==== TABLE OF CONTENTS ====
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//cycle-charging.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//data.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//geometry.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//introduction.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//module.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//paper.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//preamble.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//spectrum.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//trace.tex
An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//references.bib
===========================

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//cycle-charging.tex ====
\begin{tikzpicture}[font=\small,>=Stealth,vertex/.style={circle,fill=black,inner sep=1.5pt}]
\node[circle,draw,fill=blue!8,minimum size=15mm] (b) at (0,0) {$B_P$};
\node[ellipse,draw,fill=orange!10,minimum width=35mm,minimum height=18mm] (c1) at (4,0.8) {$C_1$};
\node[ellipse,draw,fill=orange!10,minimum width=30mm,minimum height=14mm] (c2) at (4,-1.3) {$C_2$};
\draw (b.25) to[bend left=12] node[above,sloped] {attachments} (c1.155);
\draw (b.0) to[bend right=10] (c1.195);
\draw (b.-35) -- (c2.160);
\node[align=center,text width=31mm] at (0,-1.65) {overlap tree\\contracted schematically};
\node[align=center,text width=43mm] at (4,-2.6) {exterior vertices\\charged only to $P$};
\node[align=center,text width=31mm] at (0,1.6) {all remaining copies\\meet $P$ inside $B_P$};
\end{tikzpicture}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//cycle-charging.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//data.tex ====
\section{Labelled expanders and their finite cycle spaces}\label{sec:data}

Fix a positive \(\lambda<1/24\) so small that
\begin{equation}\label{eq:lambda-choice}
12(L_0+1)\lambda\le\frac1{16},
\end{equation}
where \(L_0\) is the universal constant from
\eqref{eq:root-constants}. Put \(\eta=1/100\).

\begin{proposition}[Choice of data]\label{prop:data}
There are labelled graphs \((\Theta_i)_{i\ge1}\) satisfying all the
hypotheses of Section~\ref{sec:geometry}, with the following additional
properties. Write \(n_i=|\operatorname{Vert}(\Theta_i)|\),
\(k_i=\lfloor n_i/2\rfloor\), and let \(\sigma_i\) be the girth.
There are constants \(\rho,\kappa>0\), independent of \(i\), such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item On the orthogonal complement of constants in
\(\ell^2(\operatorname{Vert}(\Theta_i))\),
\(dI-\Adj_{\Theta_i}\ge\rho I\), and for every vertex subset \(W\),
\begin{equation}\label{eq:expansion}
|\partial W|\ge\kappa\min\{|W|,n_i-|W|\}.
\end{equation}
Here \(\partial W\) is the unoriented edge boundary in \(\Theta_i\).
\item With
\begin{equation}\label{eq:c-definition}
c=\frac\eta2\min\{\kappa,1\},
\end{equation}
we have \(n_i>100\), \(\sigma_i>\max\{16,20/c\}\),
\(\lambda\sigma_i>3\), and
\(1<\lfloor\lambda\sigma_i\rfloor<\lfloor\lambda\sigma_{i+1}\rfloor\).
\item Every induced intrinsic ball of radius at most \(\sigma_i/16\)
is a tree with fewer than \(\eta n_i\) vertices.
\item The integers \(k_i\) strictly increase and
\(k_{i+1}-k_i\to\infty\).
\end{enumerate}
Consequently every ball \(B_P\) in Lemma~\ref{lem:overlap}, for every
permitted root and chosen root geodesic, is an induced tree satisfying
\begin{equation}\label{eq:small-ball}
r_P\le\sigma(P)/16,\qquad |B_P|<\eta|P|.
\end{equation}
\end{proposition}

\begin{proof}
Start with a fixed-degree family of connected regular expanders of
unbounded order and girth at least a positive constant times logarithmic
order, as supplied by the Lubotzky--Phillips--Sarnak construction
\cite[Theorems~3.4 and~4.1]{LPS1988}; see also \cite[Section~2.4]{Osajda2020}. Discarding small
graphs leaves simple graphs. Their Laplacians have a common gap
\(\rho>0\) off the constants. Bipartiteness, if present, causes no
problem: the adjacency eigenvalue \(-d\) gives Laplacian eigenvalue
\(2d\), not another zero eigenvalue.

For \(w=|W|\), the centered indicator of \(W\) has squared norm
\(w(n-w)/n\) and Laplacian energy \(|\partial W|\). Thus
\[
|\partial W|\ge\rho\frac{w(n-w)}n
\ge\frac\rho2\min\{w,n-w\},
\]
so we may choose \(\kappa=\rho/2\). A ball containing at most half the
vertices has an outside vertex boundary of size at least \(\kappa/d\)
times its size. Balls grow by a fixed factor until they contain more than
half the graph. Any two such balls intersect, giving a uniform
\(O(\log n)\) diameter bound. Hence the diameter/girth ratio is bounded.

An induced ball of integer radius \(r\) with \(2r+1<\sigma\) is a
tree: an edge outside a shortest-path tree would create a cycle of
length at most \(2r+1\). Its order at \(r\le\sigma/16\) is at most
\[
1+d\sum_{j=0}^{\lfloor\sigma/16\rfloor-1}(d-1)^j.
\]
Regularity and the same tree property at radius
\(\lfloor(\sigma-2)/2\rfloor\) give the lower bound
\[
n\ge1+d\sum_{j=0}^{\lfloor(\sigma-2)/2\rfloor-1}(d-1)^j.
\]
The ratio tends to zero with \(\sigma\). A subsequence therefore satisfies
(ii)--(iv), including (iii); growing order allows the gaps in (iv) to be
chosen arbitrarily large.

The resulting sequence has bounded degree, growing girth, bounded
diameter/girth ratio, and the required strictly increasing floors.
Osajda's labelling theorem \cite[Theorem~2.7]{Osajda2020}, applied to the
already chosen \(\lambda\), gives a reduced labelling over one finite
alphabet with no repeated long path word. Its conclusion for
nonbacktracking paths includes the simple-path condition used here.
The associated graphical presentation has isometrically embedded
relators by \cite[Lemma~3.1 and Theorem~3.2]{Osajda2020}.
This supplies the data of Section~\ref{sec:geometry}. Finally,
\eqref{eq:lambda-choice}, \eqref{eq:overlap-radius}, and (iii) give
\eqref{eq:small-ball}.
\end{proof}

\paragraph{Order of choices.}
The constants used above are chosen in the order recorded in
Table~\ref{tab:parameters}. In particular, the geometric constant is
universal before the small-cancellation parameter is selected. Neither
the alphabet nor the code field is fixed while choosing the expanders.

\begin{table}[ht]
\centering
\small
\begin{tabularx}{\linewidth}{@{}lX@{}}
\toprule
Choice & Requirement and subsequent use \\
\midrule
$D_0,t,L_0$ & Universal thinness and root-path constants from
Proposition~\ref{prop:cone-geometry} and \eqref{eq:root-constants}.\\
$\lambda$ & $0<\lambda<1/24$ and
$12(L_0+1)\lambda\le1/16$, so every overlap ball has radius at most
one sixteenth of the girth.\\
$d,\rho,\kappa$ & One fixed-degree expander family; take
$\kappa=\rho/2$. Set $\eta=1/100$ and
$c=(\eta/2)\min\{\kappa,1\}$.\\
$(\Theta_i)$ & Pass to a subsequence with
$\sigma_i>\max\{16,20/c\}$, small tree balls, strictly increasing
$\lfloor\lambda\sigma_i\rfloor$, and diverging gaps between
$k_i=\lfloor n_i/2\rfloor$.\\
$S$ & A single finite alphabet supplied by the labelling theorem for
this sequence and this value of $\lambda$.\\
$p,C_i$ & Choose a prime $p>2$ encoding all $d$-element sets of outgoing labels;
then choose the $k_i$-dimensional codes in Lemma~\ref{lem:codes}.\\
$\varepsilon$ & The uniform gap is
$\min\{\rho,c^2/(8d)\}$, as proved in
Lemma~\ref{lem:spectral-gap}.\\
\bottomrule
\end{tabularx}
\caption{Parameter choices for $\Gsur$. Each row uses only earlier
choices; the last row records the resulting spectral estimate.}
\label{tab:parameters}
\end{table}

From now on all these data, the group \(\Gamma\), its Cayley graph
\(X\), and the coned graph \(\widehat X\) are fixed. By
Proposition~\ref{prop:cone-geometry}, \(X\) is simple. We identify a copy
with its vertex set when discussing supports and specify its edges when
discussing chains.

For a graph \(Y\), let \(Z_1(Y;\F_2)\) be its ordinary
\emph{finite-support} cycle space: its elements are finite sums of edges
with coefficients in \(\F_2\), with even incidence at every vertex.
Thus no completed chain spaces are used in the next statement.

\begin{lemma}[Direct sum of copy cycles]\label{lem:cycle-sum}
The inclusions of the copies give an internal algebraic direct sum
\begin{equation}\label{eq:cycle-sum}
Z_1(X;\F_2)=\bigoplus_P Z_1(P;\F_2).
\end{equation}
\end{lemma}

\begin{proof}
Every finite cycle is a sum of closed walks. The label of any such walk
is trivial in \(\Gamma\), so a finite van Kampen diagram over the
presentation expresses its
edge chain, modulo two, as the sum of the chains of its bounded-face
walks. Each lies in a copy. This proves spanning.

Suppose finitely many nonzero cycles in distinct copies sum to zero.
Choose a group-vertex root \(R\) and a participating copy \(P\) whose
center is farthest from \(R\). By Lemma~\ref{lem:overlap}, all its
intersections with the other participating copies lie in one ball
\(B_P\), which is an induced tree by \eqref{eq:small-ball}. Every
nonzero edge coefficient in its cycle must cancel against an edge of
another participating copy. Both endpoints therefore lie in \(B_P\),
and inducedness puts the edge itself in that tree. A tree has no nonzero
finite cycle, a contradiction.
\end{proof}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//data.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//geometry.tex ====
\section{Coned geometry of graphical presentations}\label{sec:geometry}

This section establishes the geometric control needed by the module and
spectral arguments: relative to one root, all intersections of a translated
relator with relators no farther from the root lie in one short intrinsic
ball. We first define the labelled graphs and their coned Cayley graph,
then prove an avoidance estimate and uniform thinness. The existence of
the graph data, with the required expansion properties, is established in
Section~\ref{sec:data}.

Let \((\Theta_i)\) be finite connected simple \(d\)-regular graphs,
\(d\ge3\), of girths \(\sigma_i\). Label their oriented edges by a finite
alphabet \(S^{\pm1}\), with reversal corresponding to formal inversion.
Assume that the labelling is reduced, meaning locally injective into the
bouquet with alphabet \(S\), and that
\begin{equation}\label{eq:geometric-parameters}
0<\lambda\le\frac1{12},\qquad \lambda\sigma_i>3.
\end{equation}
We require the following uniqueness property: the label word of a simple
path in \(\Theta_i\) of length at least \(\lambda\sigma_i\) occurs on no
other simple path in the disjoint family.

Define
\[
\Gamma=\langle S\mid\text{labels of all closed walks in all }\Theta_i\rangle,
\qquad X=\Cay(\Gamma,S).
\]
Initially \(X\) is the formal labelled Cayley graph: it has the specified
edge from \(h\) to \(hs\) for each \(s\in S\), and the reverse traversal
has label \(s^{-1}\). This graph is deterministic for reading words, even
before loops or multiple edges have been excluded. Assume that the based
label maps \(\phi_i:\Theta_i\to X\) are isometric embeddings on vertices.
They are well-defined by the presentation. For every \(g\in\Gamma\), write
\[
P_{g,i}=g\phi_i(\Theta_i)
\]
for the corresponding \emph{copy}, with its source identification, and put
\(\sigma(P_{g,i})=\sigma_i\). Denote intrinsic distance in a copy by
\(d_P\).

\begin{lemma}\label{lem:distinct-copies}
Different indexed copies cannot share a simple edge path of length at
least \(\lambda\sigma(P)\) in either copy \(P\). In particular, different
index pairs give different labelled subgraphs.
\end{lemma}
\begin{proof}
Read a common path in the same direction in \(X\). Its pullbacks are
simple, since the vertex maps are injective. Label uniqueness identifies
the source paths and hence their source graph. Equality at the first
target vertex then identifies the translating elements. The graphs have
simple paths longer than the threshold, by regularity and girth, so
equality of whole copies also forces equality of their indices.
\end{proof}

Adjoin a new center \(c_P\) for every copy and join it by an edge to every
vertex of \(P\). The resulting graph is denoted \(\widehat X\); all edges
have length one.

\subsection{Diagrams with adjustable sides}

We shall use one boundary estimate twice: first to control paths avoiding
a specified center, and then to obtain short fillings in \(\widehat X\).
For a passage through a center, we hold fixed its endpoints and assigned
copy, while allowing the path inside that copy to change. A minimal
filling will cut these paths into linearly many arcs, counted in terms
of the length of the coned walk. An arc separating different assigned
copies will be short relative to the girth of either copy.

Start a nonempty closed edge walk in \(\widehat X\) at a group vertex
and partition it into \(m\) sides. Each side is a generator edge in
\(X\) or a two-edge passage through a center. Inflate a center passage
to any walk between its endpoints in its assigned copy. Such a side is
\emph{adjustable}; an empty walk is allowed when the endpoints coincide.
Generator sides remain fixed. The result is a closed walk in \(X\).

For an inflated loop, use a finite connected plane multigraph diagram
whose exterior contour maps to that loop and whose bounded-face contours
are closed walks in assigned copies. Such diagrams exist by van Kampen's
lemma: the presentation allows all closed walks in the source graphs,
and the group value at a face's initial vertex selects its translated
copy. Contours may repeat vertices and edges. This is essential for the
surgeries below: deleting a common edge merges its incident regions,
but other contacts can remain as repeated vertices or bridges. The
merged bounded contour must remain a closed walk in its assigned copy;
an adjusted exterior side must remain a walk with its prescribed
endpoints and assignment.

An \emph{incidence} is one occurrence of an edge on a contour; a bridge
has two incidences on the same contour. The exterior contour retains its
partition into side occurrences. Mark the positions where sides end,
and also the corresponding plane-graph vertices. There are at most
\(m\) distinct marked vertices, although the contour may visit a marked
vertex repeatedly. Only the marked positions are side breaks: another
visit to the same vertex need not end the current side. If inflation
gives the empty walk, an isolated marked vertex is permitted.

Minimize the number \(F\) of bounded faces, then the number of edges,
over diagrams and adjustable walks with the prescribed side data. Suppress
all unmarked degree-two vertices; the resulting edges will be called
\emph{arcs}, and their number will be denoted \(b\). An arc retains its
original edge length. Loop arcs are allowed.

\begin{lemma}[Short arcs and their number]\label{lem:diagram-count}
In a diagram chosen as above, the following hold.
\begin{enumerate}[label=\textup{(\alph*)}]
\item Every bounded face assigned to \(P\) has perimeter at least
\(\sigma(P)\). Every arc incident to that face has original length less
than \(\lambda\sigma(P)\).
\item More generally, an arc with different copy assignments on its two
incidences has length less than \(\lambda\sigma(Q)\) for either
assignment \(Q\). An arc with a generator-side incidence has length one.
\item The compressed diagram satisfies
\begin{equation}\label{eq:diagram-count}
(1/\lambda-6)F\le6m-6,\qquad b\le6m.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We first record the consequences of minimality, with contour incidences
understood throughout.

\emph{Face words and mergers.}
A bounded-face word cannot be freely trivial. Indeed, the diagram
identity expresses the exterior word as a product of conjugates of the
face words, each used once. Omitting a freely trivial factor and applying
van Kampen's lemma again gives a diagram with fewer faces for the same
exterior word. Its prescribed exterior paths lift identically by
determinism of \(X\). A closed walk in \(P\) shorter than \(\sigma(P)\)
cancels to the empty walk, so every face has the claimed lower perimeter
bound.

Suppose two distinct bounded faces with the same copy assignment share
an edge. That edge is not a bridge. Delete it and join the two contours
along its former incidences. Every remaining step lies in the same copy,
so the merged contour is an admissible closed walk there. Exactly one
bounded face disappears. Other contacts between the old contours may
produce repeated vertices or bridges, which our diagram category allows.
Thus this configuration contradicts minimality of \(F\).

The same operation excludes an edge shared by a bounded face and an
adjustable exterior side with the same assignment. Replace this
particular occurrence of the edge in the side by the rest of the
bounded contour. The resulting side is still a walk in that copy with
the same endpoints. Its prescribed position among the exterior sides
is unchanged, and again one bounded face disappears.

\emph{Leaves and bounded-face bridges.}
There is no unmarked vertex of valence one. Its leaf edge gives an
immediate backtrack; on the exterior this lies within one adjustable side,
because a side break would be marked and a generator side has marked
endpoints. Erasing the leaf preserves the assignments and lowers the edge
count.

There is also no bridge whose two incidences belong to a bounded face.
After its open edge is removed, the exterior contour lies wholly in one
component. Keep that component and discard the other component together
with the bridge. On the affected bounded contour this replaces
\(q e\gamma e^{-1}q'\) by \(q q'\), where \(e\) is the bridge and
\(\gamma\) is the entire excursion around the discarded component.
The retained walk is still closed and still lies in the assigned copy;
no free triviality of \(\gamma\) is required. The exterior, including
its side data, is unchanged. The face count cannot increase---also
directly from \(F=E-V+1\)---and the edge count decreases.

\emph{Short arcs.}
The incidence type on either side of a compressed arc is constant: it is
a bounded face or an exterior side of the prescribed decomposition.
Changes of exterior side occur at marked vertices and are not suppressed.
An arc with a generator-side incidence therefore has length one. By the
preceding restrictions, an arc incident to a bounded face has either such
an incidence on the other side or a different copy assignment there.

Consider an arc with different assignments and length greater than two.
Its word is freely reduced. Otherwise a consecutive cancelling pair has
an unmarked bivalent midpoint and two distinct endpoints in the diagram.
The latter assertion holds even for a loop arc: its internal vertices are
distinct and closure occurs only at the two ends of the entire arc.
Contract the cancelling pair as a length-two tree, identifying its outer
endpoints, which have the same group value. At each incident contour the
two steps cancel, so assignments and side data are preserved. This
non-loop contraction preserves \(F\) and lowers the edge count, a
contradiction.

If the arc had length at least \(\lambda\sigma(Q)\) for one assignment
\(Q\), its first \(\lceil\lambda\sigma(Q)\rceil\) edges would be a
nonbacktracking path shorter than the girth. It is simple in \(Q\), hence
also in the other source by the vertex embeddings. This contradicts
Lemma~\ref{lem:distinct-copies}. Arcs of length at most two are already
shorter than the threshold. This proves (a) and (b).

\emph{Counting.}
There is a marked vertex, so suppression cannot leave an exceptional
unmarked circle component. Every unmarked surviving vertex has valence
at least three. If \(v_D\) is the compressed vertex count, then
\[
v_D-b+F=1,\qquad 2b\ge3(v_D-m),
\qquad b\le3F+3m-3.
\]
By (a), every bounded face has at least \(1/\lambda\) arc incidences,
counted with multiplicity. Their total is at most \(2b\), so
\[
F/\lambda\le2b\le6F+6m-6.
\]
This is the first inequality in \eqref{eq:diagram-count}. Since
\(\lambda\le1/12\), it gives \(F\le m-1\) and then \(b\le6m\).
The same formulas cover trees and the isolated marked vertex.
\end{proof}

\subsection{Avoidance and uniform thinness}

The arc bound has two geometric consequences. Closing an avoiding path
by a designated cone passage bounds its endpoints' intrinsic distance.
Replacing arcs by cone passages while retaining fixed generator edges
produces a linear filling inequality with universal constants.

\begin{proposition}\label{prop:cone-geometry}
There is an integer \(D_0\), independent of all the above data, such that
every vertex on a geodesic triangle in \(\widehat X\) is within \(D_0\) of
a vertex on another side. Moreover:
\begin{enumerate}[label=\textup{(\roman*)}]
\item\label{item:avoidance}
If an edge path of length \(L\) joins \(x,a\in P\) while avoiding
\(c_P\), then
\begin{equation}\label{eq:avoidance}
d_P(x,a)\le12(L+1)\lambda\sigma(P).
\end{equation}
\item\label{item:simple-cayley}
If \(\lambda<1/12\), the graph \(X\) is simple: no nonfreely-trivial
closed label word in \(X\) has length at most two.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof of Proposition~\ref{prop:cone-geometry}]
\emph{Avoidance.}
For \eqref{eq:avoidance}, close the given avoiding path by one designated
side through \(c_P\). Then \(m\le L+1\), and no other side has center
\(c_P\). Use a minimal diagram. An arc incidence on the designated side
is opposite a bounded face of different assignment or an exterior
incidence across a bridge. In the latter case the other incidence cannot
also lie in the designated side. If it did, the interval from that
bridge's first traversal to its retraversal in the \emph{linear} side
walk would contain no prescribed side break. Excise the visited component
and the bridge. This shortens only the designated side, preserving its
endpoints and copy assignment, and cannot increase \(F\). It contradicts
minimality. This argument concerns marked positions; additional visits to
a marked vertex do not insert a side break.

Thus every designated-side incidence has length less than
\(\lambda\sigma(P)\), either by Lemma~\ref{lem:diagram-count}(b) or
because its opposite incidence is a generator side. There are at most
\(2b\) such occurrences. The adjusted walk inside \(P\) has length at
most \(12(L+1)\lambda\sigma(P)\), giving the distance bound. Empty
walks, including the case \(x=a\), satisfy it as well.

\emph{Simplicity of the Cayley graph.}
For part~\ref{item:simple-cayley}, consider a closed word of one or two
generator sides. Strict \(\lambda<1/12\) in
\eqref{eq:diagram-count} forces \(F=0\). The exterior contour of a tree
is freely trivial. This excludes identity generators, unintended
coincidences of generator edges, and generator involutions, as claimed.

\emph{Short fillings.}
To establish uniform thinness, we first show that
every edge loop of length \(n>0\) in \(\widehat X\) has a filling with
at most \(12n\) relation faces of perimeter at most four. For its
\(m\le n\) sides, take the compressed minimal diagram. Choose one
\emph{standard path} for each arc, consistently under reversal. Keep the
unit edge if the arc has a generator-side incidence. Otherwise an
assignment at an incidence contains both endpoints, so use the two-edge
path through that copy's center.

Around a bounded face assigned to \(P\), replace each standard path by
the two-edge path through \(c_P\). Each replacement uses one closed walk
of length at most four. The resulting contour freely contracts through
that one center. Thus the standard exterior walk can be filled at cost
at most the number of bounded-face incidences. On an exterior adjustable
side, make the same replacements through its prescribed center. The
consecutive pairs reduce to the required single cone passage. If an
inflated side was empty, insert its desired backtrack freely. Generator
sides need no change. The total cost is at most the number of bounded-face
and exterior adjustable-side incidences, hence at most
\(2b\le12m\le12n\).

This filling argument works for the singular diagrams being used. More
explicitly, for directed edge paths with their endpoint values, the
plane-diagram identity expresses the exterior contour as a product of
conjugates of the bounded contours, one for each face, up to free edge
cancellations and their inverses. Substituting the indicated fillings
gives the claimed product of short relations. Conversely, the van Kampen
construction from this product gives a plane diagram over the graph:
inverse-edge folds identify only vertices with the same target value.

\emph{From fillings to thin triangles.}
The bound implies the coarse filling hypothesis in
\cite[III.H.2.1 and III.H.2.9]{BH1999} with universal constants. Subdivide a rectifiable
loop of length \(\ell\) into \(O(\ell+1)\) arcs of length at most one.
Choose nearby vertices at distance at most one and join successive ones
by paths of at most three edges. Fill this edge loop as above. In a
resulting small-face diagram with \(f\) faces and exterior length \(n\),
\[
2E=n+\sum_F|\partial F|\le n+4f,\qquad V\le E+1.
\]
Thicken its finite plane graph to a regular neighborhood, using vertex
disks and edge strips, and cap the bounded contour components. The result
is a disk. The vertex regions are mapped to vertices, the strips along
their edges, and each cap has a boundary of bounded image perimeter.
Triangulation has total cost \(O(E+V+f)\); each triangle can be assigned a
map of uniformly bounded image diameter. Coarse fillings do not require
continuous maps on their interiors. An additional collar with \(O(n)\)
triangles restores the prescribed exterior parameterization: each block
joins a unit-edge traversal to that traversal with its inserted vertex
pauses, and its image lies in the same edge. The short paths to the original
subdivision points give an annulus with \(O(\ell+1)\) further triangles
of bounded image diameter. Constant loops or empty approximating walks
are filled separately by a bounded number of triangles.

The constants in this construction are independent of the graphs,
alphabet, and \(\lambda\). A connected unit-edge graph is geodesic,
regardless of local finiteness. The cited theorem therefore gives a
hyperbolicity constant depending only on these universal coarse filling
constants. Enlarging it to an integer accounts for moving points on
triangle sides to vertices and gives the required \(D_0\).
\end{proof}

\subsection{Simultaneous overlaps}

Set, once and for all,
\begin{equation}\label{eq:root-constants}
t=D_0+3,\qquad L_0=2+2t+2D_0.
\end{equation}
The next consequence is stronger than a pairwise intersection bound: one
ball controls all the intersections in the specified distance order.

\begin{lemma}[Simultaneous overlap bound]\label{lem:overlap}
Let \(R\) be a group vertex or a center of \(\widehat X\), and let
\(A=c_P\ne R\). Choose one geodesic from \(A\) to \(R\), whose first
group vertex is \(a\in P\). Every vertex shared by \(P\) and any
\emph{different} copy \(Q\) with
\[
d_{\widehat X}(c_Q,R)\le d_{\widehat X}(c_P,R)
\]
lies in the intrinsic induced ball \(B_P\) about \(a\) of radius
\begin{equation}\label{eq:overlap-radius}
r_P=\left\lfloor12(L_0+1)\lambda\sigma(P)\right\rfloor.
\end{equation}
\end{lemma}

\begin{proof}
Take such a \(Q\) and a shared vertex \(x\), and write \(B=c_Q\).
The path \(A,x,B\) is a length-two geodesic. A geodesic from \(B\) to
\(R\) avoids \(A\), since passing through \(A\) would give length
\(2+d(A,R)>d(A,R)\). All distances in the rest of this paragraph are in
\(\widehat X\).

If \(d(A,R)\ge t\), let \(z\) be the vertex at distance \(t\) along the
chosen \(A\)-to-\(R\) geodesic. Its distance from every vertex of
\(A,x,B\) is at least \(t-2>D_0\). Thinness supplies a vertex \(y\) on
the \(B\)-to-\(R\) side with \(d(y,z)\le D_0\). A shortest bridge
between them avoids \(A\), because \(d(A,z)=t>D_0\). Follow
\(x,B,y,z,a\), using that bridge and the indicated geodesic segments.
This walk avoids \(A\), and its length is at most
\[
1+(2+t+D_0)+D_0+(t-1)=L_0.
\]
If \(d(A,R)<t\), use instead the walk from \(x\) through \(B\) to
\(R\) and back to \(a\). It avoids \(A\) and has length at most
\(1+d(B,R)+d(A,R)-1\le2d(A,R)\le L_0\).

Apply \eqref{eq:avoidance} to obtain
\(d_P(x,a)\le12(L_0+1)\lambda\sigma(P)\). Intrinsic vertex distances
are integers, so taking the floor gives the claimed radius. Crucially,
the chosen \(a\) does not depend on \(Q\) or \(x\).
\end{proof}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//geometry.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//introduction.tex ====
\section{Introduction}\label{sec:introduction}

For a countable discrete group \(G\), let \(C_r^*(G)\) be the norm
closure of its complex group ring in the left regular representation.
The reduced Baum--Connes assembly map is
\[
\mu_j^G:K_j^G(\underline EG)\longrightarrow K_j(C_r^*(G)),
\qquad j=0,1.
\]
Here \(\underline EG\) is the classifying space for proper actions:
its fixed-point sets are contractible for finite subgroups and empty for
infinite subgroups. The domain is equivariant \(K\)-homology with
\(G\)-compact supports, and assembly takes geometric classes to their
analytic indices. The conjecture asserts that \(\mu_j^G\) is an
isomorphism for \(j=0,1\). We use integral complex \(K\)-theory, with
coefficient algebra \(\C\) carrying the trivial action; no
finite-dimensional or cocompact model for \(\underline EG\) is assumed.

Baum and Connes first formulated the conjecture in a manuscript
circulated in 1982 and published in 2000 \cite{BaumConnes2000}.
Their geometric formulation connects elliptic indices with the
\(K\)-theory of reduced crossed products. The formulation of
Baum--Connes--Higson uses the universal proper space and equivariant
Kasparov theory to state this index problem for second-countable locally
compact groups \cite[Conjecture~3.15]{BCH1994}.
The stronger formulation allows a separable \(G\)-\(C^*\)-algebra \(A\)
as coefficient, with target \(K_j(A\rtimes_rG)\)
\cite[Conjecture~9.6]{BCH1994}.

For the discrete groups considered here, important positive results
cover both flexible Hilbert-space geometry and hyperbolic geometry.
Higson--Kasparov prove assembly with coefficients for groups acting
properly by affine isometries on Hilbert space, including countable
amenable groups \cite[Theorem~9.1 and Corollary~9.2]{HigsonKasparov2001}.
Using Lafforgue's Banach \(KK\)-theory, Mineyev--Yu prove the
coefficient-free conjecture for hyperbolic groups and their subgroups
\cite[Theorems~19 and~20]{MineyevYu2002}; Lafforgue later proves
the conjecture with coefficients for hyperbolic groups
\cite[Theorem~0.4]{Lafforgue2012}.

The distinction between trivial and general coefficients is central to
this history.
Higson--Lafforgue--Skandalis use expanders and spectral projections to
detect failures of exactness in reduced crossed products, obtaining
counterexamples for groupoids and for discrete groups with suitable
commutative coefficients \cite[Section~7 and Remark~12]{HLS2002}.
Those constructions do not give a counterexample to the coefficient-free
group assertion above.

Our obstruction uses the canonical trace
\(\tau_G(\sum_g a_g g)=a_1\), normalized by \(\tau_G(1)=1\).
L\"uck's trace theorem implies that its value on every degree-zero
assembly class is rational. Thus a projection in \(C_r^*(G)\) with
irrational trace would obstruct surjectivity. We construct such a
projection.

\begin{theorem}\label{thm:main}
There exist a finitely generated discrete group \(\Gsur\) and a projection
\(b_0\in C_r^*(\Gsur)\) with irrational canonical trace. Moreover,
\[
[b_0]\notin\mu_0^{\Gsur}\bigl(K_0^{\Gsur}(\underline E\Gsur)\bigr).
\]
In particular, the coefficient-free reduced Baum--Connes conjecture for
countable discrete groups is false.
\end{theorem}

The trace restriction just used holds without a surjectivity assumption.
The extension of \(\tau_G\) to matrices is the sum of the diagonal
traces, and \(\tau_{G*}\) denotes the induced homomorphism on \(K_0\).

\begin{theorem}[L\"uck]\label{thm:trace-image}
For every countable discrete group \(G\),
\[
\tau_{G*}\!\left(\mu_0^G(K_0^G(\underline EG))\right)\subseteq\Q.
\]
\end{theorem}

\begin{proof}[Deduction from the trace theorem]
Put
\(\Lambda_G=\Z[1/|F|:F\le G\text{ finite}]\subseteq\Q\).
L\"uck's Theorem~0.3 \cite{Luck2002} states that the trace image of assembly
after extension of scalars to \(\Lambda_G\) is \(\Lambda_G\).
On \(1\otimes x\), this composite has value
\(\tau_{G*}(\mu_0^G(x))\). Hence that value belongs to
\(\Lambda_G\subseteq\Q\) for every integral class \(x\), without
assuming surjectivity of assembly or injectivity of localization.

For precision, our domain is the direct limit of
\(KK_j^G(C_0(Z),\C)\) over closed invariant \(G\)-compact subspaces
of a proper \(G\)-CW model. Assembly is reduced Kasparov descent
followed by the product with a cutoff projection and this direct limit.
L\"uck uses the same proper equivariant \(KK\)-homology and analytic
index \cite[Sections~4 and~5]{Luck2002}. His cocompact subcomplexes
are cofinal among these supports: the quotient is a CW complex, and
its compact subsets lie in finite subcomplexes. Comparisons with the
Davis--L\"uck formulation are given in
\cite[Corollary~8.4]{HambletonPedersen2004} and
\cite[Theorem~5.3]{Kranz2021}.
\end{proof}

\paragraph{Kernel dimensions and the spectral gap.}
The distinction between an individual kernel and a \(K\)-theory index
already appears in Atiyah's question about irrational
\(L^2\)-Betti numbers of coverings
\cite[Section~6.4, question~(iii)]{Atiyah1976}.
Grigorchuk and \.{Z}uk's lamplighter spectral calculation led
Grigorchuk--Linnell--Schick--\.{Z}uk to examples with kernel dimension
\(1/3\), although every finite subgroup had order a power of two
\cite{GLSZ2000}. Dicks--Schick gave an elementary spectral decomposition
using finite path operators and projections \cite{DicksSchick2002}.
Austin then obtained irrational kernel dimensions by varying
translation-invariant quotients of finite-field lamp modules
\cite[Theorem~1.1 and Corollary~1.2]{Austin2013}.
Fourier transformation turns these operators into local rules on the
compact dual of the module; this viewpoint also underlies the present
construction. Pichot--Schick--\.{Z}uk refined the method to compute
explicit transcendental values \cite{PichotSchickZuk2015}.

These results concern projections in group von Neumann algebras, which
need not belong to reduced group \(C^*\)-algebras. The difference is
particularly clear in Grabowski's construction of integral group-ring
matrices with transcendental kernel dimension over the ordinary
lamplighter groups \((\Z/m\Z)\wr\Z\), for integers \(m>1\)
\cite[Theorem~2]{Grabowski2016}. Those groups are amenable and satisfy
Baum--Connes. By Theorem~\ref{thm:trace-image}, their irrational kernel
projections cannot lie in matrix algebras over their reduced group
\(C^*\)-algebras.

A uniform spectral gap at zero supplies the required \(C^*\)-algebra
property: the indicator of zero becomes continuous on the spectrum.
Li--Nowak--Pooya use this mechanism for cohomological Laplacians and
higher Kazhdan projections, relating gaps and trace restrictions to
surjectivity of assembly \cite[Propositions~1,~5 and~7]{LiNowakPooya2024}.
Here the operator comes from local finite-field rules. Its kernel
consists of constants on designated expander components, and its trace
is computed from their pattern probabilities. Beyond the gaps on those
expanders, the construction must provide a uniform positive lower bound
on the entire remaining graph.

We obtain that bound using labelled expanders as graphical relators,
following the strategy of Gromov \cite{Gromov2003}. Osajda's stronger
labelling theorem provides one finite alphabet, globally distinguishable
long paths, and isometrically embedded relators
\cite[Theorems~2.7 and~3.2]{Osajda2020}. To pass from separate
expander estimates to the full operator, we prove a geometric bound on
simultaneous intersections of translated relators and use it in a
global cycle-rank estimate.

\paragraph{Construction and proof.}
The two ingredients that distinguish the construction are exact local
module relations and a global spectral estimate. Both use the same
geometric ordering, but neither follows from pairwise small overlaps
alone.
We first choose finite \(d\)-regular expanders \(\Theta_i\) with large girth and
label them by Osajda's theorem so that they embed isometrically in a Cayley
graph \(X=\Cay(\Gamma,S)\) of a finitely generated group \(\Gamma\). A \emph{copy} is any left translate of an embedded
\(\Theta_i\). Coning off every copy gives a uniformly hyperbolic graph.
An avoidance estimate then shows that, relative to any root, all
intersections of a copy with other copies whose centers are no farther from
the root lie in one small intrinsic tree ball.

This simultaneous intersection bound has two uses. It makes the finite
cycle spaces of copies an internal direct sum, and it prevents new local
linear relations when codes on copies are imposed on a lamp module.
For one prime \(p\), codes over \(\F_p\) of large dual distance give
exact finite-pattern distributions;
large primal distance forces a positive proportion of discrepancies on
every copy whose pattern is not the designated one.

The resulting group is \(\Gsur=V\rtimes\Gamma\), with \(V\) a finite-field
module. On its compact dual, a coordinate value specifies allowed
generator labels. The operator \(T\in C_r^*(\Gsur)\) is \(d\) minus the
adjacency operator of the enabled graph. Copies carrying the designated
pattern are isolated expander components. To control everything else, we
eliminate copy coordinates in finite cycle spaces and charge their ranks
to disjoint exterior vertex sets. This yields a uniform spectral gap at
zero and hence \(b_0=\one_{\{0\}}(T)\in C_r^*(\Gsur)\) by continuous
functional calculus. Its trace is a series \(\sum_i p^{-k_i}\) whose
base-\(p\) digits are not eventually periodic; here \(k_i\) are the
code dimensions, selected to have increasingly large gaps.

The external geometric inputs are recalled where used. The diagram,
module, and spectral arguments are given in full, including the
simultaneous overlap bound needed to pass from individual copies to the
whole enabled graph.

\paragraph{Organization.}
Section~\ref{sec:geometry} proves the coned geometry and simultaneous
overlap estimate. Section~\ref{sec:data} chooses the labelled expanders
and derives the finite cycle-space decomposition.
Section~\ref{sec:module} constructs the codes, the module, and its exact
finite marginals. Section~\ref{sec:spectrum} proves the spectral gap,
with Figure~\ref{fig:cycle-charging} illustrating the cycle-rank argument.
Section~\ref{sec:trace} computes the irrational trace and applies
Theorem~\ref{thm:trace-image}.
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//introduction.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//module.tex ====
\section{Codes and the module extension}
\label{sec:module}

Fix the labelled graphs and the group \(\Gamma\) supplied by
Proposition~\ref{prop:data}.  Retain the notation
\[
  n_i=|\operatorname{Vert}(\Theta_i)|,\qquad
  k_i=\lfloor n_i/2\rfloor,\qquad \eta=\frac1{100}.
\]
In particular, \(n_i>100\).  A copy \(P=P_{g,i}\) carries its prescribed
identification with \(\Theta_i\), and its vertex set is a subset of \(\Gamma\).
For every choice of root and copy to which Lemma~\ref{lem:overlap} applies,
the ball furnished by that lemma is an induced tree with fewer than
\(\eta n_i\) vertices, by Proposition~\ref{prop:data}.

\subsection{Port patterns and codes}

We call an outgoing generator label a \emph{port}.
At a vertex \(v\) of \(\Theta_i\), let
\[
  J_i(v)\subseteq S^{\pm1}
\]
be the set of outgoing labels of its incident edges.  Reducedness of the
labelling and \(d\)-regularity give \(|J_i(v)|=d\).  Choose a prime \(p>2\)
large enough that the \(d\)-element subsets of \(S^{\pm1}\) admit an
injection
\[
  J\longmapsto a_J\in\F_p^\times.
\]
Define the decoder
\[
  \mathcal J(a)=
  \begin{cases}
    J,&a=a_J\text{ for a \(d\)-element subset }J\subseteq S^{\pm1},\\
    \varnothing,&\text{otherwise},
  \end{cases}
  \qquad a\in\F_p.
\]
The prototype pattern on \(\Theta_i\) is
\[
  w_i(v)=a_{J_i(v)}.
\]
Every coordinate of \(w_i\) is nonzero.  Write \(J_P(h)\) and \(w_P(h)\)
for the corresponding data transported to a copy \(P=P_{g,i}\), so that
\(\mathcal J(w_P(h))=J_P(h)\).

On a finite coordinate space \(\F_p^E\), orthogonal complements will
always refer to the nondegenerate bilinear form
\[
  \langle z,z'\rangle=\sum_{v\in E}z(v)z'(v)\in\F_p.
\]
For \(z\in\F_p^E\), let \(\supp z=\{v\in E:z(v)\ne0\}\).

We seek a linear code containing the prescribed word $w_i$ with two
different support guarantees. Any other codeword must differ from $w_i$
at many vertices, forcing many missing prescribed edge incidences after
decoding. Vectors in the dual code will instead supply the module
relations. Their support must be larger than an overlap ball, so that a
nonzero relation on a farthest copy cannot cancel against all the other
relations. The following Gilbert--Varshamov type counting argument
\cite{Gilbert1952,Varshamov1957} obtains both bounds while keeping $w_i$
in the code.

\begin{lemma}[Codes with both distances]
\label{lem:codes}
For every \(i\) there is a \(k_i\)-dimensional subspace
\(C_i\subseteq\F_p^{\operatorname{Vert}(\Theta_i)}\) such that \(w_i\in C_i\)
and
\[
  |\supp z|>\eta n_i
  \qquad
  \text{for every nonzero }z\in C_i\cup C_i^\perp.
\]
\end{lemma}

\begin{proof}
Write \(n=n_i\), \(k=k_i\), and \(w=w_i\), and choose \(C\) uniformly among
the \(k\)-dimensional subspaces containing \(\langle w\rangle\).
For a fixed \(z\notin\langle w\rangle\), the quotient by
\(\langle w\rangle\) gives
\begin{equation}
\label{eq:primal-probability}
  \Pr(z\in C)
  =\frac{p^{k-1}-1}{p^{n-1}-1}
  <p^{k-n}.
\end{equation}
Indeed, a uniform \(a\)-dimensional subspace of an \(N\)-dimensional
vector space contains a fixed nonzero vector with probability
\((p^a-1)/(p^N-1)\), by counting nonzero vectors and symmetry.
Nonzero vectors of \(\langle w\rangle\) have full support.

Orthogonal complementation is a bijection from the sample space just
used to the \((n-k)\)-dimensional subspaces of the \((n-1)\)-dimensional
space \(w^\perp\).  Consequently, for a fixed nonzero \(z\),
\begin{equation}
\label{eq:dual-probability}
  \Pr(z\in C^\perp)=
  \begin{cases}
    \displaystyle\frac{p^{n-k}-1}{p^{n-1}-1}<p^{1-k},
       &z\in w^\perp,\\[6pt]
    0,&z\notin w^\perp.
  \end{cases}
\end{equation}
This argument uses nondegeneracy of the form on the ambient coordinate
space, not of its restriction to \(w^\perp\); it remains valid when
\(\langle w,w\rangle=0\).

We bound the number of low-weight vectors.  Put \(r=\lfloor\eta n\rfloor\)
and \(q=\eta/(1-\eta)<1\).  The binomial theorem gives
\[
  q^r\sum_{j=0}^r\binom nj
  \le \sum_{j=0}^r\binom nj q^j
  \le (1+q)^n.
\]
It follows that
\[
  \sum_{j=0}^r\binom nj
  \le q^{-\eta n}(1+q)^n
  =2^{H_2(\eta)n},
  \qquad
  H_2(t)=-t\log_2t-(1-t)\log_2(1-t).
\]
Here \(H_2(1/100)<9/100\): use \(\log_2 100<7\),
\(-\log(1-\eta)\le\eta/(1-\eta)\), and \(\log2>1/2\).
Since \(p\ge2\), the Hamming ball therefore satisfies
\begin{equation}
\label{eq:hamming-ball}
  \bigl|\{z:|\supp z|\le r\}\bigr|
  =\sum_{j=0}^r\binom nj(p-1)^j
  \le p^{\eta n}2^{H_2(\eta)n}
  \le p^{(\eta+H_2(\eta))n}
  <p^{n/4}.
\end{equation}
Every nonzero vector in this ball lies outside \(\langle w\rangle\).
By \eqref{eq:primal-probability}--\eqref{eq:hamming-ball} and the union
bound, the probability that either \(C\) or \(C^\perp\) contains such a
vector is less than
\[
  p^{n/4}\bigl(p^{k-n}+p^{1-k}\bigr)
  \le p^{-n/4}+p^{3/2-n/4}<1,
\]
where \(k=\lfloor n/2\rfloor\) and \(n>100\).
Thus a suitable \(C\) exists.
\end{proof}

Choose a code \(C_i\) as in Lemma~\ref{lem:codes} for every \(i\).
Transporting coordinates defines subspaces \(C_P\) and \(C_P^\perp\)
on every copy \(P=P_{g,i}\).  The transport preserves the bilinear
form, so these subspaces are orthogonal complements in \(\F_p^P\).

\subsection{The quotient module}

Translation-invariant quotients of finite-field lamp modules also occur
in Austin's irrational-kernel construction
\cite[Sections~1--2]{Austin2013}. Here the relations are dual codewords on
all translated expander copies. We must show that imposing them globally
creates exactly the prescribed relations on each individual copy.

Let \(\F_p^{(\Gamma)}\) denote the finite-support coordinate space with
basis \((\mathbf e_h)_{h\in\Gamma}\), on which \(\Gamma\) acts by
\(g\mathbf e_h=\mathbf e_{gh}\).  Regard \(\F_p^P\) as its coordinate
subspace by extension by zero.  Define the algebraic sum
\begin{equation}
\label{eq:relation-module}
  W_0=\sum_{\substack{g\in\Gamma\\i\ge1}} C_{P_{g,i}}^\perp
  \subseteq\F_p^{(\Gamma)}.
\end{equation}
Every vector in this sum is a sum of finitely many terms.  Left
translation carries \(C_{P_{g,i}}^\perp\) to
\(C_{P_{hg,i}}^\perp\), so \(W_0\) is \(\Gamma\)-invariant.

\begin{lemma}[Exact coordinate relations]
\label{lem:module-intersection}
For every copy \(P\),
\begin{equation}
\label{eq:module-intersection}
  W_0\cap\F_p^P=C_P^\perp.
\end{equation}
\end{lemma}

\begin{proof}
The inclusion from right to left follows from
\eqref{eq:relation-module}.  For the converse, let
\[
  z=\sum_{Q\in\mathcal F}z_Q\in W_0\cap\F_p^P,
  \qquad 0\ne z_Q\in C_Q^\perp,
\]
where \(\mathcal F\) is finite and its copies are distinct.  Such an
expression is obtained by combining terms on the same copy and
discarding zero terms.

Suppose that \(\mathcal F\) contains a copy different from \(P\).
Root the coned graph at \(R=c_P\), and choose
\(Q\in\mathcal F\setminus\{P\}\) whose center is farthest from \(R\).
Every other copy in \(\mathcal F\), and also \(P\) itself, has center at
distance from \(R\) at most \(d_{\widehat X}(c_Q,R)\).
Lemma~\ref{lem:overlap} places all their vertex intersections with \(Q\)
in one intrinsic ball \(B_Q\subseteq Q\).
If \(Q\) has type \(j\), Proposition~\ref{prop:data} gives
\[
  |B_Q|<\eta n_j<|\supp z_Q|.
\]
Choose \(h\in\supp z_Q\setminus B_Q\).  Then \(h\notin P\) and
\(h\) belongs to no other copy in \(\mathcal F\).  The \(h\)-coordinate
of \(z\) is therefore \(z_Q(h)\ne0\), contradicting
\(z\in\F_p^P\).
Hence the expression has no term away from \(P\), and
\(z\in C_P^\perp\).
\end{proof}

\begin{definition}
Define the discrete additive group and distinguished element
\[
  V=\F_p^{(\Gamma)}/W_0,\qquad v_*=[\mathbf e_1]\in V,
\]
with the induced \(\Gamma\)-action, and put
\[
  \Gsur=V\rtimes\Gamma.
\]
We use the multiplication
\((v,g)(v',g')=(v+gv',gg')\) in \(\Gsur\).
\end{definition}

For \(h\in\Gamma\), one has \(hv_*=[\mathbf e_h]\).
The group \(\Gsur\) is countable: \(\Gamma\) is finitely generated, and the
finite-support coordinate space over the finite field \(\F_p\), its
quotient \(V\), and \(V\times\Gamma\) are countable.
In fact \(\Gsur\) is finitely generated.  The additive group of \(V\) is
generated by the \(\Gamma\)-orbit of \(v_*\), since these are the images
of the coordinate basis and multiplication by a scalar in \(\F_p\)
is repeated addition.  In the semidirect product,
\[
  (0,h)(v_*,1)(0,h)^{-1}=(hv_*,1).
\]
It follows that \((v_*,1)\) together with the finitely many lifts
\((0,s)\), \(s\in S\), generates \(\Gsur\).

\subsection{The compact dual and its finite marginals}

Write the compact dual of \(V\) as
\[
  \Omega=\operatorname{Hom}_{\F_p}(V,\F_p).
\]
The corresponding complex character is
\(v\mapsto \exp(2\pi\mathrm i\,\omega(v)/p)\), where field elements
are identified with residues modulo \(p\).
We equip \(\Omega\) with the topology of pointwise convergence,
equivalently the subspace topology from \(\F_p^V\).
It is a closed subgroup of this compact product, and hence a compact
abelian group.  Let \(m_\Omega\) be its Haar probability measure.
For \(\omega\in\Omega\) and \(h\in\Gamma\), define
\[
  X_\omega(h)=\omega(hv_*).
\]
The notation \(X_\omega\) denotes a coordinate pattern, not the
Cayley graph \(X\).

\begin{lemma}[Uniform pattern marginals]
\label{lem:marginals}
For a copy \(P=P_{g,i}\), the restriction map
\[
  \operatorname{res}_P:\Omega\longrightarrow\F_p^P,
  \qquad \omega\longmapsto X_\omega|_P
\]
has image exactly \(C_P\).  Its pushforward of \(m_\Omega\) is the
uniform measure on \(C_P\).  In particular, for every \(a\in C_P\),
\begin{equation}
\label{eq:uniform-marginal}
  m_\Omega\{\omega:X_\omega|_P=a\}=p^{-k_i},
\end{equation}
and this applies to \(a=w_P\).
\end{lemma}

\begin{proof}
Let \(q_P:\F_p^P\to V\) be the restriction of the quotient map.
Lemma~\ref{lem:module-intersection} gives
\[
  \ker q_P=C_P^\perp.
\]
For \(z\in\F_p^P\) and \(\omega\in\Omega\),
\[
  \omega(q_P(z))
  =\sum_{h\in P}z(h)\omega(hv_*)
  =\langle z,X_\omega|_P\rangle.
\]
Thus \(X_\omega|_P\) annihilates \(C_P^\perp\), and belongs to
\((C_P^\perp)^\perp=C_P\).

Conversely, if \(a\in C_P\), the rule
\[
  q_P(z)\longmapsto\langle z,a\rangle
\]
is a well-defined linear functional on \(q_P(\F_p^P)\).
Extend a basis of this subspace to a basis of \(V\), and extend the
functional linearly to obtain \(\omega\in\Omega\).
Then \(X_\omega|_P=a\), proving surjectivity.
No continuity condition is lost in this extension, since \(V\) is
discrete.

The map \(\operatorname{res}_P\) is a continuous homomorphism of compact
groups onto the finite group \(C_P\).  Its pushforward of Haar
probability is translation-invariant and hence uniform on \(C_P\).
As \(|C_P|=p^{k_i}\), formula~\eqref{eq:uniform-marginal} follows.
\end{proof}

\begin{remark}
Lemma~\ref{lem:marginals} specifies each individual copy marginal.
It makes no independence assertion about patterns on different
copies.  The exact intersection
\eqref{eq:module-intersection}, which uses simultaneous containment
of all relevant overlaps in one ball, is what guarantees these
marginals despite the relations imposed on every translated copy.
\end{remark}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//module.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{An irrational-trace counterexample to reduced Baum--Connes}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We construct a finitely generated discrete group whose reduced group
\(C^*\)-algebra contains a projection of irrational canonical trace. Its
\(K_0\)-class lies outside the image of the coefficient-free reduced
Baum--Connes assembly map. This disproves the coefficient-free reduced
Baum--Connes conjecture for countable discrete groups.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\input{introduction}
\input{geometry}
\input{data}
\input{module}
\input{spectrum}
\input{trace}
\begingroup
\small
\setlength{\parskip}{0pt}
\bibliographystyle{plain}
\bibliography{references}
\endgroup
\end{document}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//paper.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//preamble.tex ====
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\hypersetup{pdftitle={An irrational-trace counterexample to reduced Baum--Connes},pdfauthor={OpenAI}}
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==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//preamble.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//spectrum.tex ====
\section{An operator with a uniform spectral gap}
\label{sec:spectrum}

We use the data of Proposition~\ref{prop:data} and the module constructed
above.  In particular, the induced intrinsic balls supplied by
Lemma~\ref{lem:overlap} have radius at most $\sigma(P)/16$, are trees,
and contain fewer than $\eta|P|$ vertices.  All copies below retain their
indices, and $|P|=n_i$ for a copy of type $i$.

\subsection{The reduced algebra and its evaluation representations}

The action of $\Gamma$ on the compact dual $\Omega$ of $V$ is
\[
 (g\omega)(v)=\omega(g^{-1}v).
\]
Write $\alpha_g(f)(\omega)=f(g^{-1}\omega)$ for the corresponding action
on $C(\Omega)$.  Fourier transformation on the discrete abelian group $V$
identifies $C_r^*(V)$ with $C(\Omega)$.  The regular representation of the
semidirect product, or equivalently regular induction followed by this
Fourier transformation, gives
\begin{equation}
 C_r^*(\Gsur)=C_r^*(V\rtimes\Gamma)
 \cong C(\Omega)\rtimes_{\alpha,r}\Gamma.
 \label{eq:semidirect-fourier}
\end{equation}
We use this identification throughout, and write $u_g$ for the canonical
unitary corresponding to $g\in\Gamma$.  Thus
$u_gfu_g^*=\alpha_g(f)$.

For $\omega\in\Omega$, the regular representation induced by evaluation
at $\omega$ acts on $\ell^2(\Gamma)$.  After inverting the usual group
index, its formulas are
\begin{equation}
 \begin{split}
 (\pi_\omega(f)\xi)(h)&=f(h^{-1}\omega)\xi(h),\qquad f\in C(\Omega),\\
 (\pi_\omega(u_g)\xi)(h)&=\xi(hg),\qquad g\in\Gamma.
 \end{split}
 \label{eq:regular-evaluations}
\end{equation}
The direct sum of all evaluations is a faithful representation of
$C(\Omega)$; inducing it regularly shows that
$\bigoplus_{\omega\in\Omega}\pi_\omega$ is faithful for the reduced
crossed product.  In particular, the norm of an element of
\eqref{eq:semidirect-fourier} is the supremum of its norms in this family.

Let $\tau_{\Gsur}$ denote the canonical trace, normalized by $\tau_{\Gsur}(1)=1$.
For every $b\in C_r^*(\Gsur)$,
\begin{equation}
 \tau_{\Gsur}(b)=\int_\Omega
 \langle\pi_\omega(b)\delta_1,\delta_1\rangle\,
 \mathrm{d}m_\Omega(\omega).
 \label{eq:trace-integral}
\end{equation}
Indeed, for a finite crossed-product sum $b=\sum_g F_g u_g$ the diagonal
coefficient at $1$ is $F_1(\omega)$, and its Haar integral is the
canonical group trace under Fourier transformation.  Such sums are dense.
The diagonal coefficients converge uniformly under norm approximation,
so the identity extends to every $b$.  This also proves that the
integrand in \eqref{eq:trace-integral} is continuous.

For $s\in S^{\pm1}$, let $f_s\in C(\Omega)$ be the clopen-set projection
\[
 f_s(\omega)=\mathbf 1_{\{s\in\mathcal J(\omega(v_*))\}}.
\]
Define
\begin{equation}
 T=dI-\sum_{s\in S^{\pm1}}f_su_sf_{s^{-1}}
 \quad\in C_r^*(\Gsur).
 \label{eq:operator}
\end{equation}
This element is self-adjoint, since taking adjoints interchanges the
terms indexed by $s$ and $s^{-1}$.  Each $f_s$ is a function of a single
finite-valued coordinate and hence is a finite Fourier sum of characters
of $\Omega$ indexed by elements of $V$.  Thus the construction takes
place in the ordinary reduced group $C^*$-algebra in
\eqref{eq:semidirect-fourier}.

\begin{definition}
For $\omega\in\Omega$, its \emph{enabled graph} $G_\omega$ has vertex
set $\Gamma$.  At $h\in\Gamma$ the allowed outgoing labels are
$\mathcal J(X_\omega(h))$.  The Cayley edge from $h$ to $hs$ is enabled
precisely when $s$ is allowed at $h$ and $s^{-1}$ is allowed at $hs$.
A copy $P$ is \emph{valid for $\omega$} if $X_\omega|_P=w_P$.
Let $U_\omega$ be the complement in $\Gamma$ of the union of all valid
copies.
\end{definition}

The graph $G_\omega$ is undirected and simple, and all its degrees are
at most $d$, because the decoder returns either a $d$-element set or the
empty set.  Since
\[
 (h^{-1}\omega)(v_*)=\omega(hv_*)=X_\omega(h),
\]
the formulas \eqref{eq:regular-evaluations} give
\begin{equation}
 \pi_\omega(T)=dI-\Adj(G_\omega).
 \label{eq:enabled-operator}
\end{equation}
The degree bound implies $\|\Adj(G_\omega)\|\le d$, so this operator
is bounded, positive, and at most $2dI$.

On a valid copy $P$, the allowed port sets are exactly $J_P(h)$.
All copy edges are therefore enabled, and their $d$ incidences at each
vertex exhaust its allowed labels.  As $P$ is connected, it is a whole
isolated component of $G_\omega$, with precisely its copy edges.
Moreover, distinct valid copies are disjoint.  To see this, two valid
copies meeting at a vertex would both be the entire same component.
If their indices were distinct, root $\widehat X$ at a shared group
vertex.  Their centers would both have distance one from the root, so
Lemma~\ref{lem:overlap} would place their entire common vertex set in
an intrinsic ball of the first copy containing fewer than
$\eta|P|<|P|$ vertices, a contradiction.  It follows also that no enabled
edge joins $U_\omega$ to its complement.

\subsection{Deficits on invalid copies and on arbitrary finite sets}

For $B\subseteq P$, let $e_{\omega,P}(B)$ be the number of enabled
copy edges with both endpoints in $B$.

\begin{lemma}[Deficit inside an invalid copy]
\label{lem:local-deficit}
If $P$ is invalid for $\omega$, then every subset $B\subseteq P$ satisfies
\begin{equation}
 d|B|-2e_{\omega,P}(B)\ge c|B|.
 \label{eq:local-deficit}
\end{equation}
\end{lemma}

\begin{proof}
Write $n=|P|$, and let
$M=\{h\in P:X_\omega(h)\ne w_P(h)\}$.
By Lemma~\ref{lem:marginals}, the restriction $X_\omega|_P$ belongs to
$C_P$, and $w_P\in C_P$ by Lemma~\ref{lem:codes}.  Their difference is
nonzero, since $P$ is invalid.  The primal-distance bound therefore gives
$|M|>\eta n$.

At a vertex of $M$, the decoder returns either the empty set or a
$d$-element set different from the prescribed $J_P(h)$.  In either case
at least one prescribed incidence is missing.  Consequently, writing
$b=|B|$, we have both
\[
 db-2e_{\omega,P}(B)\ge |M\cap B|
 \quad\text{and}\quad
 db-2e_{\omega,P}(B)\ge |\partial_P B|.
\]
If $n-b\ge\eta n/2$, then
$\min(b,n-b)\ge(\eta/2)b$.  The edge-expansion bound in
Proposition~\ref{prop:data} yields
\[
 |\partial_P B|\ge\kappa\min(b,n-b)
 \ge(\eta\kappa/2)b\ge cb.
\]
Otherwise,
\[
 |M\cap B|\ge |M|-(n-b)>\eta n/2
 \ge(\eta/2)b\ge cb.
\]
Here we used $c=(\eta/2)\min(\kappa,1)$.  The assertion for $B=\varnothing$
is included.
\end{proof}

\begin{lemma}[Deficit on the remaining graph]
\label{lem:global-deficit}
For every $\omega\in\Omega$ and every finite set $Y\subseteq U_\omega$,
let $A$ be the graph on $Y$ consisting of all its internal enabled edges.
Then
\begin{equation}
 d|Y|-2|E(A)|\ge(c/2)|Y|.
 \label{eq:global-deficit}
\end{equation}
\end{lemma}

\begin{proof}
The local deficits cannot simply be added over copies, since their
vertices and edges can overlap.  Instead, we bound the cycle rank
$b_1(A)=\dim Z_1(A;\F_2)$.  Euler's identity
\[
 |E(A)|=|Y|-\#\pi_0(A)+b_1(A)
\]
holds also for isolated vertices and the empty graph.  It shows that
the desired deficit follows from
\[
 b_1(A)\le\bigl(d/2-1-c/4\bigr)|Y|.
\]
We will obtain this bound by charging the ranks of successive copy
coordinates to pairwise disjoint subsets of $Y$.

\emph{Eliminating cycle coordinates.}
We regard the finite-dimensional space
$\mathcal L=Z_1(A;\F_2)$ as a subspace of $Z_1(X;\F_2)$.
Lemma~\ref{lem:cycle-sum} provides the algebraic direct-sum decomposition
\[
 Z_1(X;\F_2)=\bigoplus_P Z_1(P;\F_2),
\]
with coordinate maps $\operatorname{pr}_P$.  These are the coordinates
in the algebraic decomposition, not restrictions of edge chains to $P$:
a projected cycle can contain edges absent from the original cycle.
Only finitely many
coordinates can be nonzero on $\mathcal L$: choose a finite basis of
$\mathcal L$ and take the union of its finite coordinate supports.
Call these copies $P_1,\ldots,P_N$, ordered by nonincreasing distance
of their centers from a fixed group-vertex root $R$ in $\widehat X$.
Ties may be resolved arbitrarily.

For $1\le j\le N+1$, set
\[
 \mathcal L_j=
 \{z\in\mathcal L:\operatorname{pr}_{P_i}z=0\text{ for }i<j\}.
\]
Then $\mathcal L_1=\mathcal L$, $\mathcal L_{N+1}=0$, and rank-nullity
gives
\begin{equation}
 \dim\mathcal L=
 \sum_{j=1}^N\dim\operatorname{pr}_{P_j}(\mathcal L_j).
 \label{eq:cycle-elimination}
\end{equation}
If $\mathcal L=0$, this is understood as an empty sum.

\emph{Localizing projected cycles.}
Fix a step $j$, put $P=P_j$, and choose the induced intrinsic tree ball
$B_P$ given by Lemma~\ref{lem:overlap} for this root.  It contains every
vertex of $P$ shared with any of the other remaining copies
$P_{j+1},\ldots,P_N$.  Let
\[
 D_P=B_P\cup(A\cap P),
\]
where $A\cap P$ has vertex set $Y\cap P$ and the edges common to $A$
and $P$.  We claim that
\begin{equation}
 \operatorname{pr}_P(\mathcal L_j)\subseteq Z_1(D_P;\F_2).
 \label{eq:projected-cycle-support}
\end{equation}
Indeed, let an edge $e$ occur with nonzero coefficient in
$\operatorname{pr}_P z$ for $z\in\mathcal L_j$.  If $e$ does not belong
to $A$, its coefficient must cancel with a coefficient from another
remaining copy, because the total chain $z$ is supported in $A$.
Both endpoints of $e$ are consequently shared with that copy and lie
in $B_P$.  As the ball is induced in $P$, the edge $e$ itself lies in
$B_P$.  All other edges of $\operatorname{pr}_P z$ lie in $A\cap P$.
The projection is already a cycle in $P$, so the claimed inclusion
follows.  In particular,
\begin{equation}
 \dim\operatorname{pr}_P(\mathcal L_j)\le b_1(D_P).
 \label{eq:projected-cycle-rank}
\end{equation}

Figure~\ref{fig:cycle-charging} summarizes the bookkeeping in the next
step. The projected cycle may acquire edges in the overlap tree, but its
other edges remain enabled edges on $Y$. The tree can contribute to a
cycle only through multiple attachments of an exterior component.

\begin{figure}[ht]
\centering
\input{figures/cycle-charging}
\caption{The graph $D_P=B_P\cup(A\cap P)$ used to bound a projected
cycle coordinate. The tree $B_P$ is shown as a single region, and
$C_1,C_2$ represent components after deleting its vertices. Multiple
attachments contribute $l_C-1$ to the cycle rank, where $l_C$ is the
number of attachment edges of $C$. Large girth separates
their exterior endpoints. Only vertices outside $B_P$ are charged to
$P$, so the charges for successively eliminated copies are disjoint.
The drawing is schematic and specifies no distances.}
\label{fig:cycle-charging}
\end{figure}

\emph{Bounding exterior cycles and attachments.}
If $P$ is valid, $Y\cap P=\varnothing$, so $D_P=B_P$ is a tree and the
right side is zero.  Suppose from now on that $P$ is invalid, and write
$\sigma=\sigma(P)$ and $r$ for the radius of $B_P$.  Thus
$r\le\sigma/16$ and $\sigma>\max(16,20/c)$.

Consider the connected components $C$ of the graph obtained from $D_P$
by deleting the vertices of $B_P$.  Their vertex sets partition
$(Y\cap P)\setminus B_P$.  Let $l_C$ be the number of edges joining
$C$ to $B_P$.  Since $B_P$ is a connected tree, the Euler characteristic
formula for graph cycle rank gives
\begin{equation}
 b_1(D_P)=\sum_C\bigl(b_1(C)+\max(0,l_C-1)\bigr).
 \label{eq:tree-attachment-rank}
\end{equation}
This formula includes components with no attachment: such a component
contributes only its own cycle rank.

All edges of $C$ are enabled copy edges, and all enabled copy edges
between its vertices occur in $C$, by the definition of $A$.
If $C$ is cyclic, it has at least $\sigma$ vertices.  Applying
Lemma~\ref{lem:local-deficit} to its vertex set and using connectedness,
we obtain
\begin{equation}
 \begin{split}
 b_1(C)&=|E(C)|-|C|+1\\
 &\le\bigl(d/2-1-c/2+1/\sigma\bigr)|C|.
 \end{split}
 \label{eq:exterior-cycle-rank}
\end{equation}
The same bound holds for an acyclic $C$, because then $b_1(C)=0$ and
the displayed coefficient is positive.  Here $d\ge3$ and
$0<c\le\eta/2=1/200$.

We next bound the extra rank caused by attachments to the tree.
Choose two distinct attachment edges, and denote their endpoints in
$C$ by $v,v'$ and their endpoints in $B_P$ by $b,b'$.  A shortest path
in $C$ from $v$ to $v'$, these two edges, and the tree path from $b'$ to
$b$ form a simple cycle in $P$.  This remains true if one pair of
endpoints agrees, using the corresponding path of length zero; both
pairs cannot agree because $P$ is simple.  The tree path has length at
most $2r$, since both its endpoints lie in the intrinsic ball of
radius $r$.  Girth therefore gives
\begin{equation}
 \dist_C(v,v')\ge\sigma-2r-2>\sigma/2.
 \label{eq:attachment-separation}
\end{equation}
In particular, all the attachment endpoints in $C$ are distinct.

If $l_C\ge2$, the intrinsic balls in $C$ of radius
$\lfloor\sigma/4\rfloor$ centered at these endpoints are pairwise
disjoint, by \eqref{eq:attachment-separation}.  Each contains at least
$\lfloor\sigma/4\rfloor+1>\sigma/4$ vertices: take an initial segment
of a shortest path to any other attachment endpoint.  Thus
\begin{equation}
 \max(0,l_C-1)\le(4/\sigma)|C|.
 \label{eq:attachment-packing}
\end{equation}
For $l_C=0$ or $1$ the same inequality is immediate.

Combining \eqref{eq:tree-attachment-rank},
\eqref{eq:exterior-cycle-rank}, and \eqref{eq:attachment-packing}, and
using $5/\sigma<c/4$, yields
\begin{equation}
 \begin{split}
 b_1(D_P)
 &\le\bigl(d/2-1-c/2+5/\sigma\bigr)
       |(Y\cap P)\setminus B_P|\\
 &\le\bigl(d/2-1-c/4\bigr)|(Y\cap P)\setminus B_P|.
 \end{split}
 \label{eq:exterior-rank-bound}
\end{equation}
As observed above, this last bound holds also for a valid copy, since
both sides then vanish.

\emph{Summing over disjoint vertex sets.}
The exterior sets charged in \eqref{eq:exterior-rank-bound} are
pairwise disjoint.  Indeed, if $j<k$, the center of $P_k$ is no farther
from $R$ than the center of $P_j$.  Lemma~\ref{lem:overlap} therefore
implies
\[
 P_j\cap P_k\subseteq B_{P_j}.
\]
Consequently $(Y\cap P_j)\setminus B_{P_j}$ misses all of $P_k$,
including its charged exterior.  The non-strict distance comparison
also covers ties in the elimination order.

Summing \eqref{eq:projected-cycle-rank} and
\eqref{eq:exterior-rank-bound} in \eqref{eq:cycle-elimination} now gives
\[
 \dim Z_1(A;\F_2)\le\bigl(d/2-1-c/4\bigr)|Y|.
\]
The coefficient is positive, so this statement also follows from the
empty sum when $\mathcal L=0$.  Substituting this bound into Euler's
identity proves \eqref{eq:global-deficit}.
\end{proof}

\subsection{The spectral gap and its projection}

The passage from finite-set expansion to a spectral estimate belongs to
the discrete Cheeger tradition \cite{Dodziuk1984,AlonMilman1985}. We give
the coarea and Cauchy--Schwarz calculation to account for the vertex
potential $d-\deg_\omega(x)$ and the unnormalized Laplacian used here.

\begin{lemma}[Uniform spectral gap]
\label{lem:spectral-gap}
For every $\omega\in\Omega$, the restriction of $\pi_\omega(T)$ to
$\ell^2(U_\omega)$ satisfies
\begin{equation}
 \pi_\omega(T)|_{\ell^2(U_\omega)}\ge\frac{c^2}{8d}I.
 \label{eq:remaining-spectral-bound}
\end{equation}
Put
\[
 \varepsilon=\min\{\rho,c^2/(8d)\}>0.
\]
Then
\begin{equation}
 \spec(\pi_\omega(T))\subseteq\{0\}\cup[\varepsilon,2d],
 \label{eq:uniform-spectrum}
\end{equation}
and its kernel is exactly the Hilbert-space direct sum of the
one-dimensional spaces of constants on the valid copies.
\end{lemma}

\begin{proof}
On $U_\omega$ write $\deg_\omega(x)$ for the enabled degree and put
$q(x)=d-\deg_\omega(x)\ge0$.  There are no enabled edges to the removed
set.  Thus, for finitely supported complex $\xi$ on $U_\omega$,
\begin{equation}
 \begin{split}
 \mathcal E(\xi)
 :=\langle\pi_\omega(T)\xi,\xi\rangle
 =\sum_{\{x,y\}\in E(G_\omega|_{U_\omega})}
       |\xi(x)-\xi(y)|^2
   +\sum_{x\in U_\omega}q(x)|\xi(x)|^2.
 \end{split}
 \label{eq:remaining-energy}
\end{equation}
All edge sums in the rest of this proof are over enabled unoriented
edges on $U_\omega$.  For a finite set $F\subseteq U_\omega$, its
deficit is exactly
\[
 d|F|-2|E(G_\omega|_F)|
 =|\partial_{G_\omega|_{U_\omega}}F|+\sum_{x\in F}q(x).
\]

Set $s(x)=|\xi(x)|$.  Apply Lemma~\ref{lem:global-deficit} to the finite
level sets $\{x:s(x)^2>t\}$ and integrate in $t\ge0$.  The layer-cake
identity gives
\[
 (c/2)\|\xi\|^2
 \le\sum_{\{x,y\}}|s(x)^2-s(y)^2|+\sum_xq(x)s(x)^2.
\]
Cauchy--Schwarz, applied together to the edge terms and the
vertex-potential terms, then gives
\begin{align*}
 (c/2)\|\xi\|^2
 &\le
 \left(\sum_{\{x,y\}}|s(x)-s(y)|^2+\sum_xq(x)s(x)^2\right)^{1/2}
 \left(\sum_{\{x,y\}}(s(x)+s(y))^2+\sum_xq(x)s(x)^2\right)^{1/2}\\
 &\le\mathcal E(\xi)^{1/2}\sqrt{2d}\,\|\xi\|.
\end{align*}
For the first factor we used
$||\xi(x)|-|\xi(y)||\le|\xi(x)-\xi(y)|$.  For the second we used
\[
 \sum_{\{x,y\}}(s(x)+s(y))^2+\sum_xq(x)s(x)^2
 \le\sum_x\bigl(2\deg_\omega(x)+q(x)\bigr)s(x)^2
 \le2d\|\xi\|^2.
\]
If $\xi\ne0$, division and squaring prove
$\mathcal E(\xi)\ge(c^2/(8d))\|\xi\|^2$; the zero vector satisfies
the same inequality.  Finite-support vectors are dense and the operator
is bounded, so \eqref{eq:remaining-spectral-bound} holds on all of
$\ell^2(U_\omega)$.  If $U_\omega$ is empty, this assertion is vacuous.

On each valid copy $P$ of type $i$, the operator is its graph Laplacian
$dI-\Adj(P)$.  Its kernel consists of constants, and on the orthogonal
complement of constants it is at least $\rho I$, by
Proposition~\ref{prop:data}.  The valid copies are disjoint isolated
components.  Decomposing $\ell^2(\Gamma)$ into their component spaces
and $\ell^2(U_\omega)$ therefore shows that the full kernel is exactly
\[
 \bigoplus_{P\text{ valid for }\omega}\C\mathbf 1_P,
\]
and that the operator is at least $\varepsilon I$ on its orthogonal
complement.  These conclusions remain true for infinitely many valid
components, because both lower bounds are uniform.  The earlier bound
$\pi_\omega(T)\le2dI$ proves \eqref{eq:uniform-spectrum}.
\end{proof}

\begin{proposition}[A projection in the reduced group algebra]
\label{prop:projection}
There exists a projection $b_0\in C_r^*(\Gsur)$ such that, for every
$\omega\in\Omega$, $\pi_\omega(b_0)$ is the orthogonal projection
onto $\ker\pi_\omega(T)$.  Explicitly, it is projection onto constants
on each valid copy and is zero on $\ell^2(U_\omega)$.  In particular,
\begin{equation}
 \langle\pi_\omega(b_0)\delta_1,\delta_1\rangle
 =
 \begin{cases}
  1/n_i,&\text{if $1$ belongs to a valid copy of type $i$},\\
  0,&\text{if $1\in U_\omega$}.
 \end{cases}
 \label{eq:projection-diagonal}
\end{equation}
The valid copy in the first case is unique.
\end{proposition}

\begin{proof}
Take the faithful direct sum of the representations
$\pi_\omega$.  Lemma~\ref{lem:spectral-gap} gives the same positive
constant $\varepsilon$ in every summand, so its spectrum is contained
in the closed set $\{0\}\cup[\varepsilon,2d]$.  Faithfulness preserves
the spectrum of the self-adjoint element $T$; hence
\[
 \spec_{C_r^*(\Gsur)}(T)\subseteq\{0\}\cup[\varepsilon,2d].
\]
Choose a continuous real-valued function $\chi$ on $[0,2d]$ with
$\chi(0)=1$ and $\chi=0$ on $[\varepsilon,2d]$, and set
$b_0=\chi(T)$ by continuous functional calculus.  Since $\chi^2=\chi$
on the spectrum, $b_0$ is a projection in $C_r^*(\Gsur)$.  This uses the
ordinary C*-algebra functional calculus, with no passage to a von
Neumann algebra.

Functional calculus commutes with each $\pi_\omega$.  The kernel
description in Lemma~\ref{lem:spectral-gap} thus identifies
$\pi_\omega(b_0)$ as claimed, also when there are no valid copies or
infinitely many of them.  The normalized constant vector on a valid
copy of type $i$ is $n_i^{-1/2}\mathbf 1_P$, so the diagonal of its
rank-one projection at any vertex of $P$ is $1/n_i$.  At vertices of
$U_\omega$ the diagonal is zero.  Disjointness of the valid copies
proves uniqueness and \eqref{eq:projection-diagonal}.
\end{proof}
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//spectrum.tex ====

==== BEGIN An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//trace.tex ====
\section{An irrational trace and failure of assembly}\label{sec:trace}

We now compute the trace of the projection constructed in
Proposition~\ref{prop:projection}.

\begin{proposition}\label{prop:irrational-trace}
The projection \(b_0\in C_r^*(\Gsur)\) satisfies
\begin{equation}\label{eq:irrational-trace}
\tau_{\Gsur}(b_0)=\sum_{i\ge1}p^{-k_i}\notin\Q.
\end{equation}
\end{proposition}

\begin{proof}
For a fixed \(\omega\), the diagonal entry of \(\pi_\omega(b_0)\) at
the identity is zero unless that vertex belongs to a valid copy. If it
does, the copy is unique and has some type \(i\); projection onto the
constants on that component has diagonal entry \(1/n_i\).

Exactly \(n_i\) indexed copies of type \(i\) contain the identity. Indeed,
their translating elements are precisely
\(g=\phi_i(v)^{-1}\) for the \(n_i\) distinct embedded vertices
\(v\in\operatorname{Vert}(\Theta_i)\).
Lemma~\ref{lem:distinct-copies} ensures that these indices give distinct
copies. By Lemma~\ref{lem:marginals}, each is valid on an event of Haar
measure \(p^{-k_i}\). Validity events for copies containing the identity
are mutually exclusive, including across types. The trace formula
\eqref{eq:trace-integral} and nonnegative countable summation therefore
give
\[
\tau_{\Gsur}(b_0)=\sum_{i\ge1}n_i\frac1{n_i}p^{-k_i}
=\sum_{i\ge1}p^{-k_i}.
\]
No independence of these events is needed.

Since \(p>2\) and the positive integers \(k_i\) strictly increase, this
number lies in \([0,1)\) and has base-\(p\) digits equal to one exactly
at positions \(k_i\), and zero elsewhere. There is no eventually
\((p-1)\) expansion ambiguity. The digits contain infinitely many ones,
but their successive gaps tend to infinity by
Proposition~\ref{prop:data}(iv). They cannot be eventually periodic.
Rational numbers have eventually periodic base-\(p\) expansions, so
\eqref{eq:irrational-trace} is irrational.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The module construction gives a finitely generated discrete group
\(\Gsur\), and Proposition~\ref{prop:projection} gives an actual projection
\(b_0\in C_r^*(\Gsur)\). Its trace is positive and irrational by
Proposition~\ref{prop:irrational-trace}; in particular its integral
\(K_0\)-class is nonzero. Every class in the image of the proper-action
assembly map has rational trace by Theorem~\ref{thm:trace-image}. Thus
\([b_0]\) is outside that image.

The algebra \(C(\Omega)\rtimes_r\Gamma\) was used to represent
\(C_r^*(V\rtimes\Gamma)\) and to analyze the operator. The assembly map
just obstructed is the coefficient-free reduced map for \(\Gsur\) itself,
with the full proper classifying space. Its failure of surjectivity in
degree zero refutes the universal Baum--Connes assertion.
\end{proof}

The obstruction persists after rationalization. Extend the canonical
trace to \(K_0(C_r^*(\Gsur))\otimes_{\Z}\Q\) by
\(x\otimes q\mapsto q\tau_{\Gsur*}(x)\). Every element in the image
of \(\mu_0^{\Gsur}\otimes_{\Z}\operatorname{id}_{\Q}\) has rational
trace, since it is a finite rational linear combination of assembly
classes. The trace of \([b_0]\otimes1\) remains irrational, so this class
is outside the image of rationalized assembly as well.
==== END An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes//trace.tex ====

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