\section{Folds, reflection bigraphs, and graph norms}\label{sec:prelim}
\subsection{Graphs, symmetries, and notation}\label{sec:graph-notation}
\begin{definition}\label{def:bigraph}
A \emph{bigraph} is a finite simple bipartite graph $H=(A,B;E)$
with distinguished vertex parts $A=A(H),B=B(H)$ and
$E=E(H)\subseteq A\times B$.
Simple means that there are no loops or repeated edges.
It is \emph{biregular} if all vertices of $A$ have one common
degree and all vertices of $B$ have another common degree.
\end{definition}
All maps of bigraphs preserve the specified parts. A
\emph{homomorphism} preserves edges; an \emph{isomorphism} is a
bijection preserving both edges and nonedges. An \emph{automorphism}
is an isomorphism from a graph to itself, and an \emph{endomorphism}
is a homomorphism from a graph to itself. Write $\Aut(H)$ for the
automorphism group.

Throughout the paper, our definitions and results concern connected
bigraphs with at least one edge.\footnote{For the disconnected case of
graph norms, see Garbe, Hladk\'y, and Lee~\cite[Theorem 3.1]{GHL}.}
Auxiliary graphs and subgraphs need not be connected.

We write $V(H)=A\sqcup B$, where $\sqcup$ denotes disjoint union,
$N_H(v)$ for the neighbors of $v$, and $\deg_H(v)=|N_H(v)|$.
For a vertex set $T$, put $N_H(T)=\bigcup_{v\in T}N_H(v)$.
We omit the subscript when the graph is clear.
The induced subgraph $H[U]$ keeps the vertices in $U$ and all
edges between them; $H-Z=H[V(H)\setminus Z]$.
A \emph{vertex cut} is a set whose deletion leaves at least two
connected components. The complete bigraph $K_{a,b}$ has parts of
sizes $a,b$ and every possible edge; $C_\ell$ is the cycle with
$\ell$ edges. A \emph{star} is $K_{1,d}$, with one center and $d$
leaves. The star at a vertex means the set of its incident edges.
In $\RR^{V(H)}$, $\mathbf e_v$ is one at $v$ and zero elsewhere.
Throughout, $\NN=\{0,1,2,\ldots\}$.

\begin{definition}\label{def:actions}
For a group $G$ acting on a set $U$, the \emph{orbit} of $u$ is
$Gu=\{gu:g\in G\}$ and its \emph{stabilizer} is
$\Stab_G(u)=\{g\in G:gu=u\}$.
The action is \emph{transitive} if it has one orbit.
For $G\le\Aut(H)$, the induced action on edges sends $uv$ to
$(gu)(gv)$; $G$ is \emph{edge-transitive} if any two edges are
related by this action. Transitivity on each part means that each
of $A,B$ is one vertex orbit.
\end{definition}
A permutation fixes a set \emph{pointwise} if it fixes every member,
and preserves it \emph{setwise} if it maps the set onto itself.
We put $\Fix(f)=\{v:f(v)=v\}$.
An \emph{involution} is a nonidentity permutation whose square is
the identity. A subgroup \emph{generated} by permutations consists
of all finite products of those permutations and their inverses.
Conjugating $f$ by $g$ means replacing it by $gfg^{-1}$.
The symmetric group $\Sym(U)$ consists of all permutations of $U$;
$\Sym(v)$ abbreviates $\Sym(\{1,\ldots,v\})$.
A \emph{transposition} $(u\ w)$ swaps $u,w$ and fixes everything else.

\begin{definition}\label{def:twins}
Vertices in the same part with identical neighborhoods are
\emph{twins}. A bigraph is \emph{twin-free} if each twin class is
a singleton. The \emph{uniform twin blowup} $F(a,b)$, for integers $a,b\ge1$, replaces each
left vertex of $F$ by $a$ independent copies and each right vertex
by $b$ independent copies, replacing each original edge by all
$ab$ edges between its two clone classes. These copies are also called
\emph{clones}.
\end{definition}

\subsection{Genuine folds and cut-percolation}
\begin{definition}\label{def:fold}
A \emph{fold involution} is a nonidentity automorphism $f$ with $f^2=1$ such that
no connected component of $H-\Fix(f)$ is preserved setwise by $f$.
Thus those components occur in exchanged pairs. Choose a union $L$
containing one component from each pair and set $Z=\Fix(f)$.
The associated \emph{oriented fold} $t$ is
\[
 t(v)=\begin{cases}v,&v\in L\cup Z,\\ f(v),&v\in f(L).\end{cases}
\]
We call $L$ the retained side, $L,f(L)$ the \emph{open sides},
and $Z$ their fixed boundary. The \emph{fold group} $\Fold(H)$ is
generated by all fold involutions of $H$.
\end{definition}

There is no edge from $L$ to $f(L)$, so $t$ is a graph endomorphism.
Conversely, a side partition $V(H)=L\sqcup Z\sqcup f(L)$ with
$Z=\Fix(f)$ and no edge between the two open sides gives a fold in
Definition~\ref{def:fold}. This is the fold convention of Coregliano
\cite[Definition 2.6]{Coregliano}.

\begin{definition}\label{def:percolation}
The bigraph $H$ is \emph{cut-percolating} if there are an edge $e_0$
and a possibly empty sequence of oriented folds $t_1,\ldots,t_\ell$ such that
\[
 E_0=\{e_0\},\qquad E_i=t_i^{-1}(E_{i-1}),\qquad E_\ell=E(H).
\]
Here a fold acts on an edge through its two endpoints. Equivalently,
a composition of oriented folds sends every edge to one edge.
\end{definition}

A \emph{word} of transformations means a finite sequence composed
in chronological order; it is \emph{collapsing} if its image is a
singleton. The empty word acts as the identity.
The order reversal in the equivalent formulation is useful: if
$q=t_\ell\circ\cdots\circ t_1$ is a collapsing word in chronological
order, its inverse-image percolation sequence uses
$t_\ell,t_{\ell-1},\ldots,t_1$.

\begin{definition}\label{def:atomic}
A nonidentity fold involution is \emph{atomic} if its nonfixed set
has exactly two connected components.
Its \emph{moved support} is $V(H)\setminus\Fix(f)$, the union of
these two components.
\end{definition}

\begin{lemma}\label{lem:decomposition}
Every fold involution is a product of commuting atomic fold
involutions. Every oriented fold is a composition of commuting
oriented atomic folds. Atomic folds are invariant under conjugation
by $\Aut(H)$.
\end{lemma}
\begin{proof}
For each exchanged component pair, restrict the involution to that
pair and fix all other vertices. This is an automorphism: every edge
leaving either component ends at a vertex fixed by the original
involution. It is atomic. The restrictions have disjoint moved
supports and multiply to the original involution. Orienting each
restriction toward its original retained component gives the
oriented decomposition. Conjugation preserves fixed sets and
connected components, proving the last assertion.
\end{proof}

\begin{remark}[Choices of halves]\label{rem:cl-percolation}
Conlon--Lee \cite[Section~3]{CL} fix a compatible pair of
exchanged halves for each fold involution and allow copying in
either direction. Definition~\ref{def:fold} instead allows the
choice to vary between occurrences of the same involution.
Both conventions generate exactly the same vertex maps, and
hence the same percolating sequences after inserting intermediate
steps. Indeed, Lemma~\ref{lem:decomposition} expresses every
oriented fold as a composition of oriented atomic folds.
An atomic fold has just one exchanged component pair, so its
halves are unique up to reversal and both orientations are
available under either convention. Thus
Definition~\ref{def:percolation} agrees with Conlon--Lee's
percolation notion, independently of their fixed choices.
This argument uses all fold involutions, including the atomic
restrictions; it need not hold for a prescribed smaller family.
\end{remark}

\begin{remark}[The broader fixed-cut group]\label{rem:broader-cut-group}
Conlon--Lee's original definition of a cut involution required
only that its fixed set be a vertex cut; their folding maps also
required exchanged sides. They later included the latter condition
in the definition and explained that it had been used implicitly
throughout their earlier work~\cite[Section~3, footnote~1]{CLdom}.
Our assertions about their conjecture use this clarified convention.

The literal fixed-cut condition defines a possibly larger group,
which we denote by $\operatorname{Cut}(H)$: it is generated by
all involutions $f\in\Aut(H)$ for which $\Fix(f)$ is a vertex
cut, even if $f$ preserves some component of $H-\Fix(f)$ setwise.
Edge-transitivity of this larger group does not imply
cut-percolation.

For example, let $F$ be the incidence bigraph of the Fano plane
and put $H=F(2,1)$, doubling each point vertex. Then $H$ has
21 vertices and 42 edges. Its fold group consists precisely of
the independent swaps within the seven point twin pairs, by
Corollaries~\ref{cor:design-no-atoms} and~\ref{cor:all-twins}.
These swaps have 21 edge orbits, so $H$ does not cut-percolate.
Nevertheless, $\operatorname{Cut}(H)$ is edge-transitive.

To see this, realize the Fano points as the nonzero vectors of
$\mathbb F_2^3$, where $\mathbb F_2$ is the field with two
elements, and its lines as the two-dimensional subspaces,
with adjacency given by membership.
The linear involution
$T(x_1,x_2,x_3)=(x_1+x_2,x_2,x_3)$ fixes the point
$c=(1,0,0)$ and every line through it. Lift $T$ to an
automorphism $\widehat T$ of $H$ by retaining clone indices,
and let $\tau_c$ swap the clones above $c$. The two maps
commute. The moved subgraph of $\widehat T\tau_c$ consists
of the blowup of an eight-cycle and two isolated clones
above $c$. Indeed, $T$ moves four points and four lines;
their induced graph has degree two and no four-cycle,
so it is an eight-cycle. Thus the fixed set of
$\widehat T\tau_c$ is a vertex cut, and
$\widehat T=(\widehat T\tau_c)\tau_c$ belongs to
$\operatorname{Cut}(H)$. The fixed-cut condition is invariant
under conjugation, so conjugating by invertible linear maps
supplies all elementary coordinate additions
$x_i\mapsto x_i+x_j$ with $i\ne j$. These generate the invertible linear
group over $\mathbb F_2$, which is transitive on incident
point--line pairs. Together with the twin swaps, they give
edge-transitivity on $H$.
\end{remark}

\subsection{Reflection bigraphs and tensor products}\label{sec:reflection-bigraphs}
A hyperplane is a linear subspace of codimension one.
An orthogonal reflection fixes a hyperplane pointwise and reverses
the perpendicular direction. The reflecting hyperplane is also
called a \emph{mirror}; its two \emph{closed halfspaces} are the
sets with nonnegative and nonpositive scalar product with a normal
vector, respectively. A finite real reflection group $W$
is a finite group generated by such reflections. Its reflection
hyperplanes divide the orthogonal complement of the common fixed
subspace (the \emph{essential space}) into open
cones called \emph{chambers}. Choose one, called the
\emph{fundamental chamber}; its closure is the \emph{closed
fundamental chamber}. Its bounding hyperplanes are its \emph{walls}.
The reflections in these walls form a generating set $S$ of
\emph{simple reflections}, and $(W,S)$ is a finite Coxeter system.
The word simple refers to
the chosen chamber; the other reflections need not be wall
reflections of that chamber.

The \emph{rank} of the system is $|S|$. For distinct $s,t\in S$,
let $m_{st}$ be the order of $st$. The \emph{Coxeter diagram}
has node set $S$ and an edge $st$ when $m_{st}\ge3$, labeled
by $m_{st}$; labels three are usually omitted. A system is
\emph{irreducible} when this diagram is connected.

We use the following standard facts \cite{Humphreys}. For
$I\subseteq S$, write $W_I=\langle I\rangle$ for the standard
parabolic subgroup, that is, the subgroup generated by $I$.
A point in the closed fundamental chamber has
stabilizer $W_I$, where $I$ consists of the walls containing that
point. Every $W$-orbit meets the closed chamber in exactly one point.

Write $W/W_I$ for the set of left cosets
$wW_I=\{wu:u\in W_I\}$, and similarly for $J\subseteq S$.

\begin{definition}\label{def:reflection}
For $I,J\subseteq S$ with $I\cup J=S$, the
\emph{reflection bigraph} $\mathcal R(W;I,J)$ of Conlon and Lee
\cite{CL} has distinguished
parts $W/W_I$ and $W/W_J$ and edge set
\[
 \{(wW_I,wW_J):w\in W\}.
\]
Repeated pairs are identified, so the graph is simple.
The condition $I\cup J=S$ is exactly the connectedness requirement:
paths change a coset representative by right multiplication in
$W_I$ or $W_J$. We allow $I=S$ or $J=S$, direct products of Coxeter
systems (\emph{reducible} systems), and the trivial system
$W=\{1\}, S=\varnothing$.
\end{definition}

For $\Sym(v)$, the \emph{type-$A_{v-1}$ simple system} consists of
the adjacent transpositions $(1\ 2),(2\ 3),\ldots,(v-1\ v)$.
Its geometric action permutes coordinates in $\RR^v$; the essential
space is the subspace whose coordinate sum is zero.

Geometrically, choose points $x,y$ in the closed chamber with
stabilizers $W_I,W_J$. The two parts are distinguished copies of
their orbits, and the edges are $(wx,wy)$ for $w\in W$.
This description also explains why an edge never crosses a
reflection hyperplane: both endpoints belong to the same closed
chamber $w\overline C$.

\begin{example}\label{ex:reflection-cycle}
Let $W=\Sym(3)$, $S=\{s_1,s_2\}$ with $s_1=(1\ 2)$ and
$s_2=(2\ 3)$, and take $I=\{s_1\}$, $J=\{s_2\}$.
Each part has three cosets and every vertex has degree two.
The resulting graph is $C_6$. In its planar representation the
chambers have angle $60^\circ$, and the two point orbits form
the alternating vertices of a regular hexagon; see
Figure~\ref{fig:reflection-chambers}.
\end{example}

\begin{definition}\label{def:tensor}
The \emph{part-respecting tensor product} $R\otimes F$ has parts
$A(R)\times A(F)$ and $B(R)\times B(F)$. The vertices $(v,x)$ and
$(w,y)$ are adjacent exactly when $vw\in E(R)$ and $xy\in E(F)$.
\end{definition}

Products of reflection bigraphs are reflection bigraphs, using the
direct product of the Coxeter systems and the unions of the
corresponding parabolic subsets. The complete bigraph $K_{a,b}$ is
reflection: use $W=\Sym(a)\times\Sym(b)$ with left stabilizer
$\Sym(a-1)\times\Sym(b)$ and right stabilizer
$\Sym(a)\times\Sym(b-1)$, under the standard type-$A$ simple systems.
The factors with size one are trivial. Consequently
$F(a,b)\cong F\otimes K_{a,b}$.

\subsection{Weak norming and decorated H\"older}\label{sec:analytic-formulations}
Let $(\Omega,\mu)$ and $(\Lambda,\nu)$ be probability spaces. For
bounded measurable functions of two variables, called \emph{kernels},
$W_e:\Omega\times\Lambda\to\RR$, set
\begin{equation}\label{eq:hom-density}
 t_H((W_e)_{e\in E})=
 \int_{\Omega^A\times\Lambda^B}
 \prod_{uv\in E}W_{uv}(x_u,y_v)
 \,d\mu^A(x)\,d\nu^B(y).
\end{equation}
The measures $\mu^A,\nu^B$ are products: each source vertex is
assigned an independent target variable in the corresponding space.
We call this a \emph{decorated homomorphism density}
and write $t_H(W)$ when all edge kernels are equal.

\begin{definition}\label{def:weak-norming}
The bigraph is
\emph{weakly norming} if $W\mapsto t_H(|W|)^{1/|E|}$ is a norm
on the space of bounded kernels, modulo equality almost everywhere,
for every such pair of probability spaces. This is the bigraph
formulation, with independent measures on the two vertex parts.
\end{definition}
Here being a norm means positivity away from zero, absolute
homogeneity, and the triangle inequality. Functions agreeing outside
a set of measure zero are identified.

The \emph{weak H\"older property}, also called \emph{decorated
H\"older}, is the following inequality for every family of
nonnegative bounded kernels:
\begin{equation}\label{eq:holder}
 t_H((W_e)_{e\in E})\le
 \prod_{e\in E} t_H(W_e)^{1/|E|}.
\end{equation}
The implication to weak norming is elementary. For bounded kernels
$U,V$, we have $|U+V|\le |U|+|V|$. Expanding the product defining
$t_H(|U|+|V|)$ and applying \eqref{eq:holder} term by term gives
the triangle inequality. Taking one edge kernel $W\ge0$ and all
others equal to one gives $t_H(W)^{1/|E|}\ge\int W$, ensuring
positivity away from zero; absolute homogeneity is automatic.
The converse is Hatami's theorem \cite{Hatami,Coregliano}.
We also use two known structural facts. Reflection bigraphs are
weakly norming \cite[Theorem 1.2]{CL}, and a connected weakly
norming bigraph has a part-preserving edge-transitive automorphism
group \cite{Sidorenko,Coregliano}. The appendix uses this latter
fact to obtain transitivity on point--block incidences, called
\emph{flags}; see Definition~\ref{def:design}.

\paragraph{Finite models and activities.}\label{def:finite-model}
A \emph{finite-state model} uses finite target sets $\mathcal I$
and $\mathcal J$, nonnegative vertex weights $(\alpha_i)$ and
$(\beta_j)$, and nonnegative kernels given by matrices. The weights are called
\emph{activities}. Its \emph{partition function} is the unnormalized
sum over all assignments:
\begin{equation}\label{eq:finite-model}
 Z=\sum_{x\in\mathcal I^A,\ y\in\mathcal J^B}
 \prod_{u\in A}\alpha_{x_u}\prod_{v\in B}\beta_{y_v}
 \prod_{uv\in E}W_{uv}(x_u,y_v).
\end{equation}
When $Z>0$, dividing each assignment's weight by $Z$ gives a
probability measure; expectations in that measure are denoted by
$\EE_W$ when all kernels equal $W$.
Normalizing the activities
multiplies every homomorphism integral by a common factor, which
cancels in \eqref{eq:holder}. Zero activities and zero kernel entries
are obtained by continuity from strictly positive finite models.

\paragraph{Symmetric kernels.}
A \emph{graphon} is a symmetric measurable kernel
$W:[0,1]^2\to[0,1]$. For a symmetric kernel on one probability
space, $t_H^{\mathrm{sym}}(W)$ denotes the usual graph density:
every vertex, in either part, is assigned a variable in that space.
For connected bipartite graphs, testing weak norming on symmetric
kernels gives the same class as the bigraph formulation above.
Approximating kernels by \emph{finite-step kernels}, constant on
the rectangles of finite measurable partitions, reduces the
question to weighted finite
probability spaces, represented by interval partitions.
Here is the reduction. Write $H^{\mathsf{op}}$ for $H$ with its two
parts exchanged. Given a kernel $W:\Omega\times\Lambda\to\RR$,
its transpose is $W^{\mathsf T}(y,x)=W(x,y)$. Put $W$ and
$W^{\mathsf T}$ in the two off-diagonal blocks of a
symmetric kernel $B_W$ on the equally weighted disjoint union
$\Omega\sqcup\Lambda$, with zeros in the diagonal blocks. Connectedness
gives, with $n=|V(H)|$,
\[
 t_H^{\mathrm{sym}}(|B_W|)
 =2^{-n}\bigl(t_H(|W|)+t_{H^{\mathsf{op}}}(|W|)\bigr).
\]
If the two terms on the right agree for every nonnegative kernel,
the symmetric triangle inequality immediately gives the bigraph one.
Otherwise choose a nonnegative kernel $Z$ with
$t_H(Z)>t_{H^{\mathsf{op}}}(Z)$, transposing a witness and exchanging
its probability spaces if necessary. For kernels $U,Z$ on possibly
different pairs of spaces, their tensor product is
\[
 (U\otimes Z)((x,x'),(y,y'))=U(x,y)Z(x',y'),
\]
on the product probability spaces; $Z^{\otimes r}$ is its $r$-fold
tensor power. Apply the symmetric triangle
inequality to the embeddings of $U\otimes Z^{\otimes r}$ and
$V\otimes Z^{\otimes r}$. Divide by
$2^{-n/|E(H)|}t_H(Z)^{r/|E(H)|}$ and let $r\to\infty$.
Multiplicativity of homomorphism densities makes the opposite
orientation vanish, leaving the bigraph triangle inequality.
Absolute homogeneity is immediate. For positivity, rescale a
nonzero nonnegative kernel $W$ so that $W\le1$. With
$a=|A|$ and $b=|B|$, two applications of Jensen's inequality give
\[
 t_H(W)\ge t_{K_{a,b}}(W)\ge\left(\int W\right)^{ab}>0.
\]
The converse follows by restricting the bigraph norm to symmetric
kernels. This identifies the weak-norming notion used in the
step-Sidorenko characterization \cite{DGRR} with the one used here.

\begin{definition}\label{def:step-sidorenko}
For a finite measurable partition $\mathcal P$ of $[0,1]$ into
sets of positive measure, let $W_{\mathcal P}$ be the graphon
whose value on $P\times Q$ is the average of $W$ there.
The bigraph $H$ has the \emph{step Sidorenko property} if
$t_H^{\mathrm{sym}}(W_{\mathcal P})\le t_H^{\mathrm{sym}}(W)$
for every graphon $W$ and every such partition $\mathcal P$.
\end{definition}
Weak norming implies this inequality directly. The kernel
$W_{\mathcal P}$ is an $L^1$ limit of finite averages of
$W(\phi(x),\psi(y))$, where $\phi,\psi$ independently range over
measure-preserving bijections within the partition classes.
Each such relabeling preserves the graph norm, so the triangle
inequality bounds the norm of every average by $\|W\|_{r(H)}$.
Continuity of homomorphism densities on uniformly bounded kernels
gives the same bound in the limit. The converse for connected
graphs is proved in \cite[Theorem 3.25]{DGRR}.

\subsection{One fold is one Cauchy--Schwarz inequality}\label{sec:fold-cs}
Here is the analytic meaning of the copying operation in
Figure~\ref{fig:c6-percolation}. Give the edges a coloring
$c:E(H)\to\{1,\ldots,r\}$ and choose a nonnegative kernel $W_i$ for
each color. Write $t_H(c)$ for the integral in which edge $e$
carries $W_{c(e)}$.
A coloring is \emph{monochromatic} if all edges have the same color.
A \emph{star coloring} assigns the same color to all edges at each
vertex of one specified part.

\begin{definition}\label{def:fold-stable}
An edge coloring $c$ is \emph{fold-stable} if, for every oriented
fold $t$, the colored bigraphs $(H,c)$ and $(H,c\circ t)$ are
isomorphic. Here a colored isomorphism preserves the parts and
the edge colors. Thus stability allows a relabeling of vertices;
the stronger identity $c\circ t=c$ allows none.
\end{definition}

Let $f$ exchange the open sides $L,R$ and fix $Z$, and let
$t_L,t_R$ be its two oriented folds. Keep the variables at $Z$
fixed and integrate over $L$ and $R$ separately. This gives
nonnegative functions $\Phi_L,\Phi_R$ of the variables at $Z$.
If $K_Z$ is the product of the kernels on edges with both endpoints
in $Z$, then
\[
 t_H(c)=\int K_Z\,\Phi_L\Phi_R.
\]
Apply Cauchy--Schwarz with measure $K_Z$ times the product measure
on the variables at $Z$. Since $f$ preserves the parts, the two
sides use the same product measures. Copying the decoration on
$L$ to $R$ therefore gives $\int K_Z\Phi_L^2$, and copying in the
other direction gives $\int K_Z\Phi_R^2$. Thus
\begin{equation}\label{eq:fold-cs}
 t_H(c)\le
 \bigl(t_H(c\circ t_L)\,t_H(c\circ t_R)\bigr)^{1/2}.
\end{equation}
The action on colors is exactly the inverse-image action on marked
sets: an edge has color $i$ after the fold precisely when its image
had color $i$ before the fold.

For the sequence on $C_6$, the retained open sides are
$\{0,5\}$, then $\{1,2\}$, then $\{0,1\}$.
Their fixed vertex pairs are $\{1,4\}$, $\{0,3\}$, and $\{2,5\}$.
Starting with edge $01$, copying these sides marks
$\{01,12\}$, then $\{01,12,45,50\}$, then all six edges.
Equation~\eqref{eq:fold-cs} explains each step of this process.

A single chosen branch of these inequalities does not yet prove
decorated H\"older: the other factors must also be accounted for.
Conlon--Lee's iteration keeps the full Cauchy--Schwarz tree, repeats
a percolating sequence, and makes the total weight of monochromatic
branches tend to one \cite[Section 3]{CL}. Here the tree records
both factors on the right of each application of
\eqref{eq:fold-cs}; each branch receives one half of its parent's
weight. This is the known implication from cut-percolation to
weak norming.