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\begin{document}

A pointed metric space is a metric space $(M,d)$ with a chosen basepoint $o$.
For a Lipschitz map $f:M\to\R$ put $\Lip(f)=\sup_{x\ne y}\abs{f(x)-f(y)}/d(x,y)$,
and let $\Lip_0(M)$ be the Banach space of those $f$ with $f(o)=0$, normed by
$\Lip$. The Lipschitz-free space $\F(M)$ is the closed span of the evaluations
$\delta_x(f)=f(x)$ inside $\Lip_0(M)^*$, so $\delta_o=0$. The embedding is
isometric, and a pointed $1$-Lipschitz map $u:M\to N$ induces a contraction
$\F(u)$ carrying $\delta_x$ to $\delta_{u(x)}$; metric questions thereby become
linear ones. Arens and Eells introduced these spaces, and Godefroy and Kalton
developed their linear structure.
\par\noindent{\footnotesize (Arens and Eells, \emph{Pacific J.\ Math.} 6 (1956), 397--403.) \url{https://msp.org/pjm/1956/6-3/pjm-v6-n3-p03-s.pdf}}
\par\noindent{\footnotesize (Godefroy and Kalton, \emph{Studia Math.} 159 (2003), 121--141.) \url{https://doi.org/10.4064/sm159-1-6}}

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

\begin{lemma}\label{lem:lin}
A pointed $1$-Lipschitz $u:(M,d,o)\to(N,\rho,o')$ extends from
$\sum_x c_x\delta_x\mapsto\sum_x c_x\delta_{u(x)}$ to a unique contraction
$\F(u):\F(M)\to\F(N)$, with $\F(\Id)=\Id$ and $\F(v\circ u)=\F(v)\F(u)$.
\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
\[
 \Big\langle\sum_x c_x\delta_{u(x)},\varphi\Big\rangle
 =\sum_x c_x\,\varphi(u(x))
 =\Big\langle\sum_x c_x\delta_x,\ \varphi\circ u\Big\rangle
 \le\Big\|\sum_x c_x\delta_x\Big\|.
\]
The supremum over such $\varphi$ gives
$\norm{\sum_x c_x\delta_{u(x)}}\le\norm{\sum_x c_x\delta_x}$, so the map is well
defined and extends to a contraction. Functoriality holds on evaluations, hence
everywhere by continuity.
\end{proof}

A bounded operator is \emph{finite rank} when 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 $T$ being allowed to
depend on $C$ and $\varepsilon$. For $\lambda\ge1$, $X$ has \emph{$\lambda$-BAP}
if such $T$ can always be taken with $\norm{T}\le\lambda$; \emph{BAP} means
$\lambda$-BAP for some finite $\lambda$, and \emph{MAP} is $1$-BAP. Thus AP says
approximation is always possible, BAP that one bound serves every compact set.

Kalton proved that over a uniformly discrete space the free space always has AP,
and asked whether a uniform bound always exists. The question was restated by
Godefroy and Ozawa and in Godefroy's survey. The answer turns out to be negative.
\par\noindent{\footnotesize (Kalton, \emph{Collect.\ Math.} 55 (2004), 171--217.) \url{https://kaltonmemorial.missouri.edu/assets/docs/collect2004.pdf}}
\par\noindent{\footnotesize (Godefroy and Ozawa, \emph{Proc.\ Amer.\ Math.\ Soc.} 142 (2014), 1681--1687.) \url{https://arxiv.org/pdf/1201.0847}}
\par\noindent{\footnotesize (Godefroy, \emph{Comment.\ Math.} 55 (2015), 89--118.) \url{https://doi.org/10.14708/cm.v55i2.1104}}

\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 space must be nonproper: Dalet showed that over a countable proper space
$\F(M)$ always has MAP. Uniform discreteness cannot be dropped either, since
Smith had produced countable discrete examples.
\par\noindent{\footnotesize (Dalet, \emph{Mediterr.\ J.\ Math.} 12 (2015), 973--986.) \url{https://arxiv.org/pdf/1404.3939}}
\par\noindent{\footnotesize (Smith, \emph{Lipschitz-free spaces and Bossard's reduction argument}, arXiv 2026.) \url{https://arxiv.org/pdf/2509.00722}}

Fix a candidate bound $p$. A bounded space $M_p$ is built with a finite anchor
set $A_p$ at which every finite-rank operator of norm at most $p$ moves some
$\delta_a$, $a\in A_p$, by at least $\tfrac12$. Such a block settles nothing
alone, since a bounded uniformly discrete space has BAP; it is a trap at scale
$p$. The blocks are then glued at a common basepoint into a wedge $M$. 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 force a $\lambda$-BAP of
$\F(M_p)$ for every $p$, contradicting the trap once $p\ge\lambda$. Approximation
on a compact set never needs more than finitely many blocks, each handled by its
own BAP with a norm allowed to grow, which gives AP. The work lies in one block.

\begin{theorem}\label{thm:block}
For each integer $p\ge1$ 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 points and $\diam M_p\le B_p=300p^2+150p+5$, such
that every bounded finite-rank $T_0:\F(M_p)\to\F(M_p)$ with $\norm{T_0}\le p$
satisfies
\[
 \max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\tfrac12.
\]
\end{theorem}

The block is built from three layers: sparse random-like regular colored
multigraphs whose small neighborhoods mimic a regular tree, a labeling of each
vertex by capped distances to the color classes, and a metric that routes all
vertices sharing a label through one anchor and links the anchors. The empirical
rooted-neighborhood viewpoint is that of Benjamini and Schramm, and the
independent-permutation model of random regular graphs is Friedman's.
\par\noindent{\footnotesize (Benjamini and Schramm, \emph{Electron.\ J.\ Probab.} 6 (2001), no.~23.) \url{https://doi.org/10.1214/EJP.v6-96}}
\par\noindent{\footnotesize (Friedman, \emph{A proof of Alon's second eigenvalue conjecture}, arXiv 2004.) \url{https://arxiv.org/pdf/cs/0405020}}

Multiple edges are distinct instances, and each instance has two oriented
incidences $e,\bar e$ exchanged by reversal, even for a loop, so a loop
contributes 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.

\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, with fixed points giving loops and 2-cycles giving 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 each relevant
permutation image and preimage at vertices within distance $r$, and recording
without expanding the vertices first met at distance $r+1$ so that every edge of
the induced radius-$r$ ball is seen. The number of queries and revealed vertices
is at most $C=C(\Delta,r)$, independent of $N$. Conditionally on the revealed
data, each unrevealed part of a permutation is a uniform bijection between unused
domain and range, so a fresh image or preimage is uniform among at least $N-C$
unused values, at most $C$ of which are discovered; 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, and a loop or second parallel edge occurs 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; a larger $C(\Delta,r)$
bounds the chance of any collision, and then the two balls are disjoint tree balls
with independent colors.

Let $I_v$ indicate that 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 $\E X_{\tau,N}\to q_\tau$, and the two-root
exploration gives
$\E X_{\tau,N}^2=\frac1N\E I_1+(1-\frac1N)\E(I_1I_2)\to q_\tau^2$, which also
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 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. Choosing one such realization of size $N_m$, strictly increasing in
$m$, makes every fixed radius-and-type frequency tend to $q_\tau$.
\end{proof}

The scalar tests take values in $[-L,L]$, so a Lipschitz inequality holds
automatically between opposite values at points at least $2L$ apart. The
attachment length is taken above $2L$ so no short path crosses a thread, and the
label slope $L/K$ is made tiny so that comparing 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}
Substitution 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 $2L$. Total
variation will be bounded by $V$, 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 admissible source colors, while 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 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.

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 $+\infty$, hence the coordinate $D$, if the component of
$x$ has no vertex of color $i$. 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 $a_v,a_w$, and 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 $o$, which exists because each $G_m$
is nonempty. Each 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 while keeping small balls
inside a single source graph.

\begin{lemma}\label{lem:geometry}
The construction satisfies:
\begin{enumerate}[label=\textup{(\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 at least one, and 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 join through the two threads and the anchor edge in length at
most $R+(2R+D)+R=4R+D=B$, and with a common anchor in length $2R$. A source
vertex and an anchor lie within $3R+D$, two anchors within $2R+D$. Hence 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 the color is present in the component, since the edge is appended 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, and 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 edge other than a source 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, and 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}

An operator that nearly fixes the anchors yields, at each source vertex, a signed
measure of bounded total variation with a large integral against a bump set by the
vertex's label. Finite rank does not mean finite support, so a small perturbation
first buys a common finite support.

\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 $T=0$ and $S=\{o\}$. Otherwise choose a basis $v_1,\ldots,v_r$ of
the range, with bounded coordinate functionals composed with $T_0$ giving bounded
$\phi_i$ and $T_0u=\sum_i\phi_i(u)v_i$. Take $0<\eta<\min\{p,1/(2B)\}$ and finite
evaluation combinations $w_i$ with $\sum_i\norm{\phi_i}\,\norm{w_i-v_i}<\eta$,
possible by density. Then $Tu=\sum_i\phi_i(u)w_i$ has $\norm{T-T_0}<\eta$; 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}

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 label $h$ 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 a finite set $S$ 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 affine formula \eqref{eq:pairing}.

For the variation, let $g$ equal the sign of $\mu_x(\{j\})$ on $S$ and $0$ off $S$,
with $\operatorname{sign}(0)=0$. 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 $x$, Lemma~\ref{lem:geometry} gives $d_p(x,a(x))=R$, so
$\norm{T\delta_x-\delta_{a(x)}}\le\norm{T(\delta_x-\delta_{a(x)})}
+\norm{T\delta_{a(x)}-\delta_{a(x)}}\le2pR+1$. 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}

Fix a finite set $S\subseteq M_p$ containing $o$; its points are \emph{states} and
their labels and distances stay fixed. This is the support from
Lemma~\ref{lem:support}, and the graph is chosen 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 $1\le s\le t$. Both the
states and the source vertices may repeat; a return along a different parallel
edge is allowed, only the reverse of the preceding incidence is forbidden. The
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 at most $2L$;
the inequality $K+2L+1<D$ also restricts the possible source colors.

Each state has at most $Q$ possible successors, by the radius-$2L$ ball bound,
while a graph walk meets many possible colors; this disparity bounds the fraction
of roots from which a long matched walk starts.

\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 chooses $n\ge2$ 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 $0\le s\le n$. This 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, and no colors are conditioned on
it. Along a uniform non-backtracking walk with $\Delta$ choices per step, there
are at most $\Delta^n$ paths reaching $x_n$; a fixed choice of all states gives
probability at most $(H/k)^{n+1}$ for the required color memberships, since tree
vertices are distinct and overlaps of the $C_j$ do not matter. 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 below $1/(40V)$, then $m$
with $\beta_m<1/(20V)$.
\end{proof}

Call a vertex \emph{bad} if a matched walk of length $n$ starts there, else
\emph{good}. 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 at least $n$
has a matched prefix of length $n$, so $\mathcal L_x(e,j)\subseteq\{1,\ldots,n-1\}$
and the maximum is defined; the value $0$ says no positive-length walk exists. Call
$e$ \emph{exceptional at $(x,j)$} if its record is strictly larger than every other
outgoing record, so there is at most one such incidence; at a bad vertex call 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 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}

For an incidence $e:x\to y$ the bump $b_x$ is prescribed off $E_e$ and $-b_y$ off
$E_{\bar e}$; bump bounds control pairs inside one set and far cross pairs, and
Lemma~\ref{lem:exceptional} controls close cross pairs. The finite-data extension
used is McShane's.
\par\noindent{\footnotesize (McShane, \emph{Bull.\ Amer.\ Math.\ Soc.} 40 (1934), 837--842.) \url{https://www.ams.org/journals/bull/1934-40-12/S0002-9904-1934-05978-0/S0002-9904-1934-05978-0.pdf}}

\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,
\[
 \norm{h(y)-h(j')}_\infty\le\norm{h(y)-h(x)}_\infty+\norm{h(x)-h(j)}_\infty+\norm{h(j)-h(j')}_\infty<1+K+2L.
\]
Thus $(x,j)$ and $(y,j')$ are compatible with state distance 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 $j=j'$ 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 extend and clip:
\[
 F(z)=\min_{j\in P}\{v(j)+d_p(z,j)\},\qquad f_e(z)=\max\{-L,\min\{L,F(z)\}\}.
\]
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}

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. Take $T,S$ 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} and $\Lip(f_e)\le1$ with values in
$[-L,L]$,
\begin{equation}\label{eq:upper}
 \int f_e\,d(\mu_x-\mu_y)
 =\big(T(\delta_x-\delta_y)\big)(f_e-f_e(o)),
 \qquad
 \Big|\int f_e\,d(\mu_x-\mu_y)\Big|\le2p\,d_p(x,y)\le2p,
\end{equation}
because the difference of the two affine pairings is a single pairing of
$\delta_x-\delta_y$, and the Lipschitz norm of $f_e-f_e(o)$ is at most one.

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} gives
\[
 2p>\frac{3L}{2}-\frac{4L}{10}=\frac{11L}{10}=11p,
\]
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}

A bounded uniformly discrete space has a concrete coefficient model and a
controlled BAP. The identification with $\ell_1$ appears in the work of Albiac,
Ansorena, C\'uth, and Doucha, and the coefficient bounds are stated by C\'uth and
Kalenda.
\par\noindent{\footnotesize (Albiac, Ansorena, C\'uth, Doucha, \emph{Israel J.\ Math.} 240 (2020), 65--98.) \url{https://arxiv.org/pdf/1811.01265}}
\par\noindent{\footnotesize (C\'uth and Kalenda, \emph{Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur}, arXiv 2026.) \url{https://arxiv.org/pdf/2604.01875}}

\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 let $f_c:N\to\R$ equal $\sgn(c_x)/2$ 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 $C$. 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}

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 lie in one block that block's inequality applies; if the third point $z$
lies in another block, the detour through $o$ has length
$d_p(x,o)+2d_q(z,o)+d_p(y,o)\ge d_p(x,y)$, and a third point equal to $o$ gives
equality. So $d$ is a metric. Distinct points are at distance at least one: within
a block by Theorem~\ref{thm:block}, and across blocks by \eqref{eq:wedge} because
each nonbasepoint is at distance at least one from $o$.

Were $M$ of finite diameter $C_0$, every $M_p$ would have diameter at most $C_0$,
so Proposition~\ref{prop:coeff} would give $2C_0$-BAP on every $\F(M_p)$. For an
integer $p\ge\max\{1,2C_0\}$, applying that property with accuracy $\tfrac12$ to
$\{\delta_a:a\in A_p\}$ would produce a finite-rank operator of norm at most
$2C_0\le p$ with error below $\tfrac12$ at every anchor, contradicting
Theorem~\ref{thm:block}. Hence $M$ is unbounded. The closed ball about $o$ of
radius $B_1$ contains the countably infinite $1$-separated set $M_1$; if it were
compact, its cover by balls of radius $1/3$ would have a finite subcover, each
member holding at most one point of $M_1$, which is impossible. So $M$ is
nonproper.

Let $i_p$ be the inclusion and let $r_p:M\to M_p$ fix $M_p$ and send every other
block to $o$. Both are pointed and $1$-Lipschitz: inclusion preserves distances,
and for the retraction the only case needing an inequality has $x\in M_p$ and
$y\in M_q\setminus\{o\}$ with $q\ne p$, where
$d_p(r_px,r_py)=d_p(x,o)\le d(x,y)$.
\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
$r_p\circ i_p=\Id$, one has $P_pI_p=\Id$ 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.

\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 $R_N=\F(r_{\le N})=\sum_{p\le N}I_pP_p$ 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.

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}

A nonlinear consequence follows. 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. Its $1$-separated
image $\{\delta_x\}$ extends to a maximal $1$-separated net $N_X\subseteq X$, so
every point lies within distance one of $N_X$. If $X$ were approximable, one
Lipschitz bound $\Lambda\ge1$ would serve for maps $g:N_X\to N_X$ with relatively
compact range, and separation makes such a range finite, with approximation within
distance below one becoming exact fixing.

\begin{proposition}\label{prop:nonlinear}
The space $X=\F(M)$ of Theorem~\ref{thm:main} is not approximable.
\end{proposition}
\begin{proof}
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 $M\to X$, so by the linearization property of free spaces into a Banach space it
extends to a linear $T_F:X\to X$ of norm at most $\Lambda$ with
$T_F\delta_x=g(\delta_x)$ for all $x$; its range is finite, since the relatively
compact range of $g$ lies in the $1$-separated set $N_X$ and is therefore finite. 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. These $T_F$ would be
finite-rank operators of uniformly bounded norm approximating on compacts, that is
BAP, contradicting Theorem~\ref{thm:main}.
\end{proof}

The block rests on classical and modern ingredients. The finite metric-extension
lower bounds that separate projections from general operators come from Brudnyi
and Brudnyi, and the signed-versus-probability distinction in almost extension is
Naor's; the comparison of non-backtracking continuation lengths that underlies the
parent-direction records appears in the work of Ben-David, Childs, Gily\'en,
Kretschmer, Podder, and Wang.
\par\noindent{\footnotesize (Brudnyi and Brudnyi, \emph{Amer.\ J.\ Math.} 129 (2007), 217--314.) \url{https://arxiv.org/pdf/math/0404304}}
\par\noindent{\footnotesize (Naor, \emph{Adv.\ Math.} 384 (2021), 107761.) \url{https://arxiv.org/pdf/2009.11373}}
\par\noindent{\footnotesize (Ben-David, Childs, Gily\'en, Kretschmer, Podder, Wang, \emph{SIAM J.\ Comput.} 53 (2024), FOCS20-368--415.) \url{https://doi.org/10.1137/23M1573975}}

\end{document}