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title[Arithmeticity of twisted finite correspondences]{Arithmeticity of twisted finite correspondences for lattices over local fields}




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.

maketitle

tableofcontents


# 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~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.

## 1. Symmetry, and the grids inside it

A group is a mathematical object that records symmetries. The
rotations of a square form a group with four elements; the ways to
permute three objects form a group with six elements; the translations
of the plane form an infinite group. Whenever something has symmetry,
there is a group behind it, and the group tells you exactly which
transformations leave the object unchanged.

Some of the most important groups are 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 ntimes n real matrices
of determinant one, written SL_n(R), is a continuous group of
transformations of R^n that preserve volume and orientation.
Groups like these, that are also smooth manifolds, are called Lie
groups. They are the basic objects of the geometry of symmetry.

Inside a continuous group, one can sometimes find a lattice: a
discrete grid of symmetries that still "spreads out" through the whole
group, like the integer points
[
 Z^n R^nbbR^n
]
inside the group of all translations, or the famous
[
 SL_n(Z) R)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 place is from number
theory: the real numbers R, the complex numbers C,
and the p-adic numbers 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
R, C, or 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.

## 2. Rigidity: property (T)

A group has Kazhdan's property (T) if it is rigid: it cannot be
"almost" represented by symmetries without actually being represented.
Formally, every unitary representation that has almost-invariant unit
vectors already has an exactly invariant one. Intuitively, a group with
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
K, and it consists of the countably infinite groups that are
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_iin{R,C,finitebbC,finite extension of 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.

## 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 ginGamma there is a unitary operator
lambda(g) shifting a function: (lambda(g)xi)(h)=xi(g^{-1}h). The
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 how much does the shadow remember? If two
groups Gamma and Λ have isomorphic shadows L(Gamma)~=
L(Λ), must the groups be isomorphic? If the shadow retains, along
with any isomorphism, a controlled correspondence with the original
group, the group is called W^*-superrigid.

Recall the notion of a factor: a von Neumann algebra whose center
is trivial. A factor is the noncommutative analogue of an "indecomposable
system," a quantum object that cannot be split into independent
subsystems. The condition that Gamma be ICC is exactly what guarantees
that L(Gamma) is a factor of type II_1, the cleanest
infinite-dimensional kind, on which a natural trace (a kind of
dimension function) is defined.

## 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 projective
representation, and its failure of multiplicativity is encoded in a
cocycle mu:GammatimesGammatoT. The rule is
[
 mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj)
]
with values in the unit circle T. The twisted group von
Neumann algebra L_mu(Gamma) is built from Gamma together with
this cocycle: the generators u_g now satisfy u_gu_h=mu(g,h)u_{gh}.
Scalar cocycles are unavoidable in the geometry of these problems, and a
central theme of this paper is that they must be kept, not discarded:
they appear as the exact ratio between the two endpoint twists of
the final models.

## 5. Finite correspondences: quantized relations

A relation between two sets X and Y is a subset of Xtimes Y. In
the quantum world, a relation between two von Neumann algebras M and
N is an M--N bimodule: a Hilbert space on which M acts
from the left and N from the right, with the two actions commuting.
This is the operator-algebraic analogue of a bridge or a coupling
between the two systems.

There is a notion of size, or module dimension, for such a
bimodule, generalizing the dimension of a group factor over a subgroup
(after Jones' index theory and Pimsner--Popa). A bimodule is
bifinite if it has finite module dimension on both sides. Intuitively,
a finite correspondence is a bridge that is finite and
trace-preserving at both ends, the quantum analogue of a finite
relation.

The central question becomes: where can finite correspondences come
from? One family of examples is always available. Suppose Gamma and
Λ are groups and AleGamma, BleΛ are subgroups of
finite index, with an actual isomorphism delta:Ato 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)=(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~sec:models). These are the arithmetic models.

## 6. The question, and the answer

The question: given a finite correspondence between the twisted group
factors L_mu(Gamma) and L_omega(Λ) for Gammainmathscr
KscrK and arbitrary countably infinite ICC Λ, is it built out of
arithmetic models? The answer proved here is yes.


Main theorem (Arithmetic exhaustion). Let GammainK,
let Λ be any countably infinite ICC group, and let mu,omega
be normalized scalar cocycles. Every bifinite
L_mu(Gamma)--L_omega(Λ) correspondence is unitarily
isomorphic to a closed bimodule summand of a finite direct sum of
arithmetic models E(A,B,delta,sigma), where AleGamma and
BleΛ have finite index, delta:Ato 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[Λ:B] on the left and n[Gamma:A] on the right,
where n=dim V_sigma.


Three things are worth stressing.

First the theorem is an exhaustion result: it says every finite
correspondence is contained in finitely many arithmetic models, up to a
closed summand. It does not claim every correspondence 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 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 Λ to be abstractly commensurable.

Third, the theorem is unconditional within its class. It says nothing
about general property-(T) groups, and indeed property (T) alone
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.

## 7. Why this is hard

The comparison group Λ is only assumed countably infinite
and ICC. Its von Neumann algebra is as wild as any such algebra can be.
In particular we have no geometric action of Λ on any symmetric
space or building: the comparison group is an algebraic cipher, not a
geometric object. Yet we must produce actual finite-index subgroup graphs
of Λ. The only geometry available at the start is the geometry of
Gamma and its ambient groups G_i. The proof must manufacture
geometry inside the correspondence from the source side alone, then
must read Λ through it.

The second hardness is scalar. Even in the classical, untwisted
case, the argument must be careful. With twists in the game, phases
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.

## 8. How the proof works: a tour

The proof is long, but its shape is a single idea executed carefully:
read off the arithmetic from the geometry, working in small enough
pieces that phases stay visible. Here are the main steps.

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 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~sec:setup and~sec:models).

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~sec:links and~sec:transfer). The strong point is
that the construction is ordered: the image depends on the PVM's
order of integration, and the paper proves that the ordering survives the
limit exactly. This is what keeps the phases—which would otherwise be
invisible—under control.

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 upper speed theorem, and it is
proved separately at three kinds of local places:
 Finite place (Section~sec:finite): compact-open fixed
algebras, a uniform p-adic logarithm lemma, and a Hausdorff-limit
argument on block stabilizers give the equality of source and target
dimensions over the prime field and the upper bound ell_T(g)le
d_St+O(1).
 Non-isolated archimedean place (Section~sec:panels):
optimal finite-dimensional modules of the rank-one Levi factor, and
L^2-trace ineualities on compact operators reconstructed from the
graph relation, give the analogous bound.
 Isolated real place (Sections~sec:ray--sec:scale):
ray densities on the symmetric space, a uniform radial-multiplier
(completely bounded) estimate, and a "Kazhdan tightening" that pins down
a linear subsequence give, instead of an upper bound, a 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.


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~sec:scale and~sec:heights). The height lets us
compare source and target Haar densities exactly, and calibrate the
whole correspondence to a genuine Haar-height isometry—an isometry
between the target chamber-variable space and the source chamber-variable
space, weighted by the height. This is the machine that converts "the
looks right" into "the same object" (Section~sec:heights).

Step 5. Prove locality and commutation. Boundary measurements at
different labels must commute. This is not automatic; they are
a priori noncommuting operators. Section~sec:locality proves
commutation for all labels of an isolated factor by a deep
finite-dimensional locality argument: it builds a graph domain
D,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 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.

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_Sto B_T (Sections~sec:panels and~sec:gluing). The
finite-place panels give homogeneous analytic maps with isomorphic
p-adic differential; the real panels give smooth homogeneous maps
after a Taylor-density argument; the gluing sections install reflection
substitutions that propagate monotone commutativity along a longest Weyl
word.

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 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 Ato 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~thm:arithmetic-exhaustion
follows (Section~sec:recovery).



# Introduction

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 inside the correspondence: no boundary or ambient Lie
group is available on the comparison side at the start.

## The class, conventions, and main theorem


Let 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, G_i=H_i(k_i)^+.H_i(k_i)^+.
]
Here k_i is R, C, or a finite extension of
Q_p; the primes may differ. The algebraic group 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.


Property (T) means that every continuous unitary representation with
almost invariant unit vectors admits a nonzero invariant vector. The real
factors in Definition~def:class are either higher rank or locally
Sp(n,1), nge2, 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~sec:setup; see also
[cite].

All scalar cocycles are T-valuedbbT-valued and normalized:
[
 mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj),
 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)=1_bf1_{{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 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(M_n(C) ⊗⊗  Q)p,
  (Tr_n ⊗ tau_Q)(p)=t,
]
with the unnormalized matrix trace Tr_n; the corner itself carries its
normalized trace.

For finite-index subgroups AleGamma, BleΛ, a group
isomorphism delta:Ato B, and a nonzero finite-dimensional projective
unitary representation sigma satisfying

 sigma(a)sigma(b)=(mu(a,b))/(omega(delta(a),delta(b)))sigma(ab),
  sigma(e)=I,

define the arithmetic model

 E(A,B,delta,sigma)
 =L^2(M),barbar ⊗⊗ _{L_{mu|_A}(A)}
       (V_sigma ⊗  L^2(N)),
  M=L_mu(Gamma), N=L_omega(Λ).

The right N-action is standard. The left subgroup action on the second
factor is u_a(xi ⊗ eta)=sigma(a)xi ⊗  v_{delta(a)}eta;
Equation~(eq:intro-multiplier) makes this an action with multiplier
mu|_A.

A right action is represented by rho(h)xi=xi v_h^*, so
rho(h)rho(k)=omega(h,k)barrho(hk). The left action is
pi(g)xi=u_gxi. Hence the joint action
U_{(g,h)}xi=u_gxi v_h^* has multiplier
alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar.

[Arithmetic exhaustion]
Let GammainK, let Λ be any countably infinite ICC
group, and let muin Z^2(Gamma,T),
omegain Z^2(Λ,T) be normalized scalar cocycles.
Every bifinite L_mu(Gamma)--L_omega(Λ) correspondence is
unitarily isomorphic to a closed bimodule summand of a finite direct sum
of the models E(A,B,delta,sigma) of Equation~(eq:intro-model),
where AleGamma and BleΛ have finite index, delta:Ato B
is an actual group isomorphism, and sigma is a finite-dimensional
projective unitary representation satisfying
Equation~(eq:intro-multiplier).

Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. If
n=dim_{C}, the elementary model has dimensions
[
 dim_{L_mu(Gamma)}E(A,B,delta,sigma)=n[Λ:B],
 dim E(A,B,delta,sigma)_{L_omega(Λ)}=n[Gamma:A].
]


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], which uses
Theorem~thm:arithmetic-exhaustion as its geometric input. The
elementary construction and its normality and dimensions are proved here
in Section~sec:models.

Only the matched ratio mu/delta^*omega must admit the
finite-dimensional projective representation of
Equation~(eq:intro-multiplier). Neither endpoint cocycle is
required to have finite order, finite type, or a virtually trivial
restriction, and the image of sigma may be infinite. In particular,
existence of a nonzero correspondence forces Gamma and Λ to be
abstractly commensurable. These conclusions cover nonuniform and
reducible lattices and their ICC commensurable groups.

## 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];
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.ε, Problems~1--2]{Connes1994}.
Popa's survey states a stronger recovery question that controls the given
isomorphism, up to a character and inner conjugacy, together with its
amplification scale cite[Section~3, Equations~(3.2) and~(3.2')]{Popa2007}.

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~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]. Ioana--Popa--Vaes obtained W^*-superrigid groups among
generalized wreath products [cite], and
Chifan--Ioana--Osin--Sun constructed property-(T) examples using
wreath-like products [cite]. 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], the finite-index expectation ineuality of
Pimsner--Popa [cite], and Connes' relative tensor product
cite[Chapter~V, Appendix~B]{Connes1994}. Vaes' classification for
generalized Bernoulli factors already organizes finite-index bimodules by
subgroup isomorphisms and finite-dimensional projective representations
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(Z), nge3 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.

## 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~(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 xin B: P_x(A) is the projection for
its image to lie in A⊆ B. Initially the construction supplies
positive operator-valued observations. Showing that they are PVMs, and
that observations at different labels commute, are separate steps.

To compare two source copies, use K_0 ⊗  K_1, with the left
Gamma-action in each slot and the right Λ-action acting
diagonally. The eventual joint angular law is a PVM Θ on the space
of nonsingular, essentially invertible maps Bto B. The two left actions
transport its source and target labels; diagonal rho(Λ) fixes
it. The same construction on K_0 ⊗  K_1 ⊗  K_2 compares slot
zero separately with slots one and two, and the required conclusion
includes
[
 [Θ_{01}(A),Θ_{02}(C)]=0
  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~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 Λ is assumed. To read the known
geometry from Fourier coefficients, omit one left action and put the
remaining actions, together with diagonal rho, in a finite regular
column. Inducing the omitted action to G/Gamma gives a finite tracial
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~sec:finite).

Real panels inside higher-rank components use optimal finite-dimensional
modules: trace ineualities force a nonzero scalar witness in their
section spaces, and positivity of its algebraic action excludes excess
speed (Section~sec:panels).

Isolated quaternionic and Cayley factors use uniform ray estimates and a
finite-module spectral gap to obtain positive mass in linear windows on
fixed vectors (Sections~sec:ray--sec:scale). Pair packing
then gives horizontal derivatives, whose closed module core produces an
injective positive height with its own-label covariance. The Carnot
boundary geometry belongs to the work of Cowling--Dooley--Kor'anyi--Ricci
and Pansu [cite]; the ray and finite-module
estimates used here are proved in Sections~sec:ray--sec:scale.

Full tuples bring these branches together. Comparing lattice packing with
Haar growth calibrates the speeds around the finite type permutation and
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], but the measured
operator-domain argument is proved here (Section~sec:locality).
Applying it also to the two shared-slot copies gives the isolated-type
commutation needed for the joint law.

For higher-component panels, pairing powers and exact residue graphs
supply the corresponding commutation. Simultaneous reflections and panel
galleries then glue all local types into the nonsingular, essentially
invertible chamber maps described above (Section~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~thm:arithmetic-exhaustion
(Section~sec:recovery).

[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.
 


## Organization

Section~sec:models verifies the elementary arithmetic models.
Sections~sec:setup--sec:transfer construct the finite-module
framework and the ordered sharp boundary links. The finite-place,
isolated real, and higher-component real arguments occupy
Sections~sec:finite--sec:panels; Table~tab:local-branches
records their exact inputs to Section~sec:heights.
Sections~sec:locality--sec:gluing prove locality and
simultaneous commutation. Section~sec:recovery extracts the
arithmetic models and completes Theorem~thm:arithmetic-exhaustion.
A self-contained appendix, Section~sec:appendix, gathers background
for the reader. All geometric and analytic arguments needed for the main
theorem are given in this article.

# Elementary arithmetic correspondences

The models of Theorem~thm:arithmetic-exhaustion admit an explicit
counting-space realization. We give the scalar transport and the
normality proof before starting the geometric converse; neither depends
on arithmetic exhaustion. Projective induction is classical
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,Λ are arbitrary countably infinite ICC
groups, M=L_mu(Gamma), N=L_omega(Λ), and
[
 D=GammatimesΛ,
 alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar.
]
For a subgroup Cle D the graph of an isomorphism
delta:Ato B of finite-index subgroups is
C_delta={(a,delta(a)):ain A}le D.

## Projective induction

Let Cle D be a subgroup and let gamma be a normalized unitary
projective representation on a finite-dimensional space V, with
multiplier alpha|_{Ctimes C}. Write X=D/C for the left cosets,
choose representatives r_xin D with r_C=e, and give each xin X a
copy V_x of V. Set
H(C,gamma)=⊕_{xin X}V_x, with the counting Hilbert
norm. For sin D put

 
 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→ V_{sx}.
 


[Exact projective induction]

Equation~(eq:input-A1) defines a unitary projective representation
of D with multiplier alpha. Changing the coset representatives gives
a unitarily equivalent bundle by unitary identifications of its
individual fibres. If C is the graph of a finite-index subgroup
isomorphism Ato B, the bundle has commuting normal M and right N
actions, and is bifinite.


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~(eq:input-A1) gives

 U^X_s(tx)U^X_t(x)=alpha(s,t)U^X_{st}(x).

Equivalently, all scalar factors can be verified at once in the central
extension with multiplication (z,s)(w,t)=(zwalpha(s,t),st): the
representation of the preimage of C is (z,c)|-> zgamma(c), and
sr_x=r_{sx}c(s,x) gives exactly the ratio in Equation~(eq:input-A1).
This argument retains the multiplier rather than discarding it.

For explicit independence of the choice of representatives, suppose
r'_x=r_xk_x with k_xin 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~(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_xin 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) ⊗  V_{x_0}→
 ⊕_{ginGamma}V_{(g,e)x_0},
  delta_g ⊗ xi⟼ U^X_{(g,e)}xi
]
intertwines the action with lambda_mu ⊗ 1, because
U_{(a,e)}U_{(g,e)}=mu(a,g)U_{(ag,e)}. Hence it extends normally to
L_mu(Gamma). The same argument for the second coordinate gives copies
of lambda_{overlineomega}, which are the standard normal
representations of the right generators xi|->xi v_h^*. The two
actions commute, since the product cocycle has trivial mixed-coordinate
factors.

The first-coordinate orbits are indexed by Λ/B and the
second-coordinate orbits by Gamma/A; both are finite. Each orbit is a
regular module with multiplicity dim_{C}, proving bifiniteness
and the dimension formulas below.


We next identify this construction with the relative tensor product model
of Theorem~thm:arithmetic-exhaustion. That identification also gives
a direct normality check of the subgroup action used in the definition of
the model.

[The graph-bundle unitary]

Let delta:Ato B be a finite-index subgroup isomorphism and let
sigma:AtoU(V) have multiplier
beta=mu|_{Atimes A}/delta^*omega. Set
gamma(a,delta(a))=sigma(a). Then
[
 E(A,B,delta,sigma)=
 L^2(M)mathbin{barbar ⊗⊗ }_{L_{mu|A}(A)}
       (V ⊗  L^2(N))
 congH(C_,gamma).
]
The unitary sends the vector xi in the fibre labelled x to
U_{r_x}jxi, where
jxi=hathat11 ⊗ (xi ⊗ hathat1)1) in the displayed model.


Proof. 
On V ⊗  L^2(N) the subgroup action is
[
 u_a(xi ⊗ eta)=sigma(a)xi ⊗  v_{delta(a)}eta.
]
Its multiplier is beta(a,b)omega(delta(a),delta(b))=mu(a,b). For
normality, choose representatives t for BbackslashΛ. On the
summand spanned by xi ⊗  v_{delta(a)}v_t, the map

 xi ⊗  v_{delta(a)}v_t
 ⟼delta_a ⊗ sigma(a)^*xi
 quadin ell^2(A) ⊗  V

is unitary and conjugates the subgroup action to
lambda_{mu|A} ⊗ 1. 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

 E(A,B,delta,sigma)
 =⊕_{rinGamma/A}u_r ⊗ (V ⊗  L^2(N)).

At its identity fibre the graph action is exactly
[
 U_{(a,delta(a))}jxi
 =u_a ⊗ (xi ⊗  v_{delta(a)}^*)
 =jsigma(a)xi.
]
For g=ra, balancing in Equation~(eq:calculus-coset-model) gives
the more explicit formula
[
 U_{(g,h)}jxi
 =mu(r,a)^{-1}u_r ⊗ 
       (sigma(a)xi ⊗  v_{delta(a)}v_h^*).
]
The last Fourier coordinate is delta(a)h^{-1}, up to its displayed
unit scalar. Two translated identity fibres are therefore orthogonal
unless their labels in D/C_delta agree: equality of the first cosets
and the last Fourier coordinates is exactly that coset equality.
Conversely, these fibres exhaust every summand in
Equation~(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}}jgamma(c(s,x)),
]
which shows that the stated unitary has exactly the transport in
Equation~(eq:input-A1).


[Elementary normality and dimensions]
The correspondence E(A,B,delta,sigma) is normal and bifinite. If
n=dim_{C}, then
[
 dim_M E(A,B,delta,sigma)=n[Λ:B],
 dim E(A,B,delta,sigma)_N=n[Gamma:A].
]

Proof. 
Use Lemma~lem:models-graph together with the orbit decompositions of
Lemma~lem:models-induction. Each free coordinate orbit gives n
regular copies, of module dimension n in the unnormalized matrix-trace
convention. The respective orbit counts are [Λ:B] and [Gamma:A].


Finite direct sums and closed bimodule summands of these models remain
normal and bifinite: the two actions restrict normally to a reducing
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.

# Finite-module realizations and local-field geometry


The first analytic goal is a finite regular realization of each choice of
an omitted left action on the tensor copies of the correspondence. Each
realization includes the other left actions and the diagonal right
action. These realizations supply the finite traces from which the
boundary measurements of Section~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(Λ)
correspondence K as in Theorem~thm:arithmetic-exhaustion. A right
action of L_omega(Λ) is written as the projective representation
[
 rho(h)xi=xi v_h^*,
 rho(h)rho(k)=omega(h,k)barrho(hk).
]
The left representation is pi(g)xi=u_gxi. 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.

## 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

 Adrho on
 L^2(pi(L_mu(Gamma))',tr)
       ominusC1
 quadhas no nonzero finite-dimensional subrepresentation.
 

Here the commutant is a finite factor, equipped with its normalized trace.
The following lemmas justify this choice; their final use, after the
geometric partition has been constructed, is made in
Section~sec:recovery.

[Twisted regular 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 ⊗  v~=lambda_{alphanu} ⊗ 1_V.
]
If L is ICC and V is finite-dimensional, tensoring a finite normal
L_alpha(L)-module with v gives a finite normal
L_{alphanu}(L)-module. Any operators commuting with every v_h are
preserved by the absorption identification.


Proof. 
On ell^2(L) ⊗  V use the unitary
[
 T(delta_g ⊗ xi)=delta_g ⊗  v_g^*xi.
]
Indeed
[
 T(lambda_alpha(h) ⊗  v_h)T^*(delta_g ⊗ xi)
 =alpha(h,g)nu(h,g)delta_{hg} ⊗ xi.
]
A finite normal module over a finite factor embeds in finitely many copies
of the regular module cite[Theorems~10.1.1 and~10.2.1]{Jones2009}. When
L is ICC both twisted group algebras are factors, so applying this
unitary to such an embedding proves the second assertion. The same
formula also proves the last assertion directly.


[Compact commutant reduction]

Let K be a nonzero bifinite
L_mu(Gamma)--L_omega(Λ) correspondence with both groups ICC.
There is a finite-index subgroup Λ_0leΛ such that the
restriction of K to the right subgroup factor is a finite orthogonal
sum of blocks
[
 K_c=V_c ⊗  K'_c,
 pi(g)|_{K_c}=1 ⊗ pi'_c(g),
 rho(h)|_{K_c}=v^c_h ⊗ rho'_c(h)(hinΛ_0).
]
The spaces V_c are finite-dimensional. The v^c_h are normalized
projective matrix representations with multipliers nu_c, and

 rho'_c(h)rho'_c(k)
 =omega(h,k)bar/nu_c(h,k)rho'_c(hk).
 

Each K'_c is a finite normal module on both sides and satisfies
Equation~(eq:setup-weak-mixing) for its reduced representations.
No assertion is made that any v^c has finite image.


Proof. 
Put M=L_mu(Gamma), N=L_omega(Λ), and P=pi(M)'. Finite
M-module dimension makes P a finite factor, and N^{op}subset
P has finite index. Indeed, regard K as the equivalence module for the
commuting factors pi(M) and P; it has finite nonzero P-dimension,
so finitely many copies of it contain L^2(P) as a P-module. Their
restrictions to N^{op} are finite by right finiteness of K.
Hence L^2(P) has finite N^{op}-dimension, which is the index
assertion [cite].

Let D_csubset P be the bounded elements whose Adrho-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'|_2le|x|_∞|y-y'|_2+|x-x'|_2|y'|_∞,
]
and their bounded balls are strongly closed because bounded strong
convergence in a finite von Neumann algebra implies L^2 convergence.
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)ominusC1 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^{op}}:Pto N^{op} is equivariant and
L^2-contractive, hence E_{N^{op}}(x)=tr(x)1
for xin D_c. By the finite-index ineuality there is ε>0
with E_{N^{op}}(x)geε x for xge0, so a nonzero
projection qin D_c has tr(q)geε. A finite
algebra with this lower bound has only finitely many orthogonal nonzero
projections, and is finite-dimensional.

Restrict to the finite-index subgroup Λ_0 fixing all central
blocks of D_c. A central block z reduces both endpoint actions; write
zK=V_c ⊗  K'_c and zD_c=B(V_c) ⊗ 1. The left action
commutes with D_c, hence is 1 ⊗ pi'_c. Choose normalized matrix
implementers v^c_h for Adrho(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 B(V_c), giving
the asserted tensor form and Equation~(eq:setup-reduced-right-multiplier).

Normality and finiteness of pi'_c follow by taking a matrix corner of
the finite normal left module. For rho'_c, first embed the original
restricted rho in a finite multiple of the
overlineomega|_{Λ_0}-regular representation, possible because
finite-index restriction preserves finite module dimension. Tensor with
v^cbar; Lemma~lem:setup-twisted-absorption gives a finite
normal module with multiplier nu_cbar,overlineomega. The
vector corresponding to the identity matrix is fixed by
v^c_hbar ⊗  v^c_h, so tensoring that line with K'_c
exhibits rho'_c as a summand of a normal finite representation with the
multiplier of Equation~(eq:setup-reduced-right-multiplier).

Finally, a compact vector in the reduced commutant embeds as
1_{V_c} ⊗ xi on zK and as zero on the other blocks, and is
compact for Λ_0 hence for Λ; it therefore lies in
L^2(D_c) and its intersection with the reduced commutant on this block
is scalar. This proves Equation~(eq:setup-weak-mixing).


[Finite-index restriction and return]

Let Gamma_0leGamma and Λ_0leΛ 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,

 J_Lxi&=[Gamma:Gamma_0]^{-1/2}
       sum_{rGamma_0inGamma/Gamma_0}
              u_r ⊗ pi(u_r)^*xi,
       
 J_Rxi&=[Λ:Λ_0]^{-1/2}
       sum_{Λ_0sinΛ_0backslashΛ}
              (xi v_s^*) ⊗  v_s.
       

These are bimodular isometries, with all scalar cocycles included.
Restriction and induction preserve bifiniteness. An ICC group stays ICC
on finite-index restriction.


Proof. 
The u_r in Equation~(eq:setup-left-induction) are an orthonormal
right basis for L^2(M) over the subgroup factor, since
E_{L_{mu|Gamma_0}(Gamma_0)}(u_r^*u_{r'})=delta_{r,r'}1; the
analogous left-basis identity for the v_s holds. The relative
tensor-product inner products give the isometries.

For left equivariance, if gr=r'a with ainGamma_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
ainΛ_0,
[
 v_sv_h=(omega(s,h))/(omega(a,t))v_av_t,
 v_hv_t^*=(omega(s,h))/(omega(a,t))v_s^*v_a.
]
The other endpoint action commutes with every operation, so the two maps
compose to a bimodular isometry into two-sided induction, whose range is a
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.


For a source in K, choose the actual isomorphism of finite-index
subgroups of Definition~def:class; its image is a finite-index
subgroup of a reference lattice, hence itself a lattice in the same
ambient group. Pull the scalar cocycle through that isomorphism, without
modifying it by a finite kernel. The restrictions retain property (T)
and ICC, as well as bifiniteness; property (T) permanence is used in its
finite-covolume and finite-index forms cite[Theorem~1.7.1]{BHV2008}; see
also cite[Theorem~3]{DelarocheKirillov1968}. Lemmas~lem:setup-compact-reduction
and~lem:setup-finite-index-induction now reduce the geometric work
to the actual-lattice, weakly mixing setting. Restoring V_c after a
partition of K'_c tensors every atom with V_c and restores the
original exact right multiplier, changing neither the atom permutations
nor their freeness; this is used only after the partition is proved.

## Tensor copies and complementary regular realizations

From now until Section~sec:recovery, let Gamma<G be an actual
ICC lattice satisfying Equation~(eq:setup-weak-mixing). Comparing two
source copies uses two copies of K; comparing two comparisons sharing
their first copy uses three. The right comparison group acts diagonally.
We omit one left source action from the regular model so that its induced
transports remain available on G/Gamma.

Take m=1 or 2 and put
[
 H=K^{ ⊗ (m+1)},
 R=Gamma_1times...timesGamma_mtimesΛ.
]
The slots of H are numbered 0,...,m; pi_j denotes the left
representation in slot j. Let W be the representation of R given by
the pi_j, jge1, and diagonal rho, with split multiplier

 alpha_R((g_1,...,g_m,h),(g'_1,...,g'_m,h'))
 =prod_{j=1}^mmu(g_j,g'_j)omega(h,h')bar^{,m+1}.
 


[Complementary regularity]

The representation W is normal and finite over the twisted factor for
R. The same holds with any one slot omitted instead of slot zero. The
joint commutant of all pi_j(Gamma), 0le jle m, and diagonal
rho(Λ) is scalar.


Proof. 
Embed the omitted slot in a finite regular multiple for its
overlineomega-projective right representation. At that regular index
absorb the diagonal right actions of the remaining slots by
Lemma~lem:setup-twisted-absorption. The absorbing unitaries are
formed from the other slots' rho(h) and commute with all their
pi_j(g). After absorption the Λ-action is regular with
multiplier overlineomega^{,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~(eq:setup-tensor-multiplier), symmetrically in the omitted
slot.

Write P=pi(M)'. The final joint commutant is
(P^{barbar ⊗⊗ (m+1)})^{Adrho^{ ⊗ (m+1)}}, which
Equation~(eq:setup-weak-mixing) makes scalar. For completeness: an
invariant tensor with a factor in L^2(P)ominusC1 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 C1 ⊕ (L^2(P)ominusC1) applies this to every
nonconstant summand.


Choose a column realization
[
 H=eL^2(B)^k, ein M_k(B),
 d=(Tr_k ⊗ tau_B)(e)>0.
]
Here B is the opposite of the alpha_R-twisted factor of R; using
the adjoints of the opposite canonical generators identifies it with the
alpha_Rbar-twisted group factor, whose generators we write as
w_r. Thus W_r is right multiplication by w_r^*, which has
multiplier alpha_R. Set

 Y=G/Gamma,
 H=L^2(Y;H),H=L^2(Y;H),
 N=L^∞(Y)N=L^∞(Y)overlineoverline ⊗⊗  eM_k(B)e.
 

Give Y its invariant probability measure. The trace on N is
tr_{N}
=d^{-1}int_Y(Tr_k ⊗ tau_B)(A(y)),dy, whereas the Hilbert norm of
a column uses the unnormalized entrywise trace sum. All finite-matrix
estimates permit rectangular products. Whenever a family of coefficient
algebras is varied, coefficient traces are normalized and matrix sizes
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:Yto G and write
[
 c(a,y)=d_0(ay)^{-1}ad_0(y),
 X_a(y)=pi_0(c(a,y)),
 (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.

[Induced finite algebras]

The transports U_a are strongly continuous in a, and sigma
preserves the trace on N. For 1le jle m, let R'_j be R
with Gamma_j omitted, put beta_j=Ad W_{Gamma_j}, and let
A_j be the commutant of the base multipliers and
W|_{R'_j}. Then
[
 (A_j)^{beta_j}=N,
 (A_j)^ has a faithful finite beta_j-invariant
 trace,
 N^G= C1.C1.
]


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 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 (A_j)^ is the commutant of
the complementary regular representation obtained by omitting slot j;
Lemma~lem:setup-normal-tensors makes it finite, with normalized
beta_j-invariant trace since the remaining slot acts by unitaries in
that commutant. Including all slots and using the last assertion of that
lemma gives N^G= C1.C1.


## 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].

[Factors and panel groups]

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), nge2, 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.


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.

[Product mixing and partial escape]

For the action on L^2(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(N)^{G_0},
equivalently the conditional expectation onto 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.


Proof. 
Use Howe--Moore for the noncompact isotropic almost simple root-generated
groups and their root-generated lifts from
Convention~conv:setup-panels [cite]; an independent
proof in this convention is cite[Theorem~1.1 and Definition~4.23]{Ciobotaru2015}.
We justify the type-I input for these actual groups. For an isotropic
simple factor H over k, choose a nonzero parabolic unipotent
radical U.U. The k-span of the H(k)-conjugatesbfH(k)-conjugates of
Lie(U)U) is a nonzero invariant Lie ideal, hence all
of Lie(H).H). Choose X_1,...,X_d from these
conjugate subspaces; the map
(t_1,...,t_d)|->prod_jexp(t_jX_j) has invertible differential
at zero and lands in H(k)^+.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], 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
[
 H={Shat}S^{ ⊕ }(H_xi ⊗  M_xi),deta(xi),
 
 pi(s)=int_{Shat}S^{ ⊕ }(xi(s) ⊗ 1),deta(xi).
]
In this central decomposition every unitary commuting with S acts
fibrewise as 1 ⊗  V_xi
cite[Propositions~5.4.7 and~8.6.4, Theorem~8.6.6]{Dixmier1977}; see also
cite[Theorem~7.32]{Folland2015}. For finite tensor sums in a nontrivial
fibre, uniformly over the V_xi,
[
 |bigl<(xi(s) ⊗  V_xi)sum_j e_j ⊗  u_j,
       sum_k f_k ⊗  v_kbigr>|
 lesum_{j,k}|<xi(s)e_j,f_k>|,|u_j|,|v_k|.
]
Howe--Moore makes the right side tend to zero along escape; approximation
by finite tensor sums is uniform because all operators have norm one.
Dominated convergence then proves the uniform decay off the
S-invariants.

Let P_i project onto the invariants of the ith 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~prop:setup-induced-algebras. For partial escape,
expand only over the factors of G_0; the limit is the projection onto
L^2(N)^{G_0}, i.e. the trace-preserving conditional
expectation.


## 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 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 Q_p:Q_p: after restriction
of scalars its absolutely simple factors form one Galois orbit, so a
Q_p-idealbbQ_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 indivisible relative roots of
the factors. A face of type IsubsetPi 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}g_alpha at a finite place,
 2rho=sum_{alpha>0}m_alphaalpha,
]
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.

[Exact finite-place translation coordinates]

At a finite place use the module absolute value with |varpi|=q^{-1}.
The Cartan vector is the actual dominant translation vector of a Cartan
representative in the split centralizer: its pairing with every
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.


[Haar size of Cartan shells]

For a bounded-thickness shell about a dominant vector L, its Haar
measure is at most Cexp(2rho(L)). For one fixed sufficiently large
thickness there is a lower bound cexp(2rho(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' 2rho.


Proof. 
At a real place use the Cartan density, a product of the appropriate
powers of sinhalpha(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:Kcap aKa^{-1}] times the fixed compact measure, comparable to
exp(2rho(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_alphaalpha(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.


## 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.

[Face blocks and residual projectivities]

Let a sequence have diverging simple gaps of types D and keep its
complementary gaps in a bounded window. Its compactified matrix data
determine an image D-flag, a reverse D^*-flag, and an invertible
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.


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.


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~conv:setup-exact-lengths.

[Smooth and clopen 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.


[Interpolation in an affine building]

Fix an endpoint x in one of the affine buildings above. In an apartment
containing x,y, multiply the dominant simple coordinates of y-x by
independent numbers t_iin[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.


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_ialpha_i, where
n_ige0,
[
 0lealpha(T_t(y-x))=sum_in_it_iL_ilesum_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.


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.

[Operators and Fourier coordinates]

An invariance under U is a conjugation invariance of decomposable
operators, so base scalar phases do not change it. An ``other group'' in a
commutant includes Λ unless Λ is explicitly the measured
group. A target Fourier arrow records w_{(h,·)} and the Cartan or
flag data of hinGamma in a measured source-lattice copy; it does not
record geometric data of the Λ index. A bounded column is an
operator-bounded tracial matrix column, essentially uniformly bounded when
it depends on y. The closed range of a positive effect means its support
space when a range inclusion is asserted.


# Boundary obstructions, links, and strict gap growth


An escaping source translation acts on a finite regular array. We record the
limiting flags of its measured Fourier index and compare the measurements
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~sec:setup: H=eL^2(B)^k,
H=L^2(Y;H),H=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
field write
[
 T(C)= ⊗ tau_B)(C(y)),dy,
 tau_{N}{-1}T(C).
]
Thus tau_{N} is a probability trace while the Hilbert norm of a
column is |f|_2^2=T(ff^*).sfT(ff^*). Matrix sizes in this section are
fixed and finite. Operator-bounded columns form a dense subspace. The
constant corner frame is
[
 b_r=eε_r, 1le rle k,
 sum_{r=1}^k b_rb_r^*=e.
]

## Compact matrix data and the surviving flags

Write Pi for the indivisible relative simple roots. A face of type
IsubsetPi 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.

[Matrix compactification and residual projectivities]

There is a finite family of projectively normalized matrix coordinates,
including coordinates of the inverse, with the following properties.
[label=(roman*)]
 The set DsubsetPi 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.
 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.
 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.
 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.

The matrix coordinates are real smooth coordinates at archimedean places and
projective local-field coordinates at finite places.


Proof. 
Use the highest-line modules and normalized matrices of
Section~sec:setup. At archimedean places normalize by Hilbert--Schmidt;
at finite places use projective ratio charts with a maximal coordinate.
Lemma~lem:setup-face-blocks identifies the surviving Levi blocks,
proves their invertibility on a bounded complementary-gap window, and
describes the intrinsic transverse projection followed by residual
transport. In each fundamental highest-line coordinate, rank one detects
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~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.


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.

## 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 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~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 VH
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

 J(overlineoverline b,f)b,f)
  =sqrt d,(b(ay)^*X_a(y)f(y))^{lim}.
 

Its output has its own diagonal spectral measure E. The link identities
below determine the norm of this map and the Hilbert inputs to which it
extends. Iterated ultraproducts realize the specified successive limits;
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.

[Fourier--Gram transport]

On the limiting arrays, a fixed left or right Fourier shift transports
continuous compact matrix tests by its limiting geometric map, hence
transports the associated Borel spectral projections. For a stratum and
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.


Proof. 
For a fixed shift, normalized matrix multiplication and the inverse formula
have denominators bounded away from zero by Lemma~lem:links-matrix-data.
Their action on the compact label space is continuous; a diagonal continuous
test therefore intertwines the shift in the ultralimit, and equality on
continuous functions extends to Borel spectral measures. Base multiplication
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_lto 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|_2le(|T_l|_∞+|T|_∞)|T_l-T|_2. All errors
are estimated before applying the cutoff to the columns, so the argument
applies to partial unitaries T^*T=e and assigns no operator bound to Df.

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.


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.

## 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.

[Link identities]

The one-column measurements and the forms (eq:input-1) have the
following properties.
[label=(roman*)]
 The image measurement in one measured copy lies in A_j,
is beta_j-equivariant, and is invariant under the source face parabolic.
Its gap-stratum masses belong to N and are scalar. Conjugating
the source split frame transports the measurement by sigma.
 If the path escapes in every simple component, the link is an isometry
Hbar ⊗ H K.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 ⊗ Phi(A); the reverse
rule is symmetric.
 If the path is active precisely in a subproduct G_0, put
N_0=N^{G_0} and let E_{N_0}:N N_0 be the
trace-preserving conditional expectation. On bounded columns the link has

 |J(overlineoverline b,f)b,f)|^2
  =d^2tau_{N}{N_0}(ff^*)E_{N_0}(bb^*)).
 

Its polarized extension on finite sums,
[
 Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
 =d^2sum_{i,l}tau_{N}
   (E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)),
 
]
is a positive semidefinite form. The same Gram covariances hold on this form
space; they do not assert a tensor-product isometry. Hilbert extensions are
in Lemma~lem:links-relative-boundedness.
 For any Borel event A determined by a gap stratum and its image flag,
the finite-frame identity is

 < f,Phi(A)f>
  =frac1dsum_{r=1}^k|E(A)J(b_rbar,f)|^2.
 

It holds on mixed and nonmixed paths, and more generally for events in
additional limiting coordinates unchanged by the ignored left Fourier
shifts. When a proof starts from windows that also constrain the reverse
flag or residual projectivity, those windows must first be exhausted to such
an invariant event.
 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.

All statements use the specified order of successive limits; a contracted
translate must tend to the identity in that order.


Proof. 
(i) Base multipliers and companion actions commute with the relevant
diagonal tests and intertwine the input column map, so the compressed image
effects lie in A_j. A measured target shift changes the inverse
Fourier label by the corresponding left geometric action, giving
beta_j-equivariance via Lemma~lem:links-gram-transport.

Let n be in the radical contracted by the path. Sliding U_n through
U_a puts U_{ana^{-1}} at the far end up to a common base phase; this
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~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 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(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 N^G= C1.C1.
This argument covers a pure vertex and a mixed path alike.

(ii) For bounded b,f, cyclicity and the definition of the induced
action give
[
 |J(overlineoverline b,f)b,f)|^2
 =dlim_aT(ff^*sfT(ff^*sigma_{a^{-1}}(bb^*)).
]
On a mixed path, trace mixing makes the limit
d^2tau_{N}*)tau_{N}*)=|f|_2^2|b|_2^2.
Polarization gives
[
 Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
 =d^2sum_{i,l}tau_{N}*)tau_{N}*),
]
which is the norm on Hbar ⊗ H; the factor
sqrt d in (eq:input-1) is essential for this normalization.

Multiplication of an input by a start (resp. end) base function is
represented on arrays by its value at y (resp. ay); diagonal tests
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 Nbar, and independent reverse-radical invariance makes
it scalar by the Howe--Moore argument above; the compressed effect is
1 ⊗  C, and the frame computation below identifies C=Phi(A).

(iii) For an escape in G_0, the weak limit of the trace-preserving
action on L^2(N) is the projection onto L^2(N_0), which yields
(eq:links-relative-form) and (eq:links-polarized-relative) as
limits of squared norms. The mass calculation above is separate; it does
not replace N_0 by scalars.

(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~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 (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.

(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~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 Λ-index transfers its multiplier to
the regular Λ multiplier via projective absorption, and this
conjugation commutes with the extra Gamma-action, which can be embedded in
its own finite regular multiple. Both operations commute with scalar Fourier
cutoffs on the original Gamma_j-coordinate. In original input coordinates
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.


## 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.

[Relative boundedness and Hilbert inputs]

For a path supported in G_0, call a column f relatively bounded if the
positive affiliated operator E_{N_0}(ff^*) is bounded. Then:
[label=(roman*)]
 for fixed relatively bounded f, the link extends continuously to
every Hilbert end column b, and conversely;
 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;
 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;
 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
(eq:input-1).



Proof. 
Conditional expectation extends to positive L^1 elements. Put
A=E_{N_0}(ff^*); if A is bounded, (eq:links-relative-form) gives

 |J(overlineoverline b,f)b,f)|^2le d|A|_∞|b|_2^2.
 

This proves the continuous extension from bounded b; the link limit is
always taken before this extension.

Let cin 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 cin N_0,
< Qf,cQf>le|Q|^2< f,cf>; the
characterization of the conditional expectation by these tests implies
E_{N_0}((Qf)(Qf)^*)le|Q|^2E_{N_0}(ff^*). This holds on the generated
algebra, its support projections, and after finite matrix amplification.

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=dtau_{N}-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 (eq:links-relative-form)
bounds the error by d|b_l|_∞^2|f_n-f|_2^2 on the tail while
E_{N_0}((f_n-f)(f_n-f)^*)le4E_{N_0}(ff^*) controls it uniformly. Take
nto∞ then lto∞. 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~lem:links-gram-transport. This proves the same inclusion for
bounded pairs with no mixed tensor decomposition.


## Additional invariant variables

[Additional invariant limiting variables]

Adjoin to the compact spectral data any bounded limiting tests
asymptotically unchanged by fixed left and right Fourier shifts. The
limiting effects retain base decomposability and the measured equivariance
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.


Proof. 
A fixed Fourier shift gives precisely the limiting intertwining identity
used in Lemma~lem:links-gram-transport; the finite-Fourier proof is
unchanged. Base decomposability and commutation with companions follow
before limits. An invariant one-column test gives an element of
N; face-parabolic invariance, followed by the active-factor
radical argument and the inactive-factor centralizer argument of
Proposition~prop:links-link, puts it in N^G= C1.C1.

For mixed links, independent target invariance puts the respective slices
in Nbar and 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.


## 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~sec:transfer.

[Cancelling source multipliers]

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.


Proof. 
Write the source multiplier as eta. The regular representation with
multiplier overlineeta, 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 N, not the enlarged commutant.


## Escape from bounded target coordinates

[No all-bounded stratum]

On every escaping source face path, the all-bounded gap stratum has zero
mass in a single measured lattice copy. This holds for the one-column
measurement and for bounded pairs in the link.


Proof. 
We record the finite-module Bessel estimate used here. For a bounded column
bin eL^2(B)^k and a Hilbert column xi, Parseval in regular coordinates
gives
sum_{hin R}|< bw_h^*,xi>|^2=|b^*xi|_2^2le|b|_∞^2|xi|_2^2,
projective phases being harmless. The same holds for subsets, in particular
R'_j-translates, and in finite matrix versions summing over fixed row and
column indices.

For bounded end and start columns, the squared norm of the coefficient at a
fixed measured index ginGamma_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_0times 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
2delta, 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 (eq:links-frame) for the
one-column assertion.


## An amenable-boundary obstruction

[Full-flag obstruction]

Let S be the root-generated group of a noncompact isotropic almost simple
component of a semisimple source Levi subgroup, using its root-generated
lift from Convention~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 A_j which is target-equivariant and fixed by the
source action sigma|_S.


Proof. 
Suppose Phi:C(Z)toA_j is such a map. Its mass a=Phi(1)
belongs to N^S:N^S: target equivariance makes it commute with the
measured target action, and membership in A_j the rest. Choose
ε>0 with p=1_{[ε,∞)}(a)ne0. On pH,
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 FcongF_2 S, obtained from two
sufficiently strong rank-one contractions with disjoint attracting and
repelling neighborhoods. Let D be the concrete C^*-algebra on
pH generated by cut base multipliers, the companion actions
W|_{R'_j}, and U_f, fin F. These commute with the measured target
action and every Psi value.

Use the minimal Stinespring construction with commutant lifting
[cite]. The minimal dilation has algebraic
vectors u ⊗ xi with form
<sum_iu_i ⊗ xi_i,sum_lv_l ⊗ eta_l>
=sum_{i,l}<xi_i,Psi(u_ibarv_l)eta_l>, and
Dhat(uD(u ⊗ xi)=u ⊗  Dxi satisfies
|DDhat|le|D| in the concrete input norm. The covariant lift
T_g(u ⊗ xi)=(g· u) ⊗  W_gxi 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:Ztoell^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_ieta=sum_{tinGamma_j}delta_t ⊗ pi_Z(xi_i(·)_t)eta;
these are isometries intertwining the companion algebra exactly, with
|Q_iT_g-(lambda_g ⊗  T_g)Q_i|to0 for each fixed g. Projective
absorption A(delta_t ⊗ eta)=delta_t ⊗  T_t^*eta conjugates
lambda_g ⊗  T_g to lambda_zeta(g) ⊗ 1, leaving the companion
unchanged. Taking limits on the original input gives, at every finite matrix
level,

 Bigl|sum_g W_gD_gBigr|
 leBigl|sum_glambda_zeta(g) ⊗  D_gBigr|_{
   C^*_{r,zeta}(Gamma_j) ⊗ _{min}D}
 

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_0times R'_j with split multiplier. Inducing the
Gamma_0-coordinate and using (y,h)|-> 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 (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_fnu(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)· 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_fnu(f)R_{T_{f,n}}U_f^{(k)} has S_nEto Ene0. By
(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=T(p)<∞.sfT(p)<∞.


## Strict growth under a slower Levi translation

[A slower Levi translation removes every compact window]

Let an inner source face path have a positive target gap stratum
Ω_D. After the inner limit, append a slower commuting split
translation whose projection escapes in a noncompact simple component S of
the remaining source Levi; additional split parts commuting with S are
allowed. Every fixed compact window in Ω_D has mass tending to zero
in the outer limit. The entire positive contribution of Ω_D then
passes to strata Ω_{D'} with D'⊋ D.


Proof. 
The scalar mass m_D>0 is given by Proposition~prop:links-link;
divide the stratum measurement by it, giving a unital full-data
measurement Psi. If D=Pi, its image measurement is a nonzero
full-flag map fixed by the remaining source component S, contradicting
Lemma~lem:links-obstruction. Thus DnePi.

The marginal inclusion. Let Psi(F)=V_Ω^*M_FV_Ω be a
minimal full-data dilation with image-flag map x:Ω_Dto X_D and
minimal image dilation V:H K_X.K_X. The generator map
M_uVf|-> M_{ucirc x}V_Ω f preserves inner products and extends
to an isometric inclusion i:K_X K_ΩcalK_Ω with
iV=V_Ω, intertwining the image spectral measure, base multiplication,
companions, and target lifts. Disintegrate
K_X= ⊕  K_x,dθ(x) with isometric fibre
inclusions i_x.

Positive density forms. Let 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
omegainΩ_D transports that probability to a probability
k_omega(l),deta_x(l) on F_x, positive with total mass one.
On a compact data window CsubsetΩ_D, this density has bounds
0<c_Cle k_omega(l)le C_C<∞ uniformly in omegain C and
linF_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_{√(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 VH, so vanishes by
minimality), then Tonelli and int k,deta=1 give
[
 int_{X_D}int_{F_x}{x,l}(Vf_r(x)),deta_x(l)dθ(x)
  =|f_r|^2.
]
For the associated positive D_x(l), measurability follows from the
bounded approximants, and int_{F_x}{x,l}(v),deta_x(l)=|v|^2
for vin K_x. Consequently

 (Tf)(x,l)=D_x(l)^{1/2}(Vf)(x)
 

is a bounded isometry from H to
R= ⊕ _{X_D}L^2(F_x,eta_x;K_x)dθ(x).

Equivariance. For a target gamma, set
r_{gamma,x}(l')=d(gamma_*eta_x)/deta_{gamma x}(l'). Then
k_{gammaomega}(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 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 R by
its action on K_x with induced base reindexing and identity on l. These
lifts use only the image marginal.

The finite-trace intertwiner space. Let E_0 be the bounded
intertwiners H RcalR for the base and target actions;
B^*CinN for B,CinE_0, and
< B,C>_{E}{N}*C) completes to
E with

 |B|_{E}{r=1}^k|Bb_r|^2,
 
 |Bf|^2le d|f|_∞^2|B|_{E},
 

the second by testing B^*B against the positive matrix ff^*. Source
conjugation U_sB= U_sBU_s^* is isometric because
sigma_s preserves tau_{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|_{op}le|B|_{op} converge in E; the
bounded-column estimate gives strong convergence on bounded columns, and
B_nto B_∞ with B_∞ intertwining base and target and S-fixed.
Then u|-> B_∞^*M_{u(l)}B_∞ is a nonzero positive full-flag
map into A_j, target-equivariant and S-invariant, contradicting
Lemma~lem:links-obstruction.

Howe--Moore now gives weak decay of translates of T along escape in S,
also when multiplied by commuting translations. Disintegrating into
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.

The positive compact-window comparison. For a compact window
CsubsetΩ_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

 D_x(l)^{1/2}ge√(c_C),L_C(x).
 

Write the extended path a_tb_s, inner t-limit first. The exact cocycle
formula and change of base z=b_sy show the inner full-data law is
precomposed by U_{b_s}: Psi_s(F)=U_{b_s}^*Psi(F)U_{b_s}, with
unchanged image marginal. Set T^{(s)}=widetilde U_{b_s}^*TU_{b_s}. The
marginal lift fixes (x,l), so its fibre expression is
(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X applied to Vf; ineuality
(eq:links-positive-window) persists. With J_0f(x,l)=Vf(x),
B|->< Bf,J_0f> is continuous on E by
(eq:links-intertwiner-frame). Weak decay and the positive ineuality
give
0le√(c_C)< f,Psi_s(C)f>le< T^{(s)}f,J_0f>to0;
density of bounded columns and 0lePsi_s(C)le1 extend the vanishing to
every input. This proves disappearance of the actual translated window
mass.

The entire stratum contribution. Each Psi_s has mass one on
Ω_D. Take its outer limit on Ω_Dbar. Every continuous
test compactly supported in Ω_D has zero limiting effect by the
window estimate, so a countable exhaustion shows the outer law gives no
mass to Ω_D. Lemma~lem:links-matrix-data places the remaining
support in Ω_Dbar∖Ω_D
⊆∪_{D'⊋ D}Ω_{D'}. Restoring m_D proves the
assertion for the entire positive contribution; further subpieces are
dominated by the positive total measurements.


## Gap counts and vertex quotients

[Gap count and vertex quotients]

In one measured copy, every positive stratum of a source path toward a face
of type I has at most |I| infinite gaps. A vertex path has exactly one
infinite gap. In a successive single-gap lex chain starting at a vertex,
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.


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~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~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∖{j} is 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.

# Transverse Fourier products and geometric transfer


We continue with the induced finite-module realization of
Section~sec:links. All target indices here are indices of a measured
copy of Gamma; the comparison group Λ remains in the coefficient
algebra, and no flag space carries an action of Λ. Our objective is
a sharp flag measurement at each source flag, with an exact equation for
the residual projectivity on a missing panel.

## 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

 (D_g B_0)(D_f A_0), with index product qp,
 

where A_0,B_0 are operator-bounded with Fourier expansions
A_0=sum_pa_pw_p, B_0=sum_qb_qw_q, operator-valued coefficients in the
unmeasured algebra, p an index of A_0 and q an index of B_0.
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.

[Transverse products]

Fix face types, a thickness bound, and sufficiently small compact
transverse face patches at the basepoint o. Suppose the supports of f
and g lie within that thickness of the respective positive cones, with
all specified positive gaps sufficiently large, and that the ideal
directions of po and q^{-1}o belong to the fixed transverse patches.
Then mu(qp)=mu(q)+mu(p)+O(1), uniformly on the supports. The product
in (eq:input-2) has a uniform tracial L^2 bound in either of the
following situations:
[label=(roman*)]
 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.
 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.

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

 |(D_gB_0)(D_fA_0)|_2^2
 le C|A_0|_∞|B_0|_∞
     |D_{h_A}A_0|_2|D_{h_B}B_0|_2.
 

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)|_2le Csqrt t|A_0|_∞|B_0|_∞.


Proof. 
Common apartments and addition. On compact transverse patches, the
two ideal faces admit common apartment or flat sections with origins in a
bounded set. After erasing the bounded thickness, po=z+A+O(1) and
q^{-1}o=z-B+O(1) with A,B positive and z bounded, so the vector
distance is A+B+O(1); the vector-distance ineuality 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_iin[0,1].
In an archimedean polar chart, the angular differential in a positive-root
direction is multiplied by sinh(alpha(t· L))/sinh(alpha(L))le1
and the radial differential by the bounded linear map L|-> t· 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~lem:setup-building-interpolation. The product of real and
finite-place bounds covers the whole ambient space.

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

 ko=z+B'-B+O(1).
 

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
(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'|-> kp' and
q^{-1}|-> 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|-> 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.

Gram coefficients and twists. Expand the squared norm in
(eq:input-2) and group pairs with qp=q'p' by k=p(p')^{-1}. After
trace cyclicity the summand is the pairing of two Gram coefficients, one
made from a_{p'}a_p^* and the other from b_q^*b_{q'} (adjoints changed
in the opposite orientation). The cocycle factors are exactly the factors
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 (eq:transfer-homogeneous).

For unbounded shifts, if p=kp' is the length-increased side, retain
h_A(p')bar in f(kp')f(p')bar=
h_A(p')barf(kp')f_1(p')bar and expand the rest in the
shorter argument p'. On the other side, retain h_B(q) in
g(q)g(qk)bar=h_B(q)g_1(q)g(qk)bar; its shorter geometric
argument is q^{-1}. Conjugating Gram coefficients may conjugate these
formulas, with no effect on the cut Hilbert norm; the opposite orientation
interchanges primes, retaining h_A(p) and h_B(q'). Extend with an
initial-patch cutoff with slack. For the first Gram sequence, choose
conjugated basis functions so its full weight has the form

 w_k(r)=sum_nc_n(k)h(r)u_n(r)bar,
 |u_n|_∞le1,
 sum_nsup_k|c_n(k)|<∞.
 

The strong order of sum and supremum follows from Fourier expansion on a
slightly larger real box with uniform derivative bounds; at finite places
a finite clopen mask basis replaces it. Minkowski, Plancherel, and the
Hilbert ideal ineuality give
|(sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k|_{ell^2(L^2)}
le(sum_nsup_k|c_n(k)|)|A_0|_∞|D_hA_0|_2.
The other side is symmetric. Cauchy--Schwarz in the shift variable then
proves (eq:transfer-homogeneous), and the small-factor assertion.
Smooth joint symbols follow from summable separated expansions retaining
the restrictions with slack; approximating bounded columns by uniformly
bounded Fourier polynomials with unary L^1 convergence gives the asserted
L^2 bound. No step requires operator-norm control of an angularly cut
column.


[Uniformity used below]

In case (i) the constant is insensitive to the location of the
bounded-width intervals and to arbitrary bounded separate masks. In case
(ii) all smooth seminorms and finite-place clopen levels are held fixed
during one use of the estimate. A gap threshold may be fixed first and
exhausted only after the relevant inner limit.


## 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.

[Fullness of open patches]

Let Phi_z be a nonzero family of image measurements as in
Proposition~prop:links-link, transported over its source label space
and normalized by its positive scalar stratum mass to be unital. For every
nonempty open target patch O, the closed sum, over conull source labels
z, of the ranges of

 int_O n(v)n(v)^* ⊗ Phi_z(dv)
 

is the full amplified input space. For the trivial module this is the
ordinary patch-fullness assertion; the assertion passes to normal copies
in the three-slot system.


Proof. 
Omit O first. Source covariance makes the uncut range join a projection
over (A_1)^; 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 tr(Q)=
tr(P) and ran Q strongly approximated by A^nran 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~prop:links-link.


A normalized image branch of a pure vertex path is a candidate. Its
label is a source vertex, its values are target vertices. Different
candidates may use different paths, limits, and strata. Face-parabolic
invariance makes their transport well defined on the vertex space, and
strong continuity is used only for these constructed families.

[Type comparison]

There is a unique permutation p of the full relative type list Pi
such that every candidate at source type i takes values in target type
p(i); it preserves opposition, p(i^*)=p(i)^*. For noncoaxial source
vertices of dual types, all ordered products of candidates vanish on
transverse target patches. The permutation preserves the collection of
isolated types.


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~prop:links-gap-count). Nondual transverse target types
would force two distinct large gaps. The bounded-overlap version of
Proposition~prop:transfer-transverse bounds the cut-product norm
uniformly at fixed angular patches and thickness, and the localized
quotient matrix norm tends to zero on the forbidden strata by the finite
frame columns; Cauchy--Schwarz makes the pairing zero.

If noncoaxial, the quotient has at least two gaps. Dual transverse target
vertex cones put the cut product in a fixed thickened single-vertex cone;
the smooth version of Proposition~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~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 jin T_i and kin
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~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.


## Equality at isolated types

Fix an isolated archimedean source type and its matched target boundary
Z.

[Disjoint angular forms]

For almost every independent pair of source labels, possibly from two
different candidate families at the fixed isolated type, there is no
common nonzero target-equivariant positive map dominated by the two
candidate maps (complete-positive domination). After induction, there is
no common nonzero linear or conjugate-linear functional on
C^∞(Z)odotH bounded by both angular form norms

 b⟼(int_Z< b(v),Phi_x(dv)b(v)>)^{1/2}.
 

The generic conclusion holds throughout the transverse source orbit by
source covariance.


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_xpi_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
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.
zC^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|:0lethetalephi_x, theta is z-normal}
be upper semianalytic, and apply Jankov--von Neumann
[cite] to obtain measurable dominated theta_q(x) with
norm >q on {n>q}; rational q_m(x)<n(x) with n(x)-q_m(x)<1/m
gives |phi_x^z-theta_{q_m(x)}(x)|<1/m. No simultaneous measurable
choice of second-family central supports is needed.

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 (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.


[A consequence for sharp measurements]

If the two measurements in Lemma~lem:transfer-disjointness are
sharp, the ranges P_x(U)P_y(V)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.


[Sharpness at isolated types]

At an isolated source vertex, every candidate is sharp, and all candidate
choices agree.


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

 sum_iBigl< D_{h_i(q^+)B_1}[b'(ay)^*X_a(y)f'(y)],,
 b_i(ay)^*X_a(y)(Lf)(y)Bigr>.
 

Here q^+ is the image of the ordinary index q (the reverse angular
variable in the matched component), and smooth boundary tests on the
large-gap windows extend through the simple top singular line in the
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 (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~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~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.


## Incidence range containments

Let zsim 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
I_i(O)={x:the target face x admits incident
type-p(i) vertex in O}. For an archimedean highest-line module
V, let R_x be the orthogonal projector onto
span{n(v):vsim x, type(v)=p(i)}, and
define

 L_z=ranbarint n(v)n(v)^* ⊗ Phi_z(dv).
 


[Incidence containment]

The following inclusions hold:
[label=(roman*)]
 The support space of Psi_F(I_i(O)) is contained in
overline{sum_{zsim F}ranPhi_z(O)}.
 If the path is mixed and p(i) is archimedean, in every even highest
power the support of int R_x ⊗ Psi_F(dx) is contained in
overline{sum_{zsim F}L_z}.

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 finkerPhi_z(O) and binranbarPhi_z(V_1), then
E(window)J(barbar b,f)=0.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~lem:links-relative-boundedness. In both cases b may be any
vector in the stated closed end range.


Proof. 
We prove (i) by annihilating the common kernel C=∩_calC=∩_{zsim
F}kerPhi_z(O). For mixed paths take any finC and
operator-bounded f_lto f. For nonmixed paths supported in G_0 with
N_0=N^{G_0}: if i is in another component, ordinary fullness
gives C=0 and the inclusion is immediate; otherwise
Lemma~lem:links-relative-boundedness(ii) makes C
N_0-invariant and part (iii) gives f_lto f with
E_{N_0}(f_lf_l^*)le E_{N_0}(ff^*); in either case
< f_l,Phi_z(O)f_l>to0 for each zsim F.

Fix a compact window sending a buffered transverse V_1 into O,
zsim 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
ranbarPhi_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 < f_l,Phi_z(O)f_l>^{1/2}. The main split
translation fixes z (trivially on Levi chamber refinements incident to
F), so the bound is independent of main time. Proposition~prop:transfer-transverse
gives a uniform Hilbert bound on the other cut product depending only on
the fixed data, not on |f_l|_∞. Take the main-link limits with l
fixed, then lto∞; 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~lem:links-relative-boundedness)
gives E(window)J(barbar b,f)=0b,f)=0 for all
binranbarPhi_z(V_1).

Move the end occurrence also by the independent reverse radical. As
zsim 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 I_i(O); monotone
continuity gives E(A)J(barbar b,f)=0.b,f)=0. Then A is invariant under ignored
left Fourier shifts, and the frame identity
c_F< f,Psi_F(I_i(O))f>=I_i(O))f>=frac1dsum_r
|E(A)J(b_rbar,f)|^2=0 (with b_r the constant frame of e)
combined with positivity and density gives (i). Arbitrary Borel output
subprojections are imposed after buffered tests.

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

 E(window)sum_{ij}E(S_{ij}(h))J(b_jbar,f_i)=0.
 

The auxiliary fast symbol is S(q)S(q)^* with cutoffs, and
S(h)S(q)S(q)^*=rho(h,q)S(qh)S(q)^* with rho and derivatives bounded
on transverse supports. Move S(qh)^* to the full pairing column; its
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~lem:links-matrix-data); bounded local
right inverses or finite local spanning columns transfer the annihilation
in (eq:input-5) to R_x. Compressing with mixed input independence
from Proposition~prop:links-link proves (ii).


## Sharp flags and Coxeter compatibility

[Propagation of 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.


Proof. 
The base failure rule is nonopposition of dual types
(Proposition~prop:transfer-types); at equal self-opposite vertices
it follows by continuity. Suppose all incident replacements of one or both
source vertices satisfy a previously proved failure rule. If an incident
target pair violates it, proposition~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] 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.


[An ineuality-detecting word]

Every higher-rank component has a vertex type j and an incidence word
that fails exactly when two initial type-j vertices are equal.


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_{jhat}j{1,s_j}W_{jhat}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_{jhat}js_jW_{jhat}j. Every uneual position is above
that least nontrivial double coset, so success at all uneual pairs
follows from Lemma~lem:transfer-incidence-words.


[Sharp flag system]

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.


Proof. 
Lemma~lem:transfer-ineuality-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~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~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~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 iin I. Proposition~prop:transfer-incidence(i)
puts Psi_F(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(I_i(O) I_i(O'))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 iin I
occurs. The upper bound of Proposition~prop:links-gap-count forces
the stratum type to equal p(I), with marginals P_{z_i}; sharp
marginals put the joint measurement in their common multiplicative domain,
giving P_F. Lexicographic path marginals are uniquely identified and
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.


[Coxeter compatibility and ordered galleries]

The permutation p preserves the Coxeter matrix. Along a source chamber
gallery, the ordered product of the chamber PVMs is supported on target
galleries of the permuted type, allowing stutters. Smooth functions
vanishing on that target gallery manifold annihilate the ordered product,
with arbitrary smooth multipliers; at finite places the same holds for
locally constant tests by compact clopen partitions.


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.


## 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|-> 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.

[Residue graph equation]

On compact output windows the mixed link satisfies

 E(dg)J(Q_zbar(dq) ⊗  P_z(du))
 quadis supported onquad u=gq.
 

The equation means annihilation by smooth joint tests vanishing on the
graph, with locally constant tests in finite-place angular variables;
conjugate spectral measures conjugate projections. It holds with bounded
Borel output parameters after the angular identities are established. The
joint law is invariant under simultaneous source Levi transport, and
exact input translations may be made before any further limit.


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~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 pr(v) on a transverse
window; integrating intermediate slots by identity projections preserves
the equation.

Use (eq:input-5) at z, moving only its end occurrence by the
reverse radical. Sharpness permits the start column to be
(1-n(P_z)n(P_z)^*) applied to an arbitrary column and the end column to
be the moved line projector; the link is annihilated by
(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=gpr(v), which generates its smooth vanishing ideal on
compact transverse windows (the highest-line embedding is immersive).
Output coordinates may be Borel parameters with locally uniform angular
derivative bounds.

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
pr(v). Integrate out that slot; pull the sharp
reverse-panel projection through J (Proposition~prop:links-link);
the line equation gives u=gpr(v)=gq. This proves
(eq:input-6) on those ordered ranges; no discontinuous joint symbol
has been applied to noncommuting end measurements.

The ordered ranges are total: their closed join is target invariant, fixed
by a split source subgroup escaping in every ambient component, so scalar
mass (Proposition~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~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~sec:links); it
tensors by identity and cancels the phase defect without altering any
finite-trace mixing assertion.


[Fibrewise links and polynomial fullness]

Disintegrate over the two sharp endpoint faces with genuine source
amplifications if necessary. For product-almost every pair of endpoint
fibres, the mixed isometry (eq:input-1), the graph equation
(eq:input-6), simultaneous source Levi invariance, and exact input
translations hold. On almost every individual fibre, for every even
highest power in an archimedean target module, the line ranges over z
span at least the Hilbert fibre tensored with the span of all ordinary
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.


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~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 {Deltane0}, clear a common
homogeneous denominator and call the numerator F_num; global
polynomial fullness applied to Delta F_num vanishes spectrally
on the chart by the relation and on {Delta=0} by the factor; divide
by Delta on the chart. Pluecker equations and common local
denominators place algebraic membership tests in this form.

# Finite-place panels: fixed algebras and uniform upper speed


Take a mixed lexicographic face path missing one type in a non-isolated
component, or one type in each of several. Write B_S,B_T for the source
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 Sto S acting strongly
continuously on the endpoint spaces, commuting with the sharp
endpoint-face observations, moving residue labels by z|-> sz, and
its ineffective finite center acts trivially on the residue algebras. Our
goal is a fixed homogeneous analytic homeomorphism tau:B_Sto B_T and
measurable target projectivity observations a,b so that the endpoint
families are evaluations of z|-> atau(z) and z|-> btau(z).

## The paired endpoint algebra

For a locally constant scalar f on B_T, define the paired
decomposable operator on L^2(T;H') ⊕  L^2(T;I) by

 F_{z,f}(t)=(f(tP_z),f(tQ_z)), tin T,
 

with spectral evaluation f(tP_z)=int f(tu)P_z(du) and similarly at
the end. Let D_* be the von Neumann algebra generated by these over a
countable dense set of labels and clopen tests; D_* has separable
predual.

In the intended model f would be f(tatau(z)), with frame/projectivity
entering only via x=ta. The result below says the actual paired
algebra is exactly L^∞(T) with left translation, establishing
commutation of the endpoint families and the homogeneous form.

[The fixed algebra]

For a sufficiently small compact-open U<T, the fixed algebra
D_*^U is atomic (a countable product of finite matrix algebras), and
D_*^T=C1.


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,Urin Ubackslash T, jointly faithful on the
countable set Ubackslash T. Identify
overlineoverline II ⊗  H'congHS(I,H') via
overlinexi ⊗ eta|->|eta><xi|. For fixed
cosets Ut,Ur, put Ω_{t,r}={g:Utg=Ur} and
K=J^*E(Ω_{t,r})zetainHS(I,H'). Then for every
din D_*^U,

 d_t^+K=Kd_r^-.
 

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 Ubackslash
T forces D_*^T=C1:C1: any T-fixed normal functional on the
countable Cartesian product vanishes off the identity coset, and the
resulting character is scalar.


[Separation of archimedean types]

A connected noncompact simple real source cannot be matched to a
finite-place target. Hence the type permutation preserves both the
archimedean and nonarchimedean subsets.


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_Ualpha_u(d)du converge ultraweakly to d, so the ranges
generate D_*; contradiction to Proposition~prop:finite-frame.
Finiteness and bijectivity of the type permutation give the symmetric
assertion.


## 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 ein 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

 D_0=eD_*e, D_0^U=Ce.
 

Choose decreasing compact open normal subgroups U_n△left U with
intersection {1}; then D_n=D_0^{U_n} is finite-dimensional, since
an orthogonal family of nonzero projections has at most [U:U_n] members
(their average mass forces lambda_age[U:U_n]^{-1}).

[Block stabilizers]

The simple blocks of D_n form a transitive U-set; the stabilizer
L_n<U of a chosen block acts on it by an irreducible projective unitary
representation. If G_n<Vtimes 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

 [K_n,K_n] acts trivially on D_n,
 K_n={vin V:(v,1)in G_n}.
 



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 vin 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.


Let H be any closed subgroup of the fixed product Vtimes U with
fixed faithful multiplicative coordinates. The following uniform lemma
controls its logarithms.

[Uniform logarithm saturation]

There is a constant c, depending only on Vtimes U, such that for
every closed subgroup Hsubset Vtimes U,

 exp(p^c(Lie Hcap M_H))subset H,
 

where M_H=log H in the chosen coordinates and the image is a fixed
small ball.


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 Hcap 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.


[Limiting source kernel]

The source V-action on D_0 has zero Lie-algebra kernel: for every
open subgroup V_1<V acting trivially on D_0, we have V_1={1}.


Proof. 
As nto∞, 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_1subset 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]subsetLie V_1 satisfies
[v_1,v_1,v_1]=v_1]=v_1 and hence V_1 is
nontrivial, in contradiction to the hypothesis.


## Hausdorff limits and dimension equality

The compact metrizable Vtimes U has a compact space of closed
subgroups in the Hausdorff topology. Every limit G_∞ of the G_n
still projects onto V (for vin 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_∞ would force an open source subgroup into all
sufficiently late G_n, contradicting Lemma~lem:finite-limit-kernel.
Hence:

[Dimensions and abelian corner]

dim_{Q_p} s=s=dim_{Q_p} t.t. Every
Hausdorff limit G_∞ has both projections isomorphisms, and the
target-only subgroup G_∞cap(1times U) is {1}. Consequently
D_0 is abelian and equals L^∞(U) with Haar probability.


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_Vtimes{0}subset(sum_iB_{i,n}M_V)times{0}subset M_n
uniformly at late levels. By Lemma~lem:finite-uniform-log in the
product logarithm group, exp(p^{c+a}M_V)times1subset G_n uniformly.
After enlarging a, exp(p^{c+a}M_V) is a fixed open V_1subset 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~lem:finite-limit-kernel. So Lie(G_∞)'s
source-only part has zero Lie algebra. Both projections of
Lie(G_∞) are then isomorphisms (surjections with equal dimension
dims=s=dimt).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^∞(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.


## The T-torsor coordinate

[The full frame algebra]

There is a T-equivariant normal isomorphism D_*~= L^∞(T) with
left translation. In these coordinates, the commuting source action is

 (beta_s f)(x)=f(xeta(s)), eta:Sto T,
 

where eta is a continuous homomorphism with isomorphic
Q_p-differential.Q_p-differential.


Proof. 
By Lemma~prop:finite-dimensions, e is an abelian projection. On
its central support z_e the algebra is type I, and

 Z(D_*)z_e→ eD_*e, a⟼ ae
 

is a normal isomorphism. In the type I decomposition of this central
summand, e has rank one in every factor, and full central support makes
the scalar field faithful.

Countably many T-translates of e have join one, so D_* is type I.
Its center has a positive-measure U-invariant chart isomorphic to U,
and the T-action is ergodic (Equation~(eq:finite-fixed)).
Use Mackey's Borel point realization [cite]: let X realize
the center, let phi:Uto X be measurable onto the chart.
Fubini in the two Haar variables gives x_0 with
phi(u)=ux_0 almost everywhere; for a generic u_0,
phi(vu_0)=vphi(u_0) for almost every v. Let
L=Stab_T(x_0). If hin Lcap U, comparing u and
uh on a common conull set forces h=1, so Lcap U meets only the
identity in a small neighborhood, hence L is discrete and closed. The
orbit map T/Lto X is a Borel bijection with Borel inverse, countably
many translates of the chart cover modulo null sets, and
Z(D_*)~= L^∞(T/L).

The measurable automorphisms of T/L commuting with left translation
are the right translations by N_T(L)/L (explicitly: for such F,
g|-> g^{-1}F(gL) is left-invariant, hence constant =nL, giving
n^{-1}Lnsubset L, equality by applying to F^{-1}). The injection
N_T(L)/LtoAut(T/L) has a Borel inverse. The source
action gives a measurable (hence continuous, analytic over
Q_p) homomorphism eta:Sto 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~(eq:finite-center-corner)
forces V_1 to act trivially on eD_*e, contradicting
Lemma~lem:finite-limit-kernel. Since L is discrete and
dims=s=dimt,t, the closed Lie subgroup N_T(L) has
dimension equal to T and is open. For lin 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^∞(T) with left translation.

Remove matrix fibres: the type I decomposition and ergodicity of the
center give D_*~= L^∞(T;B(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 Aut(B(V))-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(V); Equation~(eq:finite-fixed) forces dimmathcal
V=1.V=1.


## Ordinary homogeneous panel maps

[Recovery of the observed panels]

The families (P_z) and (Q_z) commute on their respective start and
end spaces, and there are a homogeneous analytic homeomorphism
tau:B_Sto B_T and measurable target projectivity observations
a,b such that these families are evaluations of

 z⟼ atau(z), z⟼ btau(z).
 

On the independent input tensor product, the output projectivity is
sharp and equals ab^{-1}:

 E(W)J=J,1_W(ab^{-1})quadfor every Borel Wsubset T.
 

All statements hold coordinatewise on many-panel links.


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^∞(T) of
Proposition~prop:finite-torsor, the spectral values give a
jointly measurable boundary map F(x,z) (countable determining
observations plus label continuity). Frame covariance F(ax,z)=aF(x,z)
and Fubini give F(x,z)=xF_0(z), and the source action gives

 F_0(sz)=eta(s)F_0(z)
 

in the ordinary measure class on B_S.

Regard each effective adjoint group over its coefficient field as a
restriction of scalars to 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=deta is an isomorphism of these adjoint
algebraic groups over Q_p, and

 AAd(s)=Ad(eta(s))A, sin S.
 

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_Sto B_T exists, intertwining S with
eta(S).

Write B_S=S/P; the measurable lift s|->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 Ω^+ of the
commutative start observation algebra, with a faithful normal
probability class. The normal representation of D_*=L^∞(T) by
its start decomposable fields is pullback via a measurable h(t,omega)
into T; the generator identities read tp_z(omega)=h(t,omega)tau(z)
for almost all (t,omega), and choosing one generic frame t_0 gives
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
(eq:finite-panel-laws) in the exact graph link gives
atau(z)=g,btau(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
(eq:finite-projectivity-sharp). The map from target
projectivities to continuous homogeneous boundary maps is continuous and
injective, with Borel inverse.


## 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 Q_p, and
let ell_R be its rank-one Cartan gap. Constants d_R>0 depend only
on the panel type:

 log|Ad(r)|=d_Rell_R(r)+O(1), rin R,
 

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 |·|_k to |·|_p^{[k:Q_p]}Q_p]
with the largest relative root multiple r_{max}in{1,2},
d_R=r_{max}/[k:Q_p].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.

[Uniform finite-place upper speed]

After the inner limit of a translated panel link, let tge0 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

 d_Tell_T(g)le d_St+O_{tight}(1),
 

with tightness on each fixed ordinary input vector, uniformly in the
slower translations with t as center. This holds coordinatewise and
simultaneously as an upper-tail estimate on many-panel links.


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_tin S and eta(s_t) is
defined). Proposition~prop:transfer-residue-graph and
Proposition~prop:finite-maps give atau(s_tz)=g,btau(z), so

 g=aeta(s_t)b^{-1},
 

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=deta, Equation~(eq:finite-adjoint-intertwining) gives
Ad(eta(s_t))=AAd(s_t)A^{-1}, hence
log|Ad(eta(s_t))|le d_St+C, and the same for s_t^{-1}. Fix
xiinoverlineoverline II ⊗  H' and ε>0; the commuting
a,b have a joint scalar spectral measure on xi. Choose compact
sets containing both observations except for squared input norm
leε|xi|^2. On that joint cut |Ad(a)| and
|Ad(b^{-1})| are uniformly bounded in t and slower parameters;
submultiplicativity with (eq:finite-translated-law) and
(eq:finite-adjoint-growth) gives d_Tell_T(g)le d_St+C_{xi,ε}
on that cut. The exact graph intertwining and link isometry bound the
excluded output by the excluded input. This proves
(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.


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~cor:finite-arch-separation are archimedean, treated
below.
# Length multipliers and ray lifts


This section supplies the length estimate needed for the isolated boundary
factors: uniform ray lifts (Proposition~prop:ray-lift) combined with
property~(T) averaging give addition of rescaled spherical length laws
(Proposition~prop:ray-addition). Addition lets us later compare
repeated source segments and rule out total superlinear growth.

The isolated components are real Lie factors (Sections~sec:transfer
and~sec:finite): quaternionic hyperbolic spaces of dimension at least
two, and the Cayley hyperbolic plane. Fix such a space X; normalize its
metric so r(g)=d(o,go) is the logarithm of the dilation in the
indivisible restricted root. The argument uses an entire ambient factor;
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.

## Coordinates and a uniform smooth calculus

The horospherical group is N_b=Vtimes Z with
[
 (v,z)(v',z')=(v+v',z+z'+1/2[v,v']),
 delta_R(v,z)=(Rv,R^2z).
]
Choose inner products with < J_zv,w>=< z,[v,w]>
and J_z^2=-|z|^2id; the horizontal layer is bracket-generating. Set
Q=dim V+2dim Z. In half-space coordinates (n,lambda)in
N_btimes(0,∞), with a fixed normalization of Haar measure,

 ds_X^2=(dlambda^2)/(lambda^2)
       +(|dv|^2)/(4lambda^2)
       +(|dz-1/2[v,dv]|^2)/(4lambda^4),
  dp=frac{dn,dlambda}{lambda^{Q+1}}.

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 ∂ X, the horizontal term in
(eq:ray-metric) determines a canonical horizontal conformal class;
we take the compact-stabilizer average of a smooth representative and let
d_{∂ 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.

[Distance and changes of scale]
With constants uniform on buffered compact charts and in normalized C^m
norms for every mge0:
[label=(roman*)]
 If (v,z)=log(n^{-1}n'), then

 cosh d((n,lambda),(n',lambda'))
 =frac{√((lambda^2+lambda'^2+|v|^2/4)^2+|z|^2)}
 {2lambdalambda'}.
 Changes between the compactly rotated horospherical boundary charts
preserve normalized bounds on boxes of every scale 0<Rle1, and in the
adjoined height 0lelambdale CR. The conversion between the outward
endpoint and e^{-r} (or finite half-space coordinates) has the form

 n=n_0(e^{-2r}U,e^{-4r}W),
 lambda=e^{-r}b,

where U,W,b are smooth in r^{-1} and e^{-r}, with b bounded above
and below.
 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}, 0le ule L, with normalized derivatives
bounded. A smaller boundary box of radius deltale cq, height at most
cdelta, has image radius at most Ce^{2u-L}delta. Exterior pieces
qge c have output upper bound Ce^{-L}.



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/(2lambda). In the
orthonormal frame the three components of nabla c are
lambda A/sqrt C-sqrt C/(2lambda), (Av+J_zv)/(2sqrt C),
lambda z/sqrt C. Since J_zvperp v and |J_zv|^2=|z|^2|v|^2, the
squared norms sum to c^2-1. Thus arcoshc has gradient
norm one off its unique zero. It is proper, so it is the distance.

(ii). One convenient Weyl inversion in these coordinates is

 j(v,z,lambda)=
 (((-A_0+J_z)v)/(B_0),-frac z{B_0},
             fraclambda{√(B_0)}),
 A_0=lambda^2+|v|^2/4, B_0=A_0^2+|z|^2,

with jdelta_R=delta_{R^{-1}}j. On a compact annulus
(|v|^4/16+|z|^2)^{1/4}+lambdaasymp1, 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_{loglambda} and Taylor expanding at o, the
horizontal and central coordinate errors are lambda^2U and
lambda^4W, giving (eq:ray-collar). Boundary transitions are
H-type contact maps, so the central horizontal linear term vanishes; the
same Taylor computation applies after substitution and proves (ii).

(iii). Compactly rotate g to delta_{e^{-L}}j; rescale input by
q^{-1} and output by (q')^{-1} to see exactly the restriction of j
to a fixed buffered annulus. The derivative and height-ratio claims
follow from (eq:ray-inversion); the smaller-box claim is (ii) with
relative radius delta/q.


The horizontal density on ∂ X used later is the product of the
horizontal volume density with the density on T∂ X/V
induced by the surjective bracket ∧^2V
T∂ X/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·Isom is exactly d^Q.

[Triangle defect]
There is C such that if x,yin X and
b=d(o,x)+d(o,y)-d(x,y)>0, then
d_{∂ X}(x^+,y^+)le Ce^{-b/2}. Consequently
r(g)+r(h)-r(gh)ge b>0 forces d_{∂ X}(g^-,h^+)le Ce^{-b/2}.


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 (eq:ray-collar)), or
homogeneous separation is comparable to delta and (eq:ray-distance)
gives d(x,y)ge d(o,x)+d(o,y)+2logdelta-C. Apply to x=g^{-1}o,
y=ho.


[Smooth separation and cells]
Fix row and column dimensions and a bound on auxiliary compact variables.
For every summable power N there are m and C such that:
[label=(roman*)]
 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.
 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.



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.


## Uniform radial multipliers

[Radial cutoffs]
Let fin C^∞([0,∞)) be constant near zero and near infinity.
There are R_f,C_f>0 such that (x,x')|-> f(d(x,x')/R) has Schur
norm at most C_f for every Rge R_f. Consequently for any
homomorphism GammatoIsom(X), any normalized scalar
cocycle eta, and any finite tracial von Neumann algebra B, the
multiplier
[
 D_{f(r/R)}: B,barbar ⊗⊗  L_eta(Gamma)to
 B,barbar ⊗⊗  L_eta(Gamma),
 sum_hY_hu_h⟼sum_hf(r(h)/R)Y_hu_h
]
has completely bounded norm at most C_f; the constants are independent
of Gamma,eta,B and of the homomorphism.


Proof. 
Subtract f(∞). Choose 0<a<b with f equal to its value on
[0,2a] and to 0 on [b,∞). Write s=-loglambda,
s'=-loglambda'. 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
ε_j=e^{-j}, equal to one on a smaller ball, and
J=lceil C'_bRrceil. The annuli chi_j-chi_{j+1} for
-Jle j<J and the innermost ball chi_J cover the support of
f(d/R). Put
[
 H_j=log(max(ε_j,lambda,lambda')^2)/(lambdalambda').
]
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_{ε_j}(V,Z) on the fixed separated cell, expanding
with compact auxiliary scale ratios), then at comparable heights by the
sign-cutoff argument in s/R-s'/R, where R^me^{-aR} is bounded on the
smooth pieces. The telescoping identity
[
 sum_{j=-J}^{J-1}phi_j(chi_j-chi_{j+1})+phi_Jchi_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.


## One-sided localization

Let D=Bbarbar ⊗⊗  L_eta(Gamma) with product trace tau and
Fourier coefficients Y_hin L^2(B); D_phi denotes coefficient
multiplication on L^2(D). For finite Fourier sums,
(YY^*)_g=sum_heta(gh,h^{-1})overline{eta(h,h^{-1})}Y_{gh}Y_h^*,
and phases have absolute value one.

[Unary Plancherel estimate]
Let phi_{i,nu}:GammatoC satisfy
sup_hsum_i|phi_{i,nu}(h)|^2le 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_{phi_{i(g,t),nu}}Y)^*]_g where
|c_{g,t,nu}|le Aa_nu, sum_nu a_nu<∞, we have

 (sum_gsum_{tin T_g}|z_{g,t}|_{2,B}^2)^{1/2}
 le bA(sum_nu a_nu)|Y|_∞|Y|_2.

The same holds with the first factor localized instead.


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_{phi_{i(g,t),nu}}Y)^*]_g|_2^2
le b^2|Y|_∞^2|Y|_2^2. Minkowski sums the absolutely summable
frequencies; adjoints give the other orientation. Only the operator norm
of the uncut factor is used.


[Angular column estimate]
Fix X and an integer m; there are c,C>0 with the following
property. Put delta=e^{-s}, sge1. Let (f_C)_C be smooth masks
supported on buffered boundary boxes of radius delta, with fixed
overlap and normalized C^m bound. If Yin D has Fourier support in
{rge l} with lge Cs+C, then with A_C=D_{f_C(h^+)}Y,

 sum_{C,D'}|A_CA_{D'}^*|_2^2
 =Bigl|sum_CA_C^*A_CBigr|_2^2
 le C|Y|_∞^2|Y|_2^2.

The constants may depend on the mask bound but not on the number of
masks, and the assertion holds with arbitrary coefficient algebras and
scalar twists.


Proof. 
Estimate the coefficients of A_CA_{D'}^* at fixed shift g; put
L=r(g); the weights in YY^* are f_C((gh)^+)overlineoverline{f_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 2ule L+C_1 (then
Lemma~lem:ray-geometry(iii) makes the map to the output boundary
have bounded normalized derivatives, and Fourier expansion makes the
pair weight a unary expansion in h with compact-coefficient decay) or
2u>L+C_1 (then inversion gives output pole distance e^{-L+u} and the
reversed unary estimate applies). Bounded L and the transition
2u=L+O(1) use larger constants. Lemma~lem:ray-unary then bounds
every nonexceptional pair; a map with bounded magnification sends one
buffered delta-box into a ball of radius Cdelta, giving the
required multiplicity.

The exceptional pairs lie within C_2delta 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

 Lge 2s-C_3.

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~lem:ray-unary; the product of differences is split into
smooth unit-width annuli in the pole coordinate e^{-u},
0le ule L/2+C_4, and on such an annulus the product seminorm obeys

 C_mmin(1,e^{s-u})min(1,e^{s-L+u})le C'_m e^{-|s-u|}.

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~lem:ray-cells
and~lem:ray-unary give Ce^{-|s-u|}|Y|_∞|Y|_2 for the
depth-u square sum; geometric summation and the reversed annuli sum
these uniformly. The written equality is the trace identity
tau(A_CA_{D'}^*A_{D'}A_C^*)=tau(A_C^*A_CA_{D'}^*A_{D'}).

For general Y, take bounded Fourier-polynomial approximations with a
smooth radial cutoff equal to one on {rge l} and zero on
{rle l/2}; Proposition~prop:ray-radial bounds the approximants
uniformly. Increase the constant in l by the slack. Each masked column
converges in L^2 and each product of two in L^1 by tracial
Cauchy--Schwarz; for a fixed finite family the diagonal-terms bound in
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.


## The ray lift and its densities

Let K_X=L^2(X,dp)K_X=L^2(X,dp) and let T_g be translation on
K_X. Define
[
 Delta:D→ B(K_X) ⊗⊗  D,
 Delta(bu_h)=T_h ⊗  bu_h,
]
a normal unital homomorphism (conjugate by the unitary sending
zeta ⊗ delta_h to T_h^{-1}zeta ⊗ delta_h).

Choose real smooth psi,F with supppsiSubset(1,2), F
compactly supported on N_b and invariant under the compact isotropy
of a ray, with unit L^2 norms. For d_1ge1, basepoint x, and
ideal endpoint v, use a chart sending v to infinity and put

 theta_{x,d_1}(p,v)=d_1^{-1/2}
 psi((log(lambda_p/lambda_x))/(d_1))
 F(delta_{lambda_p^{-1}}(n_x^{-1}n_p)).

This is equivariant in x,p,v and has unit L^2 norm, supported in a
bounded neighborhood of the ray from x to v at depths
[d_1-O(1),2d_1+O(1)]. The coefficient isometry is
V_{d_1}Y=sum_htheta_{o,d_1}(·,h^+) ⊗  Y_hu_h.

For a column xiinK_X ⊗  L^2(D), the left and right
densities rho^L_xi,rho^R_xiin L^1(D) represent the vector
functionals for Delta(D) and D^{op}:
<xi,Delta(x)xi>=tau(rho^L_xi x),
<xi,xi x>=tau(rho^R_xi x)=xi^*xi.

[Ray lift]
Take any sequence of finite tracial coefficient algebras, twists, and
countable homomorphisms to Isom(X), with
d_1to∞, l/d_1to∞, and Yin D Fourier-supported in
{rge l}.
[label=(roman*)]
 For large indices, V_{d_1}Y has left density for
Delta on D and right density on D^{op}, both in L^2
with norm at most C|Y|_∞|Y|_2; also
|V_{d_1}Y(V_{d_1}Y)^*|_{2,Tr ⊗ tau} has this
bound.
 If suppxxsubsethat{rle a} and a/d_1to0, then
|V_{d_1}(xY)-Delta(x)V_{d_1}Y|_2to0 and
|V_{d_1}(Yx)-(V_{d_1}Y)x|_2to0.
 For another long column Z and bounded x, if for every
contributing triple hinsuppYhat,Y, hkinsuppZhat,Z,
kinsuppxxhat one has
d_{∂ X}(h^+,(hk)^+)le e^{-cl'} with c>0,
l'/d_1to∞, then

 <V_{d_1}Z,(V_{d_1}Y)x>
 -< Z,Yx>→0,

uniformly with |x|_∞,|Y|_2,|Z|_2 bounded.

All statements hold for matrix versions and measurable fields.


Proof. [Right density]
Write K_{d_1}(v,w)=inttheta_{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_xdelta_{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~lem:ray-angular. Tracial H"older plus
(eq:input-7) give le C|Y|_∞|Y|_2; Fourier
coefficients are summable, chosen above the polynomial derivative
growth of masks. The matrix assertion follows from
|V_{d_1}Y(V_{d_1}Y)^*|_2^2
=|(V_{d_1}Y)^*V_{d_1}Y|_2^2 and monotone
approximation.


[Common-endpoint overlap]
For a displacement g of length L, put
w_{d_1}(g,v)=int_Xtheta_{o,d_1}(p,v)theta_{go,d_1}(p,v),dp.
There are smooth annular partitions about g^+ at integer depths
0le jle L (with exterior piece and innermost ball) on which

 
 |w_{d_1}(g,·)-a_{d_1}(j,L)|_{C^m_{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,
 

with constants uniform in L,d_1ge1, and

 |∂_ja_{d_1}(j,L)|le C/d_1,
 a_{d_1}(0,L)=1+O(min(1,L/d_1)).



Proof. 
Put go=(0,ε), ε=e^{-L}, and g^+=0. For finite
v=(V,W), use
p(v)=|V|^2/4, D_h(v)=(h^2+p(v))^2+|W|^2, and
P_h(v)=((h^2+p(v)+J_W)V/D_h,W/D_h). Then
delta_{√(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

 n=delta_{√(D_1)}(P_1^{-1}P_ε),
 e^beta=ε√(D_1/D_ε).

Put v=delta_Ru, R=e^{-j}, t=ε/R, and define
Delta_R=(1+R^2p(u))^2+R^4|W|^2,
Q_R=(R^2(1+R^2p+R^2J_W)V/Delta_R,R^4W/Delta_R). Then

 beta=2j-L+B(R,t,u),
 B=1/2log(Delta_R/D_t),
 delta_Rn=m(R,t,u)=delta_{√(Delta_R)}(Q_R^{-1}P_t).

All derivatives of B,m are bounded on the buffered annulus, and
mto P_0(u)ne e as R,tto0. The rest follows by writing
H(m)=int F(q)F(m^{-1}q),dq (smooth, compact, H(e)=1):

 w_{d_1}(g,v)=intpsi(t)psi(t-beta/d_1)H(delta_{e^{-td_1}}n),dt.

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
1_{s>j} using Taylor's formula for s>j and the support of
H for sle j. The O(1/d_1) derivative bounds follow similarly.


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

 Ce^{C'd_1-l}|Y|_∞|Y|_2,

by a compact-polar computation with auxiliary variable
e^{l-r(h)} plus fixed-derivative bounds of isometries of length
le4d_1+C (product-form derivative growth at most
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|_∞|Y|_2 for the coefficient L^2 norm by
Lemma~lem:ray-unary; telescope a_{d_1}(j,L) against single
balls using (eq:ray-overlap-increments), and observe the at
most C(1+d_1) levels. Summing plus (eq:ray-endpoint-error)
gives the bound. The stronger short-coefficient version
a=o(d_1) gives difference bounded in L^2 by

 C((1+a)/d_1+e^{C'd_1-l})|Y|_∞|Y|_2.

Polarization extends to

 <V_{d_1}Z,Delta(x)V_{d_1}Y>
 -< Z,xY>→0

uniformly when norms of Z,Y,x are bounded with
suppxxsubsethat{rle a}. Bounded approximants and
weakly convergent L^2 subsequences with slacked cutoffs handle
infinite Fourier support.


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

 |K_{d_1}-1|_{C^m_{normalized}}le C_m e^{C_m d_1-d_2},

positive-derivative terms contributing e^{-d_2}, central ones
e^{-2d_2}. Choose d_1ll d_2ll 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|_∞
(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
(eq:ray-close-pairing). For short right multiplication,
r(h),r(hk)ge l-a and r(k)le a give
d_{∂ X}(h^+,(hk)^+)le Ce^{-c(l-2a)} with fixed c>0, so
(iii) applies. Expanding
|V_{d_1}(Yx)-(V_{d_1}Y)x|_2^2 finishes (ii) for
the right; for the left use (eq:ray-left-pairing) with x^*x
and Z=xY.


[Stability under Hilbert-small changes]

If Y is long with bounded |Y|_∞,|Y|_2 and
x-x'to0 in L^2(D) with x,x' uniformly bounded, then
|Delta(x-x')V_{d_1}Y|_2to0 and
|(V_{d_1}Y)(x-x')|_2to0; the analogous amplified
statement holds in the semifinite L^2 norm.


Proof. 
Pair the corresponding density with (x-x')^*(x-x') or its
reverse; tracial Cauchy--Schwarz with z=x-x' gives

 max{|Delta(z)V_{d_1}Y|_2^2,
       |(V_{d_1}Y)z|_2^2}
 le C|Y|_∞|Y|_2|z|_∞|z|_2.

The amplified version uses V_{d_1}Y(V_{d_1}Y)^*
and the semifinite norm. For fields, integrate the pointwise bound
and apply Cauchy--Schwarz on the base.


## 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.


An admissible representation for T is a finite factor
D=Bbarbar ⊗⊗  L_eta(Gamma) together with a normalized
projective representation Θ:GammatoU(D) such that
DcapΘ(Gamma)'=C1 and the representation

 gamma⟼AdDelta_T(Θ(gamma))
 quadhbox{on}
 L^2(B(L^2(X_T))barbar ⊗⊗  D,Tr ⊗ tau)

has no invariant vector. Here
Delta_T(u_h)=lambda_{X_T}(h_T) ⊗  u_h. The scalar multiplier
of Θ need not equal eta.


For ain G put X^Θ_a(y)=Θ(c(a,y)) (the section and
cocycle of Section~sec:setup). For an isolated source factor
S and positive split translation a_t of length t, the spherical
law at t, on scale R>0, is the probability

 P_{Θ,R}(t)(E)=int_Y
 bigl|D_{1_{{r_T/Rin E}}}X^Θ_{a_t}(y)bigr|_2^2,dy,

for Borel Esubset[0,∞].

[Uniform short approximation]
Let R_nto∞ and Θ_n be projective representations in
finite tensors, with

 |D_{1_{{r_T>ε R_n}}}Θ_n(gamma)|_2to0
 (gammainGamma, ε>0).

Then there are a_n=o(R_n) and uniformly operator-bounded fields
Z_{n,s} with Fourier support in {r_Tle a_n} such that

 sup_{sin C}|X^{Θ_n}_s-Z_{n,s}|_{L^2(Y;L^2(D_n))}to0

for every compact Csubset G. Thus compact pre- and post-translations
do not affect any limiting rescaled length law.


Proof. 
For fixed compact C, outside a set of measure at most
2ε both d_0(y),d_0(sy)in K, so
c(s,y)inGammacap K^{-1}CK, a finite set; apply
(eq:ray-short-fixed) to it. Smooth cutoffs
b_n=o(R_n) with a_n=2b_n and Proposition~prop:ray-radial
give the bounds. The radial multiplier uses

 |r_T(gh)-r_T(h)|le r_T(g), |r_T(hg)-r_T(h)|le r_T(g)

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.


[Averaging with a uniform gap]
There is a compactly supported, absolutely continuous, symmetric
probability m_0 on G and 0<c_0<1 such that its convolution
operator is positive in every unitary representation and has norm at
most c_0 on the orthogonal complement of the invariant vectors.


Proof. 
Take a compact Kazhdan pair (C,ε) for the product
G; a symmetric compactly supported density nu that is bounded
below on Vcup CV and their inverses, for a compact neighborhood
V of 1, satisfies
sup_{gin C}|pi(g)xi-xi|^2le
C_1int|pi(h)xi-xi|^2dnu(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.


[Addition in distribution]
Let R_nto∞, Θ_n admissible for T satisfying
(eq:ray-short-fixed). Fix an isolated source factor and a
positive split axis. For any t_n,u_nge0, if P_{Θ_n,R_n}(t_n)
converges weakly to P_t and P_{Θ_n,R_n}(u_n) to P_u, then

 P_{Θ_n,R_n}(t_n+u_n)→ P_t*P_u.

The same holds with compact pre- and post-translations.


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~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^{Θ_n}_{a_{t_n}},
B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}·),
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

 J:L^2(P_t) ⊗  L^2(P_u)→K,
 f ⊗  g⟼(A_{n,f}(sy)X^{Θ_n}_s(y)B_{n,g}(y))^{lim}.


For a positive finite output window strictly above the sum of the
inputs, (eq:ray-triangle) plus short approximation excludes it.
For the strict-deficit case, in the shortening test
A(sy)^* pairs against X^{Θ_n}_s(y)B(y)C(y,s); convert to

 tau(zeta_n(sy)^*Delta_T(X^{Θ_n}_s(y))xi_n(y)C(y,s))

with xi_n=V_{d_n}B, zeta_n=V_{d_n}(A^*). The
exact trace ineuality (via Cauchy--Schwarz on the two density fields,
each with uniformly bounded trace-Hilbert norm by
Proposition~prop:ray-lift) is bounded by C c_0^k because the
combined amplified action has no invariant vector (admissibility).
For a short change z, the error is at most
|z|_∞|z|_2|rho^L_{xi_n}|_2, i.e.

 |Delta_T(z)xi_n|^2
 le|z|_∞|z|_2|rho^L_{xi_n}|_2.

Limit nto∞ then kto∞.

For input values 0 and +∞, use triangle ineualities and
short approximations with smooth tail cutoffs (Proposition~prop:ray-radial);
the lifted shortening argument extends because it uses positive
lower length bounds and a uniform deficit but no upper bound.

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_{in}(a^{-1}(F)^c), so
E(F)J P_{in}(a^{-1}(F)^c)=0 and
E(F)J=E(F)J P_{in}(a^{-1}(F)). A closed-set exhaustion
K_j↑ F^c gives E(K_j)JP_{in}(a^{-1}(F))=0, so
E(F^c)JP_{in}(a^{-1}(F))=0; together,
E(F)J=J P_{in}(a^{-1}(F)). Monotone classes and spectral
integration give
E(phi)J=JM_{phi(s+t)} for phiin C([0,∞]).
Convergence on the whole sequence follows by compactness of
probability measures on [0,∞].


# Linear subsequences and heights for isolated components


At an isolated type, Section~sec:transfer has already supplied
sharp boundary measurements x|-> 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|-> 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~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), nge2, 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~sec:ray.

We use the admissibility and statistical conventions of
Definition~def:ray-admissible. All varying finite factor traces
are normalized, Y=G/Gamma has probability measure, and extra finite
matrices carry the unnormalized matrix trace with fixed size. Each
application of the ray estimates has both the stated operator bounds
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~lem:setup-normal-tensors, Proposition~prop:setup-induced-algebras,
Lemma~lem:setup-product-mixing); full-gap escape on complete
vertex-origin lexicographic chains (Proposition~prop:links-gap-count);
sharp, strongly continuous pure-vertex measurements and their
covariances from Propositions~prop:transfer-types,
prop:transfer-isolated-sharpness, and~prop:transfer-sharp-flags;
scalarity of limiting rescaled one-column effects
(Lemma~lem:links-invariant-variables). The compact-change argument
for those effects uses bounded approximants of the fixed transporters
and has arbitrary fixed slack in radial windows. These are conclusions,
not new assumptions.

## An admissible model for the original module


For the two-slot model of Section~sec:setup there is a finite
tensor D=Bbarbar ⊗⊗  L_eta(Gamma), with B diffuse, and an
admissible representation delta:GammatoU(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.


Proof. 
The complementary Λ-factor in the two-slot regular model is
diffuse; in a large enough finite matrix amplification choose a
projection in that coefficient factor with the same trace as the
regular-model projection e, and identify by a partial isometry to
get the required unital tensor-coordinate representation. Scalar
relative commutant follows from Proposition~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~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∞ (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.


[Entropy lower bound]
Suppose an isolated source type i, with factor S, is matched to the
measured type p(i), with factor T. Let Q_S,Q_T be the homogeneous
dimensions of their boundaries. For the representation delta of
Lemma~lem:scale-original-model, complementary lengths are
uniformly tight on the pure source ray, and for every
0<c<Q_S/Q_T,

 P_{delta,t}(t)([0,c])→0.
 

Both hold uniformly under source pre- and post-translations in
compact sets.


Proof. 
Complementary tightness is the pure-ray conclusion of
Section~sec:transfer, transported through the corner equivalence.
Before the equivalence, complementary regularity supplies a nonzero
constant frame column f for pi_0(Gamma) with W_Λ; if
e_j are the finitely many identity columns, the frame ineuality
applied to W_he_j gives, for every hinGamma,

 sum_{gammainGamma}
 |D_{{h}}pi_0(gamma)f|_2^2le C_f.
 


Let Ω_t be a source Cartan shell of fixed positive thickness
about length t in S, with all compact rotations and complementary
coordinates in a fixed compact neighborhood; |Ω_t|asymp
e^{Q_St}. Restricting both section values to a compact K, the times
with transporter gamma lie in Kgamma d_0(y)^{-1} of Haar volume
at most |K|; integrating (eq:scale-bessel-count) therefore
bounds the averaged mass at each measured index by a constant
independent of index and t.

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 Ω_t, divided by
|Ω_t|, is at most

 Ce^{(Q_Tc'-Q_S)t}.
 


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~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 (eq:scale-entropy) would have positive averaged mass in
(eq:scale-volume-ratio), contradiction. The corner-change
argument finishes.


## Tightening a Kazhdan window

For cutoffs L_nge1, a bounded sequence x_nin D_n is
tight at scale L_n if

 lim_{Cto∞}lim_{ntoU}
 |D_{1_{{r_T>CL_n}}}x_n|_2=0
 

for a free ultrafilter U. Ultraproducts are tracial;
ordinary subsequences are extracted once all countably many
requirements are fixed.


The tight elements form a von Neumann subalgebra Q_0 of
prod_{U}, and Q_0 is a factor. If each Θ_n
is relatively irreducible, the ultraproduct representation Θ
also has scalar relative commutant.


Proof. 
Smooth radial truncations are uniformly bounded (Proposition~prop:ray-radial),
so tight elements admit Hilbert approximation by bounded sequences
supported in r_Tle CL_n. Triangle ineuality (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_nge1, 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.


## Near-identity Kazhdan tightening

[Kazhdan tightening]
Fix a finite Kazhdan set F and
ε>0. There is ε_K>0 such that, if
Θ_n are admissible and

 |D_{1_{{r_T>L_n}}}Θ_n(gamma)|_2leε_K
 (gammainF),
 

then, after passing to a subsequence, there exist unitaries
u_n with limsup|u_n-1|_2<ε such that every fixed
element of u_n^*Θ_n(Gamma)u_n is tight at scale L_n.
Admissibility is preserved.


Proof. [Sketch]
Let D_U={U} and Q_0 the tight
factor of Lemma~lem:scale-tight-factor. The Jones basic
construction < D_U,e_calU,e_{Q_0}> gives

 |Θ(gamma)e_{Q_0}Θ(gamma)^*-e_{Q_0}|_2^2
 =2|Θ(gamma)-E_{Q_0}Θ(gamma)|_2^2,
 

which is O(ε_K^2) by tightness plus smooth radial
cutoffs (Proposition~prop:ray-radial). The Hilbert-space
Kazhdan ineuality 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(ε_K)

 |p_0-e_{Q_0}|_2le2|z-e_{Q_0}|_2=O(ε_K).
 

p_0 commutes with Θ(Gamma) and has trace d' near one;
take a right Q_0-module isometry with row Vin L^2(D_U)^{ ⊕ 2}
satisfying Vq=V, E_{M_2(Q_0)}(V^*V)=q, Θ(gamma)V=Vb_gamma;

 VV^*=d'1.
 

e_1=(d')^{-1}V^*Vle 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_Tge l_n and short approximations to
q,b_gamma with support r_Tle a_n:

 L_nll a_nll d_nll l_n,
 |G_n-H_{1,n}|_2to0.
 

The ray lift sends G_n to xi_n=V_{d_n}G_n with

 Delta_T(b_{gamma,n})xi_n-xi_nb_{gamma,n}to0,
 Delta_T(q_n)xi_n-xi_nto0,quadxi_nq_n-xi_nto0
 

and xi'_n=Delta_T(e_{1,n})xi_n satisfies

 |xi'_n|^2to(d')^{-1}|H_1|_2^2>0,
 


 Delta_T(Θ'_n(gamma))xi'_n-xi'_nb_{gamma,n}to0.
 

Extend b_{gamma,n} off q_n; the normalized left vector
states are almost invariant in predual norm, whose square-root
densities are almost invariant vectors by Powers--Sto rmer

 |A^{1/2}-B^{1/2}|_2^2le|A-B|_1quad(A,Bge0).
 

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^*Θ(Gamma)usubset Q_0 and
p_0=ue_{Q_0}u^*. Aligning the polar part of E_{Q_0}u yields

 |u-1|_2^2le2-2|E_{Q_0}u|_2^2
 =|p_0-e_{Q_0}|_2^2.
 

Diagonalize countably many tightness conditions; small
ε_K gives the bound.


## Composition of admissible kernels

[Exact insertion]
Let delta:L_alpha(Gamma)to B_0barbar ⊗⊗  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 Θ at an
isolated measured factor T there is an admissible
Θcircledastdelta with source multiplier alpha, whose
spherical Fourier laws are the exact compositions of the
kernels of delta and Θ. The final measured cocycle
is that of Θ.


Proof. [Sketch]
Write xi for the source multiplier of Θ and put
T_h(y)=Θ(c(h,h^{-1}y)). The section identity gives
T_g(y)T_h(g^{-1}y)=nu(g,h)(y)T_{gh}(y) with
nu(g,h)(y)=xi(c(g,g^{-1}y),c(h,g^{-1}g^{-1}y)), a normalized
scalar cocycle for the action on Y. In the twisted crossed
product Psi(w_h)=T_hb_h, with multiplier beta/nu, is
trace-preserving on the twisted group algebra and extends
normally; apply id_{B_0} ⊗ Psi to delta. The kernel
formula

 sum_{hinGamma}|a_{gamma,h}|_2^2
 int_Ybigl|D_{1_{{r_Tin E}}}Θ(c(h,h^{-1}y))bigr|_2^2,dy
 

composes the kernels. Factoriality and admissibility come from
the induced coefficient representation

 (U_gk)(y)= T_g(y),k(g^{-1}y),
 

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

 sum_{i,j:,ih=h'j}|a_i|_2|a_j|_2
 |< k',U_i k>|,
 

which vanishes off the scalar summand by Fourier escape, Howe--Moore,
and the product invariant-space decomposition of Section~sec:setup.
The amplified assertion replaces E by
L^2(Y;L^2(B(L^2(X_T))barbar ⊗⊗  D)) and T_g by
Delta_T(T_g), and admissibility of Θ gives absence of
invariant vectors.


## A linear subsequence


Suppose P_{Θ_n,R_n}(t_n)([a,∞])gealpha with
a,alpha>0 and u_n/t_nto∞. Then

 P_{Θ_n,R_n}(u_n)→delta_∞,
 

uniformly under compact pre- and post-translations.


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~prop:ray-addition makes the
limit P^{*m}*Q with P([a,∞])gealpha and
Q supported in [0,∞]. For finite M,

 (P^{*m}*Q)([0,M])le
 P{Bin(m,alpha)le M/a}to0
 

as mto∞; diagonal extraction over countably many m
justifies the passage. Compact changes are covered by
Lemma~lem:ray-compactshort.


[Positive mass at linear speed]
For each isolated source type i, matched to p(i), there
exist C<∞, alpha>0, T_nto∞ such that for
every fixed xiinH,

 liminf_{n}inf_{T_n/2le tle T_n}
 bigl|D_{1_{{r_{p(i)}le CT_n}}}U_{a_t}xibigr|^2
 gealpha|xi|^2.
 

The statement holds for normal Hilbert amplifications and
compact source pre- and post-translations, but is not uniform
over moving Hilbert unit balls.


Proof. [Sketch]
Use delta from Lemma~lem:scale-original-model.
Suppose the matched length is totally superlinear:

 P_{delta,t}(t)([0,C])to0 (C<∞).
 

Compose the original representation around the phase cycle
i|-> p(i)|->...|-> i via
Lemma~lem:scale-insertion; the first stage increases
length faster than linearly, the entropy bound prevents later
stages from losing that, and the additive structure of
Proposition~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'_nto∞: feasibility is nonempty (identity embedding
into L_eta(Gamma) suffices), so the infimum

 inf{tge1:|D_{1_{{r_i>L}}}Θ(gamma)|_2leε_K
 (gammainF), P_{Θ,D'_nL}(t)([1,∞])ge1/2}
 

is attained up to factor two at some witness. Lemma~lem:scale-tightening
shrinks Θ_n to a near-identity conjugate still retaining
1/4 mass; t_nto∞; and Lemma~lem:scale-many-increments
gives

 u_n/t_nto∞⇒
 P_{Θ_n,R_n}(u_n)todelta_∞.
 

Insert copies of delta for each cycle step. Choose
m_nto∞ with

 t'_n=t_n/m_nto∞,
 P_{delta,t_n}(t'_n)([m_n,∞])to1,
 

which total superlinearity produces. Later insertions keep
length much larger than t_n by entropy; complementary
coordinates stay tight; the terminal kernel is tested at length
div t_nto∞, so (eq:scale-minimal-escape) (with
compact uniformity) sends the final length to infinity.
A tighter feasible witness t'_n contradicts t_nle2t'_n.

Total superlinearity is therefore impossible. There are
C_0,alpha_0>0, T_nto∞ with the scalar law mass
gealpha_0 below C_0T_n, uniform for tin[T_n/2,T_n]
(enlarge C_0 if necessary); a violating sequence plus
Proposition~prop:ray-addition on scale T_n would give
a convolution limit for T_n=t_n+(T_n-t_n), and the mass of a
nonnegative sum below C_0 is at most the mass of either
summand. Lemma~lem:links-invariant-variables makes all
limiting effects scalar; Lemma~lem:scale-original-model
identifies their scalar values, and effect boundedness plus
Hilbert density extend the bound to every fixed xi.


## 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 V_R. The
action is horizontally conformal with horizontal dilation
dil(a,x)>0, obeying

 dil(ab,x)=dil(a,bx)dil(b,x).

All isolated factors here have Q_T>2. Let E HcalH
be the maximal invariant linear domain of
Proposition~prop:scale-pair-estimate, on which
(eq:input-10) holds uniformly over each fixed collection of
normalized mask controls.


For Fin C^∞(B_Ttimes B_T), its ordered evaluation is
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|_{V_T} vanishes on the diagonal.



Fix a compactly contained source chart, a nonnegative source cutoff
chi, and a smooth nonnegative compactly supported θ on the
nilpotent group. Set y=xdelta_ε w and dnu_ε=chi(x)
θ(w),dx,dw. For small ε:
[label=(roman*)]
 Order at least one gives, for each xiinE,
|K_{x,y}(F)xi|_{L^2(dnu_ε;H)}
le C_{F,xi}ε^{Q_S/Q_T}.
 If Q_S=Q_T=Q>2 and order is at least two, the same norm is
o(ε), explicitly bounded by
ε^{Q/2} (2<Q<4), ε^2(1+|logε|) (Q=4),
ε^2 (Q>4).



Proof. 
The measure pullback under w|-> y has density at most
Cε^{-Q_S} supported where d_S(x,y)le Cε; the pair
measure has density asymp d_S(x,y)^{-2Q_S}. Thus

 int<xi,p_delta(x,y)xi>,dnu_ε
 le C_xiε^{Q_S}delta^{-Q_T},

using (eq:input-10). Decompose the target-diagonal complement
into dyadic scale-delta cells cdeltale d_T(w,v)le Cdelta;
such pairs form a bounded-degree graph. Order-m bounds normalized
derivatives by Cdelta^m. Expand in separated Fourier series, and
absorb coefficients into masks: the annular piece is bounded in root
mean square by

 C_{F,xi}delta^mmin{1,ε^{Q_S/2}delta^{-Q_T/2}},

while the entire piece at scale delta_0 has
C_Fdelta_0^m|xi|. Summing over deltageε^{Q_S/Q_T}
for m=1 gives Cε^{Q_S/Q_T} since Q_T>2. For
Q_S=Q_T=Q with m=2, sum at delta_0asympε for the rates
listed.



Every matched isolated pair has Q_S=Q_T=Q. For every smooth
horizontal source field X, every fin C^∞(B_T), and every
xiinE, the distributional derivative
D_X(f)xi=X(x|-> P_x(f)xi) lies in
L^2(B_S,dx;H), linearly in f,xi and in smooth source
function multiples of X.


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
ε^{-1}intpsi_i(w)(P_{xdelta_ε w}(f)-P_x(f))xi,dw
bounded in L^2 by Cε^{Q_S/Q_T-1}; adjoint pairing recovers
the distributional derivative -Xphi-(divX)phi.
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_Sle Q_T; cyclic permutation of the finite isolated type list
gives equality throughout. Weak compactness and the identified limit
give D_X(f)xiin L^2 on charts; a partition of unity globalizes.



Set K=L^2(B_S,dx;K=L^2(B_S,dx;H) and
C=span{x|-> a(x)P_x(g)xi:
ain L^∞(B_S), gin C^∞(B_T), xiinE}. For real
f and real horizontal X, the rule

 T_X(f)(sum_ja_jP_bullet(g_j)xi_j)
 =sum_ja_jP_bullet(g_j)D_X(f)xi_j

defines a densely defined symmetric operator on C. If real
smooth g_j,f_j satisfy sum_jg_j,d_{hor}f_j=0, then

 sum_jP_bullet(g_j)T_X(f_j)=0quadhbox{on }C.



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
=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 ε. Move Delta_{x,y}f by adjunction and
replace P_y(g) by P_x(g) (dominated convergence); the resulting
derivative of the difference quotients is

 < P_x(g)D_X(f)(x)xi,eta>
 =<xi,P_x(g)D_X(f)(x)eta>

for almost every x. Inserting scalar parameter tests gives the
pointwise form; complex weights follow by linearity. Substituting
finitely many angular tests into a formal pairing of u,vinC
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_{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 (eq:scale-horizontal-cancellation).


For a real smooth horizontal target covector field
theta=sum_jg_jd_{hor}f_j, set T_X(theta)=sum_j
P_bullet(g_j)T_X(f_j); equation (eq:scale-horizontal-cancellation)
makes this independent of expression. (Global finitely many target
coordinate differentials span, and their Gram operator is smoothly
invertible on V_T.)


There is a positive self-adjoint measurable field l_x on
H, affiliated to A_1, strongly commuting with
P_x and invariant under complementary source factors, such that

 sigma_a(l_x)=dil(a,x),l_{ax},
 beta_h(l_x)=dil(h^{-1},P_x),l_x,

and l is not identically zero.


Proof. 
Choose finite smooth Parseval horizontal source fields (X_alpha)
and target covectors (theta_r) (invert positive Gram operators on
V_R). Define the finite column
D:C K^K^{n_Sn_T},
Du=(T_{X_alpha}(theta_r)u)_{alpha,r}, closable by
C density; set l=|overlineoverline D|.D|. Parameter and angular
unitary multipliers commute with D on C, extending to
the closed graph and form, so l=int^{ ⊕ }l_x,dx with l_x
strongly commuting with P_x. Unitary invariance transfers
distributionally to D_X(f) and then closes, giving affiliation and
complementary invariance.

For covariance, note that Parseval systems embed V_S,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
dil(a,x) times a horizontal isometry and target
covectors by dil(h^{-1},P_x) times an isometry, so
(eq:scale-derivative-modulus-covariance) holds on 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~lem:scale-equal-dimension
would make P_x constant, contradiction.


[Height at an isolated type]
For matched isolated pairs Q_S=Q_T=Q. There are positive injective
self-adjoint b_x, affiliated to A_1, strongly commuting
with P_x and invariant under complementary source factors, with

 sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},
 beta_h(b_x)=dil(h^{-1},P_x)^{-Q}b_x.

The construction passes to normal copies used in amplified models.


Proof. 
Make l_x everywhere covariant: at x_0in B_S with parabolic
stabilizer J, pull back along g|-> gx_0 and set
F(g)=dil(g,x_0)sigma_g^{-1}(l_{gx_0}). Haar
disintegration over S/J and
(eq:scale-derivative-modulus-covariance) give F(ag)=F(g)
almost everywhere for each fixed ain S; Fubini over
(a,g)|->(ag,g) reduces F to an essential value L_0 (test
via bounded resolvents and countably many matrix coefficients). For
kin J, F(gk)=dil(k,x_0)sigma_k^{-1}F(g), so
sigma_kL_0=dil(k,x_0)L_0, defining
l_{gx_0}=dil(g,x_0)^{-1}sigma_gL_0 everywhere with
the same covariance. Target law follows by commuting beta_h past
sigma_g; the essential value L_0 satisfies the fixed-x_0 law
and transports.

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_0inN. 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 (Ssec:setup) makes translates converge to
tau(e_0)1, so e_0in{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 A_1; taking inverse powers in the covariance laws and
using strong commutation with the dilation gives
(eq:scale-isolated-height-covariance).

For amplification: on a normal copy H ⊗ L with
orthonormal basis (e_n), an element Xi=sum_nxi_n ⊗  e_n of
the maximal amplified estimate domain has every coordinate
xi_ninE (each coordinate energy is bounded by the
total); conversely finite sums of xi_ninE 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 C{e_n} the
column is exactly D ⊗ 1, graph-densely. So
l^{amp}=l ⊗ 1 and b^{amp}=b ⊗ 1; unitary
identification with a normal copy preserves the identities, without
any mixing on an enlarged commutant.


The isolated branch now supplies its own-label height calculus:
b_x is positive and injective, strongly commutes with P_x, and
obeys (eq:scale-isolated-height-covariance) in each required
normal copy; Section~sec:heights transports its imaginary powers
through the Fourier links to establish centered length laws. The
common graph domain for comparing heights at different labels is built
from the Haar-height isometry of Section~sec:locality.

# Upper speeds at real 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~sec:finite; the isolated-axis differentiation of
Section~sec:scale is not used here.

## The estimate and the order of limits

Fix one measured coordinate of the exact graph relation
(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.

[The real-panel upper estimate]
There are positive constants d_R, determined by each real rank-one
panel type R and unchanged at the opposite type, with the following
property. Complete the mixed path by split-chamber translations along
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

 d_Tell(g)le d_St+O_{tight}(1).
 

Here tight upper error means the event where
d_Tell(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
(eq:input-11) is the resulting shifted gap. Gap signs are the
eventual orientations on the chosen filters.


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~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 (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~prop:panels-upper. These assertions use
only the exact precomposition and angular statements of
Sections~sec:links and~sec:transfer; neither the height
calibration nor the classicality conclusion proved later is used.

The structural classification of Section~sec:setup leaves the
orthogonal, unitary, and symplectic groups as the real rank-one panel
groups. The exceptional rank-one group F_{4(-20)} cannot occur as the
noncompact simple factor of a panel Levi in a higher-rank component:
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.

## 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(dpi(H_R)-t),
 h(V)=sum_{t>0}dim V_t, 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.

[Optimal panel modules]
Put k=n+1 and let E denote the standard complex module for the
group in the relevant line. Among the nontrivial irreducible complex
Lie-algebra modules, the maximum c_R=max_Vdim V/h(V) and all
modules attaining it are
[
{ccl}
 R&c_R&maximizing modules 
 SO(n,1), nge2 & k & E
 SU(n,1), n>2 & k & E,E^*
 SU(2,1) & 3 & E,E^*,Sym^2E,Sym^2E^*
 Sp(n,1), n>2 & k & E
 Sp(2,1) & 7/2 & ∧nolimits^3_0E

]
where ∧^3_0E is the primitive exterior cube (kernel of the
symplectic contraction ∧^3Eto E). There is
ε_R>0 such that every nontrivial irreducible outside the
table satisfies dim V/h(V)le c_R-ε_R. Every displayed
module has integral split weights and a nonzero split-weight-zero
space, and admits a preserved nondegenerate Hermitian form q,
positive definite on V_0, whose matrix in a compact-form Hilbert norm
obeys Q=Q^*=Q^{-1} and dpi(H_R)^*=dpi(H_R). The form pairs
V_t perfectly with V_{-t} and pairs it trivially with all other
weight spaces. With V_{>0}=⊕_{t>0}V_t and
V_{ge0}=⊕_{tge0}V_t,
[
 V_{>0}^{perp_q}=V_{ge0},
 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=2,&R=SU(2,1) or Sp(2,1),1,&otherwise,
]
and the constants apply equally to the opposite orientation.


## 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
ineuivalent modules) replace V by the direct sum of its diagram
conjugates; every summand has the same optimal ratio, so still

 h(V)=n_0/c_T.
 

Take the compact-form Hilbert norms and preserved Hermitian forms of
Lemma~lem:panels-optimal-modules, with form symmetry Q, and in
compact projectivity frames choose local or Borel lifts pi_T(g) of
target projectivities such that

 pi_T(g)^*Qpi_T(g)=Q,
 |pi_T(g)|=exp(d_Tell(g)).
 

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.