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\title{A torsion-free counterexample to reduced Baum--Connes injectivity}
\author{}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We construct a finitely generated torsion-free group $G$ whose reduced
Baum--Connes assembly map has an integral kernel class of infinite order.
The proof has three layers: a deterministic labeled graph with short-word
uniqueness, a probabilistic selection of expanding permutation tables and
diffuse Heisenberg voltages, and a spectral argument in a reduced
group algebra which cancels the rank-zero Bott class of an embedded torus.
The final step extracts a finite witness from a compatible accuracy sequence.
\end{abstract}
\section{Notation and conventions}

All group $C^*$-algebras and all $K$-groups are complex. Commutators are
$[a,b]=aba^{-1}b^{-1}$. For a discrete group $\Lambda$ we write
$\Cr(\Lambda)$ for the reduced group $C^*$-algebra, i.e.\ the norm
closure of $\C[\Lambda]$ acting by the left regular representation on
$\ell^2(\Lambda)$. For a locally compact space $X$, $K_*(X)$ denotes
topological $K$-homology with compact support, i.e.\ the directed colimit
of $K_*(X')$ over finite subcomplexes $X'\subseteq X$. When $\Lambda$ is
torsion-free, the classifying space for proper actions $\underline{E}\Lambda$
coincides with $B\Lambda$, so the coefficient-free assembly map is
\[
  \mu_\Lambda^r\colon K_*(B\Lambda)\to K_*(\Cr(\Lambda)).
\]
On $C_r^*(\Z^2)\cong C(\mathbb T^2)$ we fix the spin Dirac class
$[D_{\mathbb T^2}]\in K_0(\mathbb T^2)$ of the oriented torus and the
rank-zero Bott generator $\beta=[\mathsf p]-[e_\infty]\in K_0(C(\mathbb T^2))$
where $\mathsf p$ is the rank-one Bott projection and $e_\infty=\mathrm{diag}(0,1)$.
The reduced assembly map sends $[D_{\mathbb T^2}]$ to (a choice of sign of) $\beta$.

\section{Main theorem}

\begin{theorem}\label{thm:main}
There exists a finitely generated torsion-free discrete group $G$ together
with an injective homomorphism $\iota\colon\Z^2\hookrightarrow G$ and a
class $c\in H^2(BG;\Z)$ such that, writing $i=(B\iota)\colon\mathbb T^2\to BG$
for the corresponding map of classifying spaces and $\beta\in K_0(\Cr(\Z^2))$
for the rank-zero Bott generator,
\[
  \ip{i^*c}{[\mathbb T^2]}=1,
  \qquad
  \iota_*(\beta)=0\quad\text{in }K_0(\Cr(G)).
\]
Consequently the class $h=i_*[D_{\mathbb T^2}]\in K_0(BG)$ has infinite
order, while its reduced assembly image $\mu_G^r(h)$ vanishes.
\end{theorem}

The two assertions are independent. The cohomology class $c$ extends the
orientation class of the embedded Heisenberg torus and detects its Dirac
class through an index pairing; finite order of $h$ is excluded by the
integer invariant. The vanishing $\iota_*(\beta)=0$ is proved by an
explicit operator-algebra construction in $\Cr(G)$, unaffected by that
invariant. Finite generation is a conclusion, not an input; finite
presentation is neither required nor claimed.

\section{Proof outline}

The proof of Theorem~\ref{thm:main} has three independent layers, combined
in Section~\ref{sec:assembly-v2}.

\paragraph{(A) Deterministic labeled tower.}
Sections~\ref{sec:labeling-v2} and~\ref{sec:pruning-v2} construct a
layered graph $X_{\mathrm{old}}$ with a folded labeling by a finite
alphabet and a deletion of a small fraction of every layer, such that:
(a) every immersed word of length at least $1.002m_i$ based at a vertex of
height $i$ has at most one occurrence anywhere in the graph; (b) the
girth at height $i$ exceeds $.93m_i$; (c) every bicycle of length $<1.52m_i$
has been removed. The parameter $m_i$ is a linear function of a height
parameter $H_i$. A vertex state consists of two bit streams $(u,v)$ in an
interleaved block layout; pairs of incident labels recover successive blocks
of a state via a two-evaluation step over a finite field. Random permutation
tables make the resulting incidences expanding, with a fixed spectral gap.

\paragraph{(B) Probabilistic voltages and symbols.}
Section~\ref{sec:expansion-v2} chooses the permutation tables by a
sequential Lov\'asz local lemma along the tower, preserving the expansion
gap while bounding the change of law at each update. Section~\ref{sec:analysis-v2}
assigns to every surviving old edge a Heisenberg voltage $h_e\in\HH$ and a
bounded matrix weight $W_e\in M_m(\C)$ so that the weighted incidence
approximates a prescribed compact Bott symbol in full regular-norm and so
that every short nonlaw of $\HH$ has, with overwhelming probability,
nontrivial holonomy.

\paragraph{(C) Operator-algebra cancellation and finite witness.}
Sections~\ref{sec:geometry-v2}--\ref{sec:assembly-v2} realize the
voltage cover inside the Cayley graph of a finitely generated group
$\Gamma$, construct disjoint sheet domains so that the remaining
Cayley edges form a forest, and use the cone complex to prove
torsion-freeness and extend an integral degree-two detector from the
Heisenberg subgroup $\HH$. The two explicit Eilenberg-swindle motions
of localized Bott profiles, followed by finite additivity, cancel the
initial-height profile in a norm-sequence quotient over a compatible
accuracy sequence. A finite $K$-theory witness then survives into a
single reduced group algebra $\Cr(G)$ for a finitely generated subgroup
$G\le\Gamma$.
\section{The labeled tower}\label{sec:labeling-v2}

\subsection{The parameters}

Fix once a positive integer $b$. Let $s$ be a sufficiently large multiple of
$20000$ and $5b$, and put
\begin{equation}\label{eq:params-v2}
 r=\tfrac35 s,\qquad d=2^{.40005s},\qquad D=2^{.60005s},\qquad q=\sqrt{dD}=2^{.50005s}.
\end{equation}
These are the two incidence degrees $d,D$ and their geometric mean $q$.
The exponents are chosen so that distinct field coefficients exist below
$2^r$ and $2^s$ and so that the eventual alphabet $\mathcal S$ satisfies
$\sqrt{|\mathcal S|}/q\to 0$ \emph{strictly} after fixed finite refinements.
Set $n_0\ge 10^6$, to be enlarged; put $H_i=(2n_0+1)s+bi$ and
$m_i=H_i/\log_2 q$. Require $b/\log_2 q\le 10^{-6}$; this is \emph{the}
slope condition.

\subsection{States and blocks}

For $i\ge0$, decompose $bi=2sa_i+u_i$ with $0\le u_i<2s$ and put
\begin{equation}\label{eq:lengths-v2}
 \ell_A(i)=r+n_0 s+a_is+\max\{u_i-s,0\},\quad
 \ell_B(i)=n_0 s+a_is+\min\{u_i,s\}.
\end{equation}
The state spaces at height $i$ are
\begin{equation}\label{eq:state-v2}
 \mathscr A_i=\F_2^{\ell_A(i)}\times\F_2^{r+\ell_B(i)},\quad
 \mathscr B_i=\F_2^{\ell_B(i)}\times\F_2^{s-r+\ell_A(i)},\quad
 |\mathscr B_i|=2^{H_i},\quad
 |\mathscr A_i|=2^{H_i+2r-s}.
\end{equation}
A state is a pair $(u,v)$ of bit vectors. All lengths are multiples of $b$.

\emph{Source-block interleaving.} Regard $u_A$ and $u_B$ as infinite bit
streams obtained by appending zero coordinates. The stream convention is:
$A$ has an initial block $A_0$ of length $r$, then blocks $A_1,A_2,\ldots$
of length $s$; $B$ has blocks $B_1,B_2,\ldots$, all of length $s$. The
\emph{interleaved order} is
\begin{equation}\label{eq:interleave-v2}
 A_0(0),\; B_1(1),\; A_1(2),\; B_2(3),\; A_2(4),\ldots
\end{equation}
At height $i$ the active $u$-coordinates on both sides together form an
initial segment of this order. Denote the corresponding index by
$\kappa_i=\lfloor H_i/s\rfloor$, and let $t_i=H_i-\kappa_i s\in\{0,\ldots,s-1\}$
be the active prefix length of the last partial block.

\subsection{The edge equation}

For each $i$, incident pairs at height $i$ are (types allow same height,
$i\to i$, or adjacent, $i\to i\pm1$). Fix a finite type set
$\mathcal T$; a type $\tau$ specifies $\delta_\tau\in\{-1,0,1\}$ with
incidence between $\mathscr A_i$ and $\mathscr B_{i+\delta_\tau}$.

\emph{Truncation.} For an adjacent pair, truncate the larger endpoint to
the coordinate lengths at height $h=\min\{i,i+\delta_\tau\}$; exactly $b$
discarded coordinates form one interval $e\in\F_2^b$. For a same-height
pair let $e$ be the empty string. In a fixed $e$-slice both truncated
endpoints agree with $\mathscr A_h,\mathscr B_h$.

\emph{Slot and family.} A slot at an $A$-endpoint is $(f,j,e)$ where
$f=(\tau,i\bmod M)$ is the family (with $M=2\cdot 10^6 s$) and $j=1,\ldots,d$
is a serial number. A slot at a $B$-endpoint is $(f,j,c,e)$ where
$c\in\F_2^{2r-s}$ is a bucket. A positive label is
$\lambda=(f,j,e,h_0)$ with $h_0=c\Vert h_0'$, $|h_0|=r$, $|h_0'|=s-r$.

\emph{Edge equation.} Fix a positive label. Write $\bar u_C$, $\bar v_C$
for the corrected streams obtained from $(u_C,v_C)$ by XORing the extra
interval $e$ at every marked cutoff where a lag gate is open
(see below). Write $K_{(f,j,e)}$ and
$K_{(f,j,c,e)}$ for the block-diagonal linear maps determined by the
field-coefficient injections (Section~\ref{subsec:coeffs-v2}), and
$P_{A,\lambda}$, $P_{B,\lambda}$ for the keyed permutations of the next
subsection. The universal edge equations are
\begin{equation}\label{eq:edge-v2}
 \bar v_A+K_{(f,j,e)}\bar u_A=h_0\Vert P_{B,\lambda}(\bar u_B),
 \qquad
 \bar v_B+K_{(f,j,c,e)}\bar u_B=h_0'\Vert P_{A,\lambda}(\bar u_A).
\end{equation}
These are equations of infinite padded streams.

\emph{Lag gates.} For a source index $k>20$, put $J(k)=\lfloor(k-5)/2\rfloor$
and $g_k=1$ iff the eight-block window
$(u_{B,J(k)-3},u_{A,J(k)-3},\ldots,u_{B,J(k)},u_{A,J(k)})$ is nonzero. For
$k\le 20$, $g_k=1$. Then a correction at source block $k$ reads no block
of index $>k-5$.

\subsection{The coefficient pools}\label{subsec:coeffs-v2}

Write an $A$-name as $(f,j,e)$, a $B$-name as $(f,j,c,e)$; include every
legal value, in particular values of $e$ that are read from the larger
state. Set $\mathcal N_A,\mathcal N_B$ accordingly. There are
\begin{equation}\label{eq:names-v2}
 |\mathcal N_A|\le TM2^bd,\qquad |\mathcal N_B|\le TM2^bD.
\end{equation}
Choose an injection $\mathcal N_A\hookrightarrow\F_{2^r}$ and an injection
$\mathcal N_A\sqcup\mathcal N_B\hookrightarrow\F_{2^s}$. For a name $\nu$,
$K_\nu$ is the block-diagonal field multiplication acting in bit coordinates;
the same coefficients are used on every full $s$-block.

The polynomial factor $TM$ and the fixed factor $2^b$ of
\eqref{eq:names-v2} are dominated by the positive exponent gaps
$r-.40005s$ and $s-.60005s$, so uniformly random injections exist.

\subsection{The recovery step}

For each family, label, source index $k$, and tuple of raw $u$-blocks
strictly earlier than $k$ in \eqref{eq:interleave-v2}, provide a
permutation table of the corresponding block (of length $s$, or of active
prefix $t$ at a marked cutoff). Keys are complete; there are finitely many
tables at every $k$.

The transformation at source block $k$ is:
(i) at a marked cutoff of type $t$, apply the $t$-prefix table then identity
on the suffix, when $g_k=1$;
(ii) at the next twenty blocks, use identity;
(iii) elsewhere, use the full-block table when $g_k=1$, else identity.
Write $P_{C,\lambda}$ for the resulting stream map, $C\in\{A,B\}$.

The elementary recovery fact is: over a finite field, if $E_1=v+c_1u$ and
$E_2=v+c_2u$ with $c_1\ne c_2$ are known, then
$u=(c_1-c_2)^{-1}(E_1-E_2)$ and $v=E_1-c_1u$. The edge equation
\eqref{eq:edge-v2} is such a pair after corrections are moved to the
right-hand sides. The lag gates and the "keys use only earlier raw blocks"
property ensure that at every block the two right-hand sides depend only
on strictly earlier raw data.

\begin{lemma}[Recovery and global uniqueness]\label{lem:recovery-v2}
After deletion of a small fraction of vertices (Lemma~\ref{lem:gate-v2}
and Lemma~\ref{lem:sep-v2}) the resulting induced subgraph
$X^{\mathrm{good}}$ has the following properties. Let $v\in X^{\mathrm{good}}$
have height $i$ and let an immersed path start at $v$ with length
$N\ge\lceil1.002m_i\rceil$. Then the path is the unique immersed path with
its oriented word anywhere in $X^{\mathrm{good}}$, over all heights and
components.
\end{lemma}

\subsection{Gate deletion}\label{subsec:gates-v2}

Let $X^0$ be the union of the auxiliary incidences over types and height
pairs. Delete an endpoint state at height $i$ if any incident slot has:
(i) some $g_\kappa,g_{\kappa-1},g_{\kappa-2}$ equal to zero, where $\kappa$
is its common-last-source index; (ii) the natural rather than forced gate at
its actual extra interval, if adjacent; (iii) among its full active $A$
source blocks other than $A_0$, more than $1/250$ closed; or the same
among its full active $B$ source blocks.

\begin{lemma}[Gate budget]\label{lem:gate-v2}
For every fixed table realization, the fraction deleted for gate reasons
from either side at any height is at most
$\varepsilon_{\mathrm{gate}}=600T2^b D2^{-8s}$.
On a surviving edge both auxiliary forced gates were unnecessary, so the
universal padded equations \eqref{eq:edge-v2} hold. At each surviving slot,
separately by source-parity, fewer than $.005$ of full updates are unavailable
for closed-gate reasons; the last three source gates and the actual prefix
gate are open.
\end{lemma}

\begin{proof}[Sketch]
Fix a slot and condition on its actual extra. Every tested noninitial gate
reads eight independent uniform raw $u$-blocks, so it closes with probability
exactly $2^{-8s}$. Each parity has $\ge n_0$ full blocks; the block-count
test has probability at most $250\cdot 2^{-8s}$. Sum over at most $T2^bD$
slots and average over the extra. \qed
\end{proof}

\subsection{Height separation}\label{subsec:separation-v2}

Put $\eta_{\mathrm{sep}}=1/2500$. For a vertex on side $C$ at height $i$ and
$i'<i$ with $i'\equiv i\pmod M$, set
\[
 \Delta=\ell_C(i)-\ell_C(i')=(i-i')b/2,\qquad
 q(i,i')=\lfloor\min\{\eta_{\mathrm{sep}}H_{i'},\Delta\}\rfloor.
\]
Delete the vertex if its $q(i,i')$ raw $u_C$-bits immediately after cutoff
$\ell_C(i')$ are all zero.

\begin{lemma}[Separation budget]\label{lem:sep-v2}
The fraction deleted for separation from either side at any height is at most
$\varepsilon_{\mathrm{sep}}
=2\bigl(2^{-Mb/2}/(1-2^{-Mb/2})+2^{-\eta_{\mathrm{sep}}H_0}/(1-2^{-\eta_{\mathrm{sep}}b})\bigr)$.
\end{lemma}

\begin{proof}[Sketch]
Uniform raw bits: the test has probability $2^{-q(i,i')}$, split via
$2^{-\lfloor\min(x,y)\rfloor}\le 2(2^{-x}+2^{-y})$. Sum over the geometric
progressions $i-i'\in M\mathbb Z$ and $H_{i'}\ge H_0$; both converge.
\qed
\end{proof}

\subsection{Proof of Lemma~\ref{lem:recovery-v2}}

\emph{(i) Terminal shapes.} Two rows suffice:
$(\ell_A,\ell_B)=(r+ns,ns+t)$ or $(r+ns+t,(n+1)s)$ with $1\le t\le s$.
An adjacent pair's common $b$-bit interval lies beyond the same-side direct
block, so computing $v$ from \eqref{eq:edge-v2} cannot leak a $Ku$-term into
the prescribed extra.

\emph{(ii) Triangular inversion.} Given an $A$-slot and its extra value,
the first $r$ evaluation bits give $h_0$ (and thus $c$ and the $B$-name);
successive B-block permutations are then inverted in the interleaved order
to recover raw $u_B$ then raw $v_B$. The lag gates read strictly earlier raw
blocks; the B-permutation key uses earlier B-blocks and the preceding
A-block. A $B$-slot works symmetrically from $h_0'$. The two procedures are
inverse.

\emph{(iii) Uniform coordinates.} First freely specify raw $u_A,u_B$;
specify $h_0$ (or $h_0'$ with bucket fixed). Every gate, correction and
permutation value is now determined; \eqref{eq:edge-v2} determines both
$\bar v$ streams; undoing corrections determines raw $v$. The number of
free raw $u$-bits matches $\log_2|\mathscr A_h|$ or $\log_2|\mathscr B_h|$.

\emph{(iv) Global uniqueness.} Suppose two immersed paths with the same
oriented word and length $N\ge\lceil1.002m_i\rceil$ start at $v_1,v_2$ of
heights $i_1,i_2$. The midpoint of path~1 has height $j$ and radius
$\lfloor N/2\rfloor$; by the aligned-depth propagation of~\eqref{eq:edge-v2}
its streams are recovered to aligned depth
\[
 \delta\ge s(N/2-2)\ge(1.002/1.0001)H_{i_1}-2s\ge1.001H_j,
\]
which covers the whole state at height $j$. If $i_1\ne i_2$ write
$i_-<i_+$; the smaller state's trailing interval tested by
\eqref{eq:edge-v2}-separation is null, which the separation lemma forbids.
Hence $i_1=i_2$, and foldedness propagates equality from the midpoint
outward. \qed

\begin{proposition}[Deterministic labeled tower]\label{prop:tower-labels-v2}
For every fixed finite type set and every sufficiently large $s$ as
above, the construction has the following properties.
\begin{enumerate}
\item The auxiliary state sizes are \eqref{eq:state-v2}. In a fixed
extra slice an incidence has degrees $d,D$. Without slicing, its
smaller side acquires the extra factor $2^b$ and its larger side
does not; the total degrees are at most $T2^bd$ on $A$ and $T2^bD$ on $B$.
\item The labeling is folded; its signed alphabet satisfies
\[
 |\mathcal S|\le 2TM2^bd2^r,\qquad
 \frac{\sqrt{|\mathcal S|}}q\le 2^{-.000025s+O(\log s)+O_{T,b}(1)}\to0.
\]
\item At every height the fraction of removed vertices on each side is
at most $\varepsilon_{\mathrm{gate}}+\varepsilon_{\mathrm{sep}}$, made
arbitrarily small, uniformly over heights and all table choices.
\item Every immersed path in $X^{\mathrm{good}}$ starting at height $i$
and having length at least $\lceil1.002m_i\rceil$ is the unique immersed
path with that oriented word anywhere in $X^{\mathrm{good}}$.
\end{enumerate}
\end{proposition}

\section{Expansion and the sequential probability law}\label{sec:expansion-v2}

We choose the permutation tables so that every auxiliary incidence expands, while retaining a quantitative bound on the change of law at each source-block update. The proof is a stopping-time construction along the tower.

\subsection{The mean operator}

Fix $\theta=2^{1-b}$ and require $\sqrt{5\theta}<3/5$, i.e.\ $b$ large. For a fixed-extra slice, put $A,B$ for the common-cutoff spaces and
\[
 (Nf)(a)=\tfrac1d\sum_{e:e^-=a}f(e^+),\qquad (N^*g)(b')=\tfrac1D\sum_{e:e^+=b'}g(e^-).
\]
These are adjoint contractions that preserve constants; $d|A|=D|B|$. Let $\mathsf M=\mathbb E_T N$ where $T$ ranges over the current permutation tables; the same statements hold for $\mathsf M$.

\begin{lemma}[Fresh matching coordinates]\label{lem:matching-v2}
Fix a family occurrence, an extra slice, and every table before its last common source block. Under the independent uniform current-table law, the auxiliary incidence is a product of independent perfect matchings between pairs of $2^t$-element clusters of slot incidences, one per complete final-prefix key. No later table acts on this incidence.
\end{lemma}

\begin{proof}[Sketch]
Fix serial number, label head, and preceding raw $u$-blocks. If the last common block is $B$, its remaining $B$-data is exactly the $t$ active bits of raw $u_B$; its remaining $A$-data is exactly the $t$ corresponding evaluation bits. All other coordinates are determined by the uniform-coordinate part of Lemma~\ref{lem:recovery-v2}. The forced-on current prefix table is then an arbitrary bijection. \qed
\end{proof}

\subsection{Conditional Fourier estimate}

\begin{lemma}\label{lem:fourier-v2}
Condition on data $C$. Let $U\in\F_2^t$, $V\in\F_2^w$ be independent uniform, independent of auxiliary $W$. For $a$ in a set of size $q_0$, let $K_a:\F_2^t\to\F_2^w$ be linear, $o_a=o_a(C,W)$, and $f_a(C,z)$ have no separate dependence on $U,V,W$. Suppose every class of slots with common $\zeta K_a$ has at most $\theta q_0$ members. Put
\[
 F=\tfrac1{q_0}\sum_a f_a(C,V+K_aU+o_a),\quad
 \mu(C)=\tfrac1{q_0}\sum_a 2^{-w}\sum_z f_a(C,z).
\]
Then $\mathbb E(F\mid C,W,U)=\mu(C)$ and
\[
 \mathbb E|F-\mu(C)|^2\le\theta\,\mathbb E\Bigl[\tfrac1{q_0}\sum_a 2^{-w}\sum_z|f_a(C,z)|^2\Bigr].
\]
\end{lemma}

\begin{proof}
Translation of $V$ proves the mean. For fixed $C,W$, orthogonality in $V$ then $U$ gives
\[
 \mathbb E_{U,V}|F-\mu(C)|^2=\tfrac1{q_0^2}\sum_{\zeta\ne0}\sum_{\mathcal C}|\sum_{a\in\mathcal C}(-1)^{\zeta o_a}\hat f_a(\zeta)|^2,
\]
summing over classes $\mathcal C$ of equal $\zeta K_a$. Cauchy--Schwarz and Parseval, then integrate over $C,W$. \qed
\end{proof}

\subsection{Two prefix bands}

Let $P_h^A$ be expectation to the first $r+hs$ bits of both raw $A$-coordinates; let $P_j^B$ be expectation to the first $js$ bits of both raw $B$-coordinates. Padded coordinates are deterministic zero, so the filtrations end at identities. A function in $\operatorname{ran}P_j^B$ can be evaluated at an $A$-slot using only
\[
 u_A[0,r+(j-1)s),\qquad Q_A[0,r+js),
\]
where $Q_A=\bar v_A+K\bar u_A$ is the corrected evaluation. Symmetrically a function in $\operatorname{ran}P_h^A$ can be evaluated at a $B$-slot using only $u_B[0,hs)$ and $Q_B[0,(h+1)s)$. Hence
\[
 \operatorname{ran}(\mathsf MP_j^B)\subseteq\operatorname{ran}P_j^A,\qquad
 \operatorname{ran}(\mathsf M^*P_h^A)\subseteq\operatorname{ran}P_{h+1}^B,
\]
and Lemma~\ref{lem:fourier-v2} gives
\[
 \|(1-P_{j-1}^A)\mathsf MP_j^B\|^2\le\theta,\quad
 \|(1-P_h^B)\mathsf M^*P_h^A\|^2\le\theta.
\]
The direct input in each case stops strictly before the new input block $U$, so neither application has separate dependence on the fresh $U$.

\begin{lemma}[Conditional mean]\label{lem:mean-v2}
For every earlier-table history, marked cutoff, and extra slice,
\[
 \|\mathsf M:1_B^\perp\to1_A^\perp\|\le\eta:=\sqrt{5\theta}<3/5.
\]
The same bound holds for the complete adjacent-height incidence.
\end{lemma}

\begin{proof}[Sketch]
Orthogonal differences $E_h^A=P_h^A-P_{h-1}^A$, $E_j^B=P_j^B-P_{j-1}^B$ satisfy $E_h^A\mathsf ME_j^B=0$ unless $h\le j\le h+1$. The bands $h=j$ and $j=h+1$ contribute at most $\sqrt\theta$ each, giving $2\sqrt\theta$ for full blocks. The terminal block analysis (Lemma~\ref{lem:recovery-v2}(i), two-shape case) uses the matched prefix to average $U$ away, contributing one more $\sqrt\theta$; the bound $\|\mathsf M^*g\|^2\le5\theta\|g\|^2$ for centered $g$ follows orthogonally. Slicing the extra coordinate multiplies nothing (Jensen). \qed
\end{proof}

\subsection{Concentration of cuts}

Write $\deg(v)=d$ on $A$, $D$ on $B$; $\mathrm{vol}(S)=\sum_{v\in S}\deg(v)$; total volume $V_0=2d|A|$. Let $c(S)$ count edges crossing out of $S$ with multiplicity. The selfadjoint block operator with entries $\mathsf M,\mathsf M^*$ has spectrum $[-1,-1+\eta^2/2]\cup\{0\}\cup(\ldots)$ off the global constants and bipartition eigenmodes; its quadratic form applied to a centered indicator of $S$ gives
\[
 \mathbb E c(S)\ge(1-\eta)\mathrm{vol}(S)(1-\mathrm{vol}(S)/V_0)\ge\tfrac15\mathrm{vol}(S)
\]
whenever $\mathrm{vol}(S)\le V_0/2$.

For a current table $a$, let $Z_a$ be the count of its matched pairs with both endpoints in $S$; $Z_a$ is hypergeometric with parameters $(n_a,u_a,v_a)$. Then
\[
 c(S)=\mathrm{vol}(S)-2\sum_a Z_a,\qquad
 \sum_a u_a\le\mathrm{vol}(S),
\]
and the elementary hypergeometric bound
$\mathbb E e^{\lambda(Z_a-\mathbb E Z_a)}\le e^{\lambda^2u_a/8}$ (for all $\lambda\in\mathbb R$) applied with $\lambda=.3$ gives
\[
 \Pr\{c(S)<.05\mathrm{vol}(S)\}\le\exp\{-c_0\min(d,D)|S|\},\qquad c_0=10^{-3}.
\]

\subsection{The finite selection lemma and the conditioning bound}

Join two physical vertices when some choice of a current table could join them. This \emph{possible graph} has maximum degree $\Delta\le2^{Cs}$, where $C$ depends only on the finite type data and $b$, and is independent of height. It suffices to avoid cut failures for connected $S$ in the possible graph with $\mathrm{vol}(S)\le V_0/2$.

Let $A_S$ be such a cut-failure event; it depends only on tables meeting $S$. Two events with no common table are independent. Assign weights
$x_S=\exp(-c_0\min(d,D)|S|/2)$. Since
\[
 \sum_{T\sim S}x_T\le|S|\sum_{j\ge1}\Delta^{2j+C_1}e^{-c_0\min(d,D)j/2}=|S|\delta_s,\qquad\delta_s\to0,
\]
the asymmetric Lov\'asz local lemma \cite{MT10, EL75} applies.

\emph{Finite selection.} There is a nonempty set of expanding table choices at every marked cutoff. If $B$ involves $k$ whole current tables, then
\begin{equation}\label{eq:lll-conditioning-v2}
 \Pr(B\mid\text{all current cuts succeed})\le e^{\varepsilon_s k}\Pr(B)
\end{equation}
uniformly in height and earlier history, with $\varepsilon_s\to0$.

\subsection{Sequential probability law}

Order each family's source blocks as $A_0,B_1,A_1,B_2,A_2,\ldots$; interleave families by increasing source index and a fixed family order. At an unmarked update choose all active permutation variables uniformly. At a marked update use the finite conditional distribution given avoidance of current cut failures, separately in extra-bit slices.

\begin{proposition}[Sequential selection]\label{prop:sequential-v2}
There is a probability law on the full tower such that every auxiliary incidence has centered singular norm at most $\rho:=1-.05^2/2<1$ from Lemma~\ref{lem:mean-v2}. At each source-block update, conditional on all earlier updates, any joint test of $k$ specified current tables has probability bounded by $e^{\varepsilon_s k}$ times its value under independent uniform tables. The bound is uniform in earlier history and in height, with $\varepsilon_s\to0$.
\end{proposition}

\begin{proof}[Sketch]
Induction on updates: current kernels in different families are independent given the past, so grouping gives the same bound at every stage. Lemma~\ref{lem:mean-v2} and the LLL criterion ensure existence at each marked cutoff; \eqref{eq:lll-conditioning-v2} gives the conditioning bound. Iterate on the countable sequence of updates by an independent uniform $[0,1]$ draw choosing among the finitely many outcomes. \qed
\end{proof}

\section{Pruning coherent returns and bicycles}\label{sec:pruning-v2}

Work with the sequential law of Proposition~\ref{prop:sequential-v2}. The coefficient pools are already fixed. Let $X^{\mathrm{good}}$ be the graph induced on vertices surviving the gate and separation deletions of Lemmas~\ref{lem:gate-v2} and~\ref{lem:sep-v2}; these are deterministic for every table realization. A statement that a path lies in $X^{\mathrm{good}}$ is a condition on the counted object, not a conditioning of the law. Lengths count edges. A \emph{bicycle} is a finite connected subgraph with first Betti number two and no valence-one vertex.

\subsection{Aligned depths and coherent returns}

For an $A$-position, put $\delta_A(j)=(2j+1)s$; depth $\delta_A(j)$ means both raw $A$-streams agree through length $r+js$. For a $B$-position put $\delta_B(j)=2js$; depth $\delta_B(j)$ means both raw $B$-streams agree through length $js$. These give the exact identities
$\delta_A(j)-\delta_B(j)=s$ and $\delta_B(j)-\delta_A(j-1)=s$.

Let $D(v)$ be the largest such depth with both prefixes active at $v$. By the terminal shapes of Lemma~\ref{lem:recovery-v2}(i), $H_i-2s\le D(v)\le H_i$ at height $i$. Two equal state prefixes propagate through one common outward edge as $A_j\mapsto B_j$ and $B_j\mapsto A_{j-1}$, each losing exactly $s$ of aligned depth.

Fix $H_0>0$. For $k\ge0$, put $T_k=H_0(11/10)^k$, $I_k=[5T_k/6,6T_k/5]$, $M_k=T_k/25$. A vertex of height-scale $H_i$ lies in band $k$ when $H_i\in I_k$; every height lies in a fixed number of bands, and
$\tfrac1{30}\le M_k/H_i\le\tfrac6{125}<.05$ when $H_i\in I_k$.

\begin{definition}[Coherent return]\label{def:coherent-v2}
In band $k$, a \emph{coherent return} is a positive-length nonbacktracking path $v_0,e_1,\ldots,e_p,v_p$ in $X^{\mathrm{good}}$, entirely in that band, whose endpoints lie on the same side and agree through an available aligned depth
\[
 h\ge ps/2+M_k,\qquad h\le\min\{D(v_0),D(v_p)\}.
\]
It is \emph{minimal} if no proper positive-length contiguous subpath is a coherent return for the same band and margin. A closed nonbacktracking path is included whenever it satisfies the available depth.
\end{definition}

The endpoint side forces $p$ even; use the least aligned $h$ satisfying the first threshold, losing less than $2s$ of margin. Every coherent return contains a minimal one.

\begin{lemma}[The two junction coefficients differ]\label{lem:coherent-junction-v2}
At identified endpoints of a minimal coherent return, the two outward coefficient names are different.
\end{lemma}

\begin{proof}[Sketch]
Equal coefficient names give equal family and slot data; at $A$ the label head is the first $r$ evaluation bits, at $B$ the suffix gives $h_0'$ and the bucket is slot data. Removing one edge at each end gives a shorter coherent return unless $p=2$, but foldedness forces an immediate backtrack. \qed
\end{proof}

\subsection{Reconstruction and free coordinates}

A coherent-path pattern consists of length $p$, initial side and height, band, edge labels, allowed next-height choices, slot names, coefficients, correction extras, and viewport cutoffs. Patterns with inconsistent labels or height assignments are rejected before querying tables; equal coefficients at an internal or branch vertex are also rejected (they identify the same incident slot). Put $L_s=2^{1.001s}$, an upper bound on label/height continuation choices per edge.

For a coherent path, identify its two endpoints only through depth $h$; after that depth choose their remaining active raw $u$-bits freely. A theta has no free endpoint bits.

\begin{lemma}[Unique reconstruction and free-coordinate count]\label{lem:pruning-reconstruction-v2}
Fix a coherent-path pattern with different junction coefficients and its free endpoint $u$-bits. For any realization of the universal tables, there is at most one assignment of vertex states. If the endpoint scales are $H_-$ and $H_+$, the number of free bits is at most $(H_-+H_+)/2-h$. For a theta pattern, there is at most one assignment and no free bits. In either case all queried inputs and keys at a source block are determined before its tables are exposed.
\end{lemma}

\begin{proof}[Sketch]
Choose two incident slots at every internal vertex (two slots per theta branch vertex; one slot plus prescribed $u$ at a free end beyond $h$). Two evaluations with distinct coefficients $c_1\ne c_2$ recover $(u,v)$:
$u=(c_1-c_2)^{-1}(E_1-E_2)$, $v=E_1-c_1u$; this extends over full padded blocks. Updates proceed in the interleaved order; each key contains strictly earlier raw blocks and each correction window ends before both coupled blocks. The interleaving inequality $r-s\le\ell_A(i)-\ell_B(i)\le r$ gives free-endpoint count $(H_-+H_+)/2-h$. \qed
\end{proof}

\subsection{Probability at one update}

By Proposition~\ref{prop:sequential-v2}, an event $B_t$ involving $k_t$ whole current tables at source block $t$ satisfies
\begin{equation}\label{eq:pruning-current-v2}
 \Pr(B_t\mid\mathcal F_{<t})\le e^{\epsilon_s k_t}\Pr_{\mathrm{prod}}(B_t\mid\mathcal F_{<t}),\qquad\epsilon_s\to0.
\end{equation}

\begin{lemma}[Predictable accumulated rank]\label{lem:rank-v2}
Suppose finitely many source-block tests are selected from $\mathcal F_{<t}$, each with product-law probability at most $2^{-R_t}$ and $R_t,k_t$ nonnegative $\mathcal F_{<t}$-measurable. If an event $A$ implies that all pass, that $\sum R_t\ge R$, and that $\sum k_t\le K$, then
$\Pr(A)\le e^{\epsilon_s K}2^{-R}$.
\end{lemma}

\begin{proof}
Give the process value $0$ on failure; otherwise multiply by $2^{R_t}e^{-\epsilon_s k_t}$ on pass. Conditional expectation is nonincreasing by \eqref{eq:pruning-current-v2}; the initial value is $1$. On $A$ the value is at least $2^Re^{-\epsilon_s K}$. \qed
\end{proof}

\subsection{Terminal dimensions}

For a vertex, set $\lambda_A(i)=r+\ell_B(i)-\ell_A(i)$ and $\lambda_B(i)=s-r+\ell_A(i)-\ell_B(i)$. These are the surplus evaluation lengths in $[0,s]$; $\lambda_A(i)+\lambda_B(i)=s$ and $|\lambda_A(i)+\lambda_B(j)-s|\le b$ when $|i-j|\le1$.

\begin{lemma}[Terminal compatibility]\label{lem:terminal-v2}
If both incident common cutoffs equal the receiving vertex's cutoff, two independent incident evaluations impose terminal constraints of product-law probability exactly $2^{-\lambda(v)}$. If each incident cutoff is at most $b$ bits below the cutoff of the receiver, the probability of an event on those two current tables is at most $2^{-\max\{0,\lambda(v)-4b\}}$. In both statements the keys must be distinct between the tested edge occurrences, the inputs must be known before the current tables are exposed, and the required gates must be open.
\end{lemma}

\begin{proof}[Sketch]
If the final active input has $t$ bits, its evaluation block is full, and the map $(u,v)\mapsto(E_1,E_2)$ has image of dimension $s+t$ (injectivity uses a nonzero coefficient difference), giving codimension $s-t$. If the input has ended and the last evaluation has $t$ bits, agreement of two uniform $t$-bit values has codimension $t$. These add to $s$. For adjacent incidences, translation by the fixed correction $C^v_a+K_aC^u_a$ costs no dimension, and projection to a common active random prefix of width $w'$ leaves an affine image of dimension $\le w'+t$ in $2w'$. Boundary cases $t=b,t=s$ charge $0$ at the surplus side. \qed
\end{proof}

\begin{lemma}[Repeated terminal key identifies prefixes]\label{lem:key-prefix-v2}
If two distinct tested terminal edge occurrences at the same source block use the same table key, their receiving vertices lie on the same side and agree in raw $(u,v)$ through available aligned depth at least $H_*-8s$, where $H_*$ is the minimum scale along the counted path or theta.
\end{lemma}

\begin{proof}[Sketch]
At a $B_j$ update the queried table is $P_{A,j-1}$; its key identifies family, full label, $A$-inputs through block $j-2$, and $B$-inputs through $j-1$, hence $B$-evaluations through $j-1$. At an $A_j$ update the table is $P_{B,j}$, giving the same conclusion reversed. The preceding complete receiver prefix has aligned depth $ks\ge H_c-2s\ge H_*-2s-b$. Straddling a boundary costs one more aligned step of $2s$. \qed
\end{proof}

\subsection{A deterministic end guard for minimal returns}

For a coherent pattern with length $p$, initial scale $H$, and margin $M=M_k$, put $H_*=\min$ vertex scale and $E_p=100(s+bp)$, $J_p=\lceil 4E_p/s\rceil+4$. The end guard retains all receiver positions within distance $J_p$ of either end.

\begin{lemma}[Key separation outside the guard]\label{lem:end-guard-v2}
In every minimal coherent return, all terminal keys tested outside the end guard are distinct at each source block.
\end{lemma}

\begin{proof}[Sketch]
Repeated keys give agreement to depth $\ge H_*-E_p$. The contiguous subpath on the same side is a shorter coherent return unless it is the entire path; in the remaining case both endpoints lie in the guards. \qed
\end{proof}

In the counting experiment, reject a pattern at an update if selected terminal keys outside the guard coincide, or if a required gate is closed. These are predictable rejections and lose no actual minimal return. Total terminal rank:
\begin{equation}\label{eq:path-rank-v2}
 R_{\mathrm{ret}}\ge ps/2-10^4(s+bp).
\end{equation}

\begin{lemma}[Expected coherent-return count]\label{lem:return-count-v2}
For sufficiently large $s/b$ then $H_0/s$, the expected number of minimal coherent returns based on either side at height $i$, counted over all bands, is at most $2^{.975H_i}$.
\end{lemma}

\begin{proof}[Sketch]
Put $H=H_i$. $p\le 2(H-M)/s\le 2H/s$; eligible bands are a fixed number; there are at most $L_s^p$ patterns and at most $2^{H-h+bp}$ free coordinates. Apply Lemma~\ref{lem:rank-v2} with \eqref{eq:path-rank-v2}: contribution is $2^{H+.001sp-M+10^5(s+bp)}$. Since $M\ge.03H$ and $sp\le 2H$, exponent $\le .972H+10^5(s+2bH/s)<.975H$ at the stated scales. \qed
\end{proof}

Delete every vertex on every minimal coherent return, and denote the resulting induced graph by $X^{(1)}$.

\begin{lemma}[First deletion gives base girth]\label{lem:base-girth-v2}
No nonempty cyclically reduced closed path based at height $i$ in $X^{(1)}$ has length at most $.93m_i$.
\end{lemma}

\begin{proof}[Sketch]
Such a path has available prefix depth $\ge H_i-2s$, and required depth $<.98H_i$. It is a coherent return and contains a minimal one, hence a deleted vertex. \qed
\end{proof}

\subsection{Theta constraints}

Fix a bicycle of length $e<1.52m_i$ with a root at height $i$ in $X^{(1)}$, and put $H=H_i$. Along it, $|H_v-H|\le be$. By Lemma~\ref{lem:base-girth-v2}, a figure-eight or barbell would require two edge-disjoint cycles with total length $>1.86(1-1.52a)m_i>1.52m_i$; hence only a theta is possible. Every theta vertex lies on a cycle of length $\le e$, so restrict to $e>.9m_i$.

At each branch vertex of a theta, three evaluations satisfy
\[
 (c_2-c_3)E_1+(c_3-c_1)E_2+(c_1-c_2)E_3=0,
\]
with nonzero $c_j$ after incorporating known corrections.

\begin{lemma}[Branch-vertex ranks]\label{lem:triple-ranks-v2}
One branch-vertex relation in an ordinary full block with three open gates has product-law probability $2^{-s}$. Two distinct three-element key sets at same-side branch vertices have joint probability $2^{-2s}$; identical triples are always chargeable only once.
\end{lemma}

\begin{proof}[Sketch]
At one branch vertex, three distinct labels yield three distinct keys and three independent uniform field elements; the relation solves for any one, giving $2^{-s}$. For two distinct triples, condition on all other whole tables, leaving two exclusive tables at one known argument each. Each is uniform with nonzero coefficient and does not enter the other branch equation, giving $2^{-2s}$; extra formal queries on the same tables integrate out. \qed
\end{proof}

\begin{lemma}[Terminal keys in a surviving theta]\label{lem:theta-keys-v2}
Let a theta in $X^{(1)}$ have $e<1.52m_i$ edges and root at height $i$. At every source-block update its terminal keys are distinct. If its two branch vertices have the same side, any key shared by their branch triples at depth beyond $.65H$ is impossible, apart from a fixed initial/final block allowance.
\end{lemma}

\begin{proof}[Sketch]
A repeated terminal key gives agreement through $H-be-8s$; the shortest theta path has length $<.76m_i$ and coherent threshold $<.81H$, so survival in $X^{(1)}$ is contradicted. For a shared branch key, the shortest branch has length $<.51m_i$ and coherent threshold $<.56H$, again impossible. \qed
\end{proof}

Separate branch and terminal tests by source index. Terminal updates of the fixed theta lie in a fixed source-index interval of width $\le 9+2be/s$ around $H/s$; enlarge by ten and omit from branch charges. Over the six branch-vertex slot sequences and both source parities, deterministic omitted blocks cost less than $.001H+O(s+be)$, and the union of bad gates at three slots per branch vertex costs at most $.015H+O(s+be)$ (gate deletion being normalized separately on $A$- and $B$-parities). Charging one branch vertex before depth $.65H$ and two after gives at least $.675H-O(s+be)$ bits; after subtracting the two omitted-block budgets,
\begin{equation}\label{eq:branch-budget-v2}
 R_{\mathrm{branch}}\ge .675H-.015H-.001H-10^4(s+be)>.64H.
\end{equation}
Terminal rank on the theta's $e-1$ vertices:
\begin{equation}\label{eq:theta-terminal-v2}
 R_{\mathrm{term}}\ge se/2-100(s+be).
\end{equation}
Every terminal update was omitted from the branch charges; the two contributions compose chronologically.

\begin{lemma}[Expected theta count]\label{lem:theta-count-v2}
For sufficiently large parameters, the expected number of rooted embedded bicycles in $X^{(1)}$ of length $<1.52m_i$ based on either side at height $i$ is at most $2^{.89H_i}$.
\end{lemma}

\begin{proof}[Sketch]
Only theta patterns matter. For fixed total $e$, at most $100(e+1)^4$ combinatorics, then at most $L_s^e$ label/height patterns. No free coordinates. Use the predictable terminal and branch tests; current-table count $\le 20e$. Combine: exponent
$\le .501se-.64H_i+10^5(s+be)$, whose main coefficient is $<.883$ for $e<1.52m_i$. Absorb polynomial and error terms into the parameter choice. \qed
\end{proof}

Delete every vertex on each bicycle of length $<1.52m_i$ for at least one of its vertices of height $i$; denote the result $X_{\mathrm{old}}$.

\subsection{From bicycle pruning to Heisenberg nonlaws}

Let $\HH=\langle x,y,z\mid[x,y]=z,[x,z]=[y,z]=1\rangle$. Orientation the edges of a graph, associate a distinct variable $t_e$ to each unoriented edge, extend $t_{\bar e}=t_e^{-1}$. A path's word is an \emph{$\HH$-law} if it evaluates to the identity for every independent assignment $t_e\in\HH$.

\begin{lemma}[Heisenberg laws contain a bicycle used four times]\label{lem:heisenberg-law-v2}
Let $W$ be a nonempty closed edge path in $X$, reduced as a based edge path (no two consecutive traversals are inverse). If its word is a $\HH$-law, its support contains a bicycle $B$ such that every edge of $B$ is traversed at least four times by $W$. Hence $\ell(W)\ge 4\ell(B)$; in particular if every bicycle in the support has length $\ge T$, then $\ell(W)\ge 4T$.
\end{lemma}

\begin{proof}[Sketch]
For an oriented edge $e$, let $n_e(V)$ be its signed traversal count in an edge word $V$. Assigning $t_e=x$, others identity forces $n_e(W)=0$ for all $e$, so every used edge has positive even multiplicity; call it \emph{paired} if exactly two, \emph{heavy} if at least four.

For a paired $e$, write cyclically $W=t_e^{\epsilon}U t_e^{-\epsilon}V$ and assign $t_e=x$, $t_f=y$, others identity. Then $x^{\epsilon}y^{n_f(U)}x^{-\epsilon}y^{-n_f(U)}=z^{\epsilon n_f(U)}$, so $n_f(U)=0$; the interval between the two occurrences of any paired edge is balanced on every edge. Such intervals cannot cross.

If paired edges exist, choose an inclusion-minimal paired interval $U$ (now heavy on every edge); if none, take $U=W$. In either case $U$ is a nonempty reduced closed path, balanced, supported on heavy edges. Write $U=PCP^{-1}$ with $C$ cyclically reduced, still balanced, nonempty (else an adjacent reversal in the reduced $U$).

Delete valence-one vertices. The remaining core $S'$ has the same cycle rank as the support $S$ of $U$, and cannot have $|E(S')|\le|V(S')|$ (resp \ $|E|=|V|$ with all-valences-two) without supporting a nonzero-extremal sum on every core edge, contradicting balance. So $S'$ has cycle rank $\ge 2$. Choosing a spanning tree and two off-tree edges yields a bicycle $B$ inside $S'$, all whose edges are heavy in $W$. Summing traversal multiplicities over $B$ gives $\ell(W)\ge 4\ell(B)$. \qed
\end{proof}

At comparison scale $m>0$, exclusion of bicycles of length $<1.52m$ gives $\ell(W)\ge 6.08m$ for any $\HH$-law $W$ supported in the region of exclusion. In particular every nonempty cyclically reduced closed path there of length $<6.03m$ is a nonlaw.

\subsection{Simultaneous small deletion in every layer}

\begin{proposition}[Pruned old graph]\label{prop:pruning-v2}
Given $\varepsilon>0$, the parameters and the universal tables can be chosen so that the additional deletions of this section remove less than an $\varepsilon$ fraction of each $\mathscr A_i$ and $\mathscr B_i$. In the surviving induced old graph $X_{\mathrm{old}}$:
\begin{enumerate}
\item folded labeling, padded recovery (Lemma~\ref{lem:recovery-v2}), and global word uniqueness persist;
\item nonempty cyclically reduced closed paths at height $i$ have length $>.93m_i$;
\item bicycles touching height $i$ have length $\ge 1.52m_i$;
\item nonempty cyclically reduced closed paths at height $i$ of length $\le 6.04m_i$ are not $\HH$-laws.
\end{enumerate}
The expansion conclusions for undeleted incidences persist, and degrees remain fixed along the tower.
\end{proposition}

\begin{proof}[Sketch]
Let $Z_i^A,Z_i^B$ be the additional deleted fractions. A counted object of length $\ell$ rooted at $j$ meets layer $i$ only if $|i-j|\le\ell$ and $\ell\le 4H_j/s$. For $s\ge 20b$ and $H_j=H_i+b(j-i)$, $\ |i-j|\le 5H_i/s$ and $\ |H_j-H_i|\le 5bH_i/s$. Combine Lemmas~\ref{lem:return-count-v2} and~\ref{lem:theta-count-v2} to get
$\mathbb E(Z_i^A+Z_i^B)\le 2^{-.01H_i}$ for large $s/b,H_0/s$. Since $\sum_i$ of these is summable, choose $H_0$ so the expected total is $<\varepsilon/2$; Markov gives positive probability. Passing to an induced subgraph preserves foldedness and word uniqueness. The last assertion uses Lemma~\ref{lem:heisenberg-law-v2}: an $\HH$-law of length $L\le 6.04m_i$ contains a bicycle $B$ with $L\ge 4|E(B)|\ge 4\cdot 1.52(1-6.04a)m_i$, and $a\le 10^{-6}$ makes $6.08(1-6.04a)>6.04$, contradiction. \qed
\end{proof}

\section{Diffuse Heisenberg voltages and voltage approximation}\label{sec:analysis-v2}

Fix a surviving tower furnished by Proposition~\ref{prop:pruning-v2}, with any finite incidence type set. All later edge draws are independent on this fixed graph. Before deletion a type between $\mathscr A_i$ and $\mathscr B_{i+\delta}$ is biregular with degrees $d_{\tau,ik},D_{\tau,ik}$, geometric mean $q_{\tau,ik}=\sqrt{d_{\tau,ik}D_{\tau,ik}}$, and surviving normalized incidence $U_{\tau,ik}$. There are positive constants $c_*,C_*$ (independent of height, and of $s$ when large) with
$c_*d\le d_{\tau,ik}\le C_*d$, $c_*D\le D_{\tau,ik}\le C_*D$, $c_*q\le q_{\tau,ik}\le C_*q$. In particular $\norm{U_{\tau,ik}}\le 1$ and the maximum degree of the base graph is $\le C_*D$.

\subsection{Compact Bott representative}

Define $\theta\colon[0,\infty)\to[0,\pi]$ as follows: $h(t)=e^{-1/t}$ for $t>0$, $h(t)=0$ for $t\le 0$; $\zeta(r)=h(r-1/4)/(h(r-1/4)+h(1-r))$; $\theta(r)=2r+(\pi-2r)\zeta(r)$ for $r\in[0,1]$ and $\theta(r)=\pi$ for $r\ge1$. For $r=(t^2+u^2)^{1/2}>0$,
\begin{equation}\label{eq:bott-v2}
 p(t,u)=\tfrac12\begin{pmatrix}1+\cos\theta(r)&\dfrac{\sin\theta(r)}{r}(t-iu)\\[3pt]\dfrac{\sin\theta(r)}{r}(t+iu)&1-\cos\theta(r)\end{pmatrix},\qquad e_\infty=\begin{pmatrix}0&0\\0&1\end{pmatrix}.
\end{equation}
This extends smoothly to $(0,0)$ (since $\sin(2r)/r$ and $\cos(2r)$ are analytic functions of $r^2$), and $p-e_\infty$ is smooth and compactly supported. Direct computation, or $(n\cdot\sigma)^2=1$, gives $p=p^*=p^2$ and $\operatorname{Tr} p=1$.

Its local unit eigenvectors near origin and near exterior are related by $e^{-i\phi}$, so the winding number is $-1$; equivalently
$(2\pi i)^{-1}\int\operatorname{Tr}(p\,dp\wedge dp)=\tfrac12\int_0^1\theta'(r)\sin\theta(r)\,dr=1$.
The rank-zero difference $[p]-[e_\infty]$ represents a Bott generator.

Fix $\alpha=(\sqrt5-1)/2$ and, at height $i$, set $N_i=2^{200H_i}$, $Z_i=2^{500H_i}$, $J_i=2^{50H_i}$. Define
$\mathsf p_i(\vartheta,\xi)=p(Z_i(\vartheta-\alpha),N_i\xi)$ for torus coordinates $(\vartheta,\xi)\in(\R/\Z)^2$ around $(\alpha,0)$, extended by $e_\infty$. Then $\beta_i=\mathsf p_i-e_\infty$ coincides with the fixed Bott class of the torus up to a $(\pm1)$ sign.

Adjacent profiles join by the explicit rescaling
$p(Z_i^{1-v}Z_k^v(\vartheta-\alpha),N_i^{1-v}N_k^v\xi)$ for $|i-k|=1$, $0\le v\le1$; and the matrix rotation $R(v)=I_2\otimes(\begin{smallmatrix}\cos(\pi v/2)&-\sin(\pi v/2)\\\sin(\pi v/2)&\cos(\pi v/2)\end{smallmatrix})$ acts by conjugation to produce the Eilenberg-swindle. All entries lie in a fixed compact smooth family with moduli independent of height when $b$ is fixed.

More generally we use profiles $f_i(\vartheta,\xi)=F(Z_i(\vartheta-\alpha),N_i\xi)$ with $F$ in a fixed compact family $C_c^\infty(\R^2;M_\ell(\C))$ supported in one fixed rectangle.

\subsection{Spreading on disjoint torus translates}

The golden-ratio separation
\[
 \norm{n\alpha}_{\R/\Z}\ge\frac1{3|n|}\qquad(n\ne0)
\]
holds because $(n\alpha-p)(n\alpha'-p)=p^2+np-n^2\in\Z\setminus\{0\}$ for $\alpha'=(-\sqrt5-1)/2$ and any nearest-integer $p$.

Choose $a\in C_c^\infty((-1,1))$ with $a\not\equiv0$. Put
\[
 a_{ij}=\frac{a(j/J_i)}{\bigl(\sum_k|a(k/J_i)|^2\bigr)^{1/2}},\qquad
 T_i=\sum_j a_{ij}\lambda(y)^j,\qquad \sum_j|a_{ij}|^2=1.
\]
The denominator is bounded above and below by positive multiples of $J_i^{1/2}$ at large $H_i$.

Let $E_i$ be the Borel spectral projection of the enlarged rectangle $|\vartheta-\alpha|\le R/Z_i$, $|\xi|\le R/N_i$ containing all support of profiles incident at height $i$.

\begin{lemma}[Supported spreading]\label{lem:spreading-v2}
For large $H_0$ the projections $\lambda(y)^jE_i\lambda(y)^{-j}$, $|j|\le J_i$, are pairwise orthogonal. Hence $E_iT_i^*T_iE_i=E_i$, $P_i=T_iE_iT_i^*$ is a projection, and
\[
 \tau_\HH(P_i)=\tau_\HH(E_i)\le C_R2^{-700H_i}.
\]
If $f,g$ are supported in the enlarged rectangles at adjacent heights as indicated, then
$(T_ifT_k^*)(T_kgT_j^*)=T_i(fg)T_j^*$, $(T_ifT_k^*)^*=T_kf^*T_i^*$, $\norm{T_ifT_k^*}\le\norm f$.
\end{lemma}

\begin{proof}[Sketch]
$[x,y]=z$ gives $y^jxy^{-j}=xz^{-j}$; conjugating the torus rectangle translates the $\xi$-coordinate by $-j\vartheta$. When $0<|j-k|\le 2J_i$ and $|\vartheta-\alpha|\le R/Z_i$,
$\ \norm{(j-k)\vartheta}_{\R/\Z}\ge\frac1{6J_i}-2RJ_i/Z_i>2R/N_i$ at large heights (as $N_i=J_i^4,Z_i=J_i^{10}$). Translated supports are disjoint. The trace on the torus algebra is Haar measure (integral against characters $\lambda(z)^c\lambda(x)^n$); $E_iT_i^*T_iE_i=E_i$ then gives $\tau(P_i)=\tau(E_i)\le 4R^2/(Z_iN_i)$. \qed
\end{proof}

\subsection{Fourier envelope}

Write $A=\sum_{n,l,c}A_{n,l,c}\lambda(x^ny^lz^c)$ and $\norm A_1=\sum\norm{A_{n,l,c}}$; the one-norm dominates the regular operator norm.

For a periodized function $g(Z(\vartheta-\alpha))$ its central Fourier coefficients are $c_k=Z^{-1}e^{-2\pi ik\alpha}\widehat g(k/Z)$, and integration by parts $A+2$ times gives
\begin{equation}\label{eq:rescaling-v2}
 \sum_k(1+|k|/Z)^A\norm{c_k}\le C_A\sum_{j=0}^{A+2}\norm{g^{(j)}}_{L^1(\R)}.
\end{equation}

\begin{lemma}[Fourier envelope]\label{lem:fourier-envelope-v2}
Let $|i-k|\le1$ and let $f$ belong to the profile family. Writing
$T_ifT_k^*=\sum_{n,l,c}A_{n,l,c}\lambda(x^ny^lz^c)$ with $N=N_i,Z=Z_i,J=J_i$, for every pair $A,P\ge 0$, $P\ge3$,
\[
 \sum_c(1+|c|/Z)^A\norm{A_{n,l,c}}\le C_{A,P}N^{-1}(1+|n|/N)^{-P}(1+J\norm{n\alpha}_{\R/\Z})^{-3}
\]
whenever $|l|\le C_bJ$ (and coefficients vanish otherwise). The constants are uniform in height and in the compact profile family; the one-norm is uniformly bounded. For every $\eta>0$ the series can be truncated to $|n|\le L_\eta N$, $|l|\le C_bJ$, $|c|\le L_\eta Z$ with one-norm error $\le\eta$.
\end{lemma}

\begin{proof}[Sketch]
Fourier transform in $\xi$: with $t=Z(\vartheta-\alpha)$, the $n$-th coefficient is $N^{-1}\Phi(t,n/N)$ where $\Phi(t,v)=\int F(t,u)e^{-2\pi ivu}du$ is Schwartz in $v$ uniformly. The $y$-power $l$ coefficient picks up $f_n(\vartheta)S_{n,l}(\vartheta)$ with $S_{n,l}=\sum_j b_{j,l}e^{-2\pi inj\vartheta}$, $b_{j,l}=a_{ij}\overline{a_{k,j-l}}$; the $b_{j,l}$ are supported on $O_b(J)$ indices, vanish unless $|l|\le C_bJ$, and satisfy $\sum|\Delta^rb_{j,l}|\le C_rJ^{-r}$. Summing by parts against $e^{-2\pi inj\alpha}$ (and retaining the residual central phase) gives the displayed corner. Applying \eqref{eq:rescaling-v2} with $g=N^{-1}\Phi(t,n/N)S_{n,l}(\alpha+t/Z)$ and $r=3$ absorbs $NJ/Z\le2^{-250H_i}$ into the Schwartz exponent. The compensating $J^{-1}$ uses golden separation applied to the integral identity $(1+Jv)^{-3}=\int_v^\infty3J(1+Ju)^{-4}du$; splitting $n$ into $\{|n|\le N\}$ and dyadic annuli gives $\sum_n N^{-1}(1+|n|/N)^{-3}(1+J\norm{n\alpha})^{-3}\le C/J$. \qed
\end{proof}

\subsection{Diffuse product sampler}

\begin{lemma}[Product sampler]\label{lem:sampler-v2}
For every fixed $\eta>0$, each symbol in Lemma~\ref{lem:fourier-envelope-v2} has a finite distribution of pairs $(W,v)$ with $W\in M_\ell(\C)$, $v=x^ny^lz^c\in\HH$, satisfying
$\norm W\le M_\eta$ and $\norm{\mathbb E[W\lambda(v)]-T_ifT_k^*}_1\le\eta$. The horizontal coordinates $n,l$ are independent with maximal atoms $O_b(J_i/N_i), O_b(J_i^{-1})$. Every realized group coordinate (including inverses) is bounded by $2^{a_0H_i}$ for a fixed $a_0$ (e.g.\ $502$). The weights can be restricted to one finite adjoint-closed net at any additional one-norm cost.
\end{lemma}

\begin{proof}[Sketch]
Truncate to the finite Fourier box of Lemma~\ref{lem:fourier-envelope-v2}. Put $w_n=N_i^{-1}(1+|n|/N_i)^{-3}(1+J_i\norm{n\alpha})^{-3}$ and $R_{n,l}=\sum_c\norm{A_{n,l,c}}\le C_1w_n$; $\sum w_n\le C_0/J_i$. Build $\mu_i(n)$ from masses $J_iw_n/(2C_0)$ plus uniform remainder on $\{-N_i,\ldots,N_i\}$, and $\nu_i(l)$ uniform on the allowed interval. Then $R_{n,l}\le M_\eta\mu_i(n)\nu_i(l)$. Draw $(n,l)\sim\mu_i\otimes\nu_i$, then $c$ conditioned on $\norm{A_{n,l,c}}/R_{n,l}$ and set $W=R_{n,l}/(\mu_i(n)\nu_i(l))\cdot A_{n,l,c}/\norm{A_{n,l,c}}$; zero coefficients yield $W=0$ with the same diffuse horizontal law. Round to a finite adjoint-closed net of the closed matrix ball of radius $M_\eta$. The coordinate bound follows from $F$'s support along with $|nl|\ll Z_i$. \qed
\end{proof}

\subsection{Polynomial avoidance}

\begin{lemma}[Polynomial avoidance]\label{lem:poly-avoidance-v2}
Use independent edge draws as in Lemma~\ref{lem:sampler-v2}. For a base walk reuse its voltage variable on every occurrence of an unoriented edge (and inverse on the reverse). If the associated word is not an $\HH$-law, the probability that its holonomy is the identity is at most $C_b2^{-50H_{\min}}$, where $H_{\min}$ is the minimum scale of an edge in the walk. The estimate is unaffected by matrix weights, including zero.
\end{lemma}

\begin{proof}[Sketch]
Schwartz--Zippel: $\kappa$-atomic real variables with nonzero polynomial of total degree $d$ have vanishing probability at most $d\kappa$. The normal-form multiplication $(n,l,c)(n',l',c')=(n+n',l+l',c+c'-ln')$ makes the horizontal coordinates of any word integral linear combinations of edge $(n_e,l_e)$ with signed occurrence counts; if any count is nonzero, one horizontal polynomial is nonzero. Otherwise the central coordinate is a polynomial of degree $\le2$ in independent horizontal variables, nonzero because otherwise the word would be an $\HH$-law. Their maximal atoms are $C_b2^{-50H_{\min}}$ (since $J_i/N_i=2^{-150H_i}$). Apply the polynomial bound. \qed
\end{proof}

\subsection{Finite tests for the full regular norm}

\begin{lemma}[Heisenberg box and norm test]\label{lem:box-v2}
Finite group elements and their inverses with coordinates bounded by $2^{a_0H}$ fit a finite box $\mathcal B_H\subset\HH$ of cardinality $\le 2^{K_0H}$ with symmetric-difference ratio $\le C2^{-BH}$ for every fixed $B$ and any fixed finite set of such elements. If $A\in B(\ell^2)\otimes M_\wedge(\C)$ is a selfadjoint finite regular convolution sum supported there, with absolute coefficient row sums $\le R$, then
\[
 \norm{Q_HAQ_H}\le\norm A\le\norm{Q_HAQ_H}+C2^{-BH}R.
\]
The rectangular case follows by off-diagonal doubling.
\end{lemma}

\begin{proof}
Take the box $\mathcal B(T)=\{x^ny^lz^c:|n|,|l|\le T,|c|\le T^2\}$ with $T=\lceil2^{(a_0+B+2)H}\rceil$. Right or left multiplication by a coordinate-bounded $v$ changes a horizontal coordinate by at most $R_0$ and the central coordinate by $R_0+R_0T$, giving strip fraction $\le CR_0/T$; use the bijectivity of translation. For the norm bound: right translation commutes with left convolution, so every right-translate compression $Q_tAQ_t$ has the same norm; averaging $\ip{AQ_t\xi}{Q_t\xi}$ over $t$ introduces boundary losses of at most $C2^{-BH}R$. Cauchy--Schwarz on the coefficient-matrix of nonnegative entry-norms with row and column sums $\le C2^{-BH}R$ completes the difference bound. \qed
\end{proof}

\subsection{Support rank and walk count}

\begin{lemma}[Length vs cycle rank]\label{lem:length-rank-v2}
Let $S$ be a finite connected metric graph of total length $\le Kg$ and girth $>g$. Then
$\dim H_1(S;\mathbb Q)\le\max\{1,\lceil20K\rceil\}^2$.
\end{lemma}

\begin{proof}[Sketch]
Set $a=g/10$ and choose a maximal $a$-separated net $P\subset S$. Disjoint $a/2$-balls give $|P|\le\max\{1,\lceil20K\rceil\}$. Join nearby net points by shortest paths; the resulting abstract graph $L$ surjects on $H_1(S;\mathbb Q)$ because every path is homotopic to connectors plus sub-cycles shorter than $g$. Hence $\dim H_1(S)\le|P|^2$. \qed
\end{proof}

\begin{lemma}[Repeated-edge walk count]\label{lem:walk-count-v2}
Let a graph have max degree $\Delta\ge1$. Consider length-$2h$ closed walks from a fixed vertex with no unoriented edge used exactly once. If every support under consideration has cycle rank $\le R$, their number is at most $[72(2R+2)^2\Delta]^h$.
\end{lemma}

\begin{proof}[Sketch]
Prune leaves and bound by the rooted core $K$; $\sum_{v\in K}(\deg_K(v)-2)=2\dim H_1(K)-2$ gives $\deg_K(v)\le 2R+2$. Root and explore by marking edges; core steps have $\le 2R+2$ choices, descents $\Delta$; a balanced push/pop word with at most $4^e$ possibilities and a $3^{2h}$ step classification combine to the displayed bound. \qed
\end{proof}

\subsection{Centered block concentration}

\begin{lemma}[Centered concentration]\label{lem:block-concentration-v2}
Fix a surviving block $X_{ik}$. Under the independent sampling law without conditioning on girth, its centered convolution matrix satisfies
\[
 \Pr\{\norm{X_{ik}}>\eta q_{\tau,ik}\}\le 2^{-6H_i}
\]
uniformly in height, after the parameters are fixed in the order $b$, profiles/net, then $s$, then $H_0$.
\end{lemma}

\begin{proof}[Sketch]
Entrywise $\norm{W_e\lambda(v_e)-\mathbb E W_e\lambda(v_e)}\le 2M_\eta$; off-diagonal doubling makes $X_{ik}$ selfadjoint; row and column coefficients $\le C_*M_\eta D$. Choose $H_0\ge s$ so $D\le 2^{.60005H_i}$ and $|\mathscr A_i|\le 2^{1.2H_i}$, absorb fixed constants in $H_0$ so the row sum is $\le 2^{2H_i}$ and the base-plus-matrix index count is $\le 2^{3H_i}$. The $\HH$-box error is $o(\eta q_{\tau,ik})$ and the compression dimension is $\le 2^{K_1H_i}$ for a fixed $K_1$. Fix $C>2(K_1+6)$ and $h=\lceil CH_i/\log_2d\rceil$. Supports in a $2h$-moment touch $\le h$ base edges in the two fixed layers, whose base girth is at least $.9H_i/\log_2q$; since $\log_2q/\log_2d<1.25$, Lemma~\ref{lem:length-rank-v2} bounds cycle rank $R=R(C)$ alone. Every summand with a singleton unoriented edge vanishes in expectation by conditioning; other summands contribute $\le 2^{K_1H_i}(C_1D)^h$ via Lemma~\ref{lem:walk-count-v2}. Markov:
$\Pr\{\norm{X_{ik}}>\eta q_{\tau,ik}\}\le 2^{K_1H_i}(C_2/\eta^2d)^h\le 2^{(K_1-C/2)H_i}\le 2^{-6H_i}$ once $C_2/\eta^2\le d^{1/2}$. $C_2$ may depend on $C$; $s$ is chosen after. \qed
\end{proof}

\subsection{Simultaneous selection of voltages}

\begin{proposition}[Simultaneously sampled symbols]\label{prop:sampled-symbols-v2}
Fix $b$, a matrix dimension, an accuracy $\eta>0$, and a finite list of compact smooth symbol profiles with height selections. Allocate a prescribed incidence type to every same- or adjacent-layer symbol. For large enough $s$ then large enough $H_0$, the surviving base graph admits voltages on every edge and finite-net matrix weights such that:
\begin{enumerate}
\item the voltage cover has no nontrivial cyclically reduced closed walk of length $\le 6.03m_i$ at an old vertex of height $i$;
\item every weighted block normalized by the common $q$ satisfies $\norm{\mathcal V_{\tau,ik}-U_{\tau,ik}\otimes S_{\tau,ik}}\le\eta$ in the full regular operator norm at every height;
\item $|\mathcal A|\le 2^{1.00005s+O(\log s)+O_{b,\eta,\mathrm{profiles}}(1)}<q^2$, so $\sqrt{|\mathcal A|}/q\to0$ after fixed finite refinements.
\end{enumerate}
Zero-weight edges are retained in the voltage cover.
\end{proposition}

\begin{proof}[Sketch]
Put $a_{\tau,ik}=q/q_{\tau,ik}$, which lies in a fixed compact interval of $(0,\infty)$. Apply Lemma~\ref{lem:sampler-v2}, with smaller tolerances, to $a_{\tau,ik}S_{\tau,ik}$; round to the finite net; normalize the block by $q$. The mean matches to the specified one-norm error, and Lemma~\ref{lem:block-concentration-v2} supplies the fluctuation error (using $q_{\tau,ik}/q\le C_*$). For girth: a forbidden closed lift projects to a cyclically nonbacktracking closed base walk of length $\le\lceil6.03m_i\rceil<6.04m_i$, which is a nonlaw by Proposition~\ref{prop:pruning-v2}(iv). Lemma~\ref{lem:poly-avoidance-v2} bounds its identity probability by $2^{-40H_i}$. The number of possible rooted walk instructions is
$\le(|\mathscr A_i|+|\mathscr B_i|)\sum_{a\le\lceil6.03m_i\rceil}(C_*D)^a\le 2^{9H_i}$
(the exponent ratio $1+6.03(.60005/.50005)<8.24$ leaves margin). Union over heights (sum of failures $<1$) gives positive probability of a simultaneous assignment, and a diagonal subsequence argument upgrades finite satisfiability to a single assignment. Refined alphabet count: original $d2^r=2^{1.00005s}$, then finite net and residue flags contribute $2^{O(\log s)+O(1)}$; $\log_2(q^2)=1.0001s$, giving the sharp margin. \qed
\end{proof}

\section{Geometrical realization}\label{sec:geometry-v2}

Throughout this section all paths and diagrams are finite, even though the labels, graphs, and presentations may be infinite. Positive edges carry one formal letter; traversing backwards reads its inverse. A labeling is \emph{folded} if at each vertex each signed letter is read on at most one outgoing incidence; then an immersed path realizes a word that is freely reduced and the path (with its start) is determined by its word whenever it exists.

\subsection{Inputs from the pruning stage}

Let $X_{\mathrm{o}}=X_{\mathrm{old}}$ from Proposition~\ref{prop:pruning-v2} be the surviving old graph, with old alphabet $S_{\mathrm{o}}$ and Heisenberg voltages $h_e\in\HH$ from Proposition~\ref{prop:sampled-symbols-v2}. Put $a=b/\log_2 q\le10^{-6}$, $L=1.003$, $m_{\min}\ge10^6$; at an old vertex of height $i$ set $m(v)=m_i$. The pruning proposition supplies three assertions, used repeatedly below.

\begin{lemma}[Three inputs]\label{lem:three-inputs-v2}
(a) $X_{\mathrm{o}}$ is countable, folded, locally finite, of maximum degree $\le K_{b,\eta}D$; edges join same or consecutive heights, so $|m(v)-m(w)|\le a$ on old edges.
(b) If an immersed old path starting at $v$ has two distinct projected occurrences of its word in $X_{\mathrm{o}}$, its length is less than $Lm(v)$; the assertion applies to either occurrence and to its reverse.
(c) In the old voltage graph, every simple cycle $C$ satisfies $|C|>6.03m(v)$ at every vertex $v$ of $C$.
\end{lemma}

Indeed (a) is Proposition~\ref{prop:tower-labels-v2}(1,2) and Lemma~\ref{lem:gate-v2}; (b) is Proposition~\ref{prop:tower-labels-v2}(4) with the bound $\lceil1.002m(v)\rceil<1.003m(v)$ for $m(v)\ge10^6$; (c) is Proposition~\ref{prop:pruning-v2}(4) after passing to the actual cover.

\subsection{Private completion}

A graph with marked old part $(X_{\mathrm{o}},S_{\mathrm{o}},\mathcal Q)$ is a folded labeled graph containing an isomorphism $X_{\mathrm{o}}\to Y|_{\mathcal Q}$ with $\mathcal Q$ connected and locally finite, whose restriction to $S_{\mathrm{o}}$ is exactly the old labeling; letters on the other edges form the private alphabet $S_{\mathrm{p}}$, and a private edge is one incident with a private vertex. A private vertex has no old coordinates.

\begin{lemma}[Private completion with controlled scale]\label{lem:completion-v2}
For $X_{\mathrm{o}}$ folded, connected, piecewise folded, there is a completion $X_{\mathrm{t}}$ with private alphabet $S_{\mathrm{p}}$ such that:
\begin{enumerate}
\item $|S_{\mathrm{p}}|$ is countable and each private letter is used on at most one abstract edge;
\item the label-preserving automorphism group equals the deck group (no nontrivial label-preserving automorphism fixes an old vertex);
\item any new simple path between old vertices has length $\ge2m_{\min}$;
\item the distance from any old vertex to any other vertex is $\ge m_{\min}$; every new cycle through an old vertex has length $\ge2m_{\min}$;
\item any old-path-reduced path of length $\le m_{\min}$ starting at an old vertex stays in the old part.
\end{enumerate}
\end{lemma}

\begin{proof}[Sketch]
Enumerate old components $(U_l)$ and their base vertices; connect them with private paths through fresh intermediate private vertices, using a fresh private letter on each abstract edge. Each $\geq2m_{\min}$ guarantee comes from a finite meridian of private letters through vertices not in old parts. The private completion preserves folding because each new letter is used once.
\end{proof}

Let $\Omega_{\mathrm{p}}$ denote the associated directed system of private letters; a \emph{private arc} is a maximal segment of an old-relative path lying entirely outside the old part.

\begin{lemma}[Closed-walk lower bound]\label{lem:closed-walk-v2}
Let $X_{\mathrm{o}}\subset X_{\mathrm{t}}$ as above, and $m_{\min}\ge10^6$. A closed walk $W$ in $X_{\mathrm{t}}$ of projective length less than $Lm_{\min}$ must contain a private arc or lie in $X_{\mathrm{o}}$. If it contains a private arc, then removing its maximal projective private arcs leaves an old-relative closed walk in $X_{\mathrm{o}}$ of length $\le 2Lm_{\min}+3$.
\end{lemma}

\subsection{The presentation group}

Put $\widetilde X=Y_{\mathrm{t}}$ for the ($\HH$-equivariant) completion of the old voltage cover $Y_{\mathrm{o}}$ of $X_{\mathrm{o}}$, with private completion applied equivariantly; this is a connected folded labeled graph, $\HH$ acts freely and label-preservingly, and is the full label-preserving automorphism group. Let $S=S_{\mathrm{o}}\sqcup S_{\mathrm{p}}$ be the full alphabet.

\begin{definition}\label{def:group-v2}
Let $S$ be as above and
\[
 \Gamma=\left\langle S\ \middle|\ \mathrm{lab}(C)=1\text{ for every finite closed walk }C\text{ in }\widetilde X\right\rangle.
\]
\end{definition}

Each occurrence of a relator in a Van Kampen diagram retains an actual closed walk of $\widetilde X$ reading it; distinct free words correspond to distinct source walks in $\widetilde X$.

\begin{lemma}[Closure of the cover]\label{lem:closure-v2}
The vertex map $j\colon\widetilde X\to\Cay(\Gamma,S)$ obtained by path-lifting from a fixed base vertex is injective, and every label-preserving automorphism of $\widetilde X$ is a deck transformation.
\end{lemma}

\begin{proof}[Sketch]
Suppose $j$ is not injective at some vertex; fix a globally minimal counterexample: a finite plane multigraph diagram $D$ embedded in the sphere, with an exterior contour reading a word $w$ that is trivial in $\Gamma$ but whose source lift has distinct endpoints, and with minimum filling cost $c$ (sum of bounded-face perimeters). All bounded faces are cyclically immersed and nonempty. We repeatedly reduce $c$:

\emph{Coincident bounded projections.} If two adjacent bounded faces project to the same base edge of $X_{\mathrm{o}}$, splice their source walks along a deck translation; new perimeter is the sum minus two.

\emph{Bridges.} A bridge with the same bounded face on both sides is isolated with its interior; the source excursion reads a word trivial in $\Gamma$ with filling cost below $c$.

\emph{Exterior coincidence.} If the exterior path and a bounded face project to the same edge, absorb the face into the exterior.

After these reductions, edges incident to bounded faces have distinct projections. Collect them as \emph{arcs} (maximal same-projection segments); each arc reads an immersed word on both occurrences, and the private bounds give length $\le Lm$ at either starting point. Every bounded face therefore has $\ge7$ incidences; otherwise \eqref{eq:arc-sum-v2} below violates $\ell\ge m/2$.

\begin{equation}\label{eq:arc-sum-v2}
 \ell(\partial f)=\sum_{\alpha\subseteq\partial f}\ell(\alpha)\le (L+3.018a+1.009a^2)\cdot 5m(v)<5.02m(v).
\end{equation}

Euler's formula then contradicts: with $V,E,F$ the compressed vertex, edge, and total face counts,
$6F-2E=12+2\sum_v(\deg(v)-3)\ge8$, while $\sum_f(6-\deg(f))\le(-1)\cdot(\text{bounded faces})+6$, giving $<8$. The retained exceptional marked vertex contributes the slack $6F-2E-6=2$ that keeps tightness. Contradiction.
\end{proof}

\subsection{Deck transport and its consequences}

Let $\phi\colon\HH\to\Gamma$ be defined by $\phi(f)=\mathrm{lab}(j(v_*,f))$: the word read on any source path from $(v_*,1)$ to $(v_*,f)$, which depends only on the endpoint by Lemma~\ref{lem:closure-v2}. This is a group homomorphism (concatenation of paths), and injective because $j$ is injective on vertices. We have the deck transport identity
\begin{equation}\label{eq:deck-transport-v2}
 j(fu)=\phi(f)j(u)\qquad(u\in\widetilde X,f\in\HH).
\end{equation}

\begin{proposition}[Graphical realization]\label{prop:graphical-v2}
$\phi\colon\HH\hookrightarrow\Gamma$ is injective. Its image is exactly the stabilizer in $\Gamma$ of the image of the lift $Y_{\mathrm{image}}=j(\widetilde X)\subseteq\Cay(\Gamma,S)$, and every translate of $Y_{\mathrm{image}}$ is a \emph{sheet}. Two sheets either coincide or meet in a vertex set whose incident Cayley edges form a forest.
\end{proposition}

\begin{proof}[Sketch]
Injectivity of $\phi$ and the stabilizer identity follow from Lemma~\ref{lem:closure-v2}. For the retraction-free intersection property, use Lemma~\ref{lem:three-inputs-v2} (b): if two sheets share a source path of length $\ge Lm(v)$ starting at $v$, folding propagates backward to identify the source vertices, contradicting distinct cosets. Since the old graph edges form a forest between translates of any boundary component, its removal leaves only those edges that lie in at most one sheet.
\end{proof}

\subsection{Fixed-contact polygons}

The next section needs a bound on the length of paths between prescribed sheet contacts. Say a source-parametrized sheet \emph{meets} another at a shared physical vertex.

\begin{lemma}[Fixed-contact arc bound]\label{lem:arc-v2}
Let a finite family of sheets be given, all sharing a fixed old base vertex $v$ at pairwise distinct cosets, and let $P_1,P_2$ be physical contacts on two (possibly equal) sheets. Then a reduced path in $X_{\mathrm{o}}$ from $P_1$ to $P_2$ whose maximal sheet-change segments alternate between at most three source paths projects to at most three source-arcs, each of length $\le L_3\,m(v)$ with $L_3<3.014$, and the total length is less than
\[
 \kappa m(v),\qquad\kappa=3.014.
\]
\end{lemma}

\begin{proof}[Sketch]
Let $P_1,P_2$ lie on sheets $Q_0,Q_1$ with source parametrizations $\rho^0,\rho^1$; fix a boundary basepoint $v$. Consider the unique reduced path in $X_{\mathrm{o}}$ joining them without backtracking in sheets. Each time it changes sheets, it reads a source path that is common to two sheets, hence short by Proposition~\ref{prop:graphical-v2}. The successive initial scale is bounded by $m(v)$, then $(1+La)m(v)$, then $(1+La)^2m(v)$; summing the three arc bounds and using $a\le10^{-6}$ gives
$\ \ell<L(3+3.018a+1.009a^2)m(v)<3.010m(v)$.
\end{proof}

\section{Disjoint domains, cone chains, and an integral detector}\label{sec:domains-v2}

The fixed-contact bound supplies disjoint analytic domains. Independently, coning the full sheets yields an exact cellular complex that proves torsion-freeness and extends an integral class from $\HH$.

\subsection{Peeling finite families and taking a diagonal limit}

For a vertex set $B\subseteq V(Q)$ let $\mathrm{Star}_Q(B)$ be the subgraph of all edges of $Q$ with at least one endpoint in $B$, together with their endpoints; this is the incidence neighborhood, not the induced subgraph. Put $\kappa=3.014$ and
$B_Q(v)=\{w\in V(Q):d_Q(v,w)\le\lfloor\kappa m_Q(v)\rfloor\}$.

\begin{lemma}[Allowed removed sets]\label{lem:allowed-domain-v2}
For an old vertex $v\in Q$, $\mathrm{Star}_Q(B_Q(v))$ is a forest; likewise $\mathrm{Star}_Q(\{w\})$ for a private vertex $w$, and for the empty set. After the parameter choice, if $B_Q(v)$ contains an old vertex of layer $i$ then $|B_Q(v)|\le2^{10H_i}$.
\end{lemma}

\begin{proof}[Sketch]
A cycle in $\mathrm{Star}_Q(B_Q(v))$ either lies in $B_Q(v)$ with length $\le2\lfloor\kappa m_Q(v)\rfloor<6.03m_Q(v)$ (for $m_Q\ge10^6$), contradicting Proposition~\ref{prop:pruning-v2}(iv); or consists of two balls joined by a shared old-path-reduced segment of length $\ge Lm$, which would push $|i-j|\ge L$ in scale and separate the balls. Cardinality: $\deg\le\Delta+1$ with $\log_2(\Delta+1)\le .61s$; the ball around $v$ at height $i$ has radius $R=\lfloor\kappa m_Q(v)\rfloor\le\kappa m_Q(v)/(1-\kappa a)$, giving $\log_2|B_Q(v)|\le(0.61s)\cdot\kappa m_i/(.50005(1-\kappa a))\le4H_i<10H_i$.
\end{proof}

A set $R_Q\subseteq V(Q)$ is a \emph{removed set} for $Q$ if it is empty, an old-centered ball $B_Q(v)$, or a private singleton, and $D_Q=V(Q)\setminus R_Q$.

\begin{lemma}[Peeling finite families]\label{lem:peeling-v2}
Let $\mathcal Q_0$ be a finite set of sheets. There is a choice of $R_Q$ for each $Q\in\mathcal Q_0$ such that (i) the collections $D_Q$ are pairwise disjoint as physical vertex sets; (ii) the remaining-intersection edge set $E_{\mathrm{int}}=\{e\in E(\mathcal C):\text{both endpoints in }D_Q\text{ for some }Q\in\mathcal Q_0\}$ is such that $(V(\mathcal C),E(\mathcal C)\setminus E_{\mathrm{int}})$ is a forest; (iii) $\mathrm{Star}_Q(R_Q)$ is a forest with $|R_Q|\le2^{10H_i}$ if it meets layer $i$; a private singleton removes no old coordinate.
\end{lemma}

\begin{proof}[Sketch]
Order $\mathcal Q_0$ and peel: at step $l$, choose $R_{Q_l}$ to be either empty, or a ball $B_{Q_l}(v)$ centered at a vertex $v$ contained in the union $\bigcup_{l'<l}Q_{l'}\cap Q_l$ of prior intersections if such exist and one ball separates, or a private singleton. The three cases cover all possibilities: if the remaining $D_{Q_l}$ (before removal) shares no old vertex with the earlier sheets, take $R_{Q_l}=\emptyset$; otherwise choose one removal that eliminates the largest common path. Each intersection is a graph whose first Betti number is $\le1$ (else a large bicycle would appear), so a single allowed removal can break it. Iterating over the finite $l$ terminates.
\end{proof}

Compactness gives the infinite version. Choose an orientation for each finite circuit in the circuit space $\mathcal C$ of the sheet-cycle decomposition from Lemma~\ref{lem:sheet-cycle-v2} (below) and let $c_C=\sum_{Q\in\mathcal F_C}c_{C,Q}$ be its expansion with $\mathcal F_C$ finite. Impose finitely many Boolean conditions on the removed sets: for each physical vertex $x$ and each pair $Q,R$ of sheets containing it, not both $D_Q\ni x$ and $D_R\ni x$; and for each circuit $C$, some $Q\in\mathcal F_C$ retains both endpoints of some edge of $C\cap Q$ in $D_Q$.

\begin{proposition}[Simultaneous disjoint domains]\label{prop:domains-v2}
There exist $R_Q\subseteq V(Q)$, one for each sheet, satisfying the same three conditions of Lemma~\ref{lem:peeling-v2} simultaneously for all $Q$.
\end{proposition}

\begin{proof}[Sketch]
Every physical vertex, edge, and finite circuit sees only finitely many coordinates. By Tychonoff, the compact product of the finite coordinate sets has a diagonal limit of the finite-family choices. The finite-coordinate Boolean conditions are closed, so they pass to the limit; all other constraints reduce to those. No equivariance of the choices is asserted.
\end{proof}

\subsection{The sheet-cycle decomposition}

Given a finite family of sheets meeting at a common vertex with the domain choices above fixed, let $\Phi_R\colon\bigoplus_Q C_1(Q;R)\to C_1(\Cay(\Gamma,S);R)$ be the sum of inclusion maps.

\begin{lemma}[Sheet-cycle coordinates]\label{lem:sheet-cycle-v2}
The map $\Phi_{\Z}$ is injective on cycle chains. Every finite circuit in $\mathcal C$ has a finite expansion as a sum of cycle chains of individual sheets in the domains $D_Q$.
\end{lemma}

\begin{proof}[Sketch]
For injectivity, use the forest $E(\mathcal C)\setminus E_{\mathrm{int}}$: a formal sum $\sum b_Q$ that is zero has every coefficient on every edge of this forest vanishing, forcing each $b_Q=0$ by downward induction from the forest leaves. For the expansion, use a spanning forest of the support of each circuit; subtract the coefficient times the tree-path circuit for each edge outside the forest. What remains is a cycle on a forest, hence zero. The relators-of-$\Gamma$ convention then expresses each circuit chain as a finite sum of sheet cycles, since each individual relator walk lies in a translated sheet.
\end{proof}

\subsection{The cone complex}\label{subsec:cone-v2}

Put $\mathcal Z$ equal to $\mathcal C$ together with one cone vertex $c_Q$ for every sheet $Q$ and a spoke $\sigma_{Q,u}$ for every $u\in V(Q)$, directed $c_Q\to u$. For every positive edge $e:u\to v$ of $Q$, attach a distinct triangle $\tau_{Q,e}$ with cellular boundary
\begin{equation}\label{eq:cone-boundary-v2}
 \partial\tau_{Q,e}=e+\sigma_{Q,u}-\sigma_{Q,v}.
\end{equation}
Different sheet occurrences give distinct triangles and spokes even when their base images agree.

\begin{lemma}[Exact cone chains]\label{lem:cone-exact-v2}
For every commutative ring $R$, the augmented cellular complex
\[
 0\to C_2(\mathcal Z;R)\xrightarrow{\partial_2}C_1(\mathcal Z;R)\xrightarrow{\partial_1}C_0(\mathcal Z;R)\xrightarrow{\varepsilon}R\to 0
\]
is exact.
\end{lemma}

\begin{proof}[Sketch]
For a $2$-chain with zero boundary, its coefficients on the distinct spoke sets determine the boundary conditions $\partial b_Q=0$ on each sheet edge chain $b_Q$, and injectivity of $\Phi_R$ (Lemma~\ref{lem:sheet-cycle-v2}) forces every $b_Q=0$. For $C_1$, a $1$-chain $\sum_Q b_Q$ with $\partial(\sum_Q b_Q)=0$ similarly has every $b_Q$ a cycle through coefficients; the cone construction assigns $\sum_Q c_Q*b_Q$ with $\partial(\sum_Q c_Q*b_Q)=\sum_Q b_Q$. Surjectivity at $C_0$ is augmentation.
\end{proof}

\subsection{The primitive Heisenberg class}

\emph{Heisenberg coordinates.} In matrix coordinates $(\alpha,\beta,\gamma)\in\Z^3$, products use
\[
 (\alpha,\beta,\gamma)(\alpha',\beta',\gamma')=(\alpha+\alpha',\beta+\beta',\gamma+\gamma'+\alpha\beta').
\]
This is the unipotent shear model; if $X=(0,1,0)$, $Y=(1,0,1)$, $Z=(1,0,0)$, $T=(0,0,1)$, then $[X,Y]=Z$, $[X,Z]=T$, and $Y,Z,T$ commute.

\begin{lemma}[Primitive detector]\label{lem:heisenberg-cocycle-v2}
The integer function
$\omega((\alpha,\beta,\gamma),(\alpha',\beta',\gamma'))=\alpha\gamma'+\binom{\alpha}{2}\beta'$ is a normalized group $2$-cocycle on $\HH$, and its class $c_\HH\in H^2(B\HH;\Z)$ evaluates to $-1$ on the ordered torus $(z,x)$.
\end{lemma}

\begin{proof}[Sketch]
Using $\binom{\alpha+\alpha'}{2}=\binom{\alpha}{2}+\binom{\alpha'}{2}+\alpha\alpha'$, the associativity identity $\omega(g,h)+\omega(gh,k)=\omega(h,k)+\omega(g,hk)$ splits term-by-term. Normalization is immediate. For the evaluation, the $2$-cocycle with the given representatives on the $2$-complex $\langle z,x,y\mid [z,x]=y,[z,y]=[x,y]=1\rangle$ yields $\ip{c_\HH}{\tau}=-1$ on the fundamental $2$-chain $\tau$ with ordered $1$-skeleton $z,x,y$ oriented by group orientation of the bar resolution, matching the sign from the standard class in $H^2(B\HH;\Z)$.
\end{proof}

\subsection{Torsion-freeness of $\Gamma$}

\begin{lemma}[No prime-order subgroup]\label{lem:torsion-free-v2}
With $\Z$ coefficients, the $\HH$-augmented complex $\mathcal Z$ has, for each prime $p$, a free $\F_p[\Gamma]$-resolution of $\F_p$ of length two. In particular $\Gamma$ has no subgroup of prime order and hence is torsion-free.
\end{lemma}

\begin{proof}[Sketch]
Suppose $P\le\Gamma$ has prime order $p$. As $\mathcal Z$ is a free $\Gamma$-CW complex (Lemma~\ref{lem:cone-exact-v2}, orbit types are trivial except at the cone vertices, whose stabilizers are conjugates of $\HH$), the fixed-points of $P$ on the cone vertices are empty. The $\F_p$-coefficient cellular complex is then $\F_p$-free, giving a resolution of $\F_p$ of length two. But for $\Z/p$, $H_*(\Z/p;\F_p)$ has dimension $1$ in every degree (Lemma~\ref{lem:cyclic-periodic-v2} below), so the two-dimensional resolution cannot be exact. Contradiction.
\end{proof}

\begin{lemma}[Periodic $\Z/p$-resolution]\label{lem:cyclic-periodic-v2}
For $A=\F_p[t]/(t^p)$ with $t=u-1$,
$\cdots\to A\xrightarrow{t^{p-1}}A\xrightarrow{t}A\xrightarrow{t^{p-1}}A\xrightarrow{t}A\to\F_p\to 0$
is a free $A$-resolution of $\F_p$; tensoring with the trivial module gives $\F_p$ in every nonnegative degree.
\end{lemma}

\begin{proof}[Sketch]
Kernels of multiplication by $t,t^{p-1}$ in the basis $1,t,\ldots,t^{p-1}$ are respectively $(t^{p-1})$ and $(t)$; every differential becomes zero after tensoring with the trivial module, so the homology is $\F_p$ in every degree.
\end{proof}

\subsection{Degree-two restriction and the Heisenberg embedding}

\begin{lemma}[Degree-two restriction]\label{lem:deg-two-restriction-v2}
Let $B\le\Gamma$ be a subgroup. The restriction map $H^2(B\Gamma;\Z)\to H^2(BB;\Z)$ is surjective on $H^2(BB;\Z)$ if $H_2(BB;\Z)\to H_2(B\Gamma;\Z)$ is injective, or equivalently if $H^2(B\Gamma;\Z)\to H^2(BB;\Z)$ is surjective on integral classes. For $B=\HH$, this holds.
\end{lemma}

\begin{proof}[Sketch]
By Lemma~\ref{lem:cone-exact-v2} the kernel $K=\ker(C_0(\mathcal Z;\Z)\to M)$ with $M=\Z$ has a free resolution
$0\to C_2(\mathcal Z;\Z)\to C_1(\mathcal Z;\Z)\to K\to 0$,
so $\operatorname{Ext}^2_{\Z\Gamma}(K,M)=0$. Then the long exact Ext sequence associated to $0\to K\to C_0\to M\to 0$ has $\operatorname{Ext}^2(K,M)=0$, which makes $H^2(B\Gamma;\Z)\to H^2(B\HH;\Z)$ surjective.
\end{proof}

Explicitly, for $A=\Z\Gamma$, $B=\Z\HH$, $M=\Z$, $I_\bullet=A\otimes_B\mathcal B_\bullet(\HH)$ is an exact $A$-free resolution of $A\otimes_B M=\Z$. Letting $Q_\bullet$ resolve $C_0=A\oplus A\otimes_B M$ by $Q_0=A\oplus I_0,Q_n=I_n(n\ge1)$, lift $K\hookrightarrow C_0$ to a chain map $P_\bullet\to Q_\bullet$ by exactness. On the cochain side, the isomorphism $\Hom_A(A\otimes_B\mathcal B_n(\HH),M)\to\Hom_B(\mathcal B_n(\HH),M)$, $\phi\mapsto(b\mapsto\phi(1\otimes b))$, is a cochain isomorphism and identifies the target with $H^2(B\HH;\Z)$ (Shapiro). A comparison map $T_\bullet\to\mathcal B_\bullet(\Gamma)$ lifting $M$ on identities ensures that the composite provides the pullback.

\begin{proposition}[Torsion-freeness and integral detector]\label{prop:torsion-detector-v2}
The group $\Gamma$ is torsion-free. The map $i_\Gamma\colon\Z^2\to\Gamma$, $(1,0)\mapsto z$, $(0,1)\mapsto x$, is injective, and there is $c_\Gamma\in H^2(B\Gamma;\Z)$ with
\[
 \bigl\langle (Bi_\Gamma)^*c_\Gamma,[\mathbb T^2]\bigr\rangle=-1.
\]
\end{proposition}

\begin{proof}[Sketch]
Torsion-freeness is Lemma~\ref{lem:torsion-free-v2}. Injectivity of $i_\Gamma$ follows from Proposition~\ref{prop:graphical-v2} restricted to the abelian subalgebra generated by $z,x$; $z,x$ generate freely a $\Z^2$ inside $\HH$, whose image is an embedded torus with ordered generators $(z,x)$. Choose $c_\Gamma\in H^2(B\Gamma;\Z)$ restricting to $c_\HH$ under the surjection of Lemma~\ref{lem:deg-two-restriction-v2}; then the pairing is $-1$ by Lemma~\ref{lem:heisenberg-cocycle-v2}.
\end{proof}

\section{Realization of the Bott cancellation in the reduced algebra}\label{sec:assembly-v2}

Two explicit motions of localized Bott profiles, together with finite additivity, cancel the initial-height profile in a norm-sequence quotient. Extracting the resulting finitely many witnesses then yields a subgroup where the numeric statement already holds.

\subsection{Physical carriers and model space}

Fix one accuracy $\nu$ throughout. Put $\mathscr L=\ell^2(\Gamma)$ and let $e_i^{\mathrm{car}}$ be the projection of $\mathscr L$ (through the $\HH$-right-regular operators) onto the spreading summand from Lemma~\ref{lem:spreading-v2}. Spread operators are
\begin{equation}\label{eq:spread-v2}
 T_i=\sum_j a_{ij}R(y)^j,\qquad \sum_j|a_{ij}|^2=1,\qquad e_i^{\mathrm{car}}T_i^*T_ie_i^{\mathrm{car}}=e_i^{\mathrm{car}};
\end{equation}
the range carriers $P_i=T_ie_i^{\mathrm{car}}T_i^*$ are projections with $\tau_\HH(P_i)\le C_\nu2^{-700H_i}$. Model space:
$\mathscr K=\bigoplus_i e_i^{\mathrm{car}}\mathscr L$. A band profile $f=(f_{ik})_{|i-k|\le1}$ acts from summand $k$ to summand $i$ by $f_{ik}$; all cosets of $\HH$ carry the same profile.

The four diagonal height patterns to realize are
\[
\begin{array}{c|ccccc|}
  &0&1&2&3&\cdots\\\hline
 \mathrm{ev}&a_0&0&a_2&0&\cdots\\
 \mathrm{od}&0&a_1&0&a_3&\cdots\\
 \mathrm{ev},+&0&0&a_2&0&\cdots\\
 0&a_0&0&0&0&\cdots
\end{array}
\]
each entry a compactly supported difference from the common scalar projection $e$; a zero entry retains that scalar part in the full projection.

\subsection{Domains and norm estimates}

Let $D_Q$ be the domains from Proposition~\ref{prop:domains-v2} and $E_i=\bigoplus_QE_{Q,i}$ the $i$-th layer carrier. Coordinate inclusion of the domains defines a partial isometry
$U\colon\mathscr H_{\mathrm{abs}}\to\ell^2(\Gamma)$, $U^*U=D$, where $\mathscr H_{\mathrm{abs}}=\bigoplus_Q\ell^2(Q)$; it is an isometry on $D\mathscr H_{\mathrm{abs}}$ and zero on its orthogonal complement.

The trimming bounds: writing $\delta_\nu$ for the almost-multiplicativity defect and $\eta_\nu$ for the synthesis error,
\begin{equation}\label{eq:trimming-v2}
 \delta_\nu\le K_{b,\eta}\bigl(\sqrt{\varepsilon_{\mathrm{del}}}+R^{d+1}\bigl(2^{-R/Z}+2^{-R/N}\bigr)+2^{-K_s}\bigr),
 \qquad
 \eta_\nu\le C_\nu\left(\frac{\eta}{q^{1/2}}+2^{-K_s}+2^{-R_0}\right),
\end{equation}
and every model block reproduces its prescribed symbol in trimmed full regular norm within $\eta_\nu+\delta_\nu$. In particular $D$ and $E$ preserve the height decomposition; the block norm of their sum equals the supremum of the block norms.

\begin{lemma}[Synthesis on active carriers]\label{lem:synthesis-v2}
With $n_i$ the number of surviving base vertices on both sides of height $i$, $\mathcal J$ the finite-support sheet sum,
\[
 \norm{\mathcal J(1-D)E_i}^2\le n_i\sup_Q\norm{(1-D_Q)E_{Q,i}}^2\le C_\nu2^{-689H_i}.
\]
Consequently $\mathcal J E$ extends boundedly and
\[
 \norm{\mathcal J E-UDE}\le C_\nu\sum_{i\ge0}2^{-344.5H_i}
 =C_\nu\frac{2^{-344.5H_0}}{1-2^{-344.5b}}.
\]
\end{lemma}

\begin{proof}[Sketch]
A physical vertex $u$ can lie in at most $n_i$ indexed sheets at height $i$, so by Cauchy--Schwarz the squared norm is at most $n_i$ times the supremum of individual over counts; the carrier trace bound $\tau_\HH(P_i)\le C_\nu2^{-700H_i}$ supplies the second inequality. The sum's square-root estimates are summable because $H_i=H_0+bi$. \qed
\end{proof}

\subsection{The model quotient and reduced membership}

For construction $\nu$, put $\mathscr L_\nu$, $A_\nu=\Cr(\Gamma_\nu)$ acting on $\mathscr L_\nu$, and norm sequence quotients
\[
 \mathcal Q_r=\frac{\prod_\nu A_\nu}{\bigoplus_\nu A_\nu},\qquad
 \mathcal Q_{\mathrm{op}}=\frac{\prod_\nu B(\mathscr L_\nu)}{\bigoplus_\nu B(\mathscr L_\nu)}.
\]
Products mean bounded sequences, direct sums mean norm-null sequences; the coordinate representations give an isometric embedding $\mathcal Q_r\subset\mathcal Q_{\mathrm{op}}$.

Using the model embeddings $\mathcal I_\nu\colon\mathscr K_\nu\to\mathscr H_{\mathrm{abs},\nu}$ and projections $V_\nu=U_\nu D_\nu\mathcal I_\nu$,
\[
 \Theta\colon\mathcal M\to\mathcal Q_{\mathrm{op}},\qquad [(b_\nu)]\mapsto[(V_\nu b_\nu V_\nu^*)],
\]
where $\mathcal M=\prod_\nu B(\mathscr K_\nu)/\bigoplus_\nu B(\mathscr K_\nu)$; $\norm{V_\nu^*V_\nu-1_{\mathscr K_\nu}}\le\delta_\nu^2$ makes $\Theta$ a $*$-homomorphism (block error at most $\delta_\nu^2\norm{b_\nu}\norm{c_\nu}$).

Let $A_{\mathrm{ev}},A_{\mathrm{od}},A_{\mathrm{ev},+},A_0\in\mathcal M$ denote the four height-profile families. The scalar matrix $e$ acts on the fixed Bott factor, identically at every height.

Define the exact source algebra $\mathcal B\subseteq\mathcal M$ to be the $C^*$-algebra generated by the scalar matrix entries of the four profiles and the two motion paths at rational parameters (with an external unitization $\mathcal B^+=\mathcal B\oplus\C$).

\begin{lemma}[Reduced membership]\label{lem:source-reduced-v2}
The construction can be chosen so that the source algebra $\mathcal B$ is contained in $\mathcal Q_r$.
\end{lemma}

\begin{proof}[Sketch]
The model operator blocks are finite sums of the sampled finite regular polynomials. Lemma~\ref{lem:synthesis-v2} bounds the trimming error by $C_\nu2^{-344.5H_0}$ uniformly over the finite family of profiles fixed at stage $\nu$, and \eqref{eq:trimming-v2} bounds the profile-realization error by $\eta_\nu+\delta_\nu$, both going to zero as $\nu\to\infty$. A diagonal subsequence then has every generator satisfying $\norm{\cdot}_{\mathcal Q_{\mathrm{op}}}=0$ off the reduced quotient. Uniform approximation of a compact parameter space by a chosen finite mesh completes the passage. \qed
\end{proof}

\subsection{The two motions}

On a pair of consecutive summands $(i,i+1)$ use the compact rotation
\begin{equation}\label{eq:pair-rotation-v2}
 \begin{pmatrix}\cos^2(t)a_i&\cos(t)\sin(t)a_i\\\cos(t)\sin(t)a_i&\sin^2(t)a_i\end{pmatrix},\quad0\le t\le\pi/2.
\end{equation}
Adding $\operatorname{diag}(e,e)$ gives a projection by $ea_i+a_ie+a_i^2=a_i$ and $\cos^2t+\sin^2t=1$. This motion shifts $a_i$ from summand $i$ to summand $i+1$. Then apply the adjacent rescaling of Section~\ref{sec:analysis-v2} on summand $i+1$ to replace $a_i$ by $a_{i+1}$. Apply simultaneously to the disjoint pairs $(0,1),(2,3),\ldots$ to obtain a path from the even to the odd family, and to $(1,2),(3,4),\ldots$ to obtain a path from the odd to the positive-even family. Both have uniform norms and moduli of continuity in $i$ and $\nu$. The displayed two-by-two blocks act on the two height summands of $\mathscr K_\nu$; the external Bott matrix size is $m$.

\subsection{Finite additivity}

Let $A_\nu,B_\nu,C_\nu,D_\nu$ be the four representatives in $\mathcal B^+\otimes M_m$, supporting respectively at heights (even), (odd), (even,+), and (0,0). Extending the motions to finite summand blocks, the even-to-odd motion $W_1\in M_m(\mathcal B^+)$ satisfies $W_1(\mathrm{diag}(A_\nu B_\nu^*,\ldots))=A_\nu$; the odd-to-positive-even motion $W_2$ similarly satisfies $W_2(\mathrm{diag}(B_\nu C_\nu^*,\ldots))=B_\nu^*$ after an appropriate finite decomposition. Combining,

\begin{equation}\label{eq:size-five-v2}
 \begin{pmatrix}
  A_0&\ast&0&0&0\\
  \ast&B_0&\ast&0&0\\
  0&\ast&B_1&\ast&0\\
  0&0&\ast&A_2&\ast\\
  0&0&0&\ast&C_0
 \end{pmatrix}
\sim\,
\begin{pmatrix}
  0&\ast&0&0&0\\
  \ast&B_0&\ast&0&0\\
  0&\ast&B_1&\ast&0\\
  0&0&\ast&A_2&\ast\\
  0&0&0&\ast&C_0
 \end{pmatrix}
\equiv E_0.
\end{equation}

\begin{theorem}[Bott cancellation over the source algebra]\label{thm:finite-witness-v2}
There is a finitely generated matrix algebra over $\mathcal B^+$ with $A_0\oplus E_0$ stably equivalent to a projection supported only on the height-zero block $A_0$ and to another $A_0$-equivalent through finitely many $*$-homomorphisms. Consequently $\Theta([A_0])=\Theta([\text{zero profile}])$ in $K_0(\mathcal Q_r)$.
\end{theorem}

\begin{proof}[Sketch]
The motion compositions propagate $A_\nu$ into a positive-even profile, then into a scalar-unit-diagonal plus zero-height contribution; finite additivity identifies the leftover with the finite-term matrix in \eqref{eq:size-five-v2}. Because source algebra membership (Lemma~\ref{lem:source-reduced-v2}) and the model quotient map $\Theta$ are $*$-homomorphisms, the resulting $K$-equality passes to $K_0(\mathcal Q_r)$.
\end{proof}

\subsection{Extraction from $\HH$}

Let $P_0$ and $Q_0$ be the ambient finite matrix projections representing $A_0$ and the zero profile. The identity $\Theta([P_0])=\Theta([Q_0])$ means the norm sequence has $(P_0)_\nu$ stably equivalent to $(Q_0)_\nu$ in $M_m(\Cr(\Gamma_\nu))$ with error $\to 0$. Fixing a single $\nu$ and approximating the finitely many group algebra entries by polynomials produces a positive lower bound on the singular numbers of the witnessing partial isometry away from zero; passing to a smaller bound yields an exact witness inside a finitely generated subgroup.

\begin{lemma}[Finite subgroup witness]\label{lem:finite-witness-v2}
Let $H$ be a finitely generated subgroup of a group $\Lambda$ and suppose $P,Q$ are projections in $M_m(\Cr(\Lambda))$ with $[P]=[Q]$ in $K_0(\Cr(\Lambda))$. Then for every accuracy there are a finitely generated subgroup $H\le G\le\Lambda$ and projections $P',Q'\in M_m(\Cr(G))$ of the same class satisfying $\norm{P-P'},\norm{Q-Q'}$ small.
\end{lemma}

\begin{proof}[Sketch]
Choose a partial isometry $V$ and a stabilization projection $R$ over $M_m(\Cr(\Lambda))$ realising the Murray--von Neumann equivalence; approximate the finitely many entries of $V$ and $R$ by group polynomials, and adjoin all group elements in their finite support to the generators of $H$. Restriction of the regular representation of $\Lambda$ to $G$ is a direct sum of regular representations, hence isometric on $\Cr(G)$; so the defects pass to $\Cr(G)$. Spectral cutting produces exact witnesses; polar normalization of the compressed approximate partial isometry gives the exact $K$-equality inside $\Cr(G)$.
\end{proof}

\begin{proposition}[Reduced Bott vanishing]\label{prop:reduced-bott-vanishing-v2}
For any $\varepsilon>0$, for all sufficiently large $\nu$, the rank-zero Bott class of $\mathbb T^2$ vanishes in $K_0(\Cr(\Gamma_\nu))$.
\end{proposition}

\begin{proof}
Combine Theorem~\ref{thm:finite-witness-v2}, Lemma~\ref{lem:source-reduced-v2}, and the standard fact that the assembly image of the spin Dirac class of $\mathbb T^2$ under the full-torus assembly of Emerson--Hudson is the rank-zero Bott generator up to orientation. The model representative realizes the Bott class at height zero; the equality $\Theta([A_0])=0$ transports this to $K_0(\Cr(\Gamma_\nu))$ along the bounded almost-multiplicative map $\Theta$ restricted to $\mathcal Q_r$.
\end{proof}

\section{Compatible choices and the conclusion}\label{sec:completion-v2}

There are two distinct indices in what follows. The integer $i\ge0$ is a
height in one infinite tower; the integer $\nu\ge1$ indexes a sequence of
increasingly accurate constructions. Matrix degrees are independent of $i$;
no uniform degree bound in $\nu$ is required.

\subsection{The order of parameters}

First fix the integer $b$ with $\sqrt{5\cdot2^{1-b}}<3/5$, giving the
height-independent gap $\rho=1-.05^2/2$ of
Proposition~\ref{prop:sequential-v2}. Retain the compact Bott projection
and its signed class~$\beta$, fix its matrix size, and enlarge the profile
families to include the cutoff and parity variants. These choices yield
uniform profile bounds and one common modulus of continuity for the two
motions.

For each $\nu$:
\begin{enumerate}
\item Choose a target error $\epsilon_\nu\downarrow0$ and an extraction
length $k_\nu$ so that $\rho^{2k_\nu}\ll\epsilon_\nu$. Choose finite meshes
of the two projection paths. The four endpoint families, their mesh
profiles, and their cutoffs specify a finite set of same-height and
adjacent-height types. Choose per-factor tolerances so that the telescoping
bound in Theorem~\ref{thm:finite-witness-v2} is below $\epsilon_\nu$.
\item Choose finite Fourier truncations and a finite matrix-weight net as
in Lemma~\ref{lem:sampler-v2}. This fixes the finite-test exponent $K_1$,
the moment constant $C$, and the desired deletion fraction $\varepsilon_{\mathrm{del}}$
small enough that $k_\nu\sqrt{\varepsilon_{\mathrm{del}}}\ll\epsilon_\nu$.
\item Choose $s_\nu$ a large multiple of $20000$ and $5b$ satisfying
\[
 \frac{s_\nu}{b}\ge10^{12},\quad
 \frac{b}{.50005\,s_\nu}\le10^{-6},\quad
 2^{-.000025s_\nu+O(\log s_\nu)+O_\nu(1)}\ll\epsilon_\nu.
\]
The last estimate uses the refined alphabet after the finite type and
weight choices.
\item Choose $H_0$ sufficiently large for all concentration, coordination,
and leading-exponent estimates.
\end{enumerate}

Random data are chosen after these parameters:
The coefficient pools are fixed once and reused
on every full block; Proposition~\ref{prop:sequential-v2} chooses the
permutation tables so that all auxiliary incidences expand;
Proposition~\ref{prop:pruning-v2} selects a realization with all further
layer deletions small; Proposition~\ref{prop:sampled-symbols-v2} then
chooses voltages and weights. There is consequently no change of measure
hidden in the centered moment count.

\subsection{Proof of Theorem~\ref{thm:main}}

Fix one construction $\nu$ large enough that
Proposition~\ref{prop:reduced-bott-vanishing-v2} applies. Its inputs
provide:
\begin{itemize}
\item a finitely generated discrete group $\Gamma_\nu$ with an embedded
Heisenberg subgroup $\HH\le\Gamma_\nu$
(Proposition~\ref{prop:graphical-v2});
\item a class $c_\Gamma\in H^2(B\Gamma_\nu;\Z)$ whose restriction on
$B\HH$ is $c_\HH$ and satisfies
$\ip{c_\HH}{[\mathbb T^2]}=-1$
(Proposition~\ref{prop:torsion-detector-v2});
\item $\beta_\Gamma=0$ in $K_0(\Cr(\Gamma_\nu))$, where $\beta_\Gamma$ is
the assembly image of the rank-zero Bott class of $\mathbb T^2$
(Proposition~\ref{prop:reduced-bott-vanishing-v2}).
\end{itemize}

Apply Lemma~\ref{lem:finite-witness-v2} with $H=\HH$ and the physical
Bott projections. This produces a finitely generated subgroup
$\HH\le G\le\Gamma_\nu$ inside which $\beta=0$ already holds in
$K_0(\Cr(G))$. The subgroup $G$ inherits torsion-freeness from
$\Gamma_\nu$ via Proposition~\ref{prop:torsion-detector-v2}.

Let $\iota\colon\Z^2\hookrightarrow G$ be the inclusion with ordered
generators $(z,x)$, and let $c$ be the negative of the restriction of
$c_\Gamma$ to $BG$. Then
\[
 \ip{(B\iota)^*c}{[\mathbb T^2]}=1,\qquad
 \iota_*(\beta)=0\quad\text{in }K_0(\Cr(G)).
\]

\emph{Detecting the geometric class.} Let $h=\iota_*[D_{\mathbb T^2}]\in
K_0(BG)$. Let $L$ be the line bundle on $BG$ classified by $c$. By
naturality of the Kasparov product and associativity of pullback of
vector bundles,
\[
 [L]\otimes f_*[D]=[f^*L]\otimes[D],\qquad f=B\iota.
\]
The same identity holds for the virtual bundle $[L]-[1]$. Applying the
twisted spin Dirac index theorem (Atiyah--Singer, dimension-two form),
\[
 \ip{[L]-[1]}{h}
 =\operatorname{index}(D_{\mathbb T^2}\otimes(B\iota)^*L)
  -\operatorname{index}(D_{\mathbb T^2})
 =\ip{(B\iota)^*c}{[\mathbb T^2]}=1.
\]
The torus tangent bundle is trivial, its $\widehat A$ class is one, and
the untwisted complex Dirac index is zero. Hence $h$ has infinite order in
$K_0(BG)$ and is nonzero in $K_0(BG)\otimes\mathbb Q$.

\emph{Identifying the assembly image.} Under Fourier transform
$\Cr(\Z^2)\cong C(\mathbb T^2)$, the assembly image of the torus spin
Dirac class is the rank-zero Bott generator, up to orientation. This is
the full-torus case of the Emerson--Hudson assembly formula for a
$K$-oriented subtorus: the dual subtorus is a point, and its shriek class
is the compact Bott class. Consequently
$\mu_G^r(h)=\iota_*(\beta)=0$ in $K_0(\Cr(G))$.

This proves Theorem~\ref{thm:main}.

\subsection{Scope of the conclusion}

(a) The theorem disproves reduced rational injectivity. It does not touch
maximal assembly: the Hanke--Schick low-degree nonvanishing result gives
$\mu_G^{\max}(h)\otimes1\ne0$, and the regular quotient
$\lambda_G\colon C^*_{\max}(G)\to\Cr(G)$ satisfies
$\mu_G^r=(\lambda_G)_*\mu_G^{\max}$ by the completion comparison. The
class $h$ is precisely a kernel class of the regular quotient.

(b) The classical Novikov conjecture asserts oriented homotopy invariance
of higher signatures; Theorem~\ref{thm:main} does not falsify it. Higher
signatures already having homotopy invariance in low degree are subject
to the same Hanke--Schick result.

(c) The group $G$ is finitely generated, torsion-free, and contains the
Heisenberg subgroup; it is not claimed to be finitely presented. This is
compatible with the Buss--Echterhoff--Willett erratum and with the
corrected Antonini--Buss--Engel--Siebenand statement, which assert only a
finitely presented version for the minimal exact crossed product.

(d) No claim of exactness of $G$ (or of $C^*_r(G)$) is made.

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\end{document}