\documentclass[11pt,a4paper]{article}
\usepackage[margin=28mm]{geometry}
\usepackage[T1]{fontenc}
\usepackage{lmodern,microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{enumitem}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,decorations.pathreplacing,positioning}
\usepackage{caption}
\captionsetup{font=small,labelfont=bf}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\hypersetup{
 pdftitle={Finite-length permutation modules for free-amalgamation structures of arbitrary arity},
 pdfauthor={Andr\'as Pongr\'acz}
}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\vsupp}{\operatorname{vsupp}}
\newcommand{\length}{\operatorname{length}}
\newcommand{\Bal}{\operatorname{Bal}}
\newcommand{\Cog}{\operatorname{Cog}}
\newcommand{\F}{\mathbb F}
\newcommand{\A}{\mathbb A}
\newcommand{\K}{\mathcal K}
\newcommand{\OO}{\mathcal O}
\newcommand{\PP}{\mathcal P}
\newcommand{\aset}[1]{\underline{#1}}
\newcommand{\Hh}{H_S}
\title{Finite-length permutation modules for\\free-amalgamation structures of arbitrary arity}
\author{Andr\'as Pongr\'acz\\[2pt]
\small University of Debrecen, Institute of Mathematics\\
\small ORCID: \href{https://orcid.org/0000-0002-2771-8974}{0000-0002-2771-8974}}
\date{10 November 2026}
\begin{document}
\maketitle

\begin{abstract}
Let $\A$ be the countable homogeneous limit of a free-amalgamation
class in a finite relational language, expanded by a generic linear order.
We prove that its tuple permutation modules have finite composition
length over every field, also after naming finitely many points.
More precisely, each orbit of increasing $d$-tuples disjoint from
the named points has a permutation module of length $2^d$. This extends the theorem of Yang--Boja\'nczyk--Klin from unary
and binary relations to relations of arbitrary arity. The main step
shows that every finitely supported array whose sums over each single
coordinate vanish is a linear combination of alternating sums obtained
by independently doubling its coordinates. Its proof combines a Ramsey separation theorem with
a construction that moves finite configurations into prescribed order
intervals. Over a finite field, we also obtain finitely many closed
invariant linear subspaces of $\F^{A^k}$; each is generated, in the
product topology, by functions definable using at most $k$ additional
parameters.
\end{abstract}

\section{Introduction}

A permutation group acting on an infinite set gives rise to an
infinite-dimensional vector space with a distinguished basis. The
subspaces preserved by the group can nevertheless have a finite and
rigid structure. For example, under the full symmetric group on an infinite set
$X$, the space $\F[X]$ of finite formal linear combinations over
a field $\F$ has exactly one nonzero proper invariant subspace:
the vectors whose coefficients sum to zero.
Indeed, swapping a point with nonzero coefficient with a point
outside the support and subtracting shows that every nonzero
invariant subspace contains all differences of basis vectors;
these span the coefficient-sum kernel, which has codimension one,
so $\F[X]$ has composition length two
\cite[Example~4.2]{BFKM}.

For an automorphism group of a homogeneous structure, tuples provide
a natural supply of permutation modules. Their invariant subspaces
encode linear relations compatible with the structure's symmetries.
Finite composition length asks whether the number of strict steps
in a chain of such subspaces is bounded. This question has arisen
both in the theory of infinite permutation groups and in the study
of computation over infinite alphabets; see Evans~\cite[Section~1.1]{Evans}
and Boja\'nczyk--Fijalkow--Klin--Moerman~\cite{BFKM}.
The latter authors prove finite length over arbitrary fields for
the pure countable set and the countable dense linear order, including
finite parameter expansions \cite[Theorems~4.8, 4.10 and
Corollary~4.9]{BFKM}.

Yang--Boja\'nczyk--Klin~\cite[Theorem~7.3]{YBK} extend these results
to homogeneous structures obtained from free-amalgamation classes in
finite unary and binary relational languages. Their theorem allows
a generic linear order and finitely many named parameters, and it
holds over every field. It covers, for instance, the random graph
and the Henson graphs, as well as their generic ordered expansions.
They also determine the exact length of an increasing tuple orbit
and describe invariant subspaces by coordinate sums
\cite[Theorem~8.14 and Corollary~8.16]{YBK}. They leave the removal
of the restriction on relation arity open \cite[Remark~7.4]{YBK}.
The purpose of this paper is to remove that restriction; see Theorem~\ref{thm:main}.

For background on homogeneous structures, $\omega$-categoricity,
automorphism groups and structural Ramsey theory, we refer to
Bodirsky's monograph \cite[Chapters~2, 4, 5, 9 and~11]{BodirskyBook}.

\subsection{Setting and main results}

Let $L$ be a finite relational language and $\K$ a Fra\"iss\'e class
of finite $L$-structures with free amalgamation. Here free amalgamation
means the following. If $B,C\in\K$ have a common induced substructure
$D$, their union over $D$ belongs to $\K$ when its basic relations
are exactly those already present in $B$ or $C$. In particular, no
relation tuple is added whose entries meet both $B\setminus D$
and $C\setminus D$. For graphs, this is the familiar operation of
gluing over an induced subgraph and adding no edges between the two
new parts. The same definition applies to relations of any arity.

Let $\K^{<}$ consist of all expansions of members of $\K$ by a
new linear order relation $<$, and let $\A$ be its countable Fra\"iss\'e limit, with
domain $A$. We call this the \emph{generically ordered limit}.
The word ``generic'' means that every linear order on every finite
member of $\K$ is admitted.

Free amalgamation is required only for the original language $L$;
the added order relation $<$ is treated separately. After forming
the free $L$-amalgam of two ordered structures over their common
ordered substructure, we equip it with a linear order extending
the orders on both pieces. Every such extension is allowed.
For example, points from opposite pieces lying in the same gap
of the common ordered substructure may be interleaved, provided
the order within each piece is preserved. The resulting amalgam
has order comparisons between the two new parts, but no basic
$L$-relation instance meeting both parts. Choosing such an order
in a particular construction places no further restriction on
the class $\K^{<}$.

No bound on the arity of the symbols of $L$ is assumed, and
relation tuples may have repeated entries. The underlying
unordered limit will be denoted by $\A_0$.

We will repeatedly use the \emph{finite extension property} of
the Fra\"iss\'e limit; see \cite{BodirskyBook}. If $B$ is a
finite induced substructure of $\A$ and $C\in\K^{<}$ contains
$B$ as an induced substructure, then there is an embedding
$C\to\A$ fixing $B$ pointwise. Indeed, universality first
gives an embedding of $C$ into $\A$. Homogeneity extends the
inverse of its restriction to $B$ to an automorphism of $\A$;
composing with that automorphism gives an embedding fixing $B$.

Let $\F$ be a field. For a finite set $S\subseteq A$, write
\[
 H_S=\{g\in\Aut(\A):g(s)=s\text{ for every }s\in S\}.
\]
For any $H_S$-set $X$, the \emph{permutation module} $\F[X]$ is the
vector space with basis $\{[x]:x\in X\}$ and action
$g[x]=[g(x)]$, extended linearly. Its \emph{composition length}
$\length_{H_S}\F[X]$ is the supremum of the set of integers $n$ for which
there is a chain of invariant linear subspaces
\[
 \{0\}=W_0<W_1<\cdots<W_n=\F[X].
\]
An \emph{increasing orbit of degree $d$ over $S$} is an $H_S$-orbit
$\OO\subseteq A^d$ consisting of tuples
$a_1<\cdots<a_d$ with all $a_i\notin S$. The degree-zero orbit
consists of the empty tuple.

\begin{theorem}\label{thm:main}
Let $\A$ be as above. For every field $\F$, every finite set
$S\subseteq A$, and every increasing orbit $\OO$ of degree $d$ over $S$,
\[
 \length_{H_S}\F[\OO]=2^d.
\]
Consequently $\F[A^k]$ has finite composition length for every $k$,
also after naming finitely many parameters. The same finite-length
conclusion holds for the pointwise stabilizer $\Aut(\A_0)_{(S)}$
acting on $\F[A^k]$.
\end{theorem}

The theorem applies, in particular, to generic ordered $r$-uniform
hypergraphs for every finite $r$. In degree two it gives length four;
in degree three it gives length eight. The independence of the answer
from the relation arities is part of the conclusion. Different tuple
orbits can have different relational diagrams, but each increasing
orbit of the same degree has the same length.

There is also a dual formulation over a finite field. Give
$\F^{A^k}$, the space of all functions $A^k\to\F$, the product
topology, with $\F$ discrete. The group action is
$(g f)(a)=f(g^{-1}a)$. A closed invariant linear subspace of this
space is often called an \emph{invariant linear code}.
A function is \emph{fixed over $S\cup T$} if it is invariant under
$H_{S\cup T}$. There are finitely many tuple orbits over any finite parameter set,
each defined by its finite relational diagram. For finite $\F$,
invariance is therefore equivalent to definability of the fibres
using parameters from $S\cup T$.

\begin{corollary}\label{cor:codes}
For every finite set $S\subseteq A$, every finite field $\F$,
and every integer $k\ge0$, there are only finitely many closed
$H_S$-invariant linear subspaces of $\F^{A^k}$.
Each such subspace is the closed linear span of its functions fixed
over $S\cup T$, where $T\subseteq A$ ranges over sets of size at most $k$.
\end{corollary}

The bound is on the number of additional parameters: different
spanning functions may use different sets $T$. The field, the
structure, $S$, and $k$ are fixed in the finiteness assertion.

This is a consequence of the marginal description of
\cite[Theorem~8.14]{YBK}, transferred to arbitrary arity below,
together with the duality in \cite[Lemma~2.2]{Evans}.

\subsection{Method and relation to earlier work}

The proof begins with coefficient sums. If $v$ is a finite linear
combination of tuples, summing its coefficients over a deleted
coordinate produces a vector on shorter tuples. Call $v$
\emph{balanced} if all these sums vanish. In degree two, this says
that a finite matrix has zero row sums and zero column sums.
Alternating sums around suitable doubled configurations are elementary
examples of balanced vectors. The terminology of balanced vectors
and \emph{cogs} is used for ordered atoms in
\cite[Section~4.1]{BFKM} and generalized to the free-amalgamation
setting with unary and binary relations in \cite[Section~8]{YBK}. Section~\ref{sec:cogs} gives the
definitions used here and illustrates them in degrees one and two.

The central step is to prove that these cogs span all balanced
vectors, in every degree. For binary structures this is
\cite[Theorem~8.12]{YBK}. Our proof adapts the slice decomposition
and the separation of coordinate positions in
\cite[Appendix~E]{YBK}. We extend these steps to relations of
arbitrary arity and obtain replacement points in prescribed
compatible order cuts. We combine this construction with Evans's
Ramsey separation theorem \cite[Corollary~1.5]{Evans}. The latter
provides a separating linear functional invariant over finitely
many parameters. We move the support of a hypothetical counterexample
into fixed order intervals over those parameters, without changing
its class modulo cogs. Free amalgamation then makes the separator's
value a sum of functions of proper coordinate subtuples. Every such
function annihilates a balanced vector, giving a contradiction.

Sections~\ref{sec:ramsey}--\ref{sec:finish-cogs} prove this
extension of cog generation. Section~\ref{sec:profiles} then
checks the higher-arity compatibility needed to transfer YBK's
reconstruction proof \cite[Theorem~8.14 and Appendix~F.1]{YBK},
and states the resulting marginal description. The exact-length
argument is also inherited from \cite[Corollary~8.16 and
Appendix~F.2]{YBK}; Section~\ref{sec:consequences} cites that
argument and derives the closed-function consequences.

For those consequences, we use the standard duality recorded in
\cite[Lemma~2.2]{Evans} and the finite-dimensional separation
method in the proof of \cite[Corollary~2.3]{Evans}.
Applied to YBK's marginal evaluations, this method gives
generating functions fixed over the entries of a projected
tuple, hence over at most $k$ additional parameters.
This bound and the finite-field finiteness assertion already
follow from YBK's marginal description in the binary setting.
Their extension here rests on the arbitrary-arity cog-generation
theorem; the duality and marginal arguments are inherited.

The model-theoretic motivation concerns first-order reducts.
A \emph{first-order reduct} of a structure $M$ is a relational
structure on the same domain whose basic relations are definable
without parameters in $M$. Two structures are \emph{first-order
interdefinable} if each is a first-order reduct of the other.
Thomas's conjecture \cite[p.~177]{Thomas} asserts that a countable
structure homogeneous in a finite relational language has only
finitely many first-order reducts up to this equivalence.

Several complete classifications concern examples within the
present setting. Cameron showed that $(\mathbb Q,<)$ has five
reducts \cite{Cameron}, and Thomas proved the corresponding
five-reduct theorem for the random graph \cite{Thomas}.
Thomas subsequently classified the reducts of the random
$r$-uniform hypergraph for each fixed $r\ge2$; see \cite{ThomasHypergraphs}. 
For the random ordered graph,
Bodirsky--Pinsker--Pongr\'acz classified all reducts, finding
42 proper nontrivial ones, besides the original structure
and its equality reduct \cite{BPP42}.
Named parameters also lead to nontrivial finite lattices:
Junker--Ziegler classified the reducts of $(\mathbb Q,<)$ expanded
by finitely many named constants \cite{JunkerZiegler}, and
Pongr\'acz classified the reducts of the Henson graphs with
one named vertex \cite{PongraczHenson}.
All these classifications are up to first-order interdefinability.

General finiteness theorems are also known for broad classes of
structures. To state one such result, let $o_n^{\ne}(M)$ be the
number of $\Aut(M)$-orbits on $n$-tuples of pairwise distinct
elements. Bodirsky--Bodor \cite[Theorem~6.37]{BBGrowth} proved that
a countable structure $M$ has finitely many first-order reducts
up to interdefinability if
\[
 o_n^{\ne}(M)\le c\,n^{\alpha n}\qquad(n\ge1)
\]
for some constants $c>0$ and $\alpha<1$.
Bodor \cite[Theorem~5.11]{BodorMonadic} extends this finiteness
result to all countable $\omega$-categorical monadically stable
structures. 
These results establish the conjectured finiteness for entire
classes, without a separate classification of the reducts of
each individual structure.

The connection with permutation modules is explained by Evans:
failure of finite length over a finite field for a structure
homogeneous in a finite relational language would yield a
counterexample to Thomas's conjecture \cite[Remark~1.3]{Evans}.
Our result establishes this finite-length condition throughout
the free-amalgamation setting. Determining all reducts of these
structures requires additional arguments.

A second motivation concerns closed normal subgroups.
Macpherson \cite[Question~2.2.7(4)]{MacphersonSurvey} asks
whether the automorphism group of every countable structure
homogeneous in a finite relational language has only finitely
many closed normal subgroups, where closure is taken in the
topology of pointwise convergence. Evans observes that a failure
of finite composition length over a finite field would also
yield a counterexample to this question
\cite[Remark~1.3]{Evans}.
Classical positive cases include the random graph, whose
automorphism group was proved simple by Truss \cite{TrussGraph}.
For $(\mathbb Q,<)$, the work of Higman \cite{Higman} and
Holland \cite{Holland}, as stated in \cite[Theorem~1]{Eisenbud},
gives exactly three proper nontrivial normal subgroups:
the automorphisms fixing some final interval pointwise,
those fixing some initial interval pointwise, and their
intersection. All three are dense, so $\Aut(\mathbb Q,<)$
is topologically simple, although it is not simple as an
abstract group.
Broader simplicity results apply to free-amalgamation limits.
Macpherson--Tent \cite[Theorem~1.1]{MacphersonTent} prove that
$\Aut(M)$ is simple when $M$ is countably infinite and
homogeneous, its age has free amalgamation, and $\Aut(M)$
is transitive on $M$ but is not the full symmetric group
$\operatorname{Sym}(M)$.
Calderoni--Kwiatkowska--Tent \cite[Theorem~1.3]{CKT}
prove simplicity for the generic linear order expansions
of these structures. These results also cover the Henson
graphs and generic $r$-uniform hypergraphs for $r\ge2$,
together with their generic ordered expansions. In all
these cases the only closed normal subgroups are the
identity subgroup and the whole automorphism group.

\section{Coordinate sums and doubled configurations}\label{sec:cogs}

We fix a finite set $S\subseteq A$, a field $\F$, and an
increasing $H_S$-orbit $\OO\subseteq A^d$ of degree $d\ge0$. For a tuple $a$, let $\aset a$ be the set of its entries.
For $v=\sum_{a\in\OO}v(a)[a]\in\F[\OO]$, distinguish the
\emph{tuple support} and the \emph{vertex support}:
\[
 \supp(v)=\{a:v(a)\ne0\},\qquad
 \vsupp(v)=\bigcup_{a\in\supp(v)}\aset a.
\]
Both are finite. Homogeneity implies that two tuples are in the same
orbit over $S$ precisely when the map between their coordinates,
together with the identity on $S$, is a finite partial isomorphism.
In particular, the full relational diagram over $S$ is fixed on $\OO$.

\subsection{Marginals and balanced vectors}

Write $[d]=\{1,\ldots,d\}$, with $[0]=\varnothing$.
For $J\subseteq[d]$ and $a\in\OO$, let $p_J(a)$ be the
subtuple in the positions indexed by $J$, in increasing order
of indices. Its image $p_J(\OO)$ is again an increasing orbit
over $S$. We also write
\[
 p_J:\F[\OO]\longrightarrow\F[p_J(\OO)]
\]
for the linear extension. For $v\in\F[\OO]$ and
$b\in p_J(\OO)$, it is given by
\begin{equation}\label{eq:marginal}
 p_J(v)=\sum_{a\in\OO}v(a)[p_J(a)],\qquad
 (p_Jv)(b)=\sum_{\substack{a\in\OO\\p_J(a)=b}}v(a).
\end{equation}
Thus coefficients with the same projected tuple are added. We call
$p_Jv$ a \emph{coordinate marginal}. The empty marginal is identified
with the scalar $p_\varnothing(v)=\sum_a v(a)$.
Put $p_{-i}=p_{[d]\setminus\{i\}}$.

\begin{definition}
For $d>0$, the space of balanced vectors is
\[
 \Bal(\OO)=\bigcap_{i=1}^d\ker p_{-i}.
\]
\end{definition}

For ordered atoms, this is the balancedness condition of
\cite[Section~4.1, equation~(4)]{BFKM}; its formulation on tuple
orbits follows \cite[Section~8.1]{YBK}.

Every proper marginal of a balanced vector vanishes: a projection
onto a proper subset of positions factors through at least one of
the one-coordinate deletions. All these maps commute with $H_S$,
so $\Bal(\OO)$ is invariant.

\begin{example}\label{ex:matrix}
For $d=1$, balanced means simply that the coefficient sum is zero.

For $d=2$, we can arrange the coefficients of a vector
\[
 v=\sum_{(x,y)\in\OO}\lambda_{x,y}[(x,y)]
\]
in a table: put $\lambda_{x,y}$ in row $x$, column $y$, and put
zero in every position $(x,y)\notin\OO$. Denote the resulting
matrix by $M$. Its rows and columns are indexed by $A$, but only
finitely many entries are nonzero. Thus a finite portion of the
matrix contains all the nonzero coefficients.

The projection $p_{\{1\}}$ keeps the first coordinate and forgets
the second. In particular, both $[(x,y)]$ and $[(x,z)]$ are sent
to $[x]$, so their coefficients are added. More generally,
\[
 p_{\{1\}}v
 =\sum_{x\in p_{\{1\}}(\OO)}
     \left(\sum_{y\in A}M_{x,y}\right)[x].
\]
The coefficient of $[x]$ is the sum of row $x$. Since distinct
basis vectors are linearly independent, $p_{\{1\}}v=0$ precisely
when every row sum is zero. Similarly, $p_{\{2\}}$ keeps the
second coordinate, and the coefficient of $[y]$ in $p_{\{2\}}v$
is the column sum $\sum_{x\in A}M_{x,y}$. Hence $v$ is balanced
exactly when every row and every column separately sums to zero.
Merely requiring the sum of all coefficients to vanish would
be weaker.

For a concrete example, take $\A=(\mathbb Q,<)$, $S=\varnothing$,
and the orbit $\OO=\{(x,y)\in\mathbb Q^2:x<y\}$. The degree-two
cog described in \cite[p.~11]{BFKM} has the following numerical
instance:
\[
 v=[(1,3)]-[(1,4)]-[(2,3)]+[(2,4)].
\]
The part of its coefficient matrix containing all nonzero entries is
\[
\begin{array}{c|rr|r}
 &y=3&y=4&\text{row sum}\\ \hline
x=1& 1&-1&0\\
x=2&-1& 1&0\\ \hline
\text{column sum}&0&0&
\end{array}
\]
and every omitted entry is zero. Thus $v$ is a nonzero balanced
vector.

The orbit specifies which matrix positions are allowed to carry
nonzero coefficients. In this example, all positions with $x\ge y$
must be zero: the pair $(3,1)$, for instance, does not belong to
$\OO$. If instead $\OO$ is the orbit of increasing edges in a
random ordered graph, then an entry at $(x,y)$ is allowed only
when both $x<y$ and $E(x,y)$ hold. The row and column conditions
are imposed on the coefficients in these allowed positions,
with zeros in all other positions.
\end{example}

\subsection{Independent doubling and cogs}

For a finite $B\subseteq A$, a \emph{cut over $B$} is a maximal
convex subset of $A\setminus B$, including the two unbounded intervals.
Two points outside $B$ have the same cut if they have the same order
comparisons with every element of $B$.

Assume $d>0$ in this subsection, and start with a tuple
$a=(a_1,\ldots,a_d)\in\OO$. We want to replace each
coordinate $a_i$ by either itself or a new point $b_i$, so that all
$2^d$ choices give tuples in $\OO$. We prescribe more than this:
the whole finite configuration is an independent doubling of the
original one.

\begin{definition}\label{def:double}
An \emph{independent doubling} of $a$ over $S$ consists of a
tuple $b=(b_1,\ldots,b_d)$ of pairwise distinct points outside
$S\cup\aset a$, with
\begin{equation}\label{eq:interleave}
 a_1<b_1<a_2<b_2<\cdots<a_d<b_d,
\end{equation}
such that $a_i,b_i$ have the same cut over $S$ and the basic
$L$-relations on $S\cup\aset a\cup\aset b$ are given as follows.
Let $r:S\cup\aset a\cup\aset b\longrightarrow S\cup\aset a$
fix $S\cup\aset a$ and send $b_i$ to $a_i$ for each $i\in[d]$.
A basic $L$-relation instance whose arguments contain both $a_i$ and $b_i$
for some $i$ is false. For every other instance $R(c_1,\ldots,c_m)$,
its truth value is that of $R(r(c_1),\ldots,r(c_m))$ in the original
structure on $S\cup\aset a$.
\end{definition}

This rule covers unary relations, nonsymmetric relations, and
repeated arguments. The order is governed separately by
\eqref{eq:interleave}.

Independent doublings can be constructed by successive free
amalgamations, adapting \cite[Lemmas~8.7--8.9]{YBK} to the
rule for relations of arbitrary arity in Definition~\ref{def:double}.
Let $C_0$ be the induced structure on
$S\cup\aset a$. At stage $i\in[d]$, take a second copy of the
entire current structure $C_{i-1}$ over
$C_{i-1}\setminus\{a_i\}$, with the copied point named $b_i$.
Form the free amalgam in the language $L$, and place $b_i$
immediately after $a_i$ in the resulting finite order.
Denote the resulting ordered structure by $C_i$.
All twins introduced at earlier stages belong to the common
part of this amalgamation.

An induction on $i$ verifies the relation rule in
Definition~\ref{def:double}. Instances containing both $a_i$
and $b_i$ are false by free amalgamation. For an instance
containing $b_i$ but not $a_i$, its truth value is copied
from the corresponding instance with $a_i$ in place of $b_i$.
Instances involving only old points are unchanged.
Thus the conditions for earlier doubled pairs are preserved,
and the new pair satisfies the same rule. This also determines
relations involving several new twins. The order choice
preserves the required interleaving and the comparisons with
$S$. After $d$ stages, the extension property of the
Fra\"iss\'e limit realizes $C_d$ in $\A$, fixing $C_0$.

For $J\subseteq[d]$, let $a^J$ denote the tuple whose $i$th entry
is $b_i$ if $i\in J$ and $a_i$ otherwise. Each $a^J$ has the same
induced diagram over $S$ as $a$, and hence belongs to $\OO$.

\begin{definition}\label{def:cog}
The \emph{cog} associated with the independent doubling $(a,b)$ is
\begin{equation}\label{eq:cog}
 c(a,b)=\sum_{J\subseteq[d]}(-1)^{|J|}[a^J].
\end{equation}
Let $\Cog(\OO)$ be the linear span of all such cogs.
\end{definition}

This extends the duo and cog definitions of
\cite[Definitions~8.3 and~8.5]{YBK}. The doubled configurations in Definition~\ref{def:double} all have
the same diagram over $S$, and thus form one $H_S$-orbit.
Every automorphism fixing $S$ sends an independent doubling
to an independent doubling and its cog to the corresponding cog.
Consequently $\Cog(\OO)$ is $H_S$-invariant.

\begin{example}\label{ex:cog2}
In degree one a cog is $[a_1]-[b_1]$, where $b_1$ is a free copy of
$a_1$ over $S$ lying to its right in the same $S$-cut. In degree two
it is
\begin{equation}\label{eq:rectangle}
 [(a_1,a_2)]-[(b_1,a_2)]-[(a_1,b_2)]+[(b_1,b_2)].
\end{equation}
For ordered atoms, this is the degree-two cog of
\cite[p.~11]{BFKM}; the same expression applies to any orbit
admitting an independent doubling as defined here.
In the matrix interpretation, these are the four corners of a
rectangle. Each row and each column has two cancelling coefficients;
see Figure~\ref{fig:cog}. For example, projection onto the first
coordinate gives $[a_1]-[b_1]-[a_1]+[b_1]=0$.
\end{example}

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\small]
 \node[draw,rounded corners,minimum width=3.3cm,minimum height=.75cm] (aa) at (0,1.3) {$+[(a_1,a_2)]$};
 \node[draw,rounded corners,minimum width=3.3cm,minimum height=.75cm] (ab) at (5,1.3) {$-[(a_1,b_2)]$};
 \node[draw,rounded corners,minimum width=3.3cm,minimum height=.75cm] (ba) at (0,-.4) {$-[(b_1,a_2)]$};
 \node[draw,rounded corners,minimum width=3.3cm,minimum height=.75cm] (bb) at (5,-.4) {$+[(b_1,b_2)]$};
 \draw[dashed,gray] (aa)--(ab);
 \node[above,text=black] at (2.5,1.9) {replace position 2};
 \draw[dashed,gray] (ba)--(bb);
 \draw[dashed,gray] (aa)--(ba);
 \node[left,align=right,text=black] at (-1.9,.45) {replace\\position 1};
 \draw[dashed,gray] (ab)--(bb);
 \draw[->] (-1.7,-1.8)--(6.8,-1.8);
 \foreach \x/\lab in {-.7/a_1,1.3/b_1,3.3/a_2,5.3/b_2}{
  \fill (\x,-1.8) circle (1.6pt);
  \node[below] at (\x,-1.8) {$\lab$};
 }
\end{tikzpicture}
\caption{A schematic of the degree-two cog described in
\cite[p.~11]{BFKM}. The four boxes represent basis tuples,
not vertices of the underlying structure. Each row and column of
the square cancels under the corresponding marginal. The strip
below records the order of the four actual vertices. The dashed
lines indicate coordinate replacement, not basic relations.}
\label{fig:cog}
\end{figure}

In general, deleting position $i$ identifies the term indexed by
$J$ with the term indexed by $J\mathbin{\triangle}\{i\}$, and their
coefficients are opposite. Therefore
\begin{equation}\label{eq:easy-inclusion}
 \Cog(\OO)\subseteq\Bal(\OO).
\end{equation}
The reverse inclusion is the substantive statement.

\begin{theorem}[Generation by cogs]\label{thm:cogs}
For every finite $S\subseteq A$, every positive integer $d$, and
every increasing orbit $\OO$ of degree $d$ over $S$,
\[
 \Bal(\OO)=\Cog(\OO).
\]
\end{theorem}

\subsection{The degree-one argument}

It is useful to prove the first case explicitly. Although every
zero-sum vector is a linear combination of differences of points,
an arbitrary difference $[x]-[y]$ need not itself be a cog: the
points might have a basic relation between them, or might occur
in the wrong order. A fresh point provides the needed bridge.

\begin{proof}[Proof of Theorem~\ref{thm:cogs} in degree one]
Let $v=\sum_{x\in F}\lambda_x[x]$ with $F\subseteq\OO$ finite
and $\sum_x\lambda_x=0$. All points of $\OO$ have the same cut
and the same relational type over $S$. Freely amalgamate the
structure on $S\cup F$ with one new copy $b$ of a point of $\OO$
over $S$. Put $b$ to the right of every point of $F$, still in
their common $S$-cut.

For each $x\in F$, the pair $(x,b)$ is an independent doubling
over $S$: its two one-point diagrams agree over $S$, and
every basic $L$-relation instance on $S\cup F\cup\{b\}$ that
uses both $x$ and $b$ is false. Hence $[x]-[b]$ is a cog. Now
\[
 \sum_{x\in F}\lambda_x([x]-[b])
 =\sum_{x\in F}\lambda_x[x]
   -\left(\sum_{x\in F}\lambda_x\right)[b]
 =v.
\]
The auxiliary coefficient at $b$ disappears because $v$ is balanced.
\end{proof}

The higher-degree proof uses the same principle: auxiliary points
allow a controlled replacement, and their unwanted contributions
cancel. We must perform these replacements without disturbing the
coordinate sums.

\section{Ramsey separation and the proof strategy}\label{sec:ramsey}

A class of finite ordered structures is \emph{Ramsey} if, for every
$B,C$ in the class and every positive integer $r$, some member $D$
has the following property: every colouring of the copies of $B$
in $D$ with $r$ colours has a copy of $C$ all of whose copies of
$B$ receive the same colour. A homogeneous ordered structure is
called Ramsey when its age, the class of its finite induced
substructures up to isomorphism, has this property.

The Ramsey theorem for ordered relational structures originates
in the work of Ne\v{s}et\v{r}il--R\"odl \cite{NR77}, with further
development for classes of set systems in \cite{NR83}.
For unrestricted finite ordered relational structures, it was
also proved independently by Abramson--Harrington \cite{AH78}. 

The free-amalgamation form of the Ne\v{s}et\v{r}il--R\"odl theorem
states that $\K^{<}$ is Ramsey; see \cite[Theorem~1.2]{EHN}
for a modern formulation covering the relational languages
used here. 
Naming finitely many constants preserves the Ramsey
property: see the elegant proof in \cite[Proposition~10]{BPT}, 
based on the theorem of Kechris--Pestov--Todorcevic \cite{KPT} describing Ramsey structures via a dynamical property. 

Homogeneity in a finite relational language also implies that $H_S$
has finitely many orbits on $A^n$ for every $n$: only finitely many
induced diagrams on $n$ entries and the fixed set $S$ can occur.
By the Ryll--Nardzewski theorem, $(\A,S)$ is $\omega$-categorical;
see \cite[Chapter~4]{BodirskyBook} and \cite[Section~1.3]{Evans}
for this background. These facts permit
the following application of Evans's theorem.

\begin{lemma}[Evans's separation theorem]\label{lem:separator}
Let $U\leq\F[\OO]$ be $H_S$-invariant and $v\in \F[\OO]\setminus U$.
Set $T=S\cup\vsupp(v)$. There is a function $f:\OO\to\F$
invariant under $H_T$ such that
\[
 \sum_{a\in\OO}u(a)f(a)=0\quad(u\in U),\qquad
 \sum_{a\in\OO}v(a)f(a)\ne0.
\]
\end{lemma}

\begin{proof}
If $d=0$, then $\OO$ consists of the empty tuple and
$\F[\OO]$ is one-dimensional. Since $v\notin U$, necessarily
$U=0$ and $v\ne0$, so the function taking value $1$ on the
empty tuple has the required properties. Henceforth assume $d\ge1$.

We apply \cite[Corollary~1.5]{Evans}, identifying its structure,
group, module, and support explicitly. Let
\[
 M=(\A,(s)_{s\in S})
\]
be the expansion obtained by naming each element of $S$ as a
constant. Its automorphism group is
$G=\Aut(M)=H_S$: an automorphism of $M$ is exactly an automorphism
of $\A$ fixing every element of $S$. The preceding discussion
shows that $M$ is both $\omega$-categorical and Ramsey, as
required by Evans's corollary.

For the set $W$ in that corollary, take $W=\OO$.
Every orbit on finite tuples under the automorphism group of an
$\omega$-categorical structure is definable without parameters.
Thus $\OO$, being a $G$-orbit, is definable without parameters
in $M$: the elements of $S$ have already been named in its
language. To check the sort hypothesis, recall that Evans
permits sorts $D/E$, where $D$ is a parameter-free definable
set and $E$ is a parameter-free definable equivalence relation
on $D$ \cite[Section~1.3.2]{Evans}. Taking $D=\OO$ and $E$
to be equality identifies $D/E$ with $\OO$.

The action of $G$ on $W=\OO$ is coordinatewise:
\[
 g(a_1,\ldots,a_d)=(g(a_1),\ldots,g(a_d))
 \qquad(g\in G,\ (a_1,\ldots,a_d)\in\OO).
\]
The free vector space denoted by $FW$ in Evans's corollary is
therefore exactly $\F[\OO]$. We take his field $F$ to be our
$\F$, his invariant subspace $V$ to be $U$, and his vector
$x$ to be $v$. The hypotheses that $V$ is $G$-invariant and
$x\notin V$ are precisely the assumptions that $U$ is
$H_S$-invariant and $v\notin U$.

The support used in Evans's conclusion is the support of $x$
as a vector with basis $W$. In our application this is
\[
 \Sigma=\supp(v)=\{a\in\OO:v(a)\ne0\}.
\]
Thus $\Sigma$ is a finite set of tuples, whereas $S$ and
$T=S\cup\vsupp(v)$ are finite sets of vertices. We use the
letter $\Sigma$ because the support is denoted by $S$ in
Evans's statement, with a different meaning from our parameter
set $S$.

Evans's corollary gives invariance under $G_{(\Sigma)}$, the
subgroup fixing each tuple in $\Sigma$ individually. Fixing
a tuple under the coordinatewise action is equivalent to fixing
each of its entries. Since every element of $G=H_S$ already
fixes $S$ pointwise, we obtain
\[
\begin{aligned}
 G_{(\Sigma)}
 &=\{g\in H_S:g(a)=a\text{ for every }a\in\Sigma\}\\
 &=H_{S\cup\bigcup_{a\in\Sigma}\aset a}
  =H_{S\cup\vsupp(v)}
  =H_T.
\end{aligned}
\]
Here it is essential that the tuples are fixed individually:
merely permuting the tuples in $\Sigma$ would not suffice.

With these choices, the corollary supplies a function
$f:\OO\to\F$ satisfying
\[
 f(g(a))=f(a)\qquad(g\in H_T,\ a\in\OO)
\]
and
\[
 \sum_{a\in\OO}u(a)f(a)=0\quad(u\in U),
 \qquad
 \sum_{a\in\OO}v(a)f(a)\ne0.
\]
These sums express Evans's separation conclusion: the linear
functional obtained by summing against $f$ vanishes on every
vector in $U$ and does not vanish on $v$. All sums are finite
because the vectors have finite support. Together with the
$H_T$-invariance, these are exactly the conclusions of the lemma.
\end{proof}

We use the same symbol $f$ for the function on tuples and its
linear extension to $\F[\OO]$: for
$w=\sum_{a\in\OO}w(a)[a]$, we set
\[
 f(w)=\sum_{a\in\OO}w(a)f(a).
\]
The sum is finite because $w$ has finite support. Ordinary linear
algebra gives a functional separating $v$ from $U$; the additional
content of Lemma~\ref{lem:separator} is invariance under $H_T$.
In particular, the value of $f$ on a tuple depends only on its
induced diagram over $T$.

\subsection{Proof outline for cog generation}\label{sec:cog-outline}

Our goal is Theorem~\ref{thm:cogs},
$\Bal(\OO)=\Cog(\OO)$. The inclusion
$\Cog(\OO)\subseteq\Bal(\OO)$ was established in
\eqref{eq:easy-inclusion}, and the degree-one case was proved in
Section~\ref{sec:cogs}. We outline the inductive step for $d\ge2$,
assuming the theorem in degree $d-1$ for every finite parameter
set. The constructions needed for this step are proved in
Sections~\ref{sec:moving} and~\ref{sec:finish-cogs}; here we
describe what they must achieve and how they yield a contradiction.

Suppose that $v\in\Bal(\OO)\setminus\Cog(\OO)$, and put
$U=\Cog(\OO)$ and $T=S\cup\vsupp(v)$. Applying
Lemma~\ref{lem:separator} and rescaling the resulting function
gives an $H_T$-invariant $f:\OO\to\F$ whose linear extension
vanishes on $U$ and satisfies $f(v)=1$.

The first construction, proved in Lemma~\ref{lem:compression}
in Section~\ref{sec:moving}, produces a balanced vector
$u\in\F[\OO]$ such that
\[
 u-v\in U,\qquad \vsupp(u)\cap T=\varnothing,
\]
and, for each position $i$, all $i$th coordinates of tuples in
$\supp(u)$ lie in one fixed cut over $T$.
The construction repeatedly replaces all occurrences of one
vertex in one coordinate position. After that coordinate is
deleted, the chosen slice is balanced in degree $d-1$ over
$S$ together with the fixed vertex. The induction hypothesis
decomposes this slice into smaller cogs.
Lemma~\ref{lem:slice} constructs a replacement point so that
each smaller cog, with the old and new points inserted,
gives a degree-$d$ cog. Consequently each replacement changes
the vector by an element of $U$.
Lemma~\ref{lem:compression} specifies the order of replacements
and their destination cuts, proving that finitely many
replacements give all the stated support conditions.
These are the constructions to be justified in
Section~\ref{sec:moving}. Once $u$ has been obtained, it satisfies
\[
 f(u)=f(v)+f(u-v)=1.
\]

The second construction concerns the individual tuples occurring
in $u$. Let $a=(a_1,\ldots,a_d)$ range over $\supp(u)$, so that
\[
 u=\sum_{a\in\supp(u)}u(a)[a].
\]
We introduce a tuple $b(a)$ to complete $a$ to a cog.
Since $f$ vanishes on every cog, this will give an identity
expressing $f(a)$ through its values on mixed tuples.
The choice of $b(a)$ must also ensure that these other values
depend only on proper subtuples of $a$.
For each such $a$, Section~\ref{sec:finish-cogs} constructs a tuple
$b=b(a)=(b_1,\ldots,b_d)$ of fresh points outside $T\cup\aset a$
forming an independent doubling of $a$ over $S$
(Definition~\ref{def:double}). Each $b_i$ lies in the same
$T$-cut as $a_i$. In the induced structure on
$T\cup\aset a\cup\aset b$, every basic $L$-relation instance
meeting both $\aset b$ and $T\setminus S$ is false.
The tuple $b(a)$ is chosen separately for each $a$.
For $J\subseteq[d]$, recall that $a^J$ is obtained from $a$ by
replacing $a_i$ with $b_i$ exactly when $i\in J$.
In particular, $a^\varnothing=a$ and $a^{[d]}=b(a)$.
The independent doubling gives $c(a,b(a))\in U$, so applying
$f$ yields the connecting identity
\[
 0=f(c(a,b(a)))
   =f(a)+\sum_{\varnothing\ne J\subseteq[d]}(-1)^{|J|}f(a^J)
 \qquad(a\in\supp(u)).
\]
Thus the auxiliary tuple $b(a)$ supplies the mixed tuples
whose values under $f$ determine $f(a)$.

The additional conditions imposed on $b(a)$ relative to $T$
ensure dependence on retained coordinates. More precisely,
Section~\ref{sec:finish-cogs} proves the following assertion.
For each nonempty $J\subseteq[d]$, put $K=[d]\setminus J$.
There is a well-defined function
\[
 g_J:p_K(\supp(u))\longrightarrow\F
 \quad\text{such that}\quad
 f(a^J)=g_J(p_K(a))\qquad(a\in\supp(u)).
\]
Thus, if two supported tuples have the same retained subtuple,
their mixed tuples have the same value under $f$. The later
proof establishes this by comparing the induced diagrams over
$T$ and using the $H_T$-invariance of $f$.

Substituting these values into the displayed cog identity
expresses $f(a)$ as a signed sum of functions of proper subtuples.
When multiplied by $u(a)$ and summed over $a\in\supp(u)$,
each such term vanishes because the corresponding proper
marginal of $u$ is zero. Hence $f(u)=0$, contrary to $f(u)=1$.
The detailed summation is given in Section~\ref{sec:finish-cogs},
after the constructions and the dependence on retained
coordinates have been proved.

\section{Moving a balanced vector into fixed intervals}\label{sec:moving}

We now prove the first construction used in the outline of
Section~\ref{sec:cog-outline}. Given a balanced vector $v$ and a
finite set $T\supseteq S$, we will construct a balanced vector
$u$ with no support vertex in $T$, with each coordinate position
confined to a specified cut over $T$, and with
$v-u\in\Cog(\OO)$. Thus the construction changes the representative
of the coset $v+\Cog(\OO)$ while controlling where its vertices lie.

Throughout this section $d\ge2$. We assume
Theorem~\ref{thm:cogs} in degree $d-1$ for every finite parameter
set and every increasing orbit of that degree. This simultaneous
induction is needed because fixing one coordinate at a vertex
$z$ leaves a vector of degree $d-1$ over the enlarged parameter
set $S\cup\{z\}$.

The local operation replaces all terms having a specified vertex
in a specified position. Lemma~\ref{lem:slice} proves that a
suitable replacement changes the vector by a sum of cogs.
Lemma~\ref{lem:compression} then arranges a finite sequence of
these replacements to reach the required cuts.

\subsection{Replacing one coordinate slice}

The slice-decomposition method comes from
\cite[proofs of Claim~E.5 and Theorem~E.6]{YBK}. The proof below
extends its lifting step to relations of arbitrary arity and
establishes the freedom to prescribe a cut for the replacement
point. We give the construction explicitly, including the
placement of the auxiliary vertices used in the decomposition.

\begin{lemma}[Slice replacement]\label{lem:slice}
Let $v\in\Bal(\OO)$, let $i\in[d]$, and let $z\in A\setminus S$
occur in position $i$ in some tuple in $\supp(v)$. Define
\begin{equation}\label{eq:slice}
 w=\sum_{\substack{a\in\OO\\a_i=z}}v(a)[p_{-i}(a)],
 \qquad P=\vsupp(w).
\end{equation}
The vector $w$ is the \emph{slice at $a_i=z$}, with its fixed
coordinate deleted. Restoring $z$ in position $i$ recovers
each original tuple, so deletion is injective on this slice
and no coefficients combine. The set $P$ consists of the vertices in
the other positions of the supported tuples in this slice;
we call these vertices the \emph{partners of $z$ in position $i$}.
In particular, $z\notin P$.

Let $D\subseteq A$ be finite with
$S\cup\vsupp(v)\subseteq D$. Let $J$ be a cut over $D$ such that
\[
 p<y\quad\Longleftrightarrow\quad p<z
 \qquad(p\in S\cup P,\ y\in J).
\]
Thus moving $z$ into $J$ preserves its order comparisons with
the permanent parameters and all partners in this slice.
There is a point $z'\in J$ such that replacing $z$ by $z'$
only in position $i$ gives a vector $v'\in\Bal(\OO)$ with
\[
 v-v'\in\Cog(\OO),\qquad
 \vsupp(v')\subseteq\vsupp(v)\cup\{z'\}.
\]
In particular, $z'\notin D$.
\end{lemma}

\begin{proof}
We first decompose the slice using the induction hypothesis.
We then construct one replacement point that works for every
cog in this decomposition. Finally, we compute the difference
between the original vector and its replacement.

\medskip\noindent
\emph{Step 1: the slice is balanced in degree $d-1$.}
Put
\[
 \Theta=\{p_{-i}(a):a\in\OO,\ a_i=z\}.
\]
This is one increasing orbit of degree $d-1$ over $S\cup\{z\}$.
Indeed, if $a,\widetilde a\in\OO$ both have $z$ in position $i$,
an element of $H_S$ taking $a$ to $\widetilde a$ must fix $z$.
Their remaining coordinates avoid $S\cup\{z\}$ because all
tuples in $\OO$ have distinct entries outside $S$.
Thus $w\in\F[\Theta]$.

We spell out the balanced-slice observation of
\cite[Lemma~E.1]{YBK}. Temporarily label the coordinates of $w$
by their original positions $[d]\setminus\{i\}$. Fix $j\ne i$
and put $K=[d]\setminus\{i,j\}$. Let $t$ be any $K$-subtuple
occurring in this slice. Let $b$ be the tuple whose positions
are indexed, in increasing order, by $[d]\setminus\{j\}$,
with entry $z$ in original position $i$ and entries $t$ in
positions $K$. The coefficient of $[t]$ in the marginal of
$w$ deleting original position $j$ is
\[
 \sum_{\substack{a\in\OO\\a_i=z,\ p_K(a)=t}}v(a)
   =(p_{-j}v)(b)=0.
\]
The equality holds because $p_{-j}(a)=b$ specifies exactly
$a_i=z$ and $p_K(a)=t$. The last expression vanishes by
balancedness of $v$. This proves that every one-coordinate
marginal of $w$ is zero.

The induction hypothesis, applied to $\Theta$ over
$S\cup\{z\}$, now gives a finite expression
\begin{equation}\label{eq:slice-decomposition}
 w=\sum_{r=1}^m\alpha_r c_r,
\end{equation}
where $\alpha_r\in\F$ and every $c_r$ is a degree-$(d-1)$ cog
over $S\cup\{z\}$. This use of cog generation is in smaller
degree.

\medskip\noindent
\emph{Step 2: place the auxiliary configurations so that $z$
can be replaced in the prescribed cut.}
Let $C_0\subseteq A$ be a finite set containing
\[
 B=S\cup\{z\}\cup P
\]
and all vertices of the doubled configurations defining
$c_1,\ldots,c_m$. We use the same letters for these sets and
their induced structures. The vertices in $C_0\setminus B$
are auxiliary: all coefficients of tuples using them cancel
in \eqref{eq:slice-decomposition}, since $w$ is supported on
tuples with entries in $P$.

We will place a copy $C$ of $C_0$ over $B$ so that no vertex of
$C\setminus\{z\}$ separates $z$ from the eventual replacement
point. This ensures that doubling $z$ will preserve the order
in every cog configuration. The vertices of $B$ must remain
fixed, but the auxiliary vertices can be copied elsewhere.

Here is the finite construction. Introduce a formal marker $*$,
placed in the prescribed cut $J$ of $D$. It is used only to
specify an order position, not as a vertex of an $L$-structure.
Take a copy of $C_0$ over $B$, with its new vertices disjoint
from $D$, and form its free $L$-amalgam with $D$ over $B$.
Thus the copy, denoted by $C$, satisfies $C\cap D=B$.

We must extend the orders on $D\cup\{*\}$ and $C$ so that,
for every copied auxiliary vertex, its side of $z$ is also
its side of $*$. This is possible as follows. By the hypothesis
on $J$, the points $z$ and $*$ lie in the same gap of
$B\setminus\{z\}=S\cup P$. In this gap, place the copied
auxiliary vertices that originally lay below $z$ below both
$z$ and $*$, and those that originally lay above $z$ above
both. Preserve their order within each of these two lists.
In every other gap of $B\setminus\{z\}$, preserve the order
of the copied vertices and interleave them arbitrarily with
the vertices of $D$. All these choices preserve the order
on $C_0$ over $B$ and the order on $D\cup\{*\}$.

Remove the marker, remembering its position as a cut over
$D\cup C$. This cut is contained in $J$. Since every linear
order on a member of $\K$ is allowed in $\K^{<}$, the ordered
free amalgam belongs to $\K^{<}$. The finite extension property
of $\A$ realizes it over the actual structure $D$.
Let $h:C_0\to C$ be the resulting copying isomorphism.
It fixes $B$ pointwise, so it fixes $w$. Applying $h$ to
\eqref{eq:slice-decomposition} therefore gives the same
decomposition of $w$ by the transported cogs $h(c_r)$.
We henceforth write $c_r$ for these transported cogs.

\medskip\noindent
\emph{Step 3: add the new doubled coordinate.}
Take a second copy of $C$ over $C\setminus\{z\}$, naming the
copy of $z$ by $z'$. Freely amalgamate this copy with the
structure on $D\cup C$ over $C\setminus\{z\}$.
The only new vertex is $z'$. Place it in the cut recorded
by the marker. The order construction in Step 2 ensures that
$z$ and $z'$ have the same comparisons with every point of
$C\setminus\{z\}$, so this order extends the order on the
second copy of $C$. The extension property again realizes
the finite ordered amalgam in $\A$, fixing $D\cup C$.

Consequently $z'\in J$ and $z'\notin D\cup C$. Replacing $z$
by $z'$ induces an isomorphism from $C$ onto its second copy.
In the induced structure on $D\cup C\cup\{z'\}$, every basic
$L$-relation instance involving both $z$ and $z'$ is false.
All configurations used below lie in this finite structure,
so these statements specify the relations of all arities
needed for the lifting step.

Let $\Xi\subseteq\Theta$ be the finite set of tuples appearing
in $c_1,\ldots,c_m$; it contains $\supp(w)$.
For $t\in\Xi$, let $I_z(t)$ be obtained by inserting $z$ in
position $i$, and define $I_{z'}(t)$ by inserting $z'$ there.
The first tuple belongs to $\OO$ by the definition of $\Theta$.
The second has the same diagram over $S$, by the copied
relations and order, and hence also belongs to $\OO$.
Extend these assignments linearly:
\[
 I_z,I_{z'}:\F[\Xi]\longrightarrow\F[\OO].
\]

For each $r$ we claim that
\[
 I_z(c_r)-I_{z'}(c_r)
\]
is a degree-$d$ cog, up to its overall sign. This is the lifting
step of \cite[Lemma~8.8]{YBK}, with the following verification
for arbitrary relation arity. The proposed doubled pairs are
the $d-1$ pairs defining $c_r$, together with $(z,z')$.
Both entries of each older pair lie on the same side of $z$,
because that pair has the same cut over $S\cup\{z\}$.
Step 2 places them on the same side of $z'$ as well.
Thus the new pair has position $i$ and the pairs have the
interleaving required in Definition~\ref{def:double}.

To verify the relation rule, consider any basic $L$-relation
instance on these vertices and $S$.
If it does not contain $z'$, its truth value is already
governed by the cog configuration over $S\cup\{z\}$.
If it contains both $z$ and $z'$, it is false by free
amalgamation.
If it contains $z'$ but not $z$, its truth value is obtained
by replacing every occurrence of $z'$ by $z$, since the
second copy of $C$ is isomorphic to the first.
In particular, instances containing both entries of any older
pair remain false. This verifies the rule even when an
instance uses several pairs or repeats an argument.
If $z'<z$, orient the new pair as $(z',z)$; reversing this
pair changes the alternating sum by its overall sign only.
Expanding the alternating sum with respect to the choice of
$z$ or $z'$ gives precisely $I_z(c_r)-I_{z'}(c_r)$, with this
possible sign.

\medskip\noindent
\emph{Step 4: compute the replacement vector.}
Set
\[
 v'=v-I_z(w)+I_{z'}(w).
\]
The vector $I_z(w)$ is exactly the part of $v$ whose $i$th
coordinate equals $z$. Thus this formula replaces that entire
slice and leaves all other terms unchanged, including terms
where $z$ occurs in a different position. By linearity and
\eqref{eq:slice-decomposition},
\begin{equation}\label{eq:replace-identity}
\begin{aligned}
 v-v'&=I_z(w)-I_{z'}(w)\\
     &=\sum_{r=1}^m\alpha_r
         \bigl(I_z(c_r)-I_{z'}(c_r)\bigr)
       \in\Cog(\OO).
\end{aligned}
\end{equation}
The formula for $v'$ uses $w$ itself, so the auxiliary tuples
in its cog decomposition cancel before and after insertion.
Only the new point $z'$ can enter the vertex support.
Moreover, replacing one coordinate by a fresh point maps the
supported tuples injectively to supported tuples: replaced
tuples remain distinct and cannot coincide with unchanged
tuples, none of which contains $z'$.
Finally, every cog is balanced by \eqref{eq:easy-inclusion}.
Hence $v'=v-(v-v')$ is balanced, as required.
\end{proof}

\begin{example}[The replacement identity in degree two]\label{ex:slice-rectangle}
Take $d=2$ and replace the first-coordinate slice at $z$.
Then
\[
 w=\sum_{\substack{y\in A\\(z,y)\in\OO}}v(z,y)[y],
 \qquad \sum_{\substack{y\in A\\(z,y)\in\OO}}v(z,y)=0.
\]
The second equality is the zero row-sum condition for $v$.
Choose a finite $D\supseteq S\cup\vsupp(v)$ and prescribe
its cut immediately to the right of $z$. This cut is
admissible, since it has the same comparisons as $z$ with
every other point of $D$. Steps 2 and 3 of the lemma's proof
give one point $z'>z$ that works for the entire cog
decomposition of $w$.
Each transported degree-one cog has the form $[b]-[c]$,
where $(b,c)$ is an independent doubling over
$S\cup\{z\}$ and $b<c$. Its lift is
\[
 I_z([b]-[c])-I_{z'}([b]-[c])
   =[(z,b)]-[(z,c)]-[(z',b)]+[(z',c)].
\]
This is the rectangle cog of Example~\ref{ex:cog2}, with
doubled pairs $(z,z')$ and $(b,c)$ and order
$z<z'<b<c$. Subtracting this rectangle cog transfers the
coefficient pattern $[b]-[c]$ from the row indexed by $z$
to the row indexed by $z'$. Subtracting the linear combination
of these rectangles with the coefficients in the decomposition
of $w$ transfers the whole row. The auxiliary
column coefficients cancel because their total contribution
to $w$ is zero.
\end{example}

The lemma already allows us to remove all support vertices
from any fixed finite set $T\supseteq S$.
Let $v_{\mathrm{current}}$ denote the vector at the current
stage, initially the given balanced vector. Whenever a vertex
$z\in T\setminus S$ occurs in position $i$, take
$D=T\cup\vsupp(v_{\mathrm{current}})$ and use its cut immediately
to the right of $z$. This cut preserves comparisons with all
partners and with $S$. The replacement removes the pair
$(z,i)$ from the occurrences of vertices of $T$ in the support
and introduces no new such pair, since $z'\notin T$.
There are only finitely many such pairs, so this process
terminates. The successive differences belong to $\Cog(\OO)$
by \eqref{eq:replace-identity}.
To obtain the additional requirement that each position
occupy one fixed cut over $T$, we use the ordering of
replacements proved next.

\subsection{Separating positions and moving from right to left}

The following lemma supplies the full conclusion needed in
Section~\ref{sec:cog-outline}. The first stage of its proof
uses the removal of shared coordinate positions from
\cite[proof of Theorem~E.6]{YBK}. The second stage uses the
prescribed-cut version of Lemma~\ref{lem:slice} to move all
entries into the specified $T$-cuts.

\begin{lemma}[Fixed-interval representative]\label{lem:compression}
Let $T\supseteq S$ be finite. For each $i\in[d]$, let $I_i$ be
the $S$-cut containing the $i$th coordinate of every tuple in
$\OO$, and let $J_i$ be the rightmost $T$-cut contained in $I_i$.
For every $v\in\Bal(\OO)$, there is $u\in\Bal(\OO)$ such that
\[
 v-u\in\Cog(\OO),\qquad \vsupp(u)\cap T=\varnothing,
\]
and
\[
 a_i\in J_i
 \quad\text{for every }a\in\supp(u)\text{ and every }i\in[d].
\]
\end{lemma}

The cut $I_i$ is well-defined because the orbit $\OO$ fixes
all order comparisons of its $i$th coordinate with $S$.
The cut $J_i$ consists of the points in $I_i$ lying above
every point of $T\cap I_i$. If $T\cap I_i$ is empty, then
$J_i=I_i$. Several coordinate positions may have the same
$S$-cut and therefore the same destination $T$-cut; their
relative order must still be preserved.

\begin{proof}
If $v=0$, take $u=0$. Otherwise, let $q$ denote the current
vector, initially $v$. Every operation below is an application
of Lemma~\ref{lem:slice}; after an operation producing $q'$,
replace $q$ by $q'$. Thus every current vector is balanced,
and every successive difference belongs to $\Cog(\OO)$.
We must specify admissible cuts for the operations and show
that finitely many of them achieve the stated support conditions.

\medskip\noindent
\emph{Stage 1: give each support vertex a unique coordinate position.}
A vertex may occur in different positions in different tuples
of $\supp(q)$. For $x\in\vsupp(q)$, define its set of positions by
\[
 R_q(x)=\{j\in[d]:a_j=x\text{ for some }a\in\supp(q)\}.
\]
If $|R_q(x)|>1$, choose $i\in R_q(x)$ and put
$D=T\cup\vsupp(q)$. Choose the cut immediately to the right
of $x$ in the finite order on $D$. Every point of
$D\setminus\{x\}$ has the same comparison with this cut as
with $x$. In particular this holds for $S$ and for all
partners of $x$ in the chosen slice, which do not include
$x$ itself. Lemma~\ref{lem:slice} therefore applies.
It replaces all occurrences of $x$ in position $i$ by one
fresh vertex $x'$, leaving the other positions unchanged.

The replacement is injective on the supported tuples, as
shown in the proof of Lemma~\ref{lem:slice}. Thus $x$ loses
exactly position $i$, the new vertex $x'$ occurs only in
position $i$, and the position sets of every other vertex
are unchanged. The nonnegative integer
\[
 E(q)=\sum_{x\in\vsupp(q)}\bigl(|R_q(x)|-1\bigr)
\]
therefore decreases by exactly one. After finitely many
steps $E(q)=0$, so every support vertex has a unique position.
This property will persist through Stage 2, since each
replacement point is fresh and is used in just one position.

\medskip\noindent
\emph{Stage 2: move the positions into their destination cuts.}
Process the coordinate positions in the order
$d,d-1,\ldots,1$. Within a position, process its support
vertices in any order. At the beginning of this stage,
declare all support vertices unmoved.
Let $N$ and $M$ denote, respectively, the sets of currently
unmoved and moved support vertices; initially
$N=\vsupp(q)$ and $M=\varnothing$. A fresh replacement is
declared moved and is never processed again.

For each cut $I$ over $S$, maintain the order condition
\begin{equation}\label{eq:order-invariant}
 x<y
 \qquad
 \bigl(x\in(N\cap I)\cup(T\cap I),\ y\in M\cap I\bigr).
\end{equation}
In words, all moved vertices in $I$ lie above all unmoved
vertices there and above all points of $T$ there.
The set $T$ stays fixed throughout: a vertex of $T$ remains
an order constraint even after it disappears from the support.
The condition holds initially because $M$ is empty.

Suppose the next vertex is $z\in N$, whose unique position
is $i$, and let $I=I_i$ be its $S$-cut. Put
$D=T\cup\vsupp(q)$. We prescribe the cut over $D$ in $I$
lying above every point of
\[
 (N\cap I)\cup(T\cap I)
\]
and below every point of $M\cap I$; see
Figure~\ref{fig:compression}.
Such a cut exists: the two finite sets are separated in
this order by \eqref{eq:order-invariant}, and every point
of $D\cap I$ belongs to one of them. If $M\cap I$ is empty,
take the gap after the lower set and before the right
endpoint of $I$, if that endpoint exists. The lower set
is nonempty because it contains $z$.
In particular the new point will lie to the right of $z$.

We now verify the hypothesis of Lemma~\ref{lem:slice}, which
concerns the partners in tuples having $z$ in position $i$.
A partner in position $j<i$ is still unmoved, because its
position has not been processed yet. It is below $z$ in
its tuple, and if it lies in $I$ it is also below the
prescribed cut. A partner in position $j>i$ has already
moved, because that entire position has been processed.
It is above $z$ in its tuple, and if it lies in $I$ it
is also above the prescribed cut.
The unique-position property from Stage 1 ensures that these
statements about moved and unmoved vertices are unambiguous.
Partners outside $I$ retain their comparisons automatically,
since $z$ and its replacement stay in the same $S$-cut.
The comparisons with $S$ are preserved for the same reason.
Thus the cut has exactly the required comparisons with
$S$ and all partners, and Lemma~\ref{lem:slice} supplies
a replacement $z'$ in it.

Replace the slice, remove $z$ from $N$, and add $z'$ to $M$.
The vertex $z$ disappears from the support because it had
only position $i$. Every other old support vertex remains,
and $z'$ is fresh. The old moved vertices still satisfy
\eqref{eq:order-invariant}; the new moved vertex $z'$ satisfies
it by the chosen cut. Hence the invariant is preserved.
Each step decreases $|N|$ by one, so Stage 2 terminates
after finitely many replacements.

\medskip\noindent
\emph{Conclusion.}
Let $u$ be the final vector. Every support vertex of $u$
was introduced in Stage 2 outside a set $D$ containing $T$,
so $\vsupp(u)\cap T=\varnothing$.
Every such vertex in position $i$ lies in $I_i$ and above
every point of $T\cap I_i$, by
\eqref{eq:order-invariant}. It therefore lies in $J_i$.
Balancedness has been preserved at every step.

Finally, list all vectors in both stages as
$q_0=v,q_1,\ldots,q_\ell=u$, where $\ell$ is the total
number of replacements. Lemma~\ref{lem:slice} gives
$q_{k-1}-q_k\in\Cog(\OO)$ for each $k=1,\ldots,\ell$.
Since this is a vector subspace,
\[
 v-u=\sum_{k=1}^{\ell}(q_{k-1}-q_k)\in\Cog(\OO).
\]
This proves the remaining conclusion.
\end{proof}

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\small]
 \draw[->] (0,0)--(12.7,0);
 \foreach \x/\lab in {0/s,12/t}{
  \draw[thick] (\x,-.12)--(\x,.12);
  \node[below] at (\x,-.12) {$\lab$};
 }
 \foreach \x/\lab in {1.6/c_1,4.6/c_2}{
  \draw[thick] (\x,-.12)--(\x,.12);
  \node[above] at (\x,.14) {$\lab\in T$};
 }
 \foreach \x in {.8,2.6,5.5,6.3}{\fill[gray] (\x,0) circle (2pt);}
 \fill (3.5,0) circle (2pt);\node[below] at (3.5,-.12) {$z$};
 \fill[orange!80!black] (7.8,0) circle (2.5pt);
 \node[below] at (7.8,-.12) {$z'$};
 \foreach \x in {9.3,10.2,11.1}{\fill[blue!65!black] (\x,0) circle (2.5pt);}
 \draw[->,thick] (3.5,.2) to[bend left=28] node[above] {replace the chosen position} (7.8,.2);
 \draw[decorate,decoration={brace,mirror,amplitude=4pt}] (.6,-.65)--(6.5,-.65);
 \node[below,align=center] at (3.55,-.85) {unmoved support and extra parameters};
 \draw[decorate,decoration={brace,mirror,amplitude=4pt}] (9.1,-.65)--(11.3,-.65);
 \node[below,align=center] at (10.2,-.85) {moved support};
\end{tikzpicture}
\caption{One step inside an $S$-interval $I=(s,t)$. The new point
$z'$ is placed after the unmoved support and the parameters in
$T\cap I$, and before the moved support. Processing larger
coordinate positions first ensures that every partner in a smaller
position stays below $z'$, while every partner in a larger position
stays above it. The same construction works in unbounded $S$-intervals.}
\label{fig:compression}
\end{figure}

For three coordinates lying in the same $S$-interval, this procedure
first puts all third-coordinate points to the right of $T$, then
places the second-coordinate points between $T$ and those third
coordinates, and finally places the first-coordinate points in the
remaining gap. The fixed order of the coordinates is preserved
throughout. Notice also that the induction always uses $S$ as its
permanent parameter set. The extra points of $T\setminus S$ may
be crossed; they constrain the desired final cuts, not the
allowed cog replacements.

\section{Completion of the cog-generation proof}\label{sec:finish-cogs}

We now carry out the argument outlined in Section~\ref{sec:ramsey}.
The essential issue is to make the other corners of a doubled tuple
depend only on the coordinates that were not replaced.

\begin{proof}[Proof of Theorem~\ref{thm:cogs}]
The degree-one case was proved in Section~\ref{sec:cogs}.
Assume the theorem in degree $d-1$ for every finite parameter set.
Suppose that $v\in\Bal(\OO)\setminus\Cog(\OO)$.
By Lemma~\ref{lem:separator}, choose an $H_T$-invariant function
$f:\OO\to\F$ with $f(\Cog(\OO))=0$ and $f(v)=1$, where
$T=S\cup\vsupp(v)$. Lemma~\ref{lem:compression} gives a balanced
$u$ in the prescribed fixed $T$-cuts, disjoint from $T$, with
$u-v\in\Cog(\OO)$. Hence $f(u)=1$.

Let $Q\subseteq\OO$ consist of the tuples avoiding $T$ with these
fixed cuts, so that $\supp(u)\subseteq Q$.
For each $a=(a_1,\ldots,a_d)\in Q$, form an independent doubling
over $S$ and freely amalgamate it with the actual structure on
$T\cup\aset a$ over $S\cup\aset a$; see Figure~\ref{fig:amalgam}.
Write $b=b(a)=(b_1,\ldots,b_d)$ for the tuple of new points in
this amalgam. This choice is made separately for each $a$.
Choose their order so that $a_i<b_i<a_{i+1}$ for $i<d$ and
$a_d<b_d$, with $b_i$ in the same $T$-cut as $a_i$. These conditions are
compatible: a twin can be inserted immediately to the right of its
original in the finite order. The finite extension is therefore
realizable in $\A$.

In the induced structure on $T\cup\aset a\cup\aset b$,
every basic $L$-relation instance containing both a new twin
and a point of $T\setminus S$ is false. This includes instances
with any number of additional arguments from this finite set.
The statement uses the definition of free amalgamation and
does not depend on the arity of the relation.

Fix a nonempty $J\subseteq[d]$ and put $K=[d]\setminus J$.
We claim that the type of the mixed tuple $a^J$ over $T$ depends
only on $p_K(a)$. To check this, consider a basic $L$-relation instance
in entries of $a^J$ and parameters from $T$.
\begin{enumerate}[label=(\roman*),leftmargin=2em]
\item If it contains no new twin, it uses only the retained entries
and $T$, whose actual induced structure is fixed by $p_K(a)$.
\item If it contains a new twin and a parameter in $T\setminus S$,
it is false by the free amalgamation just performed.
\item If it contains a new twin but no parameter in $T\setminus S$,
its truth value is prescribed by the fixed diagram of $\OO$ over
$S$, through the doubling rule.
\end{enumerate}
Repeated arguments satisfy the same three-way distinction.
Equalities are fixed because all the tuple entries and twins are
distinct and outside $T$. Comparisons with $T$ are fixed by the
chosen cuts, and comparisons among tuple entries are fixed by their
positions. These observations account for the entire finite diagram.

More explicitly, if $a,\widetilde a\in Q$ have $p_K(a)=p_K(\widetilde a)$,
the map fixing $T$ and the retained entries and sending the new
entries of $a^J$ to those of $\widetilde a^J$ is a partial
isomorphism. Homogeneity extends it to an element of $H_T$.
Consequently there is a well-defined function
$g_J:p_K(Q)\to\F$ with
\[
 f(a^J)=g_J(p_K(a)).
\]
Its value does not depend on how the individual doubling was chosen.

We now obtain the contradiction by summing the cog identities.
For every
$a\in\supp(u)$, the vector $c(a,b(a))$ is a cog, so
\[
 0=f(c(a,b(a)))=\sum_{J\subseteq[d]}(-1)^{|J|}f(a^J).
\]
The term with $J=\varnothing$ is $f(a)$. Moving the remaining
terms to the other side and substituting the functions $g_J$
therefore gives, for every $a\in\supp(u)$,
\begin{equation}\label{eq:roadmap}
 f(a)=-\sum_{\varnothing\ne J\subseteq[d]}(-1)^{|J|}
 g_J\bigl(p_{[d]\setminus J}(a)\bigr).
\end{equation}

To evaluate $f(u)$, multiply this identity by $u(a)$ and sum over
all $a\in\supp(u)$. Fix one nonempty $J$ and again set
$K=[d]\setminus J$. Grouping the supported tuples according to
their retained subtuple $t$ gives
\[
\begin{aligned}
 &\sum_{a\in\supp(u)}u(a)g_J(p_K(a))\\
 &\quad=\sum_{t\in p_K(\supp(u))}g_J(t)
       \left(\sum_{\substack{a\in\supp(u)\\p_K(a)=t}}u(a)\right)\\
 &\quad=\sum_{t\in p_K(\supp(u))}g_J(t)(p_Ku)(t)=0.
\end{aligned}
\]
The inner coefficient sum is precisely the marginal
$(p_Ku)(t)$. It vanishes because $K$ is proper and $u$ is
balanced. Applying this to each nonempty $J$ in
\eqref{eq:roadmap} yields
\[
\begin{aligned}
 f(u)
 &=\sum_{a\in\supp(u)}u(a)f(a)\\
 &=-\sum_{\varnothing\ne J\subseteq[d]}(-1)^{|J|}
       \sum_{a\in\supp(u)}
       u(a)g_J\bigl(p_{[d]\setminus J}(a)\bigr)
 =0.
\end{aligned}
\]
This contradicts $f(u)=1$. The cancellation takes place in the
weighted sum; it does not require $f(a)=0$ for individual tuples.

This completes the simultaneous induction.
\end{proof}

With the notation of the preceding proof, when $d=2$ the
identity being summed is
\[
 f(a_1,a_2)=g_{\{1\}}(a_2)+g_{\{2\}}(a_1)
              -g_{\{1,2\}}(\varnothing),
 \qquad (a_1,a_2)\in\supp(u).
\]
The contribution of $g_{\{1\}}(a_2)$ vanishes by the zero column
sums of the coefficient matrix of $u$, and that of
$g_{\{2\}}(a_1)$ vanishes by its zero row sums.
The constant term contributes zero because the total coefficient
sum is zero. This is the matrix calculation from
Example~\ref{ex:matrix}.

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
 \fill[gray!12] (-1.6,-.9) rectangle (1.6,.9);
 \draw[thick,rounded corners] (-5.6,-.9) rectangle (1.6,.9);
 \draw[thick,rounded corners] (-1.6,-.9) rectangle (5.6,.9);
 \node at (-3.6,0) {$T\setminus S$};
 \node[align=center] at (0,0) {$S\cup\aset a$\\common part};
 \node at (3.6,0) {$\aset b$};
 \node[above,align=center] at (-2,1.05) {actual finite structure\\on $T\cup\aset a$};
 \node[above,align=center] at (3.1,1.05) {independent doubling\\of $a$ over $S$};
 \node[below,align=center,text width=11cm] at (0,-1.05)
 {In this finite amalgam, no true basic $L$-relation instance meets both outside parts.};
\end{tikzpicture}
\caption{The free amalgam used for one tuple $a$. The two diagrams
agree on $S\cup\aset a$. Within the displayed finite structure,
relations between new twins and extra parameters are absent,
including higher-arity relations also using vertices of the
common part. A separate finite amalgam is chosen
for each tuple; the proof compares only the resulting types over $T$.}
\label{fig:amalgam}
\end{figure}

The separation into the three cases above is where arbitrary arity
is handled. A relation instance in this finite configuration
can involve many retained entries and many parameters, but as
soon as it uses a twin and an extra parameter, free amalgamation
determines its value. The use of separate doubled
configurations is sufficient because the separating function is
invariant over $T$.

\section{Describing invariant subspaces by marginals}\label{sec:profiles}

Theorem~\ref{thm:cogs} supplies the cog-generation input to the
reconstruction argument of Yang--Boja\'nczyk--Klin
\cite[Theorem~8.14 and Appendix~F.1]{YBK}. That argument describes
an invariant subspace by the allowed joint values of its marginal
coefficients. A strict inclusion of invariant subspaces must then
change one of these finite-dimensional spaces of allowed values,
giving a bound on chain length. We verify the remaining structural
hypotheses in arbitrary arity and state the resulting description
in the notation needed in Section~\ref{sec:consequences}.
All numbered references to \cite{YBK} refer to its expanded version.

\subsection{The higher-arity compatibility check}

The extraction arguments in \cite[Theorem~8.10 and Claim~F.5]{YBK}
use automorphisms that replace one coordinate by a twin while
fixing every other relevant vertex. To obtain them here, fix a
finite parameter set $S$, an increasing tuple
$a=(a_1,\ldots,a_r)$ of distinct points outside $S$, with $r>0$,
and any finite set $D\supseteq S\cup\aset a$.
Put $B=D\setminus\aset a$ and apply
Definition~\ref{def:double} over $B$. The resulting independent
doubling $(a,b)$ has each $b_i$ immediately after $a_i$ in the
finite order on
\[
 C=D\cup\aset b.
\]
For each $i\in[r]$, define
\[
 \sigma_i:C\setminus\{b_i\}\longrightarrow C\setminus\{a_i\},
 \qquad \sigma_i(a_i)=b_i,
\]
fixing every other point of its domain.

Each $\sigma_i$ is a partial isomorphism. To check the
$L$-relations, consider any relation instance in its domain.
Whether it contains both members of another twin pair is
unchanged by $\sigma_i$; if it does, the doubling rule makes
both the instance and its image false. Otherwise, their truth
values are read from the same tuple in $D$ under the map fixing
$D$ and sending every $b_j$ to $a_j$. This verifies preservation
and reflection for every arity, including instances with several
twins or repeated arguments. The order is preserved because
$a_i$ and $b_i$ are consecutive in $C$.
Homogeneity therefore extends $\sigma_i$ to an automorphism
$\pi_i\in H_S$. Since $D$ was arbitrary, it may include the
entire vertex support of a vector to which YBK's
successive-difference argument is applied.

One further structural fact used at the start of
\cite[Appendix~F.1]{YBK} is the absence of algebraicity:
every point outside a finite set $D$ has an infinite orbit
under its pointwise stabilizer. Indeed, for any $x\in A\setminus D$,
free amalgamation produces arbitrarily many copies of
$D\cup\{x\}$ over $D$,
with their new points distinct and in the same $D$-cut as $x$.
The extension property and homogeneity realize them as images
of $x$ over $D$. Consequently, an $H_S$-equivariant bijection
between increasing tuple orbits outside $S$ must preserve the
set of entries: each image entry is fixed by the stabilizer
of the source tuple, and the inverse gives the reverse
containment. Since the tuples are increasing, the bijection
is the identity after coordinate reindexing.

\subsection{The marginal description inherited from YBK}

We retain the notation for joint marginals because the duality
argument in Section~\ref{sec:consequences} uses the individual
coefficient evaluations. Fix a finite $S\subseteq A$ and $k\ge0$.
Let $\mathscr O$ be a finite collection of distinct increasing
$H_S$-orbits of degree at most $k$. For each $\OO\in\mathscr O$,
let $E_\OO$ be a finite-dimensional $\F$-vector space with
trivial $H_S$-action, and put
\[
 V=\bigoplus_{\OO\in\mathscr O}E_\OO\otimes_\F\F[\OO].
\]
Thus $v\in V$ has coefficients $v_\OO(a)\in E_\OO$, only
finitely many of which are nonzero. Choosing a basis of
$E_\OO$ identifies its summand with $\dim E_\OO$ labelled
copies of $\F[\OO]$, as in \cite[Remark~8.13]{YBK}.

Following \cite[Definition~F.1]{YBK}, group all projections
with the same target orbit $\PP$. Keep a separate coefficient
slot for each source and projection label:
\[
 Z_\PP=\bigoplus_{\substack{\OO\in\mathscr O,\ J\subseteq[\deg\OO]\\
                            p_J(\OO)=\PP}}E_\OO.
\]
For $v\in V$ and $b\in\PP$, the joint marginal coefficient is
\begin{equation}\label{eq:joint-profile}
 \rho_\PP:V\longrightarrow Z_\PP\otimes\F[\PP],\qquad
 \bigl(\rho_\PP(v)(b)\bigr)_{(\OO,J)}
   =\sum_{\substack{a\in\OO\\p_J(a)=b}}v_\OO(a).
\end{equation}
The target orbits $\PP$ range over all coordinate projections,
including the identity and empty projections.
Keeping the slots together records linear relations between
different marginal coefficients evaluated at the same tuple.

\begin{theorem}[Marginal description]\label{thm:profiles}
For an $H_S$-invariant subspace $W\le V$, set
\[
 E_\PP(W)=\{\rho_\PP(w)(b):w\in W\}\le Z_\PP,
\]
where $b\in\PP$ is fixed. This subspace is independent of the
choice of $b$, and
\begin{equation}\label{eq:profile-description}
 W=\{v\in V:\rho_\PP(v)(b)\in E_\PP(W)
                  \text{ for every }\PP\text{ and every }b\in\PP\}.
\end{equation}
\end{theorem}

\begin{proof}[Proof by transfer of the YBK argument]
When $k=0$, the statement is ordinary finite-dimensional linear
algebra. For general $k$, linearity makes $E_\PP(W)$ a subspace; transitivity on $\PP$
and invariance of $W$ make it independent of $b$.
These are the coefficient spaces in \cite[Theorem~8.14]{YBK},
and $\rho_\PP$ is their grouped projection from Definition~F.1,
with the multiplicities recorded by the spaces $E_\OO$.
The preceding no-algebraicity observation justifies grouping
by equality of increasing target orbits.

The proof in \cite[Appendix~F.1, Lemma~F.3]{YBK} reconstructs
membership in $W$ by successively removing assumptions that
marginals vanish. Its two steps carry over as follows.
The calculation in Claim~F.4 uses only composition of coordinate
projections and finite sums, so is unchanged. It produces a
balanced vector with coefficients in a finite-dimensional
subspace. Theorem~\ref{thm:cogs}, applied in each component of
a basis of that coefficient space, supplies the cog
decomposition used there from their Theorem~8.12.
Claim~F.5 then lifts these cogs to vectors of $W$ by successive
differences; the automorphisms required for this step were
constructed above. All these statements allow several labelled
orbit components, as in \cite[Remark~8.13 and Appendix~F.1]{YBK}.
Thus the induction of Lemma~F.3 proves
\eqref{eq:profile-description} without any further change
depending on relation arity.
\end{proof}

The exact-length argument of \cite[Corollary~8.16 and
Appendix~F.2]{YBK} now carries over as well. We give that
calculation in Section~\ref{sec:consequences}, followed separately
by the duality argument and the bound on additional parameters.

\section{Composition lengths and closed function spaces}\label{sec:consequences}

We apply the marginal description to obtain the length formula
by YBK's argument, and then use Evans's formulation of standard
duality to deduce the assertions about closed function spaces.

\subsection{The exact length}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Theorem~\ref{thm:profiles}, a strict inclusion of invariant
subspaces strictly increases at least one of the spaces
$E_\PP(W)$. The dimension count of \cite[Corollary~F.6]{YBK}
therefore gives
\begin{equation}\label{eq:length-upper}
 \length_{H_S}V\le\sum_\PP\dim Z_\PP
       =\sum_{\OO\in\mathscr O}(\dim E_\OO)2^{\deg\OO}.
\end{equation}
For a single increasing orbit $\OO$ of degree $d>0$, this is
the upper bound $2^d$. The matching lower-bound proof in
\cite[Corollary~8.16, Appendix~F.2]{YBK} constructs $2^d$
strict inclusions using partial alternating sums on an
independent doubling. The configuration and automorphisms
are available by Section~\ref{sec:profiles}; the remaining
coordinate-projection identities are unchanged in every arity
and characteristic. Thus $\length_{H_S}\F[\OO]=2^d$.
In degree zero, the orbit module is one-dimensional, giving
the same formula.

To pass from increasing orbits to $A^k$, partition $A^k$ into its
finitely many $H_S$-orbits. A tuple in one orbit is determined by
the increasing list of its distinct entries outside $S$, together
with a fixed pattern specifying repeated entries and entries from
$S$. This identifies its orbit equivariantly with an increasing
orbit of degree at most $k$. If several orbits give the same
increasing orbit, they supply a finite coefficient multiplicity.
Thus $\F[A^k]$ has the form of $V$ above, and has finite length.

The unordered stabilizer $\Aut(\A_0)_{(S)}$ contains $H_S$.
Every subspace invariant under that larger group is invariant under
$H_S$, so its chains satisfy the same finite bound. This proves
the unordered assertion as well.
\end{proof}

\subsection{Duality and the parameter bound}

The closed-function assertions follow by combining the marginal
description with standard vector-space duality. We use
\cite[Lemma~2.2]{Evans}: for the pairing
\[
 \langle v,f\rangle=\sum_{x\in A^k}v(x)f(x),
 \qquad v\in\F[A^k],\quad f\in\F^{A^k},
\]
annihilators give an inclusion-reversing correspondence between
invariant subspaces of $\F[A^k]$ and closed invariant subspaces
of $\F^{A^k}$. Here the annihilator consists of the elements
on the other side that pair to zero with the given subspace;
the double annihilator of a function space is its closure.

The deduction below follows the finite-dimensional separation
and pullback argument in the proof of
\cite[Corollary~2.3]{Evans}, using the joint marginals of
\cite[Theorem~8.14]{YBK}. Evaluating a marginal at a tuple
of length at most $k$ yields the uniform parameter bound.
The same deduction already applies to YBK's binary setting.

\begin{proof}[Proof of Corollary~\ref{cor:codes}]
Let $\F$ be finite and let $C\le\F^{A^k}$ be closed and
$H_S$-invariant. Let $\Psi:\F[A^k]\to V$ be the equivariant
orbit decomposition from the proof of Theorem~\ref{thm:main},
and put $W=\Psi(C^\perp)$. For every projected orbit $\PP$,
every $b\in\PP$, and every linear functional
$\lambda:Z_\PP\to\F$ vanishing on $E_\PP(W)$, define
\begin{equation}\label{eq:generator}
 f_{\PP,b,\lambda}(x)
 =\lambda\bigl(\rho_\PP(\Psi([x]))(b)\bigr),
 \qquad x\in A^k.
\end{equation}
These functions annihilate $C^\perp$, hence belong to $C$.
Equivariance fixes each of them over $S\cup\aset b$; since
$\deg\PP\le k$, this uses at most $k$ additional parameters.
Theorem~\ref{thm:profiles} and finite-dimensional separation
in each $Z_\PP$ show that their common annihilator is exactly
$C^\perp$. Double-annihilator duality therefore gives $C$ as
their closed linear span.

Finally, the field and all the spaces $Z_\PP$ are finite,
and there are only finitely many projected orbits $\PP$.
There are consequently only finitely many choices for the
subspaces $E_\PP(W)$. These determine $W$ by
Theorem~\ref{thm:profiles}, and duality then determines $C$.
\end{proof}

There is an immediate quotient consequence. Under the same
hypotheses, let $C_1<C_2\le\F^{A^k}$ be closed invariant
subspaces. The generators of $C_2$ from
Corollary~\ref{cor:codes} cannot all lie in the closed subspace
$C_1$. Thus some $f\in C_2\setminus C_1$ is fixed over
$S\cup T$ with $|T|\le k$, and $f+C_1$ is a nonzero fixed
element of $C_2/C_1$. This gives the uniform bound in the
present setting for the separation conclusion of
\cite[Corollary~2.3]{Evans}.

\section{Concluding remarks}

The proof uses free amalgamation to build independent copies and
to control the relations involving a new point and an extra
parameter within each finite construction. It uses the generic
order to place those copies in specified intervals. These are
the two structural requirements behind the argument.

The result settles the arbitrary-arity extension of the
free-amalgamation finite-length theorem in \cite{YBK}. It leaves
open the corresponding finite-length problem for general
structures homogeneous in finite relational languages, as posed
in \cite[Question~1.2]{Evans}. Extending the support-moving
argument to classes whose amalgamation imposes additional
relations would require a replacement for the free-amalgam type
calculation in Section~\ref{sec:finish-cogs}.

\section*{Acknowledgment}
The author was supported by the NRDI grant KKP~138270.

\begin{thebibliography}{99}
\bibitem{AH78}
F.~G.~Abramson and L.~A.~Harrington,
\emph{Models without indiscernibles},
Journal of Symbolic Logic \textbf{43} (1978), no.~3, 572--600.
\href{https://doi.org/10.2307/2273534}{doi:10.2307/2273534}.

\bibitem{BodirskyBook}
M.~Bodirsky,
\emph{Complexity of Infinite-Domain Constraint Satisfaction},
Lecture Notes in Logic \textbf{52},
Cambridge University Press, Cambridge, 2021.
\href{https://doi.org/10.1017/9781107337534}{doi:10.1017/9781107337534}.

\bibitem{BBGrowth}
M.~Bodirsky and B.~Bodor,
\emph{Permutation groups with small orbit growth},
Journal of Group Theory \textbf{24} (2021), no.~4, 643--709.
\href{https://doi.org/10.1515/jgth-2018-0220}{doi:10.1515/jgth-2018-0220}.

\bibitem{BPP42}
M.~Bodirsky, M.~Pinsker and A.~Pongr\'acz,
\emph{The 42 reducts of the random ordered graph},
Proceedings of the London Mathematical Society
\textbf{111} (2015), no.~3, 591--632.
\href{https://doi.org/10.1112/plms/pdv037}{doi:10.1112/plms/pdv037}.

\bibitem{BPT}
M.~Bodirsky, M.~Pinsker and T.~Tsankov,
\emph{Decidability of definability},
Journal of Symbolic Logic \textbf{78} (2013), no.~4, 1036--1054.
\href{https://doi.org/10.2178/jsl.7804020}{doi:10.2178/jsl.7804020}.

\bibitem{BodorMonadic}
B.~Bodor,
\emph{Classification of $\omega$-categorical monadically stable structures},
Journal of Symbolic Logic \textbf{89} (2024), no.~2, 460--495.
\href{https://doi.org/10.1017/jsl.2023.66}{doi:10.1017/jsl.2023.66}.

\bibitem{BFKM}
M.~Boja\'nczyk, J.~Fijalkow, B.~Klin and J.~Moerman,
\emph{Orbit-Finite-Dimensional Vector Spaces and Weighted Register Automata},
TheoretiCS \textbf{3} (2024), Article~13, 1--41.
\href{https://doi.org/10.46298/theoretics.24.13}{doi:10.46298/theoretics.24.13}.

\bibitem{CKT}
F.~Calderoni, A.~Kwiatkowska and K.~Tent,
\emph{Simplicity of the automorphism groups of order and tournament
expansions of homogeneous structures},
Journal of Algebra \textbf{580} (2021), 43--62.
\href{https://doi.org/10.1016/j.jalgebra.2021.03.028}{doi:10.1016/j.jalgebra.2021.03.028}.

\bibitem{Cameron}
P.~J.~Cameron,
\emph{Transitivity of permutation groups on unordered sets},
Mathematische Zeitschrift \textbf{148} (1976), no.~2, 127--139.
\href{https://doi.org/10.1007/BF01214702}{doi:10.1007/BF01214702}.

\bibitem{Eisenbud}
D.~Eisenbud,
\emph{Groups of order automorphisms of certain homogeneous ordered sets},
Michigan Mathematical Journal \textbf{16} (1969), 59--63.
\href{https://doi.org/10.1307/mmj/1029000166}{doi:10.1307/mmj/1029000166}.

\bibitem{Evans}
D.~M.~Evans,
\emph{Permutation modules for Ramsey structures},
arXiv:2603.29606v2, 2026.
\url{https://arxiv.org/abs/2603.29606v2}.

\bibitem{EHN}
D.~M.~Evans, J.~Hubi\v{c}ka and J.~Ne\v{s}et\v{r}il,
\emph{Ramsey properties and extending partial automorphisms for
classes of finite structures},
Fundamenta Mathematicae \textbf{253} (2021), 121--153.
\href{https://doi.org/10.4064/fm560-8-2020}{doi:10.4064/fm560-8-2020}.

\bibitem{Higman}
G.~Higman,
\emph{On infinite simple permutation groups},
Publicationes Mathematicae Debrecen \textbf{3} (1954), 221--226.
\href{https://doi.org/10.5486/PMD.1954.3.3-4.06}{doi:10.5486/PMD.1954.3.3-4.06}.

\bibitem{Holland}
W.~C.~Holland,
\emph{The lattice-ordered group of automorphisms of an ordered set},
Michigan Mathematical Journal \textbf{10} (1963), 399--408.
\href{https://doi.org/10.1307/mmj/1028998976}{doi:10.1307/mmj/1028998976}.

\bibitem{JunkerZiegler}
M.~Junker and M.~Ziegler,
\emph{The 116 reducts of $(\mathbb Q,<,a)$},
Journal of Symbolic Logic \textbf{73} (2008), no.~3, 861--884.
\href{https://doi.org/10.2178/jsl/1230396752}{doi:10.2178/jsl/1230396752}.

\bibitem{KPT}
A.~S.~Kechris, V.~G.~Pestov and S.~Todorcevic,
\emph{Fra\"iss\'e limits, Ramsey theory, and topological dynamics
of automorphism groups},
Geometric and Functional Analysis \textbf{15} (2005), no.~1, 106--189.
\href{https://doi.org/10.1007/s00039-005-0503-1}{doi:10.1007/s00039-005-0503-1}.

\bibitem{MacphersonSurvey}
D.~Macpherson,
\emph{A survey of homogeneous structures},
Discrete Mathematics \textbf{311} (2011), no.~15, 1599--1634.
\href{https://doi.org/10.1016/j.disc.2011.01.024}{doi:10.1016/j.disc.2011.01.024}.

\bibitem{MacphersonTent}
D.~Macpherson and K.~Tent,
\emph{Simplicity of some automorphism groups},
Journal of Algebra \textbf{342} (2011), no.~1, 40--52.
\href{https://doi.org/10.1016/j.jalgebra.2011.05.021}{doi:10.1016/j.jalgebra.2011.05.021}.

\bibitem{NR77}
J.~Ne\v{s}et\v{r}il and V.~R\"odl,
\emph{Partitions of finite relational and set systems},
Journal of Combinatorial Theory, Series A
\textbf{22} (1977), no.~3, 289--312.
\href{https://doi.org/10.1016/0097-3165(77)90004-8}{doi:10.1016/0097-3165(77)90004-8}.

\bibitem{NR83}
J.~Ne\v{s}et\v{r}il and V.~R\"odl,
\emph{Ramsey classes of set systems},
Journal of Combinatorial Theory, Series A
\textbf{34} (1983), no.~2, 183--201.
\href{https://doi.org/10.1016/0097-3165(83)90055-9}{doi:10.1016/0097-3165(83)90055-9}.

\bibitem{PongraczHenson}
A.~Pongr\'acz,
\emph{Reducts of the Henson graphs with a constant},
Annals of Pure and Applied Logic \textbf{168} (2017), no.~7, 1472--1489.
\href{https://doi.org/10.1016/j.apal.2017.01.009}{doi:10.1016/j.apal.2017.01.009}.

\bibitem{Thomas}
S.~Thomas,
\emph{Reducts of the random graph},
Journal of Symbolic Logic \textbf{56} (1991), no.~1, 176--181.
\href{https://doi.org/10.2307/2274912}{doi:10.2307/2274912}.

\bibitem{ThomasHypergraphs}
S.~Thomas,
\emph{Reducts of random hypergraphs},
Annals of Pure and Applied Logic \textbf{80} (1996), no.~2, 165--193.
\href{https://doi.org/10.1016/0168-0072(95)00061-5}{doi:10.1016/0168-0072(95)00061-5}.

\bibitem{TrussGraph}
J.~K.~Truss,
\emph{The group of the countable universal graph},
Mathematical Proceedings of the Cambridge Philosophical Society
\textbf{98} (1985), no.~2, 213--245.
\href{https://doi.org/10.1017/S0305004100063428}{doi:10.1017/S0305004100063428}.

\bibitem{YBK}
J.~Yang, M.~Boja\'nczyk and B.~Klin,
\emph{The Finite Length Property of the Rado Graph and Friends},
41st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS~2026),
LIPIcs \textbf{380}, Article~82, 82:1--82:27.
\href{https://doi.org/10.4230/LIPIcs.LICS.2026.82}{doi:10.4230/LIPIcs.LICS.2026.82}.
Expanded version: \href{https://arxiv.org/abs/2605.21681v2}{arXiv:2605.21681v2}.
\end{thebibliography}
\end{document}