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\hypersetup{pdftitle={Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras},pdfauthor={OpenAI}}
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\title{Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the quadratic strong-operator paving conjecture of Popa and Vaes.
For every $0<\eps<1$, every self-adjoint element of a von Neumann algebra
admits strong-operator paving over each maximal abelian subalgebra with at most
$5\times10^8\eps^{-2}$ projections. The bound is uniform over representations
and requires no separability or conditional-expectation assumption.
\end{abstract}

\section{Introduction}\label{sec:introduction}

Throughout, $M\subseteq\mathcal B(H)$ is a von Neumann algebra on a complex
Hilbert space, and $A\subseteq M$ is a maximal abelian unital $*$-subalgebra,
or \emph{masa}. A finite partition $P=(p_1,\ldots,p_r)$ in $A$ consists of
mutually orthogonal projections with sum $1$. Its \emph{pinching} is
\[
 S_P(x)=\sum_{i=1}^r p_i x p_i,\qquad x\in M.
\]
Paving asks how small this can be made after subtracting a diagonal element of
$A$. Norm paving, which requires $S_P(x)-a$ itself to be small, fails for
general masas \cite[Theorem~3.3]{PV15}. The strong-operator formulation instead
allows a compression whose complement is arbitrarily small in the strong
operator topology.

\begin{definition}[Strong-operator paving]\label{def:so-paving}
Let $x=x^*\in M$, $\eps>0$, and $r\ge1$. We say $x$ is \emph{$(\eps,r)$
so-pavable over $A$} if for every finite $F\subset H$ and every $\delta>0$
there are a partition $P=(p_1,\ldots,p_r)$ in $A$, an element $a=a^*\in A$
with $\|a\|\le\|x\|$, and a projection $q\in M$ such that
\[
 \|(1-q)\xi\|<\delta\quad(\xi\in F),
 \qquad \|q(S_P(x)-a)q\|\le\eps\|x\|.
\]
The integer $r$ is fixed before $F$ and $\delta$ are chosen.
\end{definition}

Thus the whole game is to produce, with a number of projections depending only
on $\eps$, a compression that captures all but an arbitrarily small piece of
the operator while making the visible piece small. The main result is that a
quadratic number of projections always suffices.

\begin{theorem}\label{thm:main}
For every $0<\eps<1$ set
\[
 r_\eps=\left\lceil400000000\eps^{-2}\right\rceil
              +\left\lceil4/\eps\right\rceil+1
          \le500000000\eps^{-2}.
\]
For every masa $A\subseteq M\subseteq\mathcal B(H)$ and every $x=x^*\in M$,
the element $x$ is $(\eps,r_\eps)$ so-pavable over $A$. The number of
projections is thus fixed before the inclusion, the operator, and the strong
neighborhood. No separability or conditional-expectation assumption is imposed.
\end{theorem}

Theorem~\ref{thm:main} proves the quadratic so-paving conjecture of Popa and
Vaes \cite[Conjecture~2.8(2)]{PV15}. Their definition of so-paving uses
arbitrary strong neighborhoods of zero; Definition~\ref{def:so-paving} gives the
equivalent finite-vector formulation \cite[Definition~2.2]{PV15}. For
self-adjoint $x$, allowing a general diagonal element is the same as allowing a
self-adjoint one, since taking the real part preserves the norm bound and the
compression estimate.

For countably decomposable expected masas, the uniform bound passes to the
associated ultrapower.

\begin{corollary}[Quadratic norm paving in Ocneanu ultrapowers]
\label{cor:ultrapower-paving}
Let $A\subseteq M$ be a masa with $A$ countably decomposable and the range of a
normal conditional expectation from $M$. Let $\omega$ be a free ultrafilter on
$\mathbb N$ and $A^\omega\subseteq M^\omega$ the inclusion in the Ocneanu
ultrapower. For every $0<\eps<1$ and every $x=x^*\in M^\omega$ there are a
partition $p_1,\ldots,p_{r_\eps}\in A^\omega$ and an $a=a^*\in A^\omega$ with
\[
 \|a\|\le\|x\|,
 \qquad
 \left\|\sum_{i=1}^{r_\eps}p_i x p_i-a\right\|\le\eps\|x\|.
\]
\end{corollary}

\begin{proof}
Theorem~\ref{thm:main} makes every self-adjoint element of $M$
$(\eps,r_\eps)$ so-pavable. The forward transfer in \cite[Theorem~2.7(3)]{PV15}
gives the stated norm paving in $M^\omega$ with the same error and count.
Taking the real part of the diagonal element preserves both estimates.
\end{proof}

\subsection{History and optimality}

The classical Kadison--Singer paving problem concerns the diagonal masa in
$\mathcal B(\ell^2\mathbb N)$, resolved by Marcus, Spielman, and Srivastava with
mixed characteristic polynomials \cite{MSS15}. For general masas norm paving is
much more restrictive: when $M$ has separable predual it holds exactly when $M$
is type~I and $A$ is the range of a normal conditional expectation
\cite[Theorem~3.3]{PV15}. Strong-operator paving keeps a uniform count while
allowing a compression approaching the identity strongly.

Popa's $\mathrm{II}_1$-factor approach gave paving for singular-masa
ultraproducts \cite{Popa14}. Popa and Vaes obtained so-paving bounds of order
$\eps^{-4}$ for arbitrary masas in type~I algebras with separable predual, for
Cartan masas in amenable algebras, and for the Cartan inclusions associated
with essentially free ergodic probability-measure-preserving profinite actions
\cite[Proposition~3.2 and Theorem~4.1]{PV15}. For tracial ultraproducts of
singular masas in finite algebras they obtained a common norm-paving partition
of order $\eps^{-2}$ for each finite family \cite[Theorem~5.1]{PV15}, and later
the optimal error for singular-masa ultraproducts \cite[Theorem~4.1]{PV16}.
They explicitly noted that the Cartan and singular arguments had to be united
\cite[Remark~2.9(ii)]{PV15}. On the finite-dimensional side, Ravichandran and
Srivastava obtained asymptotically optimal multipaving bounds from mixed
determinantal polynomials \cite{RS17}; these are the finite input used here.

A naive combination shows the quantitative difficulty. For an intermediate
inclusion $A\subseteq N\subseteq M$ with a faithful normal tracial state, Popa
and Vaes assume $L^2(M\ominus N)$ has no nonzero $A$--$N$ subbimodule finitely
generated on the right; their bound uses at most $400\eps^{-2}$ times the
so-paving count for $A\subseteq N$ at error $\eps/2$ \cite[Remark~5.3]{PV15}.
Inserting their quartic Cartan bound gives order $\eps^{-6}$. The present
argument handles the normalizer algebra through a nonsingular measured
equivalence relation and uses a single partition for both terms in the
normalizer decomposition of one operator, which retains quadratic order. (The
finite-family obstruction in \cite[Remark~5.2]{PV15} concerns the stronger
requirement of a common partition for arbitrarily many separately prescribed
operators.)

The exponent two is already forced by the $L^2$ obstruction in a $\mathrm{II}_1$
factor from \cite[Proposition~5.4]{PV15}. Fix $1\le n<\eps^{-2}$, choose
$\eps<\eta<n^{-1/2}$, and use the centered self-adjoint unitary $x$ of that
proposition at threshold $\eta$. Write $\tau$ for the normalized trace and
$\|b\|_2=\tau(b^*b)^{1/2}$. For any partition $P$ with $n$ pieces and any
$a\in A$, trace orthogonality gives
\[
 \|S_P(x)-a\|_2^2=\|S_P(x)\|_2^2+\|a\|_2^2.
\]
If $\|a\|\le1$ set $z=S_P(x)-a$, so $\|z\|\le2$. Whenever $\|qzq\|\le\eps$, the
two discarded corners give $\|S_P(x)\|_2\le\|z\|_2\le\eps+4\tau(1-q)^{1/2}$. In
the trace representation a strong neighborhood can force
$\tau(1-q)<((\eta-\eps)/4)^2$, which contradicts the lower bound
$\|S_P(x)\|_2>\eta$ valid for every such $P$. So the obstruction survives the
varying diagonals and compressions allowed here, and the exponent in
Theorem~\ref{thm:main} is optimal (the constant is not optimized).

Theorem~\ref{thm:main} controls a compression of the pinching of the original
$x$. A separate expectation-free approximation-paving theorem instead produces
self-adjoint approximants $y$ close to $x$ strongly, with $\|y\|\le3\|x\|$,
and norm-paves them with error measured against $\|y\|$
\cite[Theorem~1.1]{ApproxCompanion26}; that conclusion has a finite
$\eps$-dependent count but no quadratic rate, and the two proofs are
independent.

\subsection{Reader's guide to the proof}

We give a short map before the formal work; each step is a tool that is later
used exactly once or twice, and the reader may treat this as a table of
contents with reasons.

\begin{enumerate}
\item \emph{Reduce to one faithful central state} (Section~\ref{sec:reductions}).
The masa need not be the range of a normal conditional expectation. Split $M$
into a corner where a faithful $A$-central state exists and a corner where no
nonzero $A$-central functional exists. The first corner is where all the
measured-space work happens; the second is handled by a symmetric Haagerup
$L^2$ estimate and two randomizations with $O(\eps^{-1})$ colors. Disjoint
palettes on the corners make the pinching kill cross terms and the counts add.
This is the only place $L^p$ theory enters.

\item \emph{Atomless kernels off the normalizer algebra}
(Section~\ref{sec:kernels}). Let $N$ be generated by the groupoid normalizers
of $A$; then $A\subseteq N$ is Cartan and represented by a nonsingular measured
equivalence relation. Decompose $x-E_Ax=(E_Nx-E_Ax)+(x-E_Nx)$. For the second
term $y\in\ker E_N$, the positive map $f\mapsto E_A(y^*fy)$ has an atomless
kernel. This atomlessness is what later makes collision terms vanish.

\item \emph{Finite polynomial estimate} (Section~\ref{sec:polynomials}). The
finite engine is the mixed determinantal bound of \cite[Theorem~25]{RS17}. A
partial coloring of a Hermitian matrix defines a real-rooted expected
polynomial; the sum of its positive roots after a negative diagonal shift is a
\emph{cost}. The upper mixed bound makes the all-unassigned cost zero; the
two-sign lower bound makes excessive norm on a fully assigned block force
positive cost; and a single-site assignment inequality bounds the cost increase
of fixing one color. Rooted spectral measures spread the cost over vertices.

\item \emph{Measured graphs and the integral-change identity}
(Section~\ref{sec:invariant}). For an invariant measured graph, the total cost
changes by the integral of single-site deletion increments, in an ordered
sequence of comparison environments. A separate transform comparison bounds the
effect of deleting far-away large weights on any fixed weight band, uniformly
in the largest weight. This is what lets weight cutoffs be removed later.

\item \emph{Coloring theorem} (Section~\ref{sec:coloring}). On the relative
Bernoulli extension of a \emph{nonsingular} relation, an invariant lift with an
extra height coordinate makes mass transport available. Joint cutoff limits
produce homogeneous costs, and random assignment times give a coloring of small
total cost in which any prescribed ball is fully assigned with high probability,
with uniform conditional color marginals. Completion of the remaining sites
preserves the estimate on those balls.

\item \emph{Transfer} (Section~\ref{sec:transfer}). Put $M$ and the Bernoulli
extension of $N$ in their amalgamated free product over $N$. Uniform marginals
give a free-compression bound on $\ker E_N$, while the same color projections
give the Cartan estimate. A finite-symbol evaluation plus a shortest-pair
argument in cyclic words transfers each fixed mixed scalar moment from the model
to actual projections of $A$; the atomless kernels kill the collision terms.

\item \emph{Completion} (Section~\ref{sec:completion}). Turn the local Cartan
estimate into bounded approximants by clipping, run continuous calculus for the
resulting varying sequences, and transfer one sufficiently high fixed moment. A
spectral cut produces the projection $q$. All approximation parameters are
chosen after the number of colors, and the corner assembly gives
Theorem~\ref{thm:main}.
\end{enumerate}

\subsection{Conventions}
\label{sec:conventions}

We allow zero projections in a partition and write $1$ also for the identity of
a corner regarded as an algebra. Except when restoring scale at the very end,
operators to be paved are self-adjoint contractions. Maximal abelianness makes
$A$ weakly closed with $A'\cap M=A$: the weak closure is abelian, and any
self-adjoint relative-commutant element could otherwise be adjoined. Corners by
projections in $A$ are again masas in the corresponding corners of $M$.

For a normal positive functional $\varphi$ write
$\|z\|_\varphi=\varphi(z^*z)^{1/2}$. For a faithful normal state, the
centralizer is the fixed algebra of its modular group. Measurable statements
are understood modulo null sets; in the presence of a countable nonsingular
relation, a null exceptional set is enlarged to its null saturation.

\section{Reduction to a faithful central state}\label{sec:reductions}

We first remove the obstruction to using a normal conditional expectation: the
original masa need not be the range of one. A normal positive functional $\rho$
on $M$ is \emph{$A$-central} if $\rho(bz)=\rho(zb)$ for $b\in A$, $z\in M$. The
idea is to split $M$ by the support of all such functionals.

Let $p\in A$ be the supremum of the supports of the $A$-central normal positive
functionals; it lies in $A$ because conjugation by any unitary of $A$ preserves
such a functional and hence its support. The supports form a directed family
(the support of a sum is the join), so $p$ is the join. The corner
$eMe$ for $e\le p$ carries a faithful normal state central over $eAe$, and on
$(1-p)M(1-p)$ no nonzero central functional survives. Given finite vector tests
we pick $e\le p$ so that $p-e$ acts arbitrarily small on those vectors and
discard the corner $p-e$. Because $p$ and $e$ need not be central, we use
disjoint palettes on the two retained corners so that a pinching removes all
cross-corner terms and the projection counts add. This single assembly reduces
Theorem~\ref{thm:main} to the case of a faithful $A$-central state, proved in
Proposition~\ref{prop:expected}. All steps keep a projection count independent of
the prescribed strong neighborhood.

\subsection{The corner without central normal functionals}

For this corner we use Haagerup's $L^p$ spaces for an arbitrary von Neumann
algebra, built from a normal semifinite faithful weight; see
\cite[Section~1]{Haa79}. A normal positive functional $\varphi$ has a density
$h_\varphi\in L^1(M)_+$ under the isometric identification $L^1(M)=M_*$, paired
by $\Tr$. The space $L^2(M)$ with its left and right multiplications and its
positive cone is a standard form of $M$, and the cone vector of $\varphi$ is
$h_\varphi^{1/2}$. We use H\"older's inequality, cyclicity of the
$L^p$--$L^{p'}$ pairing, and polar decomposition, whose partial isometry lies in
$M$.

The positive-cone estimate below is Haagerup's standard-form inequality
\cite[Lemma~2.10]{Haa75}; see also \cite[Lemma~4.1]{PS70} and
\cite[Theorem~4(8) and Section~8, Remark~2]{Ara74}. We include a proof so that
no faithfulness of the two functionals is hidden. For normal positive
$\varphi,\psi$,
\begin{equation}\label{eq:positive-cone-estimate}
 \|h_\varphi^{1/2}-h_\psi^{1/2}\|_2^2\le\|\varphi-\psi\|.
\end{equation}
Indeed, put $\xi_1=h_\varphi^{1/2}$, $\xi_2=h_\psi^{1/2}$, $D=\xi_1-\xi_2$, and
let $s\in M$ be the sign of $D$, so $\|s\|\le1$ and $sD=Ds=|D|$. Writing
$D=D_+-D_-$, cyclicity gives
\begin{align*}
 \Tr\bigl(s(h_\varphi-h_\psi)\bigr)
 &=\Tr\bigl(s(D\xi_1+\xi_2D)\bigr)
   =\Tr\bigl(|D|(\xi_1+\xi_2)\bigr)\\
 &=\|D\|_2^2+2\Tr(D_-\xi_1)+2\Tr(D_+\xi_2)
 \ge\|D\|_2^2.
\end{align*}
The last two pairings are nonnegative by self-duality of the positive cone, and
the first expression is at most $\|\varphi-\psi\|$.

\begin{lemma}\label{lem:no-central}
Let $A\subseteq M$ be a masa such that $M$ has no nonzero $A$-central normal
positive functional. Let $x=x^*\in M$ with $\|x\|\le1$ and $0<\eps<1$, and put
$r_{\rm p}=\lceil4/\eps\rceil$. For every normal state $\varphi$ on $M$ and
every $\rho>0$ there are a partition $p_1,\ldots,p_{r_{\rm p}}$ in $A$, an
$a=a^*\in A$ with $\|a\|\le1$, and a projection $q\in M$ such that
\[
 \varphi(1-q)<\rho,
 \qquad
 \Bigl\|q\Bigl(\sum_{i=1}^{r_{\rm p}}p_i x p_i-a\Bigr)q\Bigr\|\le\eps.
\]
\end{lemma}

\begin{proof}
Write $\xi=h_\varphi^{1/2}$ and let $T(z)=h_\varphi^{1/4}zh_\varphi^{1/4}\colon
M\to L^2(M)$. It is bounded and takes weak-star to weak convergence: testing
against $\zeta\in L^2(M)$ uses the $L^1$ element
$h_\varphi^{1/4}\zeta^*h_\varphi^{1/4}$. For a finite partition $P$ in $A$ the
pinching $S_P=\sum_{e\in P}e(\cdot)e$ acts on $L^2(M)$ as an orthogonal
projection. These projections decrease under refinement to the projection onto
$A$-central vectors. That projection is zero: a nonzero vector in its range
would give a nonzero $A$-central normal positive functional (move a unitary of
$A$ from left to right, where it commutes with the left action of $M$). Hence
$\|S_P(\xi)\|_2\to0$ as $P$ refines.

For $z\in M$, cyclicity and H\"older give
\begin{equation}\label{eq:symmetric-block-sum}
 \sum_{e,f\in P}\|T(ezf)\|_2^2
 =\Tr\bigl(z^*S_P(\xi)zS_P(\xi)\bigr)\le\|z\|^2\|S_P(\xi)\|_2^2.
\end{equation}
In particular $\|T(eze)\|_2^2\le\|e\xi e\|_2^2$ when $\|z\|\le1$.

Fix $\kappa>0$ and a coarse $P_0$ with $\|S_{P_0}(\xi)\|_2<\kappa$. As $P$
refines $P_0$, $S_P(x)$ has a weak-star cluster point $b\in A$ (it commutes
with every partition of $A$). Since the weak and norm closures of a convex set
in a Hilbert space agree, there are refinements $P^1,\ldots,P^d$ of $P_0$ and
probabilities $t_1,\ldots,t_d$ with
$\|\sum_j t_jT(S_{P^j}(x))-T(b)\|_2<\kappa$. Within each $e\in P_0$ pick
independently a refinement $P^j$ with this distribution; the resulting partition
$R$ refines $P_0$. By independence and \eqref{eq:symmetric-block-sum} the
variance of $T(S_R(x))$ is $\sum_{e\in P_0}\|e\xi e\|_2^2=\|S_{P_0}(\xi)\|_2^2<
\kappa^2$, so one realization has $\|T(S_R(x)-b)\|_2<\sqrt2\kappa$.

Now color the pieces of $R$ independently and uniformly with $r_{\rm p}$ colors
and let $S_{\rm col}$ be the color pinching. Its mean is
$\mathbb E S_{\rm col}(x)=r_{\rm p}^{-1}x+(1-r_{\rm p}^{-1})S_R(x)$. Equal-color
indicators for distinct pairs of pieces are uncorrelated even when pairs share a
piece; with $\|u+v\|_2^2\le2\|u\|_2^2+2\|v\|_2^2$ and
\eqref{eq:symmetric-block-sum} the variance after applying $T$ is at most
$2\|S_R(\xi)\|_2^2<2\kappa^2$. Thus some coloring makes
\[
 y=S_{\rm col}(x)-\bigl(1-r_{\rm p}^{-1}\bigr)b-r_{\rm p}^{-1}x
\]
satisfy $\|T(y)\|_2<2\sqrt2\kappa$, with $\|y\|\le2$. Since $\kappa$ is
arbitrary the symmetric seminorm can be made as small as we like with the count
already fixed.

It remains to turn a small symmetric seminorm into a large compression. Set
$t=\|T(y)\|_2$ and $\psi(z)=\varphi(y^*zy)$. Since small $t$ need not control
$\|y\|_\varphi$, we first find $q_0$ with both $\varphi(1-q_0)$ and
$\psi(q_0)$ small. Polar decomposition gives $h_\psi=yh_\varphi y^*$ and
$h_\psi^{1/2}=y\xi u^*$ for a partial isometry $u\in M$, so
\[
 0\le\Tr(\xi h_\psi^{1/2})\le\|\xi y\xi\|_1
 =\|h_\varphi^{1/4}T(y)h_\varphi^{1/4}\|_1\le t,
\]
the last step using $\|h_\varphi^{1/4}\|_4=1$. Let $q_0$ support the positive
part in the Jordan decomposition of $\varphi-\psi$. Then
\eqref{eq:positive-cone-estimate} gives
\begin{equation}\label{eq:jordan-overlap}
 \varphi(1-q_0)+\psi(q_0)
 =\tfrac12\bigl(\varphi(1)+\psi(1)-\|\varphi-\psi\|\bigr)\le t,
\end{equation}
and
$\|q_0yq_0\xi\|_2\le\|q_0y\xi\|_2+\|q_0y(1-q_0)\xi\|_2\le\psi(q_0)^{1/2}+
2\varphi(1-q_0)^{1/2}\le3\sqrt t$.
Inside $q_0Mq_0$ let $q$ be the spectral projection of $q_0yq_0$ for
$[-\eps/2,\eps/2]$. The spectral Markov inequality gives
\begin{equation}\label{eq:symmetric-large-projection}
 \varphi(1-q)\le\varphi(1-q_0)+\frac4{\eps^2}\varphi((q_0yq_0)^2)
 \le\Bigl(1+\frac{36}{\eps^2}\Bigr)t.
\end{equation}
Taking $\kappa$ small makes this less than $\rho$. With
$a=(1-r_{\rm p}^{-1})b$ we have $\|a\|\le1$ and
$\|q(S_{\rm col}(x)-a)q\|\le\eps/2+r_{\rm p}^{-1}\le3\eps/4$.
\end{proof}

\subsection{Assembling the corners}

\begin{proposition}\label{prop:state-reduction}
Fix $0<\eps<1$ and an integer $R\ge1$. Suppose that for every masa $A\subseteq M$
admitting a faithful $A$-central normal state $\varphi$, every self-adjoint
contraction $x\in M$, and every $\rho>0$, there exist a partition of $1$ into
$R$ projections of $A$, a self-adjoint $a\in A$ of norm at most one, and a
projection $q\in M$ with $\varphi(1-q)<\rho$ and
\[
 \Bigl\|q\Bigl(\sum_{i=1}^{R}p_i x p_i-a\Bigr)q\Bigr\|\le\eps.
\]
Then every masa in every represented von Neumann algebra has the strong-operator
paving property of Theorem~\ref{thm:main} at error $\eps$, with
$r=R+\lceil4/\eps\rceil+1$. This number is chosen before the finite vector set
and the strong-neighborhood tolerance.
\end{proposition}

\begin{proof}
Let $p$ be the supremum of the supports of the $A$-central normal positive
functionals on $M$; as noted it lies in $A$, and the supports form a directed
family. The corner $(1-p)M(1-p)$ has no nonzero $A(1-p)$-central normal
functional, since compression would extend one to $M$.

Given finite $F\subset H$ and $\delta>0$, set
$\chi(z)=\sum_{\xi\in F}\langle z\xi,\xi\rangle$. By normality there is a
support $e\le p$ in the family with $\chi(p-e)$ arbitrarily small. On the corner
$eMe$ the corresponding functional restricts, after normalization, to a
faithful normal state central on $Ae$. (Small $\omega$-mass on projections gives
uniformly small mass for any fixed normal positive functional: a contrary
sequence in the positive unit ball has a weak-star limit of zero $\omega$-mass
but positive mass, contradicting faithfulness.) Apply the assumed result on
$eMe$, using this observation to keep the lost $\chi$-mass small. On
$(1-p)M(1-p)$ apply Lemma~\ref{lem:no-central} to the normalized restriction of
$\chi$ when it is nonzero, and take $q=0$ otherwise.

Use disjoint palettes on the two corners and the projection $p-e$ as a last
piece, taking corner sums of the two diagonal terms and the two large
projections (zero on $p-e$). Each pinching piece is supported in one corner, so
all off-corner terms of $x$ vanish, the compressed norm is at most $\eps$, and
the diagonal has norm at most one. We arrange $\chi(1-q)<\delta^2$, so
$\|(1-q)\xi\|^2\le\chi(1-q)<\delta^2$ for $\xi\in F$. Zero projections pad when
needed. The total count is $r$, independent of $F$ and $\delta$. Scaling handles
a general nonzero self-adjoint $x$; $x=0$ is immediate.
\end{proof}

\subsection{State norm and separable reduction}

Now fix a faithful $A$-central normal state $\varphi$. Write
$\|z\|_\varphi=\varphi(z^*z)^{1/2}$ and let
$M_\varphi=\{b\in M:\varphi(bz)=\varphi(zb)\ \forall z\}$ be the centralizer,
the fixed algebra of the modular group; in particular $A\subseteq M_\varphi$. We
first approximate by diagonal pinchings in the state norm, then build a
separable inclusion containing the operator.

\begin{lemma}\label{lem:pinching}
Let $A\subseteq M$ be a masa and $\varphi$ a faithful $A$-central normal state.
There is a faithful normal $\varphi$-preserving conditional expectation
$E_A\colon M\to A$. On bounded sets, convergence to zero in $\|\cdot\|_\varphi$
is equivalent to strong convergence in the faithful GNS representation. For
every $z\in M$,
\begin{equation}\label{eq:pinching-limit}
 \|S_P(z)-E_A(z)\|_\varphi\longrightarrow0
\end{equation}
as finite partitions $P$ of $A$ refine; the same holds along any increasing
sequence of finite partitions generating $A$.
\end{lemma}

\begin{proof}
Takesaki's theorem \cite{Tak72} applies because the modular group fixes $A$
pointwise. Let $\Omega$ be the GNS vector; it is separating, hence cyclic for
$M'$. If a bounded net $z_j$ has $z_j\Omega\to0$, then
$z_jc'\Omega=c'z_j\Omega\to0$ for every $c'\in M'$, so $z_j\to0$ strongly; the
converse is testing on $\Omega$.

Since the partition projections lie in the centralizer, $S_P$ induces an
orthogonal projection on the GNS space; it is the state-preserving expectation
onto the commutant in $M$ of the projections of $P$. These projections decrease
under refinement. On $z\Omega$ let the limit vector be $\eta$. A bounded
weak-star cluster point $b$ of $S_P(z)$ commutes with every partition of $A$,
hence lies in $A$; its state vector is $\eta$, and $E_AS_P(z)=E_A(z)$ forces
$b=E_A(z)$. If an increasing sequence generates $A$, its limit commutes with
the generators and the same argument applies.
\end{proof}

\begin{lemma}\label{lem:separable-reduction}
Let $A\subseteq M$ be a masa with faithful $A$-central normal state $\varphi$,
and $x=x^*\in M$. There are unital von Neumann subalgebras $A_0\subseteq A$ and
$M_0\subseteq M$ with $x\in M_0$, $M_0$ of separable predual, $A_0$ a masa in
$M_0$, and $E_A(M_0)\subseteq A_0$. The restricted $\varphi$ is faithful,
normal, and $A_0$-central. Thus a uniform faithful-state paving result in the
separable-predual case implies the general faithful-state case.
\end{lemma}

\begin{proof}
Start with $D_1=W^*(1,x)$. Inductively choose a countable
$\|\cdot\|_\varphi$-dense subset $\mathcal D_j$ of the unit ball of $D_j$ (every
countably generated $D_j$ has one, by Kaplansky density applied to its separable
generator C*-algebra; its faithful normal GNS representation is separable, so
$D_j$ has separable predual). Let $D_{j+1}$ be generated by $D_j$, all
$E_A(z)$ for $z\in\mathcal D_j$, and the projections of finite partitions
$P(z,k)\subset A$ with $\|S_{P(z,k)}(z)-E_A(z)\|_\varphi<1/k$ for
$z\in\mathcal D_j$, $k\ge1$, which exist by Lemma~\ref{lem:pinching}. Only
countably many generators are added.

Set $M_0=W^*(\bigcup_jD_j)$ and let $A_0$ be generated unitally by the adjoined
elements of $A$. Bounded Kaplansky approximation, then approximation from the
$\mathcal D_j$, shows the union is state-norm dense in the unit ball of $M_0$.
Since $E_A$ is state-norm contractive, every $E_A(z)$ for $z\in M_0$ is a
bounded state-norm limit from $A_0$, so $E_A(M_0)\subseteq A_0$ by
Lemma~\ref{lem:pinching} and strong closedness.

If a unit-ball $z\in M_0$ commutes with $A_0$, each selected partition has
$S_P(z)=z$, and for $z'\in\mathcal D_j$ and $P=P(z',k)$,
$\|z-E_A(z)\|_\varphi\le2\|z-z'\|_\varphi+\|S_P(z')-E_A(z')\|_\varphi$, which
can be made small; faithfulness gives $z=E_A(z)\in A_0$. Thus $A_0$ is maximal
abelian in $M_0$. Also $M_0$ is countably generated with separable predual by
the same GNS argument. A paving in $M_0$ has the same operator-norm bound and
lost $\varphi$-mass in $M$, giving the estimate needed by
Proposition~\ref{prop:state-reduction}.
\end{proof}

\section{The normalizing algebra and atomless kernels}\label{sec:kernels}

For the structural part we assume $M$ has separable predual, $A\subseteq M$ is
a masa, and $\varphi$ is a faithful $A$-central normal state;
Lemma~\ref{lem:separable-reduction} permits these assumptions. We separate the
algebra generated by normalizers from its orthogonal complement under a
state-preserving expectation. The former has a measured-relation description;
on the latter we prove the maps $f\mapsto E_A(y^*fy)$ have atomless kernels,
which later makes fine-partition collision terms vanish.

\subsection{Normalizers and the measured relation}

A \emph{groupoid normalizer} of $A$ is a partial isometry $v\in M$ with
$v^*v,vv^*\in A$ and $vAv^*=Avv^*$, $v^*Av=Av^*v$. Let $N$ be the von Neumann
algebra they generate. For such $v$, $\sigma_t^\varphi(v)$ induces the same
corner isomorphism as $v$ (the modular group fixes $A$ pointwise), so
$v^*\sigma_t^\varphi(v)\in Av^*v$ is unitary. These unitaries form a strongly
continuous one-parameter group (the cocycle identity becomes the group identity
because the modular group fixes $Av^*v$), hence there is a positive nonsingular
$h_v$ affiliated with $Av^*v$ with
\begin{equation}\label{eq:normalizer-modular}
 \sigma_t^\varphi(v)=v h_v^{it}\qquad(t\in\mathbb R).
\end{equation}
So $N$ is $\sigma^\varphi$-invariant, and Takesaki's theorem \cite{Tak72} gives
a faithful normal $\varphi$-preserving conditional expectation
$E_N\colon M\to N$; write $M^\circ=\ker E_N$. The cuts
$v1_{[-k,k]}(\log h_v)$ are entire analytic for $\sigma^\varphi$
(\eqref{eq:normalizer-modular} extends to $v h_v^{iz}1_{[-k,k]}(\log h_v)$ for
$z\in\mathbb C$) and converge strongly-* to $v$.

Identify $A=L^\infty(X,\mu)$ on a standard probability space with $\varphi|_A$
integration. A groupoid normalizer induces a measure-class isomorphism between
measurable subsets of $X$; on its fixed points it lies in $A$. Partition its
remaining domain into countably many measurable pieces each disjoint from its
image (use a countable separating family, then disjointize). On one such piece
the cut normalizer $v'$ has orthogonal initial and final projections, and
$v'+{v'}^*+(1-{v'}^*v'-v'{v'}^*)$ is a unitary normalizer; recovering $v'$ and
summing strongly shows full unitary normalizers also generate $N$. Thus
$A\subseteq N$ is Cartan: $A$ is maximal abelian, regular, and the range of the
faithful normal expectation $E_A|_N$.

The Feldman--Moore theorem \cite[Theorem~1]{FM77} realizes this pair as the
algebra of a countable nonsingular measured equivalence relation $\mathcal R$ on
$(X,\mu)$, possibly twisted by a scalar $2$-cocycle; separability holds in the
faithful GNS representation. We use the right counting representation
\begin{equation}\label{eq:right-counting-space}
 L^2(\mathcal R,\nu_{\rm r})
 =\int_X^\oplus\ell^2([v]_{\mathcal R})\,d\mu(v),
 \qquad
 \int f\,d\nu_{\rm r}
 =\int_X\sum_{u\sim v}f(u,v)\,d\mu(v).
\end{equation}
Here $A$ acts diagonally by the left site. With a normalized cocycle $s$, a
kernel $a$ acts in the fiber rooted at $v$ by the matrix
$(a(u,w)s(u,w,v))_{u,w\in[v]_{\mathcal R}}$. The identity
$s(u,w,v)s(u,v,v')=s(u,w,v')s(w,v,v')$ shows that rerooting conjugates these
matrices by a diagonal unitary, so their norms and principal matrix data are
root-independent. Finite sums of bounded base functions times lifted kernel
normalizers form a weakly dense $*$-algebra $N_{\rm alg}$
\cite[Section~2]{FM77}, on which the canonical expectation to $A$ is the
diagonal; it agrees with $E_A|_N$ (same value on the fixed part of a normalizer,
both zero on pieces with disjoint supports, extended by normality). Hence
$\varphi|_N$ is diagonal integration, represented in
\eqref{eq:right-counting-space} by the unit vectors $\delta_v$. All data are
taken modulo null sets, and countably many null exceptions are discarded with
their null saturations.

\subsection{A type I observation}

The kernel argument produces an intertwiner whose initial projection is not yet
known to be in $A$; the next observation turns it into a groupoid normalizer.
Its state hypothesis excludes the diffuse masas that can occur in a type I
algebra without a normal conditional expectation (compare the type I criterion
of \cite[Theorem~4.1]{AS12}).

\begin{lemma}\label{lem:type-i-normalizers}
Let $D$ be a type I von Neumann algebra with separable predual, $C\subseteq D$ a
masa, and suppose $D$ admits a faithful $C$-central normal state. Then the
groupoid normalizers of $C$ generate $D$, and every abelian projection of $D$
is Murray--von Neumann equivalent in $D$ to a projection of $C$.
\end{lemma}

\begin{proof}
Use the direct integral decomposition of a type I algebra into full operator
algebras over its center \cite[Sections~6.6 and 14.1--14.2]{KR86}. Integrating
fiber traces against a central probability measure gives a faithful normal
semifinite trace $\tau$. The state has a positive trace density $h$ of full
support with $\tau(h)=1$ (the semifinite predual identification, e.g.
\cite[Proposition~1.5 and Theorem~2.1]{Haa79}). Unitaries of $C$ preserve the
state and hence $h$, so $h$ commutes with $C$ and its spectral projections lie
in $C$. The projections $f_n=1_{[1/n,\infty)}(h)$ exhaust $1$ with
$\tau(f_n)\le n$; successive differences give a countable partition of $1$ in
$C$ into trace-finite projections, each of finite rank a.e. in the type I
fibers. Split further by central projections according to rank: each corner is
a measurable field of fixed finite matrix size with compressed $C$ a masa.

Diagonalize measurably in these finite corners. Trivialize the finite-rank
field and pick countably many commuting self-adjoint matrix fields generating
the compressed $C$. Fiberwise the algebras generated by the first $j$ fields
eventually stabilize in dimension (nondecreasing bounded integers); the first
index attaining the limit is measurable (finite dimensions are ranks of finite
polynomial lists, the limit a countable supremum), so split the center by it.
On each piece, successive eigenspace decompositions measurably diagonalize the
finite commuting list; ordered eigenvalues give measurable spectral
projections and Gram--Schmidt gives measurable orthonormal bases. The resulting
rank-one fields commute with all generators, hence lie in the masa. We obtain
rank-one fields $e_j\in C$ (zero off their central domains) with $\sum_je_j=1$.

On $e_jDe_j$ the algebra $Ce_j$ is the center scalars over that domain (the
center of $D$ lies in $C$), so measurable matrix units between $e_j,e_k$ on a
common central domain partially normalize $C$. These units and center
coefficients generate $D$: compress a bounded field by finite sums of the
$e_j$, express matrix entries in the units, and take a strong limit.

An abelian projection $e\in D$ has rank one a.e. on its central support $z$.
Choose fiberwise on $z$ the first nonzero $e_j$; this is $e'\in C$ with the same
central support and rank. Measurable unit vectors spanning the ranges give a
partial isometry $w\in D$ with $ww^*=e$, $w^*w=e'$. The case $e=0$ is trivial.
\end{proof}

\subsection{Atomless kernels off the normalizing algebra}

A positive kernel on $X$ is a measurable family of finite positive Borel
measures $K(v,du)$. It represents a positive normal map $K\colon A\to A$ if
$(Kf)(v)=\int f(u)K(v,du)$ for $f\in L^\infty(X,\mu)$, where for each fixed
$\mu$-null Borel $S$ one has $K(v,S)=0$ for a.e. $v$. This makes the formula
well defined on classes; it does not assert absolute continuity of rows.

\begin{proposition}\label{prop:atomless-kernels}
In the setting of this section, for every $y\in M^\circ$ the map
$K_y(f)=E_A(y^*fy)$ has a positive kernel of mass at most $\|y\|^2$ that is
atomless for a.e. output $v$. For $k\ge1$ and $y_1,\ldots,y_k\in M^\circ$ the
composition $K_{y_1}\circ\cdots\circ K_{y_k}$ has a positive kernel of mass at
most $\prod_j\|y_j\|^2$ that is atomless a.e. If $p_{\rm at}\in A$ is the
projection of the atomic part of $A$, then $p_{\rm at}y=yp_{\rm at}=0$ for
$y\in M^\circ$.
\end{proposition}

\begin{proof}
Let $\Omega$ be the standard vector for $\varphi$. The commuting left and right
actions of $A$ have a joint spectral measure $\lambda$ on $X\times X$ on
$y\Omega$. Because $A\subseteq M_\varphi$, the right action of $f$ gives
$yf\Omega$, so
\begin{equation}\label{eq:joint-kernel-measure}
 \lambda(S\times T)=\varphi(y^*1_Sy1_T)=\int_T E_A(y^*1_Sy)\,d\mu
\end{equation}
for Borel $S,T$. Disintegrating in the right coordinate gives a kernel for
$K_y$ of mass $E_A(y^*y)(v)\le\|y\|^2$; choose a Borel version, zeroing rows on
a null exceptional set so the mass bound holds for every row. For fixed null
$S$, \eqref{eq:joint-kernel-measure} gives $K_y(v,S)=0$ a.e. No product-measure
absolute continuity is needed.

Suppose the kernel is not a.e. atomless. The map
$(v,u)\mapsto K_y(v,\{u\})$ is measurable (a decreasing limit of masses of pieces
separating $u$). For some $k$, the set where it is at least $1/k$ has finite
sections and positive-measure projection, so the Lusin--Novikov theorem
\cite[Theorem~18.10]{Kec95} yields a measurable selection $s\colon T\to X$ on a
positive-measure Borel $T$ with
\begin{equation}\label{eq:selected-atom}
 K_y(v,\{s(v)\})\ge1/k\qquad(v\in T).
\end{equation}
For null Borel $S$ this gives $\mu(\{v\in T:s(v)\in S\})\le
k\int_TK_y(v,S)\,d\mu=0$. Hence $\theta(f)=1_T(f\circ s)$ is a well-defined
unital $*$-homomorphism $A\to A1_T$; it is normal, since the pushforward under
$s$ of a finite measure with $L^1$ density on $T$ is absolutely continuous with
density giving the preadjoint.

Take increasing finite Borel partitions $P_j$ generating $A$ and separating
points, and put
\begin{equation}\label{eq:graph-intertwiner-cuts}
 b_j=\sum_{e\in P_j}e\,y\,\theta(e).
\end{equation}
The left and right support projections are orthogonal families, so
$\|b_j\|\le\|y\|$, and $E_N(b_j)=0$ by $N$-bimodularity. Their state vectors are
the decreasing joint cuts of $y\Omega$ to the sets where the left coordinate and
$s$ of the right coordinate lie in the same piece of $P_j$, with the right
coordinate in $T$; the limit $\eta$ has
$\|\eta\|^2=\int_TK_y(v,\{s(v)\})\,d\mu>0$. A weak-star convergent subnet of the
bounded $b_j$ has limit $b\in M^\circ$ with $b\Omega=\eta$ (test against
$z\Omega$, $z\in M$). So $b\ne0$, $b=b1_T$, and
\begin{equation}\label{eq:graph-intertwiner}
 fb=b\theta(f)\qquad(f\in A),
\end{equation}
first for step functions by refinement in \eqref{eq:graph-intertwiner-cuts},
then for all $f$ by bounded strong approximation and normality of $\theta$.

We show \eqref{eq:graph-intertwiner} forces $b\in N$. In the polar
decomposition $b=u|b|$, the relations give $|b^*|\in A$,
$fu=u\theta(f)$ for $f\in A$, and $e:=u^*u\in D:=\theta(A)'\cap1_TM1_T$ (since
$bb^*$ commutes with $A$ and $b^*b$ with $\theta(A)$). For $d\in eDe$, the
element $udu^*$ commutes with $A$ and lies in $M$, hence in $A$, so $e$ is
abelian in $D$. Let $z$ be its central support in $D$. By the
abelian-projection characterization of type I algebras
\cite[Sections~6.4--6.6]{KR86}, $Dz$ is type I (every nonzero central part
meets $e$), and since $A1_T$ is a masa in $D$, its center, in particular $z$,
is in $A1_T$.

The restricted state on $Dz$ is faithful, normal, and central on the masa $Az$,
so by Lemma~\ref{lem:type-i-normalizers} $Dz$ is generated by groupoid
normalizers of $Az$. These partially normalize $A$ (supports in $Az$, and $A$
restricts onto $Az$), so $Dz\subseteq N$. The same lemma gives $w\in Dz$,
$e'\in Az$ with $ww^*=e$, $w^*w=e'$. Set $v=uw$. Its range projection
$uu^*=\supp|b^*|$ and initial projection $e'$ lie in $A$, and since $w$ commutes
with $\theta(A)$, $v^*Av=w^*u^*Auw=\theta(A)e'\subseteq Ae'$. The left side is a
masa in $e'Me'$ (conjugating the masa on the range corner), so equality holds
and $v$ is a groupoid normalizer. Hence $u=vw^*\in N$ and $b=|b^*|u\in N$,
contradicting $0\ne b\in\ker E_N$. The kernel is atomless a.e.

For compositions, let $K_1,K_2$ be two such kernels and fix a null $S_0$
outside which every row of $K_2$ is atomless. For a.e. $v$,
$K_1(v,S_0)=0$, and for such $v$ the measure
$(K_1\circ K_2)(v,E)=\int K_2(u,E)K_1(v,du)$ vanishes on every singleton $E$
simultaneously, because the integrand rows are atomless off $S_0$. The mass
bound and fixed-null-set property pass to the composition; iterate.

Finally the atomic part of a standard probability space is a countable set of
positive masses. Atomlessness gives $E_A(y^*p_{\rm at}y)=0$, and faithfulness
gives $p_{\rm at}y=0$; applying the same to $y^*$ gives $yp_{\rm at}=0$.
\end{proof}

\section{Finite polynomials and the cost of assigning a color}\label{sec:polynomials}

We build a finite-dimensional estimate for coloring the sites of a Hermitian
matrix. The polynomial of a partial coloring averages over remaining color
choices and independent signs; the sum of its positive roots after a negative
diagonal shift is a \emph{cost}. Assigning a color restricts the average and
makes the shift more negative at one site, and we show some color raises the cost
by at most a fixed multiple of a per-site quantity. The locality and uniform
bounds proved here let this be used on measured graphs.

\subsection{Finite data and mixed determinantal bounds}

Fix a finite set $I$, a zero-diagonal Hermitian $T=(T_{uv})_{u,v\in I}$ with
$\|T\|\le1$, an integer $r\ge1$, and $t>0$. Throughout the coloring argument
\begin{equation*}
 \beta=80.
\end{equation*}
A partial coloring is $c:I\to\{0,1,\ldots,r\}$, with $0$ meaning unassigned.
Give each site a weight $W_v>0$ and put
\[
 \tau_v=\begin{cases}t,&c(v)=0,\\ \beta t,&c(v)>0,\end{cases}
 \qquad a_v=\tau_vW_v.
\]
At site $v$ the allowed color--sign pairs are
\[
 \mathcal A_v=\begin{cases}
  \{1,\ldots,r\}\times\{+1,-1\},&c(v)=0,\\
  \{c(v)\}\times\{+1,-1\},&c(v)>0.
 \end{cases}
\]
Choose $\omega_v$ uniformly from $\mathcal A_v$, independently over sites, and
write $k_v=|\mathcal A_v|$. For $j=(i,\sigma)$ set
$U_j=\{v:j\in\mathcal A_v\}$ and
$B_j=\sigma D_WTD_W-\diag(a_v)_{v\in I}$ with $D_W=\diag(\sqrt{W_v})$. For a
realization $\omega$ let $I_j(\omega)=\{v:\omega_v=j\}$ and let $B_\omega$ be the
direct sum of the principal matrices $B_j|_{I_j(\omega)}$ placed on these
subsets (zero between subsets). With $Z=\diag(z_v)_{v\in I}$ define
\begin{equation}\label{eq:poly-definition}
 P_I(Z)=\mathbb E_\omega\det(Z-B_\omega),\qquad P_I(z)=P_I(z,\ldots,z).
\end{equation}
Write $P=P_I$ when $I$ is fixed; restriction is always principal with the same
laws at remaining sites. Thus $P_{-v}=P_{I\setminus\{v\}}$ and
$P_\varnothing=1$.

\begin{lemma}[Mixed determinantal bounds]\label{lem:mixed-bounds}
For Hermitian $A_1,\ldots,A_k$ on a common nonempty finite set,
\[
 \chi[A_1,\ldots,A_k](z)=\mathbb E\prod_{j=1}^k\det(zI-A_j|_{S_j}),
\]
one uniform assignment per site, empty determinant $1$, is real-rooted. If
$k\ge2$ and the $A_j$ are zero-diagonal contractions, then
\begin{equation}\label{eq:poly-mixed-upper}
 \lambda_{\max}\chi[A_1,\ldots,A_k]<\frac{3\sqrt2}{\sqrt k}.
\end{equation}
For any zero-diagonal Hermitian $L$ (no norm restriction),
\begin{equation}\label{eq:poly-mixed-lower}
 \lambda_{\max}\chi[L,-L]\ge\frac{\|L\|}{2}.
\end{equation}
Consequently, with all sites unassigned the unit-weight unshifted polynomial has
largest root $<3/\sqrt r$; with all sites assigned it equals
$\prod_{i=1}^r\chi[T|_{c^{-1}(i)},-T|_{c^{-1}(i)}]$ (empty factors $1$).
\end{lemma}

\begin{proof}
The normalization matches \cite[Definition~5]{RS17}; real-rootedness is part of
the mixed determinantal theory and also follows from
Lemma~\ref{lem:polynomial-quotients}. Equation~\eqref{eq:poly-mixed-upper} is
\cite[Theorem~25]{RS17} (zero-diagonal Hermitian contractions, $k\ge2$). By
\cite[Proposition~16 and Theorem~15]{RS17}, for zero-diagonal Hermitian matrices
replacing one member by zero cannot increase the largest root; expansion in
principal minors gives $\chi[L,0](z)=\det(zI-L/2)$ (a minor on $S$ occurs with
probability $2^{-|S|}$). Keeping either $L$ or $-L$ and using monotonicity proves
\eqref{eq:poly-mixed-lower}. All-unassigned uses $r$ copies each of $T,-T$, so
$k=2r$; fully assigned factors over independent color classes.
\end{proof}

\subsection{Stable quotients and rooted spectral measures}

For a matrix $B$ write $\operatorname{Im}B=(B-B^*)/(2i)$. The next lemma gives
the interlacing that measures the effect of one site; its half-plane part also
allows non-Hermitian matrices, for the analytic perturbations later.

\begin{lemma}[Polynomial quotients]\label{lem:polynomial-quotients}
For the finite data,
\begin{equation}\label{eq:poly-differential}
 P_I(Z)=\Bigl(\prod_{v\in I}\tfrac1{k_v!}\partial_{z_v}^{\,k_v-1}\Bigr)
            \prod_j\det\bigl((Z-B_j)|_{U_j}\bigr),
 \qquad \partial_{z_v}P_I=P_{-v}.
\end{equation}
More generally use arbitrary $B_j$ on $U_j$ in the same independent-choice
construction. If $\operatorname{Im}B_j\le hI$ then $P_I(Z)$ and every deletion
polynomial are nonzero whenever $\operatorname{Im}z_v>h$ at every remaining site,
and there
\begin{equation}\label{eq:poly-half-plane}
 \operatorname{Im}\frac{P_I(Z)}{P_{-v}(Z|_{I\setminus\{v\}})}
 \ge\operatorname{Im}z_v-h.
\end{equation}
Back in the Hermitian data, set $Q_v=P_{-v}$, $F_v=P_I/Q_v$, $G_v=Q_v/P_I$. Then
$F_v,G_v$ are analytic in the upper half-plane and
\begin{equation}\label{eq:poly-arrow-quotient}
 F_v(z)=z+a_v-\sum_\lambda\frac{b_{v,\lambda}}{z-\lambda},\qquad b_{v,\lambda}\ge0,
\end{equation}
over distinct roots $\lambda$ of $Q_v$. Letting $\Lambda_v$ list those roots with
multiplicity, there is $g_v$ with squared coordinate $b_{v,\lambda}$ on one copy
of each $\lambda$ (zero on others) such that
\begin{equation}\label{eq:poly-arrow}
 H_v=\begin{pmatrix}-a_v&g_v^*\\ g_v&\Lambda_v\end{pmatrix}
\end{equation}
has characteristic polynomial $P_I$. Its first-coordinate spectral measure
$\rho_v$ is a probability measure with
\begin{equation}\label{eq:poly-root-measures}
 G_v(z)=\int\frac{\rho_v(dx)}{z-x},\qquad
 \sum_{v\in I}\rho_v=\sum_{P_I(x)=0}\delta_x,
\end{equation}
roots counted with multiplicity, and $Q_v$ interlaces $P_I$.
\end{lemma}

\begin{proof}
Each determinant factor is affine in any diagonal variable. Differentiating
$k_v-1$ times in $z_v$, every surviving Leibniz term leaves $v$ in exactly one
of its $k_v$ factors and deletes it from the others; the factor
$(k_v-1)!/k_v!=1/k_v$ gives the outcome average, and one more derivative in
$z_v$ deletes the site, proving \eqref{eq:poly-differential}.

If $\operatorname{Im}z_v>h$, each $Z-B_j$ on $U_j$ has positive imaginary part
and is invertible. Differentiation preserves nonvanishing in a half-plane by
Gauss--Lucas; the degree cannot drop because the leading coefficient in an
unprocessed $z_v$ is the corresponding less-processed expression with $v$
removed, handled by simultaneous induction over processed vertices and principal
restrictions. So every deletion polynomial is nonzero there too. For fixed other
coordinates, $P_I/P_{-v}=z_v+\zeta_0$ (derivative identity and multiaffineness)
has zero in $\operatorname{Im}\le h$, giving \eqref{eq:poly-half-plane}.

For Hermitian data the uniform-variable polynomials are real-rooted (real
coefficients, half-plane nonvanishing with conjugation). With $h=0$,
$F_v(z)-z$ has nonnegative imaginary part and constant $a_v$ at infinity. A
nonremovable real pole of a real rational function with nonnegative imaginary
part is simple with nonpositive residue (approach the pole from different angles
to exclude higher order, then vertically for the sign), proving
\eqref{eq:poly-arrow-quotient}. The Schur complement of $z-\Lambda_v$ in $z-H_v$
is $F_v(z)$, so $\det(z-H_v)=Q_vF_v=P_I$ and its first resolvent entry is
$G_v$. The spectral theorem gives the first identity in
\eqref{eq:poly-root-measures}, principal-matrix interlacing the interlacing.
Uniform specialization of \eqref{eq:poly-differential} gives
$\sum_vG_v=P_I'/P_I$, the Cauchy transform of the root-counting measure; unique
partial fractions give the second identity.
\end{proof}

At an unassigned $v$, first condition its pair on color $i$ keeping both signs
uniform and shift $t$; call these $P^0_{v,i}$ and $F^0_{v,i}=P^0_{v,i}/Q_v$. Then
raise its shift to $\beta t$ as a permanent assignment: $P_{v,i}$, $F_{v,i}$.
Deletion of $v$ gives the same $Q_v$, so
\begin{equation}\label{eq:poly-options}
 \frac1r\sum_{i=1}^r F^0_{v,i}=F_v,\qquad
 F_{v,i}=F^0_{v,i}+(\beta-1)tW_v.
\end{equation}
The arrow representation and rooted measures apply to these too; the residues of
$F^0_{v,i}$ are nonnegative and average to those of $F_v$.

\subsection{Positive-root costs and locality}

For a monic real-rooted $P$ define $s(P)=\sum_{P(x)=0}x_+$ with
$x_+=\max\{x,0\}$ and $s(1)=0$, roots with multiplicity. For the finite data set
\begin{equation}\label{eq:poly-costs}
 m_v=\int x_+\,\rho_v(dx),\qquad d_v=s(P_I)-s(P_{-v}),
\end{equation}
so $\sum_vm_v=s(P_I)$. Superscripts $0,i$ and $i$ denote the conditioned and
permanently assigned options; e.g.\ $d_v^{\,i}=s(P_{v,i})-s(Q_v)$.

\begin{lemma}[Initial cost and detection of excessive norm]
\label{lem:finite-cost-endpoints}
(i) If all sites are unassigned and $t>3/\sqrt r$ then $P_I(x)>0$ for $x\ge0$
and $s(P_I)=0$ (including $I=\varnothing$). (ii) If $I\ne\varnothing$ and all
sites are assigned, put $T_c=\sum_i1_{\{c=i\}}T1_{\{c=i\}}$. For every
$\alpha>0$, if $\|T_c\|>2\beta t+\alpha$ then
\[
 \lambda_{\max}P_I\ge\frac\alpha2\min_{v\in I}W_v,
 \qquad s(P_I)\ge\frac\alpha2\min_{v\in I}W_v.
\]
\end{lemma}

\begin{proof}
For $J\subseteq I$ let $P_J^{\mathrm{un}}$ be the expected polynomial with same
pair laws, unit weights, no shifts. Factoring weights from each outcome
determinant,
\begin{equation}\label{eq:poly-remove-weights}
 P_J(x)=\Bigl(\prod_{v\in J}W_v\Bigr)
 P_J^{\mathrm{un}}\bigl((\tau_v+x/W_v)_{v\in J}\bigr),
\end{equation}
both sides $1$ for $J=\varnothing$. Multiaffineness and
\eqref{eq:poly-differential} also give, for real $b$ and increments $h_v\ge0$,
\[
 P_J^{\mathrm{un}}((b+h)|_J)
 =\sum_{S\subseteq J}P_{J\setminus S}^{\mathrm{un}}(b|_{J\setminus S})\prod_{v\in S}h_v,
\]
so positivity of all principal evaluations at $b$ survives raising any
coordinates.

(i) If all unassigned and $t>3/\sqrt r$, Lemma~\ref{lem:mixed-bounds} puts $t$
above the largest root of every nonempty $P_J^{\mathrm{un}}$, so all principal
evaluations at $t$ are positive; for $x\ge0$ the coordinates $t+x/W_v$ are $\ge
t$, so $P_I(x)>0$ and $s(P_I)=0$.

(ii) Suppose all assigned and $\|T_c\|>2\beta t+\alpha$. The fully assigned
factorization and \eqref{eq:poly-mixed-lower} give $P_I^{\mathrm{un}}$ a root
above $\beta t+\alpha/2$. If $\lambda_{\max}P_I<x_0:=(\alpha/2)\min_vW_v$, then
repeated interlacing puts $x_0$ above every root of every nonempty principal
restriction, so $P_J(x_0)>0$; by \eqref{eq:poly-remove-weights} the corresponding
unweighted evaluations are positive at
$b_v=\beta t+x_0/W_v\le\beta t+\alpha/2$, and raising to a common
$q\ge\beta t+\alpha/2$ keeps them positive, so $P_I^{\mathrm{un}}(q)>0$ for all
such $q$, contradicting the root above $\beta t+\alpha/2$. Hence
$\lambda_{\max}P_I\ge x_0$, giving the $s(P_I)$ bound.
\end{proof}

\begin{lemma}[Deletion costs and tails]\label{lem:positive-cost}
For the finite Hermitian data let $n_P(x)$ count roots of $P$ above $x$ with
multiplicity. For every $v\in I$,
\begin{align}
 0&\le n_{P_I}(x)-n_{Q_v}(x)\le1, \label{eq:poly-interlacing-count}\\
 d_v&=\int_0^\infty(n_{P_I}-n_{Q_v})(x)\,dx, \label{eq:poly-cost-integral}\\
 \int_B^\infty(n_{P_I}-n_{Q_v})(x)\,dx
 &\le2\int(x-B)_+\,\rho_v(dx)+(a_v+B)\rho_v((B,\infty)),\quad B\ge0.
 \label{eq:poly-cost-tail}
\end{align}
In particular $0\le d_v\le2m_v+a_v$. All hold for either modification at $v$,
with its own shift and rooted measure.
\end{lemma}

\begin{proof}
\eqref{eq:poly-interlacing-count} is principal interlacing from
Lemma~\ref{lem:polynomial-quotients}; integration gives
\eqref{eq:poly-cost-integral}. For the tail write $H=H_v$,
$\Lambda=\Lambda_v$, $a=a_v$, $L=H-BI$, and let $E$ be the first-coordinate
projection, $D=1-E$, $R=1_{(0,\infty)}(L)$. The compression $DRD$ is a positive
contraction on deletion coordinates, so by the variational formula for the
positive trace $\Tr(\Lambda-BI)_+\ge\Tr((\Lambda-BI)DRD)$. Since
$L-DLD=EL+LE-ELE$,
\begin{align*}
 \Tr(H-BI)_+-\Tr(\Lambda-BI)_+
 &\le\Tr((EL+LE-ELE)R)\\
 &=2\langle e_0,LRe_0\rangle+(a+B)\langle e_0,Re_0\rangle\\
 &=2\int(x-B)_+\,\rho_v(dx)+(a+B)\rho_v((B,\infty)),
\end{align*}
with $e_0$ the first unit vector of the arrow matrix. The left side is the tail
integral. Taking $B=0$ and $\rho_v$ a probability gives the last bound.
\end{proof}

A graph supports $T$ if it is undirected, loopless, and $T_{uv}=0$ for
nonadjacent $u\ne v$.

\begin{lemma}[Rooted locality and bounds]\label{lem:rooted-locality}
If a graph supports the finite zero-diagonal Hermitian contraction $T$, the
coefficient of $z^{-1-k}$ in $G_v(z)$ is $\int x^k\rho_v(dx)$ and is determined
by the data on the radius-$k$ ball about $v$. The Laurent coefficients of $F_v$
and of $F^0_{v,i},F_{v,i}$ at an unassigned site depend only on a radius bounded
by their order. Moreover
\begin{equation}\label{eq:poly-second-moment}
 \int x^2\,\rho_v(dx)
 =a_v^2+\sum_{u\ne v}W_vW_u|T_{vu}|^2\mathbb P(\omega_u=\omega_v),
 \quad
 \sum_\lambda b_{v,\lambda}
 =\sum_{u\ne v}W_vW_u|T_{vu}|^2\mathbb P(\omega_u=\omega_v),
\end{equation}
and the same holds for either site option with its own pair law and shift. If
neighboring weight ratios lie in $[K^{-1},K]$, $K\ge1$, then
\begin{equation}\label{eq:poly-weight-bounds}
 \int x^2\rho_v\le\bigl((\beta t)^2+K\bigr)W_v^2,\quad
 m_v\le C_0W_v,\quad d_v\le(2C_0+\beta t)W_v,
 \quad C_0=\sqrt{(\beta t)^2+K}.
\end{equation}
If all weights are at most $H_0$, every polynomial and principal restriction has
roots in
\begin{equation}\label{eq:poly-root-support}
 [-L,L],\qquad L=(1+\beta t)H_0,
\end{equation}
and the rooted measures share this support bound.
\end{lemma}

\begin{proof}
Put $u_v=z_v^{-1}$ and $\mathcal H_I(u)=P_I(Z)\prod_vu_v=\sum_{S\subset I}c_S\prod_{v\in S}u_v$,
where $c_S$ is the expected determinant of the negative outcome matrix restricted
to $S$. If $S$ splits into graph components the outcome splits and choices are
independent, so $c_S$ factors over components. Since $\mathcal H_I(0)=1$ its formal
logarithm is defined, and on a disconnected subset it is a sum over components,
so every nonconstant monomial in $\log\mathcal H_I$ has connected support. As
$G_v=\partial_{z_v}\log P_I$, apart from $1/z_v$ its monomials use connected
supports containing $v$; the coefficient of order $z^{-1-k}$ uses only the
radius-$k$ ball, matching the $k$-th rooted moment. Formal inversion gives the
same for $F_v$, and conditioning/changing the shift preserves this.

Singleton and two-point coefficients are
$c_{\{v\}}=a_v$ and
$c_{\{u,v\}}=a_ua_v-W_uW_v|T_{uv}|^2\mathbb P(\omega_u=\omega_v)$. Expanding the
logarithmic derivative to order $z^{-3}$ gives the first identity of
\eqref{eq:poly-second-moment}; the arrow matrix has first diagonal $-a_v$ and
off-diagonal squared norm $\sum_\lambda b_{v,\lambda}$, giving the second.

Under the ratio hypothesis the sum in \eqref{eq:poly-second-moment} is at most
$KW_v^2\sum_{u\ne v}|T_{vu}|^2\le KW_v^2$ (also for each conditioned law, as
collision probabilities are $\le1$), and $a_v\le\beta tW_v$.
Cauchy--Schwarz for $\rho_v$ plus Lemma~\ref{lem:positive-cost} gives
\eqref{eq:poly-weight-bounds}.

Each outcome has norm at most $(1+\beta t)H_0$, so for real $z>L$ all
characteristic determinants are positive and for $z<-L$ all have sign
$(-1)^{|I|}$; their average has no root outside $[-L,L]$, and real-rootedness
proves \eqref{eq:poly-root-support}. Restriction, site options, and the spectral
representation give the rest.
\end{proof}

\subsection{The single-site assignment estimate}

\begin{lemma}[Assigning one color]\label{lem:assignment}
For a finite zero-diagonal Hermitian contraction $T$, arbitrary positive weights,
$r\ge1$, $t>0$, a partial coloring as above, and $v$ unassigned, with
$\beta=80$ there is $i\in\{1,\ldots,r\}$ with
\begin{equation}\label{eq:poly-assignment}
 d_v^{\,i}-d_v\le24m_v.
\end{equation}
The bound is independent of the finite set and its weights.
\end{lemma}

\begin{proof}
Write $a=a_v=tW_v>0$, $m=m_v$, $d=d_v$, and
$H=\begin{pmatrix}-a&g^*\\ g&\Lambda\end{pmatrix}$ for the old arrow matrix. By
\eqref{eq:poly-options} the color-conditioned residues $b_{i,\lambda}$ are
nonnegative with average $b_\lambda$; a permanent assignment replaces them by
$b_{i,\lambda}$ and the first diagonal by $-\beta a$, with $\Lambda$ fixed.

Split the deletion coordinates into $\lambda>0$ and $\lambda\le0$ (subscripts
$+$,$-$), write $H=H_+-H_-$, and set
$e_+=\Tr(H_-)_{++}$, $e_-=\Tr(H_+)_{--}$. Comparing diagonal blocks of
$H=H_+-H_-$ and using that the first diagonal of $H_+$ is $m$,
\begin{equation}\label{eq:poly-error-decomposition}
 d=m+e_++e_-,\qquad (H_-)_{00}=m+a.
\end{equation}
We treat $m\ge a$ and $m<a$.

\emph{Case $m\ge a$.} Consider the two side arrows keeping the first coordinate
and one deletion block, first diagonal $-a$. The positive block has exactly one
negative eigenvalue $-L_0$ with
$L_0=a+\sum_{\lambda>0}b_\lambda/(L_0+\lambda)$ and increment $d^+=L_0-a$
(valid empty, $L_0=a$). The nonpositive block has at most one positive
eigenvalue $s_0$ (else $0$); its increment is $d^-=s_0$ and
\begin{equation}\label{eq:poly-nonpositive-test}
 \sum_{\lambda\le0}\frac{b_\lambda}{s_0+|\lambda|}\le s_0+a.
\end{equation}
If $s_0=0$ all residues at $\lambda=0$ vanish (otherwise a positive zero
appears); omit those terms, and if the remaining sum at zero exceeds $a$,
continuity at infinity again yields a positive zero, so
\eqref{eq:poly-nonpositive-test} holds. Pinching off the other coupling cannot
increase the positive trace, so $d^+\le d$ and $d^-\le d$. By averaging, at least
three quarters of the colors satisfy
$\sum_{\lambda>0}b_{i,\lambda}/(L_0+\lambda)\le4d^+$ and at least three quarters
satisfy $\sum_{\lambda\le0}b_{i,\lambda}/(s_0+|\lambda|)\le4(s_0+a)$, so some
color satisfies both (zero old residues stay zero).

For this color the positive-block arrow with diagonal $-a$ has increment at most
$4d^+$ (immediate if the eigenvalue magnitude is $\le L_0$, else bounded by the
test sum), and the nonpositive-block arrow with diagonal $-4a$ has increment at
most $4s_0$ (a positive root $s>4s_0$ would need a residue sum above
$4(s_0+a)$). Extend both by zero and add: the full arrow has diagonal $-5a$ and,
by subadditivity of the positive trace, increment at most $4d^++4d^-$;
decreasing the diagonal to $-\beta a$ only lowers positive trace. Hence
\[
 d_v^{\,i}\le4(d^++d^-)\le8d,\qquad
 d_v^{\,i}-d\le7d\le7(2m+a)\le21m,
\]
using Lemma~\ref{lem:positive-cost}.

\emph{Case $m<a$.} For a positive block matrix the outer product of its first
coupling column is bounded by its first diagonal times its remaining block. Apply
to $H_+$ and $H_-$ and use $(p-q)(p-q)^*\le2pp^*+2qq^*$; since
$(H_+)_{++}=\Lambda_++(H_-)_{++}$ and
$(H_-)_{--}=|\Lambda_-|+(H_+)_{--}$,
\begin{align}
 g_+g_+^*&\le 2m\Lambda_++(4m+2a)(H_-)_{++}, \label{eq:poly-positive-block}\\
 g_-g_-^*&\le 2(m+a)|\Lambda_-|+(4m+2a)(H_+)_{--}. \label{eq:poly-negative-block}
\end{align}
Conjugating \eqref{eq:poly-positive-block} by $(32a+\Lambda_+)^{-1/2}$ and taking
the operator norm (left side rank one, right error bounded by its trace, and
$4m+2a\le6a$) gives
\begin{equation}\label{eq:poly-positive-test}
 \sum_{\lambda>0}\frac{b_\lambda}{32a+\lambda}\le2m+\frac6{32}e_+.
\end{equation}
Put $L_1=e_-$. If $L_1>0$, conjugating \eqref{eq:poly-negative-block} by
$(L_1+|\Lambda_-|)^{-1/2}$ gives
\begin{equation}\label{eq:poly-negative-test}
 \sum_{\lambda\le0}\frac{b_\lambda}{L_1+|\lambda|}\le2(m+a)+(4m+2a)\le10a;
\end{equation}
if $L_1=0$, positivity gives $(H_+)_{--}=0$ and
\eqref{eq:poly-negative-block} forces zero coupling on the kernel of
$|\Lambda_-|$, so invert there and omit zero residues at $\lambda=0$, keeping
\eqref{eq:poly-negative-test}. Choose a color with both residue sums at most four
times their bounds. For it, the positive-block arrow with diagonal $-32a$ has
increment $\le8m+\tfrac{24}{32}e_+$; the nonpositive-block arrow with diagonal
$-40a$ has increment $\le L_1$ (a root $s>L_1$ would force residue sum above
$40a$, contradicting $s+40a>40a$; covers $L_1=0$ with zero residues omitted).
Adding gives diagonal $-72a$; subadditivity and lowering to $-80a$ give
\[
 d_v^{\,i}\le8m+\tfrac34e_++e_-,\qquad
 d_v^{\,i}-d\le7m-\tfrac14e_+\le7m
\]
with \eqref{eq:poly-error-decomposition}. Both cases give
\eqref{eq:poly-assignment}; all allow empty blocks, one-site matrices, and zero
residues.
\end{proof}

\section{Rooted costs on invariant measured graphs}\label{sec:invariant}

The finite estimates control assigning one site. We now pass to measured
graphs, where many vertices change at once. Two facts are needed: the integral of
the rooted cost changes by the integral of the single-vertex deletion increments
in ordered comparison environments; and deleting vertices outside a fixed weight
band has a small effect within the band, uniformly in the largest weight. The
uniformity removes weight cutoffs in the next section.

Let $(X,\nu)$ be a standard finite measure space and $\mathcal R$ a countable
measured equivalence relation preserving $\nu$, so each measurable partial
isomorphism with graph in $\mathcal R$ preserves measure. Let $G\subset\mathcal
R$ be a measurable undirected loopless graph of degree at most $D_0<\infty$.
On orbit spaces let $T$ be measurable Hermitian matrices of norm at most one,
zero diagonal, supported on $G$; changing the root conjugates them by diagonal
unitaries, and all constructions below are invariant under this. Null
exceptions are discarded along their $\mathcal R$-saturations.

Fix $r\ge1$, $t>0$, $\beta=80$. A partial coloring is $c:X\to\{0,1,\ldots,r\}$
($0$ unassigned). Use the shifts and polynomial choices of
Section~\ref{sec:polynomials} and a weight $W:X\to(0,H_0]$, $H_0<\infty$. After a
measurable set is deleted take principal restrictions, keeping the original
graph for distances. At absent vertices every rooted cost or transform is zero
and the rooted measure is zero. On a finite principal $I$ use
\[
 P_I,\quad Q_{I,v}=P_{I\setminus\{v\}},\quad
 F_{I,v}=\tfrac{P_I}{Q_{I,v}},\quad G_{I,v}=\tfrac1{F_{I,v}},\quad
 \rho_{I,v},\quad m_{I,v},\quad d_{I,v},
\]
with $m_{I,v}=\int x_+\rho_{I,v}(dx)$,
$d_{I,v}=s(P_I)-s(Q_{I,v})$, $s(P)=\sum_{P(x)=0}x_+$. Write $a_v=\tau_vW_v$ and
\begin{equation}\label{inv:spectral-bound}
 L=(1+\beta t)H_0.
\end{equation}
All roots and rooted measures of these finite data lie in $[-L,L]$, including
virtual options at an unassigned vertex.

Measure invariance gives mass transport
\begin{equation}\label{inv:mass-transport}
 \int_X\sum_{u\in[v]_{\mathcal R}}h(v,u)\,d\nu(v)
 =\int_X\sum_{v\in[u]_{\mathcal R}}h(v,u)\,d\nu(u)
\end{equation}
for nonnegative (and for integrable complex) measurable $h$: partition
$\mathcal R$ into countably many partial one-to-one maps and change variables.

\subsection{Limits on bounded weighted graphs}

\begin{lemma}\label{lem:bounded-costs}
For the above data on the finite measure space $(X,\nu)$, the rooted measures,
transforms, and costs converge as finite principal sets exhaust the graph
component of the root, to limits $\rho_v,G_v,F_v,m_v,d_v$ independent of the
exhaustion and measurable in $v$. At each present vertex $\rho_v$ is a
probability measure on $[-L,L]$ and, for $z$ in the open upper half-plane,
\[
 G_v(z)=\int\frac{\rho_v(dx)}{z-x},\qquad F_v(z)=\frac1{G_v(z)}.
\]
The same holds after conditioning the color at one unassigned vertex with either
shift. For fixed $D_0,H_0,r,t$ the costs $m_v,d_v,d_v^{\,i}$ ($1\le i\le r$) are
uniformly approximable by a radius depending only on the parameters, not on the
graph, weights, coloring, or deleted set. These statements apply to a
finite-measure restriction of an invariant relation.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:rooted-locality} each moment $\int x^k\rho_{I,v}$ is fixed by a
bounded neighborhood and unchanged once $I$ contains it. Compactness of
probabilities on $[-L,L]$ and moment uniqueness give an exhaustion-independent
weak limit $\rho_v$ with Cauchy transform $G_{I,v}\to G_v$ locally uniformly on
the upper half-plane. Since $-\operatorname{Im}G_v(z)=(\operatorname{Im}z)\int
|z-x|^{-2}\rho_v(dx)>0$, this limit inverts, so $F_{I,v}\to F_v$ there; the same
holds for altered color laws or shift at the root. The finite quotients have the
form $F_{I,v}(z)-z-a_v=-\int\kappa_{I,v}(dx)/(z-x)$ with a positive residue
measure supported on $[-L,L]$ of mass at most $H_0^2$
(Lemma~\ref{lem:rooted-locality}); compactness and Cauchy uniqueness give a
limiting residue measure, justifying Laurent-coefficient limits. Since $x_+$ is
continuous on $[-L,L]$, $m_{I,v}$ converges, uniformly approximable by finite
neighborhoods.

Deletion increments need separate treatment because they involve roots at every
vertex of a finite set. For real $f$ put
$d_{I,v}[f]=\sum_{P_I(x)=0}f(x)-\sum_{Q_{I,v}(x)=0}f(x)$; the rooted-measure
sum identity gives
\begin{equation}\label{inv:polynomial-increment}
 d_{I,v}[f]=\sum_{u\in I}\Bigl(\int f\,d\rho_{I,u}
   -1_{\{u\ne v\}}\int f\,d\rho_{I\setminus\{v\},u}\Bigr).
\end{equation}
Each summand is local (vanishes unless $u$ is within a radius depending only on
$\deg f$ of $v$), so the whole increment is a finite-neighborhood calculation
independent of the ambient $I$.

For absolutely continuous real $f$ on $[-L,L]$, interlacing gives
\begin{equation}\label{inv:ac-increment}
 d_{I,v}[f]=f(-L)+\int_{-L}^{L}f'(x)(n_{P_I}-n_{Q_{I,v}})(x)\,dx,
 \qquad 0\le n_{P_I}-n_{Q_{I,v}}\le1,
\end{equation}
$n_P$ counting roots above $x$. Hence
\begin{equation}\label{inv:increment-approximation}
 |d_{I,v}[f]-d_{I,v}[g]|\le|f(-L)-g(-L)|+\|f'-g'\|_{L^1([-L,L])}.
\end{equation}
Approximate $f'$ in $L^1$ by polynomials with prescribed initial value, giving
polynomial approximants with both function and increment error tending to zero.
Apply to $f=x_+$: the local polynomial increments are eventually constant with
uniform approximation error in every finite set, so $d_{I,v}$ has the stated
limit and uniform locality; similarly for each virtual option.

Finite balls of a measurable bounded-degree graph, their matrix data, and the
conjugacy-invariant polynomial calculations are measurable, so the limits are
measurable. Deleting vertices or restricting the relation changes nothing: use
the original graph for radii and retained vertices for each principal
calculation.
\end{proof}

\subsection{Integral change under ordered updates}

\begin{lemma}\label{lem:integral-change}
Under the finite-measure hypotheses of Lemma~\ref{lem:bounded-costs}, let old and
new data differ only by coloring changes or deletion at a measurable set $E_0$,
and fix a measurable real function injective on $E_0$ within each class,
ordering $E_0$ by its values. At $u\in E_0$ use new data at earlier vertices and
old data at later vertices; let $\delta d_u$ be the new-minus-old difference of
the single-vertex deletion cost at $u$ in this environment (zero if $u$ absent).
Then
\begin{equation}\label{inv:integral-change}
 \int_X(m_v^{\mathrm{new}}-m_v^{\mathrm{old}})\,d\nu(v)=\int_{E_0}\delta d_u\,d\nu(u).
\end{equation}
In particular deletion alone cannot increase the integral of $m_v$.
\end{lemma}

\begin{proof}
An injective measurable point code supplies such an ordering if none is given.
Start with a real polynomial $f$ for $x_+$, writing $m_v[f]=\int f\,d\rho_v$ at
present vertices and zero at absent ones. For each root $v$ this depends only on
a fixed finite-radius neighborhood; listing the finitely many $E_0$-vertices in
that neighborhood in order and telescoping, each term agrees with the change
produced at that vertex in the global environment (changes outside the
neighborhood do not affect the polynomial rooted datum). No well-ordering or
infinite telescoping at a single root is assumed. Integrate the finite local sum
and apply \eqref{inv:mass-transport} to interchange root and changed vertex; for
fixed $u$ the sum over affected roots equals the change in $d_u[f]$
(checkable on a large finite principal neighborhood via
\eqref{inv:polynomial-increment}, since the old and new deletion polynomials at
$u$ agree; a deleted vertex has both new terms zero). Bounded degree, bounded
support, and finite measure justify the integrals, giving the identity for $f$.
Choose the polynomial approximants to $x_+$ from Lemma~\ref{lem:bounded-costs};
uniform function approximation controls the left terms, the $L^1$ derivative
bound \eqref{inv:increment-approximation} the right increments, and finite
measure passes to the limit, proving \eqref{inv:integral-change}. A deletion has
comparison increment $-d_u\le0$ by interlacing, giving the last assertion.
\end{proof}

\subsection{Deletion estimates independent of the largest weight}

Earlier arguments allow constants depending on $H_0$; removing weight cutoffs
needs a fixed-band comparison uniform as $H_0$ grows. Cauchy transforms supply
it, and we must also control the conditional color choices of the single-site
inequality. At an unassigned present $v$, let $F^0_{v,i}$ condition on color $i$
with shift $t$, so the permanent option is
$F_{v,i}(z)=F^0_{v,i}(z)+(\beta-1)tW_v$. Define the auxiliary analytic field
\begin{equation}\label{inv:J-definition}
 J_i(v,z)=\bigl(F^0_{v,i}(z)-z-a_v\bigr)G_v(z),\qquad c(v)=0,\ v\text{ present},
\end{equation}
zero elsewhere. Reciprocals are nonzero by Lemma~\ref{lem:bounded-costs}, and
the residue bound of Lemma~\ref{lem:rooted-locality} makes these fields bounded
on upper-half-plane compacts when the root weight and neighbor ratios are
bounded.

\begin{lemma}\label{lem:rank-comparison}
On the above data, additionally assume neighbor weight ratios in $[K^{-1},K]$,
$K\ge1$. Fix $0<b_1<b_2<\infty$ and the band $\mathcal U=\{b_1\le W_v\le b_2\}$.
Delete a measurable set $E_{\mathrm{hi}}$ disjoint from $\mathcal U$ without
changing colors at retained vertices, and let $\delta_{\mathrm{tot}}$ be the
difference in either orientation. For each compact $\Omega$ in the open upper
half-plane there is $C$ such that for all $z\in\Omega$ and $1\le i\le r$,
\begin{equation}\label{inv:rank-comparison}
 \int_{\mathcal U}|\delta_{\mathrm{tot}}G_v(z)|\,d\nu(v)
 +\int_{\mathcal U\cap\{c=0\}}|\delta_{\mathrm{tot}}J_i(v,z)|\,d\nu(v)
 \le C\nu(E_{\mathrm{hi}}),
\end{equation}
including only band vertices present before deletion. The constant may depend on
$\Omega,r,t,\beta,D_0,K,b_1,b_2$ but not on $H_0$, total mass, coloring, graph,
or deleted set.
\end{lemma}

\begin{proof}
Discard initially absent vertices from $E_{\mathrm{hi}}$ (configurations
unchanged, measure only smaller), so every deletion vertex is present first. We
bound one deletion in a finite principal set, pass to infinite bounded graphs,
then integrate using an analytic ordered-change identity.

\emph{A single deletion and $G$.} Fix finite $I$, $u\in I$, $z$ with
$y=\operatorname{Im}z>0$. For $|\lambda_v|\le1$ perturb the diagonal variables to
$z+\zeta\lambda_v$; on $|\zeta|<y/2$ they stay in the upper half-plane. By
Lemma~\ref{lem:polynomial-quotients} $P_I/P_{I\setminus\{u\}}$ has positive
imaginary part, so its logarithm has a holomorphic branch with imaginary part in
$(0,\pi)$. A holomorphic function on a disk of radius $R$ with bounded imaginary
part has center derivative bounded by a numerical constant times the oscillation
of that imaginary part over $R$ (recover the first Taylor coefficient from the
first Fourier coefficient of the imaginary part on a concentric circle of radius
$R/2$). Applying to the logarithm bounds its derivative at $0$ by $C_0/y$.
Logarithmic differentiation of $P_I$ in a diagonal variable is
$\sum_v\lambda_v(G_{I,v}-G_{I\setminus\{u\},v})$ (second transform zero at
$v=u$); choosing phases to make summands nonnegative gives
\begin{equation}\label{inv:single-G}
 \sum_{v\in I}|G_{I,v}(z)-G_{I\setminus\{u\},v}(z)|\le C_0/y,
\end{equation}
with no weight bound.

\emph{A single deletion and the conditional color field.} Assume
$u\notin\mathcal U$ and take $\lambda_v=0$ except at unassigned band vertices.
Perturb the off-diagonal parts of both pair matrices of color $i$ by left and
right diagonal scalings with entries $1+\zeta\lambda_v$ (no conjugation on the
right; the shift is unchanged), analytic but not Hermitian for complex $\zeta$.
With $D_\lambda=\diag(\lambda_v)$ and $D_W=\diag(\sqrt{W_v})$ on the relevant
allowed-site set, the off-diagonal matrix (sign suppressed) is
$C_i=D_WTD_W$ and its perturbation is
$\zeta(D_\lambda C_i+C_iD_\lambda)+\zeta^2D_\lambda C_iD_\lambda$. In the first
term only band rows and their graph neighbors occur: row weights at most $b_2$,
neighbor column weights at most $Kb_2$, so factoring diagonal weights and using
$\|T\|\le1$ gives $\|D_\lambda C_i\|\le\sqrt Kb_2$ (same for $C_iD_\lambda$); the
quadratic term has both indices in the band, norm at most $b_2$. Hence the full
perturbation has norm at most
\begin{equation}\label{inv:perturbation-bound}
 2|\zeta|\sqrt Kb_2+|\zeta|^2b_2.
\end{equation}
Choose a disk of radius depending only on $y,b_2,K$ making this less than $y/2$;
then the perturbed pair matrices have imaginary part $\le(y/2)I$, and with
diagonal variable $z$ the non-Hermitian Lemma~\ref{lem:polynomial-quotients}
gives $P_I/P_{I\setminus\{u\}}$ positive imaginary part, its logarithm with
imaginary part in $(0,\pi)$, uniformly in $H_0$. The derivative of $\log P_I$
here is
\begin{equation}\label{inv:color-derivative}
 \frac2r\sum_{v\in I}\lambda_vJ_i(v,z).
\end{equation}
Indeed, in the permutation expansion of an outcome determinant, a nontrivial
cycle through a color-$i$ target $v$ uses one row and one column factor, giving
multiplier $2\lambda_v$ (a fixed point contributes $z+a_v$, unchanged since $T$
has zero diagonal); conditional on color $i$ at $v$ the sum of nontrivial terms
through $v$ is $P_{I\setminus\{v\}}(z)(F^0_{v,i}(z)-z-a_v)$, with color
probability $1/r$, so dividing by $P_I$ gives \eqref{inv:color-derivative},
including both signs; also after deleting $u$. Applying the derivative bound and
choosing phases as before,
\begin{equation}\label{inv:single-J}
 \sum_{v\in I\cap\mathcal U\cap\{c=0\}}|J_i^{I}(v,z)-J_i^{I\setminus\{u\}}(v,z)|
 \le C_1(y,b_2,K,r).
\end{equation}
Constants in \eqref{inv:single-G},\eqref{inv:single-J} are uniform on
upper-half-plane compacts.

\emph{Passage to bounded infinite graphs.} Exhaust the component of $u$ by
finite principal sets; transforms converge by Lemma~\ref{lem:bounded-costs}
(with conditionals in \eqref{inv:J-definition}), and Fatou passes both
single-deletion estimates to the infinite graph, over the whole class (no effect
on other components), valid in every comparison environment. These estimates are
uniform in the largest weight.

\emph{The analytic integral change identity.} Fix measurable complex directions
$\lambda_v$, $|\lambda_v|\le1$, supported on the band targets ($\mathcal U$ for
$G$, $\mathcal U\cap\{c=0\}$ for $J_i$), and let $Z_v(z)$ be the respective
field, extended by zero off them. Order $E_{\mathrm{hi}}$ measurably and
injectively along classes; at a deletion vertex $u$ let $\delta_uZ_v$ be
presence-minus-absence with earlier vertices deleted and later present. Orient
$\delta_{\mathrm{tot}}$ before-minus-after. We claim
\begin{equation}\label{inv:analytic-change}
 \int_X\lambda_v\delta_{\mathrm{tot}}Z_v(z)\,d\nu(v)
 =\int_{E_{\mathrm{hi}}}\Bigl(\sum_{v\in[u]_{\mathcal R}}
   \lambda_v\delta_uZ_v(z)\Bigr)d\nu(u).
\end{equation}
Work near infinity. Every target is present in both configurations (deleted set
disjoint from the band, initially absent band vertices excluded). At such a
target, bounded support gives
$G_v(z)=z^{-1}\sum_{k\ge0}h_k(v)z^{-k}$, $|h_k(v)|\le L^k$, and
$F^0_{v,i}(z)-z-a_v=-\sum_{k\ge0}b_k(v,i)z^{-k-1}$, $|b_k(v,i)|\le
B_{v,i}L^k$ where $B_{v,i}$ is the limiting residue mass, uniformly bounded by
Lemma~\ref{lem:rooted-locality}; these hold as the limits of negative Cauchy
transforms of residue measures supported in $[-L,L]$. Each coefficient is
eventually fixed by a finite neighborhood, the bounds are uniform, and the series
converge to the limiting transforms for large $|z|$; the product defining $J_i$
has exponentially bounded Laurent coefficients, and the coefficient at each order
depends on a radius bounded by that order (Lemma~\ref{lem:rooted-locality} plus
series multiplication).

For each coefficient the ordered local telescoping of
Lemma~\ref{lem:integral-change} plus \eqref{inv:mass-transport} proves the
coefficient version of \eqref{inv:analytic-change}; bounded degree makes the
number of affected roots at a fixed deleted vertex grow at most exponentially in
the order, and the coefficient bounds give absolute summability for large $|z|$.
Finite measure allows sum/integral interchange, proving
\eqref{inv:analytic-change} on a nonempty open region (possibly depending on
$H_0$).

Both sides are holomorphic on the full upper half-plane. On the right, the sum
over $v$ converges absolutely at each $z$ by the single-deletion estimates,
its analytic partial sums (e.g.\ over increasing neighborhoods) are locally
bounded by them, and normal-family convergence makes it holomorphic; the outer
integral over the finite-measure $E_{\mathrm{hi}}$ with uniform compact bounds is
holomorphic. On the left, $|G_v(z)|\le(\operatorname{Im}z)^{-1}$ and the band
residue bound similarly bounds $J_i$ on compacts; these and finite band measure
justify holomorphy. The identity theorem extends \eqref{inv:analytic-change} to
the whole upper half-plane.

Finally fix $z$ and choose $\lambda_v$ so that
$\lambda_v\delta_{\mathrm{tot}}Z_v(z)=|\delta_{\mathrm{tot}}Z_v(z)|$ (arbitrary
when zero). These fixed directions give a right side bounded by the
single-deletion constant times $\nu(E_{\mathrm{hi}})$. Apply separately to $G$
and $J_i$ and add; the constants were independent of $H_0$ and of comparison
environments, proving \eqref{inv:rank-comparison}.
\end{proof}

The integral-change identity controls the total positive cost under color
assignments and deletions; the transform comparison controls the effect of
distant large weights on any fixed weight band. These two estimates define
normalized costs on a nonsingular base by invariant restrictions of finite
measure.

\section{Coloring a nonsingular measured graph}\label{sec:coloring}

The polynomial and invariant-graph estimates now produce a measurable coloring on
a probability extension of a nonsingular measured graph. The conclusion concerns
the operator norm on a prescribed finite ball, with small failure probability,
and preserves uniform color marginals conditional on the base point (needed when
the coloring enters a von Neumann algebra). There are two steps: build costs on
the nonsingular space by adding a height coordinate and removing weight cutoffs;
then assign colors at a small random fraction of the remaining vertices at each
step, with the integrated change controlled where two colorings agree on a large
ball.

\subsection{The probability extension and the coloring theorem}

Let $(X,\mu)$ be a standard probability space and $\mathcal R$ a countable
nonsingular measured equivalence relation on $X$, with Borel representatives off
null sets. Write the Radon--Nikodym cocycle as
\begin{equation}\label{eq:coloring-cocycle}
 \mu(\gamma S)=\int_S\Delta(\gamma v,v)\,d\mu(v)
\end{equation}
for measurable partial isomorphisms $\gamma$ with graph in $\mathcal R$. Versions
satisfy the cocycle identity off a saturated null set (cover $\mathcal R$ by
countably many partial isomorphisms, use the chain rule for compositions, and
agree on overlaps). Null sets along the countable relation are discarded with
their saturations, which are null by nonsingularity.

Let $G$ be an undirected measurable graph in $\mathcal R$, loopless, degree at
most $D_0<\infty$, with radius-$m$ ball $B_G(v,m)$. Suppose
\begin{equation}\label{eq:coloring-edge-ratios}
 K^{-1}\le\Delta(u,v)\le K\quad\text{on every edge of }G,\qquad K\ge1.
\end{equation}
Consider a measurable field $T$ of Hermitian matrices on the orbit spaces
$\ell^2([v]_{\mathcal R})$, norm at most one, zero diagonal, supported on $G$.
Changing the root conjugates them by a diagonal unitary, and all quantities used
(principal norms, polynomial costs) are invariant.

The \emph{relative Bernoulli extension} of $(X,\mu,\mathcal R)$ is
\[
 \widetilde X=\{(v,\omega):v\in X,\ \omega\in[0,1]^{[v]_{\mathcal R}}\},
 \qquad\pi(v,\omega)=v,
\]
with probability $\widetilde\mu$ whose conditional measure over $v$ is the
product Lebesgue measure $\kappa_v$. The extension identifies $(v,\omega)$ with
$(u,\omega)$ when $(u,v)\in\mathcal R$, treating $\omega$ as one function on
the class, so each extended orbit projects bijectively onto a base orbit; graph
and matrices pull back. These are standard measurable spaces and fields
(enumerate the relation and use one coordinate per site); changing the root
reindexes a product family and preserves $\kappa_v$. Lifts of base partial
isomorphisms have the same Radon--Nikodym derivative, enumerate the extended
relation, and (by conditional integration) show the cocycle is the pullback of
$\Delta$. One Lebesgue coordinate at a site encodes countably many independent
coordinates when needed.

For a coloring $c:\widetilde X\to\{1,\ldots,r\}$, its value at a site $u$ in the
orbit of $(v,\omega)$ is $c(u,\omega)$. On a finite principal set $I$ in this
orbit let
\[
 T_{I,c}=\sum_{i=1}^r1_{\{c=i\}\cap I}\,T|_I\,1_{\{c=i\}\cap I}
\]
retain entries joining equal-colored vertices.

\begin{theorem}[Bernoulli coloring]
\label{thm:bernoulli-coloring}
Let $(X,\mu,\mathcal R)$, $G$, $\Delta$, $T$ satisfy the above assumptions.
Fix an integer $r\ge1$ and $t>3/\sqrt r$, and put $\beta=80$. For every $m\ge1$
and every $\alpha,\eta>0$ there is a measurable coloring
$c:\widetilde X\to\{1,\ldots,r\}$ on the relative Bernoulli extension with
\begin{equation}\label{eq:coloring-marginal}
 \kappa_v\{\omega:c(v,\omega)=i\}=\frac1r\quad(i\in\{1,\ldots,r\},\ \text{a.e. }v)
\end{equation}
and
\begin{equation}\label{eq:coloring-ball-conclusion}
 \widetilde\mu\{(v,\omega):\|T_{B_G(v,m),c}\|\le2\beta t+\alpha\}>1-\eta.
\end{equation}
\end{theorem}

We first build the cost, then prove the theorem. Until then the probability base
may already include some Bernoulli coordinates; keep the notation $(X,\mu)$,
$\mathcal R$ for the current base, with the same bounds.

\subsection{Homogeneous costs from an invariant lift}

A partial coloring is $c:X\to\{0,1,\ldots,r\}$ ($0$ unassigned). Recall the
finite-site data of Section~\ref{sec:polynomials}: the shift is
\[
 \tau_v=\begin{cases}t,&c(v)=0,\\ \beta t,&c(v)>0,\end{cases}
\]
allowed choices are both signs of all $r$ colors at an unassigned site, or both
signs of the assigned color otherwise, uniform and independent. With positive
weights $W_v$, the outcome matrix on a choice block is the principal restriction
of $\diag(\sqrt W)(\pm T)\diag(\sqrt W)-\diag(\tau W)$. For the expected
characteristic polynomial $P$ and deletion $Q=P_{-v}$ use
\[
 F_v=P/Q,\quad G_v=1/F_v,\quad m_v=\int x_+\rho_v(dx),\quad d_v=s(P)-s(Q).
\]
At an unassigned site the superscript $i$ means permanent assignment to color
$i$ (including the increased shift); $F^0_{v,i}$ conditions on color $i$ without
increasing the shift, and
\[
 F_{v,i}=F^0_{v,i}+(\beta-1)tW_v
\]
includes it; $d_v^{\,i}=0$ on assigned sites. On bounded invariant graphs the
same notation refers to limits from Lemma~\ref{lem:bounded-costs}.

The original measure need not be invariant, so the integral-change formula does
not yet apply. Use the Maharam invariant lift \cite{Mah64}; for the
multiplicative-coordinate formula see \cite[Section~2, before Theorem~2.1]{RoySamorodnitsky06}.
Take logarithmic height $s\in\mathbb R$ with multiplicative coordinate $y=e^s$:
\begin{equation}\label{eq:coloring-lift}
 d\nu(v,s)=d\mu(v)e^s\,ds,\qquad W(v,s)=e^{-s}.
\end{equation}
In the lifted relation the site of $u$ at root $(v,s)$ is
\begin{equation}\label{eq:coloring-height-change}
 (u,s-\log\Delta(u,v)).
\end{equation}
The measure $\nu$ is invariant: the factor $\Delta(u,v)$ in
\eqref{eq:coloring-cocycle} cancels $e^{-\log\Delta(u,v)}$ from the height
density. The destination weight is $\Delta(u,v)W(v,s)$, so neighboring weights
have ratios in $[K^{-1},K]$; coloring and matrix data are independent of height.

For $0<l<H<\infty$ retain sites $l\le W\le H$ with the restricted relation,
graph, and principal data. The measure is finite, since
$\nu\{l\le W\le H\}=l^{-1}-H^{-1}$, so Section~\ref{sec:invariant} applies. Write
the cut costs $m^{l,H},d^{l,H},(d^i)^{l,H}$.

\begin{lemma}[Homogeneous costs]
\label{lem:homogeneous-costs}
Fix $D_0,K,r,t$, $\beta=80$, and the above data and partial coloring. For each
field $f$ in $m,d,d^1,\ldots,d^r$ there is a bounded measurable $\bar f$ on $X$
such that on every band
$\mathcal U_{b_1,b_2}=\{(v,s):b_1\le W(v,s)\le b_2\}$, $0<b_1<b_2<\infty$,
\begin{equation}\label{eq:coloring-band-limit}
 \frac{f^{l,H}}{W}\longrightarrow\bar f(v)
 \quad\text{in }L^1(\mathcal U_{b_1,b_2},\nu)
 \quad\text{jointly as }l\downarrow0,\ H\uparrow\infty.
\end{equation}
The errors vanish uniformly over all such bases and configurations with the
fixed parameters. There is $C_0$ depending only on those parameters bounding all
normalized cutoff fields on retained sites and all limits; the limits are
independent of height and satisfy
\begin{equation}\label{eq:coloring-limit-assignment}
 \min_{1\le i\le r}\bar d^{i}-\bar d\le24\bar m
 \quad\text{a.e. on }\{c=0\}.
\end{equation}
\end{lemma}

\begin{proof}
The two cutoffs use different controls: weights above $H$ have invariant measure
$H^{-1}$ (rank comparison controls the upper end); with the upper cutoff fixed,
the edge-ratio bound makes sufficiently small weights invisible in any chosen
finite neighborhood of a fixed band. Prove uniform convergence on $1\le W\le e$
(the same handles any band bounded away from zero). Keep $l$ fixed and compare
$H_2\ge H_1\ge e$; passing from $H_2$ to $H_1$ deletes invariant measure at most
$1/H_1$, and Lemma~\ref{lem:rank-comparison} gives, on band sites, integrated
absolute differences of order $1/H_1$ for
\[
 G_v(z)\quad\text{and}\quad
 J_i(v,z)=\bigl(F^0_{v,i}(z)-z-a_v\bigr)G_v(z),\quad a_v=\tau_vW_v,
\]
targeting unassigned band sites for $J_i$, with constants uniform in
$l,H_1,H_2$, the configuration, and $z$ on upper-half-plane compacts.

The second-moment estimate of Lemma~\ref{lem:rooted-locality},
\begin{equation}\label{eq:coloring-second-moment}
 \int x^2\rho_v(dx)\le\bigl((\beta t)^2+K\bigr)W_v^2,
\end{equation}
also holds for the altered site choices, so these moments are uniformly bounded
on the band. They give a uniform positive lower bound for $|G_v(z)|$ on compacts:
a fixed bounded interval carries uniformly positive mass and
$-\operatorname{Im}G_v(z)=(\operatorname{Im}z)\int|z-x|^{-2}\rho_v(dx)$. The
residue bound of Lemmas~\ref{lem:polynomial-quotients},\ref{lem:rooted-locality}
bounds $J_i$ on compacts, so division is controlled in
\[
 F_v=1/G_v,\qquad F^0_{v,i}=z+a_v+J_i/G_v,\qquad
 F_{v,i}=F^0_{v,i}+(\beta-1)tW_v.
\]
The integrated differences of these quotients thus tend to zero as
$H_1\to\infty$, uniformly in $l$.

Recover costs from transform comparisons. Probabilities with the uniform
second-moment bound are weakly compact, and $\int x_+$ is continuous there by
uniform integrability; Cauchy transforms on a countable dense subset separate
them. So a prescribed accuracy of $m_v$ follows from finitely many transform
comparisons to sufficient accuracy, giving the integrated comparison for $m$.

For $d$ use the argument of the quotient. In a finite restriction let $n_P(x),
n_Q(x)$ count roots above $x$ with multiplicity. Interlacing and quotient
properties give, for $y>0$,
\begin{equation}\label{eq:coloring-argument}
 \arg F(x+iy)=\int_{\mathbb R}\frac{y}{(x-u)^2+y^2}(n_P-n_Q)(u)\,du,
 \qquad 0\le\arg F\le\pi,
\end{equation}
by writing the arguments of the root factors (branch fixed by
$\operatorname{Im}F>0$). Here $0\le n_P-n_Q\le1$, and
Lemma~\ref{lem:positive-cost} with \eqref{eq:coloring-second-moment} gives
\begin{equation}\label{eq:coloring-shift-tail}
 \int_B^\infty(n_P-n_Q)(x)\,dx=O(B^{-1})\quad(B\to\infty),
\end{equation}
uniformly on the band (including shifted options): the bound
$2\int(x-B)_+\rho_v+(a_v+B)\rho_v\{x>B\}$ is controlled by the second moment and
the band bound on $a_v$. On $[0,B]$,
\begin{equation}\label{eq:coloring-shift-poisson}
 \frac1\pi\int_0^B\arg F(x+iy)\,dx\quad\text{as }y\downarrow0
\end{equation}
approximates the integral of the spectral-shift function uniformly: with
$p_y(u)=y/(\pi(u^2+y^2))$ and
$\delta_B(y)=\|p_y*1_{[0,B]}-1_{[0,B]}\|_{L^1(\mathbb R)}\to0$ for fixed $B$, the
$L^1$ smoothing error plus \eqref{eq:coloring-shift-tail} give, uniformly,
\begin{equation}\label{eq:coloring-cut-cost-recovery}
 \Bigl|d_v-\frac1\pi\int_0^B\arg F_v(x+iy)\,dx\Bigr|\le\frac CB+\delta_B(y),
 \quad B\ge1,
\end{equation}
and the same for shifted options. For fixed $B,y$, convergence of costs and
locally uniform quotient convergence (Lemma~\ref{lem:bounded-costs}) pass this
to present band vertices of a bounded infinite cut and to present unassigned band
vertices for the options; $\operatorname{Im}F\ge y$ keeps the argument
continuous. First choose $B$ large, then $y$ small, then $H_1$ large for the
fixed segment $[0,B]+iy$, proving the high-cutoff comparison.

Remove the lower cutoff. For fixed finite upper cutoff,
Lemma~\ref{lem:bounded-costs} approximates costs uniformly by bounded
neighborhoods; a radius-$R$ neighborhood of a fixed-band root cannot contain a
vertex of weight below $K^{-R}$, so a small enough lower cutoff is invisible to
each fixed neighborhood calculation. This proves uniform convergence as
$l\downarrow0$ with that upper cutoff fixed. For joint convergence, compare both
large upper cutoffs to one fixed large cutoff by the preceding estimates, then
compare lower cutoffs there, giving uniform joint $L^1$-Cauchy estimates;
dividing by $W$ preserves convergence on a band.

The uniform bound follows from \eqref{eq:coloring-second-moment} and
Lemma~\ref{lem:positive-cost}: $m_v/W_v$ is uniformly bounded and
$d_v\le2m_v+a_v$, likewise for the options, independent of band, cutoff, and
configuration.

Identify the height dependence. Translating height by real $b$ scales weights by
$e^{-b}$ and shifts cutoffs correspondingly; all matrices and costs scale by the
same positive factor, so
\[
 \frac{f^{l,H}(v,s+b)}{W(v,s+b)}=\frac{f^{e^bl,e^bH}(v,s)}{W(v,s)}.
\]
Joint convergence makes the limit invariant a.e.\ under every rational height
translation simultaneously; a measurable function on $\mathbb R$ invariant a.e.\
under all rational translations is a.e.\ constant (by convolution). Applying on
almost every base fiber gives $\bar f(v)$, independent of height; measurability
follows from the band limits (e.g.\ integrate a representative over a fixed
height interval).

Finally Lemma~\ref{lem:assignment} gives the single-site inequality on finite
restrictions; Lemma~\ref{lem:bounded-costs} passes it to each bounded cut, and
joint band convergence gives a pointwise convergent subsequence for the finite
list of normalized fields; dividing by the positive root weight yields
\eqref{eq:coloring-limit-assignment}.
\end{proof}

Two compatibility properties follow. First, for a standard nonsingular
extension $(\widehat X,\widehat\mu,\widehat{\mathcal R})$ with $\pi:\widehat
X\to X$, $\pi_*\widehat\mu=\mu$, each class projecting bijectively onto a base
class, and cocycle $\widehat\Delta(\widehat u,\widehat v)=\Delta(\pi\widehat
u,\pi\widehat v)$: pulling back the data (including unused Bernoulli
coordinates), every bounded-cut ball is exactly a pullback, band $L^1$ errors
agree, and uniqueness of limits gives the same pullback identity a.e.\ for each
normalized limit field. Second, for a color permutation $\sigma$ (with
$\sigma(0)=0$), relabeling the finite polynomial laws gives
\[
 \bar m(\sigma\circ c)=\bar m(c),\quad \bar d(\sigma\circ c)=\bar d(c),\quad
 \bar d^{i}(\sigma\circ c)=\bar d^{\sigma^{-1}(i)}(c)
\]
a.e.; the last records the renaming of virtual options.

Define the nonnegative total cost
\begin{equation}\label{eq:coloring-total-cost}
 \mathcal S(c)=\int_X\bar m(v)\,d\mu(v),
\end{equation}
uniformly bounded by the above. The next lemma records strip averaging and
locality, which make $\mathcal S$ usable.

\begin{lemma}[Strip averages and locality]
\label{lem:strip-locality}
Under Lemma~\ref{lem:homogeneous-costs}, put
$\Sigma_n=\{e^{-n}\le W\le e^n\}$ and $f_{\Sigma_n}=f^{e^{-n},e^n}$. For every
$f$ in $m,d,d^1,\ldots,d^r$,
\begin{equation}\label{eq:coloring-strip}
 \frac1{2n}\int_{\Sigma_n}|f_{\Sigma_n}-W\bar f|\,d\nu\longrightarrow0
 \quad(n\to\infty),
\end{equation}
uniformly over configurations and bases with the fixed parameters. If
$c_1,c_2$ are partial colorings on the same base with the same graph, matrix,
and cocycle data, and $E_R$ is where they agree on the radius-$R$ ball, then
\begin{equation}\label{eq:coloring-locality}
 \int_{E_R}|\bar f(c_1)-\bar f(c_2)|\,d\mu\le\gamma(R),\qquad\gamma(R)\to0,
\end{equation}
with one $\gamma$ for the finite list of fields, uniform over configurations.
Both hold after taking a product with an auxiliary probability space keeping the
auxiliary parameter fixed along the relation; the cost fields are measurable
there.
\end{lemma}

\begin{proof}
For the strip integral, $|f_{\Sigma_n}-W\bar f|\,d\nu=|f_{\Sigma_n}/W-\bar
f|\,d\mu\,ds$. Divide $[-n,n]$ into unit intervals; translating each to a fixed
interval uses homogeneity from Lemma~\ref{lem:homogeneous-costs}. Intervals a
large fixed distance from both ends have translated cutoffs uniformly near zero
and infinity, so uniform joint convergence on the fixed band bounds their
average error by any positive number; the bounded number of end intervals has
uniformly bounded error, and its contribution over $2n$ tends to zero, proving
\eqref{eq:coloring-strip}.

For \eqref{eq:coloring-locality}, approximate both limiting normalized fields
on a fixed unit height band in $L^1$ by common fixed lower and upper cutoffs
(errors uniform by Lemma~\ref{lem:homogeneous-costs}). On this cut,
Lemma~\ref{lem:bounded-costs} approximates each field uniformly by a finite-radius
calculation, and agreement of the base colorings on a large enough ball makes
these lifted local data identical. Since the normalized limits are independent
of height, integrating on the unit band bounds the base integral; choosing cutoff
then radius for each accuracy proves \eqref{eq:coloring-locality}.

Finally, bounded-cut calculations are measurable (limits of rooted finite
calculations), remain so with measurable parameters, and an auxiliary parameter
fixed along the relation changes no degree, edge ratio, or spectral estimate; the
uniform estimates hold on its product probability space, and the $L^1$ limit
gives jointly measurable limit fields there.
\end{proof}

\subsection{The cost detects balls with excessive norm}

The all-unassigned configuration has zero cost; once a ball is fully assigned, a
violation of the norm bound forces definite positive-root cost in it, and the
invariant lift sums this without assuming invariance of the original measure.

\begin{lemma}[Initial cost and bad balls]
\label{lem:bad-balls}
Assume the hypotheses and parameters of Theorem~\ref{thm:bernoulli-coloring}. The
all-unassigned coloring has $\mathcal S=0$. Fix $m\ge1$, $\alpha>0$. For a
measurable partial coloring $c$ let $\operatorname{Bad}_m(c)$ be the set of $v$
where every vertex of $B_G(v,m)$ is assigned and
$\|T_{B_G(v,m),c}\|>2\beta t+\alpha$. Then there is $C_4$ depending only on
$m,D_0,K,\alpha$ with
\begin{equation}\label{eq:coloring-bad-balls}
 \mu(\operatorname{Bad}_m(c))\le C_4\mathcal S(c).
\end{equation}
The same holds on probability extensions with pulled-back data.
\end{lemma}

\begin{proof}
For the all-unassigned coloring Lemma~\ref{lem:finite-cost-endpoints} gives zero
cost on every finite principal restriction; finite restriction limits and
Lemma~\ref{lem:homogeneous-costs} give $\mathcal S=0$.

Partition bad centers into a bounded number of measurable classes with pairwise
graph distance greater than $2m+1$. Construction: a bounded-degree Borel graph
has a countable measurable partition into independent sets (assign to each vertex
a finite prefix of an injective standard-space code long enough to separate it
from its neighbors, including the prefix length; equal codes are not adjacent);
visiting these in sequence and giving each vertex the first color not used by an
earlier-colored neighbor from a palette of size one more than the degree bound
gives a measurable proper coloring. Apply to the graph joining vertices at
distance at most $2m+1$, and intersect its color classes with
$\operatorname{Bad}_m(c)$; there are boundedly many in terms of $D_0,m$.

Fix one class $C$. In the lift retain $\Sigma_n$ and select lifted centers above
$C$ whose height is at least $m\log K$ from both ends of $[-n,n]$; each radius-$m$
ball lies in the strip, and the balls are disjoint and nonadjacent along each
orbit (\eqref{eq:coloring-height-change} plus the cocycle law give exactly one
lifted site above each base site in a fixed orbit, and distinct centers project
to distinct centers of $C$ at distance $>2m+1$). Delete every strip site outside
these balls; by Lemma~\ref{lem:integral-change} deletion cannot increase total
integrated cost, so
\begin{equation}\label{eq:coloring-ball-deletion}
 \int_{\Sigma_n}m_{\Sigma_n}\,d\nu\ge\int m_{\mathrm{retained}}\,d\nu.
\end{equation}
Each retained component is one selected finite ball; invariant mass transport
sends rooted costs to its center, so the right side equals the integral over
selected lifted centers of $s(P_{\mathrm{ball}})$.

For a bad ball, Lemma~\ref{lem:finite-cost-endpoints} gives
$s(P_{\mathrm{ball}})\ge(\alpha/2)\min_{\mathrm{ball}}W$, and the edge-ratio
bound gives
\begin{equation}\label{eq:coloring-ball-root-cost}
 s(P_{\mathrm{ball}})\ge\frac\alpha2K^{-m}W_{\mathrm{center}}.
\end{equation}
Divide \eqref{eq:coloring-ball-deletion} by $2n$, let $n\to\infty$:
\eqref{eq:coloring-strip} makes the left side tend to $\mathcal S(c)$, while on
the right $W\,d\nu=d\mu\,ds$ and the allowed height interval has length
$2n-2m\log K$, so \eqref{eq:coloring-ball-root-cost} gives
$(\alpha/2)K^{-m}\mu(C)\le\mathcal S(c)$. Summing over the bounded number of
classes proves \eqref{eq:coloring-bad-balls}. Every step uses only the stated
probability-space, cocycle, degree, and matrix bounds, so it applies to the
extensions.
\end{proof}

\subsection{Assigning a small fraction of the remaining vertices}

At a current partial coloring $c$ with $E_0=\{c=0\}$ determined by coordinates
already used, choose at each $v\in E_0$ a color $i(v)$ minimizing $\bar d^i(v)$,
breaking ties by a fresh uniform permutation of the colors (taking the first
minimizer in the permutation). The costs of $c$ are defined first on the old
coordinates then pulled back to the tie coordinates, and bounded-cut definitions
and limits commute with this pullback, so $i(v)$ uses only old and tie
coordinates.

Add fresh uniform times $U_v\in(0,1)$ independent of old and tie coordinates.
For $0\le\theta<1$ define $c^\theta$ by assigning $i(v)$ at each $v\in E_0$ with
$U_v<\theta$, leaving other sites as in $c$. This is measurable on the relative
Bernoulli extension; null exceptions are filled by measurable defaults before
adding the next independent coordinates, with negligible saturations.

\begin{lemma}[One assignment step]
\label{lem:hazard-step}
Under Theorem~\ref{thm:bernoulli-coloring}, let $c$ be a partial coloring
determined before the fresh times, with $i(v)$ and $c^\theta$ as above. There is
a nonnegative $g$, depending only on the fixed graph and polynomial parameters,
with $g(h)\to0$ as $h\downarrow0$ and, for every $0<h<1/2$,
\begin{equation}\label{eq:coloring-step-bound}
 \mathcal S(c^h)\le(1+24h)\mathcal S(c)+hg(h),
\end{equation}
the same $g$ working for all steps and current configurations.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:integral-change} to $c\to c^h$ on the lift restricted to
$\Sigma_n$, ordering changed sites by $U_v$ (a.e.\ distinct along every countable
orbit off a saturated null set; the order need not be a well-order, as allowed).
At a comparison site $v$ the pre-change configuration is $c^{U_v}$ with $U_v$
fixed as a threshold along the orbit.

We claim
\begin{equation}
 \begin{split}
 &\int_{\Sigma_n}\bigl(m_{\Sigma_n}(c^h)-m_{\Sigma_n}(c)\bigr)\,d\nu\\
 &\quad=\int_0^h\frac{d\theta}{1-\theta}
   \int_{\Sigma_n}1_{\{v\in E_0,U_v\ge\theta\}}
   \bigl[d_{\Sigma_n}^{i(v)}(c^\theta)-d_{\Sigma_n}(c^\theta)\bigr]\,d\nu,
 \end{split}
 \label{eq:coloring-finite-strip-step}
\end{equation}
$\nu$ including label laws and the height measure, $v$ the base site of the lifted
root. To justify, condition at a root in $E_0$ on old coordinates, all tie
coordinates, every fresh time except this root's, and the height. For a fixed
threshold $\theta$, leave the root unassigned and call the bracket
$D(\theta)$; whenever the root time $u_0\ge\theta$, $D$ is independent of its
value (chosen colors precede the fresh times, other sites use their own times,
and neither the graph nor the restricted weight data use the new root
coordinate), holding for deterministic rooted finite-subset calculations and
hence for bounded-strip costs; the same at $u_0=\theta$ since the update is
strict. When $u_0<h$ the ordered comparison increment at the root is $D(u_0)$
outside the null event of a tie, and the last uniform time gives
$\int_0^hD(u_0)\,du_0=\int_0^h\frac{d\theta}{1-\theta}\int_\theta^1D(\theta)\,du_0$.
The bounded-strip computations are measurable jointly in threshold and data, so
Fubini (first a.e.\ root and height, then the rest) proves
\eqref{eq:coloring-finite-strip-step}, before choosing any homogeneous limit
version depending on $\theta$.

Pass to that limit in an integrated expression. Divide by $2n$ and use
Lemma~\ref{lem:strip-locality}. On the right adjoin $\theta$ with normalized
Lebesgue measure on $[0,h]$, fixed along the relation, and regard $c^\theta$ as
one configuration on the augmented base; the lemma applies uniformly there, its
absolute errors controlling those after the mask, after the finite selection
$i(v)$, and after the factor $(1-\theta)^{-1}\le2$. For a height-independent
limit field, $W$ cancels the height density leaving an interval of length $2n$,
so
\begin{equation}
 \begin{split}
 \mathcal S(c^h)-\mathcal S(c)
 &=\int_0^h\frac{d\theta}{1-\theta}\int_X1_{\{v\in E_0,U_v\ge\theta\}}
   \bigl[\bar d^{i(v)}(c^\theta)-\bar d(c^\theta)\bigr]\,d\mu.
 \end{split}
 \label{eq:coloring-limit-step}
\end{equation}
All fields have jointly measurable versions on the augmented space; no
evaluation of an unspecified limit at the random diagonal is used.

Compare the intermediate costs with the old ones. Let $D_R$ bound the cardinality
of a radius-$R$ ball. On the event that this ball has no update $c\to
c^\theta$, the colorings agree there; \eqref{eq:coloring-locality} controls the
integrated difference of their cost fields on that event, and summing over the
finite option list controls the selected field. Conditional on old and tie
coordinates, the probability of an update in the ball is at most $D_Rh$ (union
bound); on the update event the cost differences are uniformly bounded by
Lemma~\ref{lem:homogeneous-costs}. These hold for fixed $\theta$ or jointly with
normalized measure on $[0,h]$, so replacing the two costs in
\eqref{eq:coloring-limit-step} by their values at $c$ creates absolute error at
most
\begin{equation}\label{eq:coloring-step-error}
 Ch\bigl(\gamma(R)+D_Rh\bigr),
\end{equation}
$C$ depending only on the fixed parameters, the factor $(1-\theta)^{-1}\le2$.

The old costs and the minimizing $i(v)$ are independent of the fresh root time;
the survival probability $1-\theta$ cancels the denominator. By
\eqref{eq:coloring-limit-assignment} the old-cost contribution is at most
\[
 h\int_{E_0}\bigl(\bar d^{i(v)}(c)-\bar d(c)\bigr)\,d\mu(v)
 \le24h\int_{E_0}\bar m(c)\,d\mu\le24h\mathcal S(c).
\]
For each accuracy in \eqref{eq:coloring-step-error}, take $R$ large to control
$\gamma(R)$ then $h$ small to control $D_Rh$; the infimum over integer $R$ gives
a nonnegative $g(h)\to0$, proving \eqref{eq:coloring-step-bound} uniformly.
\end{proof}

\subsection{Completion of the coloring}

\begin{proof}[Proof of Theorem~\ref{thm:bernoulli-coloring}]
Work on the relative Bernoulli extension of the original base, reserving
independent coordinates at each site for each step. Start from all-unassigned
(cost zero by Lemma~\ref{lem:bad-balls}). Choose $T_0>0$, $h\in(0,1/2)$, and apply
Lemma~\ref{lem:hazard-step} with fresh coordinates for $N=\lceil T_0/h\rceil$
steps. With $S_j$ the cost after $j$ steps, iteration gives
\begin{equation}\label{eq:coloring-iteration}
 S_N\le\frac{g(h)}{24}\bigl((1+24h)^N-1\bigr),
\end{equation}
tending to $0$ as $h\downarrow0$ for fixed $T_0$ (since
$(1+24h)^N\le\exp(24(T_0+h))$).

An unassigned site receives a color at its first successful hazard; conditional
on the base point, the probability of remaining unassigned after $N$ steps is
$(1-h)^N\le e^{-T_0}$, independent of the selected color. If $D_m$ bounds a
radius-$m$ ball, the conditional probability that any site of it stays unassigned
is at most $D_me^{-T_0}$. The unconditional probability that the ball is fully
assigned but violates the norm bound is at most $C_4S_N$ by
Lemma~\ref{lem:bad-balls}. Take $T_0$ large then $h$ small so
$D_me^{-T_0}+C_4S_N<\eta$. Complete the partial coloring by fresh independent
uniform colors at remaining sites; every previously complete ball is unchanged,
so \eqref{eq:coloring-ball-conclusion} holds.

For the marginals, fix a permutation $\sigma$ of $\{1,\ldots,r\}$ and let
$\Theta_\sigma$ apply it to color names in every tie-breaking permutation and
every final uniform color input, fixing all times. It fixes the base point,
commutes with rerooting, preserves each $\kappa_v$, fixes the height, and acts
isometrically on band $L^1$. Inductively suppose $c\circ\Theta_\sigma=\sigma\circ
c$. Exact pullback of the finite-cut calculations and color covariance give
\[
 \bigl((d^{\sigma(i)})^{l,H}(c)\bigr)\circ\Theta_\sigma=(d^i)^{l,H}(c),
 \quad1\le i\le r,
\]
and the band $L^1$ isometry plus uniqueness of limits gives
$(\bar d^{\sigma(i)}(c))\circ\Theta_\sigma=\bar d^i(c)$ a.e. So the minimizing
set at the transformed input is the $\sigma$-image of the original, the
transformed tie-breaker chooses the corresponding color, and the update test is
unchanged (times fixed).

There are finitely many color permutations and, with $T_0,h$ fixed, finitely
many steps. Take a common conull set for the required identities, intersect its
images under the finite color action, and discard the saturation of the
complement (null by nonsingularity); the induction then holds throughout,
starting from all-unassigned. The final uniform inputs transform the same way, so
the completed coloring obeys $c\circ\Theta_\sigma=\sigma\circ c$. Since
$\Theta_\sigma$ preserves the conditional product measure over each base point,
disintegration shows all output colors at the root have equal conditional
probabilities a.e.; summing to one gives \eqref{eq:coloring-marginal} and
completes the proof.
\end{proof}

\section{A Bernoulli model and transfer of state moments}
\label{sec:transfer}

Throughout this section $A\subset M$ is a masa, $M$ has separable predual, and
$\varphi$ is a faithful normal state with $A\subset M_\varphi$. Keep the
notation of Section~\ref{sec:kernels}: $N$ is generated by the groupoid
normalizers of $A$, $E_N:M\to N$ is the $\varphi$-preserving conditional
expectation, $M^\circ=\ker E_N$, and $N_{\rm alg}$ is the algebra of finite sums
of base functions times kernel groupoid normalizers; the Cartan inclusion
$A\subset N$ is represented by a countable nonsingular measured relation
$\mathcal R$ on $(X,\mu)$ with its scalar cocycle.

The coloring of Section~\ref{sec:coloring} lives on the Bernoulli extension of
this relation. We place that extension and $M$ in a common von Neumann algebra,
get a norm bound for the part of an operator in $M^\circ$, then show every fixed
scalar moment of finitely many model cylinder functions is a limit of averaged
moments in $M$, which brings the model color projections back into $A$.

\subsection{The model and a free compression bound}

Let $\widetilde X$ be the relative Bernoulli extension of $(X,\mu,\mathcal R)$
from Section~\ref{sec:coloring} with one Lebesgue label at each site of a
relation class. Write $\widetilde\mu$ for its probability and $B=L^\infty(
\widetilde X,\widetilde\mu)$. Re-rooting preserves the label assignment, so each
extended class projects bijectively onto its base class. Let $\widetilde N$ be
the von Neumann algebra of the extended relation with the cocycle pulled back.

\begin{lemma}[The Bernoulli free-product model]
\label{lem:bernoulli-model}
Pulling back base kernel operators embeds $N$ as a von Neumann subalgebra of
$\widetilde N$, and there is a faithful normal conditional expectation
$\widetilde E_N:\widetilde N\to N$ whose restriction to $B$ integrates labels
conditional on the base point. With $\widetilde\varphi$ the diagonal integral in
$\widetilde N$, $(\varphi|_N)\circ\widetilde E_N=\widetilde\varphi$. The reduced
amalgamated free product $\mathcal K=M*_N\widetilde N$ has a faithful normal
conditional expectation $E_N^{\mathcal K}$ onto $N$ restricting to $E_N$ and
$\widetilde E_N$; the state $\Phi=(\varphi|_N)\circ E_N^{\mathcal K}$ is faithful
and normal, restricts to $\varphi$ and $\widetilde\varphi$, and satisfies
$B\subset\mathcal K_\Phi$. The two constituents are free over $N$: the
expectation of every alternating product of centered letters from
$\ker E_N$ and $\ker\widetilde E_N$ is zero.
\end{lemma}

\begin{proof}
Use the right counting representations of Section~\ref{sec:kernels}. Under the
bijection of extended and base classes, a pulled-back base kernel acts by the
same orbit operator on each fiber, so pullback is a normal faithful
representation (an amplification over conditional label spaces; matrix
coefficients disintegrate to normal functionals). The pulled-back copy of $N$
lies in $\widetilde N$ since this holds for generating kernel normalizers.

The isometry from the original space to the extension space making vector fields
constant in labels intertwines the two copies of $N$ and sends the diagonal state
vector to the new one. Compression by it maps $\widetilde N$ normally into $N$.
On finite sums of $B$-functions times lifted base groupoid normalizers,
compression integrates coefficients over labels and lands in $N$; these form a
weakly dense $*$-algebra (lifted partial isomorphisms enumerate the extended
relation, cuts by extension-measurable sets and scalar phases give its kernel
normalizers), so normality and weak closure give the range assertion for
$\widetilde N$.

The compression is unital, $N$-bimodular, identity on $N$, so it is the asserted
normal conditional expectation; it preserves the diagonal state, which is
faithful by the faithful normal diagonal expectation of a countable nonsingular
relation algebra, so $\widetilde E_N$ is faithful. The diagonal expectation is
$B$-bimodular and $B$ abelian, so $B\subset\widetilde N_{\widetilde\varphi}$.

Both expectations onto $N$ are faithful and normal and the constituents are
$\sigma$-finite, so the reduced amalgamated free-product construction gives
$\mathcal K$, $E_N^{\mathcal K}$, generation, and freeness over $N$
\cite[Section~2]{Ued99},\cite[Section~2]{Ued13} (these do not require tracial
states); faithfulness and normality of $\Phi$ follow from those of the
expectation and $\varphi|_N$.

For completeness, the product has $\Phi$-preserving expectations onto each
constituent: compress the state representation to the closure of one
constituent's state vectors. The span of $N$ and alternating centered words is
weakly dense, and every reduced word with a centered letter from the other
constituent compresses to zero (expand at the ends, absorb $N$-coefficients, use
freeness). The retained algebra acts normally and faithfully on its state-vector
space, so compression is normal, bimodular, state-preserving onto that
representation; by Takesaki \cite{Tak72} the modular group restricts to each
constituent (agreeing with \cite[eq.~(2.1)]{Ued13}). Since $B$ is in the
centralizer of $\widetilde\varphi$, it is in $\mathcal K_\Phi$.
\end{proof}

\begin{lemma}[Free compression]
\label{lem:free-compression}
In the model of Lemma~\ref{lem:bernoulli-model}, let $p\in B$ be a projection
with $\widetilde E_N(p)=\lambda1$, $0\le\lambda\le1$. For every $w\in M^\circ$,
\begin{equation}\label{eq:free-compression}
 \|pwp\|\le(\lambda+2\sqrt\lambda)\|w\|.
\end{equation}
\end{lemma}

\begin{proof}
In $L^2(\mathcal K,\Phi)$ let $P$ be the orthogonal projection onto the closed
span of reduced-word vectors whose first letter is a centered $M$-letter. The
orthogonal complement is densely spanned by $N$-vectors and reduced words whose
first letter is a centered $\widetilde N$-letter (generation and freeness over
$N$). Left multiplication by $w$ sends these generators of the complement into
$\operatorname{ran}P$, so $(1-P)w(1-P)=0$. Left multiplication by
$p-\lambda1$ on a word starting with a centered $M$-letter inserts a centered
$\widetilde N$-letter at the front, orthogonal to $\operatorname{ran}P$; hence
$PpP=\lambda P$ and $\|Pp\|^2=\|PpP\|\le\lambda$. Insert $P+(1-P)$ around $w$ in
$pwp$: the $(1-P)w(1-P)$ term vanishes and the three remaining terms have norms
at most $\lambda\|w\|$, $\sqrt\lambda\|w\|$, $\sqrt\lambda\|w\|$, proving
\eqref{eq:free-compression}.
\end{proof}

\subsection{Finite symbols and their evaluation in the masa}

Fix an integer $s\ge2$ and partition each Lebesgue label into $s$ equally likely
symbols identified with the $s$th roots of unity (dyadic, $s$ a power of two,
suffices later). A \emph{finite cylinder function} is a bounded function on
$\widetilde X$ depending measurably on the base point and on these symbols at a
fixed finite list of translated sites, each given by a partial isomorphism with
graph in $\mathcal R$ (possibly also on whether the base point lies in its
domain). Let $\mathcal Y$ be the algebra of finite sums of such functions times
lifted base kernel groupoid normalizers; covariance and composition make it a
$*$-algebra, and $\widetilde E_N(\mathcal Y)\subset N_{\rm alg}$.

Choose increasing finite Borel partitions $\mathcal P_j$ of $X$ generating $A$
and separating points off the fixed null exceptions. Write
$\mathcal P=\{e_a\}_a$; for independent uniform $s$th roots of unity $z_a$ let
$u_{\mathcal P}=\sum_az_ae_a\in A$. Replace the symbol at a translated site by
the value of $u_{\mathcal P}$ there and each lifted base normalizer by its
original in $N$; this defines an evaluation $\mathrm{ev}_{\mathcal P}:\mathcal
Y\to N_{\rm alg}$, and finite cylinder functions evaluate to elements of $A$.

Evaluation respects multiplication and adjoints and fixes $N_{\rm alg}$;
substitution is consistent under re-rooting (hence with covariance) and respects
a.e.\ identities: conditional on a.e.\ base point every assignment of symbols to
a finite list of distinct sites has positive probability, so a cylinder identity
holds for every such assignment off a base null set, including those from
$u_{\mathcal P}$; for sums with normalizers, check finitely supported root
columns in the counting representations with common cocycle phases, equality
a.e.\ giving equality by faithfulness of the diagonal state. So the definition
is independent of the expression. Evaluated coefficients have norm at most their
essential bound; for a fixed finite normalizer sum its expression bounds all
evaluations uniformly (no uniform bound over arbitrary finite expressions is
needed).

The transfer result needs a uniform approximation estimate for fixed scalar
words, which also serves the final clipping.

\begin{lemma}[Uniform right multiplication in the state norm]\label{lem:state-multipliers}
Let $\varphi$ be a faithful normal state on $M$. For every $C<\infty$, fixed
$z\in M$, and $\eta>0$ there is $\delta>0$ such that
\begin{equation}\label{eq:uniform-fixed-right}
 \|D\|\le C,\ \|D\|_\varphi<\delta\ \Longrightarrow\ \|Dz\|_\varphi<\eta,
\end{equation}
uniformly over such $D$. For $U,D\in M$, $b\in M_\varphi$,
\begin{equation}\label{eq:state-left-centralizer}
 \|UD\|_\varphi\le\|U\|\|D\|_\varphi,\qquad \|Db\|_\varphi\le\|b\|\|D\|_\varphi.
\end{equation}
The right multiplier bound is uniform when $z$ ranges over a norm-bounded set
precompact in state norm. Hence a bounded family of state-norm errors tending
uniformly to zero stays so after right multiplication by a fixed finite word
whose factors range over such precompact sets or over bounded subsets of the
centralizer; in particular any specified bounded sequence converging in state
norm can serve as a right factor uniformly over the whole sequence. These
permit bounded strong-* approximation of fixed letters in fixed-length scalar
words, uniformly in intervening bounded centralizer factors and in the varying
indices of other approximation sequences.
\end{lemma}

\begin{proof}
Let $\Omega$ be the GNS vector; faithfulness makes it separating, hence cyclic
for $M'$. Pick $c'\in M'$ with $\|z\Omega-c'\Omega\|<\alpha$; then for
$\|D\|\le C$,
$\|Dz\|_\varphi\le C\alpha+\|Dc'\Omega\|=C\alpha+\|c'D\Omega\|
\le C\alpha+\|c'\|\|D\|_\varphi$, and choosing $\alpha$ first gives
\eqref{eq:uniform-fixed-right}, including parametrized families.

The left multiplier bound is immediate. For $b\in M_\varphi$ and $D_0\ge0$,
centrality gives $\varphi(b^*D_0b)=\varphi(D_0bb^*)$, and
$c\mapsto\varphi(D_0c)$ is positive on $M_\varphi$ ($c\ge0$ gives
$\varphi(c^{1/2}D_0c^{1/2})$), so $\varphi(b^*D_0b)\le\|b\|^2\varphi(D_0)$,
proving \eqref{eq:state-left-centralizer}.

For a norm-bounded state-norm precompact $\mathcal C$ take a finite approximation
$z_1,\ldots,z_m$; if $z$ is close to $z_i$ then
$\|Dz\|_\varphi\le\|Dz_i\|_\varphi+C\|z-z_i\|_\varphi$, and the finitely many
fixed-factor moduli give one for $\mathcal C$ (a convergent sequence plus limit is
precompact, covering every bounded $z_j$ with $\|z_j-z\|_\varphi\to0$). Apply
right multiplier bounds in the order of factors; errors stay uniformly bounded in
operator norm so each modulus applies. For a scalar word with middle error $D$,
$|\varphi(UDV)|\le\|U\|\|DV\|_\varphi$ and telescoping reduce it to the preceding
bounds. Strong-* approximation ensures state-norm convergence of a replaced
letter and adjoint; for each fixed approximant the finite words may be expanded,
and its index sent to infinity after the uniform estimates.
\end{proof}

\begin{proposition}[Transfer of fixed scalar moments]
\label{prop:moment-transfer}
Let $A\subset M$, $\varphi$, $\mathcal K$ be as in Lemma~\ref{lem:bernoulli-model}.
Fix a symbol size $s$, the cylinder algebra $\mathcal Y$, and the generating
$(\mathcal P_j)$. For a fixed integer $k\ge1$ and fixed factors $Z_1,\ldots,Z_k$,
each in $M$ or $\mathcal Y$, set $Z_h^{(j)}=Z_h$ for an $M$-factor and
$Z_h^{(j)}=\mathrm{ev}_{\mathcal P_j}(Z_h)$ for a $\mathcal Y$-factor, using the
same symbols for all evaluations in a word. Then
\begin{equation}\label{eq:moment-transfer}
 \lim_{j\to\infty}\mathbb E_{\{z_a\}}
 \varphi\bigl(Z_1^{(j)}\cdots Z_k^{(j)}\bigr)=\Phi(Z_1\cdots Z_k).
\end{equation}
\end{proposition}

\begin{proof}
We record how operator approximation enters, reduce to centered cyclic words,
average over label equalities, and kill the surviving equalities by a
state-Cauchy--Schwarz shortest-pair argument fed by
Proposition~\ref{prop:atomless-kernels}.

\emph{Uniform approximation in fixed words.} Lemma~\ref{lem:state-multipliers}
controls bounded state-norm errors with the right word from fixed operators and
bounded $A$-functions. Expanding each finite normalizer sum if needed,
telescoping and $|\varphi(UDV)|\le\|U\|\|DV\|_\varphi$ bound word errors
uniformly over evaluated diagonal functions. If a fixed $N$-letter is
approximated by a bounded strongly convergent sequence from $N_{\rm alg}$, the
sequence and its limit are state-norm precompact, so the lemma applies to the
whole sequence as right factors; expand each normalizer sum only after fixing an
approximant, pass to the partition limit, then send the approximation error to
zero. The same holds in the model with $\Phi$ and bounded $B$-factors
($B\subset\mathcal K_\Phi$).

\emph{Reduction to centered alternating words.} Induct on the number of factors.
A length-one $M$-word is an identity. A length-one $\mathcal Y$-word: average its
cylinder coefficients; at a.e.\ base point distinct sites of a fixed finite
translated list eventually lie in distinct atoms of $\mathcal P_j$, so the
averaged coefficient converges boundedly a.e.\ to its conditional label average,
which is also state-norm convergence with the fixed normalizers allowed as right
multipliers; this proves \eqref{eq:moment-transfer} in length one.

Fold adjacent same-algebra factors. For an alternating word subtract and add the
$N$-expectation of each model factor (the same expansion on the evaluated side;
a centered model factor need not stay centered after evaluation). Expectations
of $\mathcal Y$-factors lie in $N_{\rm alg}$ and are fixed by evaluation; each
term with an expectation folds into a neighbor, reducing length. An arbitrary
$N$-element multiplying a $\mathcal Y$-factor is first approximated boundedly
strong-* by $N_{\rm alg}$ (generation and Kaplansky), legitimate on both sides by
the uniform word estimates. The induction leaves fully centered alternating
words, whose model expectation is zero by freeness.

Approximate each centered $M$-factor boundedly strong-* by an entire analytic
element for $\sigma^\varphi$ in $M^\circ$ (convolve its modular orbit with
Gaussian approximate identities; complex translations of the Gaussian give the
entire continuation; $E_N$ commutes with the modular group, so the convolutions
and their complex translates stay centered). Each centered $\mathcal Y$-factor is
a finite sum of $vb$, $v$ a base kernel groupoid normalizer, $b$ a cylinder
function with zero conditional label average: move each coefficient right of its
normalizer by covariance, absorb bounded base coefficients into $b$, and subtract
the summand expectation (sum zero). Cut the initial support of $v$ as in
Section~\ref{sec:kernels} to be entire analytic, preserving centering with
arbitrarily small state errors in the fixed words. Cylinder functions and their
evaluations lie in the centralizers, so complex modular shifts of $vb$ are
evaluation-compatible (the shift only changes the cut normalizer by the same
bounded base formula in both algebras). Fix these analytic approximants before the
partition limit; their complex-shift norms enter later bounds.

\emph{Odd words.} For an odd alternating word of length at least three the first
and last factors lie in the same constituent. Apply the KMS identity
$\varphi(ZY)=\varphi(Y\sigma^\varphi_{-i}(Z))$ to the first analytic factor (and
the $\Phi$-analogue), fold the last with the shifted first, and use the induction
hypothesis on the shorter word, whose model value equals the original centered
word's and is zero.

\emph{Even words.} Rotate to start in $M^\circ$; expand other-side factors as
$vb$ and absorb each $v$ into the preceding centered $M$-factor (modular shifts
only add fixed base coefficients). For word length $2m$ it remains to prove the
averaged evaluations of
\begin{equation}\label{eq:cyclic-centered-word}
 \varphi(y_1d_1\cdots y_md_m)
\end{equation}
tend to zero, where the $y_h$ are fixed bounded entire analytic elements of
$M^\circ$ and each $d_h\in A$ is a substitution in a centered finite cylinder
function; constants may depend on this fixed word and the analytic approximants.

\emph{Fourier expansion and index equalities.} Split each cylinder domain by the
domains of its translated sites and the equality pattern among them; on each
piece its distinct symbols are independent. Fourier expansion on the finite
product of cyclic groups writes the centered coefficient as a finite sum of
monomials with bounded $A$-coefficients; centering removes the constant term and
we delete zero-exponent occurrences, leaving each monomial with at least one
occurrence, all nonzero mod $s$, and within each block $d_h$ the occurrence sites
pointwise distinct on the indicated support.

After further splitting, regard each translation at an occurrence as an
automorphism of $A$ (the extension off the support need not be in $\mathcal R$).
Proposition~\ref{prop:atomless-kernels} makes the $y_h$ annihilate the atomic part
of $A$ on both sides, so only diffuse supports contribute. Partial isomorphisms
map these supports into the diffuse part off null sets; splitting a nonnull common
domain piece into two positive-measure pieces gives nonnull diffuse complements
(the other piece and its image), and the isomorphism theorem for atomless standard
probability spaces \cite[Theorem~17.41]{Kec95} extends each restricted map to a
measure-class automorphism of the diffuse part (identity on the atomic part),
giving finitely many terms with $\lambda_o(f)=f\circ\gamma_o$, $\gamma_o$ a
nonsingular bijection, at each occurrence $o$.

Expand \eqref{eq:cyclic-centered-word} in one partition index $a_o$ per
occurrence. Averaging the independent $z_a$ keeps exactly the equality patterns
with zero total Fourier exponent mod $s$ in each class; since individual
exponents are nonzero, every class has at least two occurrences. Inclusion--
exclusion over coarsenings removes inequalities between classes, so it suffices to
prove vanishing of a sum with prescribed equalities within the classes of a
partition of occurrences, each class of size at least two, no inequalities
between classes; occurrence $o$ contributes $\lambda_o(e_{a_o})$.

Keep selected pair equalities as explicit indices. If selected pairs form a forest
in each class, extend to a spanning tree and encode the other tree edges with
independent phases: for an edge between $a,b$,
$\mathbf1_{\{a=b\}}=\mathbb E_{\boldsymbol\zeta}(\zeta_a\overline{\zeta_b})$
with independent uniform circle phases (fresh vector per edge). At occurrence $o$
let $w_o(a)$ be the product of incident auxiliary edge phases (conjugates per
orientation, empty product $1$), so $|w_o(a)|=1$, and set
$U_o=\sum_aw_o(a)\lambda_o(e_a)$; since $\{\lambda_o(e_a)\}_a$ is a partition,
$U_o$ is a diagonal unitary, with $\sum_a$ unretained giving $U_o$ and at retained
$j$, $w_o(j)\lambda_o(e_j)=U_o\lambda_o(e_j)$. So each $U_o$ is independent of the
retained indices; different $U_o$ may be correlated through the phases, and the
estimates hold uniformly over their joint realizations before averaging. Products
of these factors commute within a block.

\emph{A pair in one block.} If two occurrences of a class lie in one block, retain
their pair equality, encode the rest, and sum the pair index to
$p_{\mathcal P}=\mathbf1_{\mathrm{support}}\sum_a\lambda_o(e_a)\lambda_{o'}(e_a)$;
the sites are distinct on the support and partitions separate points, so
$\mu(p_{\mathcal P})\to0$. KMS-rotate this projection to the front (only fixed
analytic letters shift; diagonal factors stay in the centralizer) to get
$\varphi(p_{\mathcal P}Z)$ with $\|Z\|$ uniformly bounded; centrality gives
$|\varphi(p_{\mathcal P}Z)|=|\varphi(p_{\mathcal P}Zp_{\mathcal P})|\le
\|Z\|\mu(p_{\mathcal P})$, so these sums tend to zero uniformly in the phases.

\emph{A shortest pair between blocks.} It remains to treat patterns with at most
one occurrence of any class per block, so $m\ge2$. Number blocks by
$\mathbb Z/m\mathbb Z$ and for distinct $h,k$ let
$d_m(h,k)=\min\{(k-h)\bmod m,(h-k)\bmod m\}$ with residues in $\{1,\ldots,m-1\}$.
Among same-class pairs choose one minimizing this cyclic distance, call the
minimum $\ell$, and choose the endpoint order so one shortest arc runs forward,
labeling its blocks $0,1,\ldots,\ell$ (either order allowed in an antipodal tie).

For interior $h=1,\ldots,\ell-1$ pick an occurrence and another of its class; its
mate lies outside the chosen arc including endpoints (a mate on the arc in block
$k$ would give $d_m(h,k)\le|h-k|<\ell$, contradicting minimality). This also makes
interior classes distinct from each other and from the endpoint class; $\ell=1$
has no interior blocks.

Retain the endpoint pair and these interior-to-mate pairs; these lie in distinct
classes, form a forest, and are included in the spanning trees. Retained indices
are $J=(j_0,j_1,\ldots,j_{\ell-1})$, $j_0$ linking the endpoints. Write
$E_0=\lambda_0(e_{j_0})$, $E_\ell'=\lambda_\ell(e_{j_0})$,
$E_h=\lambda_h(e_{j_h})$ ($1\le h<\ell$); each $j_h$ ($h\ge1$) also occurs in one
further retained projection in an exterior block (several may share a block).

KMS-rotate to begin at $E_0$ and insert $E_0$ at the far end inside the state by
centrality. Absorb fixed diagonal coefficients into neighboring fixed letters
(within a block place retained projections first, then auxiliary unitaries, then
the fixed coefficient; absorb it into the following letter). The fixed
intervening letters (including modular translates) stay bounded analytic
elements of $M^\circ$; relabel as $y_h$, and write $V_h\in A$ for the auxiliary
unitaries (independent of explicit indices). Split at $E_\ell'$ using
$(E_\ell')^2=E_\ell'$. Before averaging the unitaries the summands are
$\varphi(X_JY_J)$ with
\begin{align}
 X_J&=E_0V_0\Bigl(\prod_{h=1}^{\ell-1}y_hE_hV_h\Bigr)y_\ell E_\ell',\notag\\
 Y_J&=E_\ell'V_\ell B_{j_1,\ldots,j_{\ell-1}}E_0,
 \label{eq:shortest-pair-factors}
\end{align}
the product in $X_J$ increasing and omitted when $\ell=1$; in
$B_{j_1,\ldots,j_{\ell-1}}$ each exterior mate projection for these indices
occurs exactly once and every other factor is independent of them; diagonal
factors in a block commute. (Figure~\ref{fig:shortest-pair} shows one cyclic
pattern.)

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,
  block/.style={draw,rounded corners=1pt,minimum width=1.1cm,minimum height=0.75cm,inner sep=2pt},
  word/.style={-{Stealth[length=1.5mm]},line width=0.35pt},
  equality/.style={densely dashed,line width=0.7pt},
  every node/.style={font=\small}]
\node[block] (d0) at (0,1.5) {$d_0$};
\node[block] (d1) at (2,1.5) {$d_1$};
\node[block] (d2) at (4,1.5) {$d_2$};
\node[block] (d3) at (6,1.5) {$d_3$};
\node[block] (d4) at (6,-1) {$d_4$};
\node[block] (d5) at (4,-1) {$d_5$};
\node[block] (d6) at (2,-1) {$d_6$};
\node[block] (d7) at (0,-1) {$d_7$};
\draw[word] (d0.east) -- (d1.west);
\draw[word] (d1.east) -- (d2.west);
\draw[word] (d2.east) -- (d3.west);
\draw[word] (d3.south) -- (d4.north);
\draw[word] (d4.west) -- (d5.east);
\draw[word] (d5.west) -- (d6.east);
\draw[word] (d6.west) -- (d7.east);
\draw[word] (d7.north) -- (d0.south);
\coordinate (o51) at ($(d5.north)+(-0.2,0)$);
\coordinate (o52) at ($(d5.north)+(0.2,0)$);
\draw[equality] (d0.north) to[out=90,in=90] node[above,pos=0.5] {$j_0$} (d3.north);
\draw[equality] (d1.south) to[out=-65,in=110] node[left,pos=0.45] {$j_1$} (o51);
\draw[equality] (d2.south) to[out=-70,in=70] node[right,pos=0.45] {$j_2$} (o52);
\foreach \point in {d0.north,d3.north,d1.south,d2.south,o51,o52}
  \fill (\point) circle[radius=1.4pt];
\node[fill=white,inner sep=2pt] at (3,2.1) {short arc: $X_J$};
\node at (3,-1.7) {complementary arc: $Y_J$};
\end{tikzpicture}
\caption{Selected equality links in one fixed cyclic word with eight blocks and
$\ell=3$. The short arc $0\to1\to2\to3$ and its complement indicate the positions
of $X_J$ and $Y_J$; the two marks in block $5$ are distinct mate occurrences.
Further occurrences and links are omitted; distinct prescribed classes may use
equal numerical indices after inequalities are removed.}
\label{fig:shortest-pair}
\end{figure}

State Cauchy--Schwarz gives
\begin{equation}\label{eq:shortest-pair-cs}
 \Bigl|\sum_J\varphi(X_JY_J)\Bigr|
 \le\Bigl(\sum_J\varphi(X_JX_J^*)\Bigr)^{1/2}
   \Bigl(\sum_J\varphi(Y_J^*Y_J)\Bigr)^{1/2}.
\end{equation}
The second square sum is uniformly bounded: drop $E_\ell'$ from the positive bound
of $Y_J^*Y_J$, order the exterior mate occurrences left to right (commutativity in
a block), write $\mathbf j=(j_1,\ldots,j_{\ell-1})$, and at the first remaining
mate factor $B_{\mathbf j}=C_0F_jB'_{\mathbf j'}$ ($F_j$ its projection partition,
$C_0,B'_{\mathbf j'}$ independent of $j$), so
$\sum_jB_{\mathbf j}^*B_{\mathbf j}
={B'_{\mathbf j'}}^*(\sum_jF_jC_0^*C_0F_j)B'_{\mathbf j'}
\le\|C_0\|^2{B'_{\mathbf j'}}^*B'_{\mathbf j'}$; iterating over the remaining mates
(with the fixed intervening norm bounds) gives
$\sum_{\mathbf j}B_{\mathbf j}^*B_{\mathbf j}\le C1$ (for $\ell=1$ just the fixed
exterior word bound). Summing the $E_0$ corners in the state (a centralizer
partition) leaves a bound independent of $\mathcal P$ and the unitaries.

The first square sum in \eqref{eq:shortest-pair-cs} gives the smallness, from a
collision between the two endpoint indices; the interior pinchings converge first
at fixed coarse endpoint data, which are refined only afterward. Removing $V_0$
by centrality and summing interior indices in $X_JX_J^*$ gives
\begin{equation}\label{eq:endpoint-collision}
 \sum_{j_0}\varphi\bigl(\lambda_0(e_{j_0})
   C_{\mathcal P}(\lambda_\ell(e_{j_0}))\lambda_0(e_{j_0})\bigr),
\end{equation}
where $C_{\mathcal P}:A\to M$ is $C_{\mathcal P}(f)=L_1$ with
$L_\ell=y_\ell fy_\ell^*$ and
$L_h=y_hS_{\lambda_h(\mathcal P)}(V_hL_{h+1}V_h^*)y_h^*$ for
$h=\ell-1,\ldots,1$ (for $\ell=1$, $C_{\mathcal P}(f)=y_1fy_1^*$).

Fix a coarse $\mathcal Q$ from the generating sequence and let $\mathcal P$
refine it. Enlarging each input projection in \eqref{eq:endpoint-collision} to its
coarse piece and regrouping output corners by centrality bounds that expression by
\begin{equation}\label{eq:coarse-endpoint-bound}
 \sum_{e\in\mathcal Q}\varphi\bigl(\lambda_0(e)
   C_{\mathcal P}(\lambda_\ell(e))\lambda_0(e)\bigr).
\end{equation}
For fixed $f\in A$ the nested $C_{\mathcal P}(f)$ converges in state norm to the
expression with each pinching replaced by $E_A$ and the $V_h$ removed, uniformly
over the unitaries: work innermost outward; each translated sequence
$\lambda_h(\mathcal P_j)$ is generating, so Lemma~\ref{lem:pinching} gives
state-norm convergence to $E_A$ on the fixed input; conjugation by $V_h$ commutes
with pinching, preserves the state norm, and fixes the limiting $A$-valued datum;
pinching is state-norm contractive, and Lemma~\ref{lem:state-multipliers}
propagates the error through the fixed $y_h,y_h^*$, uniformly in the remaining
phases.

Define positive kernels $K_h(f)=E_A(y_hfy_h^*)$ ($1\le h\le\ell$). For each fixed
$f=\lambda_\ell(e)$, $e\in\mathcal Q$, the preceding convergence and finiteness of
$\mathcal Q$ give the limit of \eqref{eq:coarse-endpoint-bound}:
\begin{equation}\label{eq:coarse-kernel-collision}
 \sum_{e\in\mathcal Q}\int_X\lambda_0(e)(K_1\circ\cdots\circ K_\ell)
   (\lambda_\ell(e))\,d\mu.
\end{equation}
Proposition~\ref{prop:atomless-kernels} (applied to the adjoints of the $y_h$)
makes these kernels and their composition atomless a.e.; in compositions a
preceding kernel avoids the fixed null set where a following one may fail to be
atomless.

Refine $\mathcal Q$ along the generating sequence. The integrand sum tests
whether $\gamma_0$ of the output site and $\gamma_\ell$ of the input site lie in
the same piece; these conditions decrease to equality of the two points. Since
$\gamma_\ell$ is one-to-one, this graph has zero conditional mass for each
atomless row of the composed kernel. Nonsingularity makes fixed null exceptions
irrelevant, and bounded convergence shows \eqref{eq:coarse-kernel-collision}
tends to zero.

For each fixed coarse $\mathcal Q$, the limsup of the first square sum, even after
taking the supremum over the auxiliary unitaries at each fine $\mathcal P$, was
bounded by \eqref{eq:coarse-kernel-collision}. First take that fine-partition
limsup, then refine $\mathcal Q$: the first square sum vanishes uniformly in the
auxiliary phases. The second square sum was uniformly bounded, so
\eqref{eq:shortest-pair-cs} proves vanishing of the equality-pattern sum. Averaging
the auxiliary phases, undoing the finite inclusion--exclusion and Fourier
expansions, and then removing the fixed analytic approximations proves the required
vanishing in \eqref{eq:cyclic-centered-word}; the induction proves
\eqref{eq:moment-transfer}.
\end{proof}

The transfer statement uses a fixed finite symbol alphabet; the next
approximation applies it to the measurable color partitions of
Theorem~\ref{thm:bernoulli-coloring}.

\begin{lemma}[Cylinder approximation of projection partitions]
\label{lem:cylinder-partitions}
In the Bernoulli extension of Lemma~\ref{lem:bernoulli-model}, let
$p_1,\ldots,p_r\in B$ be projections with $\sum_ip_i=1$, $p_ip_h=0$ for $i\ne h$.
For every $\delta>0$ there is a projection partition $b_1,\ldots,b_r\in B$, each a
finite cylinder function for one common finite dyadic symbol size, with
$\|b_i-p_i\|_\Phi<\delta$ ($1\le i\le r$). Every evaluation of this cylinder
partition by the finite-symbol substitution gives a projection partition of $1$ in
$A$ off the fixed null sets of its definition.
\end{lemma}

\begin{proof}
The extension $\sigma$-algebra is generated by base measurable sets and countably
many label-coordinate readings along the partial graphings, so dyadic
approximations of label coordinates show the finite cylinder set algebra
(arbitrary base measurable sets allowed) is dense in probability. Approximate the
sets of the $p_i$ by cylinder sets, disjointize in order, and put the complement
in one piece, keeping exactly $r$ pieces covering the identity; symmetric
differences can be made arbitrarily small, giving the state-norm bounds. Finitely
many pieces and coordinates allow one common finite dyadic symbol size.

The identities $b_i=b_i^*=b_i^2$, $b_ib_h=0$ ($i\ne h$), $\sum_ib_i=1$ are
cylinder identities, and the full-support argument in the construction of
$\mathrm{ev}_{\mathcal P}$ shows substitution preserves them, so the evaluations
form a genuine projection partition in $A$.
\end{proof}

\section{Completion of the paving argument}\label{sec:completion}

We combine the coloring theorem and the moment transfer. The coloring estimates
apply to finite-support approximations of a Cartan operator, but before
transferring a fixed moment we must return to the original operator without a
rate depending exponentially on the moment order; a clipping argument does this.
Its nontracial use needs the following elementary fact about varying right
multipliers.

\subsection{Continuous calculus for varying sequences}

\begin{lemma}\label{lem:multiplier-calculus}
Let $\mathcal L$ be a von Neumann algebra with faithful normal state $\omega$, and
let $\mathscr R_\omega$ be the set of uniformly bounded sequences $(R_n)$ in
$\mathcal L$ with $\|D_nR_n\|_\omega\to0$ whenever $(D_n)$ is uniformly bounded
and $\|D_n\|_\omega\to0$. Then $\mathscr R_\omega$ is an algebra closed in
supremum norm, containing all constant sequences and all uniformly bounded
sequences in the centralizer of $\omega$, and closed under adding uniformly
bounded sequences that converge to zero in $\|\cdot\|_\omega$. If
$(U_n),(V_n)\in\mathscr R_\omega$ are self-adjoint, uniformly bounded, and
$\|U_n-V_n\|_\omega\to0$, then for every continuous $f$ on a compact interval
containing their spectra, $(f(U_n)),(f(V_n))\in\mathscr R_\omega$ and
$\|f(U_n)-f(V_n)\|_\omega\to0$.
\end{lemma}

\begin{proof}
Constant and centralizer sequences lie in $\mathscr R_\omega$ by
Lemma~\ref{lem:state-multipliers}. If $(R_n),(S_n)\in\mathscr R_\omega$ and
$(D_n)$ is bounded state-norm-null, then $(D_nR_n)$ is too, so
$\|D_nR_nS_n\|_\omega\to0$; addition is immediate. Closure in supremum norm
follows by approximating the right multiplier uniformly before the limit in $n$.
If $(E_n)$ is bounded state-norm-null, $\|D_nE_n\|_\omega\le\|D_n\|\|E_n\|_\omega
\to0$, so $(R_n+E_n)\in\mathscr R_\omega$.

For polynomial $f$, membership follows from the algebra property, and differences
from the telescoping
$U_n^j-V_n^j=\sum_{h=0}^{j-1}U_n^h(U_n-V_n)V_n^{j-1-h}$: each right multiplier is
in $\mathscr R_\omega$ and left multipliers are uniformly bounded, so each summand
tends to zero in state norm. Uniform polynomial approximation on the common
spectral interval gives both assertions for continuous $f$.
\end{proof}

\subsection{Paving in the presence of a central state}

\begin{proposition}\label{prop:expected}
Let $A\subseteq M$ be a masa and $\varphi$ a faithful normal state on $M$ with $A$
in its centralizer. For $0<\eps<1$ put
$r_{\mathrm e}=\lceil20000^2\eps^{-2}\rceil$. For every self-adjoint contraction
$x\in M$ and every $\rho>0$ there are a partition $P=(p_1,\ldots,p_{r_{\mathrm e}})$
in $A$ and a projection $q\in M$ with
\[
 \varphi(1-q)<\rho,\qquad \|q(S_P(x)-E_Ax)q\|\le\eps,
\]
$E_A$ the $\varphi$-preserving conditional expectation onto $A$.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:separable-reduction} it suffices to work with separable predual;
the reduction's conditional expectation restricts $E_A$, so a diagonal choice
there proves the stated one. Use the notation of
Sections~\ref{sec:kernels},\ref{sec:transfer}.

Fix $r=r_{\mathrm e}$ and write $a=E_Ax$, $y=E_Nx-E_Ax\in N$,
$w=x-E_Nx\in\ker E_N$. These are self-adjoint, $\|a\|\le1$,
$\|y\|,\|w\|\le2$, and $E_Ay=0$ (uniqueness of the $\varphi$-preserving
expectation onto $A$ gives $E_AE_N=E_A$).

\paragraph{Finite-support Cartan approximations.}
Choose self-adjoint contractions in $N_{\mathrm{alg}}$ converging strongly to
$E_Nx$: apply Kaplansky density to the norm closure of the finite normalizer
algebra, approximate its self-adjoint contractions in norm by self-adjoint
elements of $N_{\mathrm{alg}}$, and rescale; bounded strong approximation is
sequential in the faithful separable state representation. Each approximant
involves finitely many partial isomorphisms; we also bound their edge
Radon--Nikodym ratios: for a finite list of maps $\gamma$, delete from $X$ each
domain set where $\Delta(\gamma v,v)$ lies outside $[b^{-1},b]$, with $e_b\in A$
the indicator of the rest. As $b\to\infty$, $e_b$ increase strongly to $1$;
compressing the approximant by $e_b$ keeps it self-adjoint and contractive and
restricts each surviving edge to the required ratio range (enlarge the list by
inverses if needed). Choosing $b$ large for each approximant preserves strong
convergence.

Subtracting the diagonal gives
\begin{equation}\label{eq:cartan-approximants}
 y_n=y_n^*\in N_{\mathrm{alg}},\quad E_Ay_n=0,\quad \|y_n\|\le2,\quad
 \|y_n-y\|_\varphi\longrightarrow0.
\end{equation}
Each $y_n/2$ is a measurable zero-diagonal Hermitian contraction on orbit
$\ell^2$ spaces, supported on a finite-degree undirected graph with bounded edge
cocycle ratios (bounds may depend on $n$, which Theorem~\ref{thm:bernoulli-coloring}
allows without affecting $r$).

\paragraph{One partition for the two operator terms.}
Use the relative Bernoulli extension $\widetilde N$ and amalgamated product
$\mathcal K=M*_N\widetilde N$, $\Phi=\varphi|_N\circ E_N^{\mathcal K}$, from
Lemma~\ref{lem:bernoulli-model}. Apply Theorem~\ref{thm:bernoulli-coloring} to
$y_n/2$ with
\begin{equation}\label{eq:coloring-parameters}
 t=\frac4{\sqrt r},\qquad \alpha=\frac\eps{32},\qquad m=n,\qquad
 \eta=\Bigl(\frac\eps{16}\Bigr)^{2n}.
\end{equation}
The color projections $p_1^n,\ldots,p_r^n\in B$ form a partition, lie in the
centralizer of $\Phi$, and satisfy
\begin{equation}\label{eq:color-marginals-completion}
 \widetilde E_N(p_i^n)=r^{-1}1\qquad(1\le i\le r).
\end{equation}
With $\beta=80$ the norm bound, after restoring the factor two, obeys
\begin{equation}\label{eq:cartan-numerical-bound}
 2(2\beta t+\alpha)=\frac{1280}{\sqrt r}+\frac\eps{16}
 \le\frac{253}{2000}\eps<\frac\eps4.
\end{equation}

Put $\widehat Y_n=\sum_ip_i^ny_np_i^n\in\widetilde N$. At a root where the
radius-$n$ ball has the pinching bound, $\widehat Y_n^n$ on the root vector agrees
with the $n$th power of that principal matrix (every length-$n$ path stays in the
ball); elsewhere $\|\widehat Y_n\|\le2$. Integrating squared vector norms,
\begin{equation}\label{eq:cartan-high-moment}
 \Phi(\widehat Y_n^{2n})\le\Bigl(\frac\eps4\Bigr)^{2n}
       +2^{2n}\Bigl(\frac\eps{16}\Bigr)^{2n},
\end{equation}
using diagonal integration in the relation representation, not a trace on the
nonsingular relation algebra.

Set $Y_n=\sum_ip_i^n y p_i^n$ and $W_n=\sum_ip_i^n w p_i^n$. Pinching by a
centralizer partition is state-norm contractive, so \eqref{eq:cartan-approximants}
gives
\begin{equation}\label{eq:pinched-approximation}
 \|Y_n-\widehat Y_n\|_\Phi\longrightarrow0,
\end{equation}
and by Lemma~\ref{lem:free-compression} and \eqref{eq:color-marginals-completion},
\begin{equation}\label{eq:free-numerical-bound}
 \|W_n\|\le2\Bigl(\frac1r+\frac2{\sqrt r}\Bigr)
 \le\frac{40001}{200000000}\eps<\frac\eps4,
\end{equation}
orthogonality of the $p_i^n$ making the block-sum norm the max of block norms.
Thus the same $r$ projections control the complementary term; no second
refinement.

\paragraph{Clipping before fixing a transferred moment.}
Let $f:\mathbb R\to[-\eps/2,\eps/2]$, $f(s)=\max\{-\eps/2,\min\{s,\eps/2\}\}$.
Spectral calculus and \eqref{eq:cartan-high-moment} give
\[
 \Phi\bigl(1_{\{|\widehat Y_n|>\eps/2\}}\bigr)
 \le\Bigl(\frac2\eps\Bigr)^{2n}\Phi(\widehat Y_n^{2n})\le2^{-2n}+4^{-2n},
\]
and since $\|\widehat Y_n\|\le2$,
\begin{equation}\label{eq:approximate-clipping}
 \|\widehat Y_n-f(\widehat Y_n)\|_\Phi\longrightarrow0.
\end{equation}

Apply Lemma~\ref{lem:multiplier-calculus} in $\mathcal K$: each $(p_i^n)\in
\mathscr R_\Phi$ and $r$ is fixed, so $(Y_n)$ and $(W_n)$ (finite sums of products
of centralizer sequences and fixed operators) lie in $\mathscr R_\Phi$;
\eqref{eq:pinched-approximation} places $(\widehat Y_n)$ there too.
Equations~\eqref{eq:pinched-approximation}--\eqref{eq:approximate-clipping} and
continuous calculus give
\begin{equation}\label{eq:original-clipping}
 \|Y_n-f(Y_n)\|_\Phi\longrightarrow0.
\end{equation}

The self-adjoint $Z_n=Y_n+W_n=\sum_ip_i^n(x-a)p_i^n$ thus differ by a bounded
state-norm-null sequence from $Q_n=f(Y_n)+W_n$ with $\|Q_n\|\le3\eps/4$, and both
sequences lie in $\mathscr R_\Phi$. Telescoping powers, for each fixed $k\ge1$,
\begin{equation}\label{eq:fixed-model-moment}
 \limsup_{n\to\infty}\Phi(Z_n^{2k})\le\Bigl(\frac{3\eps}{4}\Bigr)^{2k}.
\end{equation}
Only a fixed power is used here; \eqref{eq:cartan-approximants} needed no
prescribed rate.

\paragraph{Transfer and the final spectral cut.}
Fix $\rho>0$. Choose $k$ with $(3/4)^{2k}<\rho/4$, then $n$ with
$\Phi(Z_n^{2k})<\tfrac\rho2\eps^{2k}$. With $n,k$ fixed, approximate
$(p_i^n)$ in $B$ by a finite-symbol cylinder partition (Lemma~\ref{lem:cylinder-partitions});
bounded state-norm approximation of self-adjoint projections is strong-*
approximation, so the fixed polynomial moment changes little and we keep the
strict bound with slack.

Proposition~\ref{prop:moment-transfer} applies to the expansion of this moment.
For a fine enough base partition the averaged evaluated moment stays strictly below
$\rho\eps^{2k}$. Each evaluation is a genuine projection partition in $A$ with
nonnegative even moment, so some deterministic evaluation gives a partition
$P=(p_1,\ldots,p_r)$ with
\[
 \varphi\Bigl(\bigl[\sum_{i=1}^r p_i(x-a)p_i\bigr]^{2k}\Bigr)<\rho\eps^{2k}.
\]
Let $z=\sum_ip_i(x-a)p_i=z^*$ and $q=1_{[-\eps,\eps]}(z)\in M$. Then
$\varphi(1-q)\le\eps^{-2k}\varphi(z^{2k})<\rho$ and $\|qzq\|\le\eps$. Since $a\in A$
commutes with the partition, $z=S_P(x)-a$. The number of colors is fixed before
$\rho$.
\end{proof}

\subsection{Arbitrary inclusions}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
If $x=0$, take $p_1=1$, $p_2=\cdots=p_{r_\eps}=0$, $a=0$, $q=1$. Otherwise replace
$x$ by $x/\|x\|$. Proposition~\ref{prop:expected} with
Proposition~\ref{prop:state-reduction} supplies the paving in the original
representation with
\[
 r=\Bigl\lceil400000000\eps^{-2}\Bigr\rceil+\Bigl\lceil4/\eps\Bigr\rceil+1
\]
projections, depending only on $\eps$. The supporting corner and all partitions,
diagonal elements, and compressions may depend on $F$ and $\delta$ as
Definition~\ref{def:so-paving} permits. The diagonal has norm at most one and
\[
 r\le400000000\eps^{-2}+4\eps^{-1}+3\le400000007\eps^{-2}<500000000\eps^{-2}.
\]
Multiplying the diagonal by the original $\|x\|$ restores scale and gives every
assertion of Theorem~\ref{thm:main}.
\end{proof}

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\end{document}