==== TABLE OF CONTENTS ====
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//conclusion.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//expansion.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//geometry.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//introduction.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//moving-band.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//nilpotent.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//paper.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//preamble.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//pruning.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//sampling.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//scalar.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//setup.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//spectral.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//tower.tex
A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//references.bib
===========================

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//conclusion.tex ====
\section{Completion of the construction}
\label{sec:conclusion}

\begin{proof}[Proof of Theorems~\ref{setup:matrix-target}
and~\ref{thm:main}]
Theorem~\ref{spec:matrix-projection} establishes the matrix target of
Theorem~\ref{setup:matrix-target}. It supplies a
finitely generated torsion-free group \(\Gbase\), a finite matrix
size \(N\), and a projection \(p_0\in M_N(C_r^*(\Gbase))\) with
\[
 0<\tau_{\Gbase,N}(p_0)<\frac12.
\]
Now set \(\Gproj=\Gbase*\mathbb F_N\) and apply
Proposition~\ref{scalar:compression}. With
\(U=(\lambda_{\Gproj}(t_1),\ldots,\lambda_{\Gproj}(t_N))\), it gives the
scalar projection
\[
 e=(Up_0)
 (p_0U^*Up_0)^{-1}_{p_0M_N(C_r^*(\Gproj))p_0}
 (Up_0)^*
 \in C_r^*(\Gproj).
\]
The same Proposition proves that \(\Gproj\) is finitely generated
and torsion-free and that
\(\tau_{\Gproj}(e)=\tau_{\Gbase,N}(p_0)\in(0,1/2)\).
Thus \(e\ne0,1\), proving Theorem~\ref{thm:main}.
\end{proof}

The nonintegral trace also places
\([p_0]\in K_0(C_r^*(\Gbase))\) outside the image of the degree-zero
reduced assembly map with trivial coefficients, by
\cite[Proposition~10.3]{BGW2016}.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//conclusion.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//expansion.tex ====
\section{Uniform expansion before pruning}\label{exp:section}

Fix a height pair $(i,k)$, $|i-k|\leq1$, and fix all the
permutation tables at smaller cutoffs in its family. Throughout this
section write $A=A_i$, $B=B_k$, $D'=D_{ik}$, and
$E=d|A|=D'|B|$. We use the stream conventions, coefficient choices,
and matching coordinates of
Lemma~\ref{tow:coefficient-choice} and
\eqref{tow:edge-equations}; in particular, the independent current
permutations are the cluster matchings described in
Lemma~\ref{tow:matching-clusters}. The graph considered here is $Y^0$:
its current last-block gate is open unconditionally. No assertion of
expansion after pruning is needed.

On each side use uniform probability measure. If $N$ is the slot-average
operator of any realization, then
\[
 (Nf)(a)=\frac1d\sum_{e:\,e_-=a}f(e_+),\qquad
 (N^*g)(b)=\frac1{D'}\sum_{e:\,e_+=b}g(e_-).
\]
Here slots, hence parallel edges, are counted separately. The adjoint
identity follows from $d|A|=D'|B|$. Both operators preserve constants
and are contractions: Jensen's inequality followed by counting slots
gives $\|Nf\|_2^2\leq\|f\|_2^2$, and likewise for $N^*$.
Let $M=\mathbb E N$, with expectation only over the current tables.
The same facts hold for $M,M^*$.

We first bound $M$ on centered functions by organizing them according to
stream prefixes.  The coefficient multiplicity bounds control the full-block
terms; averaging the current tables reduces the terminal contribution to
the partial-block version of the same variance estimate.
We then select actual matchings by a lower bound on every cut and turn
that bound into the required normalized incidence gap.

\paragraph{A conditional variance calculation.}
The following elementary estimate specifies, in particular, the mean
that is subtracted. Its last assertion is needed when the active input
block ends partway through a full block.

\begin{lemma}[Conditional Fourier estimate]\label{exp:fourier}
Let $C$ denote conditioned data. Conditional on $C$, let
$U\in\mathbb F_2^t$ and $V\in\mathbb F_2^m$ be independent uniform
vectors, independent also of auxiliary data $W$. For $1\leq a\leq q$
let $M_a:\mathbb F_2^t\to\mathbb F_2^m$ be linear, let
$o_a=o_a(C,W)$, and let $f_a(C,z)$ be complex functions that do not
depend separately on $U,V,W$. Set
\[
 F=\frac1q\sum_a f_a(C,V+M_aU+o_a),\qquad
 \mu(C)=\frac1q\sum_a2^{-m}\sum_z f_a(C,z).
\]
Suppose that for every nonzero linear functional
$\zeta:\mathbb F_2^m\to\mathbb F_2$, every class of indices on which
$\zeta M_a$ is the same has size at most $\theta q$. Then
\begin{align}
 \mathbb E(F\mid C,W,U)&=\mu(C),\label{exp:conditional-mean}\\
 \mathbb E|F-\mathbb E(F\mid C)|^2
 &\leq\theta\,\mathbb E\left[
    \frac1q\sum_a2^{-m}\sum_z|f_a(C,z)|^2\right].
 \label{exp:conditional-variance}
\end{align}
The conclusion applies with $\theta=10^{-3}$ for a full active
$s$-block and with $\theta=0.55$ for any nonempty active initial
part, using the coefficient multiplicity bounds of
Lemma~\ref{tow:coefficient-choice}.
\end{lemma}

\begin{proof}
Translation of the independent uniform vector $V$ proves
\eqref{exp:conditional-mean}; in particular the right-hand side is
independent of both $U$ and $W$. For fixed $C,W$, write the normalized
Fourier coefficients as
\[
 \widehat f_a(\zeta)=2^{-m}\sum_z
       f_a(C,z)(-1)^{\zeta(z)}.
\]
Orthogonality first in $V$ and then in $U$ gives the exact identity
\[
 \mathbb E_{U,V}|F-\mu(C)|^2
 =\frac1{q^2}\sum_{\zeta\ne0}\sum_{\mathcal C}
 \left|\sum_{a\in\mathcal C}
       (-1)^{\zeta(o_a)}\widehat f_a(\zeta)\right|^2,
\]
where $\mathcal C$ ranges over the classes with a common
$\zeta M_a$. Each square is at most
$|\mathcal C|\sum_{a\in\mathcal C}|\widehat f_a(\zeta)|^2$.
Sum, use Parseval's identity, and integrate over $C,W$. Because the
conditional mean is exactly $\mu(C)$, this integration creates no
additional variance term.
\end{proof}

\paragraph{Prefixes and causal dependence.}
Let $P_h^A$, $h\geq0$, be conditional expectation onto the first
$r+sh$ bits of each of $u_A,v_A$. Let $P_j^B$, $j\geq0$, be
conditional expectation onto the first $sj$ bits of each of $u_B,v_B$.
Only active bits are retained; the others are deterministic zeros.
Thus these are finite increasing filtrations, eventually equal to the
identity, and $P_0^B$ is the projection onto constants. Denote the
constant projection on either side by $C_A,C_B$.

For a fixed $A$-slot and $j\geq1$, the data needed to evaluate a
function in $\operatorname{ran}P_j^B$ are
\begin{equation}
 u_A\big|_{[0,r+s(j-1))},\qquad
 Q_A\big|_{[0,p+sj)}.\label{exp:forward-data}
\end{equation}
Indeed the first $p$ evaluation bits supply $x$, including its bucket,
and inversion of $\mathcal F$ supplies the first $j$ blocks of $u_B$.
At the $j$th $B$-block the keys use only preceding interleaved blocks,
ending with the $(j-1)$st $A$-block. To determine $v_B$ through
length $sj$, the equation for $Q_B$ uses $\mathcal J(u_A)$ through
length at most
\[
 sj-p'=r+s(j-1)-\beta\leq r+s(j-1).
\]
The diagonal and forward-spill rule for $\mathcal K_B$ uses no
$u_B$ beyond its first $j$ blocks to produce these $v_B$ bits.
This proves \eqref{exp:forward-data}, including when a prefix extends
into zero padding.

In the reverse direction, for an input in $\operatorname{ran}P_h^A$
and a fixed $B$-slot, the needed data are
\begin{equation}
 u_B\big|_{[0,sh)},\qquad
 Q_B\big|_{[0,p'+r+sh)}.\label{exp:reverse-data}
\end{equation}
Here $x'$ is read first and the bucket is already part of the slot;
inverting $\mathcal J$ supplies $u_A$ through $r+sh$. Its keys end
with the $h$th $B$-block. Computing $v_A$ through $r+sh$ then uses
$\mathcal F(u_B)$ through at most $sh-\alpha\leq sh$ bits and
the already recovered $u_A$ blocks. These observations prove the
reverse assertion also for $h=0$.

To read a few bits of a diagonal evaluation block one may need its
whole input block. Consequently \eqref{exp:forward-data} and
\eqref{exp:reverse-data} imply, for any fixed tables and also after
taking their mean,
\begin{equation}
 \operatorname{ran}(MP_j^B)\subseteq\operatorname{ran}P_{j+1}^A,
 \qquad
 \operatorname{ran}(M^*P_h^A)\subseteq\operatorname{ran}P_{h+2}^B.
 \label{exp:causality}
\end{equation}
For example $p+sj=r+sj+\alpha$ lies at most $\alpha\leq2$ bits
into the next $A$-block; $p'+r+sh=(h+1)s+\beta$ is handled
by the next $B$-block. This explains both shifts in
\eqref{exp:causality}, including $\alpha=0$ or $\beta=0$, when
the stated bounds can simply be non-sharp.

We now apply Lemma~\ref{exp:fourier} to these data. For
\eqref{exp:forward-data}, condition on $P_{j-1}^A$ and put
$R=r+s(j-1)$. All still needed data from the $A$-state occur through
its evaluation on $[R,R+s+\alpha)$. If its $j$th input block and
this entire $v_A$ interval are active, write that evaluation as
\[
 Z_a=V+M_aU+o_a.
\]
Here $U$ is the full $j$th $A$ input block and $V$ consists of the
$s+\alpha$ new $v_A$ bits. The offset contains the known spill from
the preceding block and any diagonal contribution from the next
input block to the last $\alpha$ output bits. Such next-block input
bits are the auxiliary variable $W$ of Lemma~\ref{exp:fourier}.
They enter nowhere else: \eqref{exp:forward-data} uses direct
$u_A$ only before $R$. Therefore the slot summand is a function
$f_a(C,Z_a)$ with no separate $W$-dependence. The $A$ coefficient
map does not depend on the bucket extracted from $x$; that bucket
may affect $f_a$, but does not change $M_a$. It follows that
\begin{equation}
 \|(1-P_{j-1}^A)MP_j^B\|^2\leq10^{-3}
 \quad\text{whenever these full intervals are active.}
 \label{exp:forward-variance}
\end{equation}

For completeness, the right-hand side of
\eqref{exp:conditional-variance} is bounded by the squared norm of
the original input function, rather than a larger slot-dependent
norm. A uniform output vertex with an independent uniform incident
slot gives a uniform input vertex, since every input has the same
number of slots. Thus the average over slots and vertices of a
deterministic neighbor value squared is exactly the input norm
squared. When a summand averages random table choices, Jensen's
inequality bounds its square by the corresponding average of those
squares. These are precisely the individual norms in
\eqref{exp:conditional-variance}, since $V$ makes each $Z_a$ uniform
conditional on all other data.

The identical argument in \eqref{exp:reverse-data}, conditioning
on $P_h^B$, uses the $B$ input block on $[sh,s(h+1))$ and its
extended evaluation interval $[sh,s(h+1)+\beta)$. It gives
\begin{equation}
 \|(1-P_h^B)M^*P_h^A\|^2\leq10^{-3}
 \quad\text{when that input block and extended $v_B$ interval are full.}
 \label{exp:reverse-variance}
\end{equation}
The explanation of the conditioned mean is the same in both
directions: the means of the slot functions under their uniform
evaluation argument depend only on the retained prefix, so no
uncontrolled contribution comes from the frozen next-block inputs.

\begin{lemma}[Uniform gap for the mean]\label{exp:mean-gap}
For every height pair, and every fixed choice of its lower-cutoff
tables, the mean incidence $M$ satisfies
\[
 \big\|M:\mathbf1_B^\perp\longrightarrow
                   \mathbf1_A^\perp\big\|
 \leq\sqrt{16\cdot10^{-3}+0.55}<0.8.
\]
The bound is uniform in the number of full blocks, the final active
length $1\leq t\leq s$, and $0\leq\alpha,\beta\leq2$.
\end{lemma}

\begin{proof}
Use the orthogonal difference spaces
\[
 E_0^A=P_0^A-C_A,\quad E_h^A=P_h^A-P_{h-1}^A\ (h\geq1),
 \qquad E_j^B=P_j^B-P_{j-1}^B\ (j\geq1).
\]
They decompose the centered spaces; once a filtration saturates,
its further difference spaces are zero. Equation~\eqref{exp:causality}
shows that $E_h^AME_j^B$ can be nonzero only if
\begin{equation}
 h\leq j+1,\qquad j\leq h+2.
 \label{exp:four-bands}
\end{equation}
When $h=j$ or $h=j+1$, the norm of this block is at most
$\sqrt{10^{-3}}$ by \eqref{exp:forward-variance}, provided its
full intervals exist. When $j=h+1$ or $j=h+2$, the same bound
follows by adjointing \eqref{exp:reverse-variance}. For each fixed
value of $h-j$, the input difference spaces are mutually orthogonal,
as are their output difference spaces. Hence a sum along any one
of these four bands has norm at most $\sqrt{10^{-3}}$; their sum
has norm at most $4\sqrt{10^{-3}}$.

There are exactly two forms of terminal cutoff in the interleaved
order. We check the full-interval hypotheses and the remaining
partial-block estimate in each.

\smallskip\noindent
\emph{Case B: the last active block belongs to $B$.}
Write
\[
 l_A=r+ns,\qquad l_B=ns+t,\qquad 1\leq t\leq s.
\]
The active $v$ lengths are
\[
 |v_A|=r+ns+\alpha+t,
 \qquad |v_B|=(n+1)s+\beta.
\]
For $1\leq j\leq n$, the forward estimate uses the full $j$th
$A$ block, and its extended interval ends at
$r+js+\alpha\leq|v_A|$. In the other two bands with $j\leq n$,
one has $h\leq n-1$, so the reverse estimate uses a full
$(h+1)$st $B$ block, ending at $s(h+1)+\beta\leq|v_B|$.
Thus the four-band argument gives
\begin{equation}
 \|M(P_n^B-C_B)\|\leq4\sqrt{10^{-3}}.
 \label{exp:low-B}
\end{equation}

Consider now $M^*g$ for arbitrary $g\in L^2(A)$. Given a $B$-state
and a $B$-slot, inversion through the preceding blocks determines
$u_A$ and all of $v_A$ except its last $t$ bits. In fact all this
determined data use only
\[
 u_B\big|_{[0,ns)},\qquad
 Q_B\big|_{[0,(n+1)s+\beta)}.
\]
The independent current matching averages the remaining $t$
evaluation bits of $Q_A$ uniformly. These bits occupy
$[p+ns,p+ns+t)$, and $\mathcal K_Au_A$ is zero on this interval:
its last possible spill ends at $r+ns+\alpha=p+ns$.
Thus the same bits of $v_A$ are uniformly averaged. Neither their
distribution nor the determined part depends separately on the last
$t$ bits of $u_B$. The tuple defining the matching uses only the
preceding interleaved blocks, the slot, and $x$.

After conditioning on $P_n^B$, the resulting averaged slot function
therefore has the form in Lemma~\ref{exp:fourier}, with $U$ the
$t$ active bits of the last $B$ block and $V$ the full $s+\beta$
bits of $v_B$ starting at $ns$. No current or later input bits
occur except through this evaluation. The multiplicity bound for
the nonempty input prefix, followed by the slot-stationarity and
Jensen argument already given, yields
\begin{equation}
 \|(1-P_n^B)M^*\|^2\leq0.55.
 \label{exp:terminal-B}
\end{equation}
For centered $g$, the two projections $P_n^B$ and $1-P_n^B$ are
orthogonal, and the first component is bounded using the adjoint of
\eqref{exp:low-B}. Consequently
$\|M^*g\|^2\leq(16\cdot10^{-3}+0.55)\|g\|^2$.

\smallskip\noindent
\emph{Case A: the last active block belongs to $A$.}
Write
\[
 l_B=(n+1)s,\qquad l_A=r+ns+t,\qquad1\leq t\leq s.
\]
Now
\[
 |v_A|=r+(n+1)s+\alpha,
 \qquad |v_B|=(n+1)s+\beta+t.
\]
Restrict the $A$ input differences to $0\leq h\leq n$. In the
bands $h=j,j+1$, $j\leq n$, so the forward estimate uses a full
$A$ block and an interval ending by $r+ns+\alpha$. In the bands
$j=h+1,h+2$, $h+1\leq n+1$, so the reverse estimate uses a
full $B$ block and an interval ending by $(n+1)s+\beta$.
All these intervals are active. Thus
\begin{equation}
 \|M^*(P_n^A-C_A)\|\leq4\sqrt{10^{-3}}.
 \label{exp:low-A}
\end{equation}

The current permutation in $\mathcal J$ averages the final $t$
bits of $Q_B$, starting at
$p'+r+ns=(n+1)s+\beta$. The contribution
$\mathcal K_Bu_B$ ends there, so these are exactly the final
$t$ bits of $v_B$. Once they are averaged, a slot value of $Mf$
depends only on
\[
 u_A\big|_{[0,r+ns)},\qquad
 Q_A\big|_{[0,p+(n+1)s)}.
\]
Conditioning on $P_n^A$, use the $t$ active bits of the last $A$
input block and the full $s+\alpha$ evaluation interval beginning
at $r+ns$. Lemma~\ref{exp:fourier} gives
\begin{equation}
 \|(1-P_n^A)M\|^2\leq0.55.
 \label{exp:terminal-A}
\end{equation}
Combining this orthogonally with the adjoint bound from
\eqref{exp:low-A} proves the desired estimate in this case.

The value $t=s$ is included in both arguments. In particular a
cutoff immediately after a full $A$ block is Case A, with $n$ one
less than the number of full $A$ blocks following its initial
$r$ bits; a cutoff immediately after a full $B$ block is Case B.
There is no missing zero-length final block. The actual stream
lengths start with $n_0$ full blocks on each side, so the initial
$r$-bit block is never terminal. If $n=0$ is permitted in the
displayed cases, \eqref{exp:low-B} has zero domain in Case B,
whereas in Case A the only low $A$ difference is $E_0^A$, handled
by \eqref{exp:reverse-variance} with $h=0$. The argument still
applies. When a retained prefix already includes every active bit,
the complementary difference spaces are zero and all the estimates
remain valid.
\end{proof}

\paragraph{Realizing the mean expansion by independent matchings.}
We next choose the current tables so that the actual block has a
uniform edge-isoperimetric constant. A tail estimate for each cut alone
would introduce the number of vertices when summed over all subsets,
and hence would not give the height-uniform bound needed here. Instead,
we will use connected cuts and the local dependence of the matching
variables to select the tables by the local lemma.

Give $A\sqcup B$ the degree measure $\mu(a)=d$, $\mu(b)=D'$;
write $v(Z)=\sum_{z\in Z}\mu(z)$ and
$\mu(A\sqcup B)=2E$. The mean random walk on this Hilbert space is
\[
 R=\begin{pmatrix}0&M\\M^*&0\end{pmatrix}.
\]
The constant-on-each-side subspace has eigenvalues $1,-1$;
its orthogonal complement is the sum of the two sidewise centered
spaces. Lemma~\ref{exp:mean-gap} bounds $R$ above by $0.8$ on
that complement. On the orthogonal complement of the global
constant, the remaining sidewise-constant vector has eigenvalue
$-1$, so the same upper bound $0.8$ holds there. Hence, for
$0<v(Z)\leq E$, the expected number $b(Z)$ of crossing edges obeys
\begin{align}
 \mathbb E b(Z)
 &=\langle(1-R)\mathbf1_Z,\mathbf1_Z\rangle_{\mu}\\
 &\geq0.2\left(v(Z)-\frac{v(Z)^2}{2E}\right)
 \geq0.1v(Z).
 \label{exp:mean-cut}
\end{align}

\begin{lemma}[Tail for a cut]\label{exp:cut-tail}
With lower tables fixed, for every nonempty $Z$ of volume at most
$E$,
\[
 \mathbb P\{b(Z)<v(Z)/20\}\leq\exp(-v(Z)/400).
\]
\end{lemma}

\begin{proof}
Write $X$ for the number of internal matched edges. Then
$b(Z)=v(Z)-2X$. In each independent final cluster, $X$ is a
hypergeometric count: it counts the images in the specified opposite
subset of a specified subset under a uniform bijection. Orient every
cluster from the same side, choosing globally whichever of the
$A$- and $B$-sides of $Z$ has fewer incidences.
Over all clusters the number of sampled incidences is then
\[
 N_Z=\min\{d|Z\cap A|,D'|Z\cap B|\}\leq v(Z)/2.
\]
The cluster counts are independent.

Here is an elementary exponential estimate that also covers clusters
of different sizes. For a sample of $m$ entries without replacement
from a zero-one population with success fraction $\vartheta$, its
exponential moment is at most $(1-\vartheta+\vartheta e^h)^m$.
Indeed this moment is the normalized elementary symmetric polynomial
of order $m$ in the positive entries $1$ and $e^h$.
Replacing any two entries by their average, while keeping all other
entries fixed, increases that polynomial: the only changing term
is their product times a nonnegative elementary symmetric polynomial
of order $m-2$. Repeated averaging and continuity compare with
the list having all entries equal to their mean. For $m=0,1$ the
assertion is immediate. Finally, for a Bernoulli variable $B$,
the second derivative of $\log\mathbb E e^{h(B-\mathbb EB)}$
is a tilted Bernoulli variance, at most $1/4$. Its value and first
derivative vanish at $0$, so
\[
 \mathbb E e^{h(B-\mathbb EB)}\leq e^{h^2/8}
 \qquad(h\in\mathbb R).
\]
Applying these facts cluster by cluster gives
\[
 \mathbb E e^{h(X-\mathbb EX)}\leq e^{N_Zh^2/8}.
\]
By \eqref{exp:mean-cut}, a bad cut requires
$X-\mathbb EX>v(Z)/40$. If $N_Z=0$ the event is impossible.
Otherwise exponential Markov inequality, optimized at
$h=4(v(Z)/40)/N_Z$, bounds its probability by
\[
 \exp\left(-\frac{2(v(Z)/40)^2}{N_Z}\right)
 \leq\exp(-v(Z)/400).
\]
\end{proof}

Let $\mathcal G$ be the finite simple graph of possible endpoints
of edges in this block. Each cluster has at most $2^s$ vertices on
each side. A vertex has at most $\max\{d,D'\}$ incident slots, so
\[
 \Delta(\mathcal G)\leq\Delta:=2D\,2^s,
 \qquad Q:=1+\Delta.
\]
For every nonempty connected vertex set $Z$ of $\mathcal G$ with
$v(Z)\leq E$, introduce the bad event of Lemma~\ref{exp:cut-tail}.
It is enough to avoid these events. Indeed every other set splits
into its connected components in $\mathcal G$; actual edges never
join different components, so the crossing counts and volumes add,
and each component still has volume at most $E$.

Every matching variable is supported on one pair of clusters, with
every edge between their two sides possible. Its support has
$\mathcal G$-diameter at most two. An event for $Z$ depends only
on variables whose supports meet $Z$. Thus two events at distance
greater than two use disjoint sets of independent variables.
Moreover an event is independent jointly of any collection of such
distant events, since all their variables together are disjoint from
its own. We may therefore join two events in a dependency graph
whenever their vertex sets have distance at most two.

There are at most $zQ^2$ vertices within distance two of a $z$-set.
The number of connected $m$-sets containing a specified vertex is
at most $Q^{2(m-1)}$: fix orders of the vertices and their neighbors,
choose for each such set its first spanning tree under that order,
and traverse this rooted tree in depth-first order. The traversal
is a walk of length $2(m-1)$ visiting exactly that set, and the
orders make the association a well-defined injection into such
walks. It follows that a bad event for a $z$-set has at most
\begin{equation}
 zQ^{2m}\quad\text{dependent events indexed by $m$-sets}.
 \label{exp:dependency-count}
\end{equation}
Counting events outside the volume bound or the event itself only
enlarges this upper bound.

Set $a=d/800$ and assign weights $x_Z=e^{-a|Z|}$. For all sufficiently
large $\ell$, uniformly in $i,k$,
\begin{equation}
 a\geq2,\qquad \rho:=Q^2e^{-a}\leq\frac12.
 \label{exp:LLL-parameters}
\end{equation}
Indeed $\log Q=O(\ell)$, whereas $d=2^{4\ell}$; neither bound
contains the stream lengths. Equations~\eqref{exp:dependency-count}
and \eqref{exp:LLL-parameters} give the explicit uniform sum
\begin{equation}
 \sum_{Z'\sim Z}x_{Z'}
 \leq |Z|\sum_{m\geq1}Q^{2m}e^{-am}
 =|Z|\frac{\rho}{1-\rho}\leq|Z|.
 \label{exp:dependency-sum}
\end{equation}
Since every weight is at most $1/2$, the elementary inequality
$\log(1-x)\geq-2x$ yields
\[
 x_Z\prod_{Z'\sim Z}(1-x_{Z'})
 \geq e^{-a|Z|-2|Z|}\geq e^{-2a|Z|}
 \geq\mathbb P(\text{bad }Z).
\]
The last step uses $v(Z)\geq d|Z|$ and
Lemma~\ref{exp:cut-tail}; here $D'\geq D/2\geq d$.

We include the finite asymmetric local-lemma argument for this inequality.
The local lemma originates with Erd{\H o}s and Lov{\'a}sz \cite{ErdosLovasz1975};
see \cite[Theorem~1.1]{MoserTardos2010} for the asymmetric form and
\cite[Section~5.1]{AlonSpencer2016} for its conditional-probability proof. For
any event $A_Z$ and any list $\mathcal S$ of other events, induct
on $|\mathcal S|$ to prove
\[
 \mathbb P\left(A_Z\,\middle|\,
          \bigcap_{A\in\mathcal S}A^c\right)\leq x_Z,
\]
as well as positivity of the avoidance probabilities used. Split
$\mathcal S$ into neighbors $\mathcal N$ of $A_Z$ and nonneighbors
$\mathcal T$. Independence gives
$\mathbb P(A_Z\mid\bigcap_{\mathcal T}A^c)=\mathbb P(A_Z)$.
By the induction hypothesis and the product rule,
\[
 \mathbb P\left(\bigcap_{\mathcal N}A^c\,\middle|\,
                       \bigcap_{\mathcal T}A^c\right)
 \geq\prod_{A_{Z'}\in\mathcal N}(1-x_{Z'}).
\]
If $\mathcal N$ is nonempty, each conditional probability used on
the left conditions on fewer than $|\mathcal S|$ avoided events.
Dividing by this lower bound proves the desired estimate using the
local-lemma inequality above. If $\mathcal N$ is empty, independence
alone proves it. Beginning with the empty list and multiplying
successive survival probabilities also proves their positivity.
In particular the probability of avoiding the entire finite list is
positive. We have obtained an actual block satisfying
\begin{equation}
 b(Z)\geq\frac1{20}v(Z)
 \qquad(0<v(Z)\leq E).
 \label{exp:actual-cut}
\end{equation}

\begin{proposition}[Expansion of the realized incidence]
\label{exp:realized-gap}
For all sufficiently large $\ell$, independently of the height
pair and of its previously fixed lower tables, its current tables
can be chosen so that \eqref{exp:actual-cut} holds. For this choice,
the incidence operator of the actual $Y^0$ block, with each entry
normalized by $\sqrt{dD'}$ in counting Hilbert spaces, sends normalized
constants to normalized constants, as does its adjoint, and has norm
at most $1-\chi$ between their orthogonal complements, where
\[
 \chi=\frac1{800}.
\]
\end{proposition}

\begin{proof}
Only the passage from \eqref{exp:actual-cut} to the incidence norm
remains. We give the weighted discrete Cheeger argument with our
biregular normalization; compare Dodziuk \cite{Dodziuk1984} and
Alon--Milman \cite{AM1985}. Write $R_0$ for the actual random walk, self-adjoint in
degree measure. For real $f$ its Dirichlet form is
\[
 \mathcal E(f)=\langle(1-R_0)f,f\rangle_\mu
             =\sum_{xy\ \mathrm{edge}}(f(x)-f(y))^2.
\]
Each edge in this sum is counted once, with multiplicity. If $w$
is nonnegative and its positive support has volume at most $E$,
integration of \eqref{exp:actual-cut} over the level sets of $w^2$
gives
\[
 \sum_{xy\ \mathrm{edge}}|w(x)^2-w(y)^2|
 =\int_0^\infty b(\{w^2>t\})\,dt
 \geq\frac1{20}\sum_x\mu(x)w(x)^2.
\]
On the other hand Cauchy--Schwarz and
$(w(x)+w(y))^2\leq2(w(x)^2+w(y)^2)$ show that the square of the
left side is at most
\[
 \mathcal E(w)\sum_{xy\ \mathrm{edge}}(w(x)+w(y))^2
 \leq2\mathcal E(w)\|w\|_\mu^2.
\]
Thus $\mathcal E(w)\geq\|w\|_\mu^2/800$, including $w=0$.

For an arbitrary real $f$, choose a degree-weighted median $m$:
both $\{f>m\}$ and $\{f<m\}$ have volume at most $E$.
Such a value is obtained by ordering the finitely many function
values and stopping when cumulative mass reaches half the total.
Apply the preceding estimate to
$w_+=(f-m)_+$ and $w_-=(m-f)_+$. On each edge,
\[
 (w_+(x)-w_+(y))^2+(w_-(x)-w_-(y))^2
 \leq(f(x)-f(y))^2.
\]
This follows by equality when both endpoints are on the same side
of $m$, and by $a^2+b^2\leq(a+b)^2$ otherwise. Consequently,
for a degree-centered $f$,
\[
 \mathcal E(f)\geq\frac1{800}\|f-m\|_\mu^2
 =\frac1{800}\bigl(\|f\|_\mu^2+2E m^2\bigr)
 \geq\frac1{800}\|f\|_\mu^2.
\]
For complex $f$, apply this to its real and imaginary parts. Hence
$R_0\leq1-1/800$ on the orthogonal complement of the global
constant.

Let $T$ be the counting-space incidence divided by $\sqrt{dD'}$.
Multiplication by $\sqrt d$ on $A$ and $\sqrt{D'}$ on $B$ is
a unitary from degree measure to counting measure and sends $R_0$
to
\[
 \begin{pmatrix}0&T\\T^*&0\end{pmatrix}.
\]
Biregularity and $d|A|=D'|B|$ give
$T(|B|^{-1/2}\mathbf1_B)=|A|^{-1/2}\mathbf1_A$ and the
adjoint identity. The centered spaces are invariant in the two
directions. On their direct sum the displayed self-adjoint block
operator has norm $\|T|_{\mathbf1_B^\perp}\|$ and spectrum symmetric
about zero: conjugation by $\operatorname{diag}(1,-1)$ changes its
sign. Its largest spectral value therefore equals this norm.
Every vector in that direct sum is orthogonal to the global
degree constant under the unitary identification, so the preceding
upper gap proves the assertion. This also explains why the global
anti-constant eigenvalue $-1$ does not weaken the incidence bound.
\end{proof}

There are finitely many families. In each family, fix successful
current tables successively in increasing cutoff order. A current
choice does not alter any smaller-cutoff block, whose padded inputs
turn off the higher-cutoff gates. Proposition~\ref{exp:realized-gap}
is uniform over every earlier choice, so this succession defines
all the countably many tables. For example, impose a finite ordering
on each current collection of tables and choose its first successful
element. The construction therefore yields all the required
pre-pruning incidences with the same positive constant $\chi$.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//expansion.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//geometry.tex ====
\section{Geometry of the voltage cover and torsion-freeness}
\label{geo:section}

Embedding the components of the voltage cover in a Cayley graph is not
enough for the later operator comparison, because different copies can
overlap.  We therefore seek disjoint assigned domains, with the removed
vertices controlled by one finite ball per copy and the unused Cayley
edges forming a forest.  The input is the girth condition
\eqref{geo:girth}, which the later sampling argument must establish;
no assertion about an operator's spectral distribution is used here.
The constants in this section are chosen before the voltage values and
the final alphabet are chosen.

The use of labeled graphs as relators follows the geometric strategy of
Gromov \cite[Sections~2 and~4.8]{Gromov2003}. Finite alphabets with global
separation of long path words are developed by Osajda
\cite[Section~2]{Osajda2020}.  The diagrammatic embedding argument is
closely related to Ollivier \cite[Section~2]{Ollivier2006} and Gruber
\cite[Lemma~4.1]{Gruber2015}.
Here the relator graphs are components of a voltage cover, possibly with
nontrivial stabilizers. We give the embedding, overlap, and torsion
arguments for these precise hypotheses.

\subsection{Graphs, presentations, and the conditional input}
\label{geo:conventions}

All graphs in this section are combinatorial graphs, possibly with loops or
parallel edges.  An unoriented edge has two formal orientations, interchanged
by a fixed-point-free reversal operation.  Traversing an edge and immediately
traversing its reverse is a backtrack.  A closed walk is cyclically
nonbacktracking if it has no backtrack, including at its closing corner.
Edge lengths are one, degrees count incidences, and a ball denotes the induced
subgraph on the vertices at most its stated distance from its center.  A
forest has no nonempty finite circuit, with a loop and a pair of parallel
edges counted as circuits of lengths one and two.  Chains always have finite
support.  An orientation chosen once for each unoriented edge identifies the
space of edge chains over a field $k$ with the free vector space on those
edges, with boundary $\partial[x,y]=[y]-[x]$.  The cycle space is
$Z_1(-;k)=\ker\partial$.

Let $Y$ be the surviving tower.  The properties of it needed in this section
are the following: it is countable, has maximum degree at most $6D$, and has
an integer layer function whose change across an edge is at most one.  Its
oriented labeling is folded: the labels of the oriented edges leaving any
vertex are distinct.  Positive labels and their formal inverses are disjoint
alphabets.  Finally a nonbacktracking path starting in layer $i$, of length
at least $L_i=100H_i/\log_2d$, is determined, including its location in $Y$,
by its oriented label word.  This uniqueness compares paths anywhere in $Y$.
Refining these labels by arbitrary additional tags preserves both foldedness
and this uniqueness property.

The following elementary estimate will be used with fixed constants, however
large.  If two vertices can be joined by a walk of length
$aH_i/\log_2d$, starting in layer $i$, then
\begin{equation}
 \left|\frac{H_j}{H_i}-1\right|\le\frac{a}{\log_2d}
 \label{geo:layer-comparison}
\end{equation}
at every layer $j$ encountered.  This follows from
$|H_j-H_i|\le |j-i|$.  Consequently all subsequent comparisons become uniform
in $i$ by increasing $\ell$.  Also $L_i>2$ for all $i$ with the parameters
of the construction.

Let $F$ be a countable torsion-free group.  Give each positive edge $e$ of
$Y$ a voltage $z_e\in F$, and give its reverse voltage $z_e^{-1}$.  The ordinary voltage-cover construction of Gross and Gross--Tucker
\cite{Gross1974,GT1977} gives a graph $P$ with vertices $(v,f)$ and an oriented lift
\[
 (v,f)\longrightarrow(v',fz_e)
\]
over each oriented edge $e:v\to v'$.  The left deck action is
$\delta_a(v,f)=(v,af)$ for $a\in F$.  It acts freely on vertices and
transitively on the lifts of every fixed base vertex or fixed base path;
it can carry one connected component of $P$ to another.  Labels, layers,
and the distinguished positive orientations are pulled back from $Y$.

We assume that the augmented positive alphabet $S$ is finite and that
\begin{equation}
 \begin{gathered}
 \text{there is no nonempty cyclically nonbacktracking closed walk in $P$}\\
 \text{based in layer $i$ of length at most }
             C_*H_i/\log_2d,\qquad i\in\mathbb Z.
 \end{gathered}
 \label{geo:girth}
\end{equation}
Here $C_*$ is a single absolute constant fixed below.  This is a condition
on the actual cover, not just on $Y$ and not just on simple base circuits.
It includes all lifts, although whether a base closed walk has trivial
holonomy does not depend on the starting deck coordinate.

Define
\begin{equation}
 \Gbase=\langle S\mid \text{label words of all closed walks in }P\rangle,
 \qquad X=\operatorname{Cay}_{\mathrm{right}}(\Gbase,S).
 \label{geo:presentation}
\end{equation}
The Cayley graph is formal: its positive edges are the pairs $(h,\sigma)$,
with endpoints $h,h\sigma$, for $h\in\Gbase$, $\sigma\in S$.  We do not
identify distinct such edges even if their unordered endpoints coincide.
Their reverse orientations have labels $\sigma^{-1}$.  Left translation
preserves every formal edge label and its positive orientation.

For a connected component $C$ of $P$, a vertex $v_0\in C$, and $h\in\Gbase$,
there is a unique label-preserving graph map $\phi:C\to X$ with
$\phi(v_0)=h$: the image of the endpoint of a path from $v_0$ is $h$ times
the label word of that path.  Independence of the path follows from
\eqref{geo:presentation}.  The maps are not assumed injective at this stage.

\begin{theorem}[Geometric conclusion]
\label{geo:main}
There are absolute constants $C_*,C_\bullet$ such that, for sufficiently
large $\ell$, the preceding hypotheses have the following consequences.
Every component map $\phi:C\to X$ is an embedding.  Index copies by pairs
$(C,\phi)$ modulo the equivalence
\[
 (C,\phi)\sim
 (\delta_a C,\phi\delta_a^{-1}),\qquad a\in F.
\]
Thus an indexed copy $Q$ remembers its source component and source vertices
up to this deck alignment, in addition to its embedded image.
There is a well-order of these copies and an induced intrinsic ball $B(Q)$
in each $Q$ such that
\begin{enumerate}
 \item every vertex of $Q$ shared with an earlier copy belongs to $B(Q)$;
 \item the subgraph of $Q$ consisting of all edges with at least one
 endpoint in $B(Q)$ is a forest;
 \item for every vertex of $B(Q)$ lying over layer $i$,
 \begin{equation}
 |V(B(Q))|\le\mathfrak w_i,
 \qquad \mathfrak w_i=2^{C_\bullet H_i};
 \label{geo:weight}
 \end{equation}
 \item if $D_Q$ is $V(Q)$ minus its vertices shared with earlier copies,
 the domains $D_Q$ are pairwise disjoint, and deleting every edge of $Q$
 whose two endpoints lie in $D_Q$, for all $Q$, leaves a forest in $X$.
\end{enumerate}
Furthermore $\Gbase$ is torsion-free.  None of the constants depends on
$F$, the voltage values, the number of components of $P$, or $|S|$.
\end{theorem}

\subsection{Plane diagrams and reductions before injectivity}
\label{geo:diagrams}

The same short-arc count will first rule out failures of injectivity and
later bound intrinsic distances within embedded copies.  We retain
source paths in the diagrams: merging source walks requires actual deck
alignment, not merely equality of label words.

We use the following form of the van Kampen Lemma; see
\cite[Chapter V, Section 1]{LyndonSchupp1977}.  If a word is freely equal to a
product of $J$ conjugates of defining relators and their inverses, it is
the exterior contour word of a finite connected plane labeled multigraph
with at most $J$ bounded faces.  Each bounded face contour reads a defining
relator, with a chosen cyclic start and orientation.  Conversely an exterior
contour word of such a diagram is freely a product of conjugates of its
bounded face words.  In this statement a contour is a closed walk around a
face, rather than an embedded circle: vertices and bridges may occur more
than once.  Every application below uses a finite number of relators even
though the presentation is infinite.

For completeness, the elementary operations in this version are as follows.
Draw a polygon for each relation factor and its conjugating stem, joining
the stems at a base vertex in their product order.  Empty factors are
omitted.  The exterior reads the literal product.  Exterior insertion of
$aa^{-1}$ is made by attaching a spur.  For a cancellation, let the two
consecutive exterior traversals have vertices $v,w,v'$.  If they traverse
the same edge, erase the resulting leaf.  If they are distinct and $v=v'$,
their digon, or the two lobes when both edges are loops, bounds portions
which can be discarded, together with these edges.  The other exterior
contour remains connected at $v$.  The consecutiveness at $w$ ensures that
no exterior attachment is lost away from that vertex.  If $v\ne v'$, draw
an auxiliary edge in the exterior face between the indicated occurrences
of $v$ and $v'$, so that it cuts off the two-edge portion as a triangular
face.  Contract the auxiliary edge.  The artificial face is now a digon;
delete one of its edges to merge it with the adjacent face.  In that
adjacent contour the replacement traversal has the same label.  Thus the
desired exterior cancellation has taken place without increasing the
number of relation faces.  This operation uses positions on contours and
is valid when other vertices or contour portions repeat.  Conversely,
merging an exterior-adjacent bounded face across a non-bridge edge,
repeatedly until a tree remains, proves the product assertion.  These
operations allow a prescribed literal exterior word to be constructed,
after which its required break positions are marked.  A marked position
crossed by an intermediate cancellation need not retain its previous
vertex value.  All vertices of a diagram have consistent values in $\Gbase$:
propagate from one vertex along paths; simple circuits bound subdiagrams,
so the product assertion proves path independence.

In our presentation every face comes with an actual closed walk in $P$
reading its contour.  This additional assignment matters: equal words alone
will not be treated as equal paths.  When two source incidences have been
aligned by a deck transformation, deleting an edge between their faces
splices two genuine source walks into a genuine closed source walk.  This
is again a permitted relator because the presentation uses all closed
walks, including walks with backtracks and singular contours.

Suppose an injectivity failure exists.  Choose, over \emph{all} component
maps, pairs of distinct source vertices with equal image, paths between
them, and diagrams for the resulting closed image word, a choice minimizing
first the number $J$ of bounded faces and then the number of edges.  The
exterior has one adjustable side: it is this source path, with its break
vertex marked.  Its endpoints in $P$ must remain distinct, but may vary
when comparing different failure diagrams.  Minimization over all failures,
rather than just one originally selected path, is essential below.

We will use the same minimization after injectivity is proved, in a second
setting.  Adjoin to $X$ one center $c_Q$ for every indexed copy $Q$, and one
spoke of length one from $c_Q$ to each vertex of $Q$.  Denote this graph
by $\widehat X$.  For an edge loop in $\widehat X$, started at a group
vertex, each generator edge is a fixed side and each consecutive two-spoke
passage through a center is an adjustable side.  Inflate an adjustable
side to any source path in its copy with the indicated endpoints.  Mark
all side breaks.  If there are $m$ sides there are at most $m$ distinct
marked vertices.  Among diagrams and these adjustable paths minimize
$(J,\text{number of edges})$ lexicographically.  If the inflation is empty,
retain its one marked vertex.  For both minimizations, source data along
adjustable sides and bounded contours are retained as part of the diagram.

\begin{lemma}[Reduction rules]
\label{geo:reductions}
In either minimum diagram the following statements hold.
\begin{enumerate}
 \item No bounded face has a freely trivial label word.
 \item No edge separating two distinct bounded faces has aligned source
 incidences.  Nor does an edge between a bounded face and an adjustable
 exterior side have aligned source incidences.
 \item There is no unmarked vertex of degree one, and no bridge having
 both incidences on a bounded face.
\end{enumerate}
Here alignment means that, when both incidences are read in the same
direction, a left deck transformation identifies their source edges.
\end{lemma}

\begin{proof}
A freely trivial face word can be omitted from the face product expression;
the van Kampen Lemma then supplies a diagram for the identical exterior
with fewer bounded faces.  For the second assertion, delete the separating
edge.  Distinct bounded faces merge into a face whose source walk is obtained
by splicing after deck alignment.  A bounded face adjacent to an adjustable
side can instead be absorbed into that side: replace its edge traversal by
the complementary face walk with the appropriate orientation.  The source
endpoints are unchanged.  Deleting an edge separating two faces does not
disconnect the plane graph, and repeated contacts elsewhere merely make
the new contour singular; they do not prevent the splice.  In each case
$J$ decreases.

At an unmarked leaf the contour makes a backtrack.  If it is an exterior
corner it lies within an adjustable side, since a break between fixed sides
would be marked.  In all cases foldedness makes it an actual backtrack in
the assigned source walk.  Erasing the leaf preserves the assignments and
contradicts edge minimality.

Consider now a bridge with both incidences on a bounded face.  Removing its
open edge splits the diagram in two.  The exterior contour and all its
marked positions belong to one of them; retain that component.  In the
affected bounded face contour the other component is visited in one
excursion, along the bridge, around its exterior contour, and back.  The
discarded component, with the bridge as a stem, gives a diagram for the
word of this excursion with fewer than $J$ bounded faces: the affected
face is not among them.  To see the indicated exterior designation, the
retained component is connected and accessible from infinity, so lies on
the unbounded-contour side of the discarded component.  Thus its boundary
is exactly the contour traversed in that excursion, with the bridge stem.

After injectivity, the two endpoints of the source excursion are equal
because their images are equal, so the excursion can be removed.  Before
injectivity, if those source endpoints were distinct, this excursion and
its smaller diagram would themselves constitute an injectivity failure,
contrary to the global minimal choice of $J$.  They therefore agree in
this case as well.  Delete the excursion, bridge, and discarded component.
The retained affected face is still assigned a closed source walk.  Its
face count does not increase and its edge count decreases, a contradiction.
\end{proof}

Suppress every unmarked degree-two vertex.  Call a resulting edge an
\emph{arc}; it remembers its entire original edge path and its length.
Loops among these arcs are allowed.  There is at least one marked vertex,
so an entire degree-two component cannot disappear without a retained
vertex.  Every side incidence along an arc is constant: it is a bounded
face or one exterior side.  Indeed a change of exterior side is marked,
and a change of face incidence requires branching.  Degrees and contours
in what follows are those of this compressed plane graph.

\begin{lemma}[Short arcs and Euler count]
\label{geo:arcs}
Every arc incident with a bounded face has length less than $L_i$, when
read from an endpoint in layer $i$ on that face.  Its length is also less
than $L_i$ on any adjustable-side incidence with which it is compared.
Every bounded face has at least twenty arc incidences.  If $b$ is the
number of arcs and there are at most $m$ marked vertices, then
\begin{equation}
 b\le 3J+3m-3,\qquad 20J\le 2b,\qquad b\le 6m.
 \label{geo:euler}
\end{equation}
These statements require only $C_*\ge10^4$ and sufficiently large $\ell$.
\end{lemma}

\begin{proof}
An arc of length greater than two is freely reduced.  Otherwise an inverse
pair with an unmarked bivalent midpoint can be erased, identifying its two
outer vertices.  Those outer vertices are distinct in the diagram: the
interior of a maximal arc has no repetitions, and a loop arc of length
greater than two has this property for each such pair.  The two-edge path
is therefore a tree, and its contraction preserves the number of faces
and the plane embedding.  Its outer group values agree; its bivalent
middle vertex has no other incidences.  On every assigned source walk
foldedness identifies the inverse pair as a backtrack.  The contraction
therefore preserves all assignments and decreases the number of edges,
which is impossible.

If an arc has a generator-side incidence it has length one.  Otherwise an
arc incident with a bounded face has two source assignments.  Its other
incidence cannot be on the same bounded face, since an edge with the same
face on both sides is a bridge, excluded by Lemma~\ref{geo:reductions}.
If the arc were at least $L_i$ long on either compared assignment, its
freely reduced word would determine its projected nonbacktracking path in
$Y$.  Both projections would be this same path.  The two lifts of that
path then differ by a single deck transformation, so their source
incidences are aligned.  Lemma~\ref{geo:reductions} forbids that alignment.
Arcs of length at most two already satisfy the claimed inequalities.

If a face had fewer than twenty arc incidences, select its longest arc,
of length $a$, and let $i$ be its initial layer in the face assignment.
Its perimeter would be at most $19a<20L_i=2000H_i/\log_2d$.
By \eqref{geo:layer-comparison}, every layer $j$ on it has
$H_j\ge H_i/2$ for large $\ell$.  The face word is not freely trivial.
Cancel backtracks in its closed source walk, and then cancel around its
cyclic ends.  Foldedness makes these actual source cancellations; the
remaining walk is nonempty and cyclically nonbacktracking.  Its length
is at most the displayed perimeter, while
$C_*H_j/\log_2d\ge5000H_i/\log_2d$.  This contradicts
\eqref{geo:girth}.

Let $v$ be the number of vertices of the compressed diagram.  It is
connected and satisfies $b=v+J-1$ by the plane Euler formula, also for
loops, parallel edges and bridges.  Every unmarked vertex has degree at
least three; a graph consisting of one marked vertex and no edges also
satisfies the resulting inequality.  Hence
$2b\ge3(v-m)$, which gives the first inequality of
\eqref{geo:euler}.  Each arc has two face incidences in total, counting
the exterior, so the twenty-incidence bound gives its second inequality.
Combining them gives $b\le(30/7)(m-1)\le6m$ whenever edges occur;
the same final bound holds for the trivial diagram.
\end{proof}

\begin{proposition}[Embeddings]
\label{geo:embeddings}
All component label maps $C\to X$ are injective on vertices and formal
edges.  The indexed copies of Theorem~\ref{geo:main} are therefore defined.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{geo:arcs} to a minimum injectivity-failure diagram, with
$m=1$.  Its first two inequalities imply $20J\le6J$, hence $J=0$.
The uncompressed diagram is a tree, so its exterior word is freely trivial.
A path with freely trivial word in a folded graph returns to its starting
vertex: erase successive inverse pairs, which are actual backtracks.
The chosen source endpoints consequently agree, a contradiction.

For edges, two oriented source edges with the same formal image have the
same initial image and the same oriented label.  Vertex injectivity makes
their initial vertices equal, and foldedness makes the edges equal.
This includes the possibility of parallel edges and establishes the
asserted graph embeddings.
\end{proof}

We henceforth use the post-injectivity minimum diagrams.  One additional
observation follows immediately from the indexing convention.  If paths
in two indexed copies have the same word and are deck-aligned, and their
maps agree at one source point after alignment, their maps agree everywhere
on the connected component: propagate along labeled paths.  They are then
the same indexed copy.  Therefore an arc shared by two different copy
assignments has length less than the applicable $L_i$ on either assignment.
\label{geo:distinct-copy-arcs}

\subsection{Short chain fillings and an explicit thin-triangle estimate}
\label{geo:coned}

We next prepare an ordering of the copy centers by distance from one
fixed group vertex, in order to control all earlier intersections with
a copy at once.  Thin triangles in $\widehat X$ will give uniformly
short paths that avoid the copy's own center.  The short-arc estimate
will then turn such paths into intrinsic distance bounds in the copy.
We first obtain thin triangles from a linear bound on decompositions
of loop chains into short loops.

\begin{lemma}[Short chain fillings]
\label{geo:chain-filling}
In $\widehat X$ the integral oriented chain of any edge loop of length $n$
is a sum of at most $12n$ oriented chains of edge loops of length at most
four.
\end{lemma}

\begin{proof}
For a nonempty loop, rotate its start to a group vertex and split it into
the sides specified above.  There are $m\le n$ sides.  Inflate and minimize,
and apply Lemma~\ref{geo:arcs}.  For each arc choose a standard path between
the images of its endpoints: use the generator edge if it has a fixed-side
incidence, and otherwise use the two-spoke passage through any one of its
incident source-copy centers.  Choose one direction for the arc, using the
reverse standard path for its reverse incidence.

On a bounded face, replace every incident standard path by the two-spoke
passage through that face's center.  Each replacement differs by a closed
chain of length at most four; a generator standard path only reduces that
length.  The sum of the new passages is zero as a chain, because successive
spokes telescope at the consecutive contour vertices, with their contour
multiplicities.  On an exterior adjustable side make the same replacements
using that side's center; the resulting chain telescopes to the original
two-spoke side.  Exterior fixed sides already have the required standard
path.  The sum of the standard-path chains around the bounded faces is
the exterior standard-path chain, by cancellation of the two oriented
incidences of each internal edge, also when an edge is a bridge or a
contour is singular.  Combining these identities expresses the original
loop chain as the asserted sum.  At most one short loop is used for each
bounded or exterior arc incidence, hence at most $2b\le12m\le12n$.
\end{proof}

The next lemma is a quantitative form of the homological-isoperimetric
criterion for hyperbolicity; compare Gersten
\cite[Theorem~6.3]{Gersten1996}.  The proof below gives an explicit
thin-triangle bound under our chain-filling hypothesis, without assuming
local finiteness.

\begin{lemma}[Thin triangles from short chain fillings]
\label{geo:thin}
Let a connected graph with unit edge lengths have the conclusion of
Lemma~\ref{geo:chain-filling}.  Every geodesic triangle on its vertices is
$D_0$-thin, where one may take $D_0=10^6$.  Thinness here measures distance
to the vertices on the other two sides.
\end{lemma}

\begin{proof}
Fix a geodesic triangle $[x,y],[x,w],[y,w]$, and put
\[
 n=d(x,y),\quad u=d(x,w),\quad v=d(y,w),\quad
 a=(n+u-v)/2,\quad b=u-a=v-(n-a).
\]
Let $R$ be the maximum, over its side vertices, of distance to the vertices
on the other two sides.  Relabel the sides so it is attained at
$z\in[x,y]$, at position $j=d(x,z)$.  We may assume $R\ge100$.  Then
$j,n-j\ge R$, and every vertex of either other side has distance at least
$R$ from $z$.

A position $h<a-R$ on $[x,y]$ is within $R$ of $[x,w]$.  Indeed, if it
were within $R$ of position $t$ from $y$ on $[y,w]$, then
$t\ge n-h-R$, and the triangle inequality would give
\[
 u\le h+R+v-t\le2h+2R+v-n,
\]
contrary to $h<a-R$.  The definition of $R$ supplies closeness to at least
one other side.  Symmetrically positions $h>a+R$ are within $R$ of
$[y,w]$.  Also a position $k>a+R$ on $[x,w]$ must be within $R$ of
$[y,w]$: closeness to a position $h$ on $[x,y]$ would imply $h\ge k-R$
and
$v\le u-k+R+n-h\le u+n-2k+2R$, again a contradiction.

We construct a subsegment of $[x,y]$ containing at least $R$ edges on
either side of $z$, and an alternate path between its endpoints whose
vertices all have distance at least $R$ from $z$, so that the resulting
loop has length at most $200R$.

First suppose $j<a-6R$.  Take the positions
$h_- =\max(0,j-4R)$ and $h_+=j+4R$; the latter is less than $a-2R<n$.
Bridge both to $[x,w]$ by paths of length at most $R$, taking the trivial
bridge at $x$ if $h_-=0$, and join their endpoints along $[x,w]$.
Each nontrivial bridge starts $4R$ from $z$, so all its vertices are at
least $3R$ from $z$.  The side portion on $[x,w]$ has length at most
$h_+-h_-+2R$, since its endpoint coordinates from $x$ differ from $h_-$
and $h_+$ by at most $R$.  Thus the whole loop has length at most $20R$.
The case $j>a+6R$ is symmetric.

In the remaining case $|j-a|\le6R$, take
\[
 h_- =\max(0,\lfloor a-10R\rfloor),\qquad
 h_+ =\min(n,\lceil a+10R\rceil).
\]
Bridge the left endpoint to $[x,w]$ and the right endpoint to $[y,w]$,
using paths of length at most $R$ and trivial bridges at clipped endpoints.
The preceding positional rules supply every nontrivial bridge.  Such a
bridge starts at least $4R$ from $z$, so it stays at least $3R$ from $z$.
Write $s_-$ and $t_+$ for the coordinates of the bridge endpoints from
$x$ and $y$, respectively.  They satisfy
\[
 |s_--h_-|\le R,\quad |t_+-(n-h_+)|\le R,\quad
 a-h_-\le10R+1,\quad h_+-a\le10R+1.
\]
If $b\le20R+2$, join these endpoints through $w$.  Each side portion
has length at most $31R+3$, while $h_+-h_-\le20R+2$, so the entire loop
has length at most $84R+8$.  If $b>20R+2$, take the vertex $A\in[x,w]$
at position $k=\lceil a+20R\rceil<u$.  It has a bridge of length at most
$R$ to a vertex $B\in[y,w]$.  If $t=d(y,B)$, comparison of distances to
$w$ gives
\[
 |(u-k)-(v-t)|\le R,\qquad
 |t-(n-a+20R)|\le R+1.
\]
We have $s_-\le k$ and $t_+\le t$; the intervening portions on the two
sides have lengths at most $31R+2$ and $32R+2$.  Including the three
bridges and the original segment gives a loop of length at most $86R+6$.
This last bridge stays far from $z$, since
$d(z,A)\ge k-j\ge14R$ and its length is at most $R$.
All other vertices of the alternate path lie on the two other sides or
the already controlled endpoint bridges.  In all cases the selected
segment extends at least $R$ on either side of $z$: when clipped, this
uses $j,n-j\ge R$; when not clipped it follows from the stated positions.
This completes the construction with the claimed $200R$ bound.

Put $r_0=\lfloor R/4\rfloor$.  In cyclic order partition this loop into
four vertex paths: $I_1$ is the segment from $j-r_0$ to $j+r_0$;
$I_2$ is the remaining segment toward $h_+$; $I_3$ is the alternate path
from $h_+$ to $h_-$; and $I_4$ is the remaining segment back to $j-r_0$.
Opposite pairs have vertex-set distances
\[
 d(I_1,I_3)\ge R-r_0\ge R/3,\qquad
 d(I_2,I_4)=2r_0\ge R/2-2\ge R/3.
\]
The equality uses that $I_2,I_4$ are subsegments of one geodesic.
Set $q_0=R/3$ and define $1$-Lipschitz vertex functions
\[
 f(v')=\min\{q_0,d(v',I_1)\},\qquad
 g(v')=\min\{q_0,d(v',I_2)\}.
\]
On an oriented edge $(v',v'')$ define the antisymmetric form
\[
 \omega(v',v'')=
 \frac{f(v')+f(v'')}{2}\bigl(g(v'')-g(v')\bigr).
\]
The successive arcs have $f=0$, $g=0$, $f=q_0$, and $g=q_0$,
respectively.  Consequently the integral around the loop is exactly
$q_0^2$: only $I_3$ contributes, with its $g$ values running from zero
to $q_0$.  On any loop of length at most four, subtract the initial value
of $f$ without changing the integral, since the increments of $g$ sum to
zero.  The Lipschitz bounds then bound its absolute integral by $16$.
The integral depends only on the oriented chain, so the assumed short
chain filling gives
\[
 (R/3)^2\le16\cdot12\cdot200R.
\]
Thus $R\le345600<10^6$.  No local finiteness was used: all triangles,
loops and chain decompositions here are finite.  Geodesics between vertices
exist because nonempty sets of integer path lengths have minima.
\end{proof}

\subsection{Avoidance paths and simultaneous overlap}
\label{geo:overlap}

\begin{lemma}[A short path avoiding one center]
\label{geo:avoidance}
Fix a positive integer $L$.  For sufficiently large $\ell$, uniformly in
$Q$ and $i$, if a path in $\widehat X$ of length at most $L$ joins
$u,x\in Q$, with $u$ in layer $i$, and avoids $c_Q$, then
\begin{equation}
 d_Q(u,x)\le24(L+1)L_i.
 \label{geo:avoidance-bound}
\end{equation}
Here the threshold for $\ell$ can depend on $L$, which will subsequently
be an absolute constant.
\end{lemma}

\begin{proof}
Close the given path by a designated two-spoke side through $c_Q$, inflate,
and minimize.  There are at most $L+1$ sides.  We claim every arc occurrence
on the designated adjustable side is shorter than its local $L_j$.
For an opposite bounded-face incidence this is Lemma~\ref{geo:arcs};
an opposite generator incidence has length one.  If the opposite incidence
is on another adjustable side, that side uses a different center because
the original path avoids $c_Q$.  A long arc would align the two source
paths and then identify the copies, as observed after
Proposition~\ref{geo:embeddings}, which is impossible.

It remains to exclude an arc with both incidences on the designated side.
Its edges are exterior bridges.  In the linear order on the designated
side, the interval between the two traversals of such a bridge is an
excursion through the component cut off by that bridge.  All its exterior
positions lie within that interval; in particular it contains no side
break.  Its endpoint group values agree.  Injectivity of $Q$ implies that
its assigned source endpoints agree, so this excursion can be erased from
the adjustable path.  Discard the cut-off component and bridge as well.
The bounded contours in the retained component are unchanged, and no
exterior side break is lost.  This reduces the number of faces, or keeps
it fixed and decreases the number of edges, contrary to minimality.

The designated path therefore has at most $2b\le12(L+1)$ short arc
occurrences.  For clarity their local length bounds do give a uniform
total bound, without presupposing a diameter estimate.  If a prefix has
length $t$, its next initial layer $j$ has $H_j\le H_i+t$, so the next
arc has length at most $L_i+100t/\log_2d$.  Induction over at most
$12(L+1)$ arcs bounds each encountered $H_j/H_i$ by a quantity tending
to one as $\ell\to\infty$.  In particular each arc has length at most
$2L_i$ for sufficiently large $\ell$.  Their total is at most
$24(L+1)L_i$, proving the claim.
\end{proof}

\begin{proposition}[One ball for all earlier intersections]
\label{geo:balls}
There is a well-order of the indexed copies for which the first three
assertions of Theorem~\ref{geo:main} hold.
\end{proposition}

\begin{proof}
Fix one group vertex $o\in X$.  There are countably many indexed copies:
the components of the countable graph $P$ are countable in number and a
map from a fixed component is specified by one group value.  Enumerate
them, and order the centers first by $d_{\widehat X}(c_Q,o)$ and then by
this enumeration.  This is a well-order, since these distances and the
enumeration values are nonnegative integers.  Choose for each $A=c_Q$ one
geodesic from $A$ to $o$, and let its first vertex after $A$ be $u\in Q$.
The choice of $u$ is made once for this copy.

Suppose $x\in Q$ also belongs to a distinct copy with center $B$ satisfying
$d(B,o)\le d(A,o)$.  The path $A,x,B$ is a geodesic of length two,
since distinct centers have no direct edge.  A geodesic from $B$ to $o$
avoids $A$: otherwise its length is at least $2+d(A,o)>d(B,o)$.
Let $k=d(A,o)$.  If $k<D_0+3$, the path from $x$ through $B$ and $o$
to $u$, using the chosen two geodesics, avoids $A$ and has length at most
$1+k+(k-1)=2k$.  If $k\ge D_0+3$, take the vertex $p$ at position
$t=D_0+3$ on the chosen $A,o$ geodesic.  Every vertex of $[A,B]$ is at
distance greater than $D_0$ from $p$, since its distance from $A$ is at
most two.  By Lemma~\ref{geo:thin}, $p$ has a bridge of length at most
$D_0$ to a vertex $q$ of the chosen $B,o$ geodesic.  This bridge avoids
$A$ because $d(p,A)>D_0$.  Furthermore
\[
 d(B,q)=d(B,o)-d(q,o)\le k-(k-t-D_0)=t+D_0.
\]
The path $x$ to $B$, along $[B,q]$, across this bridge to $p$, and back
along $[p,u]$ therefore avoids $A$ and has length at most $2t+2D_0$.
Both cases are bounded by the absolute constant
\begin{equation}
 L^*=2+2(D_0+3)+2D_0=4D_0+8.
 \label{geo:root-path-length}
\end{equation}

Let $i$ be the layer of the fixed $u$, and define
\[
 R_Q=\left\lceil25(L^*+1)L_i\right\rceil,
 \qquad B(Q)=\{v\in Q:d_Q(u,v)\le R_Q\},
\]
with the induced edges.  Lemma~\ref{geo:avoidance} puts every intersection
with an earlier copy into this one ball.  The root $o$, chosen geodesic,
and ball center $u$ have not been changed as the earlier copy varies.

The short chain and thin-triangle arguments needed only $C_*\ge10^4$,
so $D_0$ and $L^*$ have already been fixed independently of any further
increase in $C_*$.  We now fix, for example,
\begin{equation}
 C_*=10^7(L^*+1),\qquad C_\bullet=2\cdot10^4(L^*+1).
 \label{geo:constants}
\end{equation}
There is consequently no circular choice of constants.  Increase $\ell$
to meet all the fixed-constant comparisons made above and below.

An induced ball of radius $R_Q$ with a cycle has a circuit of length at
most $2R_Q+1$: take a shortest-path spanning tree from $u$ in the ball
and any edge not in that tree.  Such a tree exists by choosing a predecessor
at distance one less for each non-root vertex.  Likewise, if a vertex
outside the ball has two edge incidences into it, those incidences and
the tree path between their endpoints give a circuit of length at most
$2R_Q+2$, including the parallel-edge case.  All starting layers involved
differ from $i$ by at most $R_Q+1$, so have $H_j\ge H_i/2$ for large
$\ell$.  Since
$R_Q\le2500(L^*+1)H_i/\log_2d+1$, either circuit would violate
\eqref{geo:girth} with \eqref{geo:constants}.  Thus the ball is a tree,
and every outside vertex has at most one edge incidence to it.  The graph
of edges meeting the ball is precisely this tree with possible leaves
attached, and is a forest.

Finally, its number of vertices is bounded by
\[
 |V(B(Q))|\le(1+6D)^{R_Q+1}.
\]
For large $\ell$, $\log_2(1+6D)\le3\log_2d$, so the logarithm of this
bound is at most
$7500(L^*+1)H_i+6\log_2d\le8000(L^*+1)H_i$.
For every layer $j$ represented in the ball,
$H_j\ge H_i/2$.  Our choice of $C_\bullet$ therefore gives
$|V(B(Q))|\le2^{C_\bullet H_j}$, as required at every vertex in the ball.
\end{proof}

\subsection{Cycle coordinates, the residual forest, and copy incidences}
\label{geo:cycles}

Because all earlier intersections with a copy lie in one tree, a nonzero
cycle in the latest participating copy cannot be canceled by earlier copy
cycles.  The resulting cycle decomposition will control the unused edges
and later supply the
acyclic complex used to exclude torsion.

\begin{proposition}[Direct cycle coordinates]
\label{geo:cycle-directness}
For every field $k$, inclusion of the indexed copies induces an isomorphism
of vector spaces
\begin{equation}
 \bigoplus_Q Z_1(Q;k)\ \longrightarrow\ Z_1(X;k),
 \qquad (c_Q)_Q\longmapsto\sum_Q c_Q.
 \label{geo:cycle-decomposition}
\end{equation}
The direct sum is algebraic, so every tuple used has finite support.
\end{proposition}

\begin{proof}
A finite-support cycle in any graph is a linear combination of finitely
many circuit chains: choose a spanning forest of its finite support graph,
and subtract for each edge outside that forest the corresponding circuit
with its given coefficient.  The remainder is a cycle supported on a finite
forest, and hence is zero by removing leaves.  This proof works over every
field and with loop and parallel edges.

A closed walk in $X$ reads a word trivial in $\Gbase$.  By the presentation
and the van Kampen Lemma it bounds a finite diagram whose faces are assigned
closed source walks in $P$.  The propagated group values give each such
source component its label map into $X$, and therefore its indexed copy.
As oriented edge chains, the exterior is the sum of the bounded face
contours.  Reduction modulo $\operatorname{char} k$ or multiplication by
the circuit coefficients proves that every element of $Z_1(X;k)$ is a
finite sum of copy cycles.  This proves surjectivity.

For injectivity take a finite relation $\sum_Qc_Q=0$ and the latest copy
$Q$ with $c_Q\ne0$.  If an edge has nonzero coefficient in $c_Q$, it has
to occur in an earlier coordinate as well.  Its endpoints then belong
to an earlier copy, and hence both belong to $B(Q)$.  Thus $c_Q$ is a
cycle supported on the tree $B(Q)$, which forces $c_Q=0$.  This
contradiction proves directness.  No claim about independence of infinite
sums has been used.
\end{proof}

\begin{proposition}[Assigned domains and the remaining forest]
\label{geo:forest}
Put
\[
 D_Q=V(Q)\setminus\bigcup_{R<Q}V(R),\qquad
 E_{\mathrm{int}}=\bigcup_Q
 \{e\in E(Q):\text{ both endpoints of }e\text{ lie in }D_Q\}.
\]
The $D_Q$ are pairwise disjoint,
$V(Q)\setminus D_Q\subseteq V(B(Q))$, and the formal graph
$(V(X),E(X)\setminus E_{\mathrm{int}})$ is a forest.  Each edge in
$E_{\mathrm{int}}$ belongs to the displayed set for exactly one $Q$.
\end{proposition}

\begin{proof}
If $R<Q$, a vertex of $R$ cannot belong to $D_Q$; this proves disjointness.
The containment follows from Proposition~\ref{geo:balls}.  Disjointness
also ensures that an edge cannot be internal to two assigned domains.

If the remaining graph contained a circuit, its edge chain over
$\mathbb F_2$ would be a nonzero cycle $c$.  Use
Proposition~\ref{geo:cycle-directness} to write $c=\sum_Qc_Q$ with
finitely many nonzero coordinates, and take their latest copy $Q$.
An edge of $c_Q$ whose endpoints are both outside $B(Q)$ cannot occur
in an earlier coordinate, since that would put its endpoints into the
earlier overlap.  If its coefficient were nonzero it would therefore
have nonzero coefficient in $c$.  But both its endpoints lie in $D_Q$,
so it was deleted, contrary to the support of $c$.  Consequently $c_Q$
is supported on edges with at least one endpoint in $B(Q)$.  These edges
form a forest by Proposition~\ref{geo:balls}, forcing $c_Q=0$, a
contradiction.
\end{proof}

\begin{proposition}[The exact incidence count]
\label{geo:incidences}
Let
\[
 \mathcal I=\{(Q,x):Q\text{ an indexed copy},\ x\in V(Q)\}.
\]
There is a canonical bijection
\begin{equation}
 \mathcal I\longrightarrow\Gbase\times V(Y),\qquad
 (Q,x)\longmapsto(x,\operatorname{type}_Q(x)),
 \label{geo:incidence-bijection}
\end{equation}
where the type is the base vertex under the unique source vertex of the
embedding.  It intertwines left translation of $\Gbase$ with left
translation of the first coordinate, with the base type fixed.
In particular, for a fixed $x\in\Gbase$ and $v\in V(Y)$ there is exactly
one incidence at $x$ of type $v$, even when $P$ is disconnected.
\end{proposition}

\begin{proof}
Embedding makes the source vertex of $x$ unique within a representative
$(C,\phi)$; deck alignment preserves its base vertex, so the type is
well-defined.  For given $(x,v)$ choose the lift $(v,1_F)$ and its component
$C$, and choose the unique label map from $C$ sending that lift to $x$.
This constructs an incidence.  Conversely a representative of any such
incidence has its source vertex equal to $(v,f)$ for some $f\in F$.
Apply $\delta_{f^{-1}}$ and change the map by precomposition with
$\delta_f$.  This is the equivalence defining indexed copies and brings
that source vertex to $(v,1_F)$.  Its component is now $C$, and its map
is the unique one just constructed.  This proves uniqueness.  Left
translation changes $x$ and not the source data, giving equivariance.
\end{proof}

The next two estimates control different costs of removing overlaps.  The
first sums over one trimming ball, allowing repeated lifts of a base
vertex.  The second sums the losses from all copies at one fixed Cayley
vertex, retaining the size of each trimming ball.  If nonnegative numbers
$a_v$ are assigned to base vertices and
$a_Q(x)=a_{\operatorname{type}_Q(x)}$, put
\[
 \mathcal S_a=\sum_{i\in\mathbb Z}\mathfrak w_i
                   \sum_{v\text{ in layer }i}a_v.
\]
Then, whenever this sum is finite,
\begin{align}
 \sum_{x\in B(Q)}a_Q(x)&\le\mathcal S_a,
 \label{geo:ball-weighted-sum}\\
 \sum_{Q:\,x\in V(Q)\setminus D_Q}|V(B(Q))|a_Q(x)
          &\le\mathcal S_a\qquad(x\in\Gbase).
 \label{geo:lost-weighted-sum}
\end{align}
For the first inequality, a base vertex in layer $i$ occurs among the
lifts in $B(Q)$ at most $|V(B(Q))|\le\mathfrak w_i$ times.  For the
second, every incidence being summed has $x\in B(Q)$, so its ball size
is at most the weight of its type's layer; the types occur at most once
by Proposition~\ref{geo:incidences}.  In particular the equivariant
incidence Hilbert space is canonically
\[
 \bigoplus_Q\ell^2(V(Q))\otimes\mathbb C^N
 \cong \ell^2(\Gbase)\otimes\ell^2(V(Y))\otimes\mathbb C^N.
\]
For a deck-equivariant family of transported projections with base
diagonal matrix sums $a_v$, its direct sum has unnormalized identity
diagonal sum $\sum_v a_v$.  This is a statement about all indexed copies;
the well-order and assigned domains need not be equivariant.

\subsection{Cone homology and exclusion of torsion}
\label{geo:torsion-subsection}

\begin{lemma}[Stabilizers of indexed copies]
\label{geo:stabilizers}
The stabilizer in $\Gbase$ of an indexed copy embeds into the subgroup
of $F$ preserving any representative source component.  In particular
that stabilizer has no nonidentity finite-order element.
\end{lemma}

\begin{proof}
Fix a representative $(C,\phi)$.  If $h$ stabilizes the indexed copy, its
definition means that there is a deck transformation $\delta_a$ preserving
$C$ such that
\[
 h\phi=\phi\delta_a\quad\text{on }C.
\]
It is unique because $\phi$ is injective and the deck action is free on
vertices.  If $h_1,h_2$ correspond to $a_1,a_2$, the displayed equality
gives $(h_1h_2)\phi=\phi\delta_{a_1a_2}$, so the correspondence is a
homomorphism.  If $a=1_F$ it implies that $h$ fixes a group vertex, so
$h=1_{\Gbase}$.  Hence it is injective.  Torsion-freeness follows from
that of $F$.  This argument does not assert that the full stabilizer is
trivial and does not require a component of $P$ to be the entire cover.
\end{proof}

Let $Z$ be the two-dimensional cellular complex obtained by attaching the
graph cone on each indexed copy to $X$.  Its one-skeleton is $\widehat X$.
For each positive edge $e:s(e)\to t(e)$ in $Q$, attach one triangular
cell $\tau_{Q,e}$ with boundary
\begin{equation}
 \partial\tau_{Q,e}=e+\sigma_{Q,s(e)}-\sigma_{Q,t(e)},
 \label{geo:cone-boundary}
\end{equation}
where $\sigma_{Q,x}$ is the oriented spoke from $c_Q$ to $x$.  Edges
belonging to different indexed copies give distinct triangular cells,
even when their base edge in $X$ is the same.  This is a cellular
attachment, so repeated incidences would be allowed; the embeddings
already proved in fact exclude loops in individual copies in this
construction.

\begin{lemma}[Acyclicity of the cone complex]
\label{geo:acyclic}
For every field $k$, the augmented cellular chain complex
\begin{equation}
 0\longrightarrow C_2(Z;k)\xrightarrow{\partial_2}C_1(Z;k)
 \xrightarrow{\partial_1}C_0(Z;k)\xrightarrow{\varepsilon}k
 \longrightarrow0
 \label{geo:acyclic-complex}
\end{equation}
is exact.
\end{lemma}

\begin{proof}
The graph $X$ is connected and each cone is attached to it, so exactness
at $C_0$ and $k$ is immediate.  To prove exactness at $C_1$, start with
a finite chain cycle in $\widehat X$.  For each center its spoke
coefficients sum to zero, since its boundary coefficient is zero.  Only
finitely many spoke coefficients are nonzero.  As its copy is connected,
a finite zero-sum chain of these copy vertices is the boundary of a
finite edge chain in that copy: join each supporting vertex to a fixed
one by a finite path and take the corresponding linear combination.
Formula~\eqref{geo:cone-boundary} therefore allows addition of a finite
combination of cone triangle boundaries to eliminate all spoke terms.
The resulting cycle lies in $X$.  By
Proposition~\ref{geo:cycle-directness}, it is a finite sum of copy cycles,
and each copy cycle is the boundary of the same linear combination of
its cone triangles, since its spoke terms cancel.  Thus every one-cycle
is a boundary.

For injectivity at $C_2$, a finite sum of triangles determines, for each
copy, the finite edge chain $c_Q$ of its triangle coefficients.  Vanishing
of the boundary on that copy's spokes says exactly $\partial c_Q=0$,
by \eqref{geo:cone-boundary}.  The remaining boundary in $X$ is
$\sum_Qc_Q$.  Its vanishing, together with directness in
\eqref{geo:cycle-decomposition}, implies $c_Q=0$ for every $Q$.
Each triangle corresponds to a distinct oriented edge in that copy,
so all triangle coefficients vanish.  There are no cells in higher
degrees.  All these arguments concern finite-support chains, also for
infinite copies and an infinite family of cones.
\end{proof}

\begin{proposition}[Torsion-freeness]
\label{geo:torsion}
The finitely generated group $\Gbase$ in \eqref{geo:presentation} is
torsion-free.
\end{proposition}

\begin{proof}
Left translation acts cellularly on $Z$.  A nontrivial finite cyclic
subgroup acts freely on its cells.  Indeed stabilizing an original group
vertex is impossible; stabilizing a formal generator edge preserves its
positive orientation and fixes its initial vertex.  Stabilizing a center,
a spoke, or a triangle entails stabilizing the indexed copy given by
that center.  Lemma~\ref{geo:stabilizers} excludes any nonidentity
finite-order element doing so.  This verifies cell freeness, not merely
freeness at generic points, and avoids an assumption about edge inversions.

If $\Gbase$ had a nonidentity torsion element, one of its powers would
generate a subgroup $C_p$ of prime order $p$.  Over $k=\mathbb F_p$,
choose orientations and orbit representatives of its cells.  Because
the action on cells is free, every $C_j(Z;k)$ is a free module over
$A=k[C_p]$, possibly of infinite rank.  By Lemma~\ref{geo:acyclic},
\eqref{geo:acyclic-complex} would be a free resolution of the trivial
$A$-module $k$ of length two.

The obstruction is the standard periodic-resolution argument for finite
cyclic groups; see Brown \cite[Examples~1.3 and~2.4(e)]{Brown2010}.
We give it here over $k$.  Writing $t$
for a generator of $C_p$ and $z=t-1$, one has
$A=k[z]/(z^p)$, and $k=A/(z)$.  The complex
\begin{equation}
 \cdots\xrightarrow{z}A\xrightarrow{z^{p-1}}A
 \xrightarrow{z}A\xrightarrow{z^{p-1}}A
 \xrightarrow{z}A\longrightarrow k\longrightarrow0
 \label{geo:periodic-resolution}
\end{equation}
is exact: in the basis $1,z,\ldots,z^{p-1}$, the kernel of multiplication
by $z$ is $(z^{p-1})$, and the kernel of multiplication by $z^{p-1}$ is
$(z)$.  Tensoring it over $A$ with $k$ makes every differential zero,
and leaves one copy of $k$ in every nonnegative degree.

Any two free resolutions of $k$ give the same homology after this tensor
operation.  Here is the comparison argument, including infinite free
ranks.  Lift the identity of $k$ to a chain map between the resolutions:
on each free basis vector in degree zero choose a preimage of its
augmentation; inductively, the image already assigned to a boundary
is a cycle, and exactness in the other resolution supplies its preimage.
Extend linearly on the free basis.  Construct a map in the opposite
direction in the same way.  Their composites lift the identity, and
their differences from the identity are null-homotopic: first lift the
degree-zero difference, whose augmentation vanishes, and then lift the
difference in each next degree after subtracting the already chosen
homotopy term.  At each step it is a cycle by the chain-map identity.
These chain homotopies persist on tensoring with $k$, proving the claimed
equality of homology.  Applied to the length-two resolution and
\eqref{geo:periodic-resolution}, it would make its homology both zero
and equal to $k$ in degrees greater than two.  This contradiction rules
out $C_p$, and hence all nonidentity torsion.

Finite generation follows directly from the finiteness of $S$ in
\eqref{geo:presentation}.  Combining the preceding results proves
Theorem~\ref{geo:main} in full.
\end{proof}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//geometry.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

The projection form of the Kadison--Kaplansky conjecture asserts that
the reduced group \(C^*\)-algebra of a torsion-free discrete group
contains no projections other than zero and the identity.
For a discrete group \(H\), this algebra is the operator-norm closure
of its left regular group algebra on \(\ell^2(H)\), and its canonical
trace is \(\tau_H(x)=\langle x\delta_1,\delta_1\rangle\).
The conjecture concerns scalar projections in this completion.
We prove the following result.

\begin{theorem}
\label{thm:main}
There exist a finitely generated torsion-free discrete group \(\Gproj\)
and a projection \(e\in C_r^*(\Gproj)\) such that
\[
 0<\tau_{\Gproj}(e)<\frac12.
\]
In particular, \(e\) is neither zero nor the identity.
\end{theorem}

Theorem~\ref{thm:main} resolves the projection form of the
Kadison--Kaplansky conjecture negatively.
The group is given by an infinite graphical presentation followed
by a free product with a finitely generated free group.
Every parameter is finite, although no useful bound on its size
is sought. The choices of labeling tables and edge voltages are
specified by finite distributions and proved simultaneous
existence statements.

\subsection*{Context}

Kadison traces his free-group projection problem to Kaplansky's question
about idempotents in simple unital \(C^*\)-algebras
\cite[pp.~222--224]{Kadison2008}.
Pimsner and Voiculescu proved the free-group case by computing the
\(K\)-theory of reduced crossed products by free groups
\cite[p.~131]{PimsnerVoiculescu1982}.
A central route to projectionlessness is integrality of the canonical
trace on \(K_0(C_r^*(H))\), where the trace on matrices is the
\emph{unnormalized} sum of the diagonal traces. For torsion-free groups,
the trace is integral on the image of reduced assembly with trivial
coefficients \cite[Theorem~0.3]{Luck2002},
\cite[Proposition~10.3]{BGW2016}. Surjectivity therefore gives scalar
projectionlessness. The survey \cite[Section~4.5.3]{GJV2019} explains the
distinct scalar and stable formulations.

This implication covers substantial classes of groups. Higson and
Kasparov proved Baum--Connes with coefficients for groups acting properly
by affine isometries on Hilbert space, including countable amenable groups
\cite[Theorem~9.1 and Corollary~9.2]{HK2001}. Mineyev and Yu established
the coefficient-free hyperbolic case and deduced the projection conjecture
for torsion-free subgroups of hyperbolic groups
\cite[Theorems~20--21]{MY2002}; Lafforgue later established the hyperbolic
case with arbitrary coefficients \cite[Theorem~0.4]{Lafforgue2012}.
The group constructed here has a different geometry: its presentation
encodes an infinite tower through embedded nilpotent covers.

For the group \(\Gproj\) in Theorem~\ref{thm:main}, the complex group
ring \(\mathbb C[\Gproj]\) has no scalar idempotents other than zero and
one \cite[Corollary~1.2(ii)]{OpenAIBass2026}. Thus the projection \(e\)
lies in the reduced completion but not in the group ring. The companion
also gives integer trace range on algebraic \(K_0(\mathbb C[\Gproj])\)
and on its image in \(K_0(C_r^*(\Gproj))\) under the comparison map
induced by inclusion \cite[Corollary~1.2(i) and Remark~10.1]{OpenAIBass2026}.
This does not assert integrality on the whole completed \(K_0\), where
\([e]\) has nonintegral trace.

Expanders also underlie the obstructions of Higson, Lafforgue,
and Skandalis for groupoids, coarse spaces, and group actions
with coefficients \cite{HLS2002}.
Their group-action statements concern assembly with coefficients,
whereas the trace obstruction here concerns trivial coefficients.

The later analysis of expander index obstructions by Willett and Yu
\cite{WY2012} gives another relevant spectral and coarse-geometric
precedent. Its group consequences likewise use coefficients.

Austin constructed rational group-ring operators with irrational
von Neumann kernel dimension \cite{Austin2013}.
Such a dimension alone does not imply that the kernel projection
lies in the reduced \(C^*\)-algebra.
Here a \emph{uniform separated spectral band} for a finite matrix
group-algebra operator in its regular representation gives
reduced-algebra membership by continuous functional calculus.
Scalar compression is applied after this matrix projection
has been obtained.

\subsection*{The obstruction and the proof strategy}

The first task is to obtain a matrix projection whose unnormalized trace
is strictly between zero and \(1/2\). A spectral projection in the group
von Neumann algebra is insufficient. We construct a finite self-adjoint
matrix \(T\) over the group algebra of a torsion-free group \(\Gbase\)
whose regular spectrum has a nonempty band near \(+1\), separated from
all remaining spectrum. Continuous functional calculus then places its
upper projection \(p_0\) in \(M_N(C_r^*(\Gbase))\). The difficult part
is to retain that gap while making the unnormalized trace small.

Begin with finite sets \(A_i,B_i\), indexed by \(i\in\mathbb Z\),
and bipartite incidences between \(A_i\) and \(B_k\) for \(|i-k|\le1\).
Their normalized operators preserve constant vectors and contract the
orthogonal complements by a common positive amount. Two versions of the
same tower serve different purposes: \(Y^0\) provides expansion and
\(Y^\#\) provides recovery of long paths from words in a finite alphabet.
They agree after deleting a uniformly small fraction of vertices.
The survivor \(Y\) is the base of a voltage cover \(P\): its vertices
are pairs \((v,f)\), with \(f\) in a torsion-free nilpotent group, and an
edge changes the second coordinate by its assigned group element.
The voltages give large girth and will also realize the required operators.

The analytic model is indexed by a central Fourier variable \(t\).
Twisted-convolution projections with effective parameters \(t\) and
\(t-\delta\), for a small positive \(\delta\), give a nonzero fiber
projection for \(0<t<\delta\). Normalized combinations of the constant
vectors on at most two adjacent layers produce an eigenvalue \(+1\),
while expansion separates all the other positive spectrum.
As \(t\) approaches either endpoint, these supporting layers move toward
an end of the tower and their fiber trace decreases. A finite magnetic
representation is essential to obtain small trace at both endpoints.
The result is a summable trace estimate even after the geometric weights
needed for intersections of copies have been included.

One edge voltage and one matrix weight per edge approximate this model
in the full regular norm. The same finite sampling construction enforces
the girth required for the group presentation. Long-word recovery then
embeds components of \(P\) as indexed copies in the Cayley graph of
\(\Gbase\). A common-root order confines every earlier intersection
with a copy to one finite tree ball. Removing those intersections leaves
disjoint domains; the remaining Cayley edges form a forest. The trace
estimate controls the loss in the upper spaces caused by the removed
balls, and a forest norm estimate controls the unused edges. Together
they preserve the separated band for the finite matrix \(T\).
For the trace estimate, we restrict a vector to every indexed copy and
project to its local upper space. This map is equivariant, although the
assignment of domains need not be. Disjointness and the same weighted
loss estimate bound its retained and removed parts. Its injectivity on
the global upper space then bounds the unnormalized trace by the total
trace of the local upper spaces.

The final passage to a scalar projection is a separate free-product
argument. Scale and comparison in reduced free products were studied by
Dykema and R{\o}rdam \cite{DR1998,DR2000,DRII2000}.
Recent work on selfless reduced free products and comparison by trace
\cite{FKOP2026,Robert2025} also underlies a conditional argument of
Sauers \cite{Sauers2026} that turns a nonintegral matrix projection over
\(C_r^*(H)\) into a scalar projection over \(C_r^*(H*\mathbb Z)\).
Such a reduction still requires an actual nonintegral matrix projection.
Here Proposition~\ref{scalar:compression} gives a direct quantitative
estimate at trace below \(1/2\): adjoining \(N\) free generators
compresses \(p_0\) to a scalar projection of the same trace. This gives
\(\Gproj=\Gbase*\mathbb F_N\), preserving finite generation and
torsion-freeness. The same small trace also places \([p_0]\) outside
the coefficient-free reduced assembly image for \(\Gbase\).

\subsection*{Organization}

Section~\ref{sec:scalar} reduces Theorem~\ref{thm:main} to the small
unnormalized matrix-trace target stated in Section~\ref{sec:setup}.
Sections~\ref{tow:section}--\ref{prune:section} construct the auxiliary
graph \(Y^0\) with expanding incidences, the gated graph \(Y^\#\)
with path recovery, and their common pruned induced subgraph \(Y\).

Section~\ref{geo:section} proves the embedding, overlap, forest, and
torsion conclusions under a girth hypothesis on the voltage cover
\(P\). Section~\ref{nil:section} constructs the deck group and the
small-trace fiber projections. Section~\ref{sam:section} identifies
the model's upper band, then chooses voltages and matrix weights
satisfying both the girth hypothesis and the regular operator
approximation.

Section~\ref{spec:section} passes the upper spectral band from the
model to the cover and then to the Cayley operator, and bounds its
unnormalized trace using all indexed copies. Section~\ref{sec:conclusion}
applies scalar compression to complete the proof.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//introduction.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//moving-band.tex ====
\begin{tikzpicture}[>=Latex,scale=0.94,every node/.style={font=\small}]
\fill[blue!7,rounded corners] (2.55,-0.5) rectangle (7.45,2.1);
\node[anchor=east] at (-0.2,1.4) {$A$-constants};
\node[anchor=east] at (-0.2,0) {$B$-constants};
\foreach \idx/\x/\lab in {0/1/{i-1},1/3.7/{i},2/6.3/{i+1},3/9/{i+2}} {
 \node[circle,draw,fill=white,inner sep=3pt] (a\idx) at (\x,1.4) {$a_{\lab}$};
 \node[circle,draw,fill=white,inner sep=3pt] (b\idx) at (\x,0) {$b_{\lab}$};
 \node at (\x,-0.85) {layer $\lab$};
}
\draw[blue!70!black,thick] (a1)--(b1);
\draw[blue!70!black,thick] (a1)--(b2);
\draw[blue!70!black,thick] (a2)--(b1);
\draw[blue!70!black,thick] (a2)--(b2);
\node[blue!70!black] at (5,2.45) {$J(t)=\{i,i+1\}$};
\draw[<-,gray] (0.5,-1.7)--(3.7,-1.7);
\draw[->,gray] (6.3,-1.7)--(9.5,-1.7);
\node[align=center] at (5,-1.7) {active layers};
\node at (1.7,-2.2) {$t\downarrow0:\ b(t)\to-\infty$};
\node at (8.3,-2.2) {$t\uparrow\delta:\ b(t)\to+\infty$};
\end{tikzpicture}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//moving-band.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//nilpotent.tex ====
\section{The nilpotent deck group and small-trace projections}\label{nil:section}

We construct a torsion-free nilpotent deck group that serves both parts
of the argument. Its voltages must detect the short holonomy words
required by the geometric construction, while its central Fourier fibers
must carry nonzero projections of small weighted identity trace. We first
fix the word detector, then build the group and the projection field.
Section~\ref{sam:section} chooses the actual edge voltages and matrix
weights that meet the geometric and operator requirements simultaneously.

The constants \(C_*\) and \(C_\bullet\) in the geometric construction are
fixed throughout this section. Every constant below is independent of the
layer index and, unless explicitly a function of \(d\), independent of
\(\ell\). Write \(\ee(v)=\exp(2\pi\mathrm i v)\).
All discrete Hilbert spaces have counting measure. Traces on auxiliary
matrix indices use their ordinary sum, without division by their number.

\subsection{A uniform detector for short walks}

\begin{lemma}[Rank bound]\label{nil:rank}
Let \(S\) be a finite connected metric graph with positive edge lengths,
total length at most \(Cg\), and girth greater than \(g>0\).
Girth means the infimum of lengths of embedded circles, with value
\(+\infty\) for a tree. Put
\[
 B(C)=\max\{1,\lceil20C\rceil\}^{\,2}.
\]
Then \(b_1(S)\le B(C)\). Vertices of degree one and parallel edges are
allowed.
\end{lemma}

\begin{proof}
Give \(S\) its intrinsic path metric and put \(a=g/10\).
Choose a maximal \(a\)-separated set \(\mathcal P\), which is an \(a\)-net.
If it has at least two elements, each \(p\in\mathcal P\) is joined to
another by a shortest path of length at least \(a\), so its open ball
of radius \(a/2\) contains an interval of length \(a/2\). Those balls
are disjoint. Therefore
\[
 |\mathcal P|a/2\le \operatorname{length}(S)\le Cg,
 \qquad |\mathcal P|\le\max\{1,\lceil20C\rceil\}.
\]
The second bound also covers the singleton case.

Form an abstract graph \(K\) on \(\mathcal P\), with one edge for each
distinct pair at distance at most \(3a\), and map each edge to a
chosen shortest path in \(S\). This graph is connected and its map
surjects on fundamental groups, using a connector for basepoints.
Indeed subdivide a path of \(S\) into segments of length at most \(a\),
choose net points within distance \(a\) of their endpoints, and replace
each segment by the chosen path between its two net points. Each
replacement exists. The original segment, its two endpoint connectors,
and its replacement form a closed path of length at most \(6a<g\).
Such a path is null-homotopic: after subdivision and cancellation
of backtracks, a nonempty reduced closed path in a graph contains
an embedded circle no longer than itself. Thus replacement preserves
homotopy relative to the connected endpoints. Applying this to paths
and then loops proves the two assertions about \(K\), and hence
\[
 b_1(S)\le b_1(K)\le|\mathcal P|^2\le B(C).
\]
\end{proof}

\begin{lemma}[Bounded pivot words and a matrix detector]\label{nil:word}
Let \(W\) be a nonempty cyclically nonbacktracking closed walk in a
unit-edge graph of girth greater than \(g\), with length at most \(Cg\).
There is a spanning tree of its support such that, after its edge
variables are set equal to the identity, the holonomy word has a
nontrivial free reduction of length at most
\[
 L(C)=\lceil3CB(C)\rceil.
\]
If \(m>L(C)\), that word is not an identity on the group \(U_{\mathbb Z}\)
of upper unitriangular \(m\)-by-\(m\) integer matrices.
Every entry of any word in generic upper unitriangular matrices and
their inverses is a polynomial of total degree at most \(m-1\) in their
upper entries. Nonidentity is preserved if each edge variable is
multiplied on either side by a fixed upper unitriangular matrix.
\end{lemma}

\begin{proof}
Let \(S\) be the support and \(b=b_1(S)\). A degree-one vertex would
force a backtrack, including at the cyclic join, so \(S\) has minimum
degree at least two and \(b\ge1\). If all degrees are two, regard the
circle as one arc with a chosen endpoint. Otherwise suppress
degree-two vertices. Euler's formula and minimum degree three give
at most \(2b-2\) vertices and \(3b-3\) edges. In either case there
are at most \(3b\) arcs.

Arcs shorter than \(g/(3b)\) form a forest, since a circuit made of
them would have length less than \(g\). Extend this forest to a
spanning tree of the suppressed graph. From each complementary arc
delete one original edge, called its pivot. The remaining original
graph is a spanning tree \(T\). Cyclically start \(W\) at a retained
vertex. Nonbacktracking forces it to traverse each arc it enters
completely. Every pivot crossing belongs to a distinct traversal
of an arc of length at least \(g/(3b)\). There are consequently
at most \(3bC\le L(C)\) pivot crossings, by Lemma~\ref{nil:rank}.
Collapsing \(T\) is a homotopy equivalence to a bouquet of circles.
Its image word is nontrivial since a nonempty cyclically reduced
path in a graph is not null-homotopic. Free reduction cannot
increase the length.

For the matrix assertion we use a finite matrix truncation of the Magnus
power-series expansion \cite{Magnus1935}. Write a nontrivial reduced word
in maximal syllables:
\[
 w=x_{j_1}^{a_1}\cdots x_{j_k}^{a_k},\qquad
 a_s\in\mathbb Z\setminus\{0\},\quad j_s\ne j_{s+1}.
\]
In formal noncommutative power series set \(x_j=1+Z_j\), using
\((1+Z)^a=\sum_{h\ge0}\binom ah Z^h\) also for negative \(a\).
The coefficient of \(Z_{j_1}\cdots Z_{j_k}\) is
\(\prod_s a_s\ne0\). A contributing syllable supplies at most one
letter, since the target has no equal adjacent letters; thus all
\(k\) syllables must supply exactly one. For \(m>k\), put
\[
 Z_j=\sum_{\{s:j_s=j\}}E_{s,s+1}\in M_m(\mathbb Z).
\]
The \((1,k+1)\) entry detects exactly that monomial. It follows that
\(w(1+Z_j)\ne1\). Here \(k\le\operatorname{length}(w)<m\).

The inverse of \(1+X\), for strictly upper triangular \(X\), is
\(\sum_{h=0}^{m-1}(-X)^h\). Every contributing product in any
matrix entry follows strictly increasing matrix indices, and hence
uses at most \(m-1\) upper-entry variables regardless of word length.
This proves the degree bound. Multiplication by a fixed
unitriangular matrix is a polynomial coordinate bijection with
polynomial inverse. Separate such changes in the edge variables
preserve a nonzero polynomial.
\end{proof}

Fix once and for all \(m\ge2\) with
\begin{equation}\label{nil:choice-m}
 m>L(201C_*).
\end{equation}
Indeed a walk based in layer \(i\) of length at most
\(C_*H_i/\log_2d\) stays in layers \(j\) with
\(H_j\ge H_i-C_*H_i/\log_2d\). For sufficiently large \(\ell\),
the tower girth property therefore makes every circuit in its
support longer than \(H_i/(201\log_2d)\). Lemma~\ref{nil:word},
with this \(g\) and \(C=201C_*\), applies. Lemma~\ref{nil:rank}
can later be applied with any other fixed \(C\), without changing \(m\).

\subsection{An integer affine group and its central product}

The short-word test only requires a quotient onto \(U_{\mathbb Z}\),
so we can add coordinates without losing that detector. The affine
extension below is chosen for a second purpose: it admits a twisted
representation in which Schwartz coefficient functions correspond to
Schwartz integral kernels. A rank-one kernel then gives a continuous
convolution projection, which we will sample on integer coordinates.

Let \(\mathfrak n\) be the strictly upper triangular real
\(m\)-by-\(m\) matrices, let \(n_{\mathrm f}=m(m-1)/2\), and let
\(U=1+\mathfrak n\). Write \(x_{ij}\) for the coordinates on
\(\mathfrak n\) and \(\lambda_{ij}\) for their dual coordinates.
For \(b\in U\), put
\[
 A_b(x)=b(1+x)-1=(b-1)+bx,\qquad L_b(x)=bx.
\]
The maps \(A_b\) form an affine action: \(A_bA_{b'}=A_{bb'}\).
Let \(E\) be the semidirect product of additive affine functions
on \(\mathfrak n\) by \(U\), acting by \(f\mapsto f\circ A_{b^{-1}}\).
Use coordinates \((\lambda,b,z)\), with affine function
\(x\mapsto z+\lambda(x)\). Explicitly,
\begin{align}
 (\lambda,b,z)(\mu,b',w)
 &=\bigl(\lambda+\mu\circ L_{b^{-1}},bb',
             z+w+\mu(b^{-1}-1)\bigr),\label{nil:affine-law}\\
 (\lambda,b,z)^{-1}
 &=\bigl(-\lambda\circ L_b,b^{-1},
             -z-\lambda(b-1)\bigr).\label{nil:affine-inverse}
\end{align}
Constants \((0,1,z)\) are central. Their quotient \(\bar E\) has
coordinates \(\gamma=(\lambda,b)\) and section
\(\sigma(\gamma)=(\lambda,b,0)\). For \(\eta=(\mu,b')\), put
\[
 \gamma\eta=(\lambda+\mu\circ L_{b^{-1}},bb'),\qquad
 c(\gamma,\eta)=\mu(b^{-1}-1).
\]
Then
\begin{equation}\label{nil:cocycle}
 \sigma(\gamma)\sigma(\eta)
       =(0,1,c(\gamma,\eta))\sigma(\gamma\eta),
\end{equation}
and associativity gives
\[
 c(\gamma,\eta)+c(\gamma\eta,\zeta)
   =c(\eta,\zeta)+c(\gamma,\eta\zeta).
\]
The cocycle is normalized at the identity, and
\(c(\gamma,\gamma^{-1})=c(\gamma^{-1},\gamma)\).

\begin{lemma}[Grading, Haar measure, and integer coordinates]\label{nil:grading}
The laws above are polynomial with integer coefficients in
\((\lambda,b-1,z)\). Give the coordinates weights
\[
 \operatorname{wt}(b-1)_{ij}=j-i,\quad
 \operatorname{wt}\lambda_{ij}=m-(j-i),\quad
 \operatorname{wt}z=m.
\]
Scaling weight \(a\) by \(y^a\) defines an automorphism \(\Delta_y\),
for \(y>0\), and \(c(\Delta_y\gamma,\Delta_y\eta)=y^m c(\gamma,\eta)\).
Lebesgue coordinate measure is both left and right Haar measure on
\(E\) and \(\bar E\); inversion preserves it. The Jacobian of
\(\Delta_y\) on \(\bar E\) is \(y^{mn_{\mathrm f}}\).
The integer coordinate sets \(E_{\mathbb Z}\) and \(\bar E_{\mathbb Z}\)
are countable nilpotent torsion-free groups.
\end{lemma}

\begin{proof}
The finite geometric series for \(b^{-1}\) proves polynomiality
and integrality in Equations~\ref{nil:affine-law} and
\ref{nil:affine-inverse}. The usual superdiagonal dilation is
an automorphism of \(U\). On affine functions use
\(f(x)\mapsto y^mf(\Delta_{y^{-1}}x)\).
The identity \(\Delta_y A_b=A_{\Delta_y b}\Delta_y\) shows that
these maps intertwine the actions, giving the stated automorphism
and cocycle homogeneity. Each paired quotient coordinate
\((\lambda_{ij},(b-1)_{ij})\) has total weight \(m\), proving
the dilation Jacobian.

Order the entries of \(b-1\) by superdiagonal. Left and right
multiplication on \(U\) are triangular coordinate transformations
with diagonal entries one, and thus determinant one. The map
\(L_b\) on \(\mathfrak n\) has determinant one: on each column it
uses a unitriangular initial submatrix of \(b\).
For a fixed left factor in Equation~\ref{nil:affine-law}, the
\(\mu\)-coordinates are transformed by \(L_{b^{-1}}^*\), the
\(b'\)-coordinates by left multiplication, and the \(w\)-coordinate
has coefficient one. Cross terms do not alter the determinant.
For a fixed right factor, the \(\lambda,z\) coordinates are
translated by functions of \(b\), and \(b\) is right multiplied.
This gives determinant one in both cases, also on omitting \(z\).
Inversion exchanges these equally normalized left and right Haar
measures, or has absolute Jacobian one by the same triangular
calculation.

In a coordinate of weight \(a\), multiplication is additive plus
a polynomial in smaller-weight coordinates. Each correction
monomial uses both factors, by homogeneity and the identity laws.
The subgroups whose coordinates of weight less than \(a\) vanish
form a filtration \(E_{\ge a}\) with
\([E,E_{\ge a}]\subset E_{\ge a+1}\). Indeed multiplication
modulo weight \(a+1\) by an element of \(E_{\ge a}\) is simply
addition of its weight-\(a\) coordinates, so is central there.
All weights are at most \(m\), proving nilpotence, also in the
quotient and the integer groups. A positive power of an element
with first nonzero weight \(a\) multiplies its weight-\(a\) vector
by that positive integer. It cannot become zero. This proves
torsion-freeness.
\end{proof}

Take copies \(X_{\mathrm f},Y_{\mathrm f}\) of \(E_{\mathbb Z}\);
these are factor names, unrelated to the graphs. Define
\begin{equation}\label{nil:deck}
 F=(E_{\mathbb Z}\times E_{\mathbb Z})/
       \{((0,1,k),(0,1,-k)):k\in\mathbb Z\}.
\end{equation}
Equivalently its coordinates are
\((\gamma_X,\gamma_Y,z)\in\bar E_{\mathbb Z}^2\times\mathbb Z\),
with law
\[
 (\gamma_X,\gamma_Y,z)(\eta_X,\eta_Y,w)
 =(\gamma_X\eta_X,\gamma_Y\eta_Y,
       z+w+c(\gamma_X,\eta_X)+c(\gamma_Y,\eta_Y)).
\]
The coordinate map from the product sums its two central integers
and has precisely the kernel in Equation~\ref{nil:deck}.
The opposite-pair subgroup is primitive in \(\mathbb Z^2\), so
the surviving coordinate is an integer coordinate with no finite
quotient. The inherited grading and first-nonzero-weight proof
show directly that \(F\) is countable, nilpotent, and torsion-free.
Projection to the upper unitriangular coordinate of either factor
is a surjective group homomorphism to \(U_{\mathbb Z}\).
Write \(z_0=(1,1,1)\) for the common central generator and
\(\sigma_X,\sigma_Y\) for the two section maps into \(F\).

\subsection{Central Fourier transform and equivariant trace}

Use right translations \((\rho(g)\xi)(f)=\xi(fg)\).
The inversion unitary \((\mathcal I\xi)(f)=\xi(f^{-1})\) satisfies
\(\mathcal I\rho(g)\mathcal I=\lambda_F(g)\).
After inversion, Fourier transform the central coordinate by
\(\delta_j\mapsto(t\mapsto\ee(tj))\), with probability Lebesgue
measure on \(\mathbb T=\mathbb R/\mathbb Z\). This identifies
\begin{equation}\label{nil:fourier}
 \ell^2(F)\cong L^2\bigl(\mathbb T;
               \ell^2(\bar E_{\mathbb Z})\otimes
               \ell^2(\bar E_{\mathbb Z})\bigr).
\end{equation}
The central generator acts by \(\ee(t)\). In either factor, the
section acts by
\begin{equation}\label{nil:twisted-left}
 u_t(\gamma)\delta_\eta
       =\ee(t c(\gamma,\eta))\delta_{\gamma\eta}.
\end{equation}
Thus
\[
 u_t(\gamma)u_t(\eta)=\ee(t c(\gamma,\eta))u_t(\gamma\eta),
 \quad
 u_t(\gamma)^*=\ee(-t c(\gamma^{-1},\gamma))u_t(\gamma^{-1}).
\]
The two factor representations commute; the full quotient
cocycle is their sum. There are commuting right operators
\begin{equation}\label{nil:twisted-right}
 v_t(k)\delta_\eta=\ee(t c(\eta,k))\delta_{\eta k}.
\end{equation}
The cocycle identity proves that these commute with \(u_t\).
Under the Fourier transform of the central coordinate on the already
inverted copy of \(\ell^2(F)\), these are the images of right multiplication
by \(\sigma(k)\) on basis indices, namely \(\rho(\sigma(k)^{-1})\).
They permute quotient basis vectors up to unit phases.
Equivariance below means intertwining these right actions,
with the identity action on auxiliary indices. Untwisted left
or right regular equivariance obeys the same arguments.

\begin{lemma}[Equivariant trace and size]\label{nil:trace}
Let \(L\) be a countable group with a right action as above and let
\(I,J\) be finite or countable auxiliary sets. On
\(\mathcal H_I=\ell^2(L)\otimes\ell^2(I)\), define, for equivariant
positive bounded \(B\),
\[
 \operatorname{Tr}_I(B)
   =\sum_{i\in I}\langle B\delta_{(1,i)},\delta_{(1,i)}\rangle
       \in[0,\infty].
\]
For a bounded equivariant \(A:\mathcal H_I\to\mathcal H_J\),
\begin{equation}\label{nil:adjoint-square}
 \operatorname{Tr}_I(A^*A)=\operatorname{Tr}_J(AA^*).
\end{equation}
This trace is additive on positive operators, monotone, and
faithful on projections, and does not depend on the chosen
orthonormal basis of the auxiliary Hilbert space.
Equivariantly equivalent projections
have equal size. A bounded equivariant map injective on the
range of a projection \(P\), with image there contained in the
range of \(Q\), gives \(\operatorname{Tr}(P)\le\operatorname{Tr}(Q)\).
These statements apply to countable orthogonal sums and to each
Fourier fiber. Integrating a fiber trace gives the identity trace
before Fourier transformation.
\end{lemma}

\begin{proof}
For a matrix entry of \(A\), translation on the right by \(g^{-1}\)
gives
\[
 |A((g,j),(1,i))|=|A((1,j),(g^{-1},i))|.
\]
The phases have absolute value one and the auxiliary indices
are fixed. Tonelli's Theorem for countable nonnegative sums gives
\begin{align*}
 \operatorname{Tr}_I(A^*A)
 &=\sum_{i,j,g}|A((g,j),(1,i))|^2\\
 &=\sum_{j,i,g}|A((1,j),(g^{-1},i))|^2
  =\operatorname{Tr}_J(AA^*),
\end{align*}
including when the sums are infinite.
Additivity and monotonicity follow from positive diagonals.
Changing the auxiliary orthonormal basis is conjugation by an
equivariant unitary \(U\). Applying the adjoint-square identity
to \(UB^{1/2}\) gives
\(\operatorname{Tr}(UBU^*)=\operatorname{Tr}(B)\), proving
basis independence.
If a projection \(P\) has zero trace, then
\(\|P\delta_{(1,i)}\|^2=0\) for each \(i\).
Right translates span the Hilbert space, so \(P=0\).

The polar partial isometry of an equivariant bounded \(A\) is
equivariant: it is the strong limit of
\(A(|A|+1/n)^{-1}\). Its initial and final projections are those
of \((\ker A)^\perp\) and \(\overline{\operatorname{ran}A}\).
Apply Equation~\ref{nil:adjoint-square} to it. This proves
projection equivalence and, on applying it to \(AP\), the
comparison, because its initial projection is \(P\) and its
final projection is at most \(Q\). Countable orthogonal sums
are handled by summing identity diagonals.

A vector at central coordinate zero becomes a constant vector
field in Equation~\ref{nil:fourier}. Its diagonal is the integral
of its fiber diagonal. Sum nonnegative diagonals and use Tonelli
for the integrated assertion, including countably many
auxiliary indices.
\end{proof}

A measurable uniformly bounded field on a separable Hilbert
space defines a bounded operator on its circle \(L^2\) space
with norm the essential supremum of the fiber norms.
Integration proves the upper inequality; for the lower one,
use a countable dense set of unit vectors restricted to
measurable circle sets where their images exceed a given norm.
Products and adjoints act pointwise. Uniform polynomial
approximation gives pointwise continuous functional calculus
for uniformly bounded self-adjoint fields. Thus a fixed
separating continuous function or resolvent contour gives the
same projection before and after decomposition.

\subsection{A Schwartz convolution projection}

Schwartz matrix coefficients and rank-one operators for Heisenberg
representations are treated by Rieffel
\cite[Lemma~2.3, Corollary~2.4, and Theorem~2.18]{Rieffel1988}.
Here we derive the coefficient-to-kernel correspondence for the affine
group above, whose quotient \(\bar E\) is generally nonabelian.

Equip \(\bar E\) with the Haar measure of Lemma~\ref{nil:grading}.
Schwartz functions refer to its Euclidean coordinates
\((\lambda,b-1)\), or to the specified product coordinates.
For \(\epsilon\in\{1,-1\}\), define
\begin{align*}
 (J*_\epsilon K)(h)
  &=\int_{\bar E}J(a)K(a^{-1}h)
          \ee\bigl(\epsilon c(a,a^{-1}h)\bigr)\,da,\\
 J^{*_\epsilon}(h)
  &=\overline{J(h^{-1})}
          \ee\bigl(-\epsilon c(h^{-1},h)\bigr).
\end{align*}
The cocycle identity gives associativity and the involution
identities. The following kernel realization verifies the
operations and gives the idempotents we need.

\begin{lemma}[Schwartz idempotents]\label{nil:schwartz}
For each sign \(\epsilon\), there is a Schwartz function
\(J_\epsilon\) on \(\bar E\) with
\[
 J_\epsilon^{*_\epsilon}=J_\epsilon,\qquad
 J_\epsilon *_\epsilon J_\epsilon=J_\epsilon,\qquad
 \|J_\epsilon\|_{L^2(\bar E)}=1.
\]
\end{lemma}

\begin{proof}
For \(\gamma=(\lambda,b)\), define on \(L^2(\mathfrak n)\)
\[
 (\pi_\epsilon(\gamma)\phi)(x)
       =\ee(\epsilon\lambda(x))\phi(A_{b^{-1}}x).
\]
The Jacobian of \(A_b\) is one, so these operators are unitary.
Direct substitution in Equation~\ref{nil:affine-law} gives
\[
 \pi_\epsilon(\gamma)\pi_\epsilon(\eta)
       =\ee(\epsilon c(\gamma,\eta))\pi_\epsilon(\gamma\eta).
\]
For a Schwartz \(J\), its integrated operator is bounded by
\(\|J\|_1\). Denote partial Fourier transform with positive sign by
\[
 \widehat J_+(\xi,b)=\int_{\mathfrak n^*}
                         J(\lambda,b)\ee(\lambda(\xi))\,d\lambda.
\]
Changing variables
\[
 y=b^{-1}(1+x)-1,\qquad b=(1+x)(1+y)^{-1}
\]
in that integrated operator gives its kernel
\begin{equation}\label{nil:kernel}
 \mathcal K_J(x,y)=
   \widehat J_+\bigl(\epsilon x,(1+x)(1+y)^{-1}\bigr).
\end{equation}
For fixed \(x\), the change from \(b-1\) to \(y\) has absolute
Jacobian one: it is inversion followed by multiplication in \(U\),
in its Lebesgue Haar coordinates.

Partial Fourier transform is a bijection on Schwartz spaces
and an \(L^2\)-isometry in the displayed normalization
\cite[Chapter 8]{Folland1999}.
The coordinate map
\[
 (x,y)\longmapsto
       \bigl(\epsilon x,(1+x)(1+y)^{-1}-1\bigr)
\]
and its inverse are polynomial. Pullback by such a map preserves
Schwartz functions bijectively: derivatives introduce polynomial
factors, and its polynomial inverse bounds any polynomial weight
in the source by a polynomial weight in the target. Its absolute
Jacobian is one as just shown. Consequently
\(J\mapsto\mathcal K_J\) is a bijection from Schwartz coefficients
to Schwartz kernels and an \(L^2\)-isometry.
It is also injective as an operator realization. If a continuous
Schwartz kernel gives the zero operator, its pairings against
all products of compactly supported smooth test functions vanish,
and approximations to point masses show the kernel vanishes.

Twisted convolution and involution preserve Schwartz coefficients.
Indeed
\[
 (a,h)\longmapsto J(a)K(a^{-1}h)
          \ee\bigl(\epsilon c(a,a^{-1}h)\bigr)
\]
is jointly Schwartz. The map \((a,h)\mapsto(a,a^{-1}h)\) is a
polynomial bijection with polynomial inverse, and differentiation
of the phase introduces only polynomial factors. Integration in
\(a\) preserves rapid decay of all derivatives in \(h\).
The same reasoning with inversion proves the involution assertion.
Absolute integrability permits Fubini, so the integrated
representations carry these operations to products and adjoints.

Choose \(\phi(x)=2^{n_{\mathrm f}/4}\exp(-\pi|x|^2)\), of \(L^2\)
norm one. The orthogonal projection onto its span has Schwartz
kernel \(\phi(x)\overline{\phi(y)}\). Let \(J_\epsilon\) be its
unique coefficient under Equation~\ref{nil:kernel}.
Product and adjoint correspondence and injectivity give the two
identities. The \(L^2\)-isometry gives
\[
 \|J_\epsilon\|_2^2
   =\int|\phi(x)|^2|\phi(y)|^2\,dx\,dy=1.
\]
\end{proof}

\subsection{The lattice limit in both coefficient norms}

Put \(L=\bar E_{\mathbb Z}\). For real \(\theta\), define
\(C_r^*(L,\theta c)\) to be the norm closure on \(\ell^2(L)\)
of the span of the operators in Equation~\ref{nil:twisted-left}
with \(t=\theta\). For \(a\in\ell^1(L)\), the series
\(\sum_\gamma a_\gamma u_\theta(\gamma)\) converges in norm
with norm at most \(\sum_\gamma|a_\gamma|\).
Its coefficient \(\ell^2\) norm equals its norm on \(\delta_1\),
because \(c(\gamma,1)=0\).
The identity diagonal is its identity coefficient and is the
trace of Lemma~\ref{nil:trace}. In particular
\(\operatorname{tr}_\theta(A^*A)=\|A\delta_1\|^2\).

The two coefficient norms have different roles in passing from the
continuous idempotent to the lattice. The \(\ell^1\) defect controls
the operator-norm correction by functional calculus. The \(\ell^2\)
defect, compared with the sampled coefficient norm, prevents the
resulting projection from being zero, while the squared \(\ell^2\)
size bounds its identity trace. The next lemma supplies both controls.

\begin{lemma}[Weighted rectangular Riemann sums]\label{nil:riemann}
For positive integer weights \(w_1,\ldots,w_k\), set
\[
 \mathcal L_y=\prod_{a=1}^k y^{w_a}\mathbb Z,\qquad
 v_y=y^{w_1+\cdots+w_k},\qquad 0<y\le1.
\]
For Schwartz \(f\) on \(\mathbb R^k\),
\[
 v_y\sum_{a\in\mathcal L_y}f(a)\longrightarrow
                   \int_{\mathbb R^k}f(x)\,dx.
\]
The corresponding normalized sums of absolute values are
uniformly bounded. For jointly Schwartz \(G(a,h)\) on
\(\mathbb R^k\times\mathbb R^k\), and every \(M\),
\begin{equation}\label{nil:riemann-weighted}
 \sup_h(1+|h|)^M
 \left|v_y\sum_{a\in\mathcal L_y}G(a,h)
                    -\int G(a,h)\,da\right|\le C_{G,M}y.
\end{equation}
\end{lemma}

\begin{proof}
Partition Euclidean space into cells
\(Q_a=a+\prod_j[0,y^{w_j})\).
They have volume \(v_y\) and diameter at most \(\sqrt{k}y\).
For \(x\in Q_a\), the Mean Value Theorem gives
\[
 |G(a,h)-G(x,h)|
     \le \sqrt{k}y\sup_{z\in Q_a}|\nabla_aG(z,h)|
     \le C_{K,M}y(1+|a|)^{-K}(1+|h|)^{-M}.
\]
On each cell \(1+|a|\) and \(1+|x|\) are uniformly comparable.
For \(K>k\), it follows that
\[
 v_y\sum_{a\in\mathcal L_y}(1+|a|)^{-K}
     \le C\int_{\mathbb R^k}(1+|x|)^{-K}\,dx<\infty.
\]
Integrating the cellwise difference and summing proves
Equation~\ref{nil:riemann-weighted}. The same comparison
proves the absolute-value bound and the assertion without \(h\).
\end{proof}

\begin{proposition}[Nonzero discrete twisted projections]\label{nil:projection}
There are constants \(\theta_0>0,C<\infty\), depending only on
\(m\) and the two fixed Schwartz functions, with the following
properties. For \(0<|\theta|<\theta_0\), set
\[
 \epsilon=\operatorname{sign}\theta,\quad
 y=|\theta|^{1/m},\quad
 \nu=y^{mn_{\mathrm f}}=|\theta|^{n_{\mathrm f}},
\]
and define the norm-convergent series
\begin{equation}\label{nil:K}
 K_\theta=\sum_{\gamma\in L}
          \nu J_\epsilon(\Delta_y\gamma)u_\theta(\gamma).
\end{equation}
It is self-adjoint and satisfies, uniformly for both signs,
\begin{align}
 \|K_\theta\|_{\mathrm{coeff},1}&\le C,&
 \|K_\theta\|_{\mathrm{coeff},2}&\asymp\sqrt{\nu},
                                               \label{nil:Knorms}\\
 \|K_\theta^2-K_\theta\|_{\mathrm{coeff},1}&\le Cy,&
 \|K_\theta^2-K_\theta\|_{\mathrm{coeff},2}&\le Cy\sqrt{\nu}.
                                               \label{nil:defects}
\end{align}
Continuous functional calculus in \(C_r^*(L,\theta c)\) gives a
projection \(p_\theta\) with
\begin{equation}\label{nil:projection-bounds}
 \|p_\theta-K_\theta\|\le Cy,\qquad
 0<\operatorname{tr}_\theta(p_\theta)\le C|\theta|^{n_{\mathrm f}}.
\end{equation}
These fields are measurable on the intervals \(0<|\theta|<\theta_0\).
\end{proposition}

\begin{proof}
In quotient coordinates, \(\Delta_y L\) is exactly the
rectangular lattice with coordinate spacings prescribed by the
weights. Its cell volume is \(\nu\), by Lemma~\ref{nil:grading}.
Lemma~\ref{nil:riemann}, applied to \(J_\epsilon\) and
\(|J_\epsilon|^2\), proves Equation~\ref{nil:Knorms}; the
normalized squared sum tends to \(\|J_\epsilon\|_2^2=1\).
Self-adjointness is exact. We have
\[
 J_\epsilon(h)=\overline{J_\epsilon(h^{-1})}
                         \ee(-\epsilon c(h^{-1},h)),
\]
\(\Delta_y(\gamma^{-1})=(\Delta_y\gamma)^{-1}\), and
\[
 \epsilon c(\Delta_y\gamma^{-1},\Delta_y\gamma)
                         =\theta c(\gamma^{-1},\gamma).
\]
Together with the adjoint formula for \(u_\theta\), these
identities prove \(K_\theta^*=K_\theta\) coefficientwise.

For output \(\gamma\), let \(h=\Delta_y\gamma\).
The coefficient of \(K_\theta^2\), divided by \(\nu\), is
\[
 \nu\sum_{a\in\Delta_yL}
   J_\epsilon(a)J_\epsilon(a^{-1}h)
       \ee\bigl(\epsilon c(a,a^{-1}h)\bigr).
\]
Its integral counterpart is \(J_\epsilon(h)\).
The summand is a fixed jointly Schwartz function of \((a,h)\),
as proved in Lemma~\ref{nil:schwartz}. Let \(D_y(h)\) denote
the difference between this sum and \(J_\epsilon(h)\).
Equation~\ref{nil:riemann-weighted} gives
\[
 |D_y(h)|\le C_M y(1+|h|)^{-M}
\]
for every \(M\). The defect coefficient is
\(\nu D_y(\Delta_y\gamma)\). The uniform lattice-sum bound
therefore gives
\[
 \nu\sum_{h\in\Delta_yL}|D_y(h)|\le Cy,\qquad
 \nu^2\sum_{h\in\Delta_yL}|D_y(h)|^2\le Cy^2\nu,
\]
which proves both estimates in Equation~\ref{nil:defects}.

Coefficient \(\ell^1\) controls operator norm, so
\(\|K_\theta^2-K_\theta\|\le Cy\).
For small \(y\), the real spectrum of \(K_\theta\) lies in
\([-1/4,1/4]\cup[3/4,5/4]\), by
\(|\lambda^2-\lambda|\le Cy\) on its spectrum.
Apply one fixed continuous real function equal to zero on
the first interval and one on the second. It gives a
self-adjoint projection \(p_\theta\) in the actual norm-closed
twisted algebra. On these intervals the difference of this
function from \(\lambda\) is \((\lambda^2-\lambda)f(\lambda)\),
where respectively
\[
 f(\lambda)=(1-\lambda)^{-1},\qquad f(\lambda)=-\lambda^{-1}.
\]
Since \(|f|\le2\), functional calculus proves the operator
approximation in Equation~\ref{nil:projection-bounds}.

The upper part is nonempty. If \(p_\theta=0\), then
\(\|K_\theta\|\le1/4\), so
\[
 \|(K_\theta^2-K_\theta)\delta_1\|
 \ge \|K_\theta\delta_1\|-\|K_\theta^2\delta_1\|
 \ge\tfrac34\|K_\theta\delta_1\|.
\]
This contradicts Equations~\ref{nil:Knorms} and
\ref{nil:defects} for small \(y\).
On the two spectral intervals, scalar functional calculus
also gives \(0\le p_\theta\le4K_\theta^2\). Hence
\[
 \operatorname{tr}_\theta(p_\theta)
 \le4\operatorname{tr}_\theta(K_\theta^2)
 =4\|K_\theta\delta_1\|^2\le C\nu.
\]
Strict positivity follows from Lemma~\ref{nil:trace} and the
nonzero projection just established.

Every matrix entry of every finite partial sum defining
\(K_\theta\) is measurable in \(\theta\). Pointwise
coefficient \(\ell^1\) convergence yields strong measurability
on the separable Hilbert space; the common bound yields a
bounded measurable field. Uniform polynomial approximation
of the one chosen continuous spectral function on a common
compact interval gives measurability of \(p_\theta\).
\end{proof}

\begin{lemma}[Compact templates and relative scale changes]\label{nil:templates}
The functions \(J_\epsilon\) in Equation~\ref{nil:K} may be
replaced by smooth compactly supported \(J_\epsilon^\flat\)
at arbitrarily small uniform coefficient \(\ell^1\) cost
for sufficiently small \(|\theta|\).
For fixed such a function, replacing \(\alpha>0\) by \(\beta>0\),
where \(|\log(\alpha/\beta)|\le a\le1\), changes the coefficients
\[
 \alpha^{n_{\mathrm f}}
     J_\epsilon^\flat(\Delta_{\alpha^{1/m}}\gamma)
\]
by \(\ell^1\) amount at most \(Ca\), uniformly for sufficiently
small \(\alpha,\beta\). These coefficient bounds imply operator
bounds when evaluated using any family of unitaries.
\end{lemma}

\begin{proof}
Multiply \(J_\epsilon\) by a smooth cutoff equal to one on a
large coordinate ball. Schwartz decay and the cell comparison
of Lemma~\ref{nil:riemann} make the normalized lattice sum of
the discarded tail arbitrarily small, uniformly as the mesh
goes to zero. For the second assertion set \(r=\alpha/\beta\)
and \(h=\Delta_{\beta^{1/m}}\gamma\). After extracting
\(\beta^{n_{\mathrm f}}\), the difference is
\[
 r^{n_{\mathrm f}}J_\epsilon^\flat(\Delta_{r^{1/m}}h)
                       -J_\epsilon^\flat(h).
\]
For \(e^{-1}\le r\le e\) it is supported on one compact set,
and its supremum is at most \(C|\log r|\) by the Mean Value
Theorem. The volume-normalized number of lattice points there
is uniformly bounded by the cell comparison. Sum to conclude.
\end{proof}

\subsection{A finite magnetic factor and its exact multiplicity}

A single small twist makes the trace small near one endpoint of a
character interval. We need it small near both endpoints, because the
supporting layers will escape in opposite directions there. The finite
representation below shifts the second effective twist from \(t\) to
\(t-\delta\). The product projection then has trace bounded by a
constant times \(N t^{n_{\mathrm f}}(\delta-t)^{n_{\mathrm f}}\).
The prescribed scales will make this small even after the ordinary
matrix multiplicity \(N\) and the geometric weights are included.


The global constant \(h_*=5003\) satisfies \(H_0\le h_*\log_2d\).
After \(m,C_*,C_\bullet\) are fixed, choose
\begin{equation}\label{nil:parameter-order}
 \begin{split}
 &E_0>10(C_*+C_\bullet+1),\\
 &A_*\in\mathbb N,\qquad A_*/m>10E_0h_*,\\
 &\kappa>0,\qquad
        \kappa/m>10(E_0+A_*+C_\bullet+1).
 \end{split}
\end{equation}
Set
\begin{equation}\label{nil:q}
 q=d^{A_*},\qquad \delta=q^{-1},\qquad
 N=q^{n_{\mathrm f}},\qquad I=H_0.
\end{equation}
The numbers \(q,N,I\) are integers. We take \(\ell\) large enough
that \(\delta<\theta_0\) in Proposition~\ref{nil:projection}.

Let \(\mathfrak n_q=\mathfrak n_{\mathbb Z/q\mathbb Z}\), a set of
\(q^{n_{\mathrm f}}=N\) elements. On
\(\mathbb C^N=\ell^2(\mathfrak n_q)\) define a representation of
the second copy \(E_{\mathbb Z}\) by
\begin{equation}\label{nil:R}
 (R(\lambda,b,z)\psi)(x)
   =\ee\bigl(-(z+\lambda(x))/q\bigr)
               \psi\bigl(b^{-1}(1+x)-1\bigr).
\end{equation}
Use any integer representative in the phase; the exponential
is independent of the representative. All arguments on the
right are reduced modulo \(q\). The affine map is a permutation
and the phase has modulus one, so the operator is unitary.
Substitution into Equation~\ref{nil:affine-law}, exactly as for
\(\pi_\epsilon\), proves that \(R\) is a group representation.
Its central character is
\[
 R(0,1,z)=\ee(-\delta z)1_N.
\]
Write \(R_\gamma=R(\sigma(\gamma))\).

Reduction of the integer polynomial laws gives a surjective
homomorphism \(L=\bar E_{\mathbb Z}\to\bar E_{\mathbb Z/q\mathbb Z}\).
Its kernel is
\[
 \Lambda_q=\{(\lambda,b)\in L:
       \lambda_{ij}\equiv(b-1)_{ij}\equiv0\pmod q\}.
\]
It has index \(q^{2n_{\mathrm f}}\) and coset representatives
with every coordinate in \(\{0,\ldots,q-1\}\).
Equation~\ref{nil:R} gives exactly
\begin{equation}\label{nil:congruence-R}
 R_y=1_N\qquad(y\in\Lambda_q).
\end{equation}

The next lemma is a form of twisted absorption
\cite[Lemma~2.4]{EchterhoffWilliams2012}. We track the quotient-identity
block in its conjugacy to retain the exact unnormalized matrix-trace
multiplicity.

\begin{lemma}[Absorption of the finite factor]\label{nil:absorption}
On the second quotient factor, define
\[
 V_t(\gamma)=u_t(\gamma)\otimes R_\gamma.
\]
Its multiplier is \(\ee((t-\delta)c)\), and it is unitarily
equivalent to \(N\) copies of \(u_{t-\delta}\).
The equivalence fixes the quotient-identity fiber, preserving
the identity trace with its unnormalized matrix sum.
Thus every coefficient sum, and every continuous spectral
function of it, has the same operator norm as its evaluation
in \(u_{t-\delta}\); the identity trace of a projection is
multiplied by \(N\).
\end{lemma}

\begin{proof}
The section law and the central character of \(R\) give
\[
 R_\gamma R_\eta
       =\ee(-\delta c(\gamma,\eta))R_{\gamma\eta}.
\]
Combining this with the multiplier of \(u_t\) proves the
multiplier assertion. Define a diagonal unitary by
\[
 W_t(\delta_\eta\otimes v)=\delta_\eta\otimes R_\eta v.
\]
Its formula is actually independent of \(t\). Then
\begin{align*}
 V_t(\gamma)W_t(\delta_\eta\otimes v)
  &=\ee(t c(\gamma,\eta))
                 \delta_{\gamma\eta}\otimes R_\gamma R_\eta v\\
  &=\ee((t-\delta)c(\gamma,\eta))
                 \delta_{\gamma\eta}\otimes R_{\gamma\eta}v\\
  &=W_t(u_{t-\delta}(\gamma)\otimes1_N)
                 (\delta_\eta\otimes v).
\end{align*}
So \(W_t^*V_t(\gamma)W_t=u_{t-\delta}(\gamma)\otimes1_N\).
As \(R_1=1_N\), \(W_t\) fixes \(\delta_1\otimes\mathbb C^N\)
pointwise, and conjugacy preserves the entire identity matrix
block. All conclusions follow first for finite sums and then
by norm limits and continuous functional calculus.
\end{proof}

In the full Fourier quotient space, amplified by \(\mathbb C^N\),
the first factor \(u_t\) and the second \(V_t\) commute. They are
equivariant for the original right quotient actions tensored
with \(1_N\).
For \(0<t<\delta\), both \(t\) and \(t-\delta\) fall in
Proposition~\ref{nil:projection}. Define
\begin{equation}\label{nil:pt}
 p(t)=p_t\otimes p_{t-\delta}[V_t].
\end{equation}
The second-factor notation means coefficient evaluation and
continuous functional calculus through Lemma~\ref{nil:absorption}.
Set \(p(t)=0\) outside this interval.
It is a measurable bounded field of equivariant projections.
Its identity trace \(s_p(t)=\operatorname{Tr}_{\mathrm{id},N}(p(t))\)
satisfies
\begin{equation}\label{nil:two-factor-trace}
 \begin{split}
 0<s_p(t)
 &=N\,\operatorname{tr}_t(p_t)
               \operatorname{tr}_{t-\delta}(p_{t-\delta})\\
 &\le C N t^{n_{\mathrm f}}(\delta-t)^{n_{\mathrm f}}
                  \qquad(0<t<\delta).
 \end{split}
\end{equation}
Factor the identity diagonals of the tensor factors and use
the identity-block assertion of Lemma~\ref{nil:absorption}
for the equality. In subsequent coefficient formulas, an
additional shared central power \(z_0^n\) acts by \(\ee(nt)1_N\).
The matrix \(R_\gamma\) is attached to a second-factor section
term as in \(V_t\); these prescriptions fix all phases.

\subsection{The layer window and its weighted trace}

\begin{lemma}[Smooth layer windows]\label{nil:window}
There is a smooth nonnegative \(h\), supported in \((-1,1)\)
and positive on \([-0.7,0.7]\), with
\(\sum_{i\in\mathbb Z}h(v-i)^2=1\) for every \(v\in\mathbb R\).
For \(0<t<\delta\), set
\begin{equation}\label{nil:h}
 b(t)=\kappa^{-1}\log_2\frac{t}{\delta-t},\qquad
 h_i(t)=h(b(t)-i),
\end{equation}
and extend \(h_i\) by zero to the rest of the circle.
Each extension is smooth. The active set
\(J(t)=\{i:h_i(t)\ne0\}\) has one or two elements, adjacent
if two, and \(\sum_i h_i(t)^2=1\) on the open interval.
With \(w_i=q^{-1}2^{-\kappa|i|}\), its support length and
derivatives satisfy
\[
 |\operatorname{supp}h_i|\le C_\kappa w_i,\qquad
 \|\partial_t^j h_i\|_\infty\le C_{\kappa,j}w_i^{-j}.
\]
On an active index \(i\), uniformly in \(\ell,i,t\),
\begin{align}
 \log_2(t^{-1})
 &=\log_2q+\kappa\max(-i,0)+O_\kappa(1),\label{nil:log-t}\\
 \log_2((\delta-t)^{-1})
 &=\log_2q+\kappa\max(i,0)+O_\kappa(1),\label{nil:log-tau}\\
 t(\delta-t)&\le 2^\kappa q^{-2}2^{-\kappa|i|}.
                                                  \label{nil:product}
\end{align}
\end{lemma}

\begin{proof}
Choose a smooth nonnegative bump \(\beta\), supported in
\((-0.9,0.9)\) and positive on \([-0.7,0.7]\), and put
\[
 h(v)=\beta(v)\left(\sum_{j\in\mathbb Z}\beta(v-j)^2\right)^{-1/2}.
\]
The denominator is positive, smooth, and one-periodic:
some translate lies in \([-1/2,1/2]\).
This proves the square-sum property. An interval of length
\(1.8<2\) contains at most two integers, necessarily adjacent.
This proves the assertion about active indices.
For each fixed \(i\), its support in \(b\)-coordinates lies
in a compact interval strictly inside the inverse image
of \((0,\delta)\); its \(t\)-support is therefore a compact
subset of \((0,\delta)\), so extension by zero is smooth.

The inverse map and its derivative are
\[
 t(b)=\delta\frac{2^{\kappa b}}{1+2^{\kappa b}},\qquad
 \delta-t(b)=\frac{\delta}{1+2^{\kappa b}},\qquad
 t'(b)=\kappa(\log2)\frac{t(b)(\delta-t(b))}{\delta}.
\]
For \(|b-i|\le0.9\), \(t'(b)\) is bounded above and below by
positive constants depending on \(\kappa\) times \(w_i\).
Every higher \(b\)-derivative is bounded by a constant times
\(w_i\), by differentiating the displayed rational function
and separating \(b\ge0\) from \(b\le0\).
The inverse-function differentiation formulas and the fixed
derivatives of \(h\) give the stated derivative bounds.
Integration of the bound on \(t'\) proves the support bound.

The exact logarithmic identities are
\[
 \log_2(t(b)^{-1})=\log_2q+\log_2(1+2^{-\kappa b}),\quad
 \log_2((\delta-t(b))^{-1})=\log_2q+\log_2(1+2^{\kappa b}).
\]
For real \(x\), \(\log_2(1+2^x)-\max(x,0)\) lies in \([0,1]\).
Using \(|b-i|<1\) proves Equations~\ref{nil:log-t} and
\ref{nil:log-tau}. Finally
\[
 \frac{2^{\kappa b}}{(1+2^{\kappa b})^2}\le2^{-\kappa|b|}
\]
and \(|b-i|<1\) prove Equation~\ref{nil:product}.
\end{proof}

\begin{proposition}[Uniform weighted trace bound]\label{nil:weighted}
For \(\mathfrak w_i=2^{C_\bullet H_i}\), define
\[
 \mu_\ell=\sup_{\substack{0<t<\delta\\i\in J(t)}}
                          \mathfrak w_i\,s_p(t).
\]
Then \(\mu_\ell\to0\) as \(\ell\to\infty\). More explicitly,
\begin{equation}\label{nil:weighted-estimate}
 \mathfrak w_i N t^{n_{\mathrm f}}(\delta-t)^{n_{\mathrm f}}
 \le C d^{-(A_*n_{\mathrm f}-C_\bullet h_*)}
          2^{-(\kappa n_{\mathrm f}-C_\bullet)|i|}
                  \qquad(i\in J(t)).
\end{equation}
The two exponents subtracted on the right are positive.
\end{proposition}

\begin{proof}
Using \(H_i=H_0+|i|\), \(H_0\le h_*\log_2d\),
\(N=q^{n_{\mathrm f}}\), and Equation~\ref{nil:product},
\begin{align*}
 2^{C_\bullet H_i}N\bigl(t(\delta-t)\bigr)^{n_{\mathrm f}}
 &\le 2^{\kappa n_{\mathrm f}}
       2^{C_\bullet H_0}q^{-n_{\mathrm f}}
       2^{-(\kappa n_{\mathrm f}-C_\bullet)|i|}\\
 &\le 2^{\kappa n_{\mathrm f}}
       d^{-(A_*n_{\mathrm f}-C_\bullet h_*)}
       2^{-(\kappa n_{\mathrm f}-C_\bullet)|i|}.
\end{align*}
Equation~\ref{nil:parameter-order}, with \(m\ge2\) and
\(n_{\mathrm f}\ge1\), implies
\(A_*n_{\mathrm f}>C_\bullet h_*\) and
\(\kappa n_{\mathrm f}>C_\bullet\).
Take the supremum and apply Equation~\ref{nil:two-factor-trace}.
\end{proof}

The dependency order for subsequent arguments is as follows.
Geometry fixes \(C_*,C_\bullet\); Equation~\ref{nil:choice-m}
then fixes \(m\), and Lemma~\ref{nil:schwartz} fixes two functions.
Equation~\ref{nil:parameter-order} fixes \(E_0,A_*,\kappa\).
All accuracies, compact truncations, relative logarithmic mesh
sizes, and finite norm-test exponents may now be chosen as
fixed constants. Only then is \(\ell\) increased. In particular
the trace estimate requires no accuracy varying with the layer,
and the word detector does not depend on subsequent moment orders.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//nilpotent.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{A torsion-free counterexample to the\\
Kadison--Kaplansky projection conjecture}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We construct a finitely generated torsion-free discrete group whose reduced
group \(C^*\)-algebra contains a projection other than zero and the identity.
This disproves the Kadison--Kaplansky projection conjecture.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/introduction}
\input{sections/scalar}
\input{sections/setup}
\input{sections/tower}
\input{sections/expansion}
\input{sections/pruning}
\input{sections/geometry}
\input{sections/nilpotent}
\input{sections/sampling}
\input{sections/spectral}
\input{sections/conclusion}
\clearpage
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//paper.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//preamble.tex ====
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs,bm}
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\usepackage{enumitem}
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  pdftitle={A torsion-free counterexample to the Kadison--Kaplansky projection conjecture},
  pdfauthor={OpenAI}
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\usepackage{etoolbox}
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==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//preamble.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//pruning.tex ====
\section{Pruning and unique recovery of long paths}
\label{prune:section}

To pass from the two towers to one graph, we need both a small deletion
fraction in every layer and long words that determine their paths.
We first prove that a word recovers padded stream prefixes at interior
positions in \(Y^\#\).  This does not yet identify the layer: the labels
determine its sign and residue but need not determine its absolute value,
and padding can conceal a shorter stream.  The subsequent deletions
resolve this ambiguity, make the gated and auxiliary graphs agree, and
remove short cyclic paths.

A path in this section is a finite edge walk,
possibly with repeated vertices.  Its radius at a specified position
is the smaller of the numbers of edges available in the two directions
from that position.  A nonbacktracking path never follows an edge
immediately by its reverse.  A closed path is cyclically
nonbacktracking when this also holds at its cyclic junction.

\begin{lemma}[Recovery of stream prefixes]
\label{prune:prefix-recovery}
Suppose two nonbacktracking paths in \(Y^\#\) have the same oriented
label word.  At corresponding \(A\)-positions of radius at least
\(2j+1\), both padded streams agree through length \(r+sj\), for
every \(j\geq0\).  At corresponding \(B\)-positions of radius at
least \(2j\), both padded streams agree through length \(sj\), for
every \(j\geq1\).  The compared positions need not be in the same
layer.
\end{lemma}

\begin{proof}
The orientation of a letter identifies the side of either endpoint.
Reading its underlying positive label gives the family, the serial
number and \(x=b\Vert x'\), and therefore both slots.  In particular,
the same word identifies the same slot names, diagonal coefficients,
spill maps, and family rules at corresponding positions.  It also
identifies the signs and residues modulo \(K\) of the layer indices,
although their absolute values may differ.

At an interior position the two incident slots are distinct.  Otherwise
the incoming and outgoing edges would be the unique edge at the same
slot, so the path would backtrack.  Write their evaluations on a current
block as
\[
 Q^a_h=v_h+\lambda_a u_h+S_a u_{h-1},\qquad
 Q^b_h=v_h+\lambda_b u_h+S_b u_{h-1}.
\]
Here \(u_h,v_h\) are the full padded current blocks, multiplication
uses their corresponding field, and the last terms are the spills
from the preceding block, extended by zero as needed.  For the first
block there is no preceding spill.  If the two evaluation blocks and
the preceding input block have been recovered, subtracting gives
\begin{equation}
 (\lambda_a-\lambda_b)u_h
   =Q^a_h-Q^b_h-(S_a-S_b)u_{h-1}.
 \label{prune:block-inversion}
\end{equation}
The coefficient difference is nonzero and hence invertible.  Thus
\(u_h\), followed by \(v_h\), is recovered.  Different spill
lengths for the two incident slots do not affect this argument.

Call the asserted \(A\)-statement at index \(j\) the assertion
\(\mathcal A_j\), and the \(B\)-statement \(\mathcal B_j\).
Interpret \(\mathcal B_0\) as the vacuous recovery of the empty
prefix.  At radius one on side \(A\), the first \(r\) bits of each
incident evaluation are present in its label \(x\), because
\(p=r+\alpha\geq r\).  Equation~\eqref{prune:block-inversion}
for the first block proves \(\mathcal A_0\).

Suppose \(j\geq1\).  At a \(B\)-position of radius \(2j\), both
neighbors satisfy \(\mathcal A_{j-1}\); centrally we already have
\(\mathcal B_{j-1}\).  For either incident edge, the second equation
of \eqref{tow:edge-equations} shows that to know \(Q_B\) through
length \(sj\), beyond the label prefix \(x'\), it suffices to know
the \(\mathcal J\)-output through length
\[
 sj-p'=r+s(j-1)-\beta\leq r+s(j-1).
\]
All source \(A\)-blocks involved are known in full at the neighbor.
Every key for the transformation of \(A\)-block \(j-1\), or an
earlier block, uses central \(B\)-blocks at most \(j-1\), together
with earlier blocks of that neighbor's \(A\)-stream.  These are all
known.  Consequently the indicated output prefix, including every
gate decision, agrees in the two paths.  Both central evaluation
prefixes therefore agree through \(sj\).  Their preceding spill
inputs are already known by \(\mathcal B_{j-1}\), so
\eqref{prune:block-inversion} recovers the next full \(B\)-block.
This proves \(\mathcal B_j\).

At an \(A\)-position of radius \(2j+1\), both neighbors satisfy
\(\mathcal B_j\), and centrally we have
\(\mathcal A_{j-1}\).  By the first equation of
\eqref{tow:edge-equations}, the evaluation prefix through \(r+sj\)
needs only the \(\mathcal F\)-output through
\[
 r+sj-p=sj-\alpha\leq sj.
\]
The required neighbor \(B\)-blocks through block \(j\) are fully
known.  The keys for such a block use central \(A\)-blocks only
through block \(j-1\), and preceding neighbor \(B\)-blocks.
These again are known, so both evaluation prefixes agree.  Subtract
the already known preceding spill and apply
\eqref{prune:block-inversion}.  This proves
\(\mathcal A_j\) and completes the alternating induction.

This proof uses full padded blocks.  A permutation may mix all the
bits of a block, but its entire source block is already known at the
neighbor before it is used.  Inactive bits cause no difficulty because
the equations are padded-stream equations.  When a rule lies beyond
one of the compared paths' actual cutoffs, it is still the same
family rule on both paths; equality of the preceding input blocks
gives equality of its key and gate.  This is precisely why
Lemma~\ref{prune:prefix-recovery} is proved in \(Y^\#\), with its
family-wide gated rules.  The
inequalities used above also include \(\alpha=0\), \(\beta=0\),
and a final partial block of any length \(1\leq t\leq s\).
\end{proof}

\begin{corollary}[Few short cyclic paths at one layer]
\label{prune:cycle-count}
Fix a layer and a side.  A nonempty cyclically reduced oriented label
word is the word of at most one based cyclically nonbacktracking
closed path with its base on that layer and side.  Consequently the
number of vertices of \(A_i\), or of \(B_i\), supporting such a
closed path of length at most \(g_i\) is at most
\[
 \sum_{a=1}^{\lfloor g_i\rfloor}(2m_\ell)^a.
\]
For all sufficiently large \(\ell\), this is a fraction at most
\(2^{-0.9H_i}\) of either vertex set, uniformly in \(i\).
\end{corollary}

\begin{proof}
Repeat two closed paths with the same word arbitrarily many times
in both directions.  Cyclic nonbacktracking ensures that every such
repetition is nonbacktracking.  At corresponding base positions,
Lemma~\ref{prune:prefix-recovery} recovers arbitrarily long prefixes.
Since the layer and side are fixed, this eventually recovers all
active coordinates of both vertices, making their base states equal.
Foldedness then determines their entire based paths.  Conversely a
cyclically nonbacktracking closed path has a cyclically reduced word:
inverse successive letters, including at the cyclic junction, would
force a backtrack by foldedness.

There are at most \((2m_\ell)^a\) words of length \(a\); counting
all words only enlarges the bound.  Both vertex-set sizes are at least
\(2^{H_i}\), and \eqref{tow:alphabet} gives
\[
 \frac{\log_2(2m_\ell)}{100\log_2 d}<\frac1{20}
\]
for large \(\ell\).  If \(\lfloor g_i\rfloor\geq1\), the
geometric sum is at most \(2(2m_\ell)^{g_i}\), so its fraction is
at most \(2^{1-0.95H_i}\leq2^{-0.9H_i}\).  If the sum is empty
the claim is immediate.  Since \(H_i\geq H_0\to\infty\), the
threshold is uniform over all layers.
\end{proof}

We now specify the deleted sets.  They are sets of vertices of the
common vertex set of \(Y^0\) and \(Y^\#\).

\paragraph{Gate deletions.}
At each vertex and each adjacent-pair slot evaluate the gate at that
pair's own final block, whether or not the current auxiliary rule
actually consults it.  Delete the vertex if any of these gates is
closed.  The inputs before the last block are determined without
using its current permutation, so this is a well-defined test on the
slot state, identical for the gated and auxiliary prescriptions.

At a fixed \(A\)-slot, conditional on \(x\), the coordinates
\(u_A\) and the tail of \(Q_A\) are independent uniform bits,
by the parametrization in Lemma~\ref{tow:matching-clusters}.
The triangular equations turn them bijectively into the interleaved
input \(u\)-values.  Thus all preceding input coordinates are
independent uniform bits.  The same reasoning at a fixed \(B\)-slot
uses \(x'\), its fixed bucket, \(u_B\), and the tail of \(Q_B\).
In particular the four full \(s\)-blocks preceding the cutoff are
jointly zero with probability exactly \(2^{-4s}\) at either slot.
The union bound and the maximum degree \(6D\) show that these
deletions have fraction at most
\begin{equation}
 \varepsilon_{\rm gate}=6D\,2^{-4s}.
 \label{prune:gate-budget}
\end{equation}

On vertices surviving these deletions the induced graphs from
\(Y^0\) and \(Y^\#\) are identical.  Indeed, from a surviving
slot state its current final gate is open, so the two endpoint
constructions agree.  All lower rules already agree.  This argument
works from either side of an edge and proves equality of the induced
edge sets, including multiplicities and labels.

\paragraph{Separation of heights.}
The next deletion distinguishes larger-layer vertices from the zero
padding of smaller-layer vertices with the same sign and residue.
For a vertex on either side at layer \(i\), consider every layer
\(i'\) with
\[
 \operatorname{sgn}(i')=\operatorname{sgn}(i),\qquad
 |i'|\equiv|i|\pmod K,\qquad |i'|<|i|.
\]
Put \(m=(|i|-|i'|)/K\geq1\) and
\begin{equation}
 q(i,i')=\min\{H_{i'},mK/2\}.
 \label{prune:zero-window}
\end{equation}
Delete the vertex if all its own \(u\)-bits immediately after that
side's cutoff at \(i'\), for these \(q(i,i')\) positions, are zero.
This interval consists of active bits at \(i\), because on either
side the length increases by exactly \(mK/2\) between these two
layers.  Its zero probability under uniform vertex measure is
\(2^{-q(i,i')}\).  No layers of distinct signs are tested, and
zero has no smaller layer with its sign.

The union bound gives, on every side and layer,
\begin{equation}
\begin{split}
 \varepsilon_{\rm sep}
 &\leq\sum_{m\geq1}2^{-mK/2}+\sum_{i'\in\mathbb Z}2^{-H_{i'}}\\
 &=\frac{2^{-K/2}}{1-2^{-K/2}}+3\,2^{-H_0}.
\end{split}
\label{prune:separation-budget}
\end{equation}
Here \(2^{-\min(a,b)}\leq2^{-a}+2^{-b}\), and the second sum
is evaluated using \(H_{i'}=H_0+|i'|\).  Thus the estimate stays
uniform even when the larger layer has arbitrarily many smaller
layers of its residue.

\paragraph{Short-cycle deletions.}
At layer \(i\), delete every vertex supporting a nonempty cyclically
nonbacktracking closed path in \(Y^\#\) of length at most \(g_i\).
By Corollary~\ref{prune:cycle-count}, the fraction deleted on either
side is at most \(2^{-0.9H_i}\).  This test is performed in the
already specified graph \(Y^\#\), so it does not depend circularly
on the final induced graph.

Let \(Y\) be the graph induced on the vertices surviving all three
deletions.  It is an induced subgraph both of \(Y^0\) and of
\(Y^\#\).  Combining the three budgets gives a uniform bound
\begin{equation}
\begin{split}
 \varepsilon_\ell
 &=6D\,2^{-4s}
   +\frac{2^{-K/2}}{1-2^{-K/2}}
   +3\,2^{-H_0}+2^{-0.9H_0}\longrightarrow0.
\end{split}
\label{prune:total-budget}
\end{equation}
For example, \(D2^{-4s}=2^{-32\ell}\).  Fractions here and below
are always taken relative to the original sets \(A_i,B_i\), not
relative to an already pruned set.

\begin{proposition}[The tower properties]
\label{prune:tower-properties}
For all sufficiently large \(\ell\), the preceding choices give a
graph \(Y^0\) on sets of sizes
\(|A_i|=d2^{H_i}\) and \(|B_i|=2^{H_i}\), and its induced
subgraph \(Y\), with these properties.
\begin{enumerate}
\item For every \(|i-k|\leq1\), the \(Y^0\)-block between
\(A_i\) and \(B_k\) has degrees \(d,D_{ik}\) with
\(D_{ik}=D2^{|i|-|k|}\).  Its counting incidence divided by
\(\sqrt{dD_{ik}}\) maps normalized constants to normalized
constants, as does its adjoint.  On their orthogonal complements its
norm is at most \(1-\chi\), with the absolute constant \(\chi\)
of Proposition~\ref{exp:realized-gap}.
\item The fraction removed from any \(A_i\) or \(B_i\) is at
most \(\varepsilon_\ell\) of \eqref{prune:total-budget}.
\item The labels of \(Y\) are folded, with positive alphabet size
at most \(m_\ell=O(d2^r(1+s)^2)\).
\item Every nonbacktracking path in \(Y\) beginning at layer \(i\)
and having length at least \(L_i\) is determined by its oriented
word among all paths in \(Y\), including paths beginning in other
layers.
\item There is no nonempty cyclically nonbacktracking closed path
in \(Y\), based at layer \(i\), of length at most \(g_i\).
\end{enumerate}
\end{proposition}

\begin{proof}
Only the fourth item requires further proof.  The degree and norm
claims are established in \eqref{tow:degrees} and
Proposition~\ref{exp:realized-gap}.  For normalized counting
constants, for example, the constant entry after applying normalized
incidence is
\(d/(\sqrt{dD_{ik}}\sqrt{|B_k|})=1/\sqrt{|A_i|}\),
using \(d|A_i|=D_{ik}|B_k|\); the adjoint calculation is the same.
The deletion and alphabet claims have already been proved.
Any short cyclic path in \(Y\) is also one in \(Y^\#\), and its
base vertex was deleted in the third step, proving the fifth item.

Take a path beginning in layer \(i\), and compare it with a path
having the same word.  By foldedness, any path with this word is also
nonbacktracking.  It suffices to identify the segment of length
\(N=\lceil L_i\rceil\) at the beginning of each path.  Choose a
midpoint position, whose radius is at least \(\lfloor N/2\rfloor\),
and let its layer on the first path be \(j\).  Every edge changes
the layer by at most one, so
\[
 H_j\leq H_i+N.
\]
Lemma~\ref{prune:prefix-recovery} recovers both padded streams at
that position through length at least
\[
 \frac{sN}{4}-2s
 \geq\frac{25s}{\log_2d}H_i-2s
 =62.5H_i-2s.
\]
Since \(H_i\geq H_0\) and
\(N\leq100H_i/\log_2d+1\), this quantity is greater than
\(2H_j\) for all sufficiently large \(\ell\), uniformly in \(i\).
For side \(A\) the exact recovered length at radius \(R\) is
\(r+s\lfloor(R-1)/2\rfloor\); on side \(B\) it is
\(s\lfloor R/2\rfloor\).  These formulas give the displayed
lower bound on both sides, including either parity of \(N\).

The common incident labels force the two midpoint layers to have the
same sign and residue modulo \(K\).  If the layer indices are equal,
all active coordinates are recovered, since each has length at most
\(H_j\), and the midpoint vertices coincide.
Suppose instead they differ.  Write \(i_-\) and \(i_+\) for their
indices ordered by absolute value, and put \(H_-=H_{i_-}\).
Then \(H_-\leq H_j\) and \(|i_+|-|i_-|=mK\) for some
\(m\geq1\).  If \(l_-\) is the smaller vertex's own
\(u\)-length, the interval tested at the larger vertex in
\eqref{prune:zero-window} ends no later than
\[
 l_-+\min\{H_-,mK/2\}\leq 2H_-\leq2H_j.
\]
The smaller vertex's padding is zero on this entire interval.
Prefix recovery forces the larger vertex's active \(u\)-bits on
the same interval to be zero.  It would therefore have been removed
by the second deletion, a contradiction.  This reasoning also covers
the case where the first path has the larger midpoint height.  The
case of a zero midpoint index was already in the equal-layer case,
since its sign is distinguished.

Thus the midpoint vertices coincide in all cases.  Foldedness
determines an edge uniquely from its initial vertex and its label;
applying this forward and backward from the common midpoint identifies
the whole two segments.  Continuing forward identifies any remaining
part of the compared words.  This proves the fourth item.
\end{proof}

\begin{corollary}[Effect of deletion on constant spaces]
\label{prune:constant-compression}
Let \(\Pi\) be the orthogonal projection that retains the vertices
of \(Y\) in the counting Hilbert space of the full tower, with any
additional Hilbert-space tensor factor.  Let \(C\) be the subspace
of vectors that are constant on each separate \(A_i\) and \(B_i\)
in the vertex coordinate.  Then
\[
 \|(1-\Pi)|_C\|\leq\sqrt{\varepsilon_\ell}.
\]
In particular the same bound holds on every subspace of \(C\),
uniformly in the additional tensor factor.
\end{corollary}

\begin{proof}
On one normalized constant vector the squared mass removed is exactly
the fraction of vertices removed from that set.  Distinct side and
layer sets are orthogonal, so the squared losses add, and each is
bounded by \(\varepsilon_\ell\) times the corresponding squared
input norm.  Tensoring changes neither calculation.
\end{proof}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//pruning.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//sampling.tex ====
\section{The model and its simultaneous realization}
\label{sam:section}

The expanding incidences and transverse projections now give a model
operator with a separated upper band and small unnormalized trace.
We then realize its compression to \(Y\) by edge voltages and matrix
weights, while imposing the girth needed for the geometric construction.
All norms in this Section, unless a finite compression is explicitly
specified, are operator norms in the full regular representation.  In
particular, the estimates below do not merely test some finite-dimensional
representations of the deck group.

\subsection{The upper space of the model}

Use the notation and parameter choices of the preceding Sections.  The
vertex space before deletion, with its auxiliary indices, is
\[
 \mathcal H^0=
 \ell^2\!\left(\coprod_{i\in\mathbb Z}(A_i\sqcup B_i)\right)
 \otimes\ell^2(F)\otimes\mathbb C^N.
\]
Let \(\Pi\) be the orthogonal projection retaining precisely the vertices
of \(Y\).  Let \(U^0_{ik}:\ell^2(B_k)\to\ell^2(A_i)\) be the
normalized \(Y^0\) incidence from Proposition~\ref{prune:tower-properties}.
It and its adjoint preserve normalized constants, and its norm between
their orthogonal complements is at most \(1-\chi\). Decrease \(\chi\),
if necessary, so that \(0<\chi\le1/4\). Write \(\beta_\ell\) for the
supremum of the deleted fractions in the individual \(A_i,B_i\), so
\(\beta_\ell\le\varepsilon_\ell\to0\) by
Equation~\eqref{prune:total-budget}.
Equation~\eqref{nil:pt} and Lemma~\ref{nil:window} provide the transverse
projections \(p(t)\) and nonnegative layer weights \(h_i(t)\), with
\begin{equation}
 \sum_i h_i(t)^2=1,\qquad
 J(t):=\{i:h_i(t)\ne0\}\text{ consists of at most two adjacent integers}.
 \label{spec:active-layers}
\end{equation}
These assertions concern \(0<t<\delta\). The identity trace of \(p(t)\)
is finite and strictly positive there by
Equation~\eqref{nil:two-factor-trace}. All identity traces include the
ordinary, unnormalized sum over the \(N\) matrix indices.

In the central Fourier decomposition define a self-adjoint
operator field \(O(t)\) by its blocks from \(B_k\) to \(A_i\):
\begin{equation}\label{sam:model}
 O(t)_{A_iB_k}=U^0_{ik}\otimes h_i(t)h_k(t)p(t),
 \qquad |i-k|\le1.
\end{equation}
The blocks in the opposite direction are the adjoints; all remaining
blocks vanish.  The normalized incidence \(U^0_{ik}\) has norm one.
Thus the block Schur estimate bounds \(O(t)\) uniformly, and it defines
an operator on \(\mathcal H^0\).  Here and below the formula is zero off
\(0<t<\delta\).  No continuity at the endpoints is needed for defining
this bounded measurable field.

Let $a_i=|A_i|^{-1/2}\mathbf1_{A_i}$ and
$b_i=|B_i|^{-1/2}\mathbf1_{B_i}$. Denote the transverse fiber Hilbert space,
including its $N$ matrix indices, by $\mathcal H_t$, and put
\[
 \mathcal H^0_t=
 \left(\bigoplus_i\ell^2(A_i)\oplus\bigoplus_i\ell^2(B_i)\right)
       \otimes\mathcal H_t.
\]

\begin{lemma}[The model band and its trace]
\label{spec:model}
For $0<t<\delta$, define an isometry on $p(t)\mathcal H_t$ by
\begin{equation}
 V_*(t)\zeta=2^{-1/2}
 \left(\bigoplus_i h_i(t)a_i\otimes\zeta\right)
 \oplus
 2^{-1/2}\left(\bigoplus_i h_i(t)b_i\otimes\zeta\right),
 \qquad E_*(t)=V_*(t)V_*(t)^*.
 \label{spec:model-isometry}
\end{equation}
Then $O(t)E_*(t)=E_*(t)$, $\|O(t)\|\leq1$, and
\begin{equation}
 O(t)|_{(1-E_*(t))\mathcal H^0_t}\leq(1-\chi)I,
 \qquad
 \operatorname{tr}_{\mathrm{id}}E_*(t)
       =\operatorname{tr}_{\mathrm{id}}p(t).
 \label{spec:model-gap-trace}
\end{equation}
Set $E_*(t)=0$ outside the window; the same upper bound holds there.
Furthermore
\begin{equation}
       \|(1-\Pi)E_*(t)\|\leq\sqrt{\beta_\ell}
 \label{spec:model-cut}
\end{equation}
uniformly in $t$.
\end{lemma}

\begin{proof}
The direct sum of the layerwise constant spaces reduces $O(t)$, since each
$U^0_{ik}$ and its adjoint preserve the corresponding normalized constants.
On these spaces, identify both sides with $\ell^2(\mathbb Z)\otimes
\mathcal H_t$.  Every pair of active indices is an allowed pair by
Equation~\eqref{spec:active-layers}.  The upper-right block is therefore
\[
       |h(t)\rangle\langle h(t)|\otimes p(t),
       \qquad \|h(t)\|_2=1.
\]
The two-sided self-adjoint block has eigenvalue $1$ on the symmetric copy
in Equation~\eqref{spec:model-isometry}, eigenvalue $-1$ on the
antisymmetric copy, and eigenvalue zero on their orthogonal complement.

On the direct sum of the mean-zero spaces, let $\xi=(\xi_k)_k$ lie on the
$B$ side, with transverse coordinates included.  Tensoring a bounded
operator with the projection $p(t)$ does not increase its norm.  Thus the
upper-right block $C(t)$ satisfies
\[
 \|(C(t)\xi)_i\|
 \leq (1-\chi)h_i(t)\sum_k h_k(t)\|\xi_k\|.
\]
Squaring, summing over $i$, and applying the scalar Cauchy--Schwarz
inequality gives $\|C(t)\xi\|\leq(1-\chi)\|\xi\|$.  The self-adjoint
two-sided block consequently has norm at most $1-\chi$ on these spaces.
This proves the norm and spectral assertions.

At a vertex in $A_i$, the diagonal of $E_*(t)$ is
$h_i(t)^2p(t)/(2|A_i|)$, and the corresponding formula on $B_i$ is
$h_i(t)^2p(t)/(2|B_i|)$.  Summing the identity diagonals over both sides and
all layers gives
\[
 \operatorname{tr}_{\mathrm{id}}E_*(t)
 =\left(\frac12\sum_i h_i(t)^2+\frac12\sum_i h_i(t)^2\right)
        \operatorname{tr}_{\mathrm{id}}p(t)
 =\operatorname{tr}_{\mathrm{id}}p(t).
\]
The antisymmetric copy is at $-1$ and makes no contribution to this upper
trace.  Finally, for $\zeta\in p(t)\mathcal H_t$, the squared norm removed
from $V_*(t)\zeta$ is at most
$\frac12\sum_i h_i(t)^2(\beta_\ell+\beta_\ell)\|\zeta\|^2$.
This proves Equation~\eqref{spec:model-cut}.
\end{proof}

The upper space just identified is supported on at most two adjacent
layers, as shown in Figure~\ref{fig:moving-band}. Its vertex factor is a
normalized combination of the constant vectors, so its trace equals the
fiber trace rather than that trace multiplied by the number of vertices.
We must now approximate the compressed model by an actual voltage-cover
operator with finitely many matrix weights, while ensuring the cover's
girth. Section~\ref{spec:section} will use the model gap and deletion
bound to preserve its upper band through this approximation.

\begin{figure}[htbp]
\centering
\input{figures/moving-band}
\caption{The constant-vector part of the model at a parameter with two
active layers. The blue connections represent the four nonzero blocks
\(h_j(t)h_k(t)p(t)\), not individual graph edges. Their symmetric upper
space is given by Equation~\eqref{spec:model-isometry}. As the parameter
approaches either endpoint, the active layers escape in the indicated
direction and the fiber trace tends to zero at the weighted rate of
Proposition~\ref{nil:weighted}.}
\label{fig:moving-band}
\end{figure}

\subsection{Realizing the model on a voltage cover}

\begin{theorem}\label{sam:realization}
For every fixed \(\eta>0\), and for all sufficiently large \(\ell\),
there are voltages \(z_e\in F\) and matrices \(W_e\in M_N(\mathbb C)\)
on the positively oriented edges of \(Y\) with the following properties.
\begin{enumerate}
\item There is a constant \(C_\eta\), independent of \(\ell\) and the
layer, such that \(\|W_e\|\le C_\eta\).  Augmenting an original tower
label by its matrix weight gives a finite positive alphabet of size
\(o(dD)\).  The normalizing number \(D_{ik}\) is determined by the
original part of the label.
\item The voltage cover has no nonempty cyclically nonbacktracking
closed walk based in layer \(i\) of length at most
\(C_*H_i/\log_2d\).
\item Let \(T_P\), extended by zero on the deleted vertex indices, have
entries from \(B_k\) to \(A_i\)
\begin{equation}\label{sam:entries}
 (T_P)_{ab}=
 \frac1{\sqrt{dD_{ik}}}\sum_{e:a\to b}W_e\rho(z_e),
 \qquad a\in A_i,\quad b\in B_k,
\end{equation}
with the adjoint entries in the opposite direction.  Then
\begin{equation}\label{sam:approx}
 \|T_P-\Pi O\Pi\|\le\eta.
\end{equation}
Both operators have layer width one.  In particular, their central
Fourier fields satisfy the same inequality almost everywhere, including
\(\|T_P(t)\|\le\eta\) outside the open window.
\end{enumerate}
The matrices \(W_e\) are the actual weights of the individual edges of
the voltage cover, so \(T_P\) reduces on each component of that cover.
\end{theorem}

We first reduce regular-norm estimates to finite tests. Finite empirical
lists then restrict the possible matrix weights, while a block-specific
distribution approximates the model mean and retains wide independent
voltage coordinates. After fixing those lists, we draw once on each
edge. The girth and centered-moment estimates apply to these same
independent draws.

\subsection{Finite tests for the full regular norm}

\begin{lemma}\label{sam:boxes}
Let \(x\ge2\).  Suppose a finite set of elements of \(F\), including
their inverses, has every integral coordinate bounded by \(x^a\), where
\(a\) is fixed.  For every fixed \(B>0\) there is a finite coordinate
box \(\mathcal B_x\subset F\) with
\[
 |\mathcal B_x|\le x^{c},\qquad
 \frac{|\mathcal B_x\mathbin\triangle\mathcal B_x g|}
 {|\mathcal B_x|}\le Cx^{-B}
\]
for all the specified elements \(g\).  Constants \(c,C\) depend on
\(a,B,F\), but not on the number of specified elements or on \(x\).

Let \(A\) be a self-adjoint finite matrix of right convolution sums on
\(\ell^2(J)\otimes\ell^2(F)\otimes\mathbb C^n\), where \(J\) is a
finite index set, with these group supports.  Write its coefficients as
\(C_{uv}(g)\in M_n(\mathbb C)\), and suppose
\[
 \max_{u\in J}\sum_{v\in J}\sum_g\|C_{uv}(g)\|\le R.
\]
If \(P_x\) retains the deck indices in \(\mathcal B_x\) and all the
other indices, then
\begin{equation}\label{sam:finite-test}
 \|P_xAP_x\|\le\|A\|
 \le\|P_xAP_x\|+Cx^{-B}R.
\end{equation}
The same conclusion holds for a rectangular convolution matrix after
self-adjoint off-diagonal doubling, using the larger of its absolute
row and column sum bounds.  The constants do not depend on the matrix
amplification \(n\).
\end{lemma}

\begin{proof}
The integral coordinates and their positive integer weights are those
of Lemma~\ref{nil:grading}.  Let a coordinate of weight \(j\) range
between \(-\lfloor T^j\rfloor\) and \(\lfloor T^j\rfloor\), and denote
the resulting box by \(\mathcal B(T)\).  If \(f\in\mathcal B(T)\),
the weight-\(j\) coordinate of \(fg\) minus that of \(f\) is a sum of
finitely many graded monomials.  Every monomial contains a coordinate
of \(g\), and consequently the total weight contributed by coordinates
of \(f\) is at most \(j-1\).  The same observation includes the pure
\(g\) term.  Since the multiplication polynomials are fixed, there are
constants \(P,C\) such that, when \(T\ge x\), all these changes have
absolute value at most \(Cx^P T^{j-1}\).

It follows that a point whose coordinate of every weight \(j\) is at
distance more than \(Cx^P T^{j-1}\) from the two corresponding faces
stays in the box after right multiplication by \(g\).  The fraction
of points in the union of these boundary strips is at most
\(C'x^P/T\); the rounding of the side lengths changes only the
constant.  Right multiplication is a bijection, so the same bound,
with a factor two, controls the relative symmetric difference.  Take
\(T=\lceil x^{P+B+2}\rceil\).  The number of coordinates and their
weights are fixed, so \(|\mathcal B(T)|\le x^c\) after enlarging a
fixed exponent.  This proves the first assertion, uniformly for all
the possible supports.

For the norm assertion, let \(P_s\) retain deck indices in the left
translate \(s\mathcal B_x\).  Right convolution commutes with left
translation, and therefore all \(P_sAP_s\), acting on their ranges,
have the same norm as \(P_xAP_x\).  For a finitely supported vector
\(\xi\),
\[
 \sum_{s\in F}\|P_s\xi\|^2=|\mathcal B_x|\,\|\xi\|^2.
\]
Indeed any fixed deck index belongs to exactly \(|\mathcal B_x|\)
left translates.  Averaging \(\langle AP_s\xi,P_s\xi\rangle\)
over \(s\), and dividing by \(|\mathcal B_x|\), multiplies the
coefficient of \(\rho(g)\) by
\[
 \alpha_x(g)=\frac{|\mathcal B_x\cap\mathcal B_x g^{-1}|}
 {|\mathcal B_x|}.
\]
Call the resulting self-adjoint convolution matrix \(A_x\).  The
averaged quadratic form satisfies
\[
 |\langle A_x\xi,\xi\rangle|
 \le\|P_xAP_x\|\,\|\xi\|^2.
\]
Moreover, \(|1-\alpha_x(g)|\le Cx^{-B}\).  Self-adjointness gives
the same coefficient norm bound for columns as for rows.  The scalar
Schur estimate, applied to the norms of the matrix entries, gives
\(\|A-A_x\|\le Cx^{-B}R\).  Density and the self-adjoint quadratic
form formula now prove \eqref{sam:finite-test}.  No coordinate within
\(\mathbb C^n\) was estimated separately; this explains the asserted
uniformity in amplification.  For a rectangular \(S\), apply the
argument to
\(\left(\begin{smallmatrix}0&S\\S^*&0\end{smallmatrix}\right)\),
whose norm equals \(\|S\|\), before and after the indicated
compressions.
\end{proof}

\begin{lemma}\label{sam:empirical}
Set \(M=\lfloor d^{1/32}\rfloor\), with \(q=d^{A_*}\) and \(A_*\)
fixed.  Consider at most \(C(1+\log q)\) finite distributions of
operators \(W\rho(g)\) on \(\ell^2(F)\otimes\mathbb C^N\).
Suppose \(\|W\|\le B_0\), every possible group coordinate is
bounded by \(q^a\), and \(N\le q^a\), for fixed constants
\(a,B_0,C\).  For each fixed \(\varepsilon>0\), and sufficiently
large \(\ell\), each expectation has an \(M\)-term empirical
average within \(\varepsilon\) in the full regular norm.  The
averages may be chosen for all the distributions simultaneously.
\end{lemma}

\begin{proof}
For each distribution take \(M\) independent samples.  Apply
Lemma~\ref{sam:boxes} at \(x=q\) to the difference of their average
and their expectation, after off-diagonal doubling.  Its absolute
coefficient row and column sums are bounded by \(2B_0\).  Choose the
box to make the error less than \(\varepsilon/2\).  The resulting
self-adjoint test matrix has dimension at most \(q^{c_0}\), with
\(c_0\) fixed.  If \(Z_j\) is its compressed, centered \(j\)-th
sample, then \(\mathbb E Z_j=0\) and \(\|Z_j\|\le2B_0\).

Expand the trace of \((M^{-1}\sum_j Z_j)^{2k}\).  If an index occurs
exactly once in a term, conditioning on all other samples makes that
term zero.  This argument is valid for a noncommutative product: the
unique occurrence has the form \(LZ_jR\) with \(L,R\) fixed under
the conditioning.  At most \(k\) different indices occur in a
remaining term.  The number of sequences using at most \(k\)
indices is at most
\(\sum_{a=1}^k M^a a^{2k}\).  Bounding the absolute trace of each
product by its dimension times the product of its factor norms gives
\[
 \mathbb E\operatorname{Tr}
 \left(M^{-1}\sum_{j=1}^M Z_j\right)^{2k}
 \le q^{c_0}(2B_0)^{2k}M^{-2k}
       \sum_{a=1}^k M^a a^{2k}
 \le q^{c_0}k\left(\frac{4B_0^2k^2}{M}\right)^k.
\]
For \(k=\lceil\log q\rceil\), Markov's inequality therefore bounds
the probability of test norm greater than \(\varepsilon/2\) by
\[
 q^{c_0}k\left(\frac{16B_0^2k^2}{M\varepsilon^2}\right)^k.
\]
Its logarithm is at most
\(-c(\log q)^2+O(\log q\log\log q)\), since
\(\log M=(32A_*)^{-1}\log q+O(1)\).  The bound tends to zero even
after multiplication by \(C(1+\log q)\).  Thus all the tests succeed
with positive probability.  Lemma~\ref{sam:boxes} transfers their
bounds to the full regular norm.  Fix one such list for each
distribution.
\end{proof}

\subsection{Compact coefficient templates and masks}

Fix a small accuracy \(\varepsilon>0\), to be chosen in terms of
\(\eta\) at the end.  In this Subsection all auxiliary errors can be
made smaller than any prescribed fixed multiple of \(\varepsilon\).
The constants they require are fixed before increasing \(\ell\).

By Lemma~\ref{nil:templates}, choose smooth compactly supported functions
\(J_\epsilon^\flat\) and a fixed relative logarithmic mesh size
\(0<u\le1\) so that truncation and coefficient rounding have arbitrarily
small prescribed \(\ell^1\) errors.  These choices depend only on
\(\varepsilon\) and the previously fixed data.
The estimates apply uniformly on the active window because
\(t,\delta-t\in(0,\delta)\) and \(\delta\to0\) as \(\ell\to\infty\);
they give operator estimates for every unitary family used below.
Throughout rounding, the translating operators, and consequently their
multipliers, keep their actual parameter; only the coefficients are rounded.

Choose a sufficiently fine fixed log-grid spacing.  A nonnegative
smooth bump on the real line whose translates at that spacing cover
the line, divided by the positive sum of those translates, gives a
smooth partition of unity.  Pull one such partition back under
\(\log_2(t/\delta)\), and another under
\(\log_2((\delta-t)/\delta)\).  Denote them by \(\psi_\nu(t)\)
and \(\theta_\omega(t)\), and denote their associated positive
magnitude templates by \(a_\nu,b_\omega\).  On the supports in
question,
\[
 |\log(a_\nu/t)|\le u,\qquad
 |\log(b_\omega/(\delta-t))|\le u.
\]
Both partitions sum to one on \(0<t<\delta\).

For a fixed block \((i,k)\), keep precisely the template pairs whose
supports intersect the support of \(h_i h_k\), and call their set
\(\mathcal T_{ik}\).  Its cardinality is bounded by a constant
\(K_\varepsilon\), uniformly in \(i,k,\ell\).  Indeed
\(h_i h_k\ne0\) implies \(|b(t)-i|<1\); each of
\(\log_2(t/\delta)\) and
\(\log_2((\delta-t)/\delta)\) varies by at most \(2\kappa\)
there.  Only boundedly many members of either fixed-spacing grid can
meet these ranges.  The specified positivity of \(h\) ensures that
the nonzero incidence blocks have some such pairs.

For any twisted representation \(U'\) of \(\bar E_{\mathbb Z}\)
define the finite sum
\[
 \mathcal L_\epsilon(\alpha;U')=
 \sum_{\gamma\in\bar E_{\mathbb Z}}
 \alpha^{n_{\mathrm f}}
 J_\epsilon^\flat(\Delta_{\alpha^{1/m}}\gamma)U'(\gamma).
\]
By Proposition~\ref{nil:projection}, Lemma~\ref{nil:templates}, and the
chosen truncation and rounding accuracies, the block field
\begin{equation}\label{sam:template-field}
 \sum_{(\nu,\omega)\in\mathcal T_{ik}}
 f_{ik\nu\omega}(t)\,
 \mathcal L_+(a_\nu;u_t)
 \mathcal L_-(b_\omega;V_t),
 \qquad
 f_{ik\nu\omega}=h_i h_k\psi_\nu\theta_\omega,
\end{equation}
approximates \(h_i h_kp(t)\) to an arbitrarily small fixed error,
uniformly on the window, for sufficiently large \(\ell\).  The two
factors act on their separate transverse indices.  The error estimate
uses the uniformly bounded coefficient \(\ell^1\) norms, positivity
of the scalar partition, and its sum-one property.  Extend each
scalar mask by zero off the window.

The masks have a useful scale uniform in the layer.  Put
\[
 w_i=q^{-1}2^{-\kappa|i|}.
\]
On the support of \(h_i\), the identities
\(t=\delta 2^{\kappa b(t)}/(1+2^{\kappa b(t)})\) and
\(\delta-t=\delta/(1+2^{\kappa b(t)})\) show that the shorter
distance to an endpoint is comparable to \(w_i\), with constants
depending on \(\kappa\).  They also show that the support has
length at most \(Cw_i\).  The chain rule for the two logarithmic
charts gives, for every fixed integer \(j\ge0\),
\begin{equation}\label{sam:mask-derivative}
 \|f_{ik\nu\omega}^{(j)}\|_\infty\le C_jw_i^{-j},
 \qquad
 \|f_{ik\nu\omega}^{(j)}\|_1\le C_jw_i^{1-j}.
\end{equation}
The compact support of the original bumps makes the zero extension
smooth; each individual support is at positive distance from the
endpoints.  In particular, its Fourier coefficients satisfy
\begin{equation}\label{sam:fourier}
 |\widehat f_{ik\nu\omega}(n)|
 \le C_jw_i(1+|n|w_i)^{-j},\qquad n\in\mathbb Z.
\end{equation}
For \(|n|w_i\le1\) this follows from the \(L^1\) bound.  Otherwise
integrate by parts \(j\) times, use
\eqref{sam:mask-derivative}, and combine the two bounds.
Consequently
\[
 \sum_n|\widehat f_{ik\nu\omega}(n)|\le C,
 \qquad
 \sum_{|n|>L/w_i}|\widehat f_{ik\nu\omega}(n)|
 \le C_j(1+L)^{1-j}.
\]
Choose a fixed sufficiently large \(L\), and truncate at
\(K_i=\lceil L/w_i\rceil\).  These are uniform approximations on
the entire circle.  In particular, the approximating trigonometric
polynomials have small values outside the active supports; exact
vanishing there is neither claimed nor used.

\subsection{Finite matrix lists on the second factor}

The scalar masks vary through infinitely many layers, but the final group
must have a finite alphabet of matrix weights. Using a new matrix list at
every layer would not meet that requirement. The range \(i<I\) uses only finitely many second-factor templates, and hence finitely many empirical lists; in the remaining range a congruence
subgroup factorization replaces all the matrix factors by one common list.
The next estimates justify this replacement in the full regular norm.


In blocks with \(i<I\), every second-factor template satisfies
\begin{equation}\label{sam:ordinary-range}
 \log_2 b_\omega^{-1}
 \le \log_2q+\kappa(I+1)+C.
\end{equation}
This follows from the formula for \(\delta-t\) on the support of
\(h_i\), followed by the fixed log rounding.  Thus there are only
\(O(1+\log q)\) distinct such templates over all these blocks, and
their coordinate supports are bounded by a fixed power of \(q\).
Choose a coordinate rectangle containing each compact template
support, with side radii fixed multiples of
\(b_\omega^{-j/m}\) in weight \(j\).  The rectangle has
\(n_\omega\) lattice points.  Uniform sampling of \(v'\) in it,
with the term
\[
 n_\omega b_\omega^{n_{\mathrm f}}
 J_-^\flat(\Delta_{b_\omega^{1/m}}v')
 R(\sigma(v'))\rho(\sigma_Y(v')),
\]
has expectation equal to the ordinary lifted template.  The scalar
multiplier is bounded uniformly, because
\(n_\omega b_\omega^{n_{\mathrm f}}\le C\).
Lemma~\ref{sam:empirical} gives, for each such template, a fixed
\(M\)-term list approximating that expectation in the full regular
norm.  Its entries retain their original scalar multipliers and
matrices.  Fix all these lists now.

For the remaining blocks a single common matrix list suffices.  Let
\(\Lambda=\Lambda_q\) be the subgroup of \(\bar E_{\mathbb Z}\) whose
coordinates are all divisible by \(q\).  Reduction modulo \(q\)
is a surjective homomorphism by the integral polynomial group law;
its kernel is \(\Lambda\), and
\begin{equation}\label{sam:index}
 [\bar E_{\mathbb Z}:\Lambda]=q^{2n_{\mathrm f}}=:Q_q.
\end{equation}
Choose a set \(\mathcal V_q\) of representatives for the cosets
\(\Lambda v\), with each coordinate in
\(\{0,\ldots,q-1\}\).  Every \(\gamma\) is uniquely \(yv\),
with \(y\in\Lambda\), \(v\in\mathcal V_q\).
Equation~\eqref{nil:congruence-R} gives
\(R(\sigma(y))=1\) for \(y\in\Lambda\): its affine shift is
the identity modulo \(q\), and its modulation exponent is an
integer.  The finite average
\begin{equation}\label{sam:coset-average}
 C_q=Q_q^{-1}\sum_{v\in\mathcal V_q}
 R(\sigma(v))\rho(\sigma_Y(v))
\end{equation}
has an \(M\)-term empirical approximation in the full regular norm,
again by Lemma~\ref{sam:empirical}.  Fix one such list, common to
all \(i\ge I\).

For those blocks replace an ordinary second-factor template by the
product
\begin{equation}\label{sam:subgroup-product}
 \left(
 Q_q\sum_{y\in\Lambda}
 b_\omega^{n_{\mathrm f}}
 J_-^\flat(\Delta_{b_\omega^{1/m}}y)\rho(\sigma_Y(y))
 \right)C_q.
\end{equation}
We next prove the needed comparison with the ordinary template,
including its phase and normalization.

The scale estimates from the window and the choices of parameters
give, uniformly for \(i\ge I\) and active templates,
\begin{equation}\label{sam:second-width}
 b_\omega^{-1/m}/q\ge 2^{2E_0H_i}
\end{equation}
when \(\ell\) is sufficiently large.  Here is the explicit
calculation.  Log rounding contributes a bounded constant, and
\[
 \log_2(b_\omega^{-1/m}/q)
 \ge \frac\kappa m i-\left(A_*-\frac{A_*}{m}\right)\log_2d-C.
\]
Since \(i\ge I=H_0\ge\log_2d\) and \(H_i=H_0+i\le2i\),
the hypothesis
\(\kappa/m>10(E_0+A_*+C_\bullet+1)\) leaves more than the
margin needed for \eqref{sam:second-width}.  It follows that every
side length of the dilated subgroup mesh tends to zero uniformly.
Counting its cells shows that the coefficient \(\ell^1\) norm of
the first parenthesis in \eqref{sam:subgroup-product} is bounded.

In the fiber with \(\tau=t-\delta\), the relevant projective
operators satisfy
\[
 V_t(y)V_t(v)
 =\exp(2\pi\mathrm i\tau c(y,v))V_t(yv).
\]
This is the multiplier identity, not a replacement of \(t\) by a
template in the translating operators.  It can also be checked by
multiplying the actual section lifts and then using the center
parameter \(-\delta\) of \(R\).  As
\(R(\sigma(y))=1\), the first parenthesis in
\eqref{sam:subgroup-product} is precisely its prescribed factor in
this product.  The coefficient at \(yv\), before empirical
replacement of \(C_q\), is therefore
\[
 b_\omega^{n_{\mathrm f}}
 J_-^\flat(\Delta_{b_\omega^{1/m}}y)
 \exp(2\pi\mathrm i\tau c(y,v)).
\]
The factor \(Q_q\) in the subgroup sum has canceled the factor
\(Q_q^{-1}\) in the representative average; no matrix trace
normalization occurs here.

Set \(\beta=b_\omega^{1/m}\).  On all the compact supports in this
calculation the elements \(\Delta_\beta v\), with
\(v\in\mathcal V_q\), tend to the identity uniformly, since their
coordinates of weight \(j\) have absolute value at most
\(q\beta^j\le q\beta\).  The polynomial group law implies
\(\Delta_\beta(yv)=(\Delta_\beta y)(\Delta_\beta v)\) and that
this differs from \(\Delta_\beta y\) by a quantity tending to
zero uniformly on those compacts.  Homogeneity of the cocycle gives
\[
 \tau c(y,v)=\frac\tau{b_\omega}
 c(\Delta_\beta y,\Delta_\beta v).
\]
On an active template \(|\tau|/b_\omega\) is bounded, while the
last cocycle tends uniformly to zero.  Smoothness of the compact
coefficient function thus gives a uniformly vanishing difference
between the preceding coefficient and
\(b_\omega^{n_{\mathrm f}}
 J_-^\flat(\Delta_\beta(yv))\).
All possibly nonzero differences lie in one fixed enlarged compact
set in the scaled coordinates.  The normalized number of ordinary
lattice points there is bounded.  Summing the coefficient differences
therefore gives \(o(1)\), uniformly in the active blocks and
parameters.  This proves the comparison with the ordinary template
in operator norm.  Finally, replacing \(C_q\) by its empirical list
costs at most its regular-norm error times the bounded coefficient
\(\ell^1\) norm of the subgroup factor.  The same bound holds almost
everywhere in every central fiber.

All the coefficient sums used above have bounded absolute norms even
at parameters where their templates are inactive.  Thus multiplication
by the masks and replacement of those masks by their Fourier
truncations preserve the error estimates globally on the circle.
More explicitly, on a mask support use its active comparison; off
that support the untruncated mask is zero.  The error made by
substituting the truncated mask is bounded everywhere by its Fourier
tail times the coefficient norm bounds.  Sum over at most
\(K_\varepsilon\) pairs.  This controls the inactive leakage as well
as the active approximation.

\subsection{One draw per edge and the finite alphabet}

We now specify a finite distribution of a single pair \((W,z)\) for
each block \((i,k)\).  This will be sampled independently on its
undirected edges, using the positive orientation and taking the
adjoint weight and inverse voltage for the reverse orientation.
The choices in different parts of the prescription are independent
conditional on the template pair.

First choose \((\nu,\omega)\) uniformly in
\(\mathcal T_{ik}\), then choose an integer \(n\) uniformly in
\(\{-K_i,\ldots,K_i\}\).  Use the total central power \(n\)
with scalar multiplier
\[
 |\mathcal T_{ik}|(2K_i+1)\widehat f_{ik\nu\omega}(n).
\]
By \eqref{sam:fourier} this multiplier is bounded uniformly.
These additional central powers carry no matrix modulation.

Choose \(x\in\bar E_{\mathbb Z}\) uniformly in a full coordinate
rectangle containing the support of
\(J_+^\flat(\Delta_{a_\nu^{1/m}}x)\).  In each coordinate of
weight \(j\), its radius is a fixed positive constant times
\(a_\nu^{-j/m}\), rounded up to an integer.  Write \(n_\nu^+\)
for its number of lattice points.  Use \(\sigma_X(x)\), with
scalar multiplier
\[
 n_\nu^+ a_\nu^{n_{\mathrm f}}
 J_+^\flat(\Delta_{a_\nu^{1/m}}x).
\]
It is uniformly bounded by the rectangle cell-volume estimate.

If \(i<I\), choose one of the fixed \(M\) entries in the ordinary
second-factor list for \(b_\omega\), uniformly, and use its section
lift, its matrix \(R(\sigma(v'))\), and its scalar multiplier.
Uniform averaging supplies the list's normalization \(1/M\).

If \(i\ge I\), choose \(y\in\Lambda\) uniformly in the analogous
full subgroup rectangle, and independently choose one entry \(v\)
uniformly from the fixed representative list.  Use
\(\sigma_Y(y)\sigma_Y(v)\), the matrix \(R(\sigma(v))\), and
scalar multiplier
\[
 n_\omega^\Lambda Q_q b_\omega^{n_{\mathrm f}}
 J_-^\flat(\Delta_{b_\omega^{1/m}}y),
\]
where \(n_\omega^\Lambda\) is the subgroup rectangle's number of
points.  Its normalized cell volume is
\(Q_q b_\omega^{n_{\mathrm f}}\).  By
\eqref{sam:second-width}, each scaled side mesh is at most one,
so this scalar multiplier is also bounded uniformly.

Let \(z\) be the product of the prescribed group elements, in the
indicated order.  Let \(c_e\) be the product of the scalar
multipliers, and let \(R_v\) be the one indicated matrix.  The raw
weight is \(c_eR_v\), with \(|c_e|\le C_\varepsilon\).  Fix a finite
grid in the complex disk of radius \(C_\varepsilon+1\), and round
\(c_e\) deterministically to a grid value \(c_e^\flat\) at distance
at most a prescribed \(\varepsilon'\).  Define
\[
 W=c_e^\flat R_v.
\]
Since \(R_v\) is unitary, the rounding changes the expectation
norm by at most \(\varepsilon'\).  The voltage \(z\) is retained
even when the weight is zero.  In particular, the support of the
voltage distribution is not conditioned on a nonzero weight.

All distributions in this prescription are finite.  Linearity of
expectation shows, term by term, that the raw expectation is the
sum of the truncated Fourier masks times the appropriate first and
second coefficient templates.  Uniform sampling of a rectangle
cancels exactly its point-count multiplier, and uniform sampling of
a template pair and a frequency cancels their two count multipliers.
The empirical lists give their specified averages.  The preceding
Subsections and the last rounding consequently show that we can
arrange
\begin{equation}\label{sam:edge-mean}
 \mathop{\mathrm{ess\,sup}}_t
 \bigl\|\mathbb E[W\rho(z)](t)-h_i(t)h_k(t)p(t)\bigr\|
 \le\varepsilon
\end{equation}
uniformly in \((i,k)\).

Only the empirical-list matrices enter the tags.  Their total number
is at most \(C_\varepsilon(1+\log q)M\), from the ordinary lists
and the single representative list.  The final scalar grid has
fixed finite cardinality depending on \(\varepsilon\) alone.
Thus the total number of weight tags has the same stated bound.
The original tower alphabet has size
\(O(d2^r(1+s)^2)\).  As \(2^r=d^{7/4}\), their product is
\[
 O_\varepsilon\bigl(d^{89/32}(1+\log d)^3\bigr)=o(d^3)=o(dD).
\]
In particular, neither the possibly very large group-coordinate
rectangles nor the number of different first-factor templates enlarge
the label alphabet.  Voltages are separate data of the cover.

There is a fixed constant \(C\), independent of \(i,\ell\), such
that every coordinate of every possible \(z\) in a block with first
index \(i\), and of its inverse, is bounded by
\begin{equation}\label{sam:support}
 2^{CH_i}.
\end{equation}
Indeed the logarithms of the template radii and of \(K_i\) are
\(O(\log q+\kappa|i|+1)=O(H_i)\).  The representative coordinates
are bounded by \(q\); the ordinary empirical terms lie in their
specified rectangles.  Fixed polynomial multiplication and inversion
preserve a bound of the form \eqref{sam:support}.

The first factor supplies wide independent coordinates in a range
overlapping that of \eqref{sam:second-width}.  Namely, for every
active first-factor template with \(i\le2I\),
\begin{equation}\label{sam:first-width}
 a_\nu^{-1/m}\ge 2^{2E_0H_i}
\end{equation}
for sufficiently large \(\ell\).  For \(0\le i\le2I\), the
lower bound
\(\log_2a_\nu^{-1}\ge\log_2q-C\), together with
\(H_i\le3H_0\le3h_*\log_2d\) and
\(A_*/m>10E_0h_*\), proves it.  For \(i<0\), use instead
\[
 \log_2a_\nu^{-1}
 \ge\log_2q+\kappa|i|-C
\]
and \(H_i=H_0+|i|\), with the same inequality for \(A_*\) and
the stronger chosen inequality for \(\kappa\).
Every upper unitriangular entry in this first rectangle is sampled
independently on an integer interval containing at least
\(2^{E_0H_i}\) points.  For \(i\ge I\), every corresponding
entry of the second-factor subgroup rectangle has at least that
many possibilities in its step-\(q\) interval.  Fixed rectangle
radii and integer rounding are harmless because
\eqref{sam:first-width} and \eqref{sam:second-width} have an
exponential margin.

\subsection{Girth with the same independent edge draws}

We use the Schwartz--Zippel polynomial zero bound, whose short proof we include.
The sharp total-degree bound is due to Schwartz
\cite[Lemma~1, p.~702, and Corollary~1]{Schwartz1980}; Zippel's sparse-interpolation
work \cite[\S 3.1, Theorem~1]{Zippel1979} gives related coordinate-degree
zero estimates, and DeMillo and Lipton \cite{DeMilloLipton1978} earlier
used random evaluations for polynomial identity testing.
The minimum-side bound below is an immediate consequence of Schwartz's
estimate for unequal coordinate sets. If a nonzero
polynomial of total degree at most \(r\), over a field, is evaluated
on independent uniform choices from finite sets of sizes at least
\(L\), its probability of vanishing is at most \(r/L\).  This
follows by induction on the number of variables.  Regard the
polynomial as a polynomial of degree \(s\) in the last variable;
its leading coefficient is a nonzero polynomial of degree at most
\(r-s\).  The probability that this coefficient vanishes is at most
\((r-s)/L\); otherwise the one-variable root bound gives at most
\(s/L\).  The constant case starts the induction.  Arithmetic
progressions of distinct integers are allowed as the finite sets.

Fix a nonempty cyclically nonbacktracking base walk \(w\), based in
layer \(i\), of length
\(a\le C_*H_i/\log_2d\).  The layer indices along it satisfy
\(|H_j-H_i|\le a+1\), so their ratios to \(H_i\) tend uniformly
to one.  If some positive-edge block encountered has first index
\(j_0<I\), then all such first indices are at most \(2I\) when
\(\ell\) is sufficiently large.  To verify this also for very
negative \(j_0\), the length estimate and
\(H_i\le H_{j_0}+a+1\) give
\(a+1\le\epsilon_\ell H_{j_0}\), with
\(\epsilon_\ell\to0\).  The maximum first index is at most
\(j_0+\epsilon_\ell(I+|j_0|)\).  If \(j_0\ge0\), this is
at most \((1+2\epsilon_\ell)I\); if \(j_0<0\), it is at most
\(\epsilon_\ell I\).  In this case use the first-factor quotient
of \(F\) onto the upper unitriangular group.  If there is no such
block, every first index is at least \(I\), and use the second
upper unitriangular quotient.  These are the two overlapping wide
ranges just proved.

Assign an independent formal upper unitriangular matrix to each
undirected edge in the support of \(w\), using its inverse when
the edge is traversed backwards.  Lemma~\ref{nil:word} says that
the resulting word is not identically the identity.  Its hypotheses
apply because the support has the tower girth and the present walk
has length at most the fixed \(C_*\) scale.  In particular, the
integer \(m\) used for this assertion was fixed at that scale.
Some entry of this word minus the identity is therefore a nonzero
polynomial in the upper entries of those edge matrices.  Its total
degree is at most \(m-1\).  Indeed every term in the \((u,v)\)
entry of a product of upper unitriangular matrices, including inverse
matrices expanded by the finite geometric series, has total
superdiagonal weight \(v-u\).  Each variable has positive integer
superdiagonal weight, so its ordinary total degree is at most
\(v-u\le m-1\).  This bound is independent of the number of
factors in the word.

Condition on the template choices, frequencies, empirical-list
choices, the other transverse factor, and the other independently
sampled coordinates.  Do not condition on the final scalar weights.
In the first-factor case the remaining upper entries have their
independent uniform interval distributions.  In the second-factor
case the edge matrix is a random subgroup upper matrix multiplied
on the right by its fixed representative upper matrix.  Right
multiplication by that fixed matrix is an invertible polynomial
coordinate change over the reals.  Replacing the subgroup entries
by their multiples of \(q\) is likewise an invertible linear
coordinate change over the reals.  Thus the nonzero word polynomial
remains nonzero in the independently sampled entries.  This also
explains why weights that happen to vanish do not remove a voltage
variable from this argument.

Each variable set has at least \(2^{E_0H_j}\) points at its local
layer \(j\).  The polynomial estimate therefore bounds the
conditional, and hence unconditional, probability of trivial
holonomy by
\[
 (m-1)2^{-E_0\min_j H_j}.
\]
There are at most
\[
 (d+1)2^{H_i}
 \sum_{1\le a\le C_*H_i/\log_2d}(6D)^a
 \le 2^{(3C_*+3)H_i}
\]
walks to test from all vertices of layer \(i\), for sufficiently
large \(\ell\).  Here the rough bound counts all walks, and the
maximum degree is at most \(6D\).  Since
\(\min_jH_j\ge(1-o(1))H_i\) and
\(E_0>10(C_*+C_\bullet+1)\), the union bound is at most
\(C2^{-2H_i}\) after increasing \(\ell\).  Summing over all
\(i\in\mathbb Z\) gives a bound tending to zero.

A closed lift projects under a graph covering to a closed walk,
and nonbacktracking, including the cyclic condition, is preserved.
Trivial holonomy depends only on the base walk, not on which lift
of its initial vertex is used.  Consequently the tests just counted
exclude every forbidden closed walk in the voltage cover.

\subsection{Centered moments, with cycle edges included}

To control deviations from the block means, we expand even moments
of the centered edge operators. Independence and centering make a
term vanish whenever its base walk traverses an undirected edge
exactly once. The next lemma counts the remaining walks when their
supports have bounded cycle rank; its constants do not affect the
earlier nilpotence parameter.

\begin{lemma}\label{sam:walk-count}
Let a finite graph have maximum degree \(\Delta\ge1\).  Fix a
vertex \(v\) and an integer \(l\ge1\).  Suppose the support of
every closed walk under consideration has first Betti number at
most \(b\).  The number of length-\(2l\) closed walks from \(v\)
in which no undirected edge occurs exactly once is at most
\[
 C_b^l\Delta^l.
\]
Multiple edges, if present, are treated as distinct edges.
\end{lemma}

\begin{proof}
The support of such a walk is connected and has at most \(l\)
edges.  Prune leaves other than \(v\) successively, and retain the
resulting rooted core \(K\).  If the support is a tree, the core
is the single vertex \(v\).  Otherwise the core contains \(v\),
has no vertex of degree one other than possibly \(v\), and has
the same first Betti number as the support.  Write \(a\) for its
edge count.  The identity
\[
 \sum_{x\in K}(\deg_K(x)-2)=2b_1(K)-2
\]
gives a maximum-degree bound of \(2b+2\). If \(K\) has no edges,
its maximum degree is zero. Otherwise all terms in the sum are
nonnegative except possibly the root term, which is at least \(-1\).
The parts removed by pruning are trees, each
attached at exactly one core vertex.  Otherwise a remaining cycle,
or a path joining two core vertices, could not have been removed
by the leaf procedure.

There are at most \(4^a\Delta^a\) possibilities for a rooted
connected subgraph \(K\) with \(a\) edges.  We give an encoding
that includes its cycle edges.  Starting at \(v\), when an as yet
unmarked incident edge of \(K\) is available, mark it, traverse it,
and push that edge onto a stack.  Explore recursively at its other
endpoint, even if that vertex has been visited before.  When this
recursive call has exhausted its available unmarked edges, return
along the pushed edge and pop it.  Marking \emph{edges}, rather
than vertices, ensures that each edge, including every edge closing
a cycle, is pushed exactly once and popped exactly once.  The
process explores the connected subgraph, and it has a balanced
push/pop word of length \(2a\), at most \(4^a\) possibilities.
At a push, record the chosen incident edge, with at most \(\Delta\)
possibilities.  At a pop the edge is determined by the stack.
Given the starting vertex, this code reconstructs the complete
traversal and hence its edge set.  One may make the encoding
deterministic using fixed orders on incident edges.  This proves
the bound; it does not assume that the explored subgraph is a
tree.

Fix a possible core \(K\).  Encode a walk by declaring each step
to be a core step, a tree descent away from the core, or a tree
return toward the core.  There are at most \(3^{2l}\) such words.
For each core step there are at most \(2b+2\) choices.  A descent
has at most \(\Delta\) choices, while a return is forced by the
stack of the current tree excursion.  Let \(c\) be the number of
core steps and \(t\) the number of descents.  Returns and descents
are equally numerous, since each excursion begins and ends at
its attachment vertex.  Thus \(2t+c=2l\).  Every core edge is
used at least twice, by the no-singleton assumption, whence
\(c\ge2a\) and \(t\le l-a\).

The number for this fixed core is bounded by
\(3^{2l}(2b+2)^{2l}\Delta^{l-a}\).  This is an upper bound even
if some of the formal encodings fail to be valid tree excursions.
Multiply by the core count and sum over \(0\le a\le l\).  Since
\(l+1\le2^l\) and \(4^a\le4^l\), the resulting bound is
\(\bigl(72(2b+2)^2\bigr)^l\Delta^l\), as required.
\end{proof}

Fix a block \((i,k)\) of the actual surviving graph.  Before degree
normalization let \(X_{ik}\) be its centered rectangular convolution
matrix, with entries
\[
 (X_{ik})_{ab}=
 \sum_{e:a\to b}
 \bigl(W_e\rho(z_e)-\mathbb E[W_e\rho(z_e)]\bigr).
\]
Double it off the diagonal to a self-adjoint matrix.  By
\eqref{sam:support}, Lemma~\ref{sam:boxes} applies with
\(x=2^{H_i}\), simultaneously for every possible assignment of
these finitely many edges.  The row and column coefficient bounds
are \(O_\varepsilon(D)\).  Choose the boundary exponent so that
the absolute error in \eqref{sam:finite-test} tends uniformly to
zero.  Its finite test dimension, including all vertex, deck, and
matrix indices, is bounded by
\begin{equation}\label{sam:dimension}
 2^{C_1H_i}
\end{equation}
for a fixed \(C_1\).  To check this, the block has
\(O((d+1)2^{H_i+1})\) vertices, the deck box has
\(2^{O(H_i)}\) points, and
\(N=q^{n_{\mathrm f}}=2^{A_*n_{\mathrm f}\log_2d}
 \le2^{O(H_i)}\).

At this point fix a constant
\begin{equation}\label{sam:C2-order}
 C_2>2(C_1+4),\qquad
 l=\left\lceil C_2H_i/\log_2d\right\rceil.
\end{equation}
The tower girth implies that the support of every length-\(2l\)
walk with no singleton edge has cycle rank bounded by a constant
depending on \(C_2\) alone, using Lemma~\ref{nil:rank}.  Indeed it
has at most \(l\) edges and lies in the two fixed layers \(i,k\),
whose girth lower bounds are comparable to
\(H_i/(100\log_2d)\).  The ratio of the edge count to that lower
bound is bounded in terms of \(C_2\).  This use of
Lemma~\ref{nil:rank} requires no identity test for a longer word.
In particular \(m\), which was fixed for the \(C_*\) girth
tests, is not increased when \(C_2\) is chosen.

Let \(\widehat X_{ik}\) denote the finite self-adjoint compression.
Every centered compressed edge entry has norm at most
\(2C_\varepsilon\).  Expand
\(\mathbb E\operatorname{Tr}(\widehat X_{ik}^{2l})\) along the
base-vertex indices.  A summand whose walk has a singleton
undirected edge has expectation zero: condition on the other
edges, and the sole remaining factor is a centered entry or its
adjoint.  This remains true after deck compression.  For the other
walks, the absolute trace of the product on a fixed vertex's deck
and matrix indices is at most their dimension times
\((2C_\varepsilon)^{2l}\).  Lemma~\ref{sam:walk-count}, with
\(\Delta\le6D\), now gives
\begin{equation}\label{sam:block-moment}
 \mathbb E\operatorname{Tr}(\widehat X_{ik}^{2l})
 \le 2^{C_1H_i}(C_4D)^l,
\end{equation}
where \(C_4\) is fixed after \(C_2\) and the accuracies have
been chosen.  It is independent of \(i,\ell\).  The exponent in
the dimension bound has not changed in this argument.

Use the finite norm test and Markov's inequality at threshold
\(\varepsilon\sqrt{dD_{ik}}\).  For large \(\ell\), the absolute
box error is less than half this threshold.  Since
\(D/2\le D_{ik}\le2D\), \eqref{sam:block-moment} implies
\begin{equation}\label{sam:block-probability}
 \mathbb P\{\|X_{ik}\|>
                   \varepsilon\sqrt{dD_{ik}}\}
 \le 2^{C_1H_i}(C_5/d)^l.
\end{equation}
The constant \(C_5\) can be very large and can depend on \(C_2\).
Nevertheless it is fixed before increasing \(\ell\).  Increase
\(\ell\) so that \(C_5\le d^{1/2}\).  The right side is then
at most
\(2^{(C_1-C_2/2)H_i}\le2^{-4H_i}\), by
\eqref{sam:C2-order}.  Summing over the at most three blocks with
each first index gives a total failure bound tending to zero.

\subsection{A simultaneous choice and the global norm}

For sufficiently large \(\ell\), the sum of the girth failure
bounds and the block-norm failure bounds is less than one.  In the
countable independent product of the finite edge distributions,
the union bound therefore gives positive probability that all
requirements hold.  There is also an elementary compactness
formulation which avoids any need for an infinite sampling
construction.  Enumerate the countably many edge choices and the
countably many requirements.  Every requirement depends on only
finitely many edges: this is immediate for a fixed walk, and a
fixed block contains finitely many edges even though its convolution
norm acts on an infinite deck space.  Any finite collection of
requirements is satisfiable by the finite-product union bound,
using the same total budget below one.  Choose assignments
satisfying the first \(n\) requirements and extend them arbitrarily
to all other edges.  Successive subsequences make the choice at
each edge constant, since every edge has finitely many possible
outcomes.  The diagonal limit satisfies every requirement: once
its finite set of edges has stabilized, the corresponding operators
and voltages are exactly the same.  Fix such an assignment.

It remains to combine the block errors.  Because the distribution
is common to all edges in a block, the expected normalized block
equals the actual surviving normalized incidence tensored with its
single-edge expectation.  The surviving incidence is the compression
of \(U^0_{ik}\) and has norm at most one.  Equation
\eqref{sam:edge-mean} therefore bounds its difference from the
corresponding block of \(\Pi O\Pi\) by \(\varepsilon\).
The simultaneous success of \eqref{sam:block-probability} bounds
the centered normalized block by \(\varepsilon\).  A matrix with
layer width one and block norms bounded by \(\varepsilon\) has
norm at most \(3\varepsilon\): apply Cauchy--Schwarz to each row
and sum, or apply the scalar row-and-column Schur bound.  Its
self-adjoint off-diagonal doubling has the same norm.  Consequently
\[
 \|T_P-\Pi O\Pi\|\le6\varepsilon.
\]
Taking \(\varepsilon<\eta/6\) proves
\eqref{sam:approx}.  All expectation errors used above were made
smaller than this chosen \(\varepsilon\) by first fixing compact
cutoffs, log-grid spacing, empirical accuracies, Fourier cutoffs,
and scalar rounding, and then increasing \(\ell\).  The order of
the constants is thus: the geometric and nilpotence parameters;
the fixed desired accuracy and its coefficient data; the finite
test exponent \(C_1\); the moment constant \(C_2\); and finally
\(\ell\).  All estimates are uniform in the layers.  This proves
Theorem~\ref{sam:realization}.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//sampling.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//scalar.tex ====
\section{From a small matrix projection to a scalar projection}
\label{sec:scalar}

We first isolate a reduction that will be useful independently of the
construction. Its trace convention is essential.
For related results on the scale and comparison of projections in
reduced free products, see Dykema--R{\o}rdam \cite{DR1998,DR2000,DRII2000}.
The explicit row estimate used here is proved below.
For a discrete group \(\Lambda\), let
\[
 (\lambda_\Lambda(g)\xi)(h)=\xi(g^{-1}h),\qquad
 C_r^*(\Lambda)=
 \overline{\lambda_\Lambda(\mathbb C[\Lambda])}^{\,\|\cdot\|}.
\]
The canonical trace is
\(\tau_\Lambda(x)=\langle x\delta_1,\delta_1\rangle\).
It is tracial on finite group-algebra sums by taking the identity
coefficient, and hence on their norm closure. It is faithful:
if \(x\ge0\) and \(\tau_\Lambda(x)=0\), then
\(x^{1/2}\delta_1=0\). Since \(x^{1/2}\) commutes with the
right regular action, it annihilates every basis vector and is zero.
On matrices we always use the \emph{unnormalized} extension
\begin{equation}
 \tau_{\Lambda,N}(x)=\sum_{j=1}^N\tau_\Lambda(x_{jj}).
 \label{scalar:matrix-trace}
\end{equation}
The same formula applies to positive equivariant operators on finite
regular multiples. In particular,
\(\tau_{\Lambda,N}(p)>0\) for a nonzero projection \(p\).

\begin{lemma}[Reduced subgroup inclusion]
\label{scalar:subgroup}
If \(\Lambda\) is a subgroup of a discrete group \(K\), the inclusion of
group algebras extends to an isometric unital inclusion
\(C_r^*(\Lambda)\subseteq C_r^*(K)\), preserving the canonical traces
and their finite matrix extensions.
\end{lemma}

\begin{proof}
Choose one representative \(v\) from every left \(\Lambda\)-orbit
\(\Lambda v\) in \(K\). The unitary
\(\delta_\gamma\mapsto\delta_{\gamma v}\) identifies the restriction
of the left regular action to \(\ell^2(\Lambda v)\) with the left
regular action on \(\ell^2(\Lambda)\). Thus the representation
restricted to \(\mathbb C[\Lambda]\) is a direct sum of regular
representations, so its norm is precisely the reduced norm on
\(\Lambda\). Taking closures gives the inclusion. The identity
diagonal proves preservation of trace; summing diagonal entries proves
the matrix statement.
\end{proof}

\begin{proposition}[Scalar compression after a free product]
\label{scalar:compression}
Let \(\Lambda\) be a discrete group and
\(p\in M_N(C_r^*(\Lambda))\) a projection with
\[
 0<a:=\tau_{\Lambda,N}(p)<\frac12.
\]
Put \(K=\Lambda*\mathbb F_N\), where \(t_1,\ldots,t_N\) freely
generate \(\mathbb F_N\), and regard \(p\) as a matrix over
\(C_r^*(K)\) by Lemma~\ref{scalar:subgroup}. Set
\[
 U=(\lambda_K(t_1),\ldots,\lambda_K(t_N)),\qquad
 A=pU^*Up.
\]
Then \(A\) is invertible in the unital corner
\(pM_N(C_r^*(K))p\), and
\begin{equation}
 e=(Up)A^{-1}(Up)^*
 \label{scalar:projection-formula}
\end{equation}
is a projection in the scalar algebra \(C_r^*(K)\) with
\(\tau_K(e)=a\).
If \(\Lambda\) is finitely generated and torsion-free, then so is \(K\).
\end{proposition}

\begin{proof}
To bound \(U\) below on \(p\ell^2(K)^N\), we separate
noncancelling inputs, whose outputs are disjoint, from cancelling
inputs, whose size both before and after applying \(U\) is controlled
by the trace \(a\).
Use free-product normal forms, writing entries of the free factor
in reduced free letters. Let \(\mathcal R\) consist of the identity
and the normal words whose first factor is in \(\mathbb F_N\).
Every element of \(K\) has a unique expression \(\gamma v\) with
\(\gamma\in\Lambda\) and \(v\in\mathcal R\): remove its initial
\(\Lambda\)-factor, if present. Thus \(\mathcal R\) is a transversal
for the orbits used in Lemma~\ref{scalar:subgroup}.

Write \(a_j=\tau_\Lambda(p_{jj})\), so
\(a_j\ge0\) and \(\sum_j a_j=a\). If
\(\xi=p\xi\in\ell^2(K)^N\), put
\[
 m_v=\|\xi|_{\Lambda v}\|^2,\qquad
 \sum_{v\in\mathcal R}m_v=\|\xi\|^2.
\]
Restriction to an orbit commutes with \(p\).
Applying Cauchy--Schwarz to the projection on that orbit gives
\begin{equation}
 |\xi_j(v)|^2
 =|\langle \xi|_{\Lambda v},p\delta_{(v,j)}\rangle|^2
 \le a_jm_v.
 \label{scalar:pointwise}
\end{equation}
Here
\(\|p\delta_{(v,j)}\|^2
=\langle p\delta_{(v,j)},\delta_{(v,j)}\rangle=a_j\).

In coordinate \(j\), call a position bad if its reduced word starts
with \(t_j^{-1}\), and good otherwise. Split \(\xi\) accordingly as
\(\xi=\xi_{\mathrm g}+\xi_{\mathrm b}\).
Every bad position belongs to \(\mathcal R\); its first letter
determines its bad coordinate uniquely. Therefore
\begin{equation}
 \|\xi_{\mathrm b}\|^2
 \le a\sum_{v\in\mathcal R}m_v=a\|\xi\|^2.
 \label{scalar:bad-input}
\end{equation}
Left multiplication by \(t_j\) sends every good position to a word
starting with \(t_j\). These output sets are disjoint for different
\(j\), and multiplication is injective within each coordinate. Hence
\begin{equation}
 \|U\xi_{\mathrm g}\|^2
 =\|\xi_{\mathrm g}\|^2
 \ge(1-a)\|\xi\|^2.
 \label{scalar:good-output}
\end{equation}

Bad outputs may coincide, and require a separate estimate.
For an output word \(h\), let \(J(h)\) be the coordinates whose bad
input \(v_{h,j}=t_j^{-1}h\) contributes to \(h\).
By \eqref{scalar:pointwise},
\[
 |(U\xi_{\mathrm b})(h)|^2
 \le\left(\sum_{j\in J(h)}a_j\right)
      \left(\sum_{j\in J(h)}m_{v_{h,j}}\right)
 \le a\sum_{j\in J(h)}m_{v_{h,j}}.
\]
The map \((h,j)\mapsto v_{h,j}\), restricted to these bad inputs,
is injective: the first letter of \(v_{h,j}\) determines \(j\),
and then \(h=t_jv_{h,j}\). Summing over \(h\) gives
\begin{equation}
 \|U\xi_{\mathrm b}\|^2\le a\|\xi\|^2.
 \label{scalar:bad-output}
\end{equation}
Equations~\eqref{scalar:good-output} and
\eqref{scalar:bad-output}, and the triangle inequality, imply
\begin{equation}
 \|U\xi\|\ge c_a\|\xi\|,\qquad
 c_a=\sqrt{1-a}-\sqrt a>0.
 \label{scalar:lower-bound}
\end{equation}

Consequently \(A\ge c_a^2p\) in its C*-corner.
Also \(UU^*=N1\), so \(A\le Np\). In particular, its inverse belongs
to the corner itself: explicitly,
\[
 A^{-1}=\frac1N\sum_{k=0}^\infty(p-A/N)^k
\]
converges in norm because
\(\|p-A/N\|\le1-c_a^2/N<1\).
Continuous functional calculus also supplies \(A^{-1/2}\) there.
The row \(V=UpA^{-1/2}\) belongs to
\(M_{1,N}(C_r^*(K))\) and satisfies \(V^*V=p\).
Thus \(e=VV^*\) is the scalar projection in
\eqref{scalar:projection-formula}. Writing \(V=(V_1,\ldots,V_N)\)
and using traciality,
\[
 \tau_K(e)
 =\sum_j\tau_K(V_jV_j^*)
 =\sum_j\tau_K(V_j^*V_j)
 =\tau_{K,N}(p)=a.
\]
It is therefore neither zero nor one.

Finally, adjoining the finite set \(t_1,\ldots,t_N\) preserves
finite generation. A nontrivial element of a free product is
conjugate either to a nontrivial factor element or to a cyclically
reduced word with at least two alternating factor entries.
Positive powers of the latter have strictly increasing reduced
length. If both factors are torsion-free, neither case has finite
order. This proves the last assertion.
\end{proof}

It remains to construct a finitely generated torsion-free
\(\Gbase\) and a projection satisfying the small
\emph{unnormalized} trace hypothesis of
Proposition~\ref{scalar:compression}. The next sections do so.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//scalar.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//setup.tex ====
\section{Conventions, parameters, and the matrix target}
\label{sec:setup}

We will prove the following intermediate statement.

\begin{theorem}[Matrix construction]
\label{setup:matrix-target}
There exist a finitely generated torsion-free discrete group
\(\Gbase\), a finite integer \(N\ge1\), and a self-adjoint element
\(T\in M_N(\mathbb C[\Gbase])\), in its regular representation,
with a nonempty separated spectral subset whose spectral projection
\(p_0\) satisfies
\[
 p_0\in M_N(C_r^*(\Gbase)),\qquad
 0<\tau_{\Gbase,N}(p_0)<\frac12.
\]
\end{theorem}

Here and below a spectral subset is \emph{separated} if its distance
from the rest of the spectrum is positive. Its indicator is
continuous on the spectrum. The spectral projection therefore
belongs to the unital \(C^*\)-algebra containing the operator,
by continuous functional calculus. We also use the equivalent
contour-resolvent formula. These facts, together with polar
decomposition, are standard bounded-operator spectral theory
\cite[Chapters~VI--VII]{ReedSimon1980}.

In graph coordinates it is convenient to use right translations
\[
 (\rho_\Lambda(g)\xi)(h)=\xi(hg).
\]
The unitary \(\mathcal I\xi(h)=\xi(h^{-1})\) satisfies
\(\mathcal I\rho_\Lambda(g)\mathcal I=\lambda_\Lambda(g)\).
This identification fixes the identity vector and leaves finite
matrix indices unchanged, so it preserves all traces used here.
Thus a finite matrix sum of right translations specifies the same
kind of reduced-algebra element as in Theorem~\ref{setup:matrix-target}.

\subsection{Graphs and labels}

Edges have length one and may be traversed in either direction.
Whenever a positive orientation is specified, reverse traversal
carries the formal inverse label.
A labeled graph is \emph{folded} if, at each vertex, a prescribed
oriented label occurs on at most one outgoing edge.
A path is nonbacktracking when it never immediately reverses an
edge; a closed path is cyclically nonbacktracking if this also
holds across its chosen basepoint.
Closed walks and their repeated traversals are allowed as
presentation relators.

For a labeled group presentation we retain the formal Cayley
edges: a positive generator and its initial vertex specify an
edge even if different formal generators have the same group
value. All degree and forest assertions are about this graph.
This convention also specifies an unambiguous left action on
oriented edges. Intrinsic distances and balls in an embedded
copy are measured using its own graph edges.

All vertex Hilbert spaces use counting measure.
A biregular incidence from a \(B\)-side of degree \(D'\) to an
\(A\)-side of degree \(d\) is normalized by
\((dD')^{-1/2}\). It maps the normalized constant vector on
one side to the normalized constant vector on the other.
When using probabilities to prove a bound for that operator,
we pass explicitly to uniform probability on each side.

\subsection{The growing parameter}

Let \(\ell\) be a positive integer, eventually sufficiently large,
and set
\begin{equation}
\begin{aligned}
 d&=2^{4\ell},& D&=d^2,& s&=10\ell,& r&=7\ell,\\
 n_0&=1000,&
 H_i&=(2n_0+1)s+1+|i| \quad(i\in\mathbb Z).
\end{aligned}
\label{setup:parameters}
\end{equation}
Write \(H_0=(2n_0+1)s+1\), and put
\begin{equation}
 L_i=\frac{100H_i}{\log_2d},\qquad
 g_i=\frac{H_i}{100\log_2d}.
 \label{setup:lengths}
\end{equation}
Inequalities comparing integer walk lengths with these real
numbers have their usual literal meaning; a lower required
length can equivalently be rounded upward.
We may use the fixed bound
\begin{equation}
 H_0\le h_*\log_2d,\qquad h_*=5003,
 \label{setup:hstar}
\end{equation}
valid for every \(\ell\ge1\).

Edges in the base tower change the layer by at most one.
Consequently, for any fixed \(C<\infty\),
\begin{equation}
 |j-i|\le \frac{CH_i}{\log_2d}
 \quad\Longrightarrow\quad
 \left|\frac{H_j}{H_i}-1\right|
 \le\frac{C}{\log_2d}.
 \label{setup:nearby}
\end{equation}
The convergence to one is uniform in \(i\). Constants such as
\(C\) may be very large, but are fixed before \(\ell\) grows.

Unless stated otherwise, a term \(o(1)\) tends to zero as
\(\ell\to\infty\), uniformly over all allowed layers, blocks,
and choices already declared arbitrary.
Fixed constants may depend on fixed accuracies and earlier
parameters. They may not depend on \(\ell\) or a layer index.
Later sections record the order of the remaining choices:
geometric constants first, then nilpotence and mask scales,
then operator accuracies and finite test constants, and
finally a sufficiently large finite \(\ell\).
We do not choose different values of \(\ell\) for different layers.

\subsection{The order of choices}

Table~\ref{tab:projection-parameters} records which choices are fixed before
which others. It is a guide to the estimates proved below: each indicated
section supplies the corresponding inequality. All choices use one final
value of \(\ell\), uniformly over the infinitely many layers.

\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.38\textwidth}>{\raggedright\arraybackslash}p{0.29\textwidth}@{}}
\toprule
Choice & Required conclusion & Proof location \\
\midrule
Geometric constants \(C_*,C_\bullet\) &
One ball contains each earlier overlap; the unused edges form a forest &
Section~\ref{geo:section} \\[3pt]
Nilpotence and scale constants \(m,E_0,A_*,\kappa\) &
Short words are detected; both endpoint regimes have small weighted trace &
Section~\ref{nil:section} \\[3pt]
Accuracy \(\eta\), coefficient cutoffs and finite lists &
Uniform approximation in the full regular representation with a finite weight alphabet &
Section~\ref{sam:section}; Equation~\eqref{spec:eta-choice} \\[3pt]
Finite-test and moment constants \(C_1,C_2\) &
The sum of all voltage and operator failure bounds is less than one &
Equation~\eqref{sam:C2-order} and Theorem~\ref{sam:realization} \\[3pt]
One sufficiently large \(\ell\) &
Deletion loss, weighted trace, and forest error meet all spectral tolerances &
Theorem~\ref{spec:matrix-projection} \\
\bottomrule
\end{tabular}
\caption{Parameter ledger for the projection construction. Constants in a row
may depend on earlier fixed choices; only the final row increases \(\ell\).
The matrix dimension \(N\) is a prescribed function of \(\ell\), and its
unnormalized contribution is retained in the trace estimates.}
\label{tab:projection-parameters}
\end{table}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//setup.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//spectral.tex ====
\section{From the cover band to the Cayley projection}
\label{spec:section}

Lemma~\ref{spec:model} identifies the model's separated upper space;
Theorem~\ref{sam:realization} supplies a voltage-cover approximation to its
compression.
We now transfer the band to that cover and then to the Cayley graph.
Disjoint assigned domains give the spectral comparison, while all indexed
copies will give an equivariant comparison of traces.

Retain the gap $\chi$, deleted fraction $\beta_\ell$, model $O(t)$, and
upper projection $E_*(t)$ from the preceding section. With
$\mathfrak w_i=2^{C_\bullet H_i}$, Proposition~\ref{nil:weighted} gives
\begin{equation}
 \mu_\ell:=\sup_{\substack{0<t<\delta\\i\in J(t)}}
       \mathfrak w_i\,\operatorname{tr}_{\mathrm{id}}p(t)=o(1).
 \label{spec:weighted-input}
\end{equation}
The trace here and throughout this section includes the ordinary,
\emph{unnormalized} sum over the $N$ matrix indices.

Theorem~\ref{sam:realization} supplies, for any fixed accuracy $\eta>0$ and
all sufficiently large $\ell$, an actual self-adjoint weighted adjacency
$T_P$ on the voltage cover, satisfying
Equation~\eqref{sam:approx}.  Thus
\begin{equation}
  \|T_P-\Pi O\Pi\|\leq\eta,
  \label{spec:approx-input}
\end{equation}
where $\Pi$ deletes the removed base vertices, and $T_P$ is extended by zero
on those vertices.  Both operators in Equation~\eqref{spec:approx-input}
have layer width one.  Every edge weight before normalization has norm at
most a fixed $C_\eta$, the normalized weight is determined by its augmented
label, and the finite positive alphabet $S$ satisfies $|S|=o(dD)$.  The cover
satisfies the girth hypothesis of Theorem~\ref{geo:main}.  That Theorem
provides the indexed embedded copies, their disjoint assigned domains, the
finite trimming balls, the incidence count, the remaining forest, and the
torsion-freeness of $\Gbase$.  In particular these are conclusions proved
before this section, rather than additional spectral assumptions.

\subsection{Compression and norm perturbation}

We record quantitative Hilbert-space estimates, including the projection
equivalences needed later for the trace.

\begin{lemma}[Compression of a separated upper space]
\label{spec:compression}
Let $A=A^*$ on a Hilbert space, with $\|A\|\leq2$, and let $E$ be a
reducing projection such that
\[
 \|(A-I)E\|\leq a,\qquad
 A|_{(1-E)\mathcal H}\leq(1-b)I,\qquad 0<b<1.
\]
Let $J$ be an orthogonal projection with
$\varepsilon:=\|(1-J)E\|\leq1/2$.  The operator $JE$ has closed range.
Writing $E^J$ for the projection onto that range, put, on $J\mathcal H$,
\begin{equation}
 B_J=E^J+(J-E^J)A(J-E^J).
 \label{spec:compression-model}
\end{equation}
Then
\begin{equation}
 \|JAJ-B_J\|\leq a+20\varepsilon,
 \qquad B_J|_{(J-E^J)\mathcal H}\leq(1-b)I,
 \qquad \|B_J\|\leq2.
 \label{spec:compression-error}
\end{equation}
There is a partial isometry $V$ with $V^*V=E$, $VV^*=E^J$, and
$\|V-E\|\leq2\varepsilon$.  If $A,E,J$ commute with a specified unitary
or projective unitary action, then so do all the projections and maps just
constructed.  In particular, in a setting with the identity trace of
Lemma~\ref{nil:trace}, $E$ and $E^J$ have equal identity traces.  These
statements include $E=0$.
\end{lemma}

\begin{proof}
In the corner $E\mathcal H$,
\[
       (1-\varepsilon^2)E\leq EJE\leq E.
\]
Thus $JE$ is bounded below on $E\mathcal H$, and
\[
        V=JE(EJE)^{-1/2}
\]
is the asserted partial isometry.  The inverse is taken in the $E$ corner.
For $\varepsilon\leq1/2$, functional calculus gives
$\|(EJE)^{-1/2}-E\|\leq2\varepsilon^2$, whence
$\|V-E\|\leq\varepsilon+2\varepsilon^2\leq2\varepsilon$.
Every vector in $(J-E^J)\mathcal H$ is orthogonal to $E\mathcal H$:
if $Jy=y$ and $y\perp JE\mathcal H$, then
$\langle y,Ex\rangle=\langle y,JEx\rangle=0$.  This proves the
complementary upper bound in Equation~\eqref{spec:compression-error}.

Transporting the $E^J$ corner by $V$ yields
\[
 \|V^*AV-E\|
 \leq a+2\|A\|\,\|V-E\|
 \leq a+8\varepsilon.
\]
Reduction by $E$ and the preceding orthogonality also give
\[
 \|(J-E^J)AV\|
 =\|(J-E^J)A(V-E)\|\leq4\varepsilon.
\]
The difference between $JAJ$ and Equation~\eqref{spec:compression-model}
has just this pair of adjoint off-diagonal blocks and the estimated upper
diagonal block.  The norm of the paired off-diagonal blocks is the norm
of either one.  Its total norm is at most $a+12\varepsilon$, and therefore
at most the stated $a+20\varepsilon$.  The orthogonal two-part form of
$B_J$ gives $\|B_J\|\leq2$.  Equivariance follows from the formulas and
functional calculus.  Equality of the two traces follows from the
adjoint-square identity for $V$ in Lemma~\ref{nil:trace}.
\end{proof}

\begin{lemma}[An elementary contour estimate]
\label{spec:perturbation}
Suppose $B=B^*$ has a reducing projection $F$ with $BF=F$, and its
complementary spectrum is contained in $[-2,1-b]$, where $0<b<1$.
Let $r_0>0$ satisfy $r_0\leq b/2$ and $r_0<1$.  If
$A=A^*$ and $\|A-B\|\leq u<r_0$, the circle
$\mathcal C_{r_0}=\{z:|z-1|=r_0\}$ lies in the resolvent sets of both
operators.  The enclosed spectral projection $F_A$ satisfies
\begin{equation}
 \|F_A-F\|\leq\frac{u}{r_0-u}.
 \label{spec:projection-distance}
\end{equation}
Its enclosed spectrum lies in $[1-u,1+u]$; all the other spectrum of $A$
is at most $1-b+u$.  If $u<r_0/2$, the projections $F_A$ and $F$ are
equivalent by a partial isometry belonging to the algebra of bounded
operators generated by them.  This equivalence preserves any common
equivariance and its identity trace.
\end{lemma}

\begin{proof}
On $\mathcal C_{r_0}$ one has
$\|(z-B)^{-1}\|\leq r_0^{-1}$.  The resolvent Neumann series gives
$\|(z-A)^{-1}\|\leq(r_0-u)^{-1}$.  The resolvent identity and integration
around the circle then give Equation~\eqref{spec:projection-distance}.
For a real number at distance greater than $u$ from $\operatorname{spec}B$,
the same Neumann argument proves it is outside $\operatorname{spec}A$.
This yields the asserted spectral intervals.

For completeness, if projections $F_1,F_2$ have $\|F_1-F_2\|<1$, the map
$F_2F_1$ is bounded below on $F_1\mathcal H$, since
$\|F_2x\|\geq(1-\|F_1-F_2\|)\|x\|$ there.  Its range is all of
$F_2\mathcal H$: a vector in that space orthogonal to its range would
satisfy $F_1x=0$ and hence $\|(F_2-F_1)x\|=\|x\|$.  Its polar
normalization
\[
 F_2F_1(F_1F_2F_1)^{-1/2}
\]
therefore has initial projection $F_1$ and final projection $F_2$.
Apply this with $F_1=F$, $F_2=F_A$.  The equivariance and trace conclusions
follow from these formulas and Lemma~\ref{nil:trace}.
\end{proof}

Set
\begin{equation}
       \alpha=\chi/1000,\qquad r_0=\chi/3,
       \qquad C_\chi=4/\chi.
 \label{spec:accuracy-constants}
\end{equation}
Choose the fixed approximation accuracy $\eta>0$ so small that
\begin{equation}
 \eta\leq\alpha/2,\qquad
 \theta:=C_\chi\eta\leq1/2,\qquad
        2^{C_\bullet}\theta^2\leq1/2.
 \label{spec:eta-choice}
\end{equation}
The other fixed parameters of the construction have already been chosen;
Theorem~\ref{sam:realization} permits this accuracy.  We subsequently take
$\ell$ sufficiently large that
\begin{equation}
       20\sqrt{\beta_\ell}\leq\alpha/2.
 \label{spec:deletion-choice}
\end{equation}
All inequalities below concern such $\ell$ and are uniform in the layers.

\begin{proposition}[The actual cover projection and its exact fiber trace]
\label{spec:cover-projection}
The operator $T_P$ has a separated upper spectral projection $E_P$,
obtained by integration around $\mathcal C_{r_0}$.  It satisfies
\begin{equation}
 \|T_P\|\leq2,\qquad
 \|(T_P-I)E_P\|\leq\alpha,\qquad
 T_P|_{(1-E_P)\mathcal H}\leq(1-\chi+\alpha)I.
 \label{spec:cover-gap}
\end{equation}
In the central direct integral, for almost every $t$,
\begin{equation}
 \operatorname{tr}_{\mathrm{id}}E_P(t)
  =\begin{cases}
        \operatorname{tr}_{\mathrm{id}}p(t),&0<t<\delta,\\
        0,&t\notin(0,\delta).
    \end{cases}
 \label{spec:exact-fiber-trace}
\end{equation}
The projection $E_P$ is nonzero and reduces to the connected components of
$P$.  Its diagonal at a lifted vertex depends only on the base vertex.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{spec:compression} to $O(t),E_*(t),\Pi$ with $a=0$ and
$b=\chi$.  Extend its two-part comparison operator by zero on
$(1-\Pi)\mathcal H_t^0$.  The extended complementary part is still at
most $1-\chi$, because $1-\chi>0$.  Denote its upper projection by
$E_*^\Pi(t)$.  Equations~\eqref{spec:approx-input},
\eqref{spec:deletion-choice}, and \eqref{spec:eta-choice} show that $T_P(t)$
differs from this comparison operator by at most $\alpha$ for almost
every $t$.  The passage from the full regular norm in
Equation~\eqref{spec:approx-input} to this almost-everywhere statement is
the direct-integral norm formula from the fiber construction.  Lemma~\ref{spec:perturbation}
with $r_0=\chi/3$ gives a common contour, Equation~\eqref{spec:cover-gap},
and
\[
       \|E_P(t)-E_*^\Pi(t)\|
              \leq\frac{\alpha}{r_0-\alpha}<1.
\]
All these fields are measurable by the resolvent formulas, and their
uniform bounds define the corresponding bounded direct-integral
operators.  The norm bound also follows directly from
$\|\Pi O\Pi\|\leq1$ and $\eta<1$.

Both the compression equivalence and the close-projection equivalence
commute with the fiber right actions used to define the identity trace.
Lemma~\ref{nil:trace} and Lemma~\ref{spec:model} therefore imply the
equality of traces in Equation~\eqref{spec:exact-fiber-trace}, with no
approximation error.  Outside the window the model is zero and
$\|T_P(t)\|\leq\eta$ almost everywhere.  Consequently its contour
projection is exactly zero there.  This conclusion uses the small norm
of the Fourier-mask leakage supplied by Theorem~\ref{sam:realization}; it
does not require the finite Fourier masks to vanish exactly.

Since $\operatorname{tr}_{\mathrm{id}}p(t)>0$ on the interval of positive
measure $(0,\delta)$, Equation~\eqref{spec:exact-fiber-trace} makes $E_P$
nonzero.  Every component subspace reduces the actual adjacency $T_P$,
and hence its resolvent and $E_P$.  Left deck translations commute with
$T_P$, since its entries are right deck translations with the prescribed
matrix coefficients.  They therefore commute with $E_P$.  They send any
lift of a fixed base vertex to every other lift, proving the last
assertion, even when they permute different connected components.
\end{proof}

\subsection{Localization of identity trace}

For a surviving base vertex $v$, let
\begin{equation}
 e_v=\sum_{j=1}^N
  \big\langle E_P(\delta_{(v,1_F)}\otimes e_j),
                \delta_{(v,1_F)}\otimes e_j\big\rangle\geq0,
 \qquad
 \mathcal S=\sum_{i\in\mathbb Z}\mathfrak w_i
                 \sum_{v\in V(Y)\cap(A_i\cup B_i)}e_v.
 \label{spec:weighted-size}
\end{equation}
Here $e_j$ denotes the standard matrix basis; put $e_v=0$ at a deleted
base vertex when summing over $Y^0$.  These are regular identity
diagonals before the central Fourier decomposition.

The weights record the two counts in
Equations~\eqref{geo:ball-weighted-sum} and~\eqref{geo:lost-weighted-sum}.
A fixed base type may have many lifts inside one trimming ball, while
the sum of losses over copies carries a ball-size factor from
Cauchy--Schwarz. Controlling $\mathcal S$ will keep both forms of
overlap loss small.

\begin{proposition}[The weighted size tends to zero]
\label{spec:localization}
For the fixed choice of $\eta$ in Equation~\eqref{spec:eta-choice},
\begin{equation}
                \mathcal S\leq96\mu_\ell=o(1).
 \label{spec:S-small}
\end{equation}
In particular the bound contains no additional factor equal to a layer
cardinality or to the matrix dimension already included in $\mu_\ell$.
\end{proposition}

\begin{proof}
Write $A_0(t)=\Pi O(t)\Pi$ and $\Delta(t)=T_P(t)-A_0(t)$.  The comparison
operator in the proof of Proposition~\ref{spec:cover-projection} is
within $\alpha/2$ of $A_0(t)$.  Therefore on $\mathcal C_{r_0}$,
\begin{equation}
 R_0(z,t)=(z-A_0(t))^{-1},\qquad
       \|R_0(z,t)\|\leq(r_0-\alpha/2)^{-1}\leq C_\chi.
 \label{spec:model-resolvent}
\end{equation}
Let $P_i$ be the projection to both sides of layer $i$, including all
transverse and matrix coordinates.  The operator $A_0(t)$ is supported
on the active layers $J(t)$.  Thus its resolvent is $z^{-1}I$ on the
orthogonal sum of the other layers, and has no matrix entries between
that sum and the active layers.  Since $\Delta(t)$ has layer width one,
its resolvent expansion is the norm-convergent series
\begin{equation}
 (z-T_P(t))^{-1}
       =\sum_{n=0}^\infty R_0(z,t)(\Delta(t)R_0(z,t))^n.
 \label{spec:resolvent-series}
\end{equation}
If $k=\operatorname{dist}(i,J(t))$ and $n<k$, all layer paths contributing
after multiplication on the left by $P_i$ remain outside $J(t)$.  Each
resolvent factor along such a path equals $z^{-1}I$; equivalently,
by induction on the number of width-one factors,
\[
 P_iR_0(z,t)(\Delta(t)R_0(z,t))^n
                   =z^{-n-1}P_i\Delta(t)^n\qquad(n<k).
\]
Its contour integral is zero, because the circle excludes zero.  Integrate
the remaining terms of Equation~\eqref{spec:resolvent-series} and use
Equation~\eqref{spec:model-resolvent}.  On the window this gives
\begin{equation}
     \|P_iE_P(t)\|
       \leq\frac{r_0C_\chi}{1-\theta}\theta^k
       \leq4\theta^{\operatorname{dist}(i,J(t))}.
 \label{spec:layer-norm}
\end{equation}
This estimate is also valid when $k=0$; there is then no vanishing initial
segment to remove.

Let $\mathfrak t_t$ denote the identity trace, summed over all base and
matrix indices of the fiber.  It applies to the countable amplification
by Lemma~\ref{nil:trace}.  Since $P_i$ and $E_P(t)$ are equivariant,
the adjoint-square identity and positivity imply
\begin{align}
 \mathfrak t_t(P_iE_P(t)P_i)
   &=\mathfrak t_t(E_P(t)P_iE_P(t))\notag\\
   &\leq\|P_iE_P(t)\|^2\,\mathfrak t_t(E_P(t))\notag\\
   &\leq16\theta^{2\operatorname{dist}(i,J(t))}
                    \operatorname{tr}_{\mathrm{id}}p(t).
 \label{spec:layer-trace}
\end{align}
The middle inequality follows from the operator inequality
$0\leq E_P(t)P_iE_P(t)\leq\|P_iE_P(t)\|^2E_P(t)$.  Thus it is the
\emph{total trace of the projection}, rather than the dimension of the
layer, that occurs.

The elementary bound $|H_i-H_j|\leq|i-j|$ implies
$\mathfrak w_i\leq\mathfrak w_j2^{C_\bullet|i-j|}$.
Assign every integer to one of its closest points in the nonempty set
$J(t)$.  At most two integers at each positive distance can be assigned
to a given point.  Since $|J(t)|\leq2$, Equation~\eqref{spec:eta-choice}
gives
\begin{align*}
 \sum_i\mathfrak w_i\theta^{2\operatorname{dist}(i,J(t))}
 &\leq2\max_{j\in J(t)}\mathfrak w_j
          \left(1+2\sum_{k\geq1}
                       (2^{C_\bullet}\theta^2)^k\right)\\
 &\leq6\max_{j\in J(t)}\mathfrak w_j.
\end{align*}
Multiplying Equation~\eqref{spec:layer-trace} by $\mathfrak w_i$ and
summing proves a pointwise bound of $96\mu_\ell$ for the weighted fiber
trace.  The projection vanishes off the window by
Equation~\eqref{spec:exact-fiber-trace}.  Central Fourier decomposition
identifies $e_v$ with the integral of its fiber identity diagonal.
Tonelli's Theorem applies to the nonnegative diagonals and sums, so
integrating over the probability circle proves
Equation~\eqref{spec:S-small}.
\end{proof}

\subsection{An actual element of the matrix reduced group algebra}

Use the finite alphabet supplied by Theorem~\ref{sam:realization} to form
the group $\Gbase$ of Theorem~\ref{geo:main}.  For a positive label
$\sigma$, let $W_\sigma$ be its matrix weight and let $D_\sigma=D_{ik}$
be its label-determined degree normalizer.  Put
\begin{equation}
 w_\sigma=(dD_\sigma)^{-1/2}W_\sigma,\qquad
 (T\xi)(g)=\sum_{\sigma\in S}
       \big(w_\sigma\xi(g\sigma)+w_\sigma^*\xi(g\sigma^{-1})\big).
 \label{spec:Cayley-operator}
\end{equation}
This is a finite sum of bounded matrix coefficients times right regular
translations.  It is self-adjoint, and inversion of the group coordinate
identifies it with an element of $M_N(\mathbb C[\Gbase])$ in the left
regular convention.  In particular
$T\in M_N(C_r^*(\Gbase))$.  We use the right convention in the proof,
so that $T$ commutes with left translations of $\Gbase$.

For an indexed copy $Q$, let $T_Q$ be the adjacency using just its own
formal edges.  The embedding and labels identify it with the restriction
of $T_P$ to the source component.  Let $E_Q$ be its upper spectral
projection on $\mathcal C_{r_0}$.  Equation~\eqref{spec:cover-gap}
holds for $T_Q,E_Q$, and if an incidence $(Q,u)$ has base type $v$,
\begin{equation}
           \operatorname{Tr}_{\mathbb C^N}E_Q(u,u)=e_v.
 \label{spec:copy-diagonal}
\end{equation}
The equation follows from component reduction and deck equivariance in
Proposition~\ref{spec:cover-projection}; it does not require the connected
component to be invariant under every deck translation.

\begin{lemma}[Finite trimming costs little operator norm]
\label{spec:copy-cut}
Let $D_Q$ and $B(Q)$ be the assigned domain and trimming ball from
Theorem~\ref{geo:main}.  Then
\begin{equation}
             \|\mathbf1_{B(Q)}E_Q\|^2\leq\mathcal S.
 \label{spec:ball-HS}
\end{equation}
For $\mathcal S\leq1/4$, the compression of $T_Q$ to
$\ell^2(D_Q)\otimes\mathbb C^N$ is within
$\alpha+20\sqrt{\mathcal S}$ of an operator $C_Q$ which is $1$ on a
projection $F_Q$ equivalent to $E_Q$, has norm at most $2$, and is at
most $1-\chi+\alpha$ on the complementary space.  In particular
$F_Q\ne0$ whenever $E_Q\ne0$.
\end{lemma}

\begin{proof}
The ball is finite.  For the ordinary Hilbert--Schmidt norm on vertex
and matrix indices, the projection identity gives
\[
 \|\mathbf1_{B(Q)}E_Q\|^2
 \leq\|\mathbf1_{B(Q)}E_Q\|_{\mathrm{HS}}^2
 =\sum_{u\in B(Q)}\operatorname{Tr}_{\mathbb C^N}E_Q(u,u).
\]
A fixed base vertex $v$ can occur at multiple lifts in the ball.  Its
multiplicity is at most $|B(Q)|$, and whenever it occurs in layer $i$,
the ball bound in Theorem~\ref{geo:main} gives
$|B(Q)|\leq\mathfrak w_i$.  Grouping the nonnegative sum by base vertex,
using Equation~\eqref{spec:copy-diagonal}, bounds it by
$\sum_i\mathfrak w_i\sum_{v\in A_i\cup B_i}e_v=\mathcal S$.
This proves Equation~\eqref{spec:ball-HS} with repetitions included.

The missing vertices $V(Q)\setminus D_Q$ lie in $B(Q)$.  Apply
Lemma~\ref{spec:compression} with $A=T_Q$, $E=E_Q$,
$J=\mathbf1_{D_Q}$, $a=\alpha$, $b=\chi-\alpha$, and
$\varepsilon\leq\sqrt{\mathcal S}$.  Its operator $B_J$ is the required
$C_Q$ and its projection $E^J$ is $F_Q$.  The explicit partial isometry
in Lemma~\ref{spec:compression} proves the nonvanishing assertion.
\end{proof}

\begin{lemma}[Matrix adjacency on a forest]
\label{spec:forest-norm}
Suppose a countable forest has maximum degree at most $m_1$, and a
self-adjoint adjacency on it has matrix edge weights of norms at most
$w_1$.  Its norm is at most $2w_1\sqrt{m_1}$.
\end{lemma}

\begin{proof}
Choose one root in each tree and orient every edge from parent to child.
Let $B$ carry the contribution from the parent coordinate to the child
coordinate, with the corresponding matrix weight.  Every nonroot vertex
has only one parent.  Hence for a finitely supported vector $\xi$,
\[
 \|B\xi\|^2
   \leq w_1^2\sum_x\#\{\text{children of }x\}\,\|\xi(x)\|^2
   \leq m_1w_1^2\|\xi\|^2.
\]
The self-adjoint adjacency is $B+B^*$, which proves the assertion first
on finitely supported vectors and then by bounded extension.  Infinite
trees cause no extra term; each connected tree is rooted at an ordinary
vertex and its finite-distance parent relation is well-defined.
\end{proof}

\begin{proposition}[A nonzero upper band in the reduced representation]
\label{spec:global-gap}
For sufficiently large $\ell$, the operator in
Equation~\eqref{spec:Cayley-operator} satisfies
\begin{equation}
 \operatorname{spec}T
 \subset(-\infty,1-\chi+4\alpha]\,
             \cup[1-3\alpha,1+3\alpha].
 \label{spec:global-spectrum}
\end{equation}
The second spectral part is nonempty.  Its projection
\begin{equation}
          p_0=\mathbf1_{[1-\chi/4,\infty)}(T)
                    \in M_N(C_r^*(\Gbase))
 \label{spec:p0}
\end{equation}
is consequently nonzero.
\end{proposition}

\begin{proof}
Increase $\ell$ so that
\begin{equation}
 \mathcal S<\min\{1/4,1/1000\},\qquad
                    20\sqrt{\mathcal S}\leq\alpha.
 \label{spec:S-final-choice}
\end{equation}
The assigned domains are disjoint.  Form the bounded orthogonal direct
sum
\[
       C=\bigoplus_Q C_Q\oplus0,
       \qquad F_D=\bigoplus_Q F_Q\oplus0,
\]
where the extra zero acts on all unassigned vertices.  The uniform norm
bound in Lemma~\ref{spec:copy-cut} proves these are bounded operators on
the ambient $\ell^2(\Gbase)\otimes\mathbb C^N$.  The direct sum $C$ is
$1$ on $F_D$, has norm at most $2$, and is at most $1-\chi+\alpha$ on
$1-F_D$.  The zero summand satisfies this upper bound because
$1-\chi+\alpha>0$.

At least one component of $P$ has a nonzero upper projection, since
$E_P\ne0$.  Its label maps give indexed copies, and
Lemma~\ref{spec:copy-cut} preserves the nonzero upper space in every such
copy.  Thus $F_D\ne0$.  This argument uses the ordinary direct sum and
does not assert that the domain assignment is equivariant.

Let $T_D$ denote the direct sum of the compressed $T_Q$, extended by zero
on unassigned vertices.  Lemma~\ref{spec:copy-cut} gives
$\|T_D-C\|\leq2\alpha$.  On every formal edge internal to an assigned
copy domain, $T_D$ has exactly the same matrix coefficient as $T$,
because that coefficient is determined by the label.  Disjointness of
the domains and the formal-edge embeddings prevent double counting of
these internal edges.  By the forest conclusion of
Theorem~\ref{geo:main}, the remaining formal edges form a forest.
Consequently $T-T_D$ is precisely a matrix adjacency of the kind in
Lemma~\ref{spec:forest-norm}.  Its maximum degree is at most $2|S|$ and
its weight norm is at most
\[
       w_1\leq\frac{\sqrt2 C_\eta}{\sqrt{dD}}.
\]
It follows that
\begin{equation}
       f_\ell:=\|T-T_D\|
          \leq4C_\eta\sqrt{\frac{|S|}{dD}}=o(1).
 \label{spec:forest-error}
\end{equation}
Take $\ell$ still larger so that $f_\ell\leq\alpha$.  Then
$\|T-C\|\leq3\alpha$.  Lemma~\ref{spec:perturbation}, with
$b=\chi-\alpha$ and the same $r_0=\chi/3$, proves
Equation~\eqref{spec:global-spectrum}.  It also shows that the enclosed
projection of $T$ has distance at most
$3\alpha/(r_0-3\alpha)<1$ from the nonzero projection $F_D$, so it is
nonzero.  Since $\alpha=\chi/1000$, the number $1-\chi/4$ lies strictly
between the two intervals in Equation~\eqref{spec:global-spectrum}.

The represented reduced algebra is a unital $C^*$-subalgebra of the
bounded operators and contains $T$.  For clarity, its spectrum for this
self-adjoint element equals the operator spectrum: a resolvent away
from the real spectrum is obtained by continuous functional calculus,
or by uniformly approximating the reciprocal on that compact spectrum
with polynomials in $T$ and the identity.  The function that equals zero
on the lower interval and one on the upper interval is continuous on
the spectrum.  Continuous functional calculus therefore gives precisely
the spectral projection in Equation~\eqref{spec:p0} as an element of
$M_N(C_r^*(\Gbase))$.
\end{proof}

\subsection{Equivariant size of all local upper spaces}

The comparison with the assigned domains proves spectral separation, but
their ordering need not be equivariant. Thus the ordinary equivalence with
$F_D$ does not by itself compare canonical traces. For the trace estimate
we return to all indexed copies. Their upper spaces form an equivariant
space of small trace, and we will show that the common kernel of the
corresponding restrictions contains no nonzero vector of the global upper space.

Write $\mathcal H=\ell^2(\Gbase)\otimes\mathbb C^N$. For a left
$\Gbase$-equivariant positive operator $Z$ on
$\ell^2(\Gbase)\otimes\mathcal K$, with a fixed countable orthonormal
basis $(k_a)$ of $\mathcal K$, its identity trace is
\begin{equation}
 \operatorname{Tr}_{\Gbase} Z
       =\sum_a\langle Z(\delta_1\otimes k_a),
                              \delta_1\otimes k_a\rangle\in[0,\infty].
 \label{spec:equivariant-trace}
\end{equation}
The size of a closed invariant subspace is the identity trace of its
orthogonal projection. Lemma~\ref{nil:trace} proves basis independence,
the adjoint-square trace identity, and invariance under equivariant
partial isometries in these countably amplified spaces. In particular
the size of $p_0\mathcal H$ is
\[
 \tau_{\Gbase,N}(p_0)
    =\sum_{j=1}^N\tau_{\Gbase}((p_0)_{jj}),
\]
with no division by $N$.

Consider the Hilbert space and projection
\begin{equation}
 \mathcal L=\bigoplus_Q\big(\ell^2(V(Q))\otimes\mathbb C^N\big),
       \qquad\mathcal E=\bigoplus_Q E_Q.
 \label{spec:all-copies}
\end{equation}
Left translation acts simultaneously on $Q$ and its vertices. The
incidence bijection of Proposition~\ref{geo:incidences} identifies this
representation, including matrix indices, with
\begin{equation}
 \mathcal L\cong\ell^2(\Gbase)\otimes\ell^2(V(Y))\otimes\mathbb C^N.
 \label{spec:incidence-identification}
\end{equation}
More explicitly, $(Q,u)$ is sent to $(u,v)$ where $v$ is its base type.
For each $u\in\Gbase$ and $v\in V(Y)$ there is exactly one such incidence:
choose any lift of $v$, map its component with that lift sent to $u$,
and use the deck-alignment equivalence in the definition of indexed
copies. Different lift choices give the same incidence, and any
incidence arises this way. The injective label maps ensure that the
base type is unambiguous. This count is valid for disconnected covers
and for copies with nontrivial, possibly infinite stabilizers. The
action on the pairs $(Q,u)$ is free because it is free on their group
vertex $u$; freeness of the action on the set of copies is unnecessary.

The operator $\mathcal E$ is equivariant in this representation. Its
identity diagonal at the base type $v$ has matrix sum $e_v$, by
Equation~\eqref{spec:copy-diagonal}. Hence
\begin{equation}
 \operatorname{Tr}_{\Gbase}\mathcal E
     =\mathcal S_0:=\sum_{v\in V(Y)}e_v
     \leq\mathcal S<\infty.
 \label{spec:all-copies-size}
\end{equation}
Thus $\mathcal U:=\mathcal E\mathcal L$ is a closed invariant space of
finite identity trace, even though there are countably many copies and
base vertices.

\begin{lemma}[The total loss from earlier intersections]
\label{spec:lost-restrictions}
Put $I_Q=V(Q)\setminus D_Q\subset B(Q)$. For every $\xi\in\mathcal H$,
\begin{equation}
 \sum_Q\|E_Q(\xi|_{I_Q})\|^2\leq\mathcal S\|\xi\|^2,
 \label{spec:all-copy-cut-loss}
\end{equation}
where restrictions are extended by zero inside their copies.
\end{lemma}

\begin{proof}
Write $v(Q,u)=\operatorname{type}_Q(u)$ for the base type of an incidence,
and $i(v)$ for the layer of a base vertex $v$. For a vector
$z\in\mathbb C^N$ supported at $u\in V(Q)$, positivity of the diagonal
matrix and Equation~\eqref{spec:copy-diagonal} give
\[
 \|E_Q(\delta_u\otimes z)\|^2
       =\langle E_Q(u,u)z,z\rangle
       \leq e_{v(Q,u)}\|z\|^2.
\]
The set $I_Q$ is finite. Cauchy--Schwarz for its sum therefore gives
\begin{equation}
 \|E_Q(\xi|_{I_Q})\|^2
       \leq |B(Q)|\sum_{u\in I_Q}
                  e_{v(Q,u)}\,\|\xi(u)\|^2.
 \label{spec:finite-cut-CS}
\end{equation}
The right-hand side is zero for an empty cut. If $u\in I_Q$ has base
type in layer $i$, the ball bound gives $|B(Q)|\leq\mathfrak w_i$.
Rearranging nonnegative terms yields
\begin{align*}
 \sum_Q\|E_Q(\xi|_{I_Q})\|^2
 &\leq\sum_{u\in\Gbase}\|\xi(u)\|^2
       \sum_{Q:u\in I_Q}|B(Q)|e_{v(Q,u)}\\
 &\leq\sum_{u\in\Gbase}\|\xi(u)\|^2
       \sum_{Q:u\in V(Q)}\mathfrak w_{i(v(Q,u))}e_{v(Q,u)}\\
 &=\mathcal S\|\xi\|^2.
\end{align*}
The final equality is the all-copy incidence count at the fixed vertex
$u$. In particular the factor $|B(Q)|$ from Cauchy--Schwarz is paid by
the layer weight in $\mathcal S$; it is not discarded or assumed
uniformly bounded.
\end{proof}

Together with disjointness of the $D_Q$, this controls the restriction
map to all upper spaces.

\begin{lemma}[A bounded restriction operator and a size bound]
\label{spec:restriction}
The formula
\begin{equation}
    A\xi=\big(E_Q(\xi|_{V(Q)})\big)_Q
       \quad(\xi\in\mathcal H)
 \label{spec:analysis-operator}
\end{equation}
defines a bounded equivariant operator $A:\mathcal H\to\mathcal U$,
with $\|A\|\leq1+\sqrt{\mathcal S}$. Its kernel $K$ is a closed
invariant subspace, and every $\xi\in K$ satisfies
\begin{equation}
        E_Q(\xi|_{V(Q)})=0\quad\text{for every indexed }Q.
 \label{spec:kernel-restrictions}
\end{equation}
Moreover,
\begin{equation}
           \operatorname{Tr}_{\Gbase} P_{K^\perp}\leq\mathcal S_0.
 \label{spec:kernel-complement-size}
\end{equation}
\end{lemma}

\begin{proof}
Since the $D_Q$ are pairwise disjoint and each $E_Q$ is a contraction,
\[
 \sum_Q\|E_Q(\xi|_{D_Q})\|^2
       \leq\sum_Q\|\xi|_{D_Q}\|^2\leq\|\xi\|^2.
\]
Together with Lemma~\ref{spec:lost-restrictions}, this shows that both
terms on the right in
\[
 A\xi=\big(E_Q(\xi|_{D_Q})\big)_Q
             +\big(E_Q(\xi|_{I_Q})\big)_Q
\]
belong to $\mathcal L$. The triangle inequality there gives the
asserted norm bound. Each coordinate belongs to the range of $E_Q$,
so the image is in the closed space $\mathcal U$. Although the
decomposition used to prove boundedness depends on the ordering of
copies, the formula~\eqref{spec:analysis-operator} does not. Left
translation permutes the copies and intertwines their $E_Q$, so $A$
is equivariant. Its kernel is therefore closed and invariant, and
Equation~\eqref{spec:kernel-restrictions} follows directly from the
coordinate formula.

The polar partial isometry of $A$ has initial projection
$P_{K^\perp}$ and final projection onto $\overline{\operatorname{ran}A}$,
a subspace of $\mathcal U$. It is equivariant by the polar-decomposition
argument in Lemma~\ref{nil:trace}. The adjoint-square identity and
monotonicity of that trace, together with
Equation~\eqref{spec:all-copies-size}, prove
Equation~\eqref{spec:kernel-complement-size}.
\end{proof}

\begin{lemma}[The upper global space injects into the small complement]
\label{spec:kernel-form}
With the choices in Equations~\eqref{spec:S-final-choice} and
\eqref{spec:forest-error},
\begin{equation}
 \langle T\xi,\xi\rangle
       \leq(1-\chi+3\alpha)\|\xi\|^2
              \quad(\xi\in K).
 \label{spec:kernel-form-bound}
\end{equation}
Consequently the bounded equivariant map
$P_{K^\perp}:p_0\mathcal H\to K^\perp$ is injective, and
\begin{equation}
        0<\tau_{\Gbase,N}(p_0)
               \leq\mathcal S_0\leq\mathcal S<1/2.
 \label{spec:strict-trace}
\end{equation}
\end{lemma}

\begin{proof}
For $\xi\in K$, Equation~\eqref{spec:kernel-restrictions} gives
\[
       E_Q(\xi|_{D_Q})=-E_Q(\xi|_{I_Q}).
\]
Lemma~\ref{spec:lost-restrictions} therefore implies
\begin{equation}
 \sum_Q\|E_Q(\xi|_{D_Q})\|^2\leq\mathcal S\|\xi\|^2.
 \label{spec:domain-upper-loss}
\end{equation}
The local spectral estimates in Equation~\eqref{spec:cover-gap} imply,
for every vector $y$ on $Q$,
\[
 \langle T_Qy,y\rangle
   \leq(1-\chi+\alpha)\|y\|^2+\chi\|E_Qy\|^2.
\]
Apply this to the zero-extended restrictions to $D_Q$, sum over the
disjoint domains, and use Equation~\eqref{spec:domain-upper-loss}.
The coefficient $1-\chi+\alpha$ is positive, so the unassigned part can
be included in its norm bound. Equation~\eqref{spec:forest-error} gives
\[
 \langle T\xi,\xi\rangle
 \leq(1-\chi+\alpha+\chi\mathcal S+f_\ell)\|\xi\|^2
 \leq(1-\chi+3\alpha)\|\xi\|^2,
\]
because $\mathcal S\leq1/1000$ and $f_\ell\leq\alpha$. This proves
Equation~\eqref{spec:kernel-form-bound} for every $\xi\in K$ directly;
no approximation by finitely supported kernel vectors is used.

On $p_0\mathcal H$, Equation~\eqref{spec:global-spectrum} gives
$\langle T\xi,\xi\rangle\geq(1-3\alpha)\|\xi\|^2$.
Since $1-3\alpha>1-\chi+3\alpha$, this upper space intersects $K$
only at zero. The map $P_{K^\perp}$ restricted to that space is
therefore injective. Both projections are equivariant, so its polar
decomposition identifies $p_0\mathcal H$ with a closed invariant
subspace of $K^\perp$. Equations~\eqref{spec:kernel-complement-size}
and~\eqref{spec:all-copies-size} prove the non-strict upper bounds in
Equation~\eqref{spec:strict-trace}.

Finally a nonzero equivariant projection on
$\ell^2(\Gbase)\otimes\mathbb C^N$ has strictly positive identity
trace. If all its identity diagonal entries were zero, their values as
squared projection-column norms would make all columns at the identity
zero; equivariance would make every group translate of those columns
zero, and hence the projection itself zero. Proposition~\ref{spec:global-gap}
proved $p_0\ne0$, so the lower bound is strict. The final upper bound
follows from Equation~\eqref{spec:S-final-choice}.
\end{proof}

\begin{theorem}[The matrix projection supplied by the construction]
\label{spec:matrix-projection}
For the fixed choices made above and some sufficiently large integer
$\ell$, the construction defines a finitely generated torsion-free
group $\Gbase$, a positive integer $N$, and the self-adjoint finite
group-algebra matrix $T$ in Equation~\eqref{spec:Cayley-operator}.  Its
spectrum in the reduced representation has the separation in
Equation~\eqref{spec:global-spectrum}.  The resulting projection $p_0$
in Equation~\eqref{spec:p0} belongs to $M_N(C_r^*(\Gbase))$ and obeys
\[
          0<\sum_{j=1}^N\tau_{\Gbase}((p_0)_{jj})<\frac12.
\]
\end{theorem}

\begin{proof}
The alphabet is finite by Theorem~\ref{sam:realization}, so the group
presentation in Theorem~\ref{geo:main} has finitely many generators;
Theorem~\ref{geo:main} proves torsion-freeness.  Equation~\eqref{spec:Cayley-operator}
is a finite sum in the actual reduced regular representation.
Proposition~\ref{spec:global-gap} proves its spectral separation and
the reduced-algebra membership and nonvanishing of $p_0$.
Lemma~\ref{spec:kernel-form} supplies the stated unnormalized trace
inequality.  The order of choices is consistent: the geometry and
transverse parameters are fixed first; Equation~\eqref{spec:eta-choice}
then fixes a single accuracy; the uniform vanishing of
$\beta_\ell$, $\mu_\ell$, and $|S|/(dD)$ permits all subsequent displayed
inequalities by one sufficiently large $\ell$.  No limit group or
limit projection is taken in this final step.
\end{proof}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//spectral.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//tower.tex ====
\section{The finite-alphabet tower before pruning}
\label{tow:section}

This section specifies the vertices, slots, and labels.  The choice of the
permutation tables needed for expansion is made in
Proposition~\ref{exp:realized-gap}.  Section~\ref{prune:section} then makes
the deletions.  Throughout, a bit stream means an element of
\(\mathbb F _2^{\mathbb N_0}\) with a specified finite set of active
coordinates; every inactive coordinate is zero.  A prefix of length
\(a\) comprises coordinates \(0,\ldots,a-1\).  In particular, references
to a prefix longer than the active stream refer to its zero padding.

The construction separates two requirements that are awkward to impose
on one graph at once. An auxiliary graph \(Y^0\) will have expanding
bipartite blocks between neighboring layers. A graph \(Y^\#\) on the same
vertices will have a single labeling rule from which long path words
recover their vertices. Deleting a uniformly vanishing fraction of the
vertices makes the two induced graphs agree; their common survivor is
\(Y\). It retains the constant-vector spaces almost isometrically and
has no short cycles. Proposition~\ref{prune:tower-properties} collects
these outputs. The finite alphabet must also remain small enough to
control the edges outside the later embedded copies. We begin with the
bit coordinates that make all these requirements compatible.

\subsection{Stream lengths and slots}

For \(h\geq0\), write \(h=2sq+t\), where \(q\geq0\) and
\(0\leq t<2s\), and set
\begin{equation}
\begin{split}
 l_A(h)&=r+n_0s+qs+\max(t-s,0),\\
 l_B(h)&=n_0s+qs+\min(t,s).
\end{split}
\label{tow:stream-lengths}
\end{equation}
We also write \(l_A(i)=l_A(|i|)\) and \(l_B(i)=l_B(|i|)\).
Thus \(l_A(h)+l_B(h)=r+2n_0s+h\).  Increasing \(h\) by one increases
exactly one of the two lengths by one: first the \(B\)-length for
\(s\) consecutive steps, and then the \(A\)-length for \(s\) steps.
Let
\[
 \begin{split}
 A_i&=\mathbb F _2^{l_A(i)}\times
       \mathbb F _2^{r+1+l_B(i)},\\
 B_i&=\mathbb F _2^{l_B(i)}\times
       \mathbb F _2^{s-r+1+l_A(i)}.
 \end{split}
\]
The two coordinates of either vertex are denoted \((u,v)\).
Since \(2r-s=\log_2d\), their exact cardinalities are
\begin{equation}
 |A_i|=d\,2^{H_i},\qquad |B_i|=2^{H_i}.
 \label{tow:cardinalities}
\end{equation}
All four active stream lengths at layer \(i\) are at most \(H_i\).
For example, \(l_A(i)\leq r+n_0s+|i|<H_i\), and the other three
bounds follow from the displayed definitions.

Partition every \(A\)-stream into a first block of length \(r\), called
block \(0\), followed by blocks of length \(s\), called
\(1,2,\ldots\).  Every \(B\)-stream has blocks of length \(s\), also
called \(1,2,\ldots\).  Order the \(u\)-blocks from the two sides as
\begin{equation}
 A,0;\ B,1;\ A,1;\ B,2;\ A,2;\ \ldots.
 \label{tow:interleaving}
\end{equation}
For every \(|i-k|\leq1\), the active bits of \(u_A\) at \(A_i\) and
\(u_B\) at \(B_k\) form an initial segment of this order, including
possibly a proper initial part of its last block.  Indeed this is true
for equal absolute indices by \eqref{tow:stream-lengths}; adjacent
absolute indices differ by adding a single bit of the next available
block, and choosing one of the two side lengths from each of these
adjacent indices preserves the same property.  Consequently the final
cutoff has exactly one of the two descriptions
\begin{equation}
\begin{array}{ll}
 l_A(i)=r+ns,\quad l_B(k)=ns+t,&1\leq t\leq s,\\
 l_B(k)=(n+1)s,\quad l_A(i)=r+ns+t,&1\leq t\leq s.
\end{array}
\label{tow:two-cutoffs}
\end{equation}
These descriptions include cutoffs at block ends by taking \(t=s\).
The initial value of \(n_0\) ensures that the four blocks immediately
before every such final block are full blocks of length \(s\).

For a pair \((i,k)\) with \(|i-k|\leq1\), put
\begin{equation}
\begin{aligned}
 \alpha&=1+l_B(i)-l_B(k),&p&=r+\alpha,\\
 \beta&=1+l_A(k)-l_A(i),&p'&=s-r+\beta.
\end{aligned}
\label{tow:offsets}
\end{equation}
Here \(\alpha,\beta\in\{0,1,2\}\), and \(p>p'>0\) for large
\(\ell\).  There are \(d\) slots at each \(A_i\)-vertex for this
pair, indexed by \(a=(k-i,j)\), \(1\leq j\leq d\).  At a
\(B_k\)-vertex the slots are
\(c=(i-k,j,b)\), where \(b\in\mathbb F _2^{p-p'}\).
Thus the desired degree on that side is
\begin{equation}
 D_{ik}=d\,2^{p-p'}=D\,2^{|i|-|k|},
 \qquad D/2\leq D_{ik}\leq2D.
 \label{tow:degrees}
\end{equation}
The identity follows from
\(p-p'=2r-s+|i|-|k|\).  In particular,
\(d|A_i|=D_{ik}|B_k|\).  After all three adjacent pairs are included,
the degrees are \(3d\) on \(A_i\) and at most \(6D\) on \(B_i\).

Set \(K=32s\).  The \emph{family} of a pair is the tuple
\[
 (k-i,\operatorname{sgn}(i),\operatorname{sgn}(k),
                  |i|\bmod K,|k|\bmod K),
\]
where zero has its own sign.  There are at most \(27K^2\) families.
The quantities \(\alpha,\beta,p,p'\), the slot conventions, and the
phase of the final block are determined by this tuple.  For a family
that occurs more than once, both indices have the same nonzero sign.
Its consecutive occurrences increase both absolute indices by \(K\),
and hence both side lengths by \(K/2=16s\).  Their final blocks are
therefore separated by \(32\) positions in
\eqref{tow:interleaving}.  Families involving zero occur only once.

\subsection{The linear slot evaluations}

On each side and for each slot define a linear map \(\mathcal K\) on
padded streams and its evaluation
\begin{equation}
 Q=v+\mathcal K u.
 \label{tow:evaluation}
\end{equation}
On a diagonal block of length \(h\in\{r,s\}\), \(\mathcal K\)
is multiplication by a coefficient of \(\mathbb F _{2^h}\), after
choosing a linear identification with \(\mathbb F _2^h\).  Only
\(A\)-block \(0\) uses \(h=r\).  From each block it may also send a
linear image into the first \(\alpha\) bits of the following block
on side \(A\), or into the first \(\beta\) bits of the following
block on side \(B\).  There are no other off-diagonal terms.  These
images will be called spills.  The same diagonal coefficient and spill
matrix are used at every repeated block of length \(s\) for a given
family and slot.  The spill from \(A\)-block \(0\) may, and will, be
zero.

Once the preceding input block is known, remove its spills from two slot
evaluations on the current block; their difference then cancels the
\(v\)-coordinate.
If the diagonal coefficients differ, their difference is invertible and
recovers the current \(u\)-block, followed by the \(v\)-block.  This is
the local operation behind recovery from a nonbacktracking path word.

We enforce this distinctness with finite global coefficient pools.
On side \(A\), a slot name is \((\delta,j)\), with
\(\delta\in\{-1,0,1\}\) and \(1\leq j\leq d\), so there are
\(3d\) names.  On side \(B\), include in the slot name the offset,
serial number, bucket width and bucket value.  The width belongs to
\(\{\log_2d-1,\log_2d,\log_2d+1\}\), so there are at most
\((21/2)D\) names.  Choose separate injections of the two pools into
\(\mathbb F _{2^s}\), and an injection of the \(A\)-pool into
\(\mathbb F _{2^r}\).  They exist because
\(3d<2^r\) and \((21/2)D<2^s\) for large \(\ell\).
Use the resulting coefficients independently of height and family.
In particular, any two different incident slots at a vertex have
different diagonal coefficients, for every applicable block size.
Spills can additionally depend on the family.  Finite fields introduce
no group elements: they are used only for these maps of bit coordinates.
Their existence follows, for example, by taking the roots of
\(z^{2^h}-z\) in a splitting field over \(\mathbb F _2\); the roots
form a field and, since the derivative is \(-1\), there are exactly
\(2^h\) of them.

Expansion requires a second property: a uniform bound on the fraction of
slots inducing the same input functional from a nonzero output functional,
including on partial blocks.  The following choice of injections and spills
supplies the needed multiplicity bounds.

\begin{lemma}[Choice of coefficients and spills]
\label{tow:coefficient-choice}
The injections and spills can be chosen so that the following holds
for every family, separately on each side.  Let \(\mathcal I\) be
the set of slots for its one adjacent pair, let \(n_{\rm sl}=|\mathcal
I|\), and let
\[
 M_a:\mathbb F _2^s\longrightarrow\mathbb F _2^{s+\alpha}
 \quad\hbox{or}\quad
 M_c:\mathbb F _2^s\longrightarrow\mathbb F _2^{s+\beta}
\]
be diagonal multiplication followed by its spill.  For every nonzero
linear functional \(\zeta\) on the output space, every value of the
restricted input functional \(\zeta M_a\), or \(\zeta M_c\), occurs
on at most \(0.55n_{\rm sl}\) slots after restriction to any nonempty
initial segment of the \(s\) input bits.  On all \(s\) input bits,
the corresponding bound is \(10^{-3}n_{\rm sl}\).
\end{lemma}

\begin{proof}
Choose each injection uniformly among injections into its field, and,
for every family and slot, choose an independent uniform two-row
binary spill matrix, retaining just its first \(\alpha\) or \(\beta\)
rows.  Suppose first that the spill component of \(\zeta\) is
nonzero.  Conditional on all diagonal coefficients, the full input
functionals \(\zeta M_a\) are independent and uniform among the
\(2^s\) binary functionals: a nonzero linear combination of independent
uniform rows is uniform.  Their restrictions to the first input bit
are therefore independent fair bits.  If the spill component is zero,
the diagonal component is nonzero.  Multiplication identifies the
\(2^s\) field coefficients bijectively with the \(2^s\) input
functionals, so the full functionals are sampled without replacement,
and their first-bit values are sampled without replacement from an
equally divided population.

Both sampling rules obey the following form of Hoeffding's concentration
bound \cite{Hoeffding1963}; we include the short argument:
\[
 \mathbb P\{S-\mathbb ES\geq \varepsilon m\}
       \leq \exp(-c_\varepsilon m)
\]
for the number \(S\) of successes in \(m\) samples and each fixed
\(\varepsilon>0\); the same bound holds for the lower tail.
For independent zero-one variables it follows by exponential Markov
and the bound
\(\mathbb E\exp(t(X-\mathbb EX))\leq\exp(t^2/8)\).
For sampling without replacement, the exponential moment is no larger
than for independent samples with the same success proportion.  Indeed,
the mean product of any fixed number of distinct nonnegative population
entries is at most the corresponding power of their mean: replacing
two unequal entries by their average increases each elementary
symmetric sum, because their sum is unchanged and their product
increases.  Repeating these replacements proves the inequality.
Apply it to the entries \(e^{tX}\), for either sign of \(t\).

Apply these bounds to each of the two first-bit values with threshold
\(0.55m\).  Apply them also to each full functional value with
threshold \(10^{-3}m\); its expected proportion is \(2^{-s}\),
which is less than \(10^{-3}/2\) for large \(\ell\).  In the
without-replacement full-functional case the stronger deterministic
bound of one occurrence is available.  The number of tests is at most
\(2^{O(s)}(1+K)^{O(1)}\), whereas every test has
\(m\geq d\) and failure probability at most \(e^{-cd}\), for
an absolute \(c>0\).  Their sum tends to zero as \(\ell\to\infty\).
Choose an outcome for which all tests pass.  Restricting to more than
one input bit only splits first-bit multiplicity classes, proving all
the asserted prefix bounds simultaneously.
\end{proof}

The output of \(\mathcal K\) fits in the active \(v\)-length at
every pair.  In fact those lengths equal
\begin{equation}
 |v_A|=p+l_B(k),\qquad |v_B|=p'+l_A(i).
 \label{tow:output-lengths}
\end{equation}
In the first case of \eqref{tow:two-cutoffs}, the \(A\)-output ends
by \(r+ns+\alpha\) and the \(B\)-output by
\((n+1)s+\beta\).  In the second case they end by
\(r+(n+1)s+\alpha\) and \((n+1)s+\beta\), respectively.
These are bounded by \eqref{tow:output-lengths}, including the cases
\(\alpha=0\), \(\beta=0\), and \(t=s\).

\subsection{One padded permutation rule for each family}

A positive edge label is a triple consisting of a family, a serial
number \(j\), and a vector \(x\in\mathbb F _2^p\).  Write
\(x=b\Vert x'\), where \(b\) has length \(p-p'\) and \(x'\)
has length \(p'\).  The reverse orientation has the formal inverse
label.  A family and \(x\) hence determine both slots, once their
serial number is given.

We next define triangular transformations \(\mathcal F\) of
\(B\)-blocks and \(\mathcal J\) of \(A\)-blocks.  Their parameters
include the family, \(j\), and \(x\).  At each block the rule may
also depend on all input \(u\)-blocks, on both sides, strictly earlier
in \eqref{tow:interleaving}.  This is the complete list of keys.
No transformation depends on input bits in its own block or a later
block through its keys.

For each actual pair in the family, mark its last active block and
its active prefix length \(t\).  At this marked block choose, for
each possible tuple of preceding keys, a permutation of
\(\mathbb F _2^t\); apply it to the first \(t\) bits and leave
the other bits of the block unchanged.  In the universal rule, apply
this permutation only when the four immediately preceding full
\(s\)-blocks of input are not jointly zero.  If they are jointly
zero, use the identity.  At unmarked blocks also use the identity.
At a \(B\)-block the transformation contributes to \(\mathcal F\),
and at an \(A\)-block to \(\mathcal J\).

The marked positions of one family are at least \(32\) blocks apart,
so there is at most one rule at any position.  Any finite prefix uses
only finitely many tables.  On inputs padded at the actual cutoff,
every higher marked rule is inactive: the first higher one has four
previous blocks past the cutoff, and its gate is closed; the same is
then true at every later marked block.  All these preceding inputs
are zero.  Thus the rule preserves the active input lengths.  It is
triangular and invertible, since its gate and permutation key are
known before its current block is inverted.  These statements also
hold when one side's entire stream is fixed and the other is solved
successively.

The equations defining the endpoints of an edge are
\begin{equation}
 Q_A=x\Vert\mathcal F(u_B),\qquad
 Q_B=x'\Vert\mathcal J(u_A).
 \label{tow:edge-equations}
\end{equation}
They are equations of padded streams.  To solve them from a vertex
\(A_i\) and its slot \(a\), compute \(Q_A\), read its first
\(p\) bits as \(x\), and obtain \(c\) from its bucket.  Invert
\(\mathcal F\) successively to recover \(u_B\); the known
\(u_A\) supplies its other keys.  Then form \(Q_B\) by the second
equation and set \(v_B=Q_B+\mathcal K_cu_B\).  Conversely, from
\(B_k,c\), the first \(p'\) bits of \(Q_B\) give \(x'\), and
the bucket in \(c\) gives \(x\).  Successively invert
\(\mathcal J\), then compute \(v_A\) from the first equation.
The support bounds above ensure that the resulting streams have the
required lengths.  In inversion beyond the cutoff, the unmarked
blocks copy zeros until the next marked block; its preceding four
blocks are zero, so its gate is closed, and induction continues.
The two constructions are inverse: they solve the same triangular
equations with the same keys.

Let \(Y^\#\) be the union of the blocks so defined.  Every slot has
exactly one incident edge, giving the degrees
\eqref{tow:degrees}.  Multiedges are allowed and are regarded as
distinct slot incidences.  The labeling is \emph{folded}: at an
\(A\)-vertex two distinct slots differ in offset or serial number;
at a \(B\)-vertex they differ in these data or in bucket.  In every
case the labels differ.  All outgoing labels at \(A\) are positive,
and all outgoing labels at \(B\) are formal inverses, so no further
coincidences are possible.  With \(m_\ell\) denoting the positive
alphabet size, we have
\begin{equation}
 m_\ell\leq 27K^2 d\,2^{r+2}
      =O\bigl(d\,2^r(1+s)^2\bigr),\qquad
 \log_2(2m_\ell)=11\ell+O(\log\ell).
 \label{tow:alphabet}
\end{equation}

For the auxiliary graph \(Y^0\), in each actual pair use the same
rule except that its own last-block permutation is applied
unconditionally.  All rules at lower marked positions retain their
gates.  Higher rules are inactive on its padded inputs.  The slot
inversion argument still applies, so the auxiliary block has the same
degrees and label set, and is folded as well.

\begin{lemma}[The independent matching variables]
\label{tow:matching-clusters}
Fix an actual pair \((i,k)\) and every table at its lower marked
positions.  The current last-block tables can be sampled so that the
remaining random choices in its \(Y^0\)-block are independent uniform
perfect matchings between pairs of clusters, each cluster having
\(2^t\) slot incidences.  There is one matching for each tuple
consisting of the family, serial number, \(x\), and all preceding
\(u\)-blocks on both sides.  Every edge in each cluster pair is a
possible edge, and its matching is used only for that pair of clusters.
\end{lemma}

\begin{proof}
Fix \(a\) and \(x\).  Since adding \(\mathcal K_au_A\) to
\(v_A\) is invertible, the \(A_i\)-vertices with these data are
freely parametrized by \(u_A\) and the tail of \(Q_A\), of lengths
\(l_A(i)\) and \(l_B(k)\).  On the other side fix the resulting
\(c\) and \(x'\).  Its vertices are freely parametrized by
\(u_B\) and the tail of \(Q_B\), of lengths \(l_B(k)\) and
\(l_A(i)\).  Thus both parameter spaces have dimension
\(l_A(i)+l_B(k)\).

The previously fixed transformations bijectively identify all data
before the final block, successively in the interleaved order.
Fix these preceding \(u\)-values.  In the first case of
\eqref{tow:two-cutoffs}, the remaining \(B\)-incidences are
parametrized by the \(t\) final bits of \(u_B\), while the remaining
\(A\)-incidences are parametrized by the corresponding \(t\) bits
of the tail of \(Q_A\).  No extra data remain, since the fixed
preceding values and these final bits specify the whole parameter
space on each side.  The current table of \(\mathcal F\) is
precisely a bijection between these two sets of \(2^t\) values.
Its variation can change the computed \(v_B\), but that coordinate
is already determined by the other endpoint data; it is not another
independent parameter.  In the second case the roles are reversed:
the \(t\) final bits of \(u_A\) are matched to the corresponding
tail bits of \(Q_B\) by the current \(\mathcal J\)-table.
Sampling a uniform permutation for each distinct full key tuple gives
the stated independent uniform matchings.  Different tuples specify
disjoint clusters of slot incidences.  The bucket is already part of
\(x\), so it supplies neither an omitted key nor a further coupling.
\end{proof}

The coefficient maps are now fixed, but the permutation tables remain to
be chosen.  The next section proves that the current matchings can be
chosen to give a uniform incidence gap, whatever lower-cutoff tables have
already been fixed.  That result will allow a successive choice in each
family: later tables cannot alter an earlier auxiliary block, since their
gates are inactive on its padded streams.
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//tower.tex ====

==== BEGIN A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//references.bib ====

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@misc{EchterhoffWilliams2012,
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@incollection{Brown2010,
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@misc{OpenAIBass2026,
 author={{OpenAI}},
 title={{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}},
 howpublished={OpenAI Math Release preprint
               \href{https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}},
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}

@inproceedings{Zippel1979,
  author = {Zippel, Richard},
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  editor = {Ng, Edward W.},
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}
==== END A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture//references.bib ====