\documentclass[11pt]{article}
\usepackage[margin=1.1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb R}
\newcommand{\F}{\mathcal F}
\newcommand{\Lip}{\operatorname{Lip}}
\newcommand{\norm}[1]{\left\|#1\right\|}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\diam}{\operatorname{diam}}
\newcommand{\sgn}{\operatorname{sign}}
\newcommand{\E}{\mathbb E}
\newcommand{\Prob}{\mathbb P}
\newcommand{\Id}{\operatorname{Id}}
\title{A Counterexample to Bounded Approximation in Lipschitz-Free Spaces}
\author{}\date{}
\begin{document}
\maketitle
\begin{abstract}
We construct a countable, uniformly discrete, unbounded metric space $M$
whose real Lipschitz-free space $\F(M)$ has the approximation property (AP)
but fails the bounded approximation property (BAP). This answers a question
of Kalton. The proof has three parts. First, for each integer $p$ we build a
bounded space, a \emph{block}, with a finite set of anchors at which every
finite-rank operator of norm at most $p$ makes a definite error. Second, we
glue the blocks at a common basepoint; a bounded-norm approximation of the
whole space would force bounded-norm approximations of every block, which is
impossible. Third, because each block itself has BAP, the whole space still
has AP. The heart of the argument is the construction and analysis of one
block, which combines random-like regular colored graphs, distance labels
capped at a constant, signed measures, and a count of non-backtracking walks.
Every step is proved in full.
\end{abstract}

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

\subsection{Linearizing a metric space}
A \emph{pointed metric space} is a metric space $(M,d)$ with a chosen
basepoint $o$. A Lipschitz function $f:M\to\R$ has constant
$\Lip(f)=\sup_{x\ne y}\abs{f(x)-f(y)}/d(x,y)$. Let $\Lip_0(M)$ be the
Lipschitz functions with $f(o)=0$, normed by $\Lip(f)$. Its \emph{Lipschitz-free
space} is the closed span
$\F(M)=\overline{\operatorname{span}}\{\delta_x:x\in M\}\subseteq\Lip_0(M)^*$,
where $\delta_x(f)=f(x)$; thus $\delta_o=0$.

The point of $\F(M)$ is that it turns $M$ into a normed space without
distortion. The Lipschitz inequality gives $\norm{\delta_x-\delta_y}\le d(x,y)$,
while the function $g(z)=d(z,y)-d(o,y)\in\Lip_0(M)$ has $\Lip(g)\le1$ and
$g(x)-g(y)=d(x,y)$. Hence
\begin{equation}\label{eq:isometric}
 \norm{\delta_x-\delta_y}_{\F(M)}=d(x,y),
\end{equation}
so $M$ sits isometrically in $\F(M)$. Moreover $\F$ \emph{linearizes}: a
pointed $1$-Lipschitz map $u:M\to N$ induces a contraction $\F(u)$ sending
$\delta_x\mapsto\delta_{u(x)}$ (proved as Lemma~\ref{lem:lin}). Linear
questions about $\F(M)$ therefore translate back to metric questions about
$M$.

\subsection{Approximation properties}
A bounded operator is \emph{finite rank} if its image is finite-dimensional.
A Banach space $X$ has the \emph{approximation property} (AP) if for every
norm-compact $C\subseteq X$ and every $\varepsilon>0$ there is a finite-rank
$T$ with $\sup_{u\in C}\norm{Tu-u}<\varepsilon$; the norm of $T$ may depend on
$C,\varepsilon$. For $\lambda\ge1$, $X$ has \emph{$\lambda$-BAP} if such $T$
can always be chosen with $\norm{T}\le\lambda$; \emph{BAP} means $\lambda$-BAP
for some finite $\lambda$; \emph{MAP} is $1$-BAP. So AP says ``we can
approximate'', BAP says ``with a uniform norm bound''.

A metric space is \emph{uniformly discrete} if distinct points are a positive
distance apart, and \emph{proper} if closed bounded balls are compact. Kalton
proved that over uniformly discrete $M$, $\F(M)$ always has AP, and asked
whether it always has BAP. We answer no.

\subsection{The theorem}
\begin{theorem}\label{thm:main}
There is a countable, unbounded, nonproper pointed metric space $(M,d,o)$ with
$d(x,y)\ge1$ for distinct $x,y$, such that $\F(M)$ has AP but fails
$\lambda$-BAP for every finite $\lambda\ge1$.
\end{theorem}
The example must be nonproper: Dalet proved that over a countable proper
space $\F(M)$ always has MAP. Our space contains an infinite $1$-separated set
inside a bounded ball. Uniform discreteness cannot be dropped either: Smith
had already produced countable discrete (not uniformly discrete) examples.

\subsection{Strategy}
Fix a candidate norm bound $p$.
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Blocks.} In Section~\ref{sec:block} we build a bounded space
$M_p$ with a finite anchor set $A_p$ such that every finite-rank operator of
norm at most $p$ moves some $\delta_a$, $a\in A_p$, by distance
$\ge\tfrac12$ (Theorem~\ref{thm:block}). Because $M_p$ is bounded and uniformly
discrete, $\F(M_p)$ itself has BAP, so a block settles nothing alone; it is a
trap at scale $p$.
\item \textbf{Gluing.} In Section~\ref{sec:assemble} we form the wedge $M$ by
identifying the basepoints $o$ of all blocks. Each block is a $1$-Lipschitz
retract of $M$, so $\F(M_p)$ is an isometrically complemented subspace of
$\F(M)$. A $\lambda$-BAP of $\F(M)$ would then give a $\lambda$-BAP of
$\F(M_p)$ for every $p$, contradicting the trap once $p\ge\lambda$.
\item \textbf{AP.} Approximating a compact set never needs more than finitely
many blocks, and inside each we use its BAP; the operator norm may grow with
the number of blocks retained, which AP allows.
\end{itemize}
The rest of the paper constructs and analyzes one block.

\section{Preliminaries}\label{sec:prelim}

We use the function space and free space of the introduction; here are the two
facts we need repeatedly.

\begin{lemma}[evaluation distances]\label{lem:dist}
For all $x,y\in M$, $\norm{\delta_x-\delta_y}_{\F(M)}=d(x,y)$.
\end{lemma}
\begin{proof}
For $\Lip(f)\le1$ we have $\abs{f(x)-f(y)}\le d(x,y)$, giving
$\norm{\delta_x-\delta_y}\le d(x,y)$. Conversely $g(z)=d(z,y)-d(o,y)$ lies in
$\Lip_0(M)$ with $\Lip(g)\le1$ and $g(x)-g(y)=d(x,y)$, giving the reverse
inequality.
\end{proof}

\begin{lemma}[linearization]\label{lem:lin}
Let $u:(M,d,o)\to(N,\rho,o')$ be pointed with $\Lip(u)\le1$. Setting
$\F(u)(\sum_x c_x\delta_x)=\sum_x c_x\delta_{u(x)}$ on finite sums extends
uniquely to a contraction $\F(u):\F(M)\to\F(N)$, and $\F$ is functorial:
$\F(\Id)=\Id$ and $\F(v\circ u)=\F(v)\circ\F(u)$.
\end{lemma}
\begin{proof}
For $\varphi\in\Lip_0(N)$ with $\Lip(\varphi)\le1$, the composition
$\varphi\circ u$ lies in $\Lip_0(M)$ with $\Lip(\varphi\circ u)\le1$, so
\[
 \Big\langle\sum_x c_x\delta_{u(x)},\varphi\Big\rangle
 =\sum_x c_x\,\varphi(u(x))
 =\Big\langle\sum_x c_x\delta_x,\ \varphi\circ u\Big\rangle
 \le\Big\|\sum_x c_x\delta_x\Big\|.
\]
Taking the supremum over such $\varphi$ gives
$\norm{\sum_x c_x\delta_{u(x)}}\le\norm{\sum_x c_x\delta_x}$, so the definition
is independent of the representation and extends to a contraction.
Functoriality holds on evaluations and hence on all of $\F(M)$ by continuity.
\end{proof}

\section{The bounded block}\label{sec:block}

Fix $p\ge1$ and put $B_p=300p^2+150p+5$.

\begin{theorem}[the block]\label{thm:block}
There are a bounded, countably infinite pointed metric space $(M_p,d_p,o)$ and
a finite set $A_p\subseteq M_p$ containing $o$, with $d_p(x,y)\ge1$ for distinct
$x,y$ and $\diam M_p\le B_p$, such that every bounded finite-rank operator
$T_0:\F(M_p)\to\F(M_p)$ with $\norm{T_0}\le p$ satisfies
\[
 \max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\tfrac12.
\]
\end{theorem}

The construction has three layers: random-like regular colored multigraphs
whose small neighborhoods mimic a regular tree; a labeling of each vertex by
its capped distances to the color classes; and a metric routing all vertices
sharing a label through one anchor and linking anchors. Sections~\ref{sec:walks}
and~\ref{sec:tests} then prove Theorem~\ref{thm:block}.

\subsection{Regular multigraphs with a tree-like colored limit}

We use multigraphs. Multiple edges are distinct instances; each instance has
two oriented \emph{incidences} $e,\bar e$ exchanged by reversal, even for a loop,
so a loop adds two to the degree. Write $e:x\to y$ for initial and terminal
vertices and let $\mathcal D(x)$ be the set of $\Delta$ outgoing incidences at
$x$. A walk is \emph{non-backtracking} if it never immediately reverses its
preceding incidence; it may revisit vertices and retake loops. A rooted
radius-$r$ ball is the induced multigraph on vertices within graph distance $r$
of the root, retaining colors, loops, and multiplicities. Only vertices are
points of the metric spaces below.

\begin{lemma}[colored tree limit]\label{lem:locallimit}
For every even integer $\Delta\ge4$ and integer $k\ge1$ there is a deterministic
sequence of nonempty finite $\Delta$-regular multigraphs $G_m$, colored by
$\{1,\ldots,k\}$, such that for every fixed $r\ge0$ and every rooted colored
radius-$r$ type $\tau$, the fraction of vertices of $G_m$ whose ball has type
$\tau$ converges, as $m\to\infty$, to the probability of $\tau$ for the
$\Delta$-regular tree with independent uniform vertex colors.
\end{lemma}
\begin{proof}
Fix $N$ and take $\Delta/2$ independent uniform permutations
$\sigma_1,\ldots,\sigma_{\Delta/2}$ of an $N$-set. For each $a$ and $v$ create a
distinct edge instance joining $v$ to $\sigma_a(v)$; each $a$ gives every vertex
one incidence to its image and one to its preimage, so the multigraph is
$\Delta$-regular (fixed points are loops, 2-cycles parallel edges). Color
vertices independently and uniformly in $\{1,\ldots,k\}$, independently of the
permutations.

Fix $r$ and explore from a root by breadth-first search, revealing every
relevant permutation image and preimage at vertices up to distance $r$, and
recording (without expanding) vertices newly met at distance $r+1$ so that all
edges of the induced radius-$r$ ball, including boundary edges, are seen. The
number of queries and revealed vertices is at most $C=C(\Delta,r)$, independent
of $N$.

Conditionally on what is revealed, each unrevealed part of a permutation is a
uniform bijection between unused domain and range. A fresh image or preimage is
uniform among at least $N-C$ unused values, at most $C$ of which are already
discovered, so its chance of hitting a known vertex is at most $C/(N-C)$. Before
any collision each fresh query adds one vertex and one edge; loops or second
parallel edges occur exactly when a fresh query hits a known vertex. A union
bound over the at most $C$ queries gives collision probability at most
$C^2/(N-C)=O_{\Delta,r}(N^{-1})$. Off this event the radius-$r$ ball is exactly
the regular-tree ball.

For two distinct roots, run both explorations together with both roots
initially discovered; with a larger $C(\Delta,r)$ the same bound holds, counting
a hit on either exploration as a collision, so with probability $1-o(1)$ the two
radius-$r$ balls are disjoint tree balls and the colors on their distinct
vertices stay independent.

Let $I_v$ indicate the ball at $v$ has type $\tau$, and set
$X_{\tau,N}=\frac1N\sum_v I_v$, $q_\tau=\Prob(\text{tree ball has type }\tau)$.
The one-root exploration gives $\E X_{\tau,N}\to q_\tau$; the two-root
exploration gives
$\E X_{\tau,N}^2=\frac1N\E I_1+(1-\frac1N)\E(I_1I_2)\to q_\tau^2$ (this covers
non-tree types, where $q_\tau=0$). Hence $\operatorname{Var}(X_{\tau,N})\to0$
and each frequency converges in probability to $q_\tau$.

At radius at most $m$ there are only finitely many colored types, so finite
union bounds show that for all large $N$, with positive probability every such
frequency is within $1/m$ of its tree value. Choose one such realization, call
its size $N_m$, taken strictly increasing in $m$. Then every fixed
radius-and-type frequency tends to $q_\tau$, the asserted deterministic
convergence.
\end{proof}

\subsection{Parameters}

Fix $p\ge1$. The scalar tests take values in $[-L,L]$ and so satisfy a
Lipschitz inequality automatically between opposite values at points at least
$2L$ apart. We take the attachment length $R>2L$ so that no path short enough to
matter crosses a thread, and the label slope $L/K$ tiny so that the comparison
across a thread, whose norm error is at most $2pR+1$, costs less than $L$. Set
\begin{equation}\label{eq:params}
 L=10p,\ \ R=3L=30p=:R_p,\ \ K=10p(R+1),\ \ D=K+2L+5,\ \ B=4R+D,\ \ V=1+4pB.
\end{equation}
Substituting gives $B=300p^2+150p+5=B_p$, with $R>2L$ and
\[
 \frac{2pR+1}{K}=\frac{60p^2+1}{300p^2+10p}\le\frac15,\qquad
 K+2L+1=D-4<D.
\]
The last inequality leaves room, below the cap $D$, for a label gap below $K$,
one source-edge step, and one support step of length at most $2L$.

The signed-measure estimate bounds total variation by $V=1+4pB$, and a large
degree dilutes the loss from at most one exceptional direction per support
point. Choose an even integer $\Delta\ge4$ with $V/\Delta<1/20$, and set
\[
 Q=\Delta^{2L+1},\qquad H=\Delta^{D+1}.
\]
The geometry gives at most $Q$ nearby successors for a support point and at most
$H$ label coordinates below the cap, bounding the admissible source colors; a
walk has $\Delta$ choices per step. A larger degree helps the averaging but
multiplies the walks in the matching count, so after fixing $\Delta$ choose an
integer $k>2\Delta QH$, making $\Delta QH/k<1/2$. Fix once and for all a
deterministic sequence $(G_m)_{m\ge1}$ from Lemma~\ref{lem:locallimit} for this
$\Delta$ and $k$. All constants and the whole sequence are fixed before any
operator appears.

\subsection{Labels, anchors, and the metric}

Take disjoint copies of the graphs $G_m$ and call their vertices \emph{source
vertices}. For a source vertex $x\in G_m$ define $h(x)\in\{0,\ldots,D\}^k$ by
\[
 h_i(x)=\min\big\{D,\ \dist_{G_m}(x,\{\text{vertices of color }i\})\big\},
 \qquad 1\le i\le k,
\]
the graph distance being $+\infty$ if $x$'s component has no vertex of color
$i$, so the coordinate equals $D$. This uses unit graph distances, before the
metric below.

Let $\mathcal H\subseteq\{0,\ldots,D\}^k$ be the labels realized by source
vertices. For each $v\in\mathcal H$ add one new vertex $a_v$, an \emph{anchor},
with $h(a_v)=v$; set $A_p=\{a_v:v\in\mathcal H\}$ and $a(x)=a_{h(x)}$. The
edges are: length $R$ from each source vertex $x$ to its anchor $a(x)$; length
$2R+\norm{v-w}_\infty$ between distinct anchors $a_v,a_w$; length $1$ on each
source edge. Let $M_p$ be all source vertices and anchors, and let $d_p(u,v)$ be
the infimum of lengths of finite paths from $u$ to $v$ (the empty path giving
$d_p(u,u)=0$). Choose an anchor as basepoint $o$ (one exists since each $G_m$ is
nonempty).

Every source vertex hangs by a length-$R$ thread from the anchor bearing its
label, and anchors are joined by edges whose length dominates the sup-distance of
their labels; this bounds the space and separates the source vertices.

\begin{lemma}\label{lem:geometry}
The construction satisfies:
\begin{enumerate}[label=(\roman*)]
\item $(M_p,d_p,o)$ is a countably infinite pointed metric space with separation
at least one and diameter at most $B=B_p$; $A_p$ is finite and contains $o$; each
source edge has endpoints at $d_p$-distance at most one.
\item The label $h$ is $1$-Lipschitz for the sup norm; a source vertex $x$ of
color $i$ has $h_i(x)=0$; and $h(a(x))=h(x)$.
\item For every source vertex $x$ and anchor $a$, $d_p(x,a)\ge R$ and
$d_p(x,a(x))=R$; in particular $d_p(x,o)\ge R_p=30p$.
\item Every closed ball of radius $2L$ has at most $Q$ points, and at most $H$
coordinates of any $h(t)$ are strictly below $D$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) There are at most $(D+1)^k$ labels, so $A_p$ is finite. The disjoint union of
the nonempty finite graphs is countable, and adding finitely many anchors keeps
it countable. The added edges connect all vertices. Every edge length is a
positive integer: source and thread lengths are integers, and the sup-distance of
integer labels is an integer. For distinct endpoints the lengths of connecting
paths form a nonempty set of positive integers, which has a least element, so
$d_p$ is attained and is a metric; distinct points are at distance at least one
because every edge has length $\ge1$. A source edge gives distance at most one
between its endpoints.

The sup-distance of two labels is at most $D$, so two source vertices with
different anchors are joined through the two threads and the anchor edge in
length at most $R+(2R+D)+R=4R+D=B$; with a common anchor the length is $2R$. A
source vertex and an anchor are within $3R+D$, two anchors within $2R+D$, so the
diameter is at most $B$.

(iii) A path from a source vertex to an anchor must first enter the anchor set,
and the only edges from source vertices to anchors are threads of length $R$;
hence $d_p(x,a)\ge R$. The thread realizes length $R$, and $o$ is an anchor, so
$d_p(x,o)\ge R$.

(ii) Across a source edge the distances to a fixed color class differ by at most
one whenever that color is present in the component (append the edge to a
realizing path), and capping at $D$ preserves this; if the color is absent, both
coordinates equal $D$. So adjacent source labels differ by at most one in sup
norm. Along a thread the labels are equal; along an anchor edge the length
dominates the sup-distance of the labels. Summing along a shortest path gives
$\norm{h(u)-h(v)}_\infty\le d_p(u,v)$. The own-color coordinate is zero by
definition, and anchor labels are copied.

(iv) Every non-thread edge has length at least $R=3L>2L$, so a shortest path of
length at most $2L$ uses only source edges. The radius-$2L$ ball about an anchor
is a singleton; that about a source vertex lies in its graph ball of radius $2L$
inside its own $G_m$. In a $\Delta$-regular multigraph every vertex within graph
distance $s$ is reached by some walk of length at most $s$, and there are at most
$\Delta^t$ walks of length $t$ from a fixed vertex, so the ball has at most
$\sum_{t=0}^{2L}\Delta^t\le\Delta^{2L+1}=Q$ points. If $h_i(x)<D$ then $x$ is
within graph distance $D-1$ of a color-$i$ vertex; witnesses for distinct colors
are distinct, so at most $\sum_{t=0}^{D-1}\Delta^t\le\Delta^{D+1}=H$ coordinates
are below $D$, at source vertices and, by copying, at anchors.
\end{proof}

\section{From operators to signed measures}\label{sec:measures}

Fix $p\ge1$ and the block $(M_p,d_p,o)$ with anchors $A_p$ and label $h$. An
operator that approximates the anchors will give, at every source vertex, a
signed measure of bounded total variation whose integral against a bump set by
the vertex's label is large.

\subsection{A common finite support}

Finite rank need not mean finite support; a small perturbation buys one.

\begin{lemma}\label{lem:support}
Suppose a bounded finite-rank $T_0:\F(M_p)\to\F(M_p)$ satisfies $\norm{T_0}\le p$
and $\norm{T_0\delta_a-\delta_a}<\tfrac12$ for all $a\in A_p$. Then there are a
finite set $S\subseteq M_p$ containing $o$ and a bounded finite-rank
$T:\F(M_p)\to\F(M_p)$ with $T\F(M_p)\subseteq\operatorname{span}\{\delta_j:j\in S\}$,
$\norm{T}\le2p$, and $\norm{T\delta_a-\delta_a}<1$ for all $a\in A_p$.
\end{lemma}
\begin{proof}
If $T_0=0$ take $T=0$, $S=\{o\}$. Otherwise choose a basis $v_1,\ldots,v_r$ of
the range; the coordinate functionals composed with $T_0$ are bounded functionals
$\phi_i$ with $T_0u=\sum_i\phi_i(u)v_i$. Choose $0<\eta<\min\{p,1/(2B)\}$ and
finite evaluation combinations $w_i$ with
$\sum_i\norm{\phi_i}\,\norm{w_i-v_i}<\eta$ (the span of evaluations is dense).
Then $Tu=\sum_i\phi_i(u)w_i$ satisfies $\norm{T-T_0}<\eta$; let $S$ be $\{o\}$
together with the supports of the $w_i$. Then $T$ has the stated support and
$\norm{T}\le p+\eta<2p$. Since $\norm{\delta_a}\le B$,
$\norm{T\delta_a-\delta_a}\le\norm{T_0\delta_a-\delta_a}+\eta B<\tfrac12+\eta B<1$.
\end{proof}

\subsection{Measures detected by label bumps}

For a source vertex $x$ define on all of $M_p$
\begin{equation}\label{eq:bump}
 b_x(t)=L\Big(1-\frac{\norm{h(x)-h(t)}_\infty}{K}\Big)_+,\qquad s_+=\max\{s,0\}.
\end{equation}
By Lemma~\ref{lem:geometry} the map $h$ is $1$-Lipschitz, so $b_x\in[0,L]$ has
Lipschitz constant at most $L/K\le1$ and equals $L$ at the anchor $a(x)$. For a
signed measure $\nu$ on finite $S$, write $|\nu|(E)=\sum_{j\in E}|\nu(\{j\})|$ and
$\norm{\nu}_{TV}=|\nu|(S)$.

\begin{lemma}\label{lem:measures}
Let finite $S\subseteq M_p$ contain $o$ and let bounded $T$ satisfy
$T\F(M_p)\subseteq\operatorname{span}\{\delta_j:j\in S\}$, $\norm{T}\le2p$, and
$\norm{T\delta_a-\delta_a}<1$ for $a\in A_p$. Then there are signed measures
$(\mu_x)_{x\in M_p}$ on $S$, each of total mass one, with
\begin{equation}\label{eq:pairing}
 \int g\,d\mu_x=(T\delta_x)(g-g(o))+g(o)
\end{equation}
for every Lipschitz $g$, with $\norm{\mu_x}_{TV}\le V=1+4pB$, and with
$\int b_x\,d\mu_x\ge\tfrac{3L}{4}$ for every source vertex $x$.
\end{lemma}
\begin{proof}
Write $T\delta_x=\sum_{j\in S\setminus\{o\}}c_x(j)\delta_j$; set
$\mu_x(\{j\})=c_x(j)$ for $j\ne o$ and $\mu_x(\{o\})=1-\sum_{j\ne o}c_x(j)$.
These masses sum to one and give the displayed affine formula.

For the variation, let $g$ equal the sign of $\mu_x(\{j\})$ on $S$ and $0$ off
$S$. Its values lie in $[-1,1]$, so by separation $\Lip(g)\le2$, and
\[
 \norm{\mu_x}_{TV}=\int g\,d\mu_x=(T\delta_x)(g-g(o))+g(o)\le2\norm{T\delta_x}+1
 \le1+4pB,
\]
using $\norm{T}\le2p$ and $\norm{\delta_x}\le B$.

For a source vertex $x$,
$\norm{T\delta_x-\delta_{a(x)}}\le\norm{T(\delta_x-\delta_{a(x)})}
+\norm{T\delta_{a(x)}-\delta_{a(x)}}\le2pR+1$. Since $b_x(a(x))=L$,
\[
 \int b_x\,d\mu_x-L=(T\delta_x-\delta_{a(x)})(b_x-b_x(o))
 \ge-\frac{L}{K}(2pR+1)\ge-\frac{L}{5},
\]
because $(2pR+1)/K\le1/5$. Hence $\int b_x\,d\mu_x\ge\tfrac{4L}{5}\ge\tfrac{3L}{4}$.
\end{proof}

\section{Rare matched walks}\label{sec:walks}

Fix a finite set $S\subseteq M_p$ containing $o$; its points are \emph{states}
and its labels and distances stay fixed. This is the support from
Lemma~\ref{lem:support}; its points may be source vertices or anchors. We select
the graph only after $S$ is fixed.

A graph vertex $x$ and a state $j\in S$ are \emph{compatible} when
\[
 \norm{h(x)-h(j)}_\infty\le K+2L+1=D-4.
\]
A non-backtracking walk $x_0,e_1,x_1,\ldots,e_t,x_t$ in one graph $G_m$ is
\emph{matched} if there are states $j_0,\ldots,j_t\in S$ with $(x_s,j_s)$
compatible for every $s$ and $d_p(j_{s-1},j_s)\le2L$ for $1\le s\le t$. States and
source vertices may repeat; only the reverse of the previous incidence is
forbidden. The compatibility threshold allows for a label gap below $K$ (where a
bump is positive), one source-edge step of length at most one, and one state step
of length $2L$; the inequality $K+2L+1<D$ also restricts the possible colors.

\begin{lemma}\label{lem:rare}
For fixed $n\ge1$, let $\beta_m$ be the fraction of vertices of $G_m$ from which
a matched walk of length $n$ starts. Then
\begin{equation}\label{eq:bad}
 \limsup_{m\to\infty}\beta_m\le|S|\frac Hk\Big(\frac{\Delta QH}{k}\Big)^n.
\end{equation}
Hence one can choose $n\ge2$ and then $m$ so that $\beta_m<1/(20V)$.
\end{lemma}
\begin{proof}
For $j\in S$ set $C_j=\{i:h_i(j)<D\}$, so $|C_j|\le H$ by
Lemma~\ref{lem:geometry}. If $x$ has color $i$ and is compatible with $j$, then
$h_i(x)=0$ forces $h_i(j)\le K+2L+1=D-4<D$, so $i\in C_j$; this is only a
necessary condition, avoiding any independence claim about full labels.

Consider the weaker event that a non-backtracking length-$n$ walk
$x_0,\ldots,x_n$ from the root has states $j_0,\ldots,j_n$ with
$d_p(j_{s-1},j_s)\le2L$ for $1\le s\le n$ and with the color of $x_s$ in $C_{j_s}$
for $0\le s\le n$. Its occurrence depends only on the induced colored radius-$n$
ball, of which there are finitely many rooted types. With $S,n$ fixed, the sets
$C_j$ and the transition relation are fixed, so the event is a fixed finite union
of types, and Lemma~\ref{lem:locallimit} makes its frequency converge to its
probability on the independently colored tree. (The support may depend on the
whole graph sequence, but once fixed it only selects types; no colors are
conditioned on it.)

On the tree there are at most $\Delta^n$ non-backtracking length-$n$ walks, each
visiting $n+1$ distinct vertices. For each $j$, its allowed successors lie in
$S\cap\overline B_{d_p}(j,2L)$, at most $Q$ points, so there are at most
$|S|Q^n$ state sequences. For a fixed walk and state sequence, independence of the
colors gives probability at most $(H/k)^{n+1}$ for all required memberships
(repeated states and overlapping $C_j$ do not matter, since tree vertices are
distinct). A union bound gives
$\Delta^n|S|Q^n(H/k)^{n+1}=|S|(H/k)(\Delta QH/k)^n$. Every matched walk satisfies
the weaker event, proving \eqref{eq:bad}. Finally $k>2\Delta QH$ gives
$\Delta QH/k<1/2$; choose $n\ge2$ with the right side of \eqref{eq:bad} below
$1/(40V)$, then $m$ with $\beta_m<1/(20V)$.
\end{proof}

\subsection{At most one exceptional direction per state}

Fix $n\ge2$ and one graph $G=G_m$. Call a vertex \emph{bad} if a matched walk of
length $n$ starts there, else \emph{good}. The next argument is deterministic: at
a good vertex the absence of long matched walks picks out at most one direction
per state.

For a good vertex $x$, an incidence $e\in\mathcal D(x)$, and a state $j\in S$,
let $\mathcal L_x(e,j)$ be the set of positive lengths of matched walks starting
through $e$ with initial state $j$, and set
$r_x(e,j)=\max(\{0\}\cup\mathcal L_x(e,j))$. A matched walk of length $\ge n$ has
a matched prefix of length $n$, so $\mathcal L_x(e,j)\subseteq\{1,\ldots,n-1\}$ and
the maximum is well-defined; the value $0$ asserts only that no positive-length
walk exists. Declare $e$ \emph{exceptional at $(x,j)$} if its record is strictly
greater than every other outgoing record. There is at most one such incidence; at
a bad vertex declare all outgoing incidences exceptional at every state. For an
incidence $e:x\to y$, put $E_e=\{j\in S: e\text{ is exceptional at }(x,j)\}$.

\begin{lemma}\label{lem:exceptional}
Let $e:x\to y$ be an incidence and $j,j'\in S$. If $(x,j)$ and $(y,j')$ are
compatible and $d_p(j,j')\le2L$, then $j\in E_e$ or $j'\in E_{\bar e}$.
\end{lemma}
\begin{proof}
If either endpoint is bad this is the definition. Suppose both are good and put
\[
 A_x=\max_{g\in\mathcal D(x)\setminus\{e\}}r_x(g,j),\qquad
 A_y=\max_{f\in\mathcal D(y)\setminus\{\bar e\}}r_y(f,j').
\]
The maxima range over nonempty finite sets (since $\Delta\ge4$). If $A_y>0$,
choose a walk realizing it and prepend $e$ with state transition $j$ to $j'$; the
old first incidence is not $\bar e$, so the result is non-backtracking and
matched. If $A_y=0$, the edge alone with states $j,j'$ is a matched walk. The same
argument after reversing $e$ gives
\begin{equation}\label{eq:records}
 r_x(e,j)\ge1+A_y,\qquad r_y(\bar e,j')\ge1+A_x.
\end{equation}
If neither incidence were exceptional, their records would be at most $A_x$ and
$A_y$, forcing $A_x\ge1+A_y\ge2+A_x$, a contradiction.
\end{proof}

\section{Scalar tests and the block obstruction}\label{sec:tests}

Keep $S$, $n\ge2$, and $G=G_m$. For an incidence $e:x\to y$ we prescribe the bump
$b_x$ off $E_e$ and $-b_y$ off $E_{\bar e}$; the bump bounds control pairs inside
one set and far cross pairs, and Lemma~\ref{lem:exceptional} controls close cross
pairs, making the combined data $1$-Lipschitz.

\begin{lemma}\label{lem:edgetest}
For every incidence $e:x\to y$ of $G$ there is a $1$-Lipschitz $f_e:M_p\to[-L,L]$
with $f_e(j)=b_x(j)$ for $j\in S\setminus E_e$ and $f_e(j')=-b_y(j')$ for
$j'\in S\setminus E_{\bar e}$.
\end{lemma}
\begin{proof}
Within either set the Lipschitz inequality follows from the bump bound. Across
the two sets the value difference at $j,j'$ is $b_x(j)+b_y(j')\le2L$, so the
inequality holds whenever $d_p(j,j')\ge2L$.

Suppose $d_p(j,j')<2L$ and $b_x(j)>0$. Then $\norm{h(x)-h(j)}_\infty<K$, and
since $d_p(x,y)\le1$ and $h$ is $1$-Lipschitz,
\[
 \begin{aligned}
 \norm{h(y)-h(j')}_\infty
 &\le\norm{h(y)-h(x)}_\infty+\norm{h(x)-h(j)}_\infty+\norm{h(j)-h(j')}_\infty\\
 &<1+K+2L.
 \end{aligned}
\]
Thus $(x,j)$ and $(y,j')$ are compatible and their state distance is at most
$2L$, so by Lemma~\ref{lem:exceptional} both prescriptions cannot be present. The
same holds when $b_y(j')>0$. Hence whenever two prescribed states, one from each
set, are less than $2L$ apart, both prescribed values vanish; taking $j=j'$ this
also makes the data consistent on the overlap.

Let $P\subseteq S$ be the union and $v:P\to[-L,L]$ the resulting data, which is
$1$-Lipschitz. If $P=\varnothing$ take $f_e=0$; otherwise McShane extension for
finite data by the formula below, then clipping, gives a $1$-Lipschitz
$f_e:M_p\to[-L,L]$ agreeing with $v$:
\[
 F(z)=\min_{j\in P}\{v(j)+d_p(z,j)\},\qquad
 f_e(z)=\max\{-L,\min\{L,F(z)\}\}.
\]
For $z\in P$ the Lipschitz inequalities give $F(z)=v(z)$; the triangle inequality
gives $F(z)\le F(w)+d_p(z,w)$ and symmetrically, so $F$ is $1$-Lipschitz, and
clipping preserves both the bound and the values.
\end{proof}

\subsection{Proof of the block theorem}

Each test compares the measures at the ends of its incidence, losing variation
only on exceptional states. At a good vertex each state is exceptional in at
most one of the $\Delta$ directions, so averaging controls the loss by the
degree and the bad fraction.

\begin{proof}[Proof of Theorem~\ref{thm:block}]
For the integer $p$, Lemma~\ref{lem:geometry} supplies the countably infinite
block, the separation and diameter bounds, and the finite anchor set $A_p$.
Suppose a bounded finite-rank $T_0$ of norm at most $p$ has error less than
$\tfrac12$ on every anchor. Get $T,S$ from Lemma~\ref{lem:support} and
$(\mu_x)$ from Lemma~\ref{lem:measures}. With $S$ fixed, choose $n\ge2$ and then
$G=G_m$ by Lemma~\ref{lem:rare} so that the bad fraction satisfies
$\beta<1/(20V)$; then apply Lemma~\ref{lem:exceptional} and
Lemma~\ref{lem:edgetest}.

For an incidence $e:x\to y$, since $f_e-b_x$ vanishes off $E_e$ and is bounded by
$2L$ there, and $f_e+b_y$ vanishes off $E_{\bar e}$, the test prescription gives
\begin{equation}\label{eq:lower}
 \int f_e\,d(\mu_x-\mu_y)
 \ge\frac{3L}{2}-2L\big(|\mu_x|(E_e)+|\mu_y|(E_{\bar e})\big).
\end{equation}
On the other hand, by \eqref{eq:pairing} (the affine pairing) and
$\norm{T}\le2p$, $\Lip(f_e)\le1$, $d_p(x,y)\le1$,
\begin{equation}\label{eq:upper}
 \Big|\int f_e\,d(\mu_x-\mu_y)\Big|
 =\Big|(T(\delta_x-\delta_y))(f_e-f_e(o))\Big|\le2p.
\end{equation}

Choose a uniform vertex $x$ of $G$, then a uniform incidence
$e\in\mathcal D(x)$. At a good vertex each state lies in at most one exceptional
set, so
\[
 \frac1\Delta\sum_{e\in\mathcal D(x)}|\mu_x|(E_e)
 \le\frac1\Delta\sum_{j\in S}|\mu_x(\{j\})|\le\frac V\Delta,
\]
while at a bad vertex the average is at most $V$; hence
\begin{equation}\label{eq:aveerr}
 \E\,|\mu_x|(E_e)\le\frac V\Delta+V\beta<\frac1{10}.
\end{equation}
By $\Delta$-regularity this sampling is uniform over all oriented incidences, and
reversal preserves it, so $\E\,|\mu_y|(E_{\bar e})<1/10$ as well. Taking
expectations in \eqref{eq:lower} and using \eqref{eq:upper} yields
\[
 2p>\frac{3L}{2}-\frac{4L}{10}=\frac{11L}{10}=11p,
\]
a contradiction. The construction fixed $p$, all constants, the whole graph
sequence, $M_p$, and $A_p$ before $T_0$; only after $T_0$ produced $S$ were $n$
and $m$ chosen, so the same finite anchor set defeats every operator in the class.
\end{proof}

\section{Assembling the answer}\label{sec:assemble}

We now prove Theorem~\ref{thm:main}. First, a bounded uniformly discrete space
has a concrete coefficient model and a controlled BAP. Then we glue the blocks
and separate AP from BAP.

\subsection{The coefficient model of a bounded block}

\begin{proposition}\label{prop:coeff}
Let $(N,d,o)$ be a countable pointed metric space with $d(x,y)\ge1$ for distinct
points and $\diam N\le B<\infty$. The map
$J:\ell_1(N\setminus\{o\})\to\F(N)$, $Jc=\sum_{x\ne o}c_x\delta_x$, is a linear
isomorphism onto $\F(N)$ with
\begin{equation}\label{eq:coeff}
 \tfrac12\norm{c}_1\le\norm{Jc}\le B\norm{c}_1.
\end{equation}
If $N\ne\{o\}$ then $\F(N)$ has $2B$-BAP; if $N=\{o\}$ both spaces are zero.
\end{proposition}
\begin{proof}
For finitely supported $c$, $\norm{\delta_x}\le B$ gives the upper bound. For
the lower bound define $f_c:N\to\R$ to be $\sgn(c_x)/2$ on the support (off $o$)
and $0$ elsewhere: it vanishes at $o$, its values differ by at most one, so by
separation it is $1$-Lipschitz, and testing gives
$\norm{Jc}\ge(Jc)(f_c)=\tfrac12\sum_{x\ne o}|c_x|$.

The upper bound makes $Jc$ converge in $\F(N)$ for every $c\in\ell_1$; both
bounds pass to the extension by approximation with finitely supported families.
So $J$ is injective with inverse of norm at most two on its range. The range is
closed since $\norm{c_m-c_j}_1\le2\norm{Jc_m-Jc_j}$; it contains every $\delta_x$,
whose span is dense, hence it is all of $\F(N)$.

For $N\ne\{o\}$ (so $B\ge1$), choose increasing finite $E_m$ exhausting
$N\setminus\{o\}$ with coordinate projections $q_m$. Then
$S_m=Jq_mJ^{-1}$ has finite rank and norm at most $2B$, and $S_mu\to u$ for all
$u$ because $q_mc\to c$ in $\ell_1$. Convergence is uniform on compact sets: given
compact $C$ and $\varepsilon>0$, take a finite $\varepsilon/(3(2B+1))$-net; for
large $m$ the error is below $\varepsilon/3$ on the net, and since
$\norm{S_m-\Id}\le2B+1$ it is below $\varepsilon$ on $C$. So $\F(N)$ has
$2B$-BAP. If $N$ is finite, eventually $q_m=\Id$. The case $N=\{o\}$ is trivial.
\end{proof}

\begin{corollary}\label{cor:renorm}
For each integer $p\ge1$, real $\ell_1$ carries an equivalent norm
$\norm{\cdot}_p$ with $\tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1$ such
that $(\ell_1,\norm{\cdot}_p)$ has $2B_p$-BAP but fails $p$-BAP; for $p=1$ it
fails MAP.
\end{corollary}
\begin{proof}
By Theorem~\ref{thm:block}, $M_p\setminus\{o\}$ is countably infinite; enumerate
it and let $J_p$ be the coefficient map of Proposition~\ref{prop:coeff} with
$B=B_p$. Set $\norm{c}_p=\norm{J_pc}_{\F(M_p)}$, so $J_p$ is a surjective
isometry $(\ell_1,\norm{\cdot}_p)\to\F(M_p)$, and \eqref{eq:coeff} gives the
stated norm bounds. Conjugation by $J_p$ preserves norms, finite rank, and
approximation errors, so it transfers $2B_p$-BAP to the new norm. If the new norm
had $p$-BAP, conjugation would give $p$-BAP on $\F(M_p)$; applying it with
accuracy $\tfrac12$ to $\{\delta_a:a\in A_p\}$ contradicts Theorem~\ref{thm:block}.
For $p=1$ this is failure of MAP.
\end{proof}

\subsection{The wedge and its retractions}

Take disjoint copies of the spaces $M_p$ and identify only their basepoints:
\[
 M=\{o\}\cup\bigsqcup_{p\ge1}\big(M_p\setminus\{o\}\big).
\]
Keep $d_p$ within each block $M_p$, and for nonbasepoint points in distinct blocks
set
\begin{equation}\label{eq:wedge}
 d(x,y)=d_p(x,o)+d_q(y,o)\qquad(x\in M_p\setminus\{o\},\ y\in M_q\setminus\{o\},\ p\ne q).
\end{equation}

\begin{lemma}\label{lem:wedge}
$M$ is a countable uniformly discrete pointed metric space with $d(x,y)\ge1$ for
distinct points; it is unbounded and nonproper; and for each $p$ there are
pointed $1$-Lipschitz maps $i_p:M_p\to M$ and $r_p:M\to M_p$ with $r_p\circ i_p=\Id$.
\end{lemma}
\begin{proof}
Symmetry and positivity are clear. For the triangle inequality, if the three
points involve only one block that block's inequality gives it; if a third point
$z$ lies in another block, the term $2d_q(z,o)$ that appears dominates, and a
detour through $o$ gives equality. So $d$ is a metric.

Countability is clear. Distinct points are at distance at least one: within a
block by Theorem~\ref{thm:block}, across blocks by \eqref{eq:wedge} since each
distance to $o$ is at least $R_p\ge1$. It is unbounded, since a source vertex of
block $M_p$ is at distance at least $R_p=30p$ from $o$; and it is nonproper,
because the closed ball about $o$ of radius $B_1$ contains the countably infinite
$1$-separated set $M_1$, whose cover by balls of radius $1/3$ has no finite
subcover.

Let $i_p$ be the inclusion and $r_p$ fix $M_p$ and send other blocks to $o$; both
are pointed. Inclusion preserves distances. For $r_p$, the only nontrivial case is
$x\in M_p$, $y\in M_q\setminus\{o\}$ with $q\ne p$, where
$d_p(r_p(x),r_p(y))=d_p(x,o)\le d(x,y)$; the other cases are immediate. So
$r_p\circ i_p=\Id$ with both maps $1$-Lipschitz.
\end{proof}

By Lemma~\ref{lem:lin} the maps $i_p,r_p$ give contractions
$I_p=\F(i_p):\F(M_p)\to\F(M)$ and $P_p=\F(r_p):\F(M)\to\F(M_p)$. Since
$r_p\circ i_p=\Id$, we have $P_pI_p=\Id$ on $\F(M_p)$, and then for every
$v\in\F(M_p)$,
\[
 \norm{v}=\norm{P_pI_pv}\le\norm{I_pv}\le\norm{v},
\]
so $I_p$ is an isometry onto an isometrically complemented subspace.

\subsection{BAP fails, AP holds}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Suppose $\F(M)$ had $\lambda$-BAP for a finite $\lambda\ge1$. Choose an integer
$p\ge\lambda$ and apply the property with accuracy $\tfrac12$ to the finite set
$\{I_p\delta_a:a\in A_p\}$, getting $U$ with $\norm{U}\le\lambda$. The compression
$P_pUI_p$ has finite rank and norm at most $\lambda\le p$, and for every $a\in A_p$
\[
 \norm{P_pUI_p\delta_a-\delta_a}\le\norm{UI_p\delta_a-I_p\delta_a}<\tfrac12,
\]
contradicting Theorem~\ref{thm:block}. All blocks and anchors were fixed before
$\lambda$ and $U$.

For AP, define $r_{\le N}:M\to M$ to keep the first $N$ blocks and send the rest
to $o$; it is pointed and $1$-Lipschitz, so its linearization
$R_N=\F(r_{\le N})=\sum_{p\le N}I_pP_p$ is a contractive projection. On an
evaluation $\delta_x$ the sum keeps $\delta_x$ if $x$ lies in the first $N$ blocks
and sends it to $0$ otherwise, so $R_N$ fixes every finite evaluation combination
once $N$ is large. Such combinations are dense and $\norm{\Id-R_N}\le2$, so
$R_Nu\to u$ for every $u$; the finite-net argument of Proposition~\ref{prop:coeff}
makes this uniform on compact sets.

Now fix compact $C\subseteq\F(M)$ and $\varepsilon>0$. Choose $N$ with
$\sup_{u\in C}\norm{R_Nu-u}<\varepsilon/2$. For $1\le p\le N$ the image $P_p(C)$
is compact, so by Proposition~\ref{prop:coeff} pick finite-rank
$T_p:\F(M_p)\to\F(M_p)$ with $\norm{T_p}\le2B_p$ and
$\sup_{u\in C}\norm{T_pP_pu-P_pu}<\varepsilon/(2N)$. Then
$T=\sum_{p\le N}I_pT_pP_p$ has finite rank and bound $\sum_{p\le N}2B_p<\infty$,
and for $u\in C$,
\[
 \norm{Tu-u}\le\norm{R_Nu-u}+\sum_{p\le N}\norm{T_pP_pu-P_pu}<\varepsilon.
\]
The combined bound may grow with $N$, which AP permits. So $\F(M)$ has AP. With the
uniform discreteness, unboundedness, and nonproperness of Lemma~\ref{lem:wedge},
this proves Theorem~\ref{thm:main}.
\end{proof}

\subsection{A nonlinear consequence}\label{sec:nonlinear}

Kalton asked whether every separable Banach space is \emph{approximable}: is
there an equi-uniformly continuous sequence $\psi_n:X\to X$ with relatively
compact ranges and $\psi_n(x)\to x$ pointwise? The space $\F(M)$ above is not
approximable, although it has AP.

\begin{proposition}\label{prop:nonlinear}
The space $X=\F(M)$ of Theorem~\ref{thm:main} is not approximable.
\end{proposition}
\begin{proof}
$X$ is separable and $\{\delta_x:x\in M\}$ is $1$-separated. Extend it to a
maximal $1$-separated set $N_X\subseteq X$; every point of $X$ is within distance
one of $N_X$, so $N_X$ is a net. By Kalton's characterization, if $X$ were
approximable there would be one Lipschitz bound $\Lambda\ge1$ for maps
$g:N_X\to N_X$ with relatively compact range that approximate any prescribed
finite subset as closely as desired.

Separation makes such a range finite and makes approximation within distance
less than one into exact fixing. Fix a finite $F\subseteq M$ containing $o$ and
choose $g$ fixing every $\delta_x$, $x\in F$. The restriction
$x\mapsto g(\delta_x)$ is a pointed $\Lambda$-Lipschitz map $M\to X$; by the
Banach-space linearization property of free spaces it yields a finite-rank
$T_F:X\to X$ of norm at most $\Lambda$ with $T_F\delta_x=g(\delta_x)$ for all $x$.

To approximate a compact set, approximate a finite net in it by finite evaluation
combinations and put all their supports in $F$. Then $T_F$ fixes those
combinations, and $\norm{T_F-\Id}\le\Lambda+1$ controls the remaining errors.
So these $T_F$ would be finite-rank operators of uniformly bounded norm
approximating on compacts, i.e. BAP, contradicting Theorem~\ref{thm:main}. Hence
$X$ is not approximable.
\end{proof}

\begin{thebibliography}{99}
\bibitem{AACD20} F.~Albiac, J.~L. Ansorena, M.~C\'uth, M.~Doucha, \emph{Lipschitz free $p$-spaces for $0<p<1$}, Israel J. Math. 240 (2020), 65--98.
\bibitem{ArensEells56} R.~Arens, J.~Eells, \emph{On embedding uniform and topological spaces}, Pacific J. Math. 6 (1956), 397--403.
\bibitem{BenjaminiSchramm01} I.~Benjamini, O.~Schramm, \emph{Recurrence of distributional limits of finite planar graphs}, Electron. J. Probab. 6 (2001), no.~23.
\bibitem{CuthKalenda26} M.~C\'uth, O.~Kalenda, \emph{Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur}, preprint, arXiv:2604.01875 (2026).
\bibitem{Dalet15} A.~Dalet, \emph{Free spaces over some proper metric spaces}, Mediterr. J. Math. 12 (2015), 973--986.
\bibitem{Friedman04} J.~Friedman, \emph{A proof of Alon's second eigenvalue conjecture and related problems}, arXiv:cs/0405020 (2004).
\bibitem{GodefroyKalton03} G.~Godefroy, N.~Kalton, \emph{Lipschitz-free Banach spaces}, Studia Math. 159 (2003), 121--141.
\bibitem{GodefroyOzawa14} G.~Godefroy, N.~Ozawa, \emph{Free Banach spaces and the approximation properties}, Proc. Amer. Math. Soc. 142 (2014), 1681--1687.
\bibitem{Kalton04} N.~Kalton, \emph{Spaces of Lipschitz and H\"older functions and their applications}, Collect. Math. 55 (2004), 171--217.
\bibitem{Kalton12} N.~Kalton, \emph{The uniform structure of Banach spaces}, Math. Ann. 354 (2012), 1247--1288.
\bibitem{McShane34} E.~McShane, \emph{Extension of range of functions}, Bull. Amer. Math. Soc. 40 (1934), 837--842.
\bibitem{Smith26} R.~J. Smith, \emph{Lipschitz-free spaces and Bossard's reduction argument}, arXiv:2509.00722 (2026).
\end{thebibliography}

\end{document}