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\title{The weak pinned planar distance theorem}
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\begin{abstract}
Fix \(n\) points in the plane.  Each point sees some number of
distances to the other \(n-1\) points.  How few can one point see,
and how often?  We prove the weak pinned Erd\H{o}s distinct-distance
conjecture: for every fixed \(\varepsilon>0\), all but \(o(n)\) of
the \(n\) points determine at least \(n^{1-\varepsilon}\) distinct
nonzero distances.  The proof tracks the size of a distance fiber at
each pin, converts an exact arithmetic identity into an exact
additive-overlap identity, and uses the two possible behaviours of a
single nonnegative scale to reach a contradiction.
\end{abstract}

\tableofcontents

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

\subsection*{The distance problem in one sentence}
Given a finite set \(P\subset\R^2\) of \(n\) points, consider the
distances from a fixed point \(x\in P\) to the other points.
For \(P\) chosen adversarially, how few distinct distances can
\(x\) see?  And even if a single \(x\) can see only very few
distances, is that really bad for most points at once?

The weak pinned conjecture is the statement that bad pins cannot
crowd out good ones: within every \(n\)-point planar set, at least one
\(x\) sees at least \(n^{1-\varepsilon}\) distances for every
\(\varepsilon>0\).  This paper proves a strictly stronger version:
almost every pin is good.

\subsection*{Definitions and main theorem}
For \(x\in P\) put
\[
D_x(P)=\{\lVert y-x\rVert_2:y\in P\setminus\{x\}\}.
\]
Erd\H{o}s asked in 1957 whether some pin always satisfies
\(|D_x(P)|\ge n^{1-\varepsilon}\) for every fixed
\(\varepsilon>0\) \cite{Erdos1957}.  This is the weak pinned
Erd\H{o}s conjecture.

The right object to track is the \emph{fiber size}, not just the
pin's count of distances.  For distinct \(x,y\in P\) define
\[
k_P(x,y)=|\{z\in P\setminus\{x\}:\lVert z-x\rVert_2=\lVert y-x\rVert_2\}|.
\]
So \(k_P(x,y)\) counts every point at the same distance from \(x\)
as \(y\) is, including \(y\) itself.  Different pairs use their own
source point; two pairs with the same numerical distance but
different sources are treated separately.

For \(n\ge2\) and \(s>0\) put
\[
F_n(s)=\sup_{\substack{P\subset\R^2\\|P|=n}}
 \frac{|\{(x,y)\in P^2:x\ne y,\ k_P(x,y)\ge n^s\}|}{n(n-1)}.
\]

\begin{theorem}[Main theorem]\label{thm:main}
For every fixed \(s>0\),
\[
F_n(s)\longrightarrow0\qquad(n\to\infty).
\]
\end{theorem}

The convergence is uniform: no separation, general position, or
coordinate hypothesis is imposed, and the constants only depend on
\(s\).  The exponent \(s\) is fixed; the theorem supplies no explicit
rate of decay.

\begin{corollary}[All but \(o(n)\) pins are good]\label{cor:pins}
For every fixed \(\varepsilon>0\),
\[
\sup_{\substack{P\subset\R^2\\|P|=n}}
 \frac{|\{x\in P:|D_x(P)|<n^{1-\varepsilon}\}|}{n}\longrightarrow0.
\]
In particular, every sufficiently large \(n\)-point planar set has a
pin determining at least \(n^{1-\varepsilon}\) distinct nonzero
distances.
\end{corollary}

\begin{proof}
For \(\varepsilon\ge1\) the assertion is trivial; every pin with
\(n\ge2\) sees at least one distance.  Suppose \(0<\varepsilon<1\)
and take \(s=\varepsilon/2\).  A pin with fewer than
\(n^{1-\varepsilon}\) distance classes can place at most
\(n^{1-\varepsilon+s}\) neighbors in classes of size below \(n^s\),
so it contributes at least \(n-1-n^{1-\varepsilon/2}\) pairs counted
by \(F_n(s)\).  If there are \(b\) such bad pins, then, for all large
\(n\),
\[
\frac{b}{n}\le
 \frac{(n-1)F_n(\varepsilon/2)}{n-1-n^{1-\varepsilon/2}}
 \longrightarrow0.
\]
The bound is uniform in \(P\).
\end{proof}

The corollary is nearly sharp.  If \(P\) consists of the vertices of a
regular polygon together with its center, the center sees a single
nonzero distance.  Exceptional pins are unavoidable; the theorem only
says that such pins cannot fill a positive fraction of the population.

\subsection*{Background and context}
Erd\H{o}s opened both the distinct-distance and repeated-distance
questions in 1946 \cite{Erdos1946}.  The global distinct-distance
problem asks for the minimum possible size of the union
\(\bigcup_{x\in P}D_x(P)\).  The integer square lattice gives sets
with at most a constant times \(n/\sqrt{\log n}\) distances
\cite[Thm.~1]{Erdos1946}.  Guth and Katz proved the matching lower
bound that every planar \(n\)-point set determines a constant times
\(n/\log n\) distances \cite[Thm.~1.1]{GuthKatz}.  Their result
concerns the union; the pinned version fixes the source.

Pinned bounds grew historically by combining incidence geometry with
stronger and stronger counting.  Solymosi and T\'oth obtained the
order \(n^{6/7}\) \cite[Thm.~1]{SolymosiToth}.  Tardos improved the
exponent with an entropy inequality \cite{Tardos2003}.  Katz and
Tardos pushed every exponent below
\[
\frac{48-14e}{55-16e}=0.864137\ldots
\]
\cite{KatzTardos}.  Each of these papers guarantees some pin in every
configuration.  The sharper pinned conjecture asks for order
\(n/\sqrt{\log n}\) distances at a pin; see the introduction
of \cite{LundPetridis}, where the distinction between the real planar
problem and pinned-distance theorems over other fields is documented.

Theorem~\ref{thm:main} recasts the problem as one about multiplicities
of a single numerical distance at a pin.  Corollary~\ref{cor:pins} is
its almost-every-pin consequence and resolves the weak pinned
conjecture as stated in \cite[Conj.~4.7]{WeakPinned2025}.

\subsection*{How the proof works}
Assume the theorem fails.  Then a positive density of ordered
distinct pairs have sources whose distance fiber is as large as
\(n^s\).  Record these pairs as a directed graph, and let \(\theta\)
be the limiting edge density.  A deletion procedure on the graph
forces both a density bound from below and a fiber size bound from
above, giving good control of large and small subsets.

Work inside a number field \(K\subset\C\) that contains the
coordinates of every point.  Write \(x=(u,v)\) and set
\[
Z_1(x)=u+iv,\qquad Z_2(x)=u-iv,\qquad i=\sqrt{-1}.
\]
The squared distance factors exactly:
\[
\lVert x-y\rVert_2^2=(Z_1(y)-Z_1(x))(Z_2(y)-Z_2(x)).\tag{1}
\]
The product formula in \(K\), applied to both coordinates, balances
the logarithms of these differences across all places, so on a single
fiber the sum of the two coordinate depths is a fixed constant.

To turn logarithms of distances into integers we build nested
partitions from randomly shifted square grids at each complex place.
The grid depth differs from the raw logarithmic depth by a small
error \(\Delta\) which has a distribution independent of the pair and
an exponential tail.  Replacing any cell with more than half the
points by its complement keeps the cell-size balanced and produces an
exact additive identity: the total overlap of two distinct points,
summed over all levels, all coordinates and all places, equals a
function of the first point plus a function of the second point plus
a constant.

That identity produces a nonnegative scale \(W\) (the expected
overlap of two distinct uniform points plus a correction for giant
complements).  Two facts pin down \(W\).  First, the variance
estimate compares the fiber law and the uniform law cell by cell and
shows that cells of size at most \(n^{1-c}\) carry only \(o(W+1)\) of
the overlap.  Second, the same variance estimate compares the
incoming-neighbour law at each target with the uniform law.

Then \(W\) either stays bounded or grows without bound along some
subsequence.

In the unbounded case, cells of mass bounded away from zero define
weighted trees.  Transporting mass along these trees uses a classical
distance formula for trees, and comparison of the fiber law with the
uniform law turns the fiber equation into an impossibility for three
independent uniform vertices.

In the bounded case, one complex embedding is normalised so that the
two coordinate laws have nonatomic weak limits.  The variance
estimate preserves the fiber equation in the limit and gives a
family of M\"obius maps between the two limits.  A short
order-statistics argument rules out any such family when both limit
laws are nonatomic.  Both alternatives are impossible, hence the
theorem.

\section{Notation and statement of the reducible problem}\label{sec:prelim}

Most of the proof is arithmetic rather than analytic.  This section
fixes the notation once so the rest can be short.

\subsection*{The extremal threshold}
Put
\[
\Theta(t)=\limsup_{n\to\infty}F_n(t),\qquad
s_*=\sup\{t>0:\Theta(t)>0\}.
\]
Each \(F_n\), and hence \(\Theta\), is nonincreasing.  Moreover
\(F_n(t)=0\) for \(t\ge1\), because a distance class at a pin has at
most \(n-1\) points.  Assuming Theorem~\ref{thm:main} fails, we get
\(0<s_*\le1\).  Fix
\begin{equation}\label{eq:ext-parameters}
 0<c<s_*/3,\qquad
 \max\{c,(1-c)s_*\}<s<s_*,\qquad c<\gamma<s,
\end{equation}
choosing \(s\) to be a continuity point of \(\Theta\).  Such an \(s\)
exists because a monotone function has at most countably many
discontinuities.  Set \(\theta=\Theta(s)>0\); positivity follows from
\(s<s_*\) and monotonicity of \(\Theta\).  All asymptotic estimates
below are with respect to the sequence of configurations constructed
in Section~\ref{sec:extremal}; their constants may depend on
\((c,s,\gamma,\theta)\) but not on \(n\) or on the configurations.

\subsection*{Two types of probability laws at a graph}
Once the directed graph \(E\) of "large-fiber" pairs is chosen, several
probability measures attached to it will appear again and again.  It
is convenient to introduce them all at once.  Partition \(E\) into
\emph{fibers}: two edges lie in the same fiber when they share their
source and their squared distance.  Write \(x_e\) for the source of a
fiber \(e\), \(F_e\subset P\setminus\{x_e\}\) for its target set, and
\(k_e=|F_e|\).  Let \(d^-(y)\) denote the number of incoming edges at
\(y\).

\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}p{0.15\textwidth}p{0.78\textwidth}@{}}
\hline
Law & Definition \\
\hline
\(\alpha\) & Uniform on \(P\). \\
\(\pi\) & Uniform on the \(n(n-1)\) distinct ordered pairs of \(P\). \\
\(q\) & Uniform on \(E\), with marginals \(q_X\) and \(q_Y\). \\
\(\mu_e\) & Uniform on \(F_e\). \\
\(\rho_y\) & Uniform on the set of incoming neighbours of \(y\); defined only if \(d^-(y)>0\). \\
\hline
\end{tabular}
\end{center}
To sample \(q\), choose a fiber \(e\) with probability \(k_e/|E|\) and
then choose \(y\sim\mu_e\), setting \(x=x_e\).  Equivalently, sample
\(y\sim q_Y\) and then \(x\sim\rho_y\).  These laws are attached to
the finite graph; they do not depend on the number field or the grids
introduced later.


\section{Extracting a graph of large fibers}\label{sec:extremal}

We start from the assumption that Theorem~\ref{thm:main} fails and pull
out a positive-density directed graph of pairs whose fibers are large.
The extraction also controls how much of a fiber can sit inside a
small subset of the vertex set.  These two properties are what allow
us later to compare fibers with the uniform distribution.

First a uniformity consequence of the choice of \(s\) as a continuity
point of \(\Theta\).

\begin{lemma}[Tail of the extremal sequence]\label{lem:ext-tail}
If \(r_N\to s\) and \(M_N\to\infty\), then
\[
  \sup_{m\ge M_N}(F_m(r_N)-\theta)_+\longrightarrow0,
\]
where the supremum is over integers \(m\ge2\) and \((u)_+=\max\{u,0\}\).
\end{lemma}

\begin{proof}
Fix \(\epsilon>0\).  Continuity of \(\Theta\) at \(s\) gives some
\(\delta>0\) with \(\Theta(s-\delta)<\theta+\epsilon/2\).  By the
definition of limsup, \(F_m(s-\delta)\le\Theta(s-\delta)+\epsilon/2\)
for all large \(m\).  For all large \(N\), \(r_N\ge s-\delta\), and
every \(m\ge M_N\) then lands in this range.  Monotonicity of \(F_m\)
in its argument gives \(F_m(r_N)\le\theta+\epsilon\).
\end{proof}

\begin{proposition}[The extremal graph]\label{prop:graph}
Under the present supposition there is a sequence of finite sets
\(P\subset\R^2\) of real algebraic points, with \(n=|P|\to\infty\),
and nonempty directed graphs \(E\) on \(P\), with the following
properties.  Their densities \(d_n=|E|/[n(n-1)]\) tend to \(\theta\),
and every fiber has \(k_e\ge n^{s-o(1)}\), uniformly in \(e\).  For a
fixed constant \(B_0\),
\begin{equation}\label{eq:ext-domination}
  q\le B_0\pi,\qquad q_X,q_Y\le B_0\alpha,\qquad
  \frac{q_X+q_Y}{2}\ge(1-o(1))\alpha
\end{equation}
pointwise, and
\begin{equation}\label{eq:ext-small-degree}
  q_Y\{y:d^-(y)<n^\gamma\}=O(n^{\gamma-1})=o(1).
\end{equation}
Put \(\lambda_n=1/\log n\).  For every subset \(C\subset P\), writing
\(m=|C|\) and \(p_e=\mu_e(C)\),
\begin{equation}\label{eq:subset}
 \E_e\,\1_C(x_e)p_e\1_{\{p_e\ge\lambda_n\}}
 \le
 \begin{cases}
   (1+o(1))\pi(C^2),&m>n^{1-c},\\
   o(1)\pi(C^2),&m\le n^{1-c}.
 \end{cases}
\end{equation}
All errors are uniform in vertices, fibers and subsets.
\end{proposition}

\begin{proof}
The proof is in three parts: first pass to a sequence of algebraic
configurations that preserve every distance equality, then delete
low-degree vertices and small fibers, then prove the subset estimate.

\paragraph{Passing to real algebraic points.}
For fixed \(N\), the numerator defining \(F_N(s)\) takes values in the
finite set \(\{0,1,\ldots,N(N-1)\}\), so it attains its maximum on some
configuration; no compactness argument is needed.  Choose \(N\to\infty\)
with \(F_N(s)\to\theta\) and one attaining configuration for each \(N\).

That configuration can be replaced by one with rational or real
algebraic coordinates without changing any distance class.  Label
its points \(1,\ldots,N\) and write the coordinate variables as
\((u_i,v_i)\), with \(D_{ij}=(u_i-u_j)^2+(v_i-v_j)^2\).  Require
\(D_{ij}>0\) for \(i<j\), and prescribe for every pair of unordered
pairs either the equality \(D_{ij}=D_{kl}\) or the disequality
\(D_{ij}\ne D_{kl}\) that holds in the original configuration.  These
are finitely many polynomial conditions over \(\Q\): a disequality
can be encoded as \((D_{ij}-D_{kl})^2>0\).  Their existential
sentence holds over \(\R\), so by transfer for real closed
fields~\cite[Thm.~3.1 and~4.1]{RCFTransfer} it holds over the real
algebraic numbers.  The resulting points are distinct and retain the
exact distance-equality pattern, so every fiber size is unchanged
and the value of \(F_N(s)\) is preserved.  The real threshold \(N^s\)
need not enter the transferred system: it only compares against the
integer class sizes, which are preserved.  No bound on the degrees or
heights of the algebraic coordinates is imposed.

\paragraph{Deleting vertices and small fibers.}
On this algebraic configuration, keep every edge counted by
\(F_N(s)\).  Each initial fiber has at least \(N^s\) edges.  Put
\[
 K_N=\frac{N^s}{\log N},\qquad
 f_N=\frac{\log K_N}{\log N}=s-\frac{\log\log N}{\log N},\qquad
 m_0=\lceil N/2\rceil.
\]
For all large \(N\), \(f_N>0\).  Lemma~\ref{lem:ext-tail} gives
\[
 b_N:=\max_{m_0\le m\le N}(F_m(f_N)-\theta)_+\longrightarrow0.
\]
Set
\[
 t_N=b_N+|F_N(s)-\theta|+\frac1{\log N},\qquad
 \eta_N=\sqrt{t_N},
\]
so \(\eta_N\to0\) and eventually \(\theta-\eta_N>0\).

Whenever a remaining nonempty fiber has fewer than \(K_N\) edges,
discard it.  If the current vertex count is \(m\) and a vertex has
combined in- and out-degree below \(2(\theta-\eta_N)(m-1)\), delete
that vertex and all its incident edges.  Clean out small fibers after
every vertex deletion and continue until all degree lower bounds hold.

There are at most \(N(N-1)/N^s\) initial fibers.  Restricting the
vertex set never splits a source-distance fiber, and each initial
fiber can be discarded at most once, costing fewer than \(K_N\) edges
each.  All fiber cleanups together therefore cost at most
\(N(N-1)/\log N\) edges.  If the deletion process reaches \(m_0\)
vertices, write \(A_N=N(N-1)\) and \(B_N=m_0(m_0-1)\).  The vertex
deletions cost fewer than
\[
 2(\theta-\eta_N)\sum_{m=m_0+1}^{N}(m-1)
 = (\theta-\eta_N)(A_N-B_N)
\]
edges.  After all cleanups, at least
\[
 F_N(s)A_N-(\theta-\eta_N)(A_N-B_N)-\frac{A_N}{\log N}
\]
edges remain.  Their excess over \((\theta+b_N)B_N\) is at least
\[
 \eta_N(A_N-B_N)-t_NA_N
 =\eta_NA_N\left(1-\frac{B_N}{A_N}-\eta_N\right)>0
\]
for large \(N\), because \(B_N/A_N\to1/4\).  But every remaining
edge has a fiber of at least \(K_N=N^{f_N}\ge m_0^{f_N}\) edges, whose
density is at most \(F_{m_0}(f_N)\le\theta+b_N\), a contradiction.

So the process stops on a set \(P\) of size \(n>m_0\) with a graph
\(E\) whose fibers all have size at least \(K_N\).  Summing the degree
lower bounds and using the definition of \(b_N\) gives
\begin{equation}\label{eq:ext-density}
  \theta-\eta_N\le d_n\le F_n(f_N)\le\theta+b_N.
\end{equation}
The upper bound applies because \(K_N\ge n^{f_N}\).  Thus
\(d_n\to\theta\) and \(k_e\ge K_N\ge n^{s-o(1)}\) uniformly in \(e\).

\paragraph{Marginal bounds.}
For every edge \(q(x,y)/\pi(x,y)=1/d_n\), so \(B_0=2/\theta\) works in
the first inequality of~\eqref{eq:ext-domination} for large \(n\).
Summing over one coordinate gives the marginal bounds.  At every
vertex the degree lower bound gives
\[
 \frac{q_X(x)+q_Y(x)}2
 \ge\frac{\theta-\eta_N}{d_n}\alpha(x)
 \ge\frac{\theta-\eta_N}{\theta+b_N}\alpha(x)
 =(1-o(1))\alpha(x).
\]
Only the sum of the two marginals is bounded below; separate lower
bounds are not needed.  For targets of small incoming degree,
\[
 q_Y\{y:d^-(y)<n^\gamma\}
 \le\frac{n\,n^\gamma}{|E|}=O(n^{\gamma-1}),
\]
proving~\eqref{eq:ext-small-degree} since \(\gamma<s<1\).

\paragraph{Large fibers inside subsets.}
Remains~\eqref{eq:subset}.  Define
\[
 T_n=\lambda_nK_N=\frac{N^s}{\log N\log n}.
\]
Since \(N/2<n\le N\) there is a sequence \(\delta_n\to0\) with
\(T_n\ge n^{s-\delta_n}\); in particular \(T_n\to\infty\).
Multiplying the left side of~\eqref{eq:subset} by \(|E|\) counts
exactly the edges \((x_e,y)\in C^2\) with \(|F_e\cap C|\ge\lambda_nk_e\).
Every such pair satisfies
\[
 k_C(x_e,y)\ge|F_e\cap C|\ge T_n.
\]
A nonzero contribution therefore needs \(m-1\ge T_n\), because the
source sits in \(C\) and is excluded from its own distance class.
Hence all bounded subsets contribute zero for large \(n\); the cases
\(m=0,1\) give zero at every \(n\).

For \(m>n^{1-c}\), the positive exponent \(s-\delta_n\) gives
\(T_n\ge n^{s-\delta_n}\ge m^{s-\delta_n}\).  Lemma~\ref{lem:ext-tail}
gives \(F_m(s-\delta_n)\le\theta+o(1)\) uniformly in these sizes.
Divide the pair count by \(|E|\):
\[
 \E_e\,\1_C(x_e)p_e\1_{\{p_e\ge\lambda_n\}}
 \le\frac{\theta+o(1)}{d_n}\frac{m(m-1)}{n(n-1)}
 =(1+o(1))\pi(C^2).
\]

For \(m\le n^{1-c}\), choose a fixed \(s'\) with
\(s_*<s'<s/(1-c)\), possible by~\eqref{eq:ext-parameters}.  For large
\(n\),
\[
 T_n\ge n^{s-\delta_n}\ge n^{(1-c)s'}\ge m^{s'}.
\]
The definition of \(s_*\) gives \(\Theta(s')=0\), so \(F_m(s')\to0\)
as \(m\to\infty\).  Every contributing subset has \(m\ge T_n+1\), so
\[
 \sup_{m\ge T_n+1}F_m(s')\longrightarrow0,
\]
again only integer sizes considered.  Dividing this pair count by
\(|E|\) gives \(o(1)\pi(C^2)\) uniformly over all contributing small
subsets; the other subsets contribute zero.  This also covers \(s'>1\),
when the corresponding pair count is identically zero.
\end{proof}

From now on we fix the sequence \(P\) and its graph \(E\) from
Proposition~\ref{prop:graph}.  In particular every \(k_e\ge n^\gamma\)
for large \(n\).  The graph and its sampling laws are now fixed.


\section{The arithmetic model: factorization and the product
formula}\label{sec:hierarchies}

The remaining proof is arithmetic.  This section builds the two
coordinate maps and the nested grids that turn logarithms of distances
into sums of nonnegative overlaps.

\subsection*{Two coordinate maps}
Take one algebraic configuration from Proposition~\ref{prop:graph}.
Fix a number field \(K\subset\C\) containing all its coordinates and
the imaginary unit \(i=\sqrt{-1}\).  For a point \(x=(u,v)\) put
\[
Z_1(x)=u+iv,\qquad Z_2(x)=u-iv.
\]
Both maps are injective, because \(u,v\) are real in the chosen
embedding.  At another complex embedding \(K\to\C\), the two
coordinates need not be complex conjugates; their injectivity and the
algebraic factorization below are the only properties we use.  For
distinct \(x,y\) the squared distance factors exactly in \(K\):
\begin{equation}\label{eq:hier-factor}
 t(x,y)=(Z_1(y)-Z_1(x))(Z_2(y)-Z_2(x)).
\end{equation}

\subsection*{The place-by-place depth}
Let \(v\) run over the places of \(K\).  At a finite place above a
prime \(p\), use the absolute value extending \(|p|_p=p^{-1}\), and
let \(\sigma_v=[K_v:\Q_p]/[K:\Q]\).  At an archimedean place \(v\),
all of which are complex since \(i\in K\), use complex modulus and
\(\sigma_v=2/[K:\Q]\).  The normalized product formula
\cite[Thm.~7.14--7.15 and~Lem.~8.6]{ProductFormula} reads
\begin{equation}\label{eq:hier-product}
 \sum_v\sigma_v\log|z|_v=0\qquad(z\in K\setminus\{0\}),
 \qquad \sum_{v\mid\infty}\sigma_v=1.
\end{equation}
The weights convert the norm-based finite absolute values and
squared complex moduli in that reference to the values used here,
after division by \([K:\Q]\).  Here and below logarithms are natural.
For a fixed configuration only finitely many finite places make any
coordinate difference a nonunit, because each nonzero element of
\(K\) has nonzero valuation at only finitely many prime ideals.

At a place \(v\), define the \emph{raw depths}
\[
 i_j(x,y)=-\log|Z_j(x)-Z_j(y)|_v,\qquad x\ne y,\ j=1,2.
\]
The place is suppressed in the notation.  On a fiber \(F_e\) with
source \(x_e\) and squared distance \(t_e\), the factorization
yields
\begin{equation}\label{eq:hier-fiber}
 i_1(x_e,y)+i_2(x_e,y)=h_e,\qquad
 h_e=-\log|t_e|_v,\qquad y\in F_e.
\end{equation}
The number \(h_e\) does not depend on the choice of \(y\) in the
fiber.

\subsection*{Nested grids at complex places}
At a finite place put \(d_j(x,y)=i_j(x,y)\).  The ultrametric
inequality makes \(d_j(x,y)\ge r\) an equivalence relation.  At a
complex place we obtain the same nesting from random square grids,
with a controlled error.

Independently of all vertex and fiber sampling, choose a uniformly
random orientation and \(\xi\) uniform on \([0,\log2)\).  Grid side
lengths are \(l_m=e^\xi 2^m\) for \(m\in\Z\), with
\(l_0=e^\xi\in[1,2)\).  Choose a shift uniform modulo \(l_0\) in each
oriented axis.  To obtain the shift modulo \(l_{m+1}\) from that
modulo \(l_m\) for \(m\ge0\), add \(0\) or \(l_m\) with probability
\(1/2\) each, independently each step and in each axis.  For \(m<0\)
reduce the initial shift modulo \(l_m\).  Every shift is uniform
modulo its side length and the resulting half-open square grids
nested.  Use such a grid family for each coordinate \(j\).  This
construction appears in Arora's Euclidean algorithms
\cite[Sec.~2.1]{Arora1998}.

For distinct points let \(d_j(x,y)\) denote minus the logarithm of
the smallest grid side at which the two images share a cell.  Put
\(d_j(x,x)=+\infty\) and, for \(x\ne y\),
\(\Delta_j(x,y)=d_j(x,y)-i_j(x,y)\).

The three depth quantities play distinct roles:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Quantity & Meaning for a distinct pair \\
\midrule
\(i_j(x,y)\) & Raw logarithmic depth, \(-\log|Z_j(x)-Z_j(y)|_v\) \\
\(d_j(x,y)\) & Sharing depth in the nested hierarchy \\
\(\Delta_j(x,y)\) & Hierarchy error, \(d_j(x,y)-i_j(x,y)\) \\
\bottomrule
\end{tabular}
\end{center}

\begin{lemma}[Grid error]\label{lem:hier-grid}
At every complex place the depths are almost surely finite on
distinct pairs.  The distribution of \(\Delta_j(x,y)\) is independent
of the distinct pair, the place and the configuration.  Its finite
mean is an absolute constant \(\kappa\), and absolute constants
\(C_0,b_0>0\) satisfy
\begin{equation}\label{eq:grid-error}
 \Pr(|\Delta_j(x,y)|>u)\le C_0e^{-b_0u}\qquad(u\ge0).
\end{equation}
At finite places \(\Delta_j=0\).
\end{lemma}

\begin{proof}
Write \(R>0\) for the separation of the two complex values and
\((a,b)\) for their displacement along the oriented axes.  At side
length \(l\), a uniform shift gives a sharing probability
\[
 (1-|a|/l)_+(1-|b|/l)_+.
\]
Sharing is impossible if \(l<R/\sqrt2\), while separation has
probability at most \((|a|+|b|)/l\le\sqrt2R/l\).  The separation
events shrink as the grids coarsen, with probabilities going to zero,
so almost surely the pair eventually shares a cell and there is a
smallest sharing scale.

Write \(\Delta=d_j(x,y)+\log R\).  A sharing side gives
\(\Delta\le\log\sqrt2\).  For \(u\ge0\), let \(l\) be the largest
grid side no larger than \(Re^u\), so \(l>Re^u/2\).  On
\(\{\Delta<-u\}\) the pair is separated at that side, so
\[
 \Pr(\Delta<-u)\le2\sqrt2e^{-u}.
\]
This proves~\eqref{eq:grid-error} and a uniform first absolute
moment.

For any real \(u\), the event \(\{\Delta\ge u\}\) is sharing at the
largest side \(l\le Re^{-u}\).  The law of \(l/R\) depends only on
\(u\), by the uniform logarithmic phase, and the displacement
direction relative to the axes is uniform.  The sharing formula
therefore makes the entire law of \(\Delta\) independent of the
pair.  Only countably many random variables and cell tests are used,
so all these events are measurable.
\end{proof}

\subsection*{Removing the giant cell}
Discard the null event on which any distinct-pair depth is not
finite.  For a place and coordinate \(j\), let \(\Pi_j(r)\) denote
the partition with sharing criterion \(d_j(x,y)\ge r\), for
\(r\in\R\).  These partitions are nested as \(r\) grows.  Call a
cell \emph{giant} if its size exceeds \(n/2\) and denote such a cell,
when it exists, by \(G_j(r)\).  Replace the giant by its complement
and keep all other cells.  The resulting \(\mathcal C_j(r)\) is a
multiset: if the complement already occurs as another cell, it occurs
twice.  Every \(C\in\mathcal C_j(r)\) has \(\alpha(C)\le1/2\).
Figure~\ref{fig:giant} illustrates why multiplicities matter.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=0.85cm,y=0.85cm,every node/.style={font=\small}]
  \node at (2.5,1.55) {Original partition, $n=8$};
  \draw[rounded corners,fill=black!5] (0,0) rectangle (3,1);
  \draw[rounded corners,fill=blue!8] (3.4,0) rectangle (5.4,1);
  \node at (1.5,0.5) {$\{1,2,3,4,5\}$};
  \node at (4.4,0.5) {$\{6,7,8\}$};
  \node at (1.5,-0.45) {Giant $G$};
  \node at (4.4,-0.45) {$C$};
  \draw[-{Stealth},thick] (5.65,0.5) -- (6.55,0.5);
  \node at (9.3,1.55) {Transformed multiset};
  \draw[rounded corners,fill=blue!8] (6.8,0) rectangle (8.8,1);
  \draw[rounded corners,fill=blue!8] (9.4,0) rectangle (11.4,1);
  \node at (7.8,0.5) {$\{6,7,8\}$};
  \node at (10.4,0.5) {$\{6,7,8\}$};
  \node at (7.8,-0.45) {$P\setminus G$};
  \node at (10.4,-0.45) {$C$};
\end{tikzpicture}
\caption{Replacing a giant by its complement can duplicate an
existing cell.  Both occurrences count towards every overlap
formula.}
\label{fig:giant}
\end{figure}

This replacement removes the overlap of all pairs at very coarse
scales.  The cost is a function of each endpoint separately, which
will cancel when we compare pair laws with the same combined
marginals.  Define, on a fixed realized hierarchy,
\begin{align*}
 M_j&=\sup\{r:G_j(r)\text{ exists}\},\\
 S_j(x)&=\int_{\{r:G_j(r)\text{ exists}\}}
              \1_{\{x\notin G_j(r)\}}\,dr,\\
 L_j(x,y)&=\int_\R\sum_{C\in\mathcal C_j(r)}
                    \1_C(x)\1_C(y)\,dr\qquad(x\ne y).
\end{align*}

\begin{lemma}[Additive identity for a fixed hierarchy]\label{lem:hier-depth}
For each realized hierarchy on \(n\ge2\) points, the displayed
quantities are finite, and for distinct \(x,y\),
\begin{equation}\label{eq:depth}
 d_j(x,y)=M_j+L_j(x,y)-S_j(x)-S_j(y).
\end{equation}
They are integrable over the grids.  At every finite place where all
distinct coordinate differences are units, \(M_j=S_j=L_j=0\).
\end{lemma}

\begin{proof}
Suppress \(j\).  Let \(d_-\) and \(d_+\) be the minimum and maximum
of the finitely many distinct-pair depths.  Below \(d_-\) the
partition has only the cell \(P\); above \(d_+\) it has only
singletons.  A giant at any level has a giant ancestor at every
coarser level, so a giant exists on an initial ray with endpoint
\(M\in[d_-,d_+]\), up to an immaterial endpoint convention.

Let \(o(r)=\1_{\{d(x,y)\ge r\}}\), \(g(r)=\1_{\{G(r)\text{ exists}\}}\),
and \(t(r)=\sum_{C\in\mathcal C(r)}\1_C(x)\1_C(y)\).  Expanding the
giant indicator through its complement gives, pointwise,
\[
 o(r)-g(r)=t(r)
       -\1_{\{G(r)\text{ exists},\ x\notin G(r)\}}
       -\1_{\{G(r)\text{ exists},\ y\notin G(r)\}}.
\]
The integral of the left side is \(d(x,y)-M\).  Each term on the
right vanishes outside \([d_-,d_+]\) on a distinct pair, and
\(t(r)\le2\).  Integrating proves~\eqref{eq:depth}, with bounds
\[
 0\le S(x)\le d_+-d_-,\quad 0\le L(x,y)\le2(d_+-d_-),\quad
 |M|\le\max(|d_-|,|d_+|).
\]
These are grid-integrable by Lemma~\ref{lem:hier-grid}, since the
configuration has finitely many pairs and finite raw depths.  At a
finite place with all raw depths zero, \(d_-=d_+=0\), proving the
final assertion.  Ties affect neither the identity nor any integral.
\end{proof}

The empty complement of \(P\) and singleton occurrences contribute
zero to every distinct-pair overlap.  We do not define \(L_j(x,x)\);
singleton cells persist to arbitrarily fine scales, so that integral
would diverge.

\subsection*{Averaging over places and grids}
For a nonnegative or integrable quantity depending on the place and
grids, write
\[
\avg{H}=\sum_v\sigma_v\E_{\mathrm{grids}}H,\qquad
\I(g)=\avg{\sum_{j=1}^2\int_\R
                 \sum_{C\in\mathcal C_j(r)}g\,dr}.
\]
There is no grid randomness at finite places.  The integrand \(g\)
may depend on the coordinate, level and cell occurrence.  All sums
over \(\mathcal C_j(r)\) retain multiplicities.  Set
\[
 S^*(x)=\avg{S_1(x)+S_2(x)},\qquad
 L^*(x,y)=\avg{L_1(x,y)+L_2(x,y)}\quad(x\ne y).
\]
Lemma~\ref{lem:hier-depth} and the finite set of nonunit places show
that these and \(\avg{M_1+M_2}\) are finite.  For each distinct pair
the product formula and Lemma~\ref{lem:hier-grid} give
\[
 \avg{d_1(x,y)+d_2(x,y)}=2\kappa.
\]
Taking expectations in~\eqref{eq:depth} is therefore legitimate and
gives an additive identity,
\begin{equation}\label{eq:additive}
 L^*(x,y)=b_*+S^*(x)+S^*(y)\quad(x\ne y),
 \qquad b_*=2\kappa-\avg{M_1+M_2}.
\end{equation}
In particular, a signed law on distinct pairs with zero total mass
and zero sum of its two marginals integrates to zero against \(L^*\).
This consequence is exact; no diagonal sampling error is involved.

The three nonnegative finite scales used below are
\begin{equation}\label{eq:hier-scales}
 W_2=\I(\pi(C^2))=\E_\pi L^*(x,y),\qquad
 A=\E_\alpha S^*(x),\qquad W=W_2+A.
\end{equation}
The quantity \(W_2\) measures distinct-pair overlap after the giant
is replaced; \(A\) measures the average time spent outside the giant.
Equation~\eqref{eq:additive} relates these scales to the graph and
fiber laws.


\section{Small cells and fiber mass}\label{sec:smallcells}

This section combines the subset estimate of Proposition~\ref{prop:graph}
with the additive identity~\eqref{eq:additive}.  The goal is that
cells of size at most \(n^{1-c}\) carry only \(o(W+1)\) of the
off-diagonal overlap.  The subtle case is a cell that contains a
source but only a small fraction of its own fiber; there the other
coordinate provides the missing control.

From~\eqref{eq:additive} and the uniform marginals of \(\pi\),
\(W_2=b_*+2A\).  If \((q_X+q_Y)/2\ge(1-\epsilon_n)\alpha\) with
\(\epsilon_n\to0\), then
\[
\E_q L^*(x,y)=W_2+\sum_{x\in P}
   \bigl(q_X(x)+q_Y(x)-2\alpha(x)\bigr)S^*(x).
\]
Since \(S^*\ge0\) and \(q\le B_0\pi\),
\begin{equation}\label{eq:q-overlap}
 W_2-o(1)A\le\I(q(C^2))=\E_q L^*(x,y)\le B_0W_2.
\end{equation}
For a cell occurrence \(C\) write \(a=\alpha(C)\) and \(p=\mu_e(C)\);
the latter depends on the fiber \(e\).  Put
\[
 T(\lambda)=\I\left(\E_e\,\1_C(x_e)\,p\,\1_{\{p<\lambda\}}\right),
\]
which is finite since its integrand is at most \(q(C^2)\).

\begin{lemma}[Small fibers]\label{lem:small-fiber}
With the prepared graph and its hierarchies, if \(0<\lambda<1/2\),
\(2/n\le\tau\le1\), and \(H\ge1\), then
\begin{equation}\label{eq:small-fiber}
 T(\lambda)\ll
 \left(\frac{\lambda}{\tau^2}+\lambda\right)W_2
 +\tau A+\lambda H+e^{-b_1H},
\end{equation}
with \(b_1>0\) absolute.  The implicit constant depends only on \(B_0\)
and the absolute constants in~\eqref{eq:grid-error}.
\end{lemma}

\begin{proof}
Split the cells into three classes: big cells, giant complements and
small cells.  For every \(m\ge2\),
\begin{equation}\label{eq:sc-pair-comparison}
 \tfrac12\alpha(C)^2\le\pi(C^2)\le\alpha(C)^2.
\end{equation}
If \(a\ge\tau\), then \(m\ge2\), and the integrand of \(T(\lambda)\)
is at most \(\lambda\le2\lambda\pi(C^2)/\tau^2\).  These occurrences
contribute at most \(2\lambda W_2/\tau^2\).
If \(C\) is a giant complement and \(a<\tau\), its contribution is
at most \(q(C^2)\le B_0\pi(C^2)\le B_0\tau a\).  The \(a\)-integral
over all giant complements is \(A\), so this class costs at most
\(B_0\tau A\).  Both classes count multiplicities.

Remains an actual nongiant cell of mass \(<\tau\), fixed at a
particular place and coordinate \(j\).  Let \(j'\) be the other
coordinate and set \(H_v=H\) at archimedean places, \(H_v=0\)
otherwise.  Fix a fiber \(e\) with source \(x=x_e\), let \(U\) be the
\(j\)-cell of \(x\) at level \(r\), and put \(p=\mu_e(U)\).  Keep
only levels where \(U\) is nongiant, \(\alpha(U)<\tau\), and
\(p<\lambda\).

Without grid error, a fiber point outside the source cell in \(j\)
lies inside the source cell at the complementary level \(h_e-r\) in
\(j'\).  Use two nearby levels in the latter coordinate to measure
failures of this complementarity.  Define the cells
\[
 V_-=\Pi_{j'}(h_e-r-H_v)(x),\qquad V_+=\Pi_{j'}(h_e-r+H_v)(x),
\]
so that \(V_+\subseteq V_-\).  Put
\[
 D^-=\mu_e(P\setminus(U\cup V_-)),\qquad
 D^+=\mu_e(U\cap V_+).
\]

The fiber identity makes these errors small after integration.  Fix
a fiber point \(y\), write \(d=d_j(x,y)\) and \(d'=d_{j'}(x,y)\).  Its
contribution to \(D^-\) occurs when \(d<r<h_e-H_v-d'\), and to
\(D^+\) when \(h_e+H_v-d'\le r\le d\).  So
\begin{align*}
 \int D^-\,dr&=\E_{y\sim\mu_e}(h_e-d-d'-H_v)_+,\\
 \int D^+\,dr&=\E_{y\sim\mu_e}(d+d'-h_e-H_v)_+.
\end{align*}
Their sum is the expectation of
\((| \Delta_j(x,y)+\Delta_{j'}(x,y)|-H_v)_+\), since
\(i_j+i_{j'}=h_e\).  It vanishes at nonarchimedean places.  At
archimedean places the tail~\eqref{eq:grid-error} and the union bound
give
\[
 \Pr(|\Delta_j+\Delta_{j'}|>u)\le2C_0e^{-b_0u/2}.
\]
Integrating this tail, averaging over fibers and using
\(\sum_{v\mid\infty}\sigma_v=1\),
\begin{equation}\label{eq:sc-errors}
 \avg{\E_e\int(D^-+D^+)\,dr}\ll e^{-b_1H}.
\end{equation}
No independence of the two errors is needed.

Split the retained levels into three cases.  In each change of level
the fiber is fixed, so \(h_e\) is constant.

\smallskip\noindent
\emph{Case 1: \(V_-\) is nongiant.}
Since \(\mu_e(V_-)+D^-\ge1-p>1/2\),
\[
 p\le2\lambda(\mu_e(V_-)+D^-).
\]
Changing levels to \(r'=h_e-r-H_v\), sharing a nongiant cell in \(j'\)
contributes to \(L_{j'}\), so the main term is at most
\(2\lambda\E_qL_{j'}\).  Summing over coordinates and places gives at
most \(2B_0\lambda W_2\).  The error is
bounded by~\eqref{eq:sc-errors}, since \(2\lambda<1\).

\smallskip\noindent
\emph{Case 2: \(V_-\) is giant and \(V_+\) is not.}
For fixed source \(x\), the levels at which its \(j'\)-cell is giant
form an initial interval.  The two levels defining \(V_-\) and \(V_+\)
straddle its endpoint, so the allowed \(r\) have length at most
\(2H_v\).  Since \(p<\lambda\), this costs at most \(2\lambda H_v\)
per fiber, for a total of at most \(4\lambda H\).

\smallskip\noindent
\emph{Case 3: \(V_+\) is giant.}
Here
\[
 p=D^++\mu_e(U\setminus V_+).
\]
The first term is covered by~\eqref{eq:sc-errors}.  Change levels to
\(r'=h_e-r+H_v\) for each fiber.  At fixed new level every counted
pair has \(y\notin G_{j'}(r')\) and shares an actual \(j\)-cell of
size less than \(\tau n\) at some original level.

For fixed target \(y\), let \(N_j(y)\) be the union of all actual
\(j\)-cells containing \(y\) whose size is less than \(\tau n\), over
all levels.  These cells form a nested chain, so
\(|N_j(y)|<\tau n\).  Every possible source therefore lies in
\(N_j(y)\setminus\{y\}\), even when its selected level depends on the
fiber.  Each edge belongs to exactly one fiber and keeps weight
\(1/|E|\) after the level change.  At \(r'\) the mass that
contributes is at most
\begin{align*}
 \sum_{y\notin G_{j'}(r')}\sum_{x\in N_j(y)\setminus\{y\}}q(x,y)
 &\le\frac{B_0}{n(n-1)}
   \sum_{\substack{y\notin G_{j'}(r')\\N_j(y)\ne\varnothing}}
       (|N_j(y)|-1)\\
 &\le B_0\tau\alpha\bigl(P\setminus G_{j'}(r')\bigr).
\end{align*}
Integrating and summing gives \(B_0\tau A\).

The three cases, the initial split and~\eqref{eq:sc-errors} prove
\eqref{eq:small-fiber}.  All rearranged integrands are nonnegative,
so Tonelli applies.
\end{proof}

We can now drop the small cells from the overlap scale.  Recall
\(\lambda_n=1/\log n\).  Taking \(\tau=\lambda_n^{1/3}\) and
\(H=\lambda_n^{-1/2}\) in Lemma~\ref{lem:small-fiber},
\[
 T(\lambda_n)
 \ll\lambda_n^{1/3}W+\lambda_n^{1/2}+e^{-b_1\lambda_n^{-1/2}}
 =o(W+1).
\]
Define
\[
 W_{2,\mathrm{sm}}=
 \I\bigl(\pi(C^2)\1_{\{|C|\le n^{1-c}\}}\bigr).
\]
Split \(\I(q(C^2))\) according to whether \(\mu_e(C)<\lambda_n\).
Equation~\eqref{eq:subset}, whose errors are uniform in every subset
\(C\), bounds the other part by
\[
 (1+o(1))(W_2-W_{2,\mathrm{sm}})+o(1)W_{2,\mathrm{sm}}.
\]
Adding \(T(\lambda_n)=o(W+1)\) and comparing with
\eqref{eq:q-overlap},
\begin{equation}\label{eq:small-overlap}
 W_{2,\mathrm{sm}}=o(W+1).
\end{equation}
This holds before passing to a bounded or unbounded subsequence of
\(W\).

\section{Variance from off-diagonal sampling}\label{sec:variance}

The additive identity~\eqref{eq:additive} is only strong enough to
compare the uniform law with fiber laws if we keep the pairs
\emph{distinct} throughout.  This section is the exact place where
that restriction pays.

\begin{lemma}[Variance]\label{lem:variance}
Let \(\mathcal F\) be a finite family of probability measures \(\nu\),
each uniform on a subset \(B\subset P\) of size \(k\ge n^\gamma\).
Give the family nonnegative weights, independent of places and grids,
of total mass at most one, and write \(\E_{\mathcal F}\) for the
weighted sum.  Suppose \(\E_{\mathcal F}\nu\le M\alpha\) pointwise,
with \(M\) fixed.  For each transformed cell \(C\) with
\(m=|C|\), \(a=\alpha(C)\), \(p=\nu(C)\),
\begin{align}
\I\!\left(\E_{\mathcal F}\left[
  (p-a)^2\1_{\{m>n^{1-c}\}}
  +p^2\1_{\{m\le n^{1-c},\ p\ge2/k\}}
\right]\right)
&\le C_M\bigl(n^{c-\gamma}W_2+W_{2,\mathrm{sm}}\bigr)
\label{eq:variance-quant}\\
&=o(W+1).
\label{eq:variance}
\end{align}
The constant \(C_M\) depends only on \(M\).
\end{lemma}

The first term measures discrepancy from \(\alpha\) on large cells,
where finite-population corrections are negligible.  On a small cell,
the second term measures \(\nu\)-mass only when \(B\cap C\) contains
at least two points, since a single point cannot contribute a
distinct-pair overlap within \(B\).

\begin{proof}
Let \(P_B\) be the uniform law on distinct ordered pairs from \(B\).
Define another law \(R_B\) on distinct pairs as follows: take \(X\)
uniformly from \(B\); if \(k<n\), conditionally on \(X\) take \(Y\)
uniformly from \(B\setminus\{X\}\) with probability \(k/n\), and
uniformly from \(P\setminus B\) otherwise.  If \(B=P\), set
\(R_B=P_B=\pi\) directly.

The first marginal of \(R_B\) is \(\nu\) and its second marginal is
\(\alpha\).  A point \(y\in B\) receives mass
\[
 (k-1)\frac1k\frac{k}{n}\frac1{k-1}=\frac1n,
\]
and a point \(y\notin B\) receives
\(k(1/k)((n-k)/n)/(n-k)=1/n\).  The signed measure \(P_B-2R_B+\pi\)
has total mass zero and marginals \(\alpha-\nu\) and \(\nu-\alpha\),
which sum to zero.  The additive overlap identity gives
\begin{equation}\label{eq:variance-cancel}
 \I\bigl(P_B(C^2)-2R_B(C^2)+\pi(C^2)\bigr)=0.
\end{equation}
These are finite integrals: for fixed \(n\) any law on distinct pairs
is at most \(n(n-1)\pi\) pointwise, and \(W_2<\infty\).  The family
weights are independent of places and grids, so averaging over the
family keeps~\eqref{eq:variance-cancel} valid.

Compute the three cell probabilities.  Writing \(b=|B\cap C|\), so
that \(p=b/k\),
\begin{align*}
 P_B(C^2)
 &=p^2-\frac{p(1-p)}{k-1},\\
 R_B(C^2)
 &=ap-\frac{k}{n}\frac{p(1-p)}{k-1},\\
 \pi(C^2)
 &=a^2-\frac{a(1-a)}{n-1}.
\end{align*}
The middle identity also holds when \(k=n\), by the direct definition
of \(R_B\).  Hence with \(H_B(C)=P_B(C^2)-2R_B(C^2)+\pi(C^2)\),
\begin{equation}\label{eq:variance-finite}
 H_B(C)=(p-a)^2+\left(\frac{2k}{n}-1\right)
                 \frac{p(1-p)}{k-1}
                 -\frac{a(1-a)}{n-1}.
\end{equation}
Write \(\overline H=\E_{\mathcal F}H_B\) and
\(\omega=\E_{\mathcal F}1\le1\).

If \(m>n^{1-c}\), then \(1/(k-1)\le2n^{-\gamma}\),
\(1/(n-1)\le2n^{-\gamma}\), and \(\E_{\mathcal F}p\le Ma\).  So
\[
 \overline H(C)\ge\E_{\mathcal F}(p-a)^2-2(M+1)n^{-\gamma}a.
\]
Since \(a>n^{-c}\),
\(n^{-\gamma}a\le2n^{c-\gamma}\pi(C^2)\) by~\eqref{eq:sc-pair-comparison}.
Thus
\begin{equation}\label{eq:variance-large}
 \overline H(C)\ge\E_{\mathcal F}(p-a)^2
       -4(M+1)n^{c-\gamma}\pi(C^2).
\end{equation}

If \(2\le m\le n^{1-c}\) and \(p\ge2/k\), then \(b\ge2\) and
\[
 P_B(C^2)=p^2\frac{k}{k-1}\frac{b-1}{b}\ge\tfrac12p^2.
\]
Since \(P_B(C^2)\) is nonnegative this gives
\(P_B(C^2)\ge\tfrac12p^2\1_{\{p\ge2/k\}}\) for every \(B\).  Also
\(R_B(C^2)\le ap\), so
\(\E_{\mathcal F}R_B(C^2)\le Ma^2\le2M\pi(C^2)\).  Discard the
nonnegative \(\omega\pi(C^2)\) and use~\eqref{eq:sc-pair-comparison}:
\begin{equation}\label{eq:variance-small}
 \overline H(C)\ge\tfrac12\E_{\mathcal F}
                \bigl[p^2\1_{\{p\ge2/k\}}\bigr]-4M\pi(C^2).
\end{equation}

If \(m=0\) or \(m=1\), all three pair probabilities vanish exactly.
A singleton has \(p\in\{0,1/k\}\), so the small-cell term also
vanishes; a singleton cannot satisfy \(m>n^{1-c}\).  This exact
cancellation treats the infinite singleton tails: we never integrate
the separate correction terms in~\eqref{eq:variance-finite} over
those tails.

Integrate~\eqref{eq:variance-large} and~\eqref{eq:variance-small} on
their regions, apply the exact zero on empty and singleton cells,
and use~\eqref{eq:variance-cancel}:
\[
 \I\!\left(\E_{\mathcal F}\left[
 (p-a)^2\1_{\{m>n^{1-c}\}}
 +\tfrac12p^2\1_{\{m\le n^{1-c},\,p\ge2/k\}}
 \right]\right)
 \le 4(M+1)n^{c-\gamma}W_2+4M W_{2,\mathrm{sm}}.
\]
Doubling gives~\eqref{eq:variance-quant}, with \(C_M=8(M+1)\).
Finally \(c<\gamma\), \(W_2\le W\), and~\eqref{eq:small-overlap}
give~\eqref{eq:variance}.
\end{proof}

Apply Lemma~\ref{lem:variance} to two families.  For the fiber laws
\(\mu_e\), weights \(k_e/|E|\) sum to one, \(\E_e\mu_e=q_Y\le B_0\alpha\),
and \(k_e\ge n^{s-o(1)}\ge n^\gamma\) for large \(n\).

For the incoming laws \(\rho_y\), retain targets with
\(d^-(y)\ge n^\gamma\) and use weights \(q_Y(y)=d_y/|E|\).  The
weighted mean is dominated by \(q_X\):
\[
 \sum_{y:d_y\ge n^\gamma}q_Y(y)\rho_y\le q_X\le B_0\alpha.
\]
Each retained law is uniform on its \(d_y\) incoming sources and the
family weights have total mass at most one.  The omitted targets have
total mass
\begin{equation}\label{eq:variance-omitted}
 \sum_{y:d_y<n^\gamma}q_Y(y)
 \le\frac{n^{1+\gamma}}{|E|}
 =O(n^{\gamma-1})=o(1),
\end{equation}
since \(d_n\to\theta>0\) and \(\gamma<1\).  The cell-mass estimates
therefore apply to outgoing fibers and to almost all incoming
conditional laws.  These are the two distributions the remaining
argument must compare with \(\alpha\).


\section{The case of unbounded overlap scale}\label{sec:unbounded}

The scale \(W=W_2+A\) is nonnegative and finite for each configuration.
Start from any counterexample sequence; it has either a bounded or an
unbounded subsequence of \(W\).  This section excludes the unbounded
alternative, the next section the bounded one.  Throughout,
\(o(W+1)=o(W)\).

\subsection*{Forcing a positive fraction of \(W\) into \(W_2\)}

First we prove
\begin{equation}\label{eq:unb-overlap-scale}
 W_2\ge c_1W
\end{equation}
for some \(c_1>0\) and all large \(n\).  Fix a place and grids;
write \(S=S_1+S_2\) and \(L=L_1+L_2\).  Choose a fiber \(e\) with its
edge weight and then \(Y,Y'\) independently with law \(\mu_e\).
Equation~\eqref{eq:depth} and the constant raw depth sum on the fiber
give
\[
 S(Y)=M_1+M_2-S(x_e)-h_e+L(x_e,Y)
       -\Delta_1(x_e,Y)-\Delta_2(x_e,Y).
\]
All source--neighbour pairs are distinct.  The first-moment
bound~\eqref{eq:grid-error}, the normalization of the archimedean
weights, and~\eqref{eq:q-overlap} give
\[
\avg{\E|S(Y)-S(Y')|}\le 2\avg{\E_qL(x,y)}+O(1)=O(W_2)+O(1).
\]
To compare with \(A\), use the pointwise bound
\begin{equation}\label{eq:potential}
 S^*(x)\ge A-2W_2\qquad(n>2).
\end{equation}
Indeed the average of \(L^*(x,y)\) over the \(n-1\) labels \(y\ne x\)
equals \(W_2+\frac{n-2}{n-1}(S^*(x)-A)\) and is nonnegative, so
\(S^*(x)\ge A-\frac{n-1}{n-2}W_2\).  Hence
\begin{equation}\label{eq:unb-min-lower}
 \avg{\E\min\{S(Y),S(Y')\}}\ge A-O(W_2)-O(1).
\end{equation}

For two nonnegative lists \((a_1,a_2)\) and \((b_1,b_2)\),
\[
 \min\{a_1+a_2,b_1+b_2\}\le\sum_{i,j=1}^2\min\{a_i,b_j\}.
\]
Indeed if the first sum is smaller, then \(\sum_j\min\{a_i,b_j\}\ge a_i\)
for each \(i\).  Nesting of the giants gives, for almost every \(u>0\),
\[
 \{z:S_i(z)>u\}=P\setminus G_i(M_i-u).
\]
A vertex leaving the giant at level \(r_z<M_i\) has
\(S_i(z)=M_i-r_z\); one that never leaves has \(S_i(z)=0\).  Conditional
independence of \(Y,Y'\) and \(ab\le(a^2+b^2)/2\) give
\begin{equation}\label{eq:unb-min-upper}
 \avg{\E\min\{S(Y),S(Y')\}}
 \le 2\avg{\sum_{i=1}^2\E_e\int_{r<M_i}
          \mu_e(P\setminus G_i(r))^2\,dr}.
\end{equation}
Bound the right side as follows.  Write \(p=\mu_e(C)\) and \(a=\alpha(C)\)
for a giant complement.  If \(|C|>n^{1-c}\), use
\(p^2\le2(p-a)^2+2a^2\), the variance estimate~\eqref{eq:variance} and
\(a^2\le2\pi(C^2)\).  For smaller complements with \(p\ge2/k_e\), use
the second term of~\eqref{eq:variance}.  The remaining terms satisfy
\(p^2\le 2n^{-\gamma}p\); since \(\E_ep=q_Y(C)\le B_0a\), their
integral is \(O(n^{-\gamma}A)\).  In total
\[
 \avg{\E\min\{S(Y),S(Y')\}}\le O(W_2)+o(W)+O(n^{-\gamma}A).
\]
With~\eqref{eq:unb-min-lower}, \(A\le O(W_2)+o(W)+O(1)\), so
enabling~\eqref{eq:unb-overlap-scale}.  Equation~\eqref{eq:q-overlap}
also gives \(\I(q(C^2))\gg W\).

\subsection*{Restricting cell masses}

Fix \(0<\delta<1/4\) and split the contribution of \(a\le\delta\)
according to \(p<2\delta\) or \(p\ge2\delta\).  In the latter case
\(p\le p^2/(2\delta)\) and \(p-a\ge p/2\); on large cells this is
controlled by the first variance term, and on small cells
\(p\ge2\delta\ge2/k_e\) eventually, so the second variance term
applies.  So
\[
 \I\bigl(\1_{\{a\le\delta\}}q(C^2)\bigr)
 \le T(2\delta)+o_\delta(W)\le O(\delta^{1/3}W)+o_\delta(W),
\]
using Lemma~\ref{lem:small-fiber} with \(\lambda=2\delta\),
\(\tau=\lambda^{1/3}\) and any fixed \(H\ge1\).  The constant in
\(O(\delta^{1/3}W)\) is independent of \(\delta\).  Therefore
\begin{equation}\label{eq:cut-mass}
 \I(\1_{\{a>\delta\}})\gg W,\qquad
 \I(\1_{\{a\ge\delta/4\}})\ll_\delta W.
\end{equation}
The first inequality uses \(q(C^2)\le1\); for the second, for large
\(n\), \(a\ge\delta/4\) implies \(\pi(C^2)\ge a^2/2\ge\delta^2/32\).
A bound \(O_\delta(1)\) is thus \(o(W)\) along the present
subsequence.

Choose a Lipschitz \(w:[0,1]\to[0,1]\) with
\[
 w(u)=w(1-u),\qquad
 w=0\text{ on }[0,\delta/2]\cup[1-\delta/2,1],\qquad
 w=1\text{ on }[\delta,1-\delta].
\]
For the original partitions define
\[
 \ell_j=\int_\R\sum_{B\in\Pi_j(r)}w(\alpha(B))\,dr.
\]
Replacing a giant by its complement preserves every summand by
symmetry, occurrences still counted separately, so
\begin{equation}\label{eq:unb-total-length}
 W\ll\avg{\ell_1+\ell_2}\ll_\delta W.
\end{equation}

For either family in Lemma~\ref{lem:variance} we will use
\begin{equation}\label{eq:cut-approx}
 \avg{\E_{\mathcal F}\sum_{j=1}^2\int_\R
 \sum_{B\in\Pi_j(r)}
 \left[|w(\nu(B))-w(\alpha(B))|
       +w(\alpha(B))|\nu(B)-\alpha(B)|\right]dr}=o(W).
\end{equation}
The first summand replaces cut weights of fiber laws with those of the
uniform law; the second controls the weighted mass imbalance that
will show up in the transport comparison.  The family of incoming
laws retains only targets with \(d^-\ge n^\gamma\).  Both sides are
unchanged under \((p,a)\mapsto(1-p,1-a)\), so work with the transformed
cells.  On \(a\ge\delta/4\) all cells are large for large \(n\); the
Lipschitz bound, Cauchy--Schwarz, \eqref{eq:cut-mass}
and~\eqref{eq:variance} give \(o(W)\).  On \(a<\delta/4\) the second
summand vanishes, and the first can be nonzero only if \(p>\delta/2\).
Then \(1\le16\delta^{-2}(p-a)^2\) on large cells, and
\(1\le4\delta^{-2}p^2\) with \(p\ge2/k\) on small cells, and the same
variance estimate applies.

\subsection*{Distances obtained from the hierarchy}

For \(\nu=\alpha\) or \(\mu_e\), put
\[
 D_j^\nu(a_0,a_1)=\int_\R\sum_{B\in\Pi_j(r)}
 |\1_B(a_0)-\1_B(a_1)|\,w(\nu(B))\,dr.
\]
Also allow a formal endpoint \(\infty\) with \(\1_B(\infty)=0\) for
every cell.  These are finite for all large \(n\): on the initial ray
the only cell is \(P\) with \(w(1)=0\); on the final ray every cell
is a singleton of \(\nu\)-mass \(0\), \(1/n\) or \(1/k_e\) and thus
has weight zero.

For distinct \(a_0,a_1\in P\),
\begin{equation}\label{eq:projection}
 D_j^\nu(a_0,a_1)=\int_\R
 w\bigl(\nu\{z:d_j(a_1,z)-d_j(a_0,z)\ge u\}\bigr)\,du.
\end{equation}
Indeed, nestedness makes the events \(\{d_j(a_1,z)>d_0\}\) and
\(\{d_j(a_0,z)=d_0\}\) for \(d_0=d_j(a_0,a_1)\) exact complements.
For \(u>0\) the event in~\eqref{eq:projection} is the \(a_1\)-cell at
level \(d_0+u\), and for \(u<0\) its complement is the \(a_0\)-cell at
level \(d_0-u\), up to strict-vs-weak threshold ties that involve only
finitely many levels.  Splitting the integral at \(0\) and using
\(w(t)=w(1-t)\) recovers the two cell contributions.  Similarly,
\begin{equation}\label{eq:unb-root-projection}
 D_j^\nu(\infty,y)=\int_\R
 w\bigl(\nu\{z:d_j(y,z)\ge u\}\bigr)\,du.
\end{equation}

For an edge \((x,y)\) in fiber \(e\), put \(j'=3-j\).  On the fiber,
for \(z\in F_e\),
\[
 Z_{j'}(z)=Z_{j'}(x)+\frac{t_e}{Z_j(z)-Z_j(x)}.
\]
Consequently, for \(z\ne y\) in that fiber,
\[
 Z_{j'}(z)-Z_{j'}(y)
 =\frac{t_e\bigl(Z_j(y)-Z_j(z)\bigr)}
 {(Z_j(z)-Z_j(x))(Z_j(y)-Z_j(x))},
\]
hence
\begin{equation}\label{eq:unb-raw-fiber}
 i_{j'}(y,z)=i_j(y,z)-i_j(x,z)+h_e-i_j(x,y).
\end{equation}
This turns one projected distance into a root distance in the other
coordinate.  Under \(z\sim\mu_e\), compare
\[
 U(z)=d_{j'}(y,z),\qquad
 V(z)=d_j(y,z)-d_j(x,z)+h_e-i_j(x,y).
\]
For \(z\ne y\), equation~\eqref{eq:unb-raw-fiber} gives
\[
 |U(z)-V(z)|\le |\Delta_{j'}(y,z)|+|\Delta_j(y,z)|+|\Delta_j(x,z)|.
\]
At \(z=y\) both variables are \(+\infty\), so their indicators agree
at every finite threshold; \(z=x\) has zero \(\mu_e\)-mass.  Tonelli's
theorem therefore gives
\[
 \int_\R|\Pr(U\ge u)-\Pr(V\ge u)|\,du
 \le \sum_{z\in F_e\setminus\{y\}}\mu_e(z)|U(z)-V(z)|.
\]
Translation of the threshold by \(h_e-i_j(x,y)\) leaves the integral
of the weighted tail unchanged.  Apply the Lipschitz bound for \(w\),
\eqref{eq:projection} and~\eqref{eq:unb-root-projection}; after
averaging over \(e\), \(y\sim\mu_e\), grids and places,
\[
 \avg{\E_q|D_j^{\mu_e}(x,y)-D_{j'}^{\mu_e}(\infty,y)|}=O_\delta(1).
\]
The error estimate holds uniformly for these sampling laws
because~\eqref{eq:grid-error} bounds the grid first moment; all vertex
laws are independent of the grids; nonarchimedean errors vanish and
the archimedean weights sum to one.

Changing \(\mu_e\) to \(\alpha\) changes any cut distance by at most
\(\int\sum_B|w(\mu_e(B))-w(\alpha(B))|\,dr\), independently of the
endpoints.  Thus~\eqref{eq:cut-approx} with \(D_j=D_j^\alpha\) yields
\begin{equation}\label{eq:fiber-tree}
 \avg{\E_q|D_j(x,y)-D_{j'}(\infty,y)|}=o(W).
\end{equation}
The remaining task is to replace the edge law by independent uniform
vertices and exploit the geometry of the cut distances.

\subsection*{Transport to independent uniform vertices}

Each \(D_j\) is the path distance on a finite weighted tree, allowing
zero-length edges.  Take the distinct cells of the nested partitions
as vertices, root the inclusion tree at \(P\), and connect each proper
cell \(B\) to its least strict supercell.  Give this edge length
\[
 l_j(B)=\int_\R\1_{\{B\in\Pi_j(r)\}}w(\alpha(B))\,dr.
\]
The points of \(P\) are singleton leaves.  An edge is crossed by the
path joining two leaves exactly when its cell separates them, so the
cut formula holds for \(D_j\).  The total edge length is \(\ell_j\).
The formal endpoint is the root, since \(w(1)=0\).

For probabilities \(\rho,\alpha\) on the leaves, let
\(\mathsf T_j(\rho,\alpha)\) be the least expected \(D_j\)-distance
among couplings of the two laws.  The tree transport formula is
\begin{equation}\label{eq:unb-transport}
 \mathsf T_j(\rho,\alpha)
 =\sum_{B\ne P}l_j(B)|\rho(B)-\alpha(B)|
 =\int_\R\sum_{B\in\Pi_j(r)}
       w(\alpha(B))|\rho(B)-\alpha(B)|\,dr,
\end{equation}
the finite-tree form of \cite[Sec.~2, Eq.~(5)]{TreeTransport}.  A
short proof: across each edge above \(B\), every coupling moves at
least \(|\rho-\alpha|\) units of mass.  Conversely first match common
mass at each leaf, work upward, and route surplus from a child subtree
to deficits in other child subtrees; across each edge the moved mass is
exactly the absolute imbalance.  Decomposing into leaf-to-leaf paths
gives a coupling attaining the displayed cost.

Since \(q_Y=\E_e\mu_e\), convexity and~\eqref{eq:cut-approx} give
\begin{equation}\label{eq:unb-two-costs}
 \avg{\mathsf T_j(q_Y,\alpha)}=o(W),\qquad
 \avg{\E_{q_Y}\mathsf T_j(\rho_y,\alpha)}=o(W).
\end{equation}
The second estimate restricts to targets with \(d^-\ge n^\gamma\).  The
omitted \(q_Y\)-mass is \(O(n^{\gamma-1})=o(1)\); since every transport
cost is at most \(\ell_j\), the omission costs \(o(1)\avg{\ell_j}=o(W)\)
by~\eqref{eq:unb-total-length}.

The conditional source law in~\eqref{eq:fiber-tree} is \(\rho_y\).  Fix
one hierarchy and put \(F(x,x',y)=|D_j(x,y)-D_j(x',y)|\).  If
\(Y\sim q_Y\) and \(X,X'\) are independent with conditional law
\(\rho_Y\), then
\[
 \E F(X,X',Y)\le2\E_q|D_j(x,y)-D_{j'}(\infty,y)|.
\]
For each \(y\), use two independent copies of an optimal coupling of
\(\rho_y\) to \(\alpha\), replacing \(X,X'\) by \(U,U'\).  Conditional
on \(y\), the new points are independent with law \(\alpha\), and
since that law does not depend on \(y\), they are also independent of
\(Y\).  The triangle inequality bounds the change in \(F\) by the sum
of two transport distances.  Then optimally couple \(Y\sim q_Y\) to
\(V\sim\alpha\), independent of \(U,U'\).  Moving the last argument
of \(F\) costs at most twice its transport distance.  Hence pointwise
\begin{align*}
 \E_{\alpha^{\otimes3}}F
 &\le 2\E_q|D_j(x,y)-D_{j'}(\infty,y)|\\
 &\quad+2\E_{q_Y}\mathsf T_j(\rho_y,\alpha)
       +2\mathsf T_j(q_Y,\alpha).
\end{align*}
Integrating and using~\eqref{eq:fiber-tree} and~\eqref{eq:unb-two-costs}
gives
\begin{equation}\label{eq:iid-tree}
 \avg{\E_{x,x',y\ \mathrm{iid}\ \alpha}
 |D_j(x,y)-D_j(x',y)|}=o(W).
\end{equation}

\subsection*{A lower bound for tree distance fluctuations}

\begin{lemma}\label{lem:unb-tree-fluctuation}
Let a finite tree have nonnegative edge lengths summing to \(L\),
with path pseudometric \(d\), and let \(\alpha\) be a probability on
its vertices.  Suppose every edge of positive length separates two
sets each having \(\alpha\)-mass at least \(\delta/2\), \(0<\delta<1\).
If \(X,X',Y\) are independent with law \(\alpha\),
\[
 \E|d(X,Y)-d(X',Y)|\ge
 \frac{\delta^2}{32(\lceil16/\delta\rceil+1)}L.
\]
\end{lemma}

\begin{proof}
Nothing to prove if \(L=0\).  For every \(y\), the expected distance
from \(X\) to \(y\) is at least \(\delta L/2\): each positive edge
separates \(y\) from mass at least \(\delta/2\).  All distances are
at most \(L\), so
\[
 \Pr\{d(X,y)\ge\delta L/4\}\ge\delta/4.
\]
Traverse the tree in a closed walk crossing every edge twice, of total
length \(2L\).  Partition this walk into \(M=\lceil16/\delta\rceil+1\)
intervals of length at most \(\delta L/8\), and assign each vertex to
an interval containing one of its visits.  The resulting classes have
diameter at most \(\delta L/8\), and
\[
 \Pr\{d(X',Y)\le\delta L/8\}\ge\sum_{r=1}^M\alpha(C_r)^2\ge1/M.
\]
Conditional on \(Y\), the far event for \(X\) and the near event for
\(X'\) are independent.  The far probability is at least
\(\delta/4\) uniformly, so both events occur with probability at
least \(\delta/(4M)\).  On their intersection the absolute distance
difference is at least \(\delta L/8\).
\end{proof}

In the tree for \(D_j\), every positive edge separates two sides of
\(\alpha\)-mass greater than \(\delta/2\), by the support condition
on \(w\).  Apply Lemma~\ref{lem:unb-tree-fluctuation} with
\(L=\ell_j\), sum over \(j=1,2\), and average over places and grids.
Equations~\eqref{eq:unb-total-length} and~\eqref{eq:iid-tree} then give
\(W\ll o(W)\), a contradiction.  This excludes every subsequence with
\(W\to\infty\).


\section{The case of bounded overlap scale}\label{sec:bounded}

It remains to exclude a subsequence on which \(W\) is bounded.  Choose
one complex embedding and normalize both coordinates so the uniform
laws have nonatomic limits.  The variance estimate gives the same
limits for the fiber laws and for the conditional incoming laws.  The
exact fiber equation then forces a family of M\"obius maps between
these limits, which cannot exist.

\subsection*{A selected embedding and normalized limits}

At an archimedean place \(v\) define the nonnegative local scale
\[
 T_v=\E_{\mathrm{grids}}\sum_{j=1}^2
       \bigl(\E_\pi L_j+\E_\alpha S_j\bigr).
\]
Let \(V_v\) be the sum of the local integrals on the left
of~\eqref{eq:variance}, for the fiber family and the retained incoming
family, including both coordinates and grid expectations but omitting
the place weight.  Since \(W\) is bounded, Lemma~\ref{lem:variance}
gives
\[
 \sum_{v\mid\infty}\sigma_vT_v\le C,\qquad
 \sum_{v\mid\infty}\sigma_vV_v=\eta_n\longrightarrow0.
\]
The archimedean weights sum to one.  The places with \(T_v\le2C+1\)
have total weight at least \(1/2\), so one of them satisfies
\(V_v\le2\eta_n\).  Choose such a place for each \(n\).  No lower
bound on an individual place weight is needed.  All coordinates and
grids in this section refer to the selected embedding.

From~\eqref{eq:depth} and the grid error bound~\eqref{eq:grid-error},
for \(j=1,2\),
\[
 \E_{\mathrm{grids}}\E_\pi|i_j(x,y)-M_j|
 \le \E_{\mathrm{grids}}\E_\pi
       \bigl(L_j(x,y)+S_j(x)+S_j(y)+|\Delta_j(x,y)|\bigr)=O(1).
\]
Set \(m_j=\E_\pi i_j(x,y)\).  For each realized grid,
\(|m_j-M_j|\le\E_\pi|i_j-M_j|\), hence
\(\E_\pi|i_j-m_j|=O(1)\).  Normalize the coordinates,
\[
 U_j(x)=e^{m_j}Z_j(x)+b_j,
\]
with translation \(b_j\) to be chosen shortly.  Then
\begin{equation}\label{eq:bounded-log}
 \E_\pi\bigl|\log|U_j(x)-U_j(y)|\bigr|\le C_1.
\end{equation}
Only distinct pairs occur.

Write \(\alpha_n^j=(U_j)_*\alpha\).  Equation~\eqref{eq:bounded-log}
implies
\[
 (\alpha_n^j\otimes\alpha_n^j)\{|z-z'|>R\}
 \le \frac{C_1}{\log R}\qquad(R>1).
\]
Choose a fixed \(R_0>1\) with the right side below \(1/2\).
Averaging over a center at an empirical point gives a radius-\(R_0\)
disk containing at least half the mass.  Choose \(b_j\) so that this
center moves to zero.  For \(R>2R_0\), pairs between this disk and the
complement of the radius-\(R\) disk have distance at least \(R/2\), so
\begin{equation}\label{eq:bounded-tail}
 \alpha_n^j\{|z|>R\}\le \frac{2C_1}{\log(R/2)}.
\end{equation}
For any center \(z_0\in\C\) and \(0<r<1/2\), a similar use
of~\eqref{eq:bounded-log} gives
\begin{equation}\label{eq:bounded-ball}
 \alpha_n^j(B(z_0,r))^2
 \le \frac1n+\frac{C_1}{\log(1/(2r))}.
\end{equation}
The diagonal mass of independent pairs in the ball is at most \(1/n\),
and every remaining pair has distance at most \(2r\).

Let \(S=\widehat{\C}\) be the Riemann sphere with chordal metric and
write \(\mathcal P(S)\) for its probability measures.  This space is
compact and metrizable for weak convergence
\cite[Prop.~5.3 and Thm.~4.2]{WeakCompactness}.  Also use the
open-set inequality in the Portmanteau theorem
\cite[Thm.~3.2]{WeakCompactness}.  Along a common subsequence,
\(\alpha_n^j\Rightarrow\beta_j\) on \(S\) for \(j=1,2\).
Apply Portmanteau to the open spherical sets
\((\{|z|>R\}\cup\{\infty\})\) in~\eqref{eq:bounded-tail} and let
\(R\to\infty\), then to the open disks in~\eqref{eq:bounded-ball} and
let \(r\downarrow0\).  Each \(\beta_j\) is a nonatomic probability
measure on \(\C\).

\subsection*{The variance estimate determines the limiting laws}

For either family in Lemma~\ref{lem:variance}, put
\(\nu_n^j=(U_j)_*\nu\) and write \(\mu\varphi=\int\varphi\,d\mu\).
We claim every \(\varphi\in C(S)\) satisfies
\begin{equation}\label{eq:test-convergence}
 \E_{\mathcal F}
 \left|\int\varphi\,d\nu_n^j-\int\varphi\,d\beta_j\right|
 \longrightarrow0.
\end{equation}
The family weights may sum to less than one; each member \(\nu\) is
still a probability measure, and \(\E_{\mathcal F}\nu\le B\alpha\) for
a fixed \(B\).

Convert the integrated cell variance into~\eqref{eq:test-convergence}.
Push every realized grid through \(z\mapsto e^{m_j}z+b_j\); this
shifts levels by \(-m_j\) and leaves integration over all levels
unchanged.  Fix \(R>0\) and \(0<\epsilon<1\).  At normalized levels in
\[
 I_\epsilon=[-\log\epsilon,-\log\epsilon+1],
\]
the actual square cells have sides between \(\epsilon/(2e)\) and
\(\epsilon\).  At most \(N=N(R,\epsilon)\) squares meet the radius-\(R\)
disk, uniformly in the grid and the level.  For each actual cell
\(B'\), let \(C\) be its transformed occurrence: \(C=B'\) unless
\(B'\) is the giant, in which case \(C=P\setminus B'\).  Both measures
have total mass one, so
\[
 |\nu(B')-\alpha(B')|=|\nu(C)-\alpha(C)|.
\]
Occurrences are retained with multiplicity.  For those with
\(|C|>n^{1-c}\), Cauchy--Schwarz over the family, grids, unit level
interval and at most \(N\) occurrences gives an integrated sum of mean
absolute discrepancies at most \(\sqrt{NV_v}\).  For the others,
including an empty giant complement,
\[
 \E_{\mathcal F}|\nu(C)-\alpha(C)|
 \le(B+1)\alpha(C)\le(B+1)n^{-c}.
\]

Approximate \(\varphi\) on each square by its value at any point of the
square.  The oscillation is at most
\(\omega_\varphi(\sqrt2\epsilon)\), with \(\omega_\varphi\) the
uniform modulus of continuity in the spherical metric.  The remaining
squares lie outside the radius-\(R\) disk with mean family mass at most
\(B\alpha_n^j\{|z|>R\}\).  Integrating the test-function estimate over
grids and the unit interval gives
\begin{align}
 \E_{\mathcal F}|\nu_n^j\varphi-\alpha_n^j\varphi|
 \le{}&
 2\omega_\varphi(\sqrt2\epsilon)
 +(B+1)\|\varphi\|_\infty\alpha_n^j\{|z|>R\}\notag\\
 &+\|\varphi\|_\infty
   \bigl(\sqrt{NV_v}+(B+1)Nn^{-c}\bigr).
 \label{eq:bounded-test-bound}
\end{align}
For fixed \(R,\epsilon\) let \(n\to\infty\), then let
\(R\to\infty\) and \(\epsilon\downarrow0\).
Equation~\eqref{eq:bounded-tail} and \(\alpha_n^j\Rightarrow\beta_j\)
prove~\eqref{eq:test-convergence}.  For the incoming laws, restoring
all omitted targets changes a mean test discrepancy by at most
\(2\|\varphi\|_\infty\) times their \(o(1)\) total \(q_Y\)-mass, so
\eqref{eq:test-convergence} holds in full \(q_Y\)-expectation for
\(\rho_y\) as well.

The two conclusions give the same nonatomic coordinate limits for
typical fibers and typical incoming laws.  The first preserves the
fiber equation in the limit; the second lets the source coordinate
vary independently of the target.

\subsection*{The joint limit and its fiber equation}

For a \(q\)-edge \((x,y)\) in fiber \(e\), consider the joint random
object
\[
 \bigl(U_1(x),U_2(x),U_1(y),U_2(y),
       (U_1)_*\mu_e,(U_2)_*\mu_e\bigr)
 \ \in\ S^4\times\mathcal P(S)^2.
\]
Take a subsequential weak limit \(Q\); write its four coordinate
entries as \((a,b,a',b')\).  The last two entries equal
\((\beta_1,\beta_2)\) almost surely.  To see this uniformly in all
tests, choose a countable dense \((f_r)\) in the unit ball of
\(C(S)\).  The metric
\[
 d(\mu,\beta)=\sum_{r\ge1}2^{-r}|\mu f_r-\beta f_r|
\]
metrizes weak convergence.
Equation~\eqref{eq:test-convergence} on finitely many tests, then a
bound on the series tail, gives convergence of this metric to zero in
mean for each fiber projection.  Passing to \(Q\) proves the
assertion.

The inequality \(q\le B_0\pi\le2B_0\alpha\otimes\alpha\) implies
domination of each limiting same-coordinate pair law by
\(2B_0\beta_j\otimes\beta_j\), because the inequality passes to the
limit against nonnegative continuous tests on \(S^2\).  Since
\(\beta_j\) is nonatomic and gives infinity mass zero,
\[
 a,b,a',b'\in\C,\quad a\ne a',\quad b\ne b',\quad
 a'\in\supp\beta_1,\quad b'\in\supp\beta_2,
\]
\(Q\)-almost surely.

Moreover \(a\) has law \(\beta_1\) independently of \((a',b')\).
For \(f\in C(S)\) and \(g\in C(S^2)\), conditioning on the actual
target vertex gives
\begin{align*}
 &\left|\E_q f(U_1(x))g(U_1(y),U_2(y))
     -(\beta_1f)\E_{q_Y}g(U_1(y),U_2(y))\right|\\
 &\hspace{20mm}\le
 \|g\|_\infty\E_{q_Y}
       \left|\int f\,d(U_1)_*\rho_y-\beta_1f\right|
 \longrightarrow0
\end{align*}
by~\eqref{eq:test-convergence}.  Products of this form determine the
joint law on \(S\times S^2\), proving independence.  Keeping both
target coordinates together in \(g\) allows the entire target pair to
be fixed in the final obstruction.

On the open set of finite quadruples with \(a\ne a'\) and \(b\ne b'\),
put \(t'=(a'-a)(b'-b)\).  We next obtain
\begin{equation}\label{eq:mobius}
 \beta_2=\Phi_*\beta_1,\qquad
 \Phi(z)=b+\frac{t'}{z-a}\quad(z\in S),
 \qquad Q\text{-almost surely}.
\end{equation}
For each prelimit edge, its two fiber projection measures are related
by this M\"obius formula with its four coordinates in place of
\((a,b,a',b')\).  Normalization multiplies the original fiber product
by \(e^{m_1+m_2}\), and evaluation at \(y\) identifies the new
product as \((U_1(y)-U_1(x))(U_2(y)-U_2(x))\).

To pass this equality to the limit, observe that the matrix
\[
 \begin{pmatrix}b&t'-ab\\1&-a\end{pmatrix}
\]
represents \(\Phi\) on the sphere with determinant \(-t'\ne0\).  The
map is jointly continuous in its parameters and the sphere variable,
including at its pole.  Its pushforward acts continuously on
probability measures with the weak topology.  The equality graph is
relatively closed over the open domain, and its complement inside
that domain has zero mass in every prelimit law; Portmanteau gives it
zero \(Q\)-mass.  The domain has full \(Q\)-mass, so~\eqref{eq:mobius}
holds.

\subsection*{A nonatomic obstruction to the fiber maps}

The following elementary observation closes the argument.  The second
source coordinate \(b\) may depend on the other coordinates, but it
enters only as an additive constant in a logarithmic distance, so the
independence already proved for \(a\) suffices.

\begin{lemma}\label{lem:bounded-mobius}
Let \(\beta_1,\beta_2\) be nonatomic probability measures on \(\C\).
There is no probability law on finite quadruples \((a,b,a',b')\)
such that, almost surely,
\[
 a\ne a',\qquad b\ne b',\qquad
 a'\in\supp\beta_1,\qquad b'\in\supp\beta_2,
\]
\(a\) has law \(\beta_1\) independently of \((a',b')\), and
\[
 \beta_2=\left(z\mapsto
 b+\frac{(a'-a)(b'-b)}{z-a}\right)_*\beta_1
\]
as measures on the Riemann sphere.
\end{lemma}

\begin{proof}
For a finite real random variable \(X\) and a continuous nonnegative
\(w\) supported in a compact subinterval of \((0,1)\), define
\[
 J_w(X)=\int_\R w\bigl(\Pr(X\ge u)\bigr)\,du.
\]
This is finite: the tail probability tends to one at \(-\infty\) and
zero at \(+\infty\), so the integrand vanishes outside a bounded
interval.  It is unchanged when a constant is added to \(X\).  Also,
\begin{equation}\label{eq:bounded-collapse}
 X_k\longrightarrow0\text{ in probability}
 \quad\Longrightarrow\quad J_w(X_k)\longrightarrow0.
\end{equation}
Indeed choose \(\delta>0\) with \(w\) vanishing outside
\([\delta,1-\delta]\).  For any \(\epsilon>0\), eventually
\(\Pr(|X_k|\ge\epsilon)<\delta\).  For \(u\ge\epsilon\) the upper tail
is below \(\delta\), and for \(u\le-\epsilon\) it is above
\(1-\delta\).  Hence \(0\le J_w(X_k)\le2\epsilon\|w\|_\infty\),
proving~\eqref{eq:bounded-collapse}.

Choose continuous weights \(w_l:[0,1]\to[0,1]\), \(l\ge3\), each
supported inside \((0,1)\) and equal to one on
\([1/l,1-1/l]\).  Let \(\Phi\) be the map in the statement.  For
\(z\ne a,a'\),
\[
 \Phi(z)-b'=(b'-b)\frac{a'-z}{z-a},\qquad
 -\log|\Phi(z)-b'|
 =\log\frac{|z-a|}{|z-a'|}-\log|b'-b|.
\]
Nonatomicity removes the exceptional points.  The pushforward identity
and translation invariance of \(J_{w_l}\) give
\begin{equation}\label{eq:tail-width}
 \int_\R w_l\!\left(
   \beta_1\!\left\{z:\log\frac{|z-a|}{|z-a'|}\ge u\right\}\right)\,du
 =
 \int_\R w_l\!\left(
   \beta_2\{z:-\log|z-b'|\ge u\}\right)\,du.
\end{equation}
All random variables are finite almost surely, so the integrals are
finite.  They are measurable in their parameters: define the logarithms
arbitrarily at coincidences, where the measures vanish, and apply
measurability of parameter integrals to the resulting Borel functions.

The equality in~\eqref{eq:tail-width} depends only on \((a,a',b')\).
Intersect the full-measure sets for all \(l\).  Independence and
Fubini then give that, for almost every target pair \((a',b')\), all
these equalities hold for \(\beta_1\)-almost every \(a\).  Fix such a
target pair in the stated supports.

For \(Z\) of law \(\beta_2\), the variable \(-\log|Z-b'|\) is
nonconstant.  Indeed, nonatomicity gives \(r_0>0\) with
\(p=\beta_2\{|z-b'|\ge r_0\}>0\), and support membership gives
\(0<r<r_0\) with \(q=\beta_2\{|z-b'|<r\}>0\).  For
\(-\log r_0<u<-\log r\) its upper tail lies in \([q,1-p]\).  Choose
\(l\) with \(1/l\le\min(p,q)\).  The right side
of~\eqref{eq:tail-width} is then strictly positive.

The full-\(\beta_1\)-measure equality set contains a sequence
\(a_k\ne a'\) tending to \(a'\), because \(a'\in\supp\beta_1\) and
\(\beta_1\{a'\}=0\).  For \(z\) outside the null countable set
\(\{a',a_1,a_2,\ldots\}\),
\[
 \log\frac{|z-a_k|}{|z-a'|}\longrightarrow0.
\]
These variables converge to zero in \(\beta_1\)-probability.
By~\eqref{eq:bounded-collapse} the left side of~\eqref{eq:tail-width},
for the \(l\) just chosen, tends to zero.  The right side is a fixed
positive number, a contradiction.
\end{proof}

The law \(Q\) constructed above satisfies every hypothesis of
Lemma~\ref{lem:bounded-mobius}, so a bounded-\(W\) subsequence is
impossible.  Section~\ref{sec:unbounded} excluded a subsequence with
\(W\to\infty\).  Every sequence of nonnegative reals has a bounded
subsequence or a subsequence tending to infinity, so the graph
sequence from Proposition~\ref{prop:graph} cannot exist.  The
assumption that \(\Theta(s)>0\) somewhere was false, hence
\(F_n(s)\to0\) for each fixed \(s>0\), which is
Theorem~\ref{thm:main}.

\section{What the argument actually does}\label{sec:recap}

For a reader who has followed the proof but wants a compact summary:

\begin{itemize}
\item The weak pinned conjecture fails if a positive density of sources
carry a distance fiber of size at least \(n^s\).  Proposition~\ref{prop:graph}
extracts that graph without losing subset control.
\item Squared distance factors through two complex coordinates, and
the product formula in a number field equalizes their logarithmic
depths.  Random nested grids convert those logarithms into integrated
cell overlaps up to a pair-independent error.
\item Removing the giant cell gives an exact additive identity: the
overlap of a distinct pair is a constant plus a function of the first
and a function of the second point.
\item The variance estimate uses the exact signed identity
\(P_B-2R_B+\pi\) on distinct pairs to compare fiber laws and the
uniform law.  Small cells contribute negligibly.
\item The scale \(W\) then either grows or stays bounded.  Growth gives
a tree-transport contradiction involving three independent uniform
points.  Boundedness gives a family of M\"obius maps between nonatomic
limit laws, and a short order-statistics argument rules that out.
\end{itemize}

Both alternatives are impossible, so the graph sequence from a
counterexample cannot exist.  The main theorem follows, and its
corollary is the almost-every-pin statement this paper advertised in
the introduction.

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\end{document}