==== TABLE OF CONTENTS ====
The-l1-Bass-Conjecture-for-Discrete-Groups//macros.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//main.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/01-introduction.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/02-reduction.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/03-cyclic.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/04-channels.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/05-connection.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/06-curvature.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//sections/07-conclusion.tex
The-l1-Bass-Conjecture-for-Discrete-Groups//references.bib
===========================
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//macros.tex ====
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==== END The-l1-Bass-Conjecture-for-Discrete-Groups//macros.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//main.tex ====
\documentclass[11pt]{article}
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\input{macros}
\title{The $\ell^1$-Bass Conjecture for Discrete Groups}
\author{OpenAI}
\date{October 5, 2026}
\hypersetup{pdftitle={The l1-Bass Conjecture for Discrete Groups},pdfauthor={OpenAI}}
\begin{document}
\maketitle
\begin{abstract}
We prove the $\ell^1$-Bass conjecture for every discrete group. The
Hattori--Stallings trace of every idempotent matrix over the complex
$\ell^1$ group algebra is supported on finitely many conjugacy classes
of finite-order elements.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-reduction}
\input{sections/03-cyclic}
\input{sections/04-channels}
\input{sections/05-connection}
\input{sections/06-curvature}
\input{sections/07-conclusion}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//main.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $G$ be a discrete group. Its complex Banach group algebra
$\ell^1(G)$ consists of functions $a\colon G\to\C$ with
$\|a\|_1=\sum_{g\in G}|a(g)|<\infty$, equipped with convolution
\[
(a*b)(g)=\sum_{h\in G}a(h)b(h^{-1}g).
\]
Write $\mathcal C(G)$ for the set of conjugacy classes, and
$\mathcal C_{\rm fin}(G)$ for those consisting of finite-order elements.
The subscript refers to element order, not to the size of a conjugacy
class.
For an idempotent matrix $e=(e_{ij})\in M_q(\ell^1(G))$, its
Hattori--Stallings trace \cite{Hattori1965,Stallings1965} has coefficients
\begin{equation}\label{intro:trace}
\tau_C(e)=\sum_{g\in C}\sum_{i=1}^q e_{ii}(g),
\qquad C\in\mathcal C(G).
\end{equation}
These sums converge absolutely, and
$\sum_C|\tau_C(e)|\le\sum_i\|e_{ii}\|_1$.
The trace is additive under direct sums and depends only on the
isomorphism class of the projective module $e\ell^1(G)^q$.
It therefore defines
\[
\operatorname{HS}^1\colon K_0(\ell^1(G))\longrightarrow
\ell^1(\mathcal C(G)),
\]
where $K_0$ is the Grothendieck group of finitely generated projective
right modules. The $\ell^1$-Bass conjecture
\cite[Conjecture~2.2]{BCM2004} asserts that this map takes values in the
algebraic direct sum over $\mathcal C_{\rm fin}(G)$.
We prove this assertion for all discrete groups.
\begin{theorem}\label{thm:main}
Let $G$ be a discrete group, let $q\ge1$, and let
$e\in M_q(\ell^1(G))$ be idempotent.
There is a finite set $F\subset\mathcal C_{\rm fin}(G)$ such that
$\tau_C(e)=0$ for every $C\notin F$. Equivalently,
\[
\operatorname{HS}^1\bigl(K_0(\ell^1(G))\bigr)
\subseteq \bigoplus_{C\in\mathcal C_{\rm fin}(G)}\C[C].
\]
\end{theorem}
The finite-support assertion is part of the conclusion, in addition to
vanishing on infinite-order classes. It is not an automatic consequence
of absolute summability. In particular, the proof must make its
geometric cutoff independent of the conjugacy class under consideration.
\subsection{History and relation to earlier methods}
The Hattori--Stallings trace refines the rank of a projective module by
taking values in the quotient of its ring by additive commutators
\cite{Hattori1965,Stallings1965}. For the algebraic group rings $\Z G$
and $\C G$, this quotient is indexed by conjugacy classes. Bass's
conjecture for the integral group
ring asks that this trace be supported at the identity
\cite{Bass1976}; the complex-group-ring formulation permits
finite-order classes. These support questions connect the algebraic
structure of projective modules with the conjugacy structure of a group.
Cyclic homology gives one route to these questions. Burghelea's
computation separates group-ring cyclic homology into contributions
from finite- and infinite-order conjugacy classes
\cite{Burghelea1985}. Eckmann used this description and the Chern
character to prove Bass-type vanishing for several classes of groups
\cite{Eckmann1986}. Emmanouil extended this method through nilpotence
properties of the cyclic periodicity operator \cite{Emmanouil1998}.
Our argument retains cyclic symmetry but works directly with weighted
configurations and differential forms.
Berrick, Chatterji, and Mislin formulated the $\ell^1$-Bass conjecture
in the finite-support form used here. They proved it for countable
groups whose Bost assembly map is rationally surjective in degree zero
\cite[Theorem~1.4]{BCM2004}. Together with Lafforgue's Banach-algebra
$K$-theory \cite{Lafforgue2002}, this includes amenable groups;
the passage from countable subgroups to arbitrary groups is explained
in \cite[Lemma~1.5]{BCM2004}. Theorem~\ref{thm:main} removes the
assembly hypothesis while retaining the algebraic finite-support
conclusion.
Ji, Ogle, and Ramsey developed related Bass conjectures for
rapid-decay completions and higher Chern characters
\cite{JOR2010,JOR2014}. Their work highlights the growth estimates
needed when passing from group rings to analytic completions.
Here the summability needed in the proof comes from a similarity
reduction to an exponentially decaying idempotent. The subsequent
estimates use a spatial scale uniform over all classes to be excluded.
The weighted cyclic configurations and Pfaffian detection strategy
build on the complex-group-ring argument in
\cite[Sections~7--8]{OpenAIBass2026}. In the group-ring setting,
nonzero products of coefficients involve group elements from a fixed
finite set. For an $\ell^1$ idempotent these elements may have
arbitrarily large word length, even after reduction to exponential
decay. The estimates must retain summable length factors without
losing control as the number of factors increases. We obtain them
from smooth spatial summaries whose derivative bounds are independent
of the size of the input support, multiscale time averaging with a
positive translation component, and a final ordered-kernel bound
uniform in the closing group element. A single spatial radius then
excludes all but finitely many finite-order classes. All ingredients
used here are proved below; the complex-group-ring theorem is not an
input. Standard concentration and transgression arguments are
credited at their points of use.
\subsection{Proof strategy}
The argument uses Banach algebra power series, differential forms on
finite-dimensional simplices, and elementary estimates for integral
operators. We give the needed estimates in full.
First, a finite-support approximation followed by an analytic
idempotent correction replaces $e$ by a similar idempotent $p$ with an
exponential moment in a finitely generated subgroup
(Section~\ref{sec:reduction}). This step preserves every trace
coefficient. It does not assert that $p$ has finite support.
For a fixed $g\in G$, consider cyclic lists of group elements whose
closing edge is twisted by multiplication by $g$. Products of the
coefficients of $p$ assign complex weights to these lists. Their total
weight is $\tau_{[g]}(p)$, and idempotency makes the weights compatible
with deleting vertices. The ordered times of a list form a simplex.
On these simplices, a one-form invariant under simultaneous time
translation and equal to one on the translation vector will be called
a connection form. A cyclic Stokes argument shows that the weighted
average of its curvature Pfaffian equals the trace coefficient, up to
sign (Section~\ref{sec:cyclic}).
The task is then to construct compatible connection forms with small
averaged curvature. Interpret a timed list as a step function with
values in $G$, and average its Dirac masses on a sequence of time scales.
The resulting probability distributions are summarized by smooth
subprobability distributions on auxiliary labels. These summaries
record spatial concentration while controlling derivatives caused by
moving one boundary of the step function
(Section~\ref{sec:channels}). Pairing their differentials with bounded
odd time kernels gives one-forms. Their contractions with time
translation telescope across the averaging scales. A final equality
kernel detects the twist and leaves a strictly positive contraction,
which allows normalization to a connection
(Section~\ref{sec:connection}).
The curvature estimates use three different features of this
construction. At the initial scale, a curvature entry vanishes unless
its two times are close; an occupancy estimate controls the resulting
sparse Pfaffians on average. At intermediate scales, the short support
of the time kernel gives a Hilbert--Schmidt bound. At the last scale,
the equality kernel reduces to weighted order kernels, whose singular
values satisfy $s_\nu\le C/\nu$. Expanding the Pfaffian across these
stages gives exponential decay in the simplex dimension
(Section~\ref{sec:curvature}). A single spatial radius works for every
infinite-order class and every finite-order class avoiding a fixed
finite word ball. This uniformity proves the finite-support conclusion
in Section~\ref{sec:conclusion}.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/01-introduction.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/02-reduction.tex ====
\section{Reduction to exponential decay}\label{sec:reduction}
The first step replaces a general $\ell^1$ idempotent by an idempotent
whose coefficients have exponential decay in a finitely generated
subgroup. The replacement preserves every trace coefficient exactly.
This permits the polynomially weighted sums that arise from differential
forms in arbitrarily large, but fixed, dimensions.
For a complex matrix $A=(A_{ij})$, write $|A|_1=\sum_{i,j}|A_{ij}|$.
On $M_q(\ell^1(G))$ we use
\[
\|a\|_1=\sum_{h\in G}|a(h)|_1.
\]
This norm is submultiplicative. The identity matrix has norm $q$;
none of the power-series arguments below requires its norm to be one.
\begin{lemma}[Traciality]\label{red:trace}
For every conjugacy class $C$ of a discrete group $G$, the functional
\[
T_C(a)=\sum_{h\in C}\tr a(h),\qquad a\in M_q(\ell^1(G)),
\]
is continuous and satisfies $T_C(ab)=T_C(ba)$. It is therefore invariant
under similarity by invertible matrices over $\ell^1(G)$.
\end{lemma}
\begin{proof}
The bound $|T_C(a)|\le\|a\|_1$ gives continuity. Expand $T_C(ab)$
as the absolutely convergent sum
\[
\sum_{\substack{x,y\in G\\xy\in C}}\tr\bigl(a(x)b(y)\bigr).
\]
The conditions $xy\in C$ and $yx\in C$ are equivalent, and the scalar
matrix trace is cyclic. Interchanging $x$ and $y$ proves the identity.
If $u$ is invertible, apply it to $ua$ and $u^{-1}$.
\end{proof}
The same rearrangement applies to rectangular matrices: if
$a\in M_{q,r}(\ell^1(G))$ and $b\in M_{r,q}(\ell^1(G))$, then
$T_C(ab)=T_C(ba)$, with the trace in the appropriate size on each side.
An isomorphism between two finitely generated projective modules,
together with its inverse, can be extended by their defining
idempotents to such matrices $a,b$ with products equal to the two
idempotents. This proves the module invariance used in the definition
of $\operatorname{HS}^1$.
\begin{proposition}[Exponentially decaying representative]\label{red:reduction}
Let $e\in M_q(\ell^1(G))$ satisfy $e^2=e$. There are a finitely generated
subgroup $\Gamma\le G$, a word length $|\cdot|$ on $\Gamma$, a number
$c>0$, and an idempotent $p\in M_q(\ell^1(\Gamma))$ such that
\begin{equation}\label{red:exponential}
\sum_{h\in\Gamma}|p(h)|_1e^{c|h|}<\infty,
\end{equation}
and $p$ is similar to $e$ in $M_q(\ell^1(G))$.
If the trace of every such $p$ has finite support on finite-order
conjugacy classes of $\Gamma$, the same conclusion holds for every $e$
over every discrete group.
\end{proposition}
\begin{proof}
Choose a finite-support matrix $a$ close to $e$ and put
$b=4(a-a^2)$. We may require $\|b\|_1<1$. Let
\[
Q=(1-b)^{-1/2}=\sum_{j=0}^\infty
\frac{\binom{2j}{j}}{4^j}b^j,
\qquad
p=\frac12 I+\left(a-\frac12 I\right)Q.
\]
The series converges absolutely in the matrix Banach algebra.
All its factors commute with $a$, and the scalar power-series identity
gives $Q^2(1-b)=I$. Since
$(a-I/2)^2=(I-b)/4$, we obtain $(p-I/2)^2=I/4$, hence $p^2=p$.
Moreover $p\to e$ as $a\to e$.
Let $\Gamma$ be generated by the finite union of the supports of the
entries of $a$. Fix a finite symmetric generating set and its word
length. The norm
\[
\|v\|_c=\sum_{h\in\Gamma}|v(h)|_1e^{c|h|}
\]
is submultiplicative by $|xy|\le |x|+|y|$. The element $b$ has finite
support, so $\|b\|_c\to\|b\|_1$ as $c\downarrow0$.
Choose $c>0$ with $\|b\|_c<1$. The same series defining $Q$ then
converges in this weighted algebra, proving \eqref{red:exponential}.
Finally, set
\[
u=pe+(1-p)(1-e).
\]
Direct multiplication gives $ue=pu$ and
$u-I=(p-e)(2e-I)$. For $a$ sufficiently close to $e$, the latter
has norm less than one, so $u$ is invertible and $p=ueu^{-1}$.
Lemma~\ref{red:trace} preserves all the trace coefficients.
For a conjugacy class $C$ of $G$, its intersection with $\Gamma$
is a union of conjugacy classes of $\Gamma$. Absolute convergence gives
\[
T_C(p)=\sum_{\substack{D\in\mathcal C(\Gamma)\\D\subset C}}T_D(p).
\]
Thus passing from $\Gamma$ to $G$ merges classes and cannot enlarge
a finite set of surviving classes. The order of an element does not
change under subgroup inclusion. This proves the final assertion.
\end{proof}
Henceforth $G$ is finitely generated, $d(x,y)=|x^{-1}y|$ is a fixed
left-invariant word metric, and $p$ satisfies \eqref{red:exponential}.
We fix a constant $\kappa_G$ such that
\begin{equation}\label{red:balls}
|B(x,s)|\le e^{\kappa_G s}\qquad(x\in G,\ s\ge1).
\end{equation}
Such a bound follows by counting words in the generators. It imposes
no additional growth hypothesis on $G$.
Constants without a subscript may depend on this metric and on fixed
construction choices. A constant denoted $C_R$ may also depend on the
radius parameter $R$ introduced later, but will not depend on a tuple,
its twist $g$, or the dimension $n$.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/02-reduction.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/03-cyclic.tex ====
\section{Cyclic simplex formulas for the trace}\label{sec:cyclic}
We now express a conjugacy-class trace coefficient as a weighted average
of curvature Pfaffians. Idempotency provides compatibility of the
weights under deletion of a vertex; cyclicity supplies the cancellation
that makes the average independent of the connection form.
The tuple weights have the same algebraic origin as the weighted
paths in \cite[Section~7]{OpenAIBass2026}; we give the identities
needed here directly.
\subsection{Tuples and their weights}
Fix $g\in G$ and write $H=C_G(g)$ for its centralizer. For $k\ge0$ let
\[
\mathcal T_k(g)=H\backslash G^{k+1},
\]
where $H$ acts by simultaneous left multiplication. For a tuple
$h=(h_0,\ldots,h_k)$ extend its indices to all integers by
$h_{i+k+1}=gh_i$ and define
\begin{equation}\label{cy:weights}
v_{k+1}(h)=\tr\left(\prod_{i=0}^k p(h_i^{-1}h_{i+1})\right),
\qquad l_i=1+d(h_{i-1},h_i).
\end{equation}
The product is in the displayed order. Both expressions descend to
$\mathcal T_k(g)$, and $l_i$ is periodic with period $k+1$.
\begin{lemma}[Weight identities]\label{cy:weights-lemma}
The following properties hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The weights have total sum $\tau_{[g]}(p)$, in every dimension.
\item Twisted rotation
$(h_0,\ldots,h_k)\mapsto(h_1,\ldots,h_k,gh_0)$ preserves the weights.
\item For $k\ge1$, summing the weights over all insertions of one entry
into a fixed shortened tuple gives its weight in $\mathcal T_{k-1}(g)$.
\end{enumerate}
All these sums converge absolutely, even after multiplication by any
fixed polynomial in the edge lengths. In particular, with
$M_p=\sum_{w\in G}|p(w)|_1(1+|w|)$,
\begin{equation}\label{cy:moment}
\sum_{h\in\mathcal T_k(g)}|v_{k+1}(h)|\prod_{i=0}^k l_i
\le M_p^{k+1}.
\end{equation}
\end{lemma}
\begin{proof}
Put $w_i=h_i^{-1}h_{i+1}$. Their product is $h_0^{-1}gh_0$.
Conversely, any list $(w_0,\ldots,w_k)$ whose product lies in $[g]$
determines a tuple: choose $h_0$ realizing this conjugate and successively
put $h_{i+1}=h_iw_i$. Two choices of $h_0$ differ by left multiplication
by an element of $H$, and this is the only ambiguity. We therefore have
a bijection between $\mathcal T_k(g)$ and such increment lists.
The inequality
$|\tr(A_0\cdots A_k)|\le\prod_i|A_i|_1$ gives
\eqref{cy:moment} upon dropping the restriction on the product of the
increments. The exponential moment gives the same conclusion for any
fixed polynomial in their lengths. The unrestricted convolution sum
with product in $[g]$ is the trace coefficient of $p^{k+1}=p$, proving
\textup{(i)}. Rotation is cyclic permutation of the matrix factors,
so \textup{(ii)} follows from their trace.
To delete an interior entry, sum the two adjacent factors over that
entry. Their convolution is $p^2=p$. The other deletions follow by
rotation. One may fix a representative of the shortened tuple because
its simultaneous left $H$-action has trivial stabilizer. Thus no
multiplicity enters the insertion sum, and \textup{(iii)} follows.
\end{proof}
\subsection{Compatible forms}
For each tuple use the ordered time domain
\[
\widetilde\Delta_k
=\{(t_0,\ldots,t_k):t_0\le t_1\le\cdots\le t_k\le t_0+1\}.
\]
Extend times by $t_{i+k+1}=t_i+1$, and set
$x_i=t_{i+1}-t_i$ for $0\le i\le k$. Thus $x_i\ge0$ and
$\sum_i x_i=1$. Simultaneous time translation is generated by
$V=\sum_{i=0}^k\partial_{t_i}$. Its quotient is the simplex
\[
\Delta_k=\{0\le t_1\le\cdots\le t_k\le1\},
\]
using the section $t_0=0$, oriented by $dt_1\wedge\cdots\wedge dt_k$.
We regard $[t_i,t_{i+1})$ as carrying the label $h_i$.
The face $x_i=0$ corresponds to deleting $(h_i,t_i)$: the interval
carrying the label $h_i$ has length zero. At the closing face use the
periodic lifts to make this same convention.
\begin{definition}\label{cy:connection}
A compatible connection family through dimension $n$ assigns a real
one-form $\alpha_h$ to each $h\in\mathcal T_k(g)$, $0\le k\le n$,
on $\widetilde\Delta_k$, with the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item It is invariant under time translation, and $\alpha_h(V)=1$.
\item It is invariant under simultaneous left multiplication by $H$
and under twisted rotation of both labels and times.
\item Its tangential restriction to $x_i=0$ agrees with the form of
the shortened tuple obtained by deleting $(h_i,t_i)$.
\item The form is smooth up to every face, by restriction of a smooth
extension, and the coefficients of $\alpha_h$ and $d\alpha_h$ are bounded
by polynomials in the edge lengths, for each fixed dimension and fixed
construction parameters.
\end{enumerate}
\end{definition}
Translation invariance and $\alpha(V)=1$ imply
$\iota_Vd\alpha=0$. Thus $d\alpha$ is a form on $\Delta_k$ independent
of the choice of translation section. A translation-invariant form
whose contraction with $V$ is zero will be called \emph{basic}.
For a real skew-symmetric $2m\times2m$ matrix $A$, our convention is
\[
\omega=\sum_{i0$ and let $\alpha$ be a compatible connection family through
dimension $n$. Write $A_h$ for the matrix of $d\alpha_h$ on the section
$t_0=0$. If $\mathbb E_{\Delta_n}$ denotes uniform probability on
$\Delta_n$, then
\begin{equation}\label{cy:trace-formula}
\sum_{h\in\mathcal T_n(g)}v_{n+1}(h)\,
\mathbb E_{\Delta_n}\Pf(A_h)
=(-1)^m\tau_{[g]}(p).
\end{equation}
The series converges absolutely.
\end{proposition}
\begin{proof}
First compare two compatible families $\alpha$ and $\alpha'$.
Their difference is basic. The following transgression form is the
elementary abelian instance of the connection-invariance mechanism in
Chern--Weil theory \cite[Sections~2--3]{ChernSimons1974}:
\[
\eta=(\alpha-\alpha')\wedge
\sum_{j=0}^{m-1}(d\alpha)^j\wedge(d\alpha')^{m-1-j}
\]
is basic, has degree $n-1$, and satisfies
$d\eta=(d\alpha)^m-(d\alpha')^m$. It respects all the face and
rotation identifications. Stokes' theorem and
Lemma~\ref{cy:weights-lemma}\textup{(iii)} express the weighted sum of
$\int_{\Delta_n}d\eta$ as a signed sum of copies of
\[
I=\sum_{h\in\mathcal T_{n-1}(g)}v_n(h)
\int_{\Delta_{n-1}}\eta_h.
\]
Each face identification has a fixed orientation sign, independent
of the labels. The polynomial bounds and the exponential moment
justify Stokes term by term and all the rearrangements.
On the simplex of dimension $n-1$, rotation cyclically permutes its
$n$ gap coordinates. It has orientation sign $(-1)^{n-1}=-1$.
It preserves the weights and the forms. Basicness ensures that
returning to the section $t_0=0$ has no additional effect on the
integral. Consequently $I=-I$, so $I=0$. The weighted integral of
$(d\alpha)^m$ is therefore independent of the compatible family.
Use now the particular connection
\[
\alpha^{\rm can}=\sum_{i=0}^k x_i\,dt_i.
\]
Its contraction with $V$ is one, and all other conditions are immediate
from the gap coordinates: on a zero gap the corresponding summand
vanishes and adjacent time differentials agree tangentially.
On $\widetilde\Delta_k$ its curvature is
\[
d\alpha^{\rm can}=\sum_{i=0}^k dt_{i+1}\wedge dt_i.
\]
For $k=n$ in the section $t_0=0$, the terms at either end vanish,
leaving the path of adjacent pairs. Its Pfaffian is $(-1)^m$:
the only complete pairing uses $(1,2),(3,4),\ldots,(n-1,n)$,
each with coefficient $-1$.
Finally, $\Delta_n$ has Euclidean coordinate volume $1/n!$, so
\begin{equation}\label{cy:factorials}
\frac{n!}{m!}\int_{\Delta_n}(d\alpha_h)^m
=\mathbb E_{\Delta_n}\Pf(A_h).
\end{equation}
Evaluation at $\alpha^{\rm can}$ and the total-weight identity give
\eqref{cy:trace-formula}.
\end{proof}
The proposition reduces trace vanishing to a quantitative construction.
We shall choose connections, in arbitrarily large even dimensions,
for which the average of the absolute Pfaffian is at most a small
exponential factor times $\prod_{i=1}^n l_i$.
Equation~\eqref{cy:moment} then permits summation against the tuple
weights. The next section constructs the spatial summaries used to
obtain these connections.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/03-cyclic.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/04-channels.tex ====
\section{Smooth summaries at a spatial scale}
\label{sec:channels}
The connection forms will be built from smooth summaries of probability
distributions on the group. A summary must retain information about spatial
concentration while changing in a controlled way when mass moves between two
group elements. We construct two versions. The first allows the spatial
radius to increase by a factor of six and has a derivative bound independent
of that radius. The second retains the radius and has a polynomial dependence
on it. Their different bounds will be used at different stages of the
averaging construction.
Throughout this section, $G$ is a finitely generated group with a fixed left
invariant word metric $d$. Write $B(y,s)=\{h:d(h,y)\le s\}$. Counting words
in a finite symmetric generating set gives a constant $c_d$ such that
\begin{equation}
|B(y,s)|\le \exp(c_d s)\qquad(s\ge1).
\label{ch:ball-growth}
\end{equation}
Indeed, if the generating set has $m$ elements, the ball is covered by words
of length at most $\lfloor s\rfloor$, and their number is at most
$(m+1)^{\lfloor s\rfloor+1}$. Increasing $c_d$ also covers the trivial
group. Constants denoted by $C$ may depend on this word metric but not on a
radius, an accuracy parameter, or the size of an input support.
Let $\mathcal P_f(G)$ be the set of finitely supported probability
distributions on $G$. For $r\ge1$ set
\begin{equation}
\begin{split}
\chi_r(s)&=\min\{1,\max\{0,2-s/r\}\},\qquad s\ge0,\\
E_r(P,y)&=\sum_{h\in G}P_h\chi_r(d(h,y)),\\
q_r(P)&=\max_{y\in G}E_r(P,y),\qquad
\phi_r(P)=(q_r(P)-a_*)_+,\qquad a_*=0.80.
\end{split}
\label{ch:concentration}
\end{equation}
The maximum exists: the score vanishes outside the finite $2r$ neighborhood
of $\supp P$. It lies in $[0,1]$, so $0\le\phi_r\le0.20$. The functions
$q_r$ and $\phi_r$ need not be smooth; they will only be used in inequalities.
\begin{definition}
\label{ch:channel-definition}
A \emph{channel} with label set $\Lambda$, on which $G$ acts, assigns to
each $P\in\mathcal P_f(G)$ a finitely supported subprobability
$Y(P)=(Y(P)_\omega)_{\omega\in\Lambda}$. Thus $Y(P)_\omega\ge0$ and
$\sum_\omega Y(P)_\omega\le1$. Translation of weights means
$(sP)_h=P_{s^{-1}h}$ and $(sY)_\omega=Y_{s^{-1}\omega}$.
We require left equivariance,
$Y(sP)=sY(P)$ for $s\in G$, and the following smoothness property. For each
finite $S\subset G$, the restriction to probabilities supported in $S$ has
its labels in a fixed finite subset of $\Lambda$, and its coordinate
functions extend smoothly to a neighborhood of that probability simplex in
$\R^S$.
\end{definition}
In a transfer direction $\dot P=\delta_h-\delta_{h'}$, derivatives are
computed after enlarging $S$ to contain $h,h'$. The channels we construct
will have compatible formulas on these enlarged spaces, and their
derivatives will vanish at zero output coordinates. For such a channel
define its Fisher seminorm in
this direction by
\begin{equation}
\mathcal F_Y(P;h,h')
=\left(\sum_{\omega:Y(P)_\omega>0}
\frac{|\dot Y(P)_\omega|^2}{Y(P)_\omega}\right)^{1/2}.
\label{ch:fisher-definition}
\end{equation}
The zero-coordinate assertion is part of the construction, and will be
proved; it is not a convention for discarding a nonzero derivative.
If $A$ has label set $G$ and $L$ has label set $\Lambda$, a \emph{cross
kernel} is a function $\kappa:G\times\Lambda\to[0,1]$ satisfying
$\kappa(sy,s\omega)=\kappa(y,\omega)$. It is fixed independently of $P$.
It gives the predictions
\begin{equation}
\operatorname{pred}(P,y)
=\sum_{\omega\in\Lambda}\kappa(y,\omega)L(P)_\omega.
\label{ch:prediction-definition}
\end{equation}
Exact predictions alone would be easy: $L(P)=P$ and
$\kappa(y,h)=\chi_r(d(h,y))$ give $\operatorname{pred}(P,y)=E_r(P,y)$.
When $G$ has at least two elements, this choice has no uniform Fisher bound:
a transfer between distinct elements, adding mass to an entry of size
$t>0$, contributes $1/t$ to the squared seminorm, which diverges as
$t\downarrow0$. The approximation allowed in the next proposition permits
uniform derivative control, including at probability faces.
\begin{proposition}[Spatial summary channels]
\label{ch:summary}
Fix $r\ge1$ and $0<\epsilon\le10^{-3}$. There are channels $A,L$ and a
cross kernel $\kappa$ with either of the following two choices of radius:
the early version has $r'=6r$, and the later version has $r'=r$.
In both versions $A$ is labelled by $G$, and for every
$P\in\mathcal P_f(G)$ and $y\in G$,
\begin{align}
\sum_{z\in G}A(P)_z\bigl(E_r(P,z)-a_*\bigr)
&\ge\phi_r(P)-10\epsilon,
\label{ch:selection-bound}\\
\operatorname{pred}(P,y)
&\le E_{r'}(P,y)+10\epsilon,
\label{ch:prediction-upper}\\
\operatorname{pred}(P,y)
&\ge E_r(P,y)-10\epsilon
\qquad\text{if }A(P)_y>0.
\label{ch:prediction-lower}
\end{align}
For $Y=A$ or $Y=L$, derivatives in every transfer direction vanish at zero
coordinates and satisfy
\begin{equation}
\mathcal F_Y(P;h,h')\le C(1+d(h,h'))
\begin{cases}
\epsilon^{-10},&\text{early version},\\
r^{10}\epsilon^{-10},&\text{later version}.
\end{cases}
\label{ch:summary-fisher}
\end{equation}
Here $C$ depends only on the fixed word metric. The numerical error constant
$10$ in \eqref{ch:selection-bound}--\eqref{ch:prediction-lower} is independent
of the group and of the parameters.
\end{proposition}
To see why these guarantees are suited to averaging, assume them for the
moment. Take a finite family $P^{(\nu)}$ of input probabilities and weights
$\theta_\nu\ge0$ with $\sum_\nu\theta_\nu=1$, and write
\[
\overline P=\sum_\nu\theta_\nu P^{(\nu)},\qquad
\overline A_y=\sum_\nu\theta_\nu A(P^{(\nu)})_y,\qquad
\overline{\operatorname{pred}}_y
=\sum_\nu\theta_\nu\operatorname{pred}(P^{(\nu)},y).
\]
The selection and lower prediction bounds give
$\sum_yA(P)_y(\operatorname{pred}(P,y)-a_*)\ge\phi_r(P)-20\epsilon$.
On the other hand, the upper prediction bound and linearity of $E_{r'}$
give
$\overline{\operatorname{pred}}_y
\le E_{r'}(\overline P,y)+10\epsilon$.
Since $\overline A$ is a subprobability, its pairing with
$E_{r'}(\overline P,\cdot)-a_*$ is at most
$\phi_{r'}(\overline P)$; if the maximal centered score is negative, that
pairing is nonpositive. Finally,
$\sum_\nu\theta_\nu\sum_yA(P^{(\nu)})_y=\sum_y\overline A_y$, so
the terms containing $a_*$ cancel and
\begin{equation}
\begin{split}
&\sum_\nu\theta_\nu\sum_y
A(P^{(\nu)})_y\operatorname{pred}(P^{(\nu)},y)
-\sum_y\overline A_y\overline{\operatorname{pred}}_y\\
&\hspace{2em}\ge
\sum_\nu\theta_\nu\phi_r(P^{(\nu)})
-\phi_{r'}(\overline P)-30\epsilon.
\end{split}
\label{ch:averaging-drop}
\end{equation}
Thus loss of concentration under averaging gives a lower bound for the
difference of the two pairings. The same calculation applies to time
averages of fields with a common finite support, which is how these
estimates will enter the connection construction.
We first construct $A$ and a second distribution $D$ on possible centers.
The channel $A$ concentrates near a maximizing center when the maximal score
exceeds $a_*$. The thresholds for $D$ are lower, so that $D$ already has
almost full mass whenever $A$ is nonzero. The two versions of $L$ will then
use this same $D$ to predict scores at the selected centers.
\subsection{Smooth selection of centers}
The number of centers that can carry substantial score has a bound
independent of $|\supp P|$.
\begin{lemma}
\label{ch:center-geometry}
For $P\in\mathcal P_f(G)$, the set
$\{y:E_r(P,y)>1/2\}$ has diameter at most $4r$ and cardinality at most
$\exp(\gamma r)$, where $\gamma$ depends only on the word metric.
\end{lemma}
\begin{proof}
The inequality $E_r(P,y)>1/2$ implies
$P(\{h:d(h,y)<2r\})>1/2$. Any two such sets intersect, since their
masses add to more than one. Their centers consequently have distance less
than $4r$. If the set of centers is nonempty, choose one of them. All the
others lie in its $4r$ ball, and \eqref{ch:ball-growth} gives the claimed
bound, for example with $\gamma=4c_d$. The empty case is immediate.
\end{proof}
Choose smooth functions $\psi_A,\psi_D:\R\to[0,1]$ such that
\begin{equation}
\begin{array}{c|cc}
&\text{identically zero on}&\text{identically one on}\\ \hline
\psi_A&(-\infty,0.70]&[0.75,\infty)\\
\psi_D&(-\infty,0.58]&[0.62,\infty).
\end{array}
\label{ch:cutoffs}
\end{equation}
For completeness, a smooth increasing cutoff from zero to one on $[0,1]$
is $\eta(t)/(\eta(t)+\eta(1-t))$, where
$\eta(t)=e^{-1/t}$ for $t>0$ and $\eta(t)=0$ for $t\le0$.
Every right derivative of $\eta$ at zero vanishes: away from zero it is
$e^{-1/t}$ times a polynomial in $1/t$, which tends to zero as $t\downarrow0$.
The denominator in the displayed quotient is everywhere positive, and
affine changes of variable give \eqref{ch:cutoffs}. Their first derivatives
are bounded by a fixed numerical constant, since they vanish outside fixed
compact transition intervals.
Take $N=M_0r\epsilon^{-2}$ and define
\begin{equation}
\begin{split}
w^A_y&=\psi_A(E_r(P,y))^2
e^{N(E_r(P,y)-0.80)},\\
w^D_y&=\psi_D(E_r(P,y))^2
e^{N(E_r(P,y)-0.66)},\\
A(P)_y&=\frac{w^A_y}{1+\sum_z w^A_z},\qquad
D(P)_y=\frac{w^D_y}{1+\sum_z w^D_z}.
\end{split}
\label{ch:selector-formulas}
\end{equation}
Here $M_0$ is fixed in terms of the metric, large enough that for all allowed
$r,\epsilon$,
\begin{equation}
(1+e^{\gamma r})e^{-N\epsilon}\le\epsilon,
\qquad e^{\gamma r-0.04N}\le1.
\label{ch:exponential-choices}
\end{equation}
For example, any $M_0\ge25\gamma+4$ suffices. Indeed,
$\log(1+e^{\gamma r})+\log(1/\epsilon)
\le\gamma r+\log2+1/\epsilon\le(\gamma+2)r/\epsilon$,
and $N\ge25\gamma r$.
The additional $1$ in each denominator may be regarded as the weight of an
unused null label. It allows the channel to have small total mass when no
center reaches the exponential threshold. Each displayed sum is finite by
Lemma~\ref{ch:center-geometry}.
\begin{lemma}[Selector estimates]
\label{ch:selector-estimates}
The formulas \eqref{ch:selector-formulas} define smooth equivariant
channels. They satisfy
\begin{equation}
\sum_y A(P)_y(E_r(P,y)-a_*)\ge\phi_r(P)-3\epsilon.
\label{ch:selector-concentration}
\end{equation}
Whenever $q_r(P)>0.70$,
\begin{equation}
\sum_zD(P)_z\ge1-\epsilon,
\qquad
\sum_zD(P)_zE_r(P,z)\ge q_r(P)-2\epsilon.
\label{ch:D-concentration}
\end{equation}
Both channels have zero derivatives at zero coordinates, and
\begin{equation}
\mathcal F_A(P;h,h')+\mathcal F_D(P;h,h')
\le C\epsilon^{-2}d(h,h').
\label{ch:selector-fisher}
\end{equation}
\end{lemma}
\begin{proof}
Fix $P$ and abbreviate $E_r(P,y)$ to $E_y$ and $q_r(P)$ to $q$.
Suppose first that $q>a_*+\epsilon$. At a maximizing center the cutoff
$\psi_A$ is one, and its weight is $e^{N(q-a_*)}$.
The normalized mass of centers with $E_y0.70$. The cutoff $\psi_D$ is one at a maximizing center.
The same comparison, with threshold $0.66$, shows that the total mass of
the null label and the centers with $E_z0.70$, its mass is at least
$1-\epsilon\ge1/2$. Its transfer derivatives vanish at zero coordinates
and satisfy
\[
\mathcal F_D(P;h,h')\le C\epsilon^{-2}d(h,h')
\le C\epsilon^{-10}(1+d(h,h')).
\]
In particular, if $\phi_r(P)>0$, then
\begin{equation}
\sum_zD(P)_z^2\ge\frac14e^{-\gamma r}.
\label{ch:terminal-square-mass}
\end{equation}
\end{lemma}
\begin{proof}
Every center in the support of $D$ has score greater than $0.58>1/2$.
Apply Lemma~\ref{ch:center-geometry} and
Lemma~\ref{ch:selector-estimates}. If $\phi_r(P)>0$, then
$q_r(P)>0.80$, so $\sum_zD_z\ge1/2$. The Cauchy--Schwarz inequality on
the finite support gives
$\sum_zD_z^2\ge(\sum_zD_z)^2/|\supp D|$, proving
\eqref{ch:terminal-square-mass}.
\end{proof}
\subsection{Predictions with an enlarged radius}
The early channel can use the average score of the centers chosen by $D$.
All these centers lie near any center selected by $A$. Enlarging the radius
allows us to bound each score contributing at $y$ by $E_{6r}(P,y)$.
Define
\begin{equation}
\Lambda=G,\qquad
L(P)_z=D(P)_zE_r(P,z),\qquad
\kappa(y,z)=\mathbf1_{\{d(y,z)\le4r\}}.
\label{ch:early-channel}
\end{equation}
This is a subprobability channel, since $0\le E_r(P,z)\le1$.
When $d(y,z)\le4r$, every $h$ with $\chi_r(d(h,z))>0$ has
$d(h,y)<6r$. On this set $\chi_{6r}(d(h,y))=1$, and therefore
\begin{equation}
E_r(P,z)\le E_{6r}(P,y).
\label{ch:radius-domination}
\end{equation}
After summing against $D$, this proves
$\operatorname{pred}(P,y)\le E_{6r}(P,y)$.
If $A(P)_y>0$, then $E_r(P,y)>0.70$ and $q_r(P)>0.70$. For every
$z$ with $D(P)_z>0$, Lemma~\ref{ch:center-geometry} gives
$d(y,z)\le4r$. Hence every such $z$ contributes to the prediction, and
\eqref{ch:D-concentration} gives
\[
\operatorname{pred}(P,y)
=\sum_zD(P)_zE_r(P,z)
\ge q_r(P)-2\epsilon
\ge E_r(P,y)-2\epsilon.
\]
To estimate derivatives, write $E_z=E_r(P,z)$ again. Where $D_z>0$,
$E_z>0.58$. Differentiating $L_z=D_zE_z$ yields
\begin{equation}
\sum_{z:L_z>0}\frac{|\dot L_z|^2}{L_z}
\le 2\sum_{z:D_z>0}E_z\frac{|\dot D_z|^2}{D_z}
+2\sum_{z:D_z>0}D_z\frac{|\dot E_z|^2}{E_z}
\le C\epsilon^{-4}d(h,h')^2.
\label{ch:early-fisher}
\end{equation}
In the second sum only $D_z>0$ is relevant; its denominators are thus bounded
below by $0.58$. We used \eqref{ch:score-derivative},
\eqref{ch:selector-fisher}, and $r\ge1$. At a zero coordinate $D_z=0$
and $\dot D_z=0$, so $\dot L_z=0$. The same finite formulas prove
smoothness at faces. This establishes all the early-version claims of
Proposition~\ref{ch:summary}.
\subsection{Accurate finite sample strings}
To keep the radius equal to $r$, an average of center scores is insufficient:
we must predict the score separately at each $y$. A finite string of sample
points approximates all scores in a bounded region simultaneously. Random
sampling will prove the existence of an accurate string. We will then put
smooth positive weights on \emph{all} strings in a fixed finite alphabet.
This distinction is useful at probability faces: the defining weights will
not be products of possibly vanishing input probabilities.
Fix a center $z\in G$. Introduce a null symbol $\partial$, fixed by the
group action, and the alphabet
\[
\mathcal A_z=B(z,20r)\cup\{\partial\}.
\]
Define a probability $Q^z$ on this alphabet by
\begin{equation}
P^z_h=P_h\chi_{10r}(d(h,z)),\qquad
Q^z_h=P^z_h\ (h\ne\partial),\qquad
Q^z_\partial=1-\sum_hP^z_h.
\label{ch:truncated-probability}
\end{equation}
The notation $E_r(P^z,y)$ has the same linear meaning as in
\eqref{ch:concentration}, although $P^z$ can have mass less than one.
This truncation provides a fixed finite alphabet while preserving every
score needed for the lower prediction bound. Indeed, if
$A(P)_yD(P)_z>0$, Lemma~\ref{ch:center-geometry} gives $d(y,z)\le4r$.
Every $h$ contributing to $E_r(P,y)$ then has $d(h,z)<6r$, where the
factor $\chi_{10r}(d(h,z))$ is one. Consequently
\begin{equation}
E_r(P^z,y)=E_r(P,y)
\qquad\text{if }A(P)_yD(P)_z>0.
\label{ch:truncation-exactness}
\end{equation}
For arbitrary $y$, we still have $E_r(P^z,y)\le E_r(P,y)$, which is the
direction needed for the upper bound. Thus accurate predictions for $P^z$
will suffice. The Lipschitz taper in \eqref{ch:truncated-probability} will
also keep their transfer derivatives proportional to $d(h,h')/r$.
For a positive integer $b$ and a string $w=(w_1,\ldots,w_b)\in\mathcal A_z^b$
let
\begin{equation}
H_y(w)=\frac1b\sum_{i:w_i\ne\partial}\chi_r(d(w_i,y)),
\qquad
\Delta(P;z,w)=\max_{y\in B(z,22r)}
|H_y(w)-E_r(P^z,y)|.
\label{ch:string-discrepancy}
\end{equation}
Both scores vanish when $d(y,z)>22r$, by the triangle inequality. Thus
$\Delta$ also controls the discrepancy at every $y\in G$. Always
$0\le\Delta\le1$.
\begin{lemma}
\label{ch:accurate-string}
There is a constant $C$, depending only on the word metric, and an integer
$b$ depending only on $r,\epsilon$, with
$1\le b\le Cr\epsilon^{-4}$, such that for every $P$ and $z$ some
$w\in\mathcal A_z^b$ satisfies $\Delta(P;z,w)\le\epsilon$.
\end{lemma}
\begin{proof}
Take $b$ independent samples with distribution $Q^z$. For a fixed test
center $y$, each summand in $H_y$ lies in $[0,1]$ and has mean
$E_r(P^z,y)$. The bounded-variable concentration estimate of
Hoeffding~\cite{Hoeffding1963} gives the needed probability bound. We
include a direct exponential-moment proof, with sufficient constants.
If a real random variable $X$ has mean zero and $|X|\le1$, then, for
$|t|\le1$,
\[
\mathbf E e^{tX}
\le1+\sum_{k\ge2}\frac{|t|^k}{k!}
\le1+t^2\le e^{t^2}.
\]
For independent copies $X_1,\ldots,X_b$, Markov's inequality and
independence, with $t=\epsilon/2$, imply
\[
\mathbf P\!\left(\frac1b\sum_iX_i>\epsilon\right)
\le e^{-tb\epsilon}\prod_i\mathbf E e^{tX_i}
\le e^{-b\epsilon^2/4}.
\]
Applying the same argument to $-X_i$ gives the bound
$2e^{-b\epsilon^2/4}$ for absolute error exceeding $\epsilon$.
By \eqref{ch:ball-growth}, there is a fixed $c_1$ such that the number of
test centers is at most $e^{c_1r}$. A union bound therefore gives
\[
\mathbf P\bigl(\Delta(P;z,w)>\epsilon\bigr)
\le2e^{c_1r-b\epsilon^2/4}.
\]
Choose $b=\lceil M_1r\epsilon^{-4}\rceil$ with a sufficiently large
fixed $M_1$. The right side is strictly less than one for all permitted
$r,\epsilon$. At least one string is accurate. Since
$r\epsilon^{-4}\ge1$, rounding preserves the asserted upper bound on $b$.
\end{proof}
\subsection{Smooth weights on strings}
Fix the integer $b$ supplied by Lemma~\ref{ch:accurate-string}. To assign
smooth weights, replace the maximum of the signed errors by the logarithm
of a sum of exponentials. Put
\begin{equation}
S(P;z,w)=\frac1\lambda\log
\left(\sum_{y\in B(z,22r)}\sum_{\sigma\in\{-1,1\}}
\exp\bigl(\lambda\sigma(H_y(w)-E_r(P^z,y))\bigr)\right).
\label{ch:smooth-discrepancy}
\end{equation}
Choose $\lambda>0$, depending only on $r,\epsilon$, so large that the
logarithm of the number of signed tests is at most $\lambda\epsilon$.
The number of tests is independent of $z$, since left translation identifies
the balls. The elementary inequalities
$e^{\lambda\max x_i}\le\sum_i e^{\lambda x_i}
\le M e^{\lambda\max x_i}$ give
\begin{equation}
\Delta(P;z,w)\le S(P;z,w)\le\Delta(P;z,w)+\epsilon.
\label{ch:smoothing-error}
\end{equation}
The first derivative of this smoothed maximum has a particularly simple
bound. The test function
$h\mapsto\chi_{10r}(d(h,z))\chi_r(d(h,y))$ is
$11/(10r)$-Lipschitz: both factors have absolute value at most one, and
their Lipschitz constants add. Thus every signed error in
\eqref{ch:smooth-discrepancy} has transfer derivative of absolute value
at most $11d(h,h')/(10r)$. Differentiation of the logarithm gives a convex
combination of these derivatives; the factor $\lambda$ cancels.
Consequently
\begin{equation}
|\dot S(P;z,w)|\le\frac{11}{10r}d(h,h').
\label{ch:smooth-discrepancy-derivative}
\end{equation}
In particular this estimate does not grow with the number of tests or the
smoothing parameter.
Now define a probability distribution on the entire fixed set of strings by
\begin{equation}
\pi_z(P)_w
=\frac{e^{-N_1S(P;z,w)}}{\sum_{v\in\mathcal A_z^b}e^{-N_1S(P;z,v)}},
\qquad N_1=M_2rb\epsilon^{-2}.
\label{ch:gibbs-strings}
\end{equation}
Every coordinate is strictly positive. The constant $M_2$ will be fixed
in terms of the metric.
\begin{lemma}
\label{ch:string-estimates}
For $M_2$ sufficiently large, uniformly in $P,z,y$,
\begin{align}
\sum_w\pi_z(P)_w\Delta(P;z,w)&\le5\epsilon,
\label{ch:mean-discrepancy}\\
\left|\sum_w\pi_z(P)_wH_y(w)-E_r(P^z,y)\right|
&\le5\epsilon.
\label{ch:mean-score}
\end{align}
The transfer derivatives satisfy
\begin{equation}
\left(\sum_w\frac{|\dot\pi_z(P)_w|^2}{\pi_z(P)_w}\right)^{1/2}
\le C N_1\frac{d(h,h')}{r}.
\label{ch:string-fisher}
\end{equation}
\end{lemma}
\begin{proof}
There are at most $e^{c_2rb}$ strings, for a constant $c_2$ fixed by the
metric. By Lemma~\ref{ch:accurate-string} and
\eqref{ch:smoothing-error}, at least one string has $S\le2\epsilon$.
Every string with $\Delta>4\epsilon$ has $S>4\epsilon$, and hence a
weight at most $e^{-2N_1\epsilon}$ times the weight of that particular
string. Therefore
\[
\sum_{w:\Delta(P;z,w)>4\epsilon}\pi_z(P)_w
\le e^{c_2rb-2N_1\epsilon}\le\epsilon
\]
once $M_2$ is sufficiently large. One fixed choice works for all
$r\ge1$, $b\ge1$, and $\epsilon\le10^{-3}$, as is seen by taking
logarithms and using $\log(1/\epsilon)\le1/\epsilon$.
Since $\Delta\le1$, splitting the expectation at $4\epsilon$ proves
\eqref{ch:mean-discrepancy}. The discrepancy controls each test center,
and both scores vanish outside the test ball, proving
\eqref{ch:mean-score} for every $y$.
Differentiating the normalized exponential in \eqref{ch:gibbs-strings}
gives
\[
\frac{\dot\pi_z(P)_w}{\pi_z(P)_w}
=-N_1\left(\dot S(P;z,w)
-\sum_v\pi_z(P)_v\dot S(P;z,v)\right).
\]
Equation~\eqref{ch:smooth-discrepancy-derivative} bounds its absolute value
by $C N_1d(h,h')/r$. Squaring and summing against the probability
$\pi_z(P)$ proves \eqref{ch:string-fisher}.
\end{proof}
\subsection{Predictions at the original radius}
We can now finish the later channel. Its label set consists of pairs
$(z,w)$, with $z\in G$ and $w\in\mathcal A_z^b$. Translation acts on
the center and on every nonnull entry of the string. Define
\begin{equation}
L(P)_{z,w}=D(P)_z\pi_z(P)_w,
\qquad \kappa(y,(z,w))=H_y(w).
\label{ch:later-channel}
\end{equation}
The total mass of $L$ is $\sum_zD(P)_z\le1$, and the kernel takes values
in $[0,1]$. All formulas commute with left translation.
By \eqref{ch:mean-score}, and since $P^z_h\le P_h$,
\[
\operatorname{pred}(P,y)
\le\sum_zD(P)_z\bigl(E_r(P^z,y)+5\epsilon\bigr)
\le E_r(P,y)+5\epsilon.
\]
Suppose next that $A(P)_y>0$. For every $z$ with $D(P)_z>0$,
\eqref{ch:truncation-exactness} identifies the predicted truncated score
with $E_r(P,y)$. Moreover $q_r(P)>0.70$, and
\eqref{ch:D-concentration} gives
$\sum_zD(P)_z\ge1-\epsilon$. It follows that
\[
\operatorname{pred}(P,y)
\ge E_r(P,y)\sum_zD(P)_z-5\epsilon\sum_zD(P)_z
\ge E_r(P,y)-6\epsilon.
\]
These are the required prediction bounds with $r'=r$.
For the derivative bound it is useful to retain the product structure
$L_{z,w}=D_z\pi_z(w)$. Since $\sum_w\pi_z(w)=1$ and
$\sum_w\dot\pi_z(w)=0$, expansion of the square gives the exact identity
\begin{equation}
\sum_{z,w:L_{z,w}>0}\frac{|\dot L_{z,w}|^2}{L_{z,w}}
=\sum_{z:D_z>0}\frac{|\dot D_z|^2}{D_z}
+\sum_zD_z\sum_w\frac{|\dot\pi_z(w)|^2}{\pi_z(w)}.
\label{ch:product-fisher}
\end{equation}
All weights and derivatives here are real; the mixed term vanishes by the
displayed zero-sum identity. Using \eqref{ch:selector-fisher},
\eqref{ch:string-fisher}, and $b\le Cr\epsilon^{-4}$ yields
\begin{equation}
\mathcal F_L(P;h,h')
\le C\bigl(\epsilon^{-2}+N_1/r\bigr)d(h,h')
\le Cr\epsilon^{-6}d(h,h').
\label{ch:later-fisher}
\end{equation}
It remains to check the regularity promised in the channel definition.
For a fixed finite input support $S$, only finitely many centers $z$ can
have $D_z\ne0$, all in the $2r$ neighborhood of $S$. For each such
center, the alphabet and string set are finite and independent of $P$.
The functions $S(P;z,w)$ and $\pi_z(P)_w$ are smooth functions of the
coordinates of $P$: the scores are linear, and each exponential sum has a
strictly positive denominator. Thus \eqref{ch:later-channel} is a fixed
finite smooth formula on each enlarged support space. If $L_{z,w}=0$,
strict positivity of $\pi_z(w)$ implies $D_z=0$; also $\dot D_z=0$ by
Lemma~\ref{ch:selector-estimates}. Consequently $\dot L_{z,w}=0$.
This proves smoothness at faces and the required behavior at zero entries.
\begin{proof}[Completion of the proof of Proposition~\ref{ch:summary}]
Use the same selector $A$ in both versions. Inequality
\eqref{ch:selector-concentration} gives
\eqref{ch:selection-bound}. The early construction
\eqref{ch:early-channel} gives the two prediction inequalities with
errors $0$ and $2\epsilon$; the later construction
\eqref{ch:later-channel} gives errors $5\epsilon$ and $6\epsilon$.
All are bounded by $10\epsilon$.
The Fisher estimates \eqref{ch:selector-fisher},
\eqref{ch:early-fisher}, and \eqref{ch:later-fisher} imply
\eqref{ch:summary-fisher}, since $r\ge1$ and $\epsilon\le1$.
The preceding constructions also verify the channel and cross-kernel
requirements, including the assertions at zero coordinates.
\end{proof}
For use with differential forms, regard the labels of $A$ and $L$ as
disjoint even when their underlying sets happen to coincide. On their
disjoint union define the symmetric kernel $K$ by
\begin{equation}
\begin{split}
K(y,\omega)=K(\omega,y)&=\tfrac12\kappa(y,\omega)
\quad(y\in G,\ \omega\in\Lambda),\\
K&=0\quad\text{within either channel}.
\end{split}
\label{ch:symmetric-kernel}
\end{equation}
For vectors on this union, write $K(U,V)=\sum_{\xi,\eta}K(\xi,\eta)U_\xi V_\eta$.
The combined field $Y=A\oplus L$ has mass at most two, and
\begin{equation}
K(Y(P),Y(P))
=\sum_yA(P)_y\operatorname{pred}(P,y).
\label{ch:kernel-diagonal}
\end{equation}
The kernel is bounded and invariant under simultaneous left translation.
The expression in \eqref{ch:fisher-definition} also makes sense for this
field of mass at most two, and its square is
$\mathcal F_A(P;h,h')^2+\mathcal F_L(P;h,h')^2$.
Thus Proposition~\ref{ch:summary} controls its derivatives as well.
These facts permit both channels to enter a single bilinear pairing in the
averaging construction.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/04-channels.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/05-connection.tex ====
\section{A connection from successive time averages}
\label{sec:connection}
Proposition~\ref{cy:trace-identity} allows us to compute the trace with any
connection satisfying Definition~\ref{cy:connection}. We now construct a
connection whose curvature can be estimated at several time scales. The
construction first gives a one-form $B$ with uniformly positive contraction
$B(V)$; division by this contraction will produce the connection. The
summary channels of Proposition~\ref{ch:summary} make the contributions of
successive scales telescope. The final contribution uses the twist by $g$.
Throughout this section $G$ is finitely generated, $d$ is its fixed left
invariant word metric, and $g\in G$. Constants denoted by $C$ may depend on
the word metric and on the fixed smooth functions used below. After the
parameter choice at the end of the construction, a subscript $R$ will
permit dependence on the radius $R$, but never on the tuple, its
dimension, or the element $g$.
\subsection{Time averages and spatial summaries}
Begin with a finite number $b_*\geq2$ of averaging stages, an even ambient
dimension $n$, and positive time scales $\delta_1<\cdots<\delta_{b_*}$
satisfying
\begin{equation}
\sum_{i=1}^j\delta_i\leq2\delta_j,
\qquad 10\delta_{b_*}<1.
\label{co:scale-support}
\end{equation}
The construction through positivity uses only these conditions on the
time scales. Their precise dependence on $R$ and $n$ will be selected
afterwards, when preparing the quantitative estimates.
\emph{The same scales are kept fixed in every dimension $0\leq k\leq n$.}
Thus passing to a face never changes the averaging operation.
Choose a nonnegative, even $C^\infty$ function $\mu$, supported in
$[-1,1]$, with integral one, and put
$\mu_j(s)=\delta_j^{-1}\mu(s/\delta_j)$. On a tuple of dimension $k$,
write its step function and its averages as
\begin{equation}
\begin{split}
f(u)&=h_i\quad(t_i\leq u0$, for a fixed
constant $C_D$. Choose a fixed $C_*\geq C_D$ and set
\begin{equation}
W=\exp(C_*r).
\label{co:accuracy}
\end{equation}
Since $0\leq\phi_r\leq1-a_*=1/5$, this gives, for every accuracy
parameter in \eqref{co:error-budget},
\begin{equation}
W\sum_zD(P)_z^2\geq\phi_r(P).
\label{co:terminal-multiplier}
\end{equation}
We shall construct the connection under either of the assumptions
\begin{equation}
\begin{array}{ll}
\textnormal{(i)}&g\text{ has infinite order},\\[2pt]
\textnormal{(ii)}&g\text{ has finite order and }
\displaystyle\min_{h\in G}d(h,gh)>4r.
\end{array}
\label{co:twist-assumption}
\end{equation}
Only the final time-lag kernel and its contraction distinguish these two
cases.
\subsection{Lag pairings and their compatibility}
We next turn the fields into one-forms. For $0\leq j0$. On a fixed tuple the smoothed probabilities have
the explicit expression
\begin{equation}
P_j(u)=\sum_{a\in\Z}\sum_{i=0}^k
\left(\int_{t_i+a}^{t_{i+1}+a}\eta_j(u-v)\,dv\right)\delta_{g^ah_i}.
\label{co:smooth-intervals}
\end{equation}
For $u$ in a bounded interval and times in a neighborhood of any point of
the closed ordered domain, only a fixed finite set of summands occurs.
Each coefficient is smooth in $u$ and the times, including when the
endpoints coincide. The channel maps are smooth on the finite
probability simplices, with smooth extensions at their faces, so $Y_j$
has the same regularity. Its possible labels on one period lie in a
fixed finite set locally in the time variables.
When $U_j$ has compact support, the two integrals in
\eqref{co:forms} can therefore be differentiated under fixed integration
domains. The final full-sign kernel in case (i) requires an additional
observation. Write $v=w+a$ with $w$ in one period. If $y,z$ are possible
labels in that period, then
\[
K_{b_*}(y,g^az)\ne0\quad\Longleftrightarrow\quad y=g^az.
\]
Since $g$ has infinite order, this equation has at most one solution
$a\in\Z$ for each pair $(y,z)$. There are only finitely many such pairs
locally. The full-sign integral is therefore locally a finite sum of
smooth integrals over bounded domains, even though the possible matching
integers need not be uniformly bounded as the tuple varies.
To check the period cut, regard
\[
F_j(u)=\int_{\R}U_j(v-u)K_j(dY_j(u),Y_j(v))\,dv
\]
as a one-form in the time parameters. The identities
$Y_j(u+1)=gY_j(u)$ and $dY_j(u+1)=g\,dY_j(u)$, followed by the change of
variable $v\mapsto v+1$, give $F_j(u+1)=F_j(u)$. Hence its integral over
one period is independent of the cut. This also shows why a moving cut
does not introduce an extra exterior-derivative term: if the cut is
$c=c(t)$, the endpoint contribution is
\begin{equation}
dc\wedge\bigl(F_j(c+1)-F_j(c)\bigr)=0.
\label{co:cut-cancellation}
\end{equation}
For the raw stage, work first away from the faces and choose a fixed
period cut away from its finitely many jumps. Evaluating the impulse
measure gives \eqref{co:raw-coefficient}; moving an impulse to another
period leaves this coefficient unchanged, by equivariance and a change
of variable. To prove one-sided smoothness explicitly, let
\[
Q(s)=\int_0^sU_0(w)\,dw
=|s|-2\int_0^s\int_0^w\rho_0(a)\,da\,dw.
\]
Then $(B_0)_i$ is a locally finite sum of terms
\begin{equation}
-\frac12K_0(J_i,\delta_{g^ah_q})
\bigl(Q(t_{q+1}+a-t_i)-Q(t_q+a-t_i)\bigr),
\quad 0\leq q\leq k.
\label{co:raw-primitive}
\end{equation}
Although $Q$ is constant rather than zero outside the lag support, each
displayed endpoint difference vanishes when its interval misses that
support. Since all boundaries lie in one period and $2\delta_1<1$, only
the shifts $a=-1,0,1$ can contribute, with the same finite choice valid
in a sufficiently small neighborhood of any point of the closed domain.
On the entire closed ordered domain, each affine expression
$t_q+a-t_i$ has a fixed weak sign: the order of the boundaries and all
their period translates is prescribed. The same is true for
$t_{q+1}+a-t_i$. Thus every absolute value in
\eqref{co:raw-primitive} equals a fixed signed affine function on that
domain. The remaining part of $Q$ is smooth on the whole line.
Consequently the coefficient has a smooth extension from the prescribed
side at each face, including intersections of faces. Compact support of
$U_0$ allows the finite set of summands to be fixed in a neighborhood.
It remains to identify the restrictions. On the face $t_i=t_{i+1}$,
where $0\leq i200$. For sufficiently large $R$ retain the spatial
radii, accuracies, and multiplier from
\eqref{co:spatial-radii}, \eqref{co:error-budget}, and
\eqref{co:accuracy}, and set
\begin{equation}
\beta_1=R,\qquad \beta_{j+1}=\beta_j^{3/2}.
\label{co:ladder}
\end{equation}
Choose $b_*$ to be the smallest integer such that
\begin{equation}
b_*\geq J_*,\qquad \beta_{b_*}\geq W^2R^{100}.
\label{co:last-scale}
\end{equation}
For each even integer $n>10\beta_{b_*}$, use
\begin{equation}
\delta_j=\beta_j/n\qquad(1\leq j\leq b_*).
\label{co:time-scales}
\end{equation}
The ratio $\beta_{j+1}/\beta_j$ is at least two when $R$ is sufficiently
large, so these scales satisfy \eqref{co:scale-support}. Thus the
construction and positivity proved above apply. For each such $n$, the
same actual scales $\beta_j/n$ are used in every dimension $k\leq n$;
face restriction never changes the denominator to $k$.
The stage count, spatial radii, accuracies, multiplier, and numbers
$\beta_j$ now depend only on $R$ and the fixed choices. Since
$\log\beta_j=(3/2)^{j-1}\log R$ and
$\log(W^2R^{100})=2C_*r+100\log R$, with $r$ a fixed multiple of $R$,
\begin{equation}
b_*=O(1+\log R).
\label{co:stage-count}
\end{equation}
The particular exponents in \eqref{co:ladder} and
\eqref{co:last-scale} will enter the curvature bounds. They leave room
for the polynomial losses in the channel derivatives.
Choose a sufficiently large fixed constant $C$ and set
\begin{equation}
a_j=C\epsilon_j^{-10}
\begin{cases}
1,&1\leq j10\beta_{b_*}$
be even, with time scales \eqref{co:time-scales}. Suppose that $g$
satisfies \eqref{co:twist-assumption}. The forms $B_j$ and $B$ in
\eqref{co:forms}, constructed in all dimensions $0\leq k\leq n$ with the
same parameters, have the following properties. For every smooth stage
$1\leq j\leq b_*$ and every coordinate $t_i$,
\begin{equation}
\left(\int_{\rm per}\sum_\omega
\frac{|\partial_iY_j(u)_\omega|^2}{Y_j(u)_\omega}\,du
\right)^{1/2}
\leq\frac{a_jl_i}{\sqrt{\delta_j}}.
\label{co:fisher}
\end{equation}
Terms with zero denominator are defined to be zero. Moreover,
\begin{equation}
|B(\partial_i)|+|\partial_iC_{\rm nor}|\leq C_Rl_i.
\label{co:coefficients}
\end{equation}
The scalar $C_{\rm nor}$ is at least $c_0=1/10$, and
\begin{equation}
\alpha=\frac{B}{C_{\rm nor}}
\label{co:normalized}
\end{equation}
is a compatible connection family in the sense of
Definition~\ref{cy:connection}. Its coefficients and those of its
exterior derivative are bounded by polynomials in the lengths, uniformly
over the tuples. In particular, the family may be used in
Proposition~\ref{cy:trace-identity}.
\end{proposition}
\begin{proof}
Differentiating \eqref{co:smooth-intervals} at a boundary gives the smeared
impulse formula
\begin{equation}
\partial_iP_j(u)
=\sum_{a\in\Z}\eta_j(u-t_i-a)g^aJ_i.
\label{co:smeared-impulse}
\end{equation}
For each $u$, at most one summand can be nonzero, because
$4\delta_j<1$. The derivative is supported within $2\delta_j$ of
$t_i$ modulo periods. The distance between the two labels in $g^aJ_i$
is $d(h_{i-1},h_i)$, independently of $a$. The derivative estimate of
Proposition~\ref{ch:summary}, or Lemma~\ref{ch:terminal} at the last
stage, therefore bounds the square root of the pointwise sum in
\eqref{co:fisher} by
$C\epsilon_j^{-10}l_i\eta_j(u-t_i-a)$ for $j0$. Indeed, each differentiation
puts a derivative on one of the two factors. The other factor has
bounded total mass, the measure $\rho_j$ has mass at most one, and
\eqref{co:integrated-derivative} is unchanged by a time shift. The last
observation follows because the sum of absolute derivatives is a
periodic scalar function.
For the raw stage, \eqref{co:raw-coefficient} bounds
$|(B_0)_i|\leq C$, since $\|J_i\|_1\leq2$ and the lag support has length
at most $4\delta_1<1$. Its contraction satisfies the exact identity
\[
C_0=1-\int_{\rm per}K_0(P_1(u),P_1(u))\,du.
\]
Formula \eqref{co:smeared-impulse} for $P_1$, before passing through any
channel map, gives
$\int_{\rm per}\|\partial_iP_1(u)\|_1\,du\leq2$.
Hence $|\partial_iC_0|\leq4$. Summing these estimates with the
coefficients $1,\ldots,1,W$ proves \eqref{co:coefficients}, because
\[
1+\sum_{j=1}^{b_*-1}a_j+Wa_{b_*}
\]
depends only on $R$ and the fixed choices.
For completeness, polynomial control of exterior derivatives can be
obtained here without using the finer estimates of the next section.
For a smooth stage, differentiation at a fixed period cut gives
\begin{equation}
dB_j=\frac12\int_{\rm per}du\int_{\R}dv\,
U_j(v-u)K_j(dY_j(u)\wedge dY_j(v)).
\label{co:exterior-pairing}
\end{equation}
There is no term from the cut by \eqref{co:cut-cancellation}. For a
compact lag kernel, reduction of both time variables to one period
introduces at most three relevant period shifts. Boundedness of the
kernel and \eqref{co:integrated-derivative} therefore imply
\[
|dB_j(\partial_i,\partial_{i'})|
\leq Ca_j^2l_il_{i'}.
\]
For the full-sign equality kernel, at most one shift contributes for
each pair of labels, and the same estimate follows. In the raw stage,
differentiate \eqref{co:raw-coefficient} with respect to a distinct
boundary. The resulting terms have factors
$U_0(t_{i'}+a-t_i)K_0(J_i,g^aJ_{i'})$; boundedness of $U_0,K_0$ and the
jump masses gives a uniform bound. Differentiation of a coefficient in
its own time variable contributes no exterior two-form. These
calculations hold in the interior and extend to the closed simplex by
Lemma~\ref{co:regularity}. Summation over stages gives
\begin{equation}
|dB(\partial_i,\partial_{i'})|\leq C_Rl_il_{i'}.
\label{co:coarse-curvature}
\end{equation}
Finally Lemma~\ref{co:positivity} permits division by $C_{\rm nor}$.
The scalar normalizer inherits all symmetries and face restrictions of
$B$, so \eqref{co:normalized} does also, and $\alpha(V)=1$.
Equations \eqref{co:coefficients}, \eqref{co:coarse-curvature}, and
\[
d\alpha=C_{\rm nor}^{-1}dB
-C_{\rm nor}^{-2}dC_{\rm nor}\wedge B
\]
give the required polynomial coefficient bounds. The resulting family
is therefore a compatible connection family. Its curvature is basic, as
also follows directly from translation invariance and $\alpha(V)=1$.
\end{proof}
The construction separates two estimates that the trace argument will
use differently. The bounds in Proposition~\ref{co:connection} justify
the weighted simplex identities, while the scale-dependent bound
\eqref{co:fisher} retains the information needed to make high-dimensional
curvature Pfaffians small. We turn to those sharper estimates next.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/05-connection.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/06-curvature.tex ====
\section{Curvature estimates and the Pfaffian bound}
\label{sec:curvature}
We estimate the curvature of the connection constructed in
Proposition~\ref{co:connection}. The smooth stages are controlled by
operators on weighted Hilbert spaces. The raw stage is instead controlled
by the number of nearby simplex coordinates. These two estimates will be
combined only after averaging over the coordinate subsets in the
Pfaffian expansion.
Throughout this section, the word metric and the fixed construction
constants are understood. The constant $C$ may depend on these data but
not on $R,n,g$, or the tuple. A constant $C_R$ may additionally depend
on $R$. All parameters $r,W,b_*,\beta_j,\delta_j$, and $a_j$ retain
their definitions from Section~\ref{sec:connection}. In particular,
$\delta_j=\beta_j/n$ and $10\delta_{b_*}<1$.
We work in the section $t_0=0$, on the simplex
\[
\Delta_n=\{0\le t_1\le\cdots\le t_n\le1\},
\qquad n=2m.
\]
Expectation with respect to uniform simplex probability will be denoted
by $\mathbb E_{\Delta_n}$; its density in these coordinates is $n!$.
If $Q$ is a two-form, its matrix is defined by
$Q=\sum_{i10\beta_{b_*}$. Suppose either that $g$ has infinite
order, or that $g$ has finite order and
\[
\min_{h\in G}d(h,gh)>4r.
\]
For every tuple $(h_0,\ldots,h_n)$, the connection $\alpha=B/C_{\rm nor}$
of Proposition~\ref{co:connection} satisfies
\begin{equation}
\mathbb E_{\Delta_n}|\Pf(d\alpha)|
\le C_R\bigl(CS^{3/2}R^{-1/10}\bigr)^n
\prod_{i=1}^n l_i,
\qquad l_i=1+d(h_{i-1},h_i).
\label{pf:main-estimate}
\end{equation}
The constants are independent of $n,g$, and the tuple. Moreover,
$S=O(1+\log R)$ as $R\to\infty$.
\end{theorem}
We prove the theorem in three steps: estimates for principal Pfaffians
of the smooth stages, an averaged estimate for the raw stage, and the
expansion of the normalized curvature.
\subsection{Smooth stages as weighted operators}
Fix a stage $j>0$ and a point of the simplex. Let $\Lambda_j$ be its
label set and equip $(0,1)\times\Lambda_j$ with the measure
\[
d\vartheta_j(u,y)=Y_j(u)_y\,du.
\]
We use the real Hilbert space $\mathcal H_j=L^2(\vartheta_j)$. Recall that
the total mass of $Y_j(u)$ is at most two, and is at most one at the
last stage. Define the operator $T_j$ by the kernel
\begin{equation}
T_j(u,y;v,z)
=\sum_{a\in\Z}U_j(v+a-u)K_j(y,g^a z),
\label{pf:kernel}
\end{equation}
where the action of $g$ on labels is the one used in the channel
construction. Oddness of $U_j$, symmetry of $K_j$, and simultaneous
equivariance of $K_j$ show that this kernel is skew. Indeed, exchange
$(u,y)$ with $(v,z)$ and replace $a$ by $-a$ in the sum.
For $1\le i\le n$, put
\[
X_i(u,y)=\frac{\partial_iY_j(u)_y}{Y_j(u)_y}
\quad\text{when }Y_j(u)_y>0,
\]
and give $X_i$ the value zero at the remaining points. The derivatives
of the zero weights vanish, and \eqref{co:fisher} gives
\begin{equation}
\|X_i\|_{\mathcal H_j}
\le \frac{a_jl_i}{\sqrt{\delta_j}}.
\label{pf:vector-norm}
\end{equation}
The exterior-derivative identity \eqref{co:exterior-pairing} is
\[
dB_j=\frac12\int_{\rm per}du\int_{\R}dv\,
U_j(v-u)K_j\bigl(dY_j(u)\wedge dY_j(v)\bigr).
\]
The integral over a period can be taken with fixed endpoints because
its integrand is periodic. Reducing $v$ to $(0,1)$ gives
\eqref{pf:kernel}; the two antisymmetric terms then agree. Consequently,
\begin{equation}
(dB_j)_{ii'}=\langle X_i,T_jX_{i'}\rangle_{\mathcal H_j}.
\label{pf:curvature-pairing}
\end{equation}
Here is the finite-dimensional estimate we shall apply to this pairing.
Let $I$ have even size $k>0$, let $E$ be the span of the vectors $X_i$,
$i\in I$, and let $s_1\ge s_2\ge\cdots$ be the singular values of the
compression $P_ET_j|_E$, completed by zeros. Then
\begin{equation}
|\Pf_I(dB_j)|
\le \prod_{i\in I}\|X_i\|
\left(\prod_{\nu=1}^k s_\nu\right)^{1/2}.
\label{pf:compression}
\end{equation}
To see this, the matrix in \eqref{pf:curvature-pairing} has rank less
than $k$ if $\dim Et},
\]
where $t$ is the output variable and $s$ the input variable. For each
integer $N\ge1$, there is an operator $A_N$ of rank at most $N$ such
that
\[
\|A-A_N\|\le \frac{M}{2N}.
\]
Also $\|A\|\le\sqrt{\mu(\R)\nu(\R)}\le M/2$.
\end{lemma}
\begin{proof}
If $M=0$, the assertion is immediate. Otherwise partition $\R$ into
$N$ successive intervals $I_1,\ldots,I_N$, each having
$(\mu+\nu)$-measure $M/N$. Such a partition exists because the measure
is nonatomic. Endpoint conventions do not affect the operators.
Let $A_N$ retain the kernel of $A$ when $t$ and $s$ lie in different
intervals, and set the kernel to zero when they lie in the same
interval. On $I_a\times I_b$ with $a\ne b$, the order of $t$ and $s$
is fixed. The range of $A_N$ is therefore contained in the span of the
$N$ functions $\mathbf1_{I_a}$ in $L^2(\mu)$.
The remainder $A-A_N$ maps $L^2(\nu|_{I_a})$ to
$L^2(\mu|_{I_a})$ for each $a$, without mixing different intervals.
On that block its kernel is bounded by one, so Cauchy--Schwarz gives
the operator norm bound
\[
\sqrt{\mu(I_a)\nu(I_a)}
\le \frac{\mu(I_a)+\nu(I_a)}2=\frac{M}{2N}.
\]
Both the source blocks and the target blocks are orthogonal. Hence
the norm of their direct sum is the maximum of their norms, proving
the approximation claim. Applying the same Cauchy--Schwarz estimate
on the whole line gives the asserted bound on $\|A\|$.
\end{proof}
\begin{lemma}[Singular values at the last stage]
\label{pf:terminal-singular}
For the operator $T_{b_*}$, every finite-dimensional orthogonal
compression has singular values satisfying
\begin{equation}
s_\nu\le \frac C\nu,\qquad \nu\ge1.
\label{pf:singular-decay}
\end{equation}
The constant is independent of $n,R,g$, and the tuple, in either of
the two cases of Theorem~\ref{pf:bound}.
\end{lemma}
\begin{proof}
We will approximate $T_{b_*}$, for every $x>0$, by an operator of rank
at most $C/x$ and norm error at most $x$.
First suppose that $g$ has infinite order. The final label set is $G$
and the label kernel is equality. It follows that
$\mathcal H_{b_*}$ splits orthogonally according to the left
$g$-orbits in $G$. In one orbit, choose a representative $y_0$ and
write a label as $g^a y_0$, $a\in\Z$. Associate to the pair
$(u,g^a y_0)$ the real coordinate $u-a$. If the second label is
$g^b y_0$, the equality kernel in \eqref{pf:kernel} selects precisely
the shift $a-b$. The resulting kernel is
\[
\operatorname{sign}(v+a-b-u)
=\operatorname{sign}\bigl((v-b)-(u-a)\bigr).
\]
Thus this orbit gives a sign operator on a nonatomic measure on the
line. Let $b$ be the mass of that measure. The orbit masses sum to
at most one, because the final channel is a subprobability channel.
Lemma~\ref{pf:order-kernel}, with the same measure in source and
target, gives norm at most $b$ and a rank-$N$ approximation with error
at most $b/N$.
Discard each orbit block with $b\le x$. On every other block choose
$N=\lceil b/x\rceil$. The error of the resulting direct-sum
approximation is at most $x$. Moreover,
\[
\sum_{b>x}\lceil b/x\rceil
\le \frac1x\sum_{b>x}b+\#\{b>x\}
\le \frac2x.
\]
This proves the desired rank bound in the infinite-order case.
Now suppose that $g$ has finite order. Put
$d\mu_y(u)=Y_{b_*}(u)_y\,du$ on $(0,1)$, extend this measure by zero
to $\R$, and write $\mathcal H_y=L^2(\mu_y)$.
Since the final lag kernel is
$\operatorname{sign}(s)\mathbf1_{|s|<1}$, the only possible shifts for
$0u}\mathbf1_{y=g^{-1}z}.
\end{split}
\label{pf:three-shifts}
\end{equation}
The shifted contributions are direct sums of maps
$\mathcal H_z\to\mathcal H_{gz}$ and
$\mathcal H_z\to\mathcal H_{g^{-1}z}$, respectively. After reindexing
the target summands, each is an orthogonal direct sum: its source
blocks are mutually orthogonal and its target blocks are mutually
orthogonal. The identity contribution has the same property.
This remains true even though the measures in a paired block need
not coincide; the shifted summands need not be invariant subspaces.
For a paired block, let $b=\mu_y(\R)+\mu_z(\R)$. By
Lemma~\ref{pf:order-kernel}, its norm is at most $b$, and it has a
rank-$N$ approximation with error at most $b/N$. The sums of these
$b$'s over a single contribution are at most two: every label mass
appears once as a source mass and once as a target mass. Discarding
blocks of mass at most $\varepsilon$ and taking
$N=\lceil b/\varepsilon\rceil$ on the others therefore gives error at
most $\varepsilon$ and total rank at most $4/\varepsilon$. Use
$\varepsilon=x/3$ for each contribution in \eqref{pf:three-shifts}
and add the three approximations. Their total error is at most $x$
and their total rank is at most $C/x$. If $g$ has order two, the two
shift permutations coincide; the separate estimates still apply and
are added in exactly the same way.
Finally, orthogonal compression cannot increase either the rank or
the error of an approximation. Singular values beyond the dimension
of the compressed space are zero, so consider an index $\nu$ no larger
than that dimension. If the compressed matrix has an approximation
of rank less than $\nu$ with error $x$, its $\nu$th singular value is
at most $x$. Indeed, the span of its first $\nu$
right singular vectors contains a unit vector annihilated by the
approximating matrix, and the original matrix has norm at least
$s_\nu$ on that vector. If the rank bound above is $C_0/x$, take
$x=2C_0/\nu$. The approximating rank is then at most $\nu/2<\nu$,
which proves \eqref{pf:singular-decay}.
\end{proof}
For an even principal set $I$ of size $k>0$, insert
\eqref{pf:singular-decay} in \eqref{pf:compression}. Multiplication
of the two-form by $W$ multiplies its size-$k$ Pfaffian by $W^{k/2}$.
We obtain
\[
|\Pf_I(WdB_{b_*})|
\le W^{k/2}\left(\frac{a_{b_*}}{\sqrt{\delta_{b_*}}}\right)^k
\frac{C^{k/2}}{\sqrt{k!}}\prod_{i\in I}l_i.
\]
For $k\ge1$, integration of $\log x$ gives
$\log(k!)\ge\int_1^k\log x\,dx\ge k\log k-k$, and hence
$k!\ge(k/e)^k$. We conclude that
\begin{equation}
|\Pf_I(WdB_{b_*})|
\le\bigl(Ca_{b_*}\sqrt W\,\beta_{b_*}^{-1/2}\bigr)^k
(n/k)^{k/2}\prod_{i\in I}l_i.
\label{pf:terminal}
\end{equation}
The estimates \eqref{pf:intermediate} and \eqref{pf:terminal} control
every smooth stage at every point of the simplex. It remains to
control the raw stage in the average required for the full Pfaffian.
\subsection{The raw stage and nearby coordinates}
Recall $J_i=\delta_{h_{i-1}}-\delta_{h_i}$. The coefficient of $dt_i$
in $B_0$ is
\[
-\frac12\int_{\R}U_0(v-t_i)K_0(J_i,Y_0(v))\,dv.
\]
For distinct $i,i'$ in the interior of the simplex, differentiating
at the boundary $t_{i'}$ and antisymmetrizing gives
\[
(dB_0)_{ii'}
=\sum_{a\in\Z}U_0(t_{i'}+a-t_i)K_0(J_i,g^aJ_{i'}).
\]
There is at most one nonzero summand. Since
$K_0(h,h')=\chi_R(d(h,h'))$ and $\chi_R$ is $1/R$-Lipschitz,
\[
|K_0(J_i,g^aJ_{i'})|
\le\frac{2d(h_{i-1},h_i)}R
\le\frac{2l_i l_{i'}}R.
\]
The support of $U_0$ also shows that this matrix entry is zero if the
circular distance between $t_i$ and $t_{i'}$ exceeds $2R/n$.
For $I\subset\{1,\ldots,n\}$ and $i\in I$, let $N_i(I)$ be the
number of indices in $I$, including $i$ itself, whose times have
circular distance at most $2R/n$ from $t_i$. If $k=|I|$ is even,
divide row and column $i$ of the principal curvature matrix by $l_i$.
Each resulting row has Euclidean norm at most
$(C/R)\sqrt{N_i(I)}$. Hadamard's inequality, followed by the
Pfaffian--determinant identity, yields
\begin{equation}
|\Pf_I(dB_0)|
\le (C/R)^{k/2}\prod_{i\in I}l_i
\prod_{i\in I}N_i(I)^{1/4}.
\label{pf:raw}
\end{equation}
We next average the last product. The needed assertion is an average
over subsets of a fixed size; it is not a bound asserted separately
for each fixed subset of the ordered coordinates.
\begin{lemma}[Average of the neighbor counts]
\label{pf:occupancy}
Let $R\ge1$, $n\ge8R$, and $0\le k\le n$. Choose
$(t_1,\ldots,t_n)$ with uniform probability on $\Delta_n$, and
independently choose $I$ uniformly among the $k$-element subsets of
$\{1,\ldots,n\}$. Then
\begin{equation}
\mathbb E_{\Delta_n,I}\prod_{i\in I}N_i(I)^{1/4}
\le C^k R^{k/4},
\label{pf:count-average}
\end{equation}
with the empty product interpreted as one.
\end{lemma}
\begin{proof}
The statement is immediate for $k=0$. Uniform simplex coordinates
are the order statistics of $n$ independent uniform points in
$(0,1)$. Selecting a uniform set of $k$ ranks is equivalent, as an
unordered set of points, to selecting a uniform set of $k$ of those
independent points. The selected unordered set therefore has the
law of $k$ independent uniform points on the circle. Notice that the
fixed endpoint $t_0=0$ is not one of the selectable coordinates.
Partition the circle into $m'=\lfloor n/(2R)\rfloor$ equal cells.
Their length is at least $2R/n$, and $m'\ge n/(4R)$. Let $p_a$ be
the number of selected points in cell $a$, with cell indices read
cyclically. A point in cell $a$ can have counted neighbors only in
that cell or its two adjacent cells. Consequently
\[
\prod_{i\in I}N_i(I)^{1/4}
\le\prod_{a:p_a>0}(p_a+p_{a-1}+p_{a+1})^{p_a/4}.
\]
The comparison with the product of the occupancies themselves follows
from
\[
\begin{split}
\sum_{a:p_a>0}p_a
\log\frac{p_a+p_{a-1}+p_{a+1}}{p_a}
&\le\sum_{a:p_a>0}(p_{a-1}+p_{a+1})\\
&\le2k.
\end{split}
\]
Thus the preceding product is at most
$e^{k/2}\prod_a p_a^{p_a/4}$, with $0^0=1$.
The occupancy vector has multinomial probabilities
$k!(m')^{-k}\prod_a(p_a!)^{-1}$. The factorial lower bound implies
$p^{p/4}\le e^{p/4}(p!)^{1/4}$, so
\[
\mathbb E\prod_a p_a^{p_a/4}
\le e^{k/4}\frac{k!}{(m')^k}
\sum_{\sum p_a=k}\left(\prod_a\frac1{p_a!}\right)^{3/4}.
\]
There are $\binom{k+m'-1}{k}$ nonnegative occupancy vectors. Applying
H\"older's inequality to the last sum, and then the multinomial
theorem, gives
\[
\sum_{\sum p_a=k}\left(\prod_a\frac1{p_a!}\right)^{3/4}
\le
\left(\frac{(m')^k}{k!}\right)^{3/4}
\binom{k+m'-1}{k}^{1/4}.
\]
After substitution, the remaining factor apart from $C^k$ is
\[
\left[\frac{k!}{(m')^k}\binom{k+m'-1}{k}\right]^{1/4}
=\left[\prod_{q=0}^{k-1}(1+q/m')\right]^{1/4}
\le(1+k/m')^{k/4}.
\]
Finally, $k\le n$ and $m'\ge n/(4R)$ give $k/m'\le4R$.
Since $R\ge1$, all factors are bounded as in
\eqref{pf:count-average}.
\end{proof}
Combining \eqref{pf:raw} with Lemma~\ref{pf:occupancy} gives a factor
of size $(CR^{-1/4})^k$, once the length factors have been extracted
and the coordinate subset has been averaged. The remaining task is
to combine this subset average with the pointwise estimates for the
smooth stages. We first make their scale dependence uniform.
\subsection{Expansion of the normalized curvature}
First record the dependence of the preceding estimates on $R$.
Since $r$ is a fixed multiple of $R$, the stopping condition compares
\[
\log\beta_j=(3/2)^{j-1}\log R,
\qquad
\log(W^2R^{100})=O(R+\log R).
\]
Taking $j$ of order $1+\log R$ makes the first expression at least
the second. As in \eqref{co:stage-count}, minimality therefore gives
\begin{equation}
b_*=O(1+\log R).
\label{pf:stage-count}
\end{equation}
For $j0$ from
Proposition~\ref{co:connection} changes the bases in
\eqref{pf:small-base} only by a fixed factor; enlarge $C_1$
accordingly. The last term in \eqref{pf:normalized-curvature} is
decomposable, so $Q\wedge Q=0$. Moreover,
\eqref{co:coefficients} implies
\begin{equation}
|Q_{ii'}|\le C_Rl_i l_{i'}.
\label{pf:rank-two}
\end{equation}
In particular, this stage can contribute to a Pfaffian partition
only on zero or two indices. The constant $C_R$ in
\eqref{pf:rank-two} is independent of $n$.
There are $S=b_*+2$ stages in \eqref{pf:normalized-curvature}:
the raw stage, the $b_*$ smooth stages, and $Q$. Denote their matrices
by $M_0,\ldots,M_{S-1}$, with $M_{S-1}=Q$. The Pfaffian expansion is
\begin{equation}
\Pf\left(\sum_{j=0}^{S-1}M_j\right)
=\sum_{I_0\sqcup\cdots\sqcup I_{S-1}=\{1,\ldots,n\}}
\sigma(I_0,\ldots,I_{S-1})
\prod_{j=0}^{S-1}\Pf_{I_j}(M_j),
\label{pf:partition-expansion}
\end{equation}
where all $|I_j|$ are even and each sign $\sigma$ is $1$ or $-1$.
There is no multiplicity factor for a fixed ordered partition. To
check this normalization, expand the divided exterior power
$(\sum_jM_j)^m/m!$ as
$\sum_{\sum m_j=m}\prod_j M_j^{m_j}/m_j!$, and then extract the
coordinate volume coefficient using \eqref{pf:normalization} on
each coordinate subset.
Take absolute values in \eqref{pf:partition-expansion}. In each
term the length factors multiply to $\prod_{i=1}^nl_i$, independently
of the partition. Fix now the even sizes $k_j=|I_j|$. For any given
raw subset $I_0$ of size $k_0$, the number of completions to a partition
of these sizes is
\[
\frac{(n-k_0)!}{\prod_{j=1}^{S-1}k_j!},
\]
which is independent of $I_0$. Every factor other than the raw
neighbor counts has already been bounded independently of the
simplex point and of the actual subset. Thus, upon summing over
these partitions and integrating over the simplex, the count product
in \eqref{pf:raw} is averaged uniformly over all raw subsets of
size $k_0$. Lemma~\ref{pf:occupancy} applies without any assertion
about an individual fixed subset.
The smooth estimates also contain factors
$(n/k_j)^{k_j/4}$ or $(n/k_j)^{k_j/2}$. Their product is at most
\begin{equation}
\prod_{j:k_j>0}(n/k_j)^{k_j/2}\le S^{n/2}.
\label{pf:entropy}
\end{equation}
For completeness, put $p_j=k_j/n$ on the positive sizes. Concavity
of the logarithm gives
\[
\sum_jp_j\log(1/p_j)
\le\log\left(\sum_jp_j/p_j\right)\le\log S;
\]
exponentiating $n/2$ times this inequality proves
\eqref{pf:entropy}. Factors for the raw and decomposable stages may
be inserted here because they are at least one.
Let $q=k_{S-1}$. A nonzero term has $q=0$ or $q=2$. The small bases
in the other stages give $\varepsilon_R^{n-q}$, while the
decomposable stage costs at most $C_R$. Since
$0<\varepsilon_R\le1$, both possibilities are bounded by
$C_R\varepsilon_R^n$ after absorbing the single factor
$\varepsilon_R^{-2}$ into $C_R$. This absorption introduces no
dependence on $n$.
For fixed sizes, the number of ordered partitions is
$n!/\prod_j k_j!$. Summed over all admissible sizes this is at most
$S^n$, the number of all assignments of the $n$ indices to $S$
stages. Combining this count, \eqref{pf:entropy}, and the preceding
uniform subset average proves
\[
\mathbb E_{\Delta_n}|\Pf(d\alpha)|
\le C_R\varepsilon_R^n S^{3n/2}\prod_{i=1}^n l_i.
\]
Substitution of \eqref{pf:small-base} proves
\eqref{pf:main-estimate}, and \eqref{pf:stage-count} supplies the
assertion about $S$. This completes the proof of
Theorem~\ref{pf:bound}.
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/06-curvature.tex ====
==== BEGIN The-l1-Bass-Conjecture-for-Discrete-Groups//sections/07-conclusion.tex ====
\section{Proof of the trace-support theorem}\label{sec:conclusion}
We now sum the curvature estimate against the cyclic tuple weights.
The order of the two limiting choices is essential: the spatial radius
is fixed first, uniformly in the conjugacy class, and only then does
the simplex dimension tend to infinity.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Proposition~\ref{red:reduction}, it suffices to treat a finitely
generated group with a fixed word metric and an idempotent $p$ having
an exponential moment. The zero idempotent is immediate, so assume
$M_p>0$, with $M_p$ as in Lemma~\ref{cy:weights-lemma}.
For the construction of Sections~\ref{sec:connection}
and~\ref{sec:curvature}, write
\[
Q_R=C S^{3/2}R^{-1/10},\qquad S=b_*+2.
\]
The fixed constants here may depend on the word metric, but not on
$g$, the tuple, or $n$. Since $S=O(1+\log R)$, we have $Q_R\to0$.
Choose $R$ once and for all so large that the construction applies and
\begin{equation}\label{final:radius-choice}
Q_RM_p<1.
\end{equation}
This fixes the final spatial radius $r=6^{J_*-1}R$, the scale ladder,
and all constants denoted $C_R$.
Let $g$ have infinite order, or have finite order with
$\min_h d(h,gh)>4r$. Proposition~\ref{co:connection} provides a
compatible connection family through every sufficiently large even
dimension $n$. Combining Proposition~\ref{cy:trace-identity},
Theorem~\ref{pf:bound}, and \eqref{cy:moment}, we obtain
\begin{align}
|\tau_{[g]}(p)|
&\le \sum_{h\in\mathcal T_n(g)}|v_{n+1}(h)|
\mathbb E_{\Delta_n}|\Pf(d\alpha_h)|\notag\\
&\le C_RQ_R^n\sum_{h\in\mathcal T_n(g)}|v_{n+1}(h)|
\prod_{i=1}^n l_i\notag\\
&\le C_RM_p\,(Q_RM_p)^n.\label{final:decay}
\end{align}
In the last step we used $l_0\ge1$ to insert the missing closing-edge
factor. The constant $C_R$ is independent of $n$ and $g$.
Letting $n$ tend to infinity through even integers in
\eqref{final:decay} gives $\tau_{[g]}(p)=0$.
It remains to identify the possible surviving classes. Every
infinite-order class has just been eliminated. If a finite-order class
does not satisfy the displacement condition, some conjugate of its
representative has word length at most $4r$, because
\[
d(h,gh)=|h^{-1}gh|.
\]
Thus each surviving class meets the finite ball $B(1,4r)$. There are
only finitely many such classes. This proves the result for $p$.
Proposition~\ref{red:reduction} transfers it to the original $e$ and
group. Finally, an element of $K_0(\ell^1(G))$ is a difference of
idempotent classes; taking the union of their two finite supports
proves the asserted inclusion for $\operatorname{HS}^1$.
\end{proof}
==== END The-l1-Bass-Conjecture-for-Discrete-Groups//sections/07-conclusion.tex ====
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