==== TABLE OF CONTENTS ====
Radius-of-comparison-equals-half-the-mean-dimension//figures/block-allocation.tex
Radius-of-comparison-equals-half-the-mean-dimension//paper.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/01-introduction.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/02-compression-interface.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/03-dynamical-comparison.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/04-cover-reconstruction.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/05-lower-bound.tex
Radius-of-comparison-equals-half-the-mean-dimension//sections/06-regularity.tex
Radius-of-comparison-equals-half-the-mean-dimension//references.bib
===========================
==== BEGIN Radius-of-comparison-equals-half-the-mean-dimension//figures/block-allocation.tex ====
\begin{figure}[tbp]
\centering
\begingroup
\begin{tikzpicture}[
x=.22\linewidth,y=8mm,
font=\small,
every node/.style={inner sep=2pt},
original/.style={draw=black!70,fill=black!8,line width=.55pt},
reconstructed/.style={draw=blue!55!black,fill=blue!9,line width=.55pt},
overlap/.style={fill=black!14,draw=none},
guide/.style={draw=black!28,densely dotted,line width=.45pt}
]
\def\blockoffset{.38}
\draw[line width=.55pt] (0,5.15) -- (4,5.15);
\foreach \x/\lab in {0/0,1/k,2/2k,3/3k,4/{e=4k}} {
\draw (\x,5.08) -- (\x,5.22);
\node[above] at (\x,5.23) {$\lab$};
\draw[guide] (\x,4.98) -- (\x,3.78);
}
\node[anchor=west,font=\small\bfseries] at (0,4.65)
{Fallback: $H_0$ unchanged blocks};
\foreach \D in {0,1,2,3} {
\draw[original] (\D,3.85) rectangle ({\D+1},4.30);
\node at ({\D+.5},4.075) {$D=\D$};
}
\node[anchor=west,font=\small\bfseries] at (0,3.12)
{Aligned: two edges and $H_0-1$ reconstructed blocks};
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\path[overlap] (3,1.13) rectangle ({3+\blockoffset},2.65);
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rectangle ({\blockoffset+\D+1},1.63);
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}
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-- ({\blockoffset+\D},1.18);
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{$\lab$};
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\draw[<->,>=stealth,line width=.5pt]
(\blockoffset,1.88) -- (1,1.88)
node[midway,font=\footnotesize,fill=white,inner sep=1pt] {$k-i$};
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(3,1.88) -- ({3+\blockoffset},1.88)
node[midway,font=\footnotesize,fill=white,inner sep=1pt] {$i$};
\node[anchor=west,font=\footnotesize] at (0,.40)
{Shaded columns mark the two edge--middle overlaps.};
\end{tikzpicture}
\endgroup
\caption{Aligned and fallback coverage of one macro-interval, translated
to $[0,e)$. The schematic uses $H_0=4$ and $02r$.
A subordinate nerve represents one long orbit block by a point in a
finite simplicial complex of dimension $m=D(\mathcal U^{[0,k)})$.
After barycentric subdivision, its vertices can be colored with $m+1$
colors, distinct on every simplex. The barycentric weight of a color
is the sum of the weights of vertices of that color.
For a large positive integer $p$, group colors into packets of $2p+2$.
Their proportions give points in a $(2p+1)$-dimensional simplex, which
we identify with a cube. Since proportions are undefined at zero packet
mass, we regularize them to obtain continuous cube-valued maps. We call
each cube factor a \emph{slot}. To replace slot maps while retaining
their cover information, we must preserve every originally absent color. The compression theorem
from the companion paper
\emph{Filtered products and boundary-preserving compression in complex
cobordism} \cite{OpenAICompression2026} supplies precisely the needed
control: after passing to arbitrarily large product powers, it puts a
positive fraction of the slots on their boundaries and fixes the entire
vector in every original boundary slot. Its full statement and bundle
hypothesis appear in \Cref{sec:compression-interface}.
Here is how comparison supplies that hypothesis. In each slot fix a
small framed torus $(\mathbb T^2)^p$ and a rank-$p$ bundle formed from
one degree-one line bundle on each two-torus factor. The compression
hypothesis tests the slot maps after translations by
parameters supported in only a small fraction of the slots. The resulting
torus loci carry a sum of these rank-$p$ bundles.
In \Cref{sec:dynamical-comparison}, an open tower allocates orbit blocks
with continuous alignment labels on most starting points. A single
finite-propagation Cuntz witness embeds the sum of the aligned bundle
fibers. Two fixed linear maps handle the varying coordinate supports,
including sparse parameters and the few blocks without full alignment.
This produces one continuous embedding over the entire parameter locus,
with target rank less than twice the source rank
(\Cref{prop:comparison-bundle}). The continuity through changes of sparse
support is needed by the compression theorem.
It remains to interpret its output as an actual smaller cover.
\Cref{sec:cover-reconstruction} proves a packet reconstruction lemma.
It discards packets of sufficiently small total weight; on retained
packets, exact boundary preservation prevents any originally absent
color from appearing. Together these operations preserve every original
zero vertex coordinate, while each boundary output leaves at least one
color absent from its packet. \Cref{sec:lower-bound} applies this
reconstruction on aligned orbit blocks. The two edge blocks retain the beginning and
end of each interval; a join with an unchanged description handles
points where alignment disappears. A count of the cells containing the
resulting image places it in one finite complex of smaller dimension.
Its open-star cover then refines the entire orbit cover, contradicting
the rate $d_{\mathcal U}$.
The conversion from $p$ complex lines to $2p+2$ colors has ratio
$p/(2p+2)$, which tends to one half. We choose all tower, sparse-parameter,
rounding, and compression losses within the initial strict margin
$d_{\mathcal U}>2r$. In particular, one orbit-window length is fixed
before invoking the arbitrarily large product powers. The final passage
takes the supremum over covers and the infimum over comparison bounds;
when mean dimension is infinite, the individual-cover bound excludes
every finite comparison bound. Niu's upper bound then completes
\Cref{thm:main}.
==== END Radius-of-comparison-equals-half-the-mean-dimension//sections/01-introduction.tex ====
==== BEGIN Radius-of-comparison-equals-half-the-mean-dimension//sections/02-compression-interface.tex ====
\section{The boundary-preserving compression input}
\label{sec:compression-interface}
This section states the complete topological input to the lower bound.
Its proof is given in OpenAI, \emph{Filtered products and
boundary-preserving compression in complex cobordism}, September 25,
2026, Theorem~1.3 \cite{OpenAICompression2026}. The input is one
bundle embedding on a compact parameter locus; the output is a uniform
number of boundary slots, with exact preservation of the boundary data.
\subsection{Slots, sparse parameters, and the fixed torus bundle}
For positive integers $t,n$, set
\[
C_t=[-1,1]^t,\qquad C(t,n)=C_t^n,
\qquad T(t,n)=([-2,2]^t)^n.
\]
Points have $n$ ordered slots in $\R^t$; all distances use the maximum
norm. A boundary slot belongs to the whole boundary $\partial C_t$.
For a real number $s$ define
\begin{equation}
\label{eq:sparse-slots}
Z_s(t,n)=\{z\in T(t,n):\#\{j:z_j\ne0\}\le s\}.
\end{equation}
Here $z_j\ne0$ means that the whole $t$-vector is nonzero. Thus the
count is at most $\lfloor s\rfloor$; $Z_s$ is empty for $s<0$ and
is $T(t,n)$ for $s\ge n$. For $0\le s0$, there is an open
set $V\subset X$ such that $V,hV,\ldots,h^{L-1}V$ are pairwise disjoint
and, writing $\mathcal T=\bigcup_{j=0}^{L-1}h^jV$,
\[
\lim_{m\to\infty}\frac1m\sup_{x\in X}
\sum_{s=0}^{m-1}\mathbf 1_{X\setminus\mathcal T}(h^sx)
<\varepsilon.
\]
In particular, $\mu(\mathcal T)>1-\varepsilon$ for every
$\mu\in\mathcal M_h(X)$.
\end{theorem}
Niu's theorem assumes the marker property. That hypothesis holds here:
there are no periodic points, since a finite orbit would be a nonempty
closed invariant subset of $X$. Given a finite collection of nonzero
integers, freeness provides a nonempty open set disjoint from its
translates by those integers, and minimality makes all its translates
cover $X$. This is the marker property. Thus
\Cref{thm:niu-open-tower} requires no restriction on the covering
dimension of $X$. The assertion about invariant measures follows by
integrating each finite orbit average before taking the limit.
For positive integers $k,H_0$ with $H_0\ge2$, put $e=kH_0$.
Whatever the offset $0\le i0$. Suppose that $\gamma>0$ and $r\geq0$ satisfy
$\gamma/k>r$. There exist an open set $B\subset X$, pairwise disjoint
open sets $O_0,\ldots,O_{k-1}\subset X$, a continuous function
$w:X\to[0,1]$, a compact set $K\subset B$, and an integer $J_0\geq1$
with the following properties:
\begin{enumerate}
\item $B,hB,\ldots,h^{k-1}B$ are pairwise disjoint, and
$\gamma\mu(B)>r$ for every $\mu\in\mathcal M_h(X)$.
\item For $0\leq i0$ so small that
\begin{equation}
\label{eq:allocation-error-budget}
\frac\gamma k(1-\eta)>r,
\qquad 3\eta<\frac{b_0}{4e},
\qquad \eta<\frac18.
\end{equation}
Choose a multiple $L$ of $k$ with $L>e$ and $e/L<\eta$.
Apply \Cref{thm:niu-open-tower} with height $L$ and error $\eta$, and
denote its base by $V$. Set
\[
B=\bigcup_{a=0}^{L/k-1}h^{ak}V.
\]
Its first $k$ translates partition the tower. Hence they are pairwise
disjoint, and
\[
k\mu(B)=\mu\left(\bigcup_{s=0}^{L-1}h^sV\right)>1-\eta.
\]
The first requirement follows from
\Cref{eq:allocation-error-budget}.
For each integer $s$ with $0\leq s1-\eta-\frac eL>1-2\eta.
\]
We next shrink $O$ uniformly over invariant measures. Take continuous
functions $\varphi_j:X\to[0,1]$ increasing pointwise to
$\mathbf 1_O$, each with compact support in $O$. Such functions are
obtained from increasing cutoffs of the distance to $X\setminus O$;
if $O=X$, use the constant function one. For each invariant measure,
monotone convergence gives
$\int\varphi_j\,d\mu\to\mu(O)>1-2\eta$.
The sets
\[
\left\{\mu\in\mathcal M_h(X):
\int\varphi_j\,d\mu>1-3\eta\right\}
\]
are increasing open sets covering the compact space
$\mathcal M_h(X)$. One of them therefore equals that whole space.
Fix the corresponding function $\varphi$ and set $U=\{\varphi>0\}$.
The continuous function $\mu\mapsto\int\varphi\,d\mu$ attains its
minimum, which is still strictly greater than $1-3\eta$.
Since $\mu(U)\geq\int\varphi\,d\mu$, we obtain
$\overline U\subset O$ and
\begin{equation}
\label{eq:closed-exception-measure}
C=X\setminus U\text{ is closed},\qquad
\sup_{\mu\in\mathcal M_h(X)}\mu(C)<3\eta<\frac{b_0}{4e}.
\end{equation}
Choose $w:X\to[0,1]$ equal to one on $\overline U$ and with
$\operatorname{supp}(w)\subset O$.
For completeness, the uniform orbit estimate for the closed set $C$ is
\begin{equation}
\label{eq:closed-set-uniform-orbit-estimate}
\limsup_{m\to\infty}\sup_{x\in X}\frac1m
\sum_{s=0}^{m-1}\mathbf 1_C(h^sx)
\leq\sup_{\mu\in\mathcal M_h(X)}\mu(C).
\end{equation}
Indeed, a sequence violating this estimate has a subsequence of its
orbit empirical measures converging weakly to an invariant probability
measure. The closed-set inequality for weak convergence bounds the
limsup of their masses on $C$ by the mass of that limit measure, a
contradiction. Thus, by
\Cref{eq:closed-exception-measure,eq:closed-set-uniform-orbit-estimate},
there is $m_0$ such that every orbit interval of length $m\geq m_0$
spends less than $b_0/(2e)$ of its times in $C$.
The set $\{w<1\}$ is contained in $C$. For $eJ\geq m_0$ and any
$y\in X$,
\[
\begin{split}
\#\{0\leq q0$, and an
integer $R$ such that
\begin{equation}
\label{eq:comparison-bundle-budgets}
\frac{RH_0}{pg(H_0-1)}+2(a+b_0)<2,
\qquad \frac{R-pg}{k}>r.
\end{equation}
Let $F_1,\ldots,F_g:X\to[-1,1]^t$ be arbitrary continuous maps.
Choose $B,(O_i)_{i=0}^{k-1},w,K$ as in
\Cref{lem:uniform-block-allocation}, with $\gamma=R-pg$ and
$e=kH_0$.
For all sufficiently large integers $J$, put
\[
N=eJ,\qquad n=J(H_0-1)g.
\]
There is a continuous map
$f_J:X\to[-1,1]^{tn}$, with slots indexed by
\[
(q,D,j),\qquad
0\leq q0.
\end{equation}
The agreement holds, more strongly, on a neighborhood of each compact
set $h^{-qe}(\operatorname{supp}(w)\cap O_i)$.
Over the compact space
\begin{equation}
\label{eq:dynamical-torus-locus}
\mathcal L_J=
\{(x,z)\in X\times Z_{an}(t,n): f_J(x)-z\in L^n\},
\end{equation}
the direct sum of the $n$ pullbacks of $H$, by the slot maps
$(x,z)\mapsto(f_J)_\alpha(x)-z_\alpha$, embeds continuously in a
trivial complex bundle of rank strictly less than $2pn$.
The same allocation $B,(O_i),w,K$ works for every such $J$.
\end{proposition}
The two inequalities in \Cref{eq:comparison-bundle-budgets} control
different parts of the construction. The tracial comparison will use
source rank at most $pg$ and target rank $R$ on $B$. Since the tower
can make $\mu(B)$ uniformly as close to $1/k$ as needed, the excess
$(R-pg)/k>r$ supplies the required strict trace gap.
For the bundle embedding, each of the $J$ intervals contributes
$(H_0-1)g$ source slots, giving rank $pn=pg(H_0-1)J$.
The comparison witness will have a fixed propagation bound $M$;
the $k$-separation of its active output times will give a target of
rank $RH_0J+O(1)$. A second map for the sparse slots and the starts
where $w<1$ will use at most $2p(a+b_0)n+O(1)$ further dimensions.
The bounded terms are independent of $J$. Dividing these target
costs by $pn$ explains the first inequality: it leaves a strict margin
below the factor two required by \Cref{thm:torus-compression}.
\begin{proof}
We retain the allocation data throughout the proof. In particular, all
cutoffs and the Cuntz witness below are fixed before $J$ is chosen.
\paragraph{Extension of the aligned functions.}
Put $W_i=\operatorname{supp}(w)\cap O_i$ and choose open sets $V_i$
with
\[
W_i\subset V_i\subset\overline V_i\subset O_i;
\]
an empty $W_i$ may be assigned $V_i=\varnothing$. For fixed $D,j$,
the maps $x\mapsto F_j(h^{i+Dk}x)$ on the disjoint closed sets
$\overline V_i$ combine to a continuous cube-valued map. Coordinatewise
extension gives a continuous map
$\widehat F_{D,j}:X\to[-1,1]^t$ agreeing with it on their union.
For every $J$, set
\begin{equation}
\label{eq:extended-aligned-functions}
(f_J)_{q,D,j}(x)=\widehat F_{D,j}(h^{qe}x).
\end{equation}
This proves the asserted agreement, including the neighborhood
assertion. The extension maps $\widehat F_{D,j}$ are independent of
$J$.
\paragraph{Positive matrix functions.}
We now encode the $g$ torus bundles by positive elements of the crossed
product. Their tracial rank gap will supply a single witness for all the
aligned orbit blocks.
Take a tubular neighborhood $\mathcal V$ of $L$ inside $(-1,1)^t$,
with retraction $\pi_L:\mathcal V\to L$, and write
$\widetilde Q=Q\circ\pi_L$. Choose a continuous function
$\psi:[-1,1]^t\to[0,1]$ with compact support in $\mathcal V$ and
equal to one on a neighborhood of $L$. The matrix function
$\psi\widetilde Q$ extends by zero outside $\mathcal V$.
By compactness there is $\delta>0$ such that, for $y\in L$ and
$v\in[-1,1]^t$,
\begin{equation}
\label{eq:nearby-bundle-projections}
\|v-y\|_\infty<\delta
\quad\Longrightarrow\quad
\psi(v)=1,\quad v\in\mathcal V,\quad
\|\widetilde Q(v)-Q(y)\|<\frac12.
\end{equation}
Consequently $\widetilde Q(v)$ maps $H_y$ injectively onto
$\operatorname{ran}\widetilde Q(v)$ in this situation.
Choose continuous scalar functions $\rho_0,\rho_1:X\to[0,1]$ such that
\[
\rho_0|_K=1,\qquad \operatorname{supp}(\rho_0)\subset B,
\qquad \{\rho_1>0\}=B.
\]
The compact containment $K\subset B$ gives $\rho_0$, and a cutoff of
the distance to $X\setminus B$ gives $\rho_1$; if $B=X$, take
$\rho_1=1$. Define positive matrix functions
$P_0\in M_{2pg}(C(X))_+$ and $P_1\in M_R(C(X))_+$ by
\begin{equation}
\label{eq:comparison-positive-functions}
\begin{split}
P_0(x)&=\rho_0(x)\bigoplus_{j=1}^g
\psi(F_j(x))\widetilde Q(F_j(x)),\\
P_1(x)&=\rho_1(x)1_R.
\end{split}
\end{equation}
The summands of $P_0$ are understood to be zero where $F_j(x)$ lies
outside $\mathcal V$.
If $\tau$ is a normalized trace on $A$, its restriction to $C(X)$ is
integration against some $\mu\in\mathcal M_h(X)$. Functional calculus
and bounded convergence give, for every positive matrix function $P$,
\[
d_\tau(P)=\int_X\operatorname{rank}(P(x))\,d\mu(x).
\]
The unnormalized matrix trace is essential in this formula. The rank
of $P_0(x)$ is at most $pg$ on $B$ and is zero off $B$, whereas $P_1$
has rank $R$ precisely on $B$. Therefore
\[
d_\tau(P_0)+r\leq pg\mu(B)+r0$, there is a unique $i$ with $h^{qe}x\in O_i$.
Put $s_\alpha=qe+i+Dk$. By
\Cref{eq:aligned-slot-agreement,eq:nearby-bundle-projections},
\[
\widetilde Q(F_j(h^{s_\alpha}x)):
H_{y_\alpha}\longrightarrow
\operatorname{ran}\widetilde Q(F_j(h^{s_\alpha}x))
\]
is injective. Furthermore, $h^{s_\alpha}x\in K$, so the indicated
range is a genuine projection range of the $j$th block of
$P_0(h^{s_\alpha}x)$.
Define a bundle map
\[
\Xi:E\longrightarrow
\mathcal L_J\times\bigoplus_{s=0}^{N-1}\mathbb C^{2pg}
\]
by sending $v_\alpha\in H_{y_\alpha}$, where
$\theta_\alpha>0$, to
\[
\theta_\alpha\widetilde Q(F_j(h^{s_\alpha}x))v_\alpha
\]
in the $j$th block of the row at time $s_\alpha$, and by setting that
contribution to zero where $\theta_\alpha=0$.
All such times lie in the $q$th interval $[qe,(q+1)e)$.
Within an interval the pairs $(i+Dk,j)$ are distinct, and intervals
are disjoint. Thus different active slots occupy different
time/block pairs.
This map is continuous on the entire space $\mathcal L_J$.
Indeed, on a region with positive weight the label is locally constant
and the formula is continuous. The projection factor has norm at most
one, so the norm of its weighted contribution is at most
$\theta_\alpha\|v_\alpha\|$ and tends to zero wherever the weight
vanishes. This also handles a change of label, which can occur only
outside the positive-weight region. More formally, one can use the
matrix $\widetilde Q(F_j(h^{s_\alpha}x))Q(y_\alpha)$ on the ambient
$\mathbb C^{2p}$ and extend its weighted entries by zero; the same
bound proves their continuity. No nearby-projection map is needed at
zero weight.
The image of $\Xi_{(x,z)}$ lies in the subspace on which
$\pi_x(P_0)$ is the identity. The first bundle map is therefore
\[
\Phi_1(x,z)=C_JW_J(x)\Xi_{(x,z)}:
E_{(x,z)}\longrightarrow\mathbb C^{Rq_J}.
\]
By \Cref{eq:whole-projection-sum-injection}, the defining property of
$C_J$, and the distinct time/block placements,
\begin{equation}
\label{eq:first-map-kernel}
\ker\Phi_1(x,z)=
\bigoplus_{\alpha:\theta_\alpha(x,z)=0} H_{y_\alpha}.
\end{equation}
\paragraph{The exceptional summands.}
The first map now controls every positive-weight summand. To complete
one embedding over the full parameter locus, we give its kernel a second
map whose target rank is charged only to the sparse slots and the
infrequent starts at which $w<1$.
Using the given embeddings $H_y\subset\mathbb C^{2p}$, define
\[
\Psi_{(x,z)}:E_{(x,z)}\longrightarrow
\bigoplus_\alpha\mathbb C^{2p},
\qquad
(v_\alpha)_\alpha\longmapsto
((1-\theta_\alpha)v_\alpha)_\alpha.
\]
If $1-\theta_\alpha\ne0$, then either $z_\alpha\ne0$ or
$w(h^{qe}x)<1$: when both $z_\alpha=0$ and $w(h^{qe}x)=1$,
\Cref{eq:slot-weights} gives $\theta_\alpha=1$.
The sparse condition bounds the first kind by $an$ slots. For
$J\geq J_0$, \Cref{eq:uniform-bad-block-count} bounds the second kind
by
\[
b_0J(H_0-1)g=b_0n
\]
slots. The range of $\Psi_{(x,z)}$ is therefore supported on at most
$(a+b_0)n$ coordinate groups, each of dimension $2p$.
Put $s_J=\lceil(a+b_0)n\rceil$. Another application of
\Cref{lem:finite-coordinate-compression} gives a fixed linear map
\[
D_J:\bigoplus_\alpha\mathbb C^{2p}\longrightarrow
\mathbb C^{2ps_J}
\]
injective on all the relevant coordinate subspaces. Set
$\Phi_2(x,z)=D_J\Psi_{(x,z)}$.
The direct sum $\Phi_1\oplus\Phi_2$ is fiberwise injective. In fact,
\Cref{eq:first-map-kernel} says that a vector killed by $\Phi_1$ has
components only where $\theta_\alpha=0$. On those components
$1-\theta_\alpha=1$. Injectivity of $D_J$ on the range support of
$\Psi$ shows that a vector also killed by $\Phi_2$ must be zero.
Both maps are continuous in $(x,z)$, including when the support of $z$
changes. They thus give a continuous bundle embedding into the
trivial bundle of rank
\begin{equation}
\label{eq:comparison-bundle-target-rank}
K_J=Rq_J+2ps_J.
\end{equation}
One may also verify the bundle assertion locally by representing the
map in a trivialization of $E$: its Gram matrix is positive definite,
so its image is a continuous rank-$pn$ subbundle. The base
$\mathcal L_J$ is compact.
\paragraph{The strict rank estimate.}
Let
\[
\zeta=2-\frac{RH_0}{pg(H_0-1)}-2(a+b_0)>0.
\]
Since $n=J(H_0-1)g$, \Cref{eq:comparison-bundle-target-rank} gives
\[
\frac{K_J}{pn}
\leq \frac{RH_0}{pg(H_0-1)}+2(a+b_0)
+\frac{Rc_M+2p}{pJ(H_0-1)g}
=2-\zeta+\frac{Rc_M+2p}{pJ(H_0-1)g}.
\]
For all sufficiently large $J\geq J_0$ this is strictly less than
two. Hence $K_J<2pn$, which proves the proposition.
\end{proof}
==== END Radius-of-comparison-equals-half-the-mean-dimension//sections/03-dynamical-comparison.tex ====
==== BEGIN Radius-of-comparison-equals-half-the-mean-dimension//sections/04-cover-reconstruction.tex ====
\section{Colored covers and exact packet reconstruction}
\label{sec:cover-reconstruction}
We prepare the passage from the boundary slots supplied by
\Cref{thm:torus-compression} to a decrease of covering order.
The reconstruction must retain the original open-cover information
exactly. A coloring of a subdivided nerve makes this possible: a
boundary output leaves at least one color absent from its packet.
To preserve the original zero coordinates, reconstruction discards
packets of small total weight. On every retained packet, fixing the
original boundary value prevents an originally absent color from
appearing.
\subsection{Polyhedral models for finite covers}
For a finite simplicial complex $S$, write $\lambda_v:S\to[0,1]$
for the barycentric coordinate of a vertex $v$. The \emph{carrier}
of $y\in S$ is the unique simplex whose relative interior contains
$y$; its vertices are precisely those for which $\lambda_v(y)>0$.
In particular, its dimension is the number of positive vertex
coordinates minus one.
We will use the following elementary refinement principle.
\Needspace{6\baselineskip}
\begin{lemma}[Polyhedral refinement]
\label{lem:polyhedral-refinement}
Let $Y$ be a compact space, let $K$ be a finite polyhedron of
dimension at most $q$, and let $\Phi:Y\to K$ be continuous.
Let $\mathcal V$ be a finite open cover of $Y$. Suppose that every
point of $\Phi(Y)$ has an open neighborhood $W$ in $K$ such that
$\Phi^{-1}(W)$ is contained in a member of $\mathcal V$.
Then $D(\mathcal V)\le q$.
\end{lemma}
\begin{proof}
Choose finitely many such neighborhoods covering the compact set
$\Phi(Y)$, and add $K\setminus\Phi(Y)$ to obtain a finite open
cover of $K$. A sufficiently fine simplicial subdivision has its
open vertex stars subordinate to this cover. These stars have order
at most $q$: a collection of stars with nonempty intersection
corresponds to vertices of a single simplex. Their pullbacks under
$\Phi$, with empty sets omitted, form a finite open refinement of
$\mathcal V$ of order at most $q$.
\end{proof}
\begin{lemma}[Colored subordinate nerve]
\label{lem:colored-subordinate-nerve}
Let $\mathcal V$ be a finite open cover of a compact metrizable
space $Y$, and put $m=D(\mathcal V)$. There are a finite simplicial
complex $S$ of dimension at most $m$, a continuous map
$\phi:Y\to S$, a coloring of the vertices of $S$ by
$1,\ldots,m+1$ that is injective on every simplex, and, for each
vertex $v$, a member $V(v)\in\mathcal V$ such that
\begin{equation}
\label{eq:vertex-subordination}
\{y\in Y:\lambda_v(\phi(y))>0\}\subset V(v).
\end{equation}
\end{lemma}
\begin{proof}
Choose a finite open refinement of $\mathcal V$ of order $m$ and
a subordinate partition of unity. The partition defines a map
to the nerve $S_0$ of that refinement, whose dimension is at most
$m$. Assign each vertex of $S_0$ a containing member of
$\mathcal V$.
Take $S$ to be the barycentric subdivision of $S_0$, and regard
the same map as a map to $S$ under the canonical identification
of their realizations. A vertex of $S$ is a nonempty face $F$ of
$S_0$; give it color $\dim F+1$ and choose one original vertex
$v_F\in F$. Faces in a simplex of $S$ form a strictly increasing
chain, so their colors are distinct. If the coordinate at the
barycenter of $F$ is positive, then the original coordinate at
$v_F$ is positive: the contribution of that barycenter to this
coordinate is its positive coefficient divided by the number of
vertices of $F$. Assign to the new vertex the member of
$\mathcal V$ already assigned to $v_F$. This proves
\Cref{eq:vertex-subordination}.
\end{proof}
For completeness, these models also explain the finite-cover
limits used below. Given two finite covers, take subordinate
nerve maps and their product. Around every point of the product,
choose a positive vertex in each factor. The corresponding product
of positive-coordinate neighborhoods pulls back into a member of
the joined cover. By \Cref{lem:polyhedral-refinement},
\begin{equation}
\label{eq:cover-subadditivity}
D(\mathcal V\vee\mathcal W)
\le D(\mathcal V)+D(\mathcal W).
\end{equation}
Here a product of finite polyhedra has dimension at most the sum
of their dimensions, as is also seen from its product cells.
For a finite open cover $\mathcal U$ of $X$, write
\[
\mathcal U^{[a,b)}=\bigvee_{s=a}^{b-1}h^{-s}\mathcal U,
\qquad m_k=D(\mathcal U^{[0,k)}).
\]
Invariance of $D$ under homeomorphisms and
\Cref{eq:cover-subadditivity} give $m_{k+l}\le m_k+m_l$.
Consequently
\begin{equation}
\label{eq:finite-cover-rate}
d_{\mathcal U}
=\lim_{k\to\infty}\frac{m_k}{k}
=\inf_{k\ge1}\frac{m_k}{k}
\le D(\mathcal U)<\infty.
\end{equation}
This finiteness concerns one cover and does not require finite
mean dimension or finite covering dimension of $X$.
\subsection{Color packets and exact reconstruction}
Recall $C_t=[-1,1]^t$ and fix a homeomorphism
$\kappa:\Delta^t\to C_t$ carrying the whole boundary of the
simplex onto the whole boundary of the cube. Such a map may, for
example, be obtained by radial extension from interior points of
these two convex bodies.
\begin{lemma}[Packet reconstruction]
\label{lem:packet-reconstruction}
Let $S$ be a finite simplicial complex of dimension at most $m$
whose vertices are colored by $1,\ldots,m+1$, with distinct colors
on each simplex. Fix an integer $t\ge1$ and put
\[
g=\left\lfloor\frac{m+1}{t+1}\right\rfloor.
\]
There exist continuous maps $\mathcal F_j:S\to C_t$,
$1\le j\le g$, with the following property.
Let $Y$ be any topological space, let $y:Y\to S$ be continuous,
and let $v_j:Y\to C_t$ be continuous replacements satisfying
\begin{equation}
\label{eq:packet-boundary-condition}
v_j(z)=\mathcal F_j(y(z))
\quad\text{whenever }\mathcal F_j(y(z))\in\partial C_t.
\end{equation}
There is then a continuous reconstructed map
$\widetilde y:Y\to S$ such that, for every $z\in Y$,
\begin{enumerate}
\item $\lambda_v(\widetilde y(z))>0$ implies
$\lambda_v(y(z))>0$ for every vertex $v$;
\item if $s(z)=\#\{j:v_j(z)\in\partial C_t\}$, then the carrier
of $\widetilde y(z)$ has dimension at most $m-s(z)$.
\end{enumerate}
In particular, reconstruction preserves every vertex
subordination relation such as \Cref{eq:vertex-subordination}.
\end{lemma}
\begin{proof}
For a color $\ell$, let
\[
c_\ell(y)=\sum_{v\text{ of color }\ell}\lambda_v(y).
\]
These continuous color weights sum to one. At any point of $S$
at most one vertex of a given color has positive weight.
Partition the first $(t+1)g$ colors into packets
\[
I_j=\{(j-1)(t+1)+1,\ldots,j(t+1)\},
\qquad 1\le j\le g,
\]
and retain the remaining colors individually. Let $w_j(y)$ be
the vector of the $t+1$ color weights in $I_j$, and let
$M_j(y)$ be their sum. At positive mass, the proportions $w_j/M_j$
record each missing color by a zero coordinate, but division by
$M_j$ is undefined at zero mass. We first regularize these
proportions near zero to obtain continuous slot maps. During
reconstruction we discard packets in this range, so any colors
introduced by regularization do not enter the reconstructed point.
Choose
\begin{equation}
\label{eq:packet-threshold}
0<\tau<\frac{1}{4(m+1)},
\qquad b=\left(\frac{1}{t+1},\ldots,\frac{1}{t+1}\right)
\in\Delta^t.
\end{equation}
Define a continuous simplex-valued function by
\begin{equation}
\label{eq:regularized-proportions}
q_j(y)=
\begin{cases}
w_j(y)/M_j(y),&M_j(y)\ge\tau,\\[2mm]
b+\bigl(w_j(y)-M_j(y)b\bigr)/\tau,&M_j(y)\le\tau.
\end{cases}
\end{equation}
The formulas agree at $M_j=\tau$. For $00.
\]
The last inequality follows from \Cref{eq:packet-threshold}.
Thus division by $Z$ is defined and continuous, even if some
packets have been discarded completely.
We first verify that a missing original color stays missing.
For a leftover color this is immediate. For a packet whose
factor $\eta(M_j)M_j$ is zero it is also immediate. In the
remaining case $M_j>\tau$, so $q_j=w_j/M_j$ exactly. If one
original color in that packet has weight zero, $q_j$ lies on
$\partial\Delta^t$ and $\mathcal F_j$ lies on $\partial C_t$.
\Cref{eq:packet-boundary-condition} forces
$\kappa^{-1}(v_j)=q_j$, and the new weight of that color is
zero. We have proved
\begin{equation}
\label{eq:no-new-colors}
c_\ell(y(z))=0\quad\Longrightarrow\quad
\widehat c_\ell(z)=0.
\end{equation}
Assign the normalized color weights to the originally present
vertices. More explicitly, for a vertex $v$ of color $\ell$,
put
\begin{equation}
\label{eq:reconstructed-vertices}
\widetilde\lambda_v(z)=
\begin{cases}
\displaystyle
\frac{\lambda_v(y(z))}{c_\ell(y(z))}
\frac{\widehat c_\ell(z)}{Z(z)},
&c_\ell(y(z))>0,\\[3mm]
0, &c_\ell(y(z))=0.
\end{cases}
\end{equation}
This function is continuous at points where $c_\ell=0$ as well:
the first ratio lies in $[0,1]$, while the second tends to zero
by \Cref{eq:no-new-colors} and the positive lower bound for $Z$.
The new vertex weights are nonnegative, sum to one, and have
support contained in the original carrier. They therefore
define a continuous map $\widetilde y$ into $S$, with the first
asserted property.
Finally, if $v_j(z)\in\partial C_t$, the vector
$\kappa^{-1}(v_j(z))$ has a zero coordinate. Thus packet $j$
contributes at most $t$ positive colors in
\Cref{eq:reconstructed-packet}; if the packet is discarded,
it contributes none. The $s(z)$ successful packets use disjoint
sets of colors. Hence at most $m+1-s(z)$ colors, and therefore
at most $m+1-s(z)$ vertices, are positive in the reconstructed
point. Its carrier has dimension at most $m-s(z)$.
\end{proof}
Notice that the loss in \Cref{lem:packet-reconstruction} is measured
from the common bound $m$, not from the dimension of the original
carrier. A packet already missing a color may already have a
boundary input, and is counted correctly by this formulation.
Nor do discarded packets cause a loss of the asserted gain: a
discarded packet has all of its colors absent.
==== END Radius-of-comparison-equals-half-the-mean-dimension//sections/04-cover-reconstruction.tex ====
==== BEGIN Radius-of-comparison-equals-half-the-mean-dimension//sections/05-lower-bound.tex ====
\section{The individual-cover bound and the equality}
\label{sec:lower-bound}
All ingredients are now available. We apply the continuous bundle
embedding of \Cref{prop:comparison-bundle} to the packet maps, use
\Cref{thm:torus-compression}, and reconstruct one finite target complex
for the full orbit cover. The proof keeps the rank margin and the
cover-order saving in the same strict parameter budget.
\subsection{A comparison bound controls every cover rate}
We use comparison bounds in the sense of
\Cref{eq:comparison-bound}. In particular, traces on $A$ are
normalized and their matrix extensions use the unnormalized
matrix trace.
\Needspace{6\baselineskip}
\begin{theorem}
\label{thm:cover-bound}
Let $h$ be a minimal homeomorphism of an infinite compact
metrizable space $X$, and let $A=C(X)\rtimes_h\Z$.
For every finite comparison bound $r$ for $A$ and every finite
open cover $\mathcal U$ of $X$,
\[
\lim_{k\to\infty}\frac{D(\mathcal U^{[0,k)})}{k}\le2r.
\]
\end{theorem}
\begin{proof}
Suppose, to the contrary, that a cover has rate $d>2r$.
By \Cref{eq:finite-cover-rate}, $d$ is finite. We choose the
parameters so that the bundle embedding and the covering-order
decrease have simultaneous strict margins.
\paragraph{Choice of parameters.}
First choose an integer $p\ge1$ such that, for $t=2p+1$,
\[
\frac{pd}{t+1}>r.
\]
This is possible because $p/(2p+2)\to1/2$. Set
$c_0=d/(t+1)>0$, and choose $0r.
\end{equation}
Choose an integer $H_0\ge2$ so large that
\begin{equation}
\label{eq:cover-height-choice}
\frac{2-10a}{H_0-1}<2a,
\qquad
\frac{d}{H_0}<\frac{ac_0}{32}.
\end{equation}
Next choose $b_0>0$ satisfying
\begin{equation}
\label{eq:cover-bad-choice}
b_0<\frac{a}{32},
\qquad b_0(d+c_0)<\frac{ac_0}{64}.
\end{equation}
These choices precede the choice of $k$.
For $m=m_k$, put
\begin{equation}
\label{eq:cover-g-r-choice}
g=\left\lfloor\frac{m+1}{t+1}\right\rfloor,
\qquad R=\lfloor(2-10a)pg\rfloor.
\end{equation}
As $k\to\infty$,
\[
\frac{m}{k}\longrightarrow d,
\qquad \frac{g}{k}\longrightarrow c_0,
\qquad \frac{R-pg}{k}\longrightarrow(1-10a)pc_0>r.
\]
For every $k$ with $g>0$, the first inequality in
\Cref{eq:cover-height-choice} gives
\begin{align}
\label{eq:cover-comparison-parameters}
\frac{RH_0}{pg(H_0-1)}+2(a+b_0)
&\le 2-8a+\frac{2-10a}{H_0-1}+2b_0<2.
\end{align}
The strict inequality follows from $b_0r,
\qquad E_kpg$. All of $p,t,a,H_0,b_0,k,m,g,R$ are now
fixed. Put $e=kH_0$.
\paragraph{The subordinate functions and their compression.}
Apply \Cref{lem:colored-subordinate-nerve} to
$\mathcal U^{[0,k)}$, and write $\phi:X\to S$ for the resulting
map. For every vertex $v$ of $S$, fix members
$U(v,s)\in\mathcal U$, $0\le s0
\quad\Longrightarrow\quad
h^s x\in U(v,s)\quad(0\le s0$ and $h^{qe}y\in O_i$, these coordinates
satisfy
\begin{equation}
\label{eq:cover-aligned-f}
f_{q,D,j}(y)=F_j(h^{qe+i+Dk}y).
\end{equation}
Moreover the rank-$pn$ torus bundle over the sparse test locus
for $f$ embeds in a trivial bundle of rank strictly less than
$2pn$.
By \Cref{thm:torus-compression}, for arbitrarily large integers
$u$ there is a continuous map
\[
G:X^u\longrightarrow C_t^{un}
\]
with at least $(a/8)un$ boundary slots at every point, agreeing
with each original slot of $f^{\times u}$ whenever that slot is on its
boundary. Pull it back along the map of actual orbit starts
\[
\iota_u:X\longrightarrow X^u,
\qquad
\iota_u(x)=(x,h^Nx,\ldots,h^{(u-1)N}x).
\]
Denote the resulting slot functions by
$\widetilde F_{b,q,D,j}(x)$, with $0\le b0$, let $i$ be the unique
label with $h^{T_{b,q}}x\in O_i$. The label is locally constant
there. For each $0\le D0$, including the value $\infty$, their contrapositives
give the final assertion.
\end{proof}
==== END Radius-of-comparison-equals-half-the-mean-dimension//sections/06-regularity.tex ====
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