. The proof includes reducible
+products, mixed real and finite places, and quaternionic and Cayley
+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}
+\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
+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
+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}
+
+A \emph{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
+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.
+
+Inside a continuous group, one can sometimes find a \emph{lattice}: a
+discrete grid of symmetries that still "spreads out" through the whole
+group, like the integer points
+\[
+ \mathbb Z^n\subset\mathbb R^n
+\]
+inside the group of all translations, or the famous
+\[
+ \SL_n(\mathbb Z)\subset\SL_n(\mathbb R)
+\]
+of integer matrices of determinant one inside all real matrices of
+determinant one. A lattice is discrete, so its elements do not
+accumulate, and yet it is large enough that the continuous group is
+"almost" covered by translates of it. Lattices are hard, rich objects,
+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
+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
+each place. A "local field" here means a finite extension of
+$\mathbb R$, $\mathbb C$, or $\mathbb Q_p$. Real place, complex place,
+$p$-adic place: these are the three kinds of ground fields on which our
+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 $(T)$}
+
+A group has \emph{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
+property $(T)$ has a spectral gap: any attempt to wiggle a
+representation produces a genuine, non-negligible displacement.
+
+Rigidity is powerful. Many strong structural theorems in operator
+algebras and geometry hold only for groups with property $(T)$ (often,
+conjecturally, for all higher-rank lattices, an expectation this paper
+does not need). Property $(T)$ is the main structural hypothesis that
+the "source" side of our theorem carries. It is what lets us pin down
+the geometry of the correspondence rather than chase a wild object that
+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
+$G=\prod_{i=1}^r G_i$, where each factor $G_i$ is the group of local
+points of a connected, adjoint, absolutely simple algebraic group over a
+local field $k_i\in\{\mathbb R,\mathbb C,\text{finite extension of }\mathbb Q_p\}$,
+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,
+cocompactness, product decomposition, or torsion hypothesis is imposed
+on the lattice. The class is large; it contains the usual arithmetic
+lattices in products of higher-rank real and $p$-adic groups, the
+quaternionic and Cayley rank-one lattices, reducible lattices, and
+nonuniform ones.
+
+The "ICC" condition appears everywhere in the paper: every nonidentity
+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}
+
+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
+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
+with any isomorphism, a controlled correspondence with the original
+group, the group is called \emph{$W^*$-superrigid}.
+
+Recall the notion of a \emph{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
+dimension function) is defined.
+
+\subsection*{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
+\[
+ \mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj)
+\]
+with values in the unit circle $\mathbb T$. The \emph{twisted group von
+Neumann algebra} $L_\mu(\Gamma)$ is built from $\Gamma$ together with
+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
+the final models.
+
+\subsection*{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
+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
+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
+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
+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
+\[
+ \sigma(a)\sigma(b)=\frac{\mu(a,b)}{\omega(\delta(a),\delta(b))}\sigma(ab).
+\]
+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}.
+
+\subsection*{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
+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 $\mu,\omega$
+be normalized scalar cocycles. Every bifinite
+$L_\mu(\Gamma)$--$L_\omega(\Lambda)$ 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
+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,
+where $n=\dim V_\sigma$.
+\end{quote}
+
+Three things are worth stressing.
+
+First the theorem is an \emph{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
+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
+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.
+
+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
+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}
+
+The comparison group $\Lambda$ 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
+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}
+geometry inside the correspondence from the source side alone, then
+must \emph{read} $\Lambda$ through it.
+
+The second hardness is scalar. Even in the classical, untwisted
+case, the argument must be careful. With twists in the game, phases
+scatter everywhere and must be controlled exactly; an argument that only
+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}
+
+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.
+
+\emph{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
+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}).
+
+\emph{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
+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
+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
+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
+proved separately at three kinds of local places:
+\begin{itemize}
+\item \emph{Finite place} (Section~\ref{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}):
+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}):
+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}
+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
+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
+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}).
+
+\emph{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
+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 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,
+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
+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
+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
+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}).
+
+\subsection*{9. A preview of the technical skeleton}
+
+The paper's structure mirrors the story above. The following table is a
+map. (Section numbers refer to the body.)
+
+\begin{center}
+\begin{tabular}{@{}p{44mm}p{96mm}@{}}
+\hline
+Section & Contents\\
+\hline
+1 & Introduction, conventions, main theorem.\\
+2 & Arithmetic models; exact normality and dimensions.\\
+3 & Compact reduction; finite-index return; complementary regular
+realizations; product mixing; local-field geometry.\\
+4 & Boundary obstruction; link forms; strict gap growth.\\
+5 & Transverse Fourier products; geometric transfer.\\
+6 & Finite-place panels: fixed algebras and uniform upper speed.\\
+7 & Ray estimates: radial multipliers, the ray lift, spherical laws.\\
+8 & Isolated scales: linear subsequences and heights.\\
+9 & Real panels: optimal modules and upper speed.\\
+10 & Product calibration; tuple heights; Haar-height isometry.\\
+11 & Locality and commutation at isolated types.\\
+12 & Higher-component pencils and gluing.\\
+13 & Arithmetic recovery; proof of the main theorem.\\
+\hline
+\end{tabular}
+\end{center}
+
+\subsection*{10. What to read if you are curious, and in what order}
+
+If you want the shortest path to the main theorem without the full
+weight of the proof, read: the main theorem statement
+(Section~\ref{sec:intro}); the arithmetic-models section
+(Section~\ref{sec:models}), where the objects are written down; the
+statement of the arithmetic-recovery section
+(Section~\ref{sec:recovery}), which closes the loop; and the guide above
+once more, in that order.
+
+If you want to see the central machinery, read Section~\ref{sec:links}
+(the boundary obstruction and the gap count) together with
+Section~\ref{sec:transfer} (the transverse product and transfer). Those
+two sections together contain the analytic seed that the rest of the
+paper waters.
+
+If you want one isolated, self-contained piece of analysis, read
+Section~\ref{sec:ray}. The ray lift and the spherical law are the
+calculus trick that turns a limit of matrix coefficients into a measure
+of a boundary distance; the rest of the isolated-factor argument hangs on
+it.
+
+If you are a student who has never seen von Neumann algebras, the
+Appendix (Section~\ref{sec:appendix}) gives a brief introduction to the
+objects and to the two classical theorems (the group factor and its
+trace) that everything rests on.
+
+\subsection*{11. Two remarks on style and honesty}
+
+The paper is a proof. In several places one finds claims like
+"one can verify by a direct calculation", "compactness makes the
+constants uniform", "a finite net suffices". These are not hand-waves in
+the ordinary sense; they are exact facts with short verifications that
+would interrupt the geometry if spelled out in the place where they are
+used. The reader should treat them as exercises with clear local
+statements rather than as gaps.
+
+Second, the paper is not a survey and does not attempt to be complete
+on the literature. Several cited items and results are recent or
+concurrent work (for instance, the counterexamples to Connes' rigidity
+conjecture). Where a recent result is quoted, the reference is exact
+and the use is limited to what the statement needs.
+
+\subsection*{12. On the reader this guide is written for}
+
+This guide assumes no operator algebras and only a handful of
+facts about groups: that groups are made of symmetries, that some are
+continuous, and that a lattice is a discrete grid inside a continuous
+group. Everything else, including the words factor, bimodule,
+cocycle, property $(T)$, is defined where it appears. If you are a
+curious high-school student, the honest situation is this: the paper
+itself is a research artifact and cannot be read without a background in
+graduate algebra and analysis. But the shape of the argument, the reason
+each step is needed, and the meaning of the theorem are all legible
+through this guide, and that is what the guide is for.
+
+\section{Introduction}\label{sec:intro}
+
+A finite correspondence between two group factors is a finite-dimensional
+analytic relation, in the sense of von Neumann module dimension. For
+lattice groups, we prove that such an analytic relation always comes from
+a finite collection of concrete subgroup isomorphisms. The scalar
+phases survive, and are not lost: the finite fibres are projective
+representations whose multiplier is exactly the ratio of the two endpoint
+cocycles along the subgroup isomorphism.
+
+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
+group is available on the comparison side at the start.
+
+\subsection{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
+\[
+ G=\prod_{i=1}^r G_i,\qquad G_i=\mathbf H_i(k_i)^+.
+\]
+Here $k_i$ is $\mathbb R$, $\mathbb C$, or a finite extension of
+$\mathbb Q_p$; the primes may differ. The algebraic group $\mathbf H_i$ is
+connected, adjoint, and absolutely simple over $k_i$. The superscript $+$
+denotes the subgroup generated by the $k_i$-points of the unipotent
+radicals of proper $k_i$-parabolic subgroups. Each $G_i$ is required to be
+noncompact and to have Kazhdan's property $(T)$.
+
+A lattice is discrete and has finite Haar covolume. Abstract
+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
+$\Sp(n,1)$, $n\ge2$, 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}.
+
+All scalar cocycles are $\mathbb T$-valued and normalized:
+\[
+ \mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj),\qquad
+ \mu(e,g)=\mu(g,e)=1.
+\]
+The twisted group factor $L_\mu(\Gamma)$ has canonical unitaries $u_g$
+with $u_gu_h=\mu(g,h)u_{gh}$ and trace $\tau(u_g)=\mathbf 1_{\{e\}}(g)$.
+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
+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
+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
+\[
+ Q^t=p\bigl(M_n(\mathbb C)\bar\otimes Q\bigr)p,
+ \qquad (\Tr_n\otimes\tau_Q)(p)=t,
+\]
+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
+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),
+ \qquad \sigma(e)=I,
+\end{equation}
+define the \emph{arithmetic model}
+\begin{equation}\label{eq:intro-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}
+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
+$\mu|_A$.
+
+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
+$\pi(g)\xi=u_g\xi$. Hence the joint action
+$U_{(g,h)}\xi=u_g\xi v_h^*$ has multiplier
+$\alpha((g,h),(g',h'))=\mu(g,g')\overline{\omega(h,h')}$.
+
+\begin{theorem}[Arithmetic exhaustion]\label{thm:arithmetic-exhaustion}
+Let $\Gamma\in\mathscr K$, let $\Lambda$ 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
+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, $\delta:A\to B$
+is an actual group isomorphism, and $\sigma$ is a finite-dimensional
+projective unitary representation satisfying
+Equation~\eqref{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
+\[
+ \dim_{L_\mu(\Gamma)}E(A,B,\delta,\sigma)=n[\Lambda:B],\qquad
+ \dim E(A,B,\delta,\sigma)_{L_\omega(\Lambda)}=n[\Gamma:A].
+\]
+\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
+elementary construction and its normality and dimensions are proved here
+in Section~\ref{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
+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
+abstractly commensurable. These conclusions cover nonuniform and
+reducible lattices and their ICC commensurable groups.
+
+\subsection{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};
+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}.
+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}.
+
+Property $(T)$ alone does not ensure group reconstruction: independent,
+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 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
+Chifan--Ioana--Osin--Sun constructed property-$(T)$ examples using
+wreath-like products \cite{CIOS2023}. 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
+\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
+of cocycle ratios, and computes their fusion by projective induction
+\cite[Theorem~2.2 and Section~6.2]{Vaes2008}. Donvil--Vaes prove rigidity
+under arbitrary scalar twists and virtual isomorphism for certain
+left-right wreath products, including the projective-ratio constraint and
+reconstruction, up to amplification, of finite-index factor neighbours
+\cite[Theorems~A and~6.6]{DonvilVaes2025}.
+Chifan--Fern\'andez Quero--Osin--Tan prove stable canonical recovery for
+twists of certain property-$(T)$ wreath-like products when the source
+cocycle is trivial on the abelian base \cite[Theorem~F]{CFOT2026}. The
+exhaustion theorem proved here treats arbitrary source cocycles in the
+local-field lattice class.
+
+A geometric line of operator-algebraic rigidity for lattices begins with
+character and finite-factor representation results. Bekka classified the
+characters of $\SL_n(\mathbb Z)$, $n\ge3$ \cite[Theorem~3]{Bekka2007};
+Peterson established a finite-factor representation dichotomy for
+irreducible higher-rank property-$(T)$ lattices
+\cite[Theorem~A]{Peterson2014}. Boutonnet--Houdayer developed stationary
+and noncommutative boundary methods that recover character rigidity
+\cite[Theorems~A--C]{BoutonnetHoudayer2021}. Their noncommutative factor
+theorem subsequently classified intermediate von Neumann algebras of
+specified lattice boundary crossed-product inclusions by parabolic
+quotients \cite[Theorem~B and Theorem~3.3]{BoutonnetHoudayer2023}.
+Related results recover Weyl groups from the distinguished inclusion
+associated with the action on the quotient by a split-torus centralizer
+\cite[Theorem~A and Corollary~B]{HI2026}, and classify intermediate
+$C^*$-algebras for boundary crossed products of irreducible lattices in
+products of real rank-one groups \cite[Corollary~B]{HoudayerLeBars2026}.
+These results show how boundary data retains the ambient geometry. For
+the finite correspondence considered here, compatible boundary
+measurements must be constructed in several module copies; their
+simultaneous law then produces finite-rank subgroup-graph fibres.
+
+The search for a classical equivariant map from an operator-valued
+coupling is motivated in part by the self-coupling strategy in Furman's
+measure-equivalence rigidity theorem
+\cite[Sections~3--4]{Furman1999}. The noncommutative classicalization and
+finite-module reconstruction used below are proved here; they are not
+consequences of that classical theorem.
+
+\subsection{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
+action is weakly mixing. In this reduced setting the goal is a fixed
+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}.
+
+The geometric construction produces this partition through boundary
+measurements. Let $B$ be the compact full flag space of the known ambient
+group $G$. A projection-valued measure (PVM) assigns an orthogonal
+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
+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
+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
+zero separately with slots one and two, and the required conclusion
+includes
+\[
+ [\Theta_{01}(A),\Theta_{02}(C)]=0
+ \qquad\text{for all Borel sets of boundary maps }A,C.
+\]
+This common-slot commutation has a concrete purpose. Normal slices in the
+separate branch slots generate an abelian algebra on $K_0$. Disintegrating
+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
+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
+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
+algebra even when the lattice is reducible. Escape in all ambient factors
+then creates an isometric boundary link. Escape in a subproduct instead
+gives a pairing through the fixed algebra's conditional expectation.
+Ordered Fourier transfer and incidence relations make the positive
+observations sharp and identify exact residue graphs, while unrelated
+measurements still retain their original order.
+
+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}).
+
+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}).
+
+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
+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}.
+
+Full tuples bring these branches together. Comparing lattice packing with
+Haar growth calibrates the speeds around the finite type permutation and
+fixes the exact Jacobians. Ordered imaginary powers then produce a
+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}).
+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
+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}).
+
+\begin{figure}[tbp]
+ \centering
+ \input{figures/proof-map}
+ \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}
+
+\subsection{Organization}
+
+Section~\ref{sec:models} verifies the elementary arithmetic models.
+Sections~\ref{sec:setup}--\ref{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
+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}
+
+The models of Theorem~\ref{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
+\cite[Section~4]{Mackey1958}, and these counting-space coordinates also
+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
+\[
+ D=\Gamma\times\Lambda,\qquad
+ \alpha((g,h),(g',h'))=\mu(g,g')\overline{\omega(h,h')}.
+\]
+For a subgroup $C\le D$ the \emph{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}
+
+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
+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}
+
+\begin{lemma}[Exact projective induction]
+\label{lem:models-induction}
+Equation~\eqref{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}
+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}
+ 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}.
+This argument retains the multiplier rather than discarding it.
+
+For explicit independence of the choice of representatives, suppose
+$r'_x=r_xk_x$ with $k_x\in C$. The identification from the new
+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}
+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
+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
+\[
+ \ell^2(\Gamma)\otimes V_{x_0}\longrightarrow
+ \bigoplus_{g\in\Gamma}V_{(g,e)x_0},
+ \qquad \delta_g\otimes\xi\longmapsto U^X_{(g,e)}\xi
+\]
+intertwines the action with $\lambda_\mu\otimes1$, 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
+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
+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
+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}
+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
+\[
+ E(A,B,\delta,\sigma)=
+ L^2(M)\mathbin{\bar\otimes}_{L_{\mu|A}(A)}
+       \bigl(V\otimes L^2(N)\bigr)
+ \cong\mathcal H(C_\delta,\gamma).
+\]
+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}
+
+\begin{proof}
+On $V\otimes L^2(N)$ the subgroup action is
+\[
+ u_a(\xi\otimes\eta)=\sigma(a)\xi\otimes v_{\delta(a)}\eta.
+\]
+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}
+is unitary and conjugates the subgroup action to
+$\lambda_{\mu|A}\otimes1$. To check the phase, applying $u_b$ before this
+map produces
+\[
+ \omega(\delta(b),\delta(a))\sigma(ba)^*\sigma(b)
+ =\omega(\delta(b),\delta(a))\beta(b,a)\sigma(a)^*
+ =\mu(b,a)\sigma(a)^*.
+\]
+Thus the left subgroup action is normal. The right $N$-action is standard
+on the second factor and commutes with it.
+
+Choose representatives $r$ for $\Gamma/A$. As a right
+$L_{\mu|A}(A)$-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}
+At its identity fibre the graph action is exactly
+\[
+ U_{(a,\delta(a))}j\xi
+ =u_a\otimes(\xi\otimes v_{\delta(a)}^*)
+ =j\sigma(a)\xi.
+\]
+For $g=ra$, balancing in Equation~\eqref{eq:calculus-coset-model} gives
+the more explicit formula
+\[
+ U_{(g,h)}j\xi
+ =\mu(r,a)^{-1}u_r\otimes
+       \bigl(\sigma(a)\xi\otimes v_{\delta(a)}v_h^*\bigr).
+\]
+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,
+transitive system with graph action $\gamma$ at the basepoint. Finally
+\[
+ U_sU_{r_x}j
+ =\frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))}
+       U_{r_{sx}}j\gamma(c(s,x)),
+\]
+which shows that the stated unitary has exactly the transport in
+Equation~\eqref{eq:input-A1}.
+\end{proof}
+
+\begin{corollary}[Elementary normality and dimensions]\label{cor:models-dimensions}
+The correspondence $E(A,B,\delta,\sigma)$ is normal and bifinite. If
+$n=\dim_{\mathbb C}V_\sigma$, then
+\[
+ \dim_M E(A,B,\delta,\sigma)=n[\Lambda:B],\qquad
+ \dim E(A,B,\delta,\sigma)_N=n[\Gamma:A].
+\]
+\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 $n$
+regular copies, of module dimension $n$ in the unnormalized matrix-trace
+convention. The respective orbit counts are $[\Lambda:B]$ and $[\Gamma:A]$.
+\end{proof}
+
+Finite direct sums and closed bimodule summands of these models remain
+normal and bifinite: the two actions restrict normally to a reducing
+subspace, and module dimension is additive and monotone under orthogonal
+projections in the commutant \cite[Theorem~10.2.1]{Jones2009}. The
+geometric problem is the converse: starting from an arbitrary bifinite
+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}
+
+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
+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
+\[
+ \rho(h)\xi=\xi v_h^*,\qquad
+ \rho(h)\rho(k)=\overline{\omega(h,k)}\rho(hk).
+\]
+The left representation is $\pi(g)\xi=u_g\xi$. Thus $\pi$ and $\rho$
+commute, although their multipliers need not be trivial. Conjugation by
+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}
+
+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)
+       \ominus\mathbb C1
+ \quad\text{has no nonzero finite-dimensional subrepresentation}.
+ \label{eq:setup-weak-mixing}
+\end{equation}
+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}.
+
+\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
+\[
+ \lambda_\alpha\otimes v\simeq\lambda_{\alpha\nu}\otimes1_V.
+\]
+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
+\[
+ T(\delta_g\otimes\xi)=\delta_g\otimes v_g^*\xi.
+\]
+Indeed
+\[
+ T(\lambda_\alpha(h)\otimes v_h)T^*(\delta_g\otimes\xi)
+ =\alpha(h,g)\nu(h,g)\delta_{hg}\otimes\xi.
+\]
+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}
+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
+restriction of $K$ to the right subgroup factor is a finite orthogonal
+sum of blocks
+\[
+ K_c=V_c\otimes K'_c,\qquad
+ \pi(g)|_{K_c}=1\otimes\pi'_c(g),\qquad
+ \rho(h)|_{K_c}=v^c_h\otimes\rho'_c(h)\quad(h\in\Lambda_0).
+\]
+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}
+Each $K'_c$ is a finite normal module on both sides and satisfies
+Equation~\eqref{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 $N^{\mathrm{op}}\subset
+P$ 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}.
+
+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
+\[
+ \|xy-x'y'\|_2\le\|x\|_\infty\|y-y'\|_2+\|x-x'\|_2\|y'\|_\infty,
+\]
+and their bounded balls are strongly closed because bounded strong
+convergence in a finite von Neumann algebra implies $L^2$ convergence.
+They are dense in the compact Hilbert subspace: the $L^2$-nearest-point
+projection onto each closed operator-norm ball is equivariant and
+nonexpansive. Thus $D_c$ is a von Neumann subalgebra.
+
+Twisted conjugation on $L^2(N)\ominus\mathbb C1$ has no nonzero
+finite-dimensional subrepresentation: the diagonal of a finite-rank
+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
+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
+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
+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}.
+
+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
+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
+vector corresponding to the identity matrix is fixed by
+$\overline{v^c_h}\otimes v^c_h$, 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}.
+
+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
+$L^2(D_c)$ and its intersection with the reduced commutant on this block
+is scalar. This proves Equation~\eqref{eq:setup-weak-mixing}.
+\end{proof}
+
+\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
+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}
+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
+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
+tensor-product inner products give the isometries.
+
+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
+$a\in\Lambda_0$,
+\[
+ v_sv_h=\frac{\omega(s,h)}{\omega(a,t)}v_av_t,\qquad
+ v_hv_t^*=\frac{\omega(s,h)}{\omega(a,t)}v_s^*v_a.
+\]
+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
+closed summand. The finite basis formulas show preservation of finite
+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
+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
+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}
+
+From now until Section~\ref{sec:recovery}, let $\Gamma<G$ be an actual
+ICC lattice satisfying Equation~\eqref{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
+transports remain available on $G/\Gamma$.
+
+Take $m=1$ or $2$ and put
+\[
+ H=K^{\otimes(m+1)},\qquad
+ R=\Gamma_1\times\cdots\times\Gamma_m\times\Lambda.
+\]
+The slots of $H$ are numbered $0,\ldots,m$; $\pi_j$ 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}
+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}
+
+\begin{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
+formed from the other slots' $\rho(h)$ and commute with all their
+$\pi_j(g)$. After absorption the $\Lambda$-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
+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
+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
+nonconstant summand.
+\end{proof}
+
+Choose a column realization
+\[
+ H=eL^2(B)^k,\qquad e\in M_k(B),\qquad
+ d=(\Tr_k\otimes\tau_B)(e)>0.
+\]
+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
+$w_r$. 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}
+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
+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
+are fixed, or the Hilbert-norm factors are retained explicitly; an
+operator-norm bound alone is not a bound independent of trace rescaling.
+
+Choose a Borel section $d_0:Y\to G$ and write
+\[
+ c(a,y)=d_0(ay)^{-1}ad_0(y),\qquad
+ X_a(y)=\pi_0(c(a,y)),\qquad
+ (U_af)(ay)=X_a(y)f(y).
+\]
+These transports compose up to a scalar function of modulus one on the
+base. More explicitly,
+$X_a(by)X_b(y)=\mu(c(a,by),c(b,y))X_{ab}(y)$. Conjugation on
+base-decomposable operators is therefore a genuine action, denoted
+$\sigma=\Ad U$.
+
+\begin{proposition}[Induced finite algebras]
+\label{prop:setup-induced-algebras}
+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
+$\mathcal A_j$ be the commutant of the base multipliers and
+$W|_{R'_j}$. Then
+\[
+ (\mathcal A_j)^{\beta_j}=\mathcal N,\qquad
+ (\mathcal A_j)^\sigma\ \text{has a faithful finite $\beta_j$-invariant
+ trace},\qquad
+ \mathcal N^G=\mathbb C1.
+\]
+\end{proposition}
+
+\begin{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
+first on compact sets where the section is continuous, by Lusin
+approximation, then remove sets of small measure. The corresponding
+projective unitaries are bounded, so their multiplication operators vary
+strongly on $\mathcal H$. Conjugation of the corner is inner at each fibre
+and translation preserves base probability, so $\sigma$ preserves the
+trace. The first identity is the commutant of all of $W$ in the
+decomposable algebra.
+
+For the second, undo induction. If $A(y)$ is $\sigma$-invariant, lift it,
+for $g=d_0(y)\gamma$, as $\widetilde A(g)=\pi_0(\gamma)^*A(y)\pi_0(\gamma)$.
+The cocycle covariance makes this field left $G$-invariant, hence almost
+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
+$\beta_j$-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 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}.
+
+\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
+a finite place the factors have relative rank at least two. For
+higher-rank property $(T)$ see \cite[Theorem~6]{DelarocheKirillov1968};
+the real rank-one classification is in \cite[Section~3.3 and
+Theorem~3.5.4]{BHV2008}. Relative rank one at a finite place gives an
+unbounded action on the Bruhat--Tits tree and is incompatible with
+property $(T)$ \cite[Remark~1.6.3]{BHV2008}.
+
+In a rank-one semisimple Levi position exactly one almost simple factor is
+isotropic. Its minimal boundary is the full panel residue. The compact and
+toric normal factors of the Levi act trivially on this boundary. Source
+transports use its root-generated group. When an algebraic central cover
+is needed, use the subgroup generated by its lifted unipotent radicals.
+The root-group isomorphisms below map this subgroup onto the effective
+root-generated group with finite central kernel. Every application of
+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
+generated by the unipotent radicals \cite[Theorem~2.20(iii) and
+Proposition~2.24(ii)]{BT1972Complements}. For a central kernel $C$, the
+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}
+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
+$G_0$, its weak limit is the projection onto $L^2(\mathcal N)^{G_0}$,
+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}
+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
+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
+conjugate subspaces; the map
+$(t_1,\ldots,t_d)\mapsto\prod_j\exp(t_jX_j)$ 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
+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
+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
+\[
+ \mathcal H=\int_{\widehat S}^{\oplus}(H_\xi\otimes M_\xi)\,d\eta(\xi),
+ \qquad
+ \pi(s)=\int_{\widehat S}^{\oplus}(\xi(s)\otimes1)\,d\eta(\xi).
+\]
+In this central decomposition every unitary commuting with $S$ acts
+fibrewise as $1\otimes V_\xi$
+\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 $V_\xi$,
+\[
+ \Bigl|\bigl\langle(\xi(s)\otimes V_\xi)\sum_j e_j\otimes u_j,
+       \sum_k f_k\otimes v_k\bigr\rangle\Bigr|
+ \le\sum_{j,k}|\langle\xi(s)e_j,f_k\rangle|\,\|u_j\|\,\|v_k\|.
+\]
+Howe--Moore makes the right side tend to zero along escape; approximation
+by finite tensor sums is uniform because all operators have norm one.
+Dominated convergence then proves the uniform decay off the
+$S$-invariants.
+
+Let $P_i$ project onto the invariants of the $i$th simple factor. These
+commute with every group action and each other. Expand the identity as the
+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,
+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}
+
+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
+coordinates survive.
+
+An archimedean panel inside a higher-rank component cannot have noncompact
+Lie algebra $\mathfrak f_{4(-20)}$: its complexification would have to
+occur as a component of a proper absolute Dynkin subdiagram, and the
+Dynkin list has no such occurrence. At a finite place the effective
+rank-one Lie algebra is simple also over $\mathbb Q_p$: after restriction
+of scalars its absolutely simple factors form one Galois orbit, so a
+$\mathbb Q_p$-ideal is either zero or the whole algebra. An adjoint
+automorphism centralizing an open subgroup centralizes all sufficiently
+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
+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
+decompositions can be made on the simply connected covers and descended.
+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
+\[
+ m_\alpha=\dim_{k}\mathfrak g_\alpha\ \text{at a finite place},\qquad
+ 2\rho=\sum_{\alpha>0}m_\alpha\alpha,
+\]
+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}
+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
+centralizer character is the exact logarithm of that character's module.
+We do not replace this by a coarsely equivalent length in identities
+involving heights. All displacements and offsets along $e_i$ range on a
+sufficiently divisible mesh in this logarithmic translation lattice.
+Cocharacter rationality and common multiples of relative coroot
+translations and central isogenies ensure these elements and their
+semisimple projections to every standard Levi have lifts in the
+corresponding root-generated covering groups. There are only finitely
+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}
+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}
+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
+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
+below by a fixed multiple of its exponential, and the box has fixed
+positive Euclidean volume.
+
+At a finite place the Cartan double coset has measure
+$[K:K\cap aKa^{-1}]$ times the fixed compact measure, comparable to
+$\exp(2\rho(L))$. In compact root charts a sufficiently small analytic
+compact open subgroup has product coordinates in negative-root,
+centralizer, and positive-root directions; under conjugation by the
+dominant translation its intersection shrinks the expanding root
+lattices, and the index is the product of their module indices with
+exponent $\sum_{\alpha>0}m_\alpha\alpha(L)$. Equivariant exponentials and
+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}
+
+For each relative vertex type choose a highest-line representation: take
+the determinant line of the relevant parabolic nilradical in an exterior
+power and the cyclic submodule it generates. Over a splitting field choose
+a compatible positive system. This line is a highest line, with weight
+strictly positive on the omitted absolute nodes and zero on the
+complementary coroots. Its relative weight is a positive multiple of the
+relevant fundamental weight. The cyclic highest-weight module is absolutely
+irreducible with one-dimensional top split space. Lowering along an omitted
+absolute simple root separates a weight by a positive multiple of the
+corresponding relative simple root; every other weight drops by a
+nonnegative combination involving that root. Consequently a normalized
+Cartan matrix has rank-one limit exactly when that simple gap diverges.
+
+At a finite place these are $k$-rational modules, used only to form compact
+projective matrix data: use compact-invariant lattice norms and compact
+sup-norm frames, recording both the matrix and its inverse up to projective
+normalization. At an archimedean target, for Hilbert highest-line
+amplifications use a real proximal module irreducible on complexification
+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}
+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
+residual Levi transport between their transverse refinement spaces,
+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}
+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
+the complementary Levi. Bounded complementary gaps keep the surviving
+block and its inverse bounded in those frames, so the block acts by an
+invertible Levi transformation. The same analysis of the inverse matrix
+identifies the reverse face.
+
+Equivalently, write an ordinary flag in the open Bruhat cell transverse to
+the reverse face. Conjugating its radical coordinates by the Cartan
+translation contracts the coordinates of the diverging gaps, leaving the
+complementary Levi coordinates. Project to the reverse refinement space,
+then apply the surviving Levi transformation: this is the intrinsic
+residual projectivity. A change of compact frame conjugates these
+coordinates by the corresponding compact Levi projectivity and leaves the
+intrinsic map unchanged. Passing to a deeper stratum preserves all
+preceding image vertices. In a face omitting one type in a higher-rank
+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}.
+
+\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}
+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
+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}
+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
+$n_i\ge0$,
+\[
+ 0\le\alpha(T_t(y-x))=\sum_in_it_iL_i\le\sum_in_iL_i=\alpha(y-x).
+\]
+Thus the new point lies in every affine root half-apartment containing both
+endpoints, i.e. in their root enclosure. Apartment intersections are
+enclosed; the transition maps agree on this enclosure, and the Weyl
+stabilizer of a wall displacement fixes its transformed displacement. This
+proves independence. On any apartment and Weyl sector $T_t$ has uniformly
+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
+repelling data and sufficiently large powers; the ping-pong domains give a
+discrete Schottky subgroup. The full chamber action is a topologically
+amenable homogeneous-space action because its minimal parabolic is
+amenable; restriction to a closed subgroup preserves this amenability
+\cite[Propositions~2.2 and~2.5, Example~2.7(5)]{AnantharamanDelaroche2002}.
+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}
+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
+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
+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}
+
+An escaping source translation acts on a finite regular array. We record the
+limiting flags of its measured Fourier index and compare the measurements
+seen from the two ends of the translation. The two tasks are to construct
+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
+field write
+\[
+ \mathsf T(C)=\int_Y(\Tr\otimes\tau_B)(C(y))\,dy,\qquad
+ \tau_{\mathcal N}(C)=d^{-1}\mathsf T(C).
+\]
+Thus $\tau_{\mathcal N}$ is a probability trace while the Hilbert norm of a
+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
+\[
+ b_r=e\varepsilon_r,\qquad 1\le r\le k,\qquad
+ \sum_{r=1}^k b_rb_r^*=e.
+\]
+
+\subsection{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 $I$;
+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}
+There is a finite family of projectively normalized matrix coordinates,
+including coordinates of the inverse, with the following properties.
+\begin{enumerate}[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
+between their Levi refinement spaces. Its closure has only gap sets
+containing $D$.
+\item On a bounded complementary-gap window the residual blocks are
+invertible in compact frames. They send a full flag transverse to the
+reverse face, by transverse parabolic projection followed by Levi transport,
+to a full flag incident with the image face.
+\item Fixed left and right group multiplications preserve gap sets. Right
+multiplication leaves the image flag unchanged; left multiplication
+transports it by the usual target action. Reverse data obey the opposite
+rule, and residual projectivities change by the corresponding compositions.
+\item In an archimedean highest-line module each surviving top face space is
+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}
+Use the highest-line modules and normalized matrices of
+Section~\ref{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,
+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
+divergence of the corresponding simple gap; the finitely many coordinates
+record the gap set, its image and reverse flags, and residual maps. If an
+additional gap diverges the previous image vertices persist by that lemma.
+
+For the multiplication rule, both $a$ and $a^{-1}$ have finite norm; thus
+projectively normalized multiplication by $a$ has denominator bounded away
+from zero, so fixed multiplication changes each simple Cartan gap by a
+bounded amount. Left multiplication carries the image top space by $a$;
+right multiplication leaves it unchanged. The same calculation for inverse
+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
+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
+with slack: a slightly larger window retains the same large-gap separation
+and lower bounds for the invertible residual blocks. At real places these
+tests are smooth functions of matrix coordinates; at finite places they are
+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 tests here act on full regular arrays; they are not followed by a
+corner projection before the next operation. This permits a diagonal
+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}
+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}.
+
+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
+coefficient indices. On full regular arrays, a scalar test of this label is
+a diagonal Hilbert multiplier; these tests are not followed by corner
+compression before another array operation.
+
+The one-column map
+\[
+ (V_a f)(y)=X_a(y)f(y)
+\]
+is an isometry with the integrated tracial Hilbert norm, so its Hilbert
+ultralimit $V$ is an isometry. On the minimal spectral span of $V\mathcal H$
+the continuous diagonal tests give a PVM $E^{(1)}$; the compression
+\[
+ \Phi(A)=V^*E^{(1)}(A)V
+\]
+is a positive effect on the original input. For $A$ recording only a gap
+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}
+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;
+restrict to the minimal closed spectral span of the input ranges and a
+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}
+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
+its image flag, a left Fourier matrix multiplier can be removed from a
+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}
+For a fixed shift, normalized matrix multiplication and the inverse formula
+have denominators bounded away from zero by Lemma~\ref{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
+and the reindexing commute with diagonal tests.
+
+If a Fourier index $h$ is changed to $rh$ by a left multiplier, the recorded
+inverse changes to $h^{-1}r^{-1}$; the gap set and image flag are unchanged
+in the limit. Let $D$ be a bounded test of those data and $T$ a bounded left
+Fourier multiplier with uniformly bounded finite Fourier approximants
+$T_l\to T$ in trace $L^2$. In a form tested on uncut operator-bounded
+columns, replacing either occurrence of $T$ by $T_l$ before moving $D$
+costs at most $C\|D\|\,\|T-T_l\|_2$, uniformly in the fixed column bounds
+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
+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 $Df$.
+
+For a field reindexed at $ay$, invariant probability on $Y$ gives
+$\int\|T(ay)-T_l(ay)\|_2^2dy=\|T-T_l\|_2^2$, so the same estimates hold after
+path transport. For a compact family of fixed translations, the cocycle
+formula and continuity of translations in measure make their matrix fields
+compact in trace $L^2$; a finite net gives uniform finite-Fourier
+approximation of the compact family. No common finite-Fourier approximation
+of the escaping paths themselves is used.
+
+Finally, the group-valued section cocycle satisfies its exact composition
+identity. Applying the projective source representation gives a scalar
+function of the base in each composition formula; pulling a fixed source
+translation from one end of an array to the other therefore multiplies the
+whole array by one scalar of modulus one, which cancels between the two
+entries of a sesquilinear form. The same observation applies to a quotient
+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}
+
+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
+following properties.
+\begin{enumerate}[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
+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
+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}
+Its polarized extension on finite sums,
+\[
+ \Bigl\|\sum_iJ(\overline{b_i},f_i)\Bigr\|^2
+ =d^2\sum_{i,l}\tau_{\mathcal N}
+   \bigl(E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)\bigr),
+ \label{eq:links-polarized-relative}
+\]
+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}.
+\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}
+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
+flag or residual projectivity, those windows must first be exhausted to such
+an invariant event.
+\item The image measurements are independent of the finite regular
+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
+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}.
+
+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
+tends strongly to the identity. Its discrete section cocycle is the identity
+off a set of measure tending to zero, after its pure base shift is undone,
+so on bounded inputs both descriptions give a vanishing Hilbert error even
+with a bounded diagonal test inserted. This proves radical invariance of
+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};
+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.
+
+The mass of a gap stratum is target invariant, hence in $\mathcal N$, and is
+invariant under the entire face parabolic. Every inactive simple factor is
+in the centralizer; in an active factor the contracted radical contains an
+escaping sequence. By Howe--Moore applied to the representation on
+$L^2(\mathcal N)$, a radical-invariant vector orthogonal to the
+factor-invariant subspace would have a matrix coefficient both constant and
+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
+action give
+\[
+ \|J(\overline b,f)\|^2
+ =d\lim_a\mathsf T(ff^*\sigma_{a^{-1}}(bb^*)).
+\]
+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
+\[
+ \Bigl\|\sum_iJ(\overline{b_i},f_i)\Bigr\|^2
+ =d^2\sum_{i,l}\tau_{\mathcal N}(f_lf_i^*)\tau_{\mathcal N}(b_ib_l^*),
+\]
+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.
+
+Multiplication of an input by a start (resp.\ end) base function is
+represented on arrays by its value at $y$ (resp.\ $ay$); diagonal tests
+commute with both, so their limiting forms are decomposable at the two
+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
+it scalar by the Howe--Moore argument above; the compressed effect is
+$1\otimes C$, and the frame computation below identifies $C=\Phi(A)$.
+
+\emph{(iii)} For an escape in $G_0$, the weak limit of the trace-preserving
+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
+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
+$b_r(ay)^*$ 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.
+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}.
+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.
+
+For a window involving residual or reverse data, the left shifts may change
+the window, so that argument does not apply to it individually. Suppose
+instead that all end vectors have been annihilated on each member of an
+exhaustion of the desired image-and-stratum event. Monotone spectral
+convergence first gives annihilation on that event, and only then may the
+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
+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
+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
+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
+the one-ended path acts only in slot zero, so its compressed tests are the
+original tests tensored with identity on the added slot. The
+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}
+
+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
+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}
+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*)]
+\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
+$N_0$; their generated algebra, matrix effects, and support projections
+preserve relative boundedness;
+\item a closed $N_0$-invariant subspace has a dense relatively bounded part,
+and a relatively bounded column has operator-bounded approximants with the
+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}
+
+\begin{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}
+This proves the continuous extension from bounded $b$; the link limit is
+always taken before this extension.
+
+Let $c\in N_0$. Along a path in $G_0$, $c$ is a fixed source-invariant left
+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
+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.
+
+For a closed subspace invariant under $N_0$, the spectral projections
+$p_n=1_{[0,n]}(E_{N_0}(ff^*))$ preserve it; $p_nf$ is relatively bounded and
+converges to $f$ since
+$\|(1-p_n)f\|_2^2=d\tau_{\mathcal N}((1-p_n)E_{N_0}(ff^*))\to0$. For
+relatively bounded $f$, $f_n=1_{[0,n]}(ff^*)f$ gives operator-bounded
+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
+$E_{N_0}((f_n-f)(f_n-f)^*)\le4E_{N_0}(ff^*)$ controls it uniformly. Take
+$n\to\infty$ then $l\to\infty$. The same argument with inputs reversed
+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
+bounded pairs with no mixed tensor decomposition.
+\end{proof}
+
+\subsection{Additional invariant variables}
+
+\begin{lemma}[Additional invariant limiting variables]
+\label{lem:links-invariant-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
+established above. If a one-column effect is also source face-parabolic
+invariant, including its inactive simple factors, and its event or test is
+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}
+A fixed Fourier shift gives precisely the limiting intertwining identity
+used in Lemma~\ref{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 $\mathcal N^G=\mathbb C1$.
+
+For mixed links, independent target invariance puts the respective slices
+in $\overline{\mathcal N}$ and $\mathcal N$; independent
+contracted-radical invariance makes them scalar, and polarization and
+normal slice separation give the scalar joint effect.
+
+Fixed left and right multiplications change Cartan coordinates by bounded
+amounts; a radial variable divided by a scale tending to infinity has
+bounded continuous tests in its compactification that are asymptotically
+unchanged. The source face-parabolic invariance follows because a
+contracted radical preserves the enlarged data law (sliding it to the far
+end, the near-identity transporter changes any bounded input by a
+Hilbert-small error against bounded diagonal tests) and a centralizer
+element slides to the same end, its reindexing removed by invariant
+probability and its left multiplier by finite-Fourier Gram. The additional
+radial test adds no term because fixed Fourier shifts change its limiting
+coordinate by zero. Thus image-and-stratum tests together with these radial
+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}
+
+Two kinds of spectral operation are used. A single limiting compact
+variable has its usual Borel functional calculus. For several possibly
+noncommuting compressed effects, an ordered integral initially means a
+product of single-slot continuous tests; smooth joint functions on compact
+real charts are inserted by absolutely summable separated expansions, and
+fixed-level finite-place functions by finite separated partitions. Support
+of an ordered product on a closed relation means vanishing on open
+rectangles disjoint from it; support alone supplies no general ideal of
+discontinuous joint functions.
+
+On a buffered transverse projective box, normalized matrix multiplication
+has denominators bounded below. Over a finite local field, choose ratio
+charts with a maximal coordinate; matrix operations are equicontinuous, and
+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}.
+
+\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}
+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
+fibre transporters satisfies the exact group composition law, and the
+section continuity-in-measure argument gives strong continuity of the
+resulting induced representation.
+
+The added coordinate is unmeasured. At one base its two transported vectors
+have exactly their original inner product, so a base-decomposable form is
+tensored pointwise with the identity. For a two-ended form the two base
+algebras commute, and the same construction is performed independently at
+the two inputs. The generator identification of minimal spectral dilations
+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}
+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}
+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
+$\sum_{h\in R}|\langle bw_h^*,\xi\rangle|^2=\|b^*\xi\|_2^2\le\|b\|_\infty^2\|\xi\|_2^2$,
+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.
+
+For bounded end and start columns, the squared norm of the coefficient at a
+fixed measured index $g\in\Gamma_j$ in $b(ay)^*X_a(y)f(y)$ is the sum over
+$R'_j$ of squared frame coefficients of the pair; equivalently these are
+inner products of $X_a(y)f(y)$ with partial translates of $b(ay)$, with $g$
+absorbed into one column. Use now the complementary regular realization for
+$\Gamma_0\times R'_j$. Project each column to a finite $\Gamma_0$-coordinate
+window; these projections commute with partial $R'_j$-translates, preserving
+Bessel bounds. Source transport is unitary and commutes with partial
+translates, and a fixed measured shift preserves their bounds. Both
+projected columns converge in integrated Hilbert norm as windows increase,
+and the Bessel estimate on the other column bounds the sum of squared
+coefficient errors by that Hilbert error. Reindexing the end column at $ay$
+does not change its averaged Hilbert error.
+
+For fixed finite windows the translated supports are disjoint unless the
+source cocycle $c(a,y)$ belongs to a fixed finite subset of $\Gamma_0$. For
+any $\delta>0$, choose a compact set containing $d_0(y)$ outside a set of
+measure $\delta$; by invariance, the same holds for $d_0(ay)$. On the
+intersection of the two good sets, $c(a,y)$ in a fixed finite set forces
+$a=d_0(ay)c(a,y)d_0(y)^{-1}$ to lie in a fixed compact subset of $G$,
+impossible eventually on an escaping path. Exceptional measure is at most
+$2\delta$, arbitrary.
+
+Off that set, every partial-translate coefficient of the finite-window pair
+vanishes; on it, their summed squares are uniformly bounded by Bessel and
+column bounds. Integration gives convergence to zero, and the finite-window
+approximation is removed afterward. We have proved vanishing of the norm at
+every fixed measured $\Gamma_j$-index.
+
+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
+one-column assertion.
+\end{proof}
+
+\subsection{An amenable-boundary obstruction}
+
+\begin{lemma}[Full-flag obstruction]
+\label{lem:links-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
+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}
+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 $p\mathcal H$,
+$\Psi(u)=pa^{-1/2}\Phi(u)a^{-1/2}p$ 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 $\Phi(u)$.
+
+Choose a discrete free $F\cong\mathbb F_2\subset S$, obtained from two
+sufficiently strong rank-one contractions with disjoint attracting and
+repelling neighborhoods. Let $\mathcal D$ be the concrete $C^*$-algebra on
+$p\mathcal H$ generated by cut base multipliers, the companion actions
+$W|_{R'_j}$, 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
+$\|\widehat D\|\le\|D\|$ in the concrete input norm. The covariant lift
+$T_g(u\otimes\xi)=(g\cdot u)\otimes W_g\xi$ has the same multiplier
+$\zeta$ as $W$.
+
+By \cite[Proposition~2.5]{AnantharamanDelaroche2002}, the action of
+$\Gamma_j$ on $Z$ is topologically amenable: choose continuous
+$\eta_i:Z\to\ell^2(\Gamma_j)$ with finite coordinate support, norms tending
+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
+$Q_i\eta=\sum_{t\in\Gamma_j}\delta_t\otimes\pi_Z(\xi_i(\cdot)_t)\eta$;
+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
+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}
+The multiplier is that of $W$ and is conjugated in the right-action
+convention.
+
+Complementary regularity makes $H$ a subrepresentation of a finite regular
+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},
+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
+probability $\nu$ on $F$, the operator in this model has norm at most
+$\|\lambda_F(\nu)\|$: pointwise Hilbert norms are bounded by
+$\sum_f\nu(f)\|\eta(f^{-1}g)\|$, and disintegrating Haar measure along left
+$F$-orbits gives the ordinary positive regular convolution bound on each
+orbit.
+
+We contradict this on one matrix vector. Regard $p$ as $k$ columns and put
+$T_f(y)=p(x)X_f(x)^*p(y)$ with $x=f^{-1}y$; this is a contraction. Since
+$p$ is $\sigma_f$-invariant, $X_f(x)p(x)X_f(x)^*=p(y)$, and right
+multiplication after the shift gives
+$(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)\cdot p(x)X_f(x)^*p(y)=p(y)$. For the
+finitely many $f$ in the support of $\nu$, choose contraction fields
+$T_{f,n}$ 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,
+$\|S_n\|\le\|\lambda_F(\nu)\|$; 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}
+
+\subsection{Strict growth under a slower Levi translation}
+
+\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
+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}
+
+\begin{proof}
+The scalar mass $m_D>0$ is given by Proposition~\ref{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 $D\ne\Pi$.
+
+\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
+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,
+companions, and target lifts. Disintegrate
+$\mathcal K_X=\int^\oplus K_x\,d\vartheta(x)$ with isometric fibre
+inclusions $i_x$.
+
+\emph{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
+$k_\omega(l)\,d\eta_x(l)$ 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
+$l\in\mathcal F_x$, 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
+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
+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
+\[
+ \int_{X_D}\int_{\mathcal F_x}q_{x,l}(Vf_r(x))\,d\eta_x(l)d\vartheta(x)
+  =\|f_r\|^2.
+\]
+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
+$\mathcal R=\int^\oplus_{X_D}L^2(\mathcal F_x,\eta_x;K_x)d\vartheta(x)$.
+
+\emph{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
+$D_{\gamma x}(\gamma l)=r_{\gamma,x}(\gamma l)u_{\gamma,x}D_x(l)u_{\gamma,x}^*$
+for the fibre-unitary $u_{\gamma,x}$; the target lift on $\mathcal R$
+includes $r_{\gamma,x}^{1/2}$ in addition to the image direct integral, so
+$T$ intertwines the target action. Base and companion operators commute
+with full-data multiplication and intertwine $i$, so $T$ intertwines them
+and all of $W_R$.
+
+For a source centralizer $s$, the invariant image marginal gives a
+canonical lift on its minimal dilation defined by
+$U_s^X(M_uVf)=M_uVU_sf$; 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
+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
+$\mathscr E$ 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}
+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 $B$
+intertwines the base algebra; strong continuity follows from the frame
+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
+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
+map into $\mathcal A_j$, target-equivariant and $S$-invariant, contradicting
+Lemma~\ref{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
+type-I irreducibles of $S$ (none trivial), commuting unitaries act on
+multiplicity spaces, and approximation by finite tensor sums reduces to
+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,
+$V^*L_CV=\Psi(C)$. 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}
+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
+give
+$0\le\sqrt{c_C}\langle f,\Psi_s(C)f\rangle\le\langle T^{(s)}f,J_0f\rangle\to0$;
+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
+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
+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}
+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,
+the stage with $k$ source gaps has exactly $k$ target gaps, so each such
+extension adds exactly one.
+
+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}
+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
+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
+$r-|I|$; 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
+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 $k$.
+
+Finally choose the common apartment and let the faster vertex be a dominant
+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 $\mathbb Rv$.
+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}
+
+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
+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}
+
+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}
+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 $B_0$.
+Matrix sizes are fixed during every limit; a trace of mass $t$ rescales an
+operator-norm-only $L^2$ bound by $\sqrt t$, but the homogeneous estimate
+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}
+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
+following situations:
+\begin{enumerate}[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.
+\item One of the face types is a single vertex. The cutoff factors are
+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
+     \|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
+function (or finite clopen level at finite places) with uniformly bounded
+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}
+
+\begin{proof}
+\emph{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
+distance is $A+B+O(1)$; the vector-distance inequality under bounded
+endpoint perturbations gives the Cartan addition. This uses the whole
+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 $\sinh(\alpha(t\cdot L))/\sinh(\alpha(L))\le1$
+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
+finite-place bounds covers the whole ambient space.
+
+\emph{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
+least one side; the relation between $A+B$ and $A'+B'$ bounds the
+corresponding coordinate of $B'-B$, and outside the overlap one side is
+bounded by thickness. Thus $k$ lies in a fixed compact subset of the whole
+product $G$, whose intersection with $\Gamma$ is finite, uniformly in the
+moving-width positions.
+
+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
+$ko=z+\delta e_i+O(1)$ 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,
+and buffered larger patches $U^+,V^+$ with every pair opposite, for
+sufficiently large positive $\delta$ the forward vertex of $k$ lies in
+$U^+$ and its reverse vertex in $V^+$, reversed for negative $\delta$;
+the patches are disjoint in the visual boundary, so a fixed large $k$
+cannot have both orientations. Bounded $\delta$ is in the bounded-shift
+case.
+
+In the positive orientation the shorter arguments are $p'$ and $q^{-1}$,
+with length-increasing substitutions $p'\mapsto kp'$ and
+$q^{-1}\mapsto k^{-1}q^{-1}$. The reverse vertex of the first multiplier is
+in $V^+$ (transverse to the entire buffered positive input patch); of the
+second, in $U^+$ (transverse to the negative input patch). For a fixed
+finite-dimensional representation, put $S(g)=R(g)/\|R(g)\|$. On either
+substitution $r\mapsto ur$, transverse Cartan addition plus highest-weight
+comparison gives $\|S(u)S(r)\|\ge c>0$ and $\|S(r^{-1})S(u^{-1})\|\ge c>0$
+uniformly; the inverse coordinates are separately normalized inverse
+matrices, no inverse of a possibly singular limit. Compactness preserves
+a common lower bound on fixed larger coordinate boxes, so normalized
+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
+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
+in these adjoint products; we do not replace the operator-valued
+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}.
+
+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
+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
+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}
+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
+$\|(\sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k\|_{\ell^2(L^2)}
+\le(\sum_n\sup_k|c_n(k)|)\|A_0\|_\infty\|D_hA_0\|_2$.
+The other side is symmetric. Cauchy--Schwarz in the shift variable then
+proves \eqref{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}
+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}
+
+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}
+Let $\Phi_z$ be a nonzero family of image measurements as in
+Proposition~\ref{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}
+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}
+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
+renormalization of $n(v)$. Inducing the target lattice action with its
+finite invariant trace gives an induced range field intertwining a
+trace-isometric ambient action with the inverse linear transport on $V$.
+
+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 $A^n\ran P$.
+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
+consecutive orbit points is constant and tends to zero). Invariance under
+all positive split matrices forces the range to split off the whole matrix
+factor; the remainder descends to the scalar joint fixed algebra. Thus
+the uncut join is full.
+
+Restore $O$: the finite trace dimension of the patch join is constant
+along target lattice translates. For any finite scalar test measure on
+$Z$, choose a regular contracting direction with attractor in $O$ and
+repeller transverse to almost every point of the measure; all-factor
+mixing supplies lattice elements whose two compact frames approximate
+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}
+
+A normalized image branch of a pure vertex path is a \emph{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}
+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}
+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
+pairing of $(D_AX_tf)(D_BX_af')^*$ with the quotient-path matrix $X_tX_a^*$
+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
+would force two distinct large gaps. The bounded-overlap version of
+Proposition~\ref{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
+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
+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
+possible target type set at source $i$, every $j\in T_i$ and $k\in
+T_{i^*}$ satisfy $k=j^*$. Since the $T_i$ are nonempty, each is a
+singleton, and the resulting map is injective and hence a permutation.
+
+At higher-rank self-opposite source types, extend nonopposition by
+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
+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}
+
+Fix an isolated archimedean source type and its matched target boundary
+$Z$.
+
+\begin{lemma}[Disjoint angular forms]
+\label{lem:transfer-disjointness}
+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}
+The generic conclusion holds throughout the transverse source orbit by
+source covariance.
+\end{lemma}
+
+\begin{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
+assertion follows from the lattice cocycle formula or Haar
+disintegration, scalar phases disappearing in conjugation. The source
+lattice action on pairs of this boundary is ergodic (the open-pair
+stabilizer contains a split subgroup escaping in the active factor and
+all complementary factors, so all-factor mixing applies).
+
+Encode $F_x$ by its minimal covariant Stinespring dilation and the vector
+obtained from the embedded columns of $e$, giving a positive functional
+$\phi_x$ on the separable matrix-amplified full twisted crossed product
+$\mathfrak C$ with norm $d$. The group and matrix operators act on these
+columns by right multiplication; the regular-normal central summand of
+the group algebra bidual supplies canonical right multipliers, so the
+vector functional is supported on the corner of the corner, and the
+positive subfunctionals of $\phi_x$ encode exactly the corresponding
+equivariant subordinate maps (covariant Arveson domination
+\cite[Theorem~1.4.2]{Arveson1969}).
+
+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
+$n(x)=\sup\{\|\theta\|:0\le\theta\le\phi_x,\ \theta\text{ is }z\text{-normal}\}$
+be upper semianalytic, and apply Jankov--von Neumann
+\cite{Kechris1995} 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.
+
+The distance between $\phi_x^z$ and $\phi_y^z$ is invariant under the
+diagonal source action, hence essentially constant by double ergodicity.
+A separable metric-valued measurable map cannot have strictly positive
+constant pair distance. So $\phi_x^z$ is constant a.e.; if nonzero, the
+encoding gives a nonzero source-invariant, target-equivariant positive
+map, whose invariant finite trace yields an invariant flag probability,
+contradicting target contraction.
+
+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
+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
+for angular multiplication, base multipliers, and target covariances.
+Angular multiplication descends; compression at constant input vectors
+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
+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}
+At an isolated source vertex, every candidate is sharp, and all candidate
+choices agree.
+\end{proposition}
+
+\begin{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
+thickness. Let $A$ be the fast cutoff and $L$ its one-ended limiting
+effect; let $B_1$ be the slow cutoff used on slow-link columns, with start
+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}
+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
+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 $p,q^{-1}$.
+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
+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
+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
+form on $b$ at $x$. Cauchy--Schwarz gives the desired bound.
+
+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}.
+
+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}
+
+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
+$V$, let $R_x$ be the orthogonal projector onto
+$\operatorname{span}\{n(v):v\sim x,\ \operatorname{type}(v)=p(i)\}$, 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}
+The following inclusions hold:
+\begin{enumerate}[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
+$\overline{\sum_{z\sim F}L_z}$.
+\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
+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
+vector in the stated closed end range.
+\end{proposition}
+
+\begin{proof}
+We prove (i) by annihilating the common kernel $\mathcal C=\bigcap_{z\sim
+F}\ker\Phi_z(O)$. 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 $\mathcal C$
+$N_0$-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 $z\sim F$.
+
+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 $l$,
+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
+translation fixes $z$ (trivially on Levi chamber refinements incident to
+$F$), so the bound is independent of main time. Proposition~\ref{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
+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
+$b\in\overline{\ran}\Phi_z(V_1)$.
+
+Move the end occurrence also by the independent reverse radical. As
+$z\sim F$ and these moves vary, the resulting labels cover a conull
+transverse open cell; fullness and strong continuity make these end ranges
+total, giving annihilation on the window for every end vector.
+
+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 $e$)
+combined with positivity and density gives (i). Arbitrary Borel output
+subprojections are imposed after buffered tests.
+
+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}
+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
+squared symbol tends in the auxiliary limit to the line-projector effect
+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
+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}
+
+\subsection{Sharp flags and Coxeter compatibility}
+
+\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
+opposite vertices. Success is increasing in the initial Bruhat position.
+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}
+The base failure rule is nonopposition of dual types
+(Proposition~\ref{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
+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.
+By induction on the word, failure propagates.
+
+For the combinatorics, extending a vertex to a chamber permits its
+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
+of success.
+
+If a word is successful at the least target position it is successful at
+every target position. If the inverse-permuted word failed at a least
+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}
+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}
+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
+diagrams, choose a self-opposite leaf $j$ with neighbor $k$ and round
+trips from $j$ to $k$ and back; each permits the block
+$W_{\widehat j}\{1,s_j\}W_{\widehat j}$. With $r$ the minimum number of
+$s_j$ 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}
+
+\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
+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
+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
+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
+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
+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
+incidence family contains the maximal parabolic and is proper, so it is
+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)
+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
+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
+commute; no additional observed vertex type appears (an extra sharp
+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}
+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}
+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
+equations define the gallery manifold as a successive fibre product of
+submersions; they generate its smooth vanishing ideal, including at
+stutters. Fullness gives a nonzero opposite-target restriction at some
+generic, hence opposite, source pair; covariance transports it over the
+opposite orbit. Inserting all intermediate PVM identities along any
+reduced longest source gallery: nonzero opposite endpoints require a
+permuted-type gallery connecting opposite target chambers. Both ends use
+the same full spherical Weyl system, so their longest elements have equal
+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}
+
+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
+Levi boundary parametrizes matched opposite-panel completions at the two
+source ends, denoted $z$. Refining each of the two opposite source faces
+by that same flag of their common Levi gives its start and ordinary-end
+missing vertices, with the ordinary-end types relabeled by opposition.
+Write $P_z,Q_z$ for the sharp missing-vertex PVMs at these vertices. Their
+common source parameter transports by $z\mapsto sz$ under simultaneous
+Levi action. Frame the two sharp endpoint faces separately by compact
+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}
+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}
+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}
+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
+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
+window; integrating intermediate slots by identity projections preserves
+the equation.
+
+Use \eqref{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
+$(1-n(u)n(u)^*)S(h)n(v)$. 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
+compact transverse windows (the highest-line embedding is immersive).
+Output coordinates may be Borel parameters with locally uniform angular
+derivative bounds.
+
+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
+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
+(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
+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.
+
+Mixed target coordinates use finite partitions in clopen slots and
+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
+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}
+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
+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
+incident highest lines. Consequently a homogeneous polynomial relation
+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}
+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
+and disintegrate as well (continuous fibre representations via a countable
+dense convolution algebra, standard direct integral theory
+\cite[Theorem~1.G.6]{BekkaHarpe2019}).
+
+Choose countable dense residue labels, compact windows, graph generators,
+angular Fourier tests, fixed-level clopen tests, and total input vectors
+first. All identities hold on one conull set. Strong source transport and
+the uniform angular bounds extend them to every residue label and to
+smooth or locally constant angular tests. Stabilizer invariance
+disintegrates, first on a countable dense subgroup and then by strong
+continuity, so the family is well defined on the residue quotient. Bounded
+Borel output multipliers are applied through the fibre's spectral calculus
+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
+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
+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}
+
+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
+and target panel boundaries. Let $S$ be the effective root-generated
+source panel group and $T$ the effective target adjoint point group, both
+over finite extensions of a prime local field. The prime-field Lie algebra
+of $T$ is simple, $S$ is simple after removal of its ineffective center,
+and $T$ embeds faithfully in the automorphism group of its prime-field Lie
+algebra, so the centralizer in $T$ of an open subgroup of its
+root-generated group is trivial.
+
+There is an actual source Levi cover $\widetilde S\to S$ acting strongly
+continuously on the endpoint spaces, commuting with the sharp
+endpoint-face observations, moving residue labels by $z\mapsto sz$, and
+its ineffective finite center acts trivially on the residue algebras. Our
+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}
+
+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}
+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
+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
+commutation of the endpoint families and the homogeneous form.
+
+\begin{proposition}[The fixed algebra]
+\label{prop:finite-frame}
+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}
+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
+$d\in D_*^U$,
+\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
+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}
+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}
+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).
+On each finite matrix block the action is a continuous homomorphism into
+a projective unitary group. A noncompact simple real Lie algebra has no
+nonzero homomorphism into a compact Lie algebra (simplicity would make it
+injective, and a positive definite invariant inner product would make
+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}.
+Finiteness and bijectivity of the type permutation give the symmetric
+assertion.
+\end{proof}
+
+\subsection{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
+compact open subgroups $V<S$ and $U<T$, respectively pro-$p_S$ and
+pro-$p_T$, in faithful adjoint matrix coordinates. Choose a minimal
+projection $e\in D_*^U$; shrink $V$ to fix $e$ (an atomic central block
+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
+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}
+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=\{v\in V:(v,1)\in G_n\}.
+ \label{eq:finite-block-commutators}
+\end{equation}
+\end{lemma}
+
+\begin{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
+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}
+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}
+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
+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}
+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
+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 $D_n$.
+Choose small logarithmic $V_1\subset V$ and $U$-compatible $D_n$. Because
+$G_n$ projects onto $V$, every block's stabilizer contains an analytic
+subgroup; a limiting source-only kernel would force one fixed open $V_1$
+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}
+
+The compact metrizable $V\times U$ has a compact space of closed
+subgroups in the Hausdorff topology. Every limit $G_\infty$ of the $G_n$
+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}.
+Hence:
+
+\begin{lemma}[Dimensions and abelian corner]
+\label{prop:finite-dimensions}
+$\dim_{\mathbb Q_p}\mathfrak s=\dim_{\mathbb Q_p}\mathfrak t$. 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}
+
+\begin{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
+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
+source-only part has zero Lie algebra. Both projections of
+$\Lie(G_\infty)$ 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
+$\{1\}$.
+
+Now fix $m$; for large $n$, every block stabilizer lies in $U_m$ and
+fixes the image of $D_m$ in the block pointwise. Its irreducible
+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
+$U$-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
+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
+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}
+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
+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
+$uh$ 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
+$Z(D_*)\cong L^\infty(T/L)$.
+
+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
+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}
+forces $V_1$ to act trivially on $eD_*e$, contradicting
+Lemma~\ref{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.
+
+Remove matrix fibres: the type I decomposition and ergodicity of the
+center give $D_*\cong L^\infty(T;B(\mathcal V))$ 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 $\operatorname{Aut}(B(\mathcal V))$-
+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 $\dim\mathcal
+V=1$.
+\end{proof}
+
+\subsection{Ordinary homogeneous panel maps}
+
+\begin{proposition}[Recovery of the observed panels]
+\label{prop:finite-maps}
+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}
+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}
+All statements hold coordinatewise on many-panel links.
+\end{proposition}
+
+\begin{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
+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
+restriction of scalars to $\mathbb Q_p$; in characteristic zero, all
+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,
+the descent condition identifies with a single parabolic over the
+original field); thus $\tau:B_S\to B_T$ exists, intertwining $S$ with
+$\eta(S)$.
+
+Write $B_S=S/P$; the measurable lift $s\mapsto\eta(s)^{-1}F_0(sP)$ is
+essentially left invariant, hence constant; right-$P$ consistency makes
+it fixed by $\eta(P)$, and a minimal parabolic fixes exactly its point on
+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
+commutative start observation algebra, with a faithful normal
+probability class. The normal representation of $D_*=L^\infty(T)$ 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
+$a(\omega)=t_0^{-1}h(t_0,\omega)$, so $p_z(\omega)=a(\omega)\tau(z)$.
+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
+$a\tau(z)=g\,b\tau(z)$ 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
+projectivities to continuous homogeneous boundary maps is continuous and
+injective, with Borel inverse.
+\end{proof}
+
+\subsection{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 $|\cdot|_p^{[k:\mathbb Q_p]}$
+with the largest relative root multiple $r_{\max}\in\{1,2\}$,
+$d_R=r_{\max}/[k:\mathbb Q_p]$. Bounded centralizer and compact Cartan
+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}
+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}
+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}
+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}
+ 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
+$\Ad(\eta(s_t))=A\Ad(s_t)A^{-1}$, 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
+$a,b$ 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
+$\|\Ad(b^{-1})\|$ are uniformly bounded in $t$ and slower parameters;
+submultiplicativity with \eqref{eq:finite-translated-law} and
+\eqref{eq:finite-adjoint-growth} gives $d_T\ell_T(g)\le d_St+C_{\xi,\varepsilon}$
+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.
+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
+below.
+\section{Length multipliers and ray lifts}
+\label{sec:ray}
+
+This section supplies the length estimate needed for the isolated boundary
+factors: uniform ray lifts (Proposition~\ref{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
+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
+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;
+a rank-one residue in a larger factor does not supply this axis. All other
+measured coordinates, including all finite places, stay in the coefficient
+algebra.
+
+Constants may depend on $X$ and the specified smooth functions but not on
+the number of Fourier indices or coefficient-algebra dimensions. The
+analytic estimates are stated homogeneously in operator and Hilbert norms,
+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}
+
+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 $\langle J_zv,w\rangle=\langle z,[v,w]\rangle$
+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},
+ \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;
+we take the compact-stabilizer average of a smooth representative and let
+$d_{\partial X}$ 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
+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}
+With constants uniform on buffered compact charts and in normalized $C^m$
+norms for every $m\ge0$:
+\begin{enumerate}[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}}
+ {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
+ \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
+input annulus of size $q=e^{-u}$ (including height) maps to an output
+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}
+(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
+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=\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
+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
+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
+of the bracket makes it the compact angular probability, and its Jacobian
+under a horizontal dilation $d\cdot\mathrm{Isom}$ is exactly $d^Q$.
+
+\begin{lemma}[Triangle defect]\label{lem:ray-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}
+
+\begin{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}
+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}
+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*)]
+\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
+the larger boxes with absolute value at most one, and unary symbols with
+compact external parameters work the same way.
+\item If the pairs of boxes supporting a kernel form a graph of degree at
+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}
+(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
+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.
+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
+cocycle $\eta$, and any finite tracial von Neumann algebra $B$, the
+multiplier
+\[
+ D_{f(r/R)}:\ B\,\bar\otimes L_\eta(\Gamma)\to
+ B\,\bar\otimes L_\eta(\Gamma),\qquad
+ \sum_hY_hu_h\longmapsto\sum_hf(r(h)/R)Y_hu_h
+\]
+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 $s=-\log\lambda$,
+$s'=-\log\lambda'$. 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
+$J=\lceil C'_bR\rceil$. 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
+\[
+ H_j=\log\frac{\max(\epsilon_j,\lambda,\lambda')^2}{\lambda\lambda'}.
+\]
+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
+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
+\[
+ \sum_{j=-J}^{J-1}\phi_j(\chi_j-\chi_{j+1})+\phi_J\chi_J
+ =\phi_{-J}\chi_{-J}+\sum_{j=-J+1}^J(\phi_j-\phi_{j-1})\chi_j
+\]
+with $\phi_j=f(H_j/R)$ and $\phi_{-J}=0$ finishes: each ball kernel has
+uniformly bounded Schur norm by smooth separation, and there are
+$O(R)$ pieces of size $O(R^{-1})$ each.
+
+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
+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}
+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}
+The same holds with the first factor localized instead.
+\end{lemma}
+
+\begin{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
+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}
+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}
+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}
+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.
+
+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
+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
+every nonexceptional pair; a map with bounded magnification sends one
+buffered $\delta$-box into a ball of radius $C\delta$, giving the
+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
+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
+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
+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
+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}
+
+Let $\mathcal K_X=L^2(X,dp)$ and let $T_g$ be translation on
+$\mathcal K_X$. Define
+\[
+ \Delta:D\longrightarrow B(\mathcal K_X)\bar\otimes D,\qquad
+ \Delta(bu_h)=T_h\otimes bu_h,
+\]
+a normal unital homomorphism (conjugate by the unitary sending
+$\zeta\otimes\delta_h$ to $T_h^{-1}\zeta\otimes\delta_h$).
+
+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}
+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
+$\mathbb V_{d_1}Y=\sum_h\theta_{o,d_1}(\cdot,h^+)\otimes Y_hu_h$.
+
+For a column $\xi\in\mathcal K_X\otimes L^2(D)$, the left and right
+densities $\rho^L_\xi,\rho^R_\xi\in L^1(D)$ represent the vector
+functionals for $\Delta(D)$ and $D^{\mathrm{op}}$:
+$\langle\xi,\Delta(x)\xi\rangle=\tau(\rho^L_\xi x)$,
+$\langle\xi,\xi x\rangle=\tau(\rho^R_\xi x)=\xi^*\xi$.
+
+\begin{proposition}[Ray lift]\label{prop: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
+$\{r\ge l\}$.
+\begin{enumerate}[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
+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
+$\|\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\|_2\to0$.
+\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}
+All statements hold for matrix versions and measurable fields.
+\end{proposition}
+
+\begin{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
+boxes of radius $e^{-s}$ with normalized $C^m$ bounds independent of
+$d_1,s$. The substitution $n_p=n_x\delta_{\lambda_p}q$,
+$t=\log(\lambda_p/\lambda_x)/d_1$ fixes the normalization; polar volume
+$J(s)\asymp e^{Qs}$ compensates the angular shadow $O(e^{-Qs})$, and
+normalized derivatives are bounded by translating back along rays and
+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
+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}
+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
+$v=(V,W)$, 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 $v$
+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
+$\Delta_R=(1+R^2p(u))^2+R^4|W|^2$,
+$Q_R=(R^2(1+R^2p+R^2J_W)V/\Delta_R,R^4W/\Delta_R)$. 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}
+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
+$H$ for $s\le j$. The $O(1/d_1)$ derivative bounds follow similarly.
+\end{proof}
+
+\begin{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}
+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
+$e^{C_m d_1}$) 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}
+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}
+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}
+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)]
+$K_{d_1}(v,v)=1$; 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 $\|t\|\|x\|_\infty
+(\sum\|Z_C\|_2^2)^{1/2}(\sum\|Y_{D'}\|_2^2)^{1/2}$, and the
+matrix norm times the frequency sum vanishes. This proves
+\eqref{eq:ray-close-pairing}. For short right multiplication,
+$r(h),r(hk)\ge l-a$ and $r(k)\le a$ give
+$d_{\partial X}(h^+,(hk)^+)\le Ce^{-c(l-2a)}$ with fixed $c>0$, so
+(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 $x^*x$
+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
+$x-x'\to0$ 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}
+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}
+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}
+
+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))
+ \quad\hbox{on}\quad
+ L^2\bigl(B(L^2(X_T))\bar\otimes D,\Tr\otimes\tau\bigr)
+\end{equation}
+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}
+
+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
+$S$ 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
+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}
+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}
+for every compact $C\subset G$. Thus compact pre- and post-translations
+do not affect any limiting rescaled length law.
+\end{lemma}
+
+\begin{proof}
+For fixed compact $C$, outside a set of measure at most
+$2\varepsilon$ 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}
+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}
+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
+$G$; 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
+$\sup_{g\in C}\|\pi(g)\xi-\xi\|^2\le
+C_1\int\|\pi(h)\xi-\xi\|^2d\nu(h)$, giving
+$\|\pi(\nu)\|\le1-\delta$ on the complement of invariants. Then
+$(1+\pi(\nu))/2$ 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}
+The same holds with compact pre- and post-translations.
+\end{proposition}
+
+\begin{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}
+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
+$A_{n,f}=D_{f(r_T/R_n)}X^{\Theta_n}_{a_{t_n}}$,
+$B_{n,g}=D_{g(r_T/R_n)}X^{\Theta_n}_{a_{u_n}}(a_{u_n}^{-1}\cdot)$,
+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}
+
+For a positive finite output window strictly above the sum of the
+inputs, \eqref{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}
+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
+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}
+ \|\Delta_T(z)\xi_n\|^2
+ \le\|z\|_\infty\|z\|_2\|\rho^L_{\xi_n}\|_2.
+\end{equation}
+Limit $n\to\infty$ then $k\to\infty$.
+
+For input values $0$ and $+\infty$, use triangle inequalities and
+short approximations with smooth tail cutoffs (Proposition~\ref{prop:ray-radial});
+the lifted shortening argument extends because it uses positive
+lower length bounds and a uniform deficit but no upper bound.
+
+The output-measurement identification uses the exact spectral
+argument: with $a(s,t)=s+t$ on the compact square, product tests
+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
+$E(F^c)JP_{\rm in}(a^{-1}(F))=0$; 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 $\varphi\in C([0,\infty])$.
+Convergence on the whole sequence follows by compactness of
+probability measures on $[0,\infty]$.
+\end{proof}
+
+\section{Linear subsequences and heights for isolated components}
+\label{sec:scale}
+
+At an isolated type, Section~\ref{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
+Jacobians. The construction takes an inverse power of the modulus of a
+horizontal derivative of $x\mapsto P_x$; the main analytic requirement
+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
+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
+ordered products at opposite source labels controls small horizontal
+differences; we then build the closed derivative modulus and prove its
+kernel is zero.
+
+The source ray lies in an entire simple factor. Isolated factors are
+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}.
+
+We use the admissibility and statistical conventions of
+Definition~\ref{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
+and bounded integrated Hilbert norms; a lower effect bound concerns a
+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});
+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};
+scalarity of limiting rescaled one-column effects
+(Lemma~\ref{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}
+
+\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
+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
+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}.
+
+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
+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
+interval). This gives $\gamma_n$.
+
+A fixed matrix corner equivalence preserves escape: approximate its
+entries and adjoints in trace Hilbert norm by uniformly bounded
+Fourier matrices, whose fixed shifts change each Cartan coordinate by
+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
+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}
+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
+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}
+Both hold uniformly under source pre- and post-translations in
+compact sets.
+\end{lemma}
+
+\begin{proof}
+Complementary tightness is the pure-ray conclusion of
+Section~\ref{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
+$e_j$ 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
+about length $t$ in $S$, with all compact rotations and complementary
+coordinates in a fixed compact neighborhood; $|\Omega_t|\asymp
+e^{Q_St}$. 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
+bounds the averaged mass at each measured index by a constant
+independent of index and $t$.
+
+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}
+ 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
+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
+argument finishes.
+\end{proof}
+
+\subsection{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}
+ \|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 $\Theta$
+also has scalar relative commutant.
+\end{lemma}
+
+\begin{proof}
+Smooth radial truncations are uniformly bounded (Proposition~\ref{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}
+gives closure under multiplication; length-invariance under inversion
+gives closure under adjoint; tightness is closed in Hilbert norm on
+each bounded ball.
+
+All coefficient-unitary sequences lie in $Q_0$. A central $z$
+commutes asymptotically, uniformly, with those unitaries (else choose
+a violating unitary per index). The Hilbert closed convex hull of
+the coefficient conjugation orbit gives a conditional expectation
+onto its commutant, so $z$ is represented, up to Hilbert-null
+sequence, on the group factor alone. All fixed group labels lie in
+$Q_0$ since $L_n\ge1$, and conjugation by those labels acts on the
+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}
+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
+ \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 $L_n$.
+Admissibility is preserved.
+\end{lemma}
+
+\begin{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
+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}
+ 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 $b_\gamma$
+
+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}
+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
+$p_0=ue_{Q_0}u^*$. 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}
+
+\subsection{Composition of admissible kernels}
+
+\begin{lemma}[Exact insertion]\label{lem:scale-insertion}
+Let $\delta:L_\alpha(\Gamma)\to B_0\bar\otimes L_\beta(\Gamma)$
+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
+isolated measured factor $T$ there is an admissible
+$\Theta\circledast\delta$ 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}
+
+\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
+$T_g(y)T_h(g^{-1}y)=\nu(g,h)(y)T_{gh}(y)$ 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
+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}
+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}
+which vanishes off the scalar summand by Fourier escape, Howe--Moore,
+and the product invariant-space decomposition of Section~\ref{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
+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}
+uniformly under compact pre- and post-translations.
+\end{lemma}
+
+\begin{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}
+ (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 $m$
+justifies the passage. Compact changes are covered by
+Lemma~\ref{lem:ray-compactshort}.
+\end{proof}
+
+\begin{proposition}[Positive mass at linear speed]\label{prop:scale-linear-subsequence}
+For each isolated source type $i$, matched to $p(i)$, there
+exist $C<\infty$, $\alpha>0$, $T_n\to\infty$ such that for
+every fixed $\xi\in\mathcal H$,
+\begin{equation}
+ \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}.
+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}
+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
+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
+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
+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}
+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}
+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}
+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
+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
+$\ge\alpha_0$ 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
+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}
+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}
+
+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}
+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
+normalized mask controls.
+
+\begin{definition}\label{def:scale-ordered-kernel}
+For $F\in C^\infty(B_T\times B_T)$, 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*)}]
+\item Order at least one gives, for each $\xi\in\mathscr E$,
+$\|\mathcal K_{x,y}(F)\xi\|_{L^2(d\nu_\epsilon;\mathcal H)}
+\le C_{F,\xi}\epsilon^{Q_S/Q_T}$.
+\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}
+
+\begin{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
+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
+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}
+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
+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
+$\xi\in\mathscr E$, 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}
+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 $-X\varphi-(\operatorname{div}X)\varphi$.
+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
+$f$ 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)
+ =\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}
+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
+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}
+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 $u,v\in\mathcal C$
+proves symmetry and, when a formal expression represents zero, shows
+its image is orthogonal to a dense subspace, hence zero. This gives
+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}
+
+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}
+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}
+and $l$ is not identically zero.
+\end{lemma}
+
+\begin{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
+$D:\mathcal C\to\mathcal K^{n_Sn_T}$,
+$Du=(T_{X_\alpha}(\theta_r)u)_{\alpha,r}$, 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 $l_x$
+strongly commuting with $P_x$. Unitary invariance transfers
+distributionally to $D_X(f)$ and then closes, giving affiliation and
+complementary invariance.
+
+For covariance, note that Parseval systems embed $\mathcal V_S,\mathcal V_T$
+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 $\mathcal C$;
+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}
+would make $P_x$ constant, contradiction.
+\end{proof}
+
+\begin{proposition}[Height at an isolated type]\label{prop:scale-isolated-height}
+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}
+The construction passes to normal copies used in amplified models.
+\end{proposition}
+
+\begin{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
+disintegration over $S/J$ and
+\eqref{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
+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.
+
+Let $e_0=\supp L_0$. Target dilation is bounded, positive, invertible,
+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
+$\tau(e_0)1$, so $e_0\in\{0,1\}$. Nonzero column forces $e_0=1$ (this
+uses complementary source invariance essentially).
+
+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}.
+
+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
+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
+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
+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}.
+
+\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.
+
+\subsection{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
+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
+of an omitted source position and its matched target position; compact
+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}
+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
+its missing coordinates, taking their times to infinity successively
+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
+$d_T\ell(g)-d_St$ 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
+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
+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
+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
+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
+only the exact precomposition and angular statements of
+Sections~\ref{sec:links} and~\ref{sec:transfer}; neither the height
+calibration nor the classicality conclusion proved later is used.
+
+The structural classification of Section~\ref{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:
+its complexification would be a component of a proper absolute Dynkin
+subdiagram, whereas no proper subdiagram of an irreducible Dynkin
+diagram has a component of type $F_4$. Equivalently, in the real
+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}
+
+Let $R$ be a noncompact rank-one panel factor with split generator
+$H_R$, normalized so the simple restricted gap is one. For a
+finite-dimensional complex module $V$ extended to the complexification,
+write
+\[
+ V_t=\ker(d\pi(H_R)-t),\qquad
+ h(V)=\sum_{t>0}\dim V_t,\qquad z(V)=\dim V_0.
+\]
+For a nontrivial irreducible $V$, $\dim V/h(V)$ is the inverse
+proportion of positive-weight vectors. We extract finite-dimensional
+source representations from the measured graph relation and compare
+their stage dimensions with those of a target module: a sharp ratio
+and a uniform gap away from the maximizers bound the dimension lost
+when retaining only maximizing source summands, whose largest split
+weights then bound source growth. These are the different roles of the
+constants $c_R$ and $d_R$.
+
+We allow representations of the simply connected covering group, so
+the weight lattices below are full and spin representations are
+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}
+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
+\[
+\begin{array}{ccl}
+ R&c_R&\text{maximizing modules}\\ \hline
+ SO(n,1),\ n\ge2 & k & E\\
+ SU(n,1),\ n>2 & k & E,E^*\\
+ SU(2,1) & 3 & E,E^*,\operatorname{Sym}^2E,\operatorname{Sym}^2E^*\\
+ Sp(n,1),\ n>2 & k & E\\
+ Sp(2,1) & 7/2 & \bigwedge\nolimits^3_0E
+\end{array}
+\]
+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
+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
+$V_{\ge0}=\bigoplus_{t\ge0}V_t$,
+\[
+ V_{>0}^{\perp_q}=V_{\ge0},\qquad
+ V_{\ge0}^{\perp_q}=V_{>0},
+\]
+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
+\[
+ d_R=\begin{cases}2,&R=SU(2,1)\text{ or }Sp(2,1),\\1,&\text{otherwise,}\end{cases}
+\]
+and the constants apply equally to the opposite orientation.
+\end{lemma}
+
+\subsection{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
+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
+they are unique up to a real scalar, and positivity on the nonempty
+zero-weight space fixes the sign. Fix Borel lifts for the coefficient
+calculations; changing a covering lift acts by a modulus-one scalar on
+each summand and so preserves the weight planes, form and norm
+identities, and limiting top split-weight projector. The calculations
+use fixed lifts and do not require a multiplication law for them.