\section{The secondary operation and its normalization}\label{op:section}

The definition of $h$ in Section~\ref{alg:section} is a diagram chase:
for a class $[c]\in\ker s$, choose a class $u\in\J(K_2)$ with
$r(u)=[c]$, apply $b$, and take its image in $\coker\gamma$. To use an
index calculation we need to express this procedure as a count on
specified cobordisms and one-parameter families. The primitive constructed
in the preceding section provides one part of this expression. The other
part is the chain homotopy used in the proof of the exact triangle.

There is a normalization to check. A chain homotopy between a composite
map and zero does not by itself ensure that the class it produces maps
back to $[c]$ under $r$. For the particular families in
\cite[Sections~6--8]{KM-tri}, this follows from the two-parameter
pentagon and the identification of one of its boundary operations with
the identity. We will spell out that argument and thereby identify our
geometric count with the map $h$ already defined.

In this section the geometric maps and homotopies are assumed to satisfy
the indicated boundary identities. Theorem~\ref{en:simultaneous-data}
constructs compatible auxiliary data for them. The algebra below states
precisely how those identities are used.

\subsection{Conventions and geometric reductions}
For complexes over $\F$, write $\delta U=dU+Ud$ for the differential on a
space of maps. All sums and formulas below are in characteristic two. We do
not impose an absolute integer grading on $\J$; the dimension of each
parametrized moduli space is supplied separately by its deformation index.

We use the exact triangle in which all three local webs are of
$L$-type, calling it the \emph{all-$L$ triangle} to distinguish it from
the triangles involving two $K$-type webs. Write its geometric complexes
and cobordism maps as
\[
 \widehat Z\xrightarrow{\widehat H}\widehat X
 \xrightarrow{\widehat F}\widehat Y
 \xrightarrow{\widehat G}\widehat Z.
\]
Choose the cyclic order so that the local reductions give
\[
 X=A\oplus B,\qquad Y=W\oplus B,\qquad Z=C.
\]
The hats distinguish the complexes before these local reductions.
The complexes without hats are the geometric Floer complexes on the
reduced webs; we have not replaced them by their homology. The triangle
expansion and collapse give maps $I_Z:C\to\widehat Z$ and
$P_Z:\widehat Z\to C$. Similarly, the two square resolutions give maps
$I_X,P_X,I_Y,P_Y$ between the other complexes and the displayed direct
sums. These are the geometric maps inducing the isomorphisms on homology
in \cite[Section~8]{KM-tri}. In the
notation of the split diagram in \cite[Section~8]{KM-tri}, the cyclic
relabelling is
\[
 K_2^{\mathrm{printed}}\mapsto K_0,\quad
 K_1^{\mathrm{printed}}\mapsto K_2,\quad
 K_0^{\mathrm{printed}}\mapsto K_1,\quad
 L_0^{\mathrm{printed}}\mapsto L_1.
\]
On homology, therefore, the three maps become $(s,0)^t$,
$\operatorname{diag}(a,1_B)$ and $(r,0)$, respectively. The crossing
conventions for the embedded foams are specified in
Section~\ref{geo:section}.

In particular, the comparison in \cite[Section~8]{KM-tri} gives
\begin{equation}\label{op:homology-comparison}
 H(P_ZI_Z)=1,\qquad H(I_YP_Y)=1,\qquad
 H(P_Z\widehat G I_Y)=(r,0).
\end{equation}
Using $H(I_YP_Y)=1$ in the last identity also gives
\begin{equation}\label{op:projection-comparison}
 H(P_Z\widehat G)=(r,0)H(P_Y).
\end{equation}
No chain-level equality between the transported map $\widehat G$ and
$(r,0)$ is required.

\subsection{A primitive on the unreduced complex}

We begin with a source cycle $c\in C$ for which $[sc]=0$. To carry out
the diagram chase on the all-$L$ triangle, we need a chain
$\widehat x\in\widehat X$ with
$d\widehat x=\widehat H I_Zc$. We obtain it by transporting the
primitive $T_Asc$ through the local reductions, keeping the comparison
homotopies explicitly.

Suppose, as in Section~\ref{loc:section}, that geometric homotopies
$T_X$ and $L_H$ have been chosen with
\begin{equation}\label{op:source-comparison}
 \delta T_X=1+I_XP_X,\qquad
 \delta L_H=P_X\widehat H I_Z+(s,0)^t.
\end{equation}
The first uses the square relation; the second has a theta local boundary
on its $A$ component and a handcuff local boundary on its $B$ component.
Their geometric construction is included in Section~\ref{loc:section}.
We use these equalities for the geometric chain maps themselves;
knowing only the corresponding identities on homology would not suffice
for the following formula.

Let $c$ be either of the full source cycles $c_1,c_3$ specified in
Section~\ref{alg:section}. The terminal reductions there and in
Section~\ref{loc:section} give
\begin{equation}\label{op:terminal}
 \delta T_A=1+\Pi,\qquad \Pi s c=0.
\end{equation}
In particular $y=T_As c$ satisfies $dy=s c$. Define
\begin{equation}\label{op:primitive}
 \widehat c=I_Zc,\qquad
 \widehat x=T_X\widehat H\widehat c+
 I_X\bigl(L_Hc+(T_As c,0)^t\bigr).
\end{equation}

\begin{lemma}\label{op:primitive-lemma}
Under \eqref{op:source-comparison} and \eqref{op:terminal},
$d\widehat x=\widehat H\widehat c$.
\end{lemma}
\begin{proof}
Since $c$ is a cycle and all $I,P,H,s$ maps in this calculation are chain
maps, differentiating \eqref{op:primitive} gives
\begin{align*}
 d\widehat x
 &=\widehat H\widehat c+I_XP_X\widehat H\widehat c
   +I_X\bigl(P_X\widehat H I_Zc+(s c,0)^t\bigr)
   +I_X\bigl(s c+\Pi s c,0\bigr)^t\\
 &=\widehat H\widehat c+I_X(\Pi s c,0)^t.
\end{align*}
The last term vanishes by \eqref{op:terminal}. Only its value on the full
source $c$ is used. It need not vanish on an individual half-foam summand.
\end{proof}

\subsection{Normalization by the pentagon}

In the proof of the all-$L$ exact triangle, the composite of two
successive foam cobordisms is equipped with a one-parameter family of
metrics. One end stretches along the intermediate web; the other
stretches along a local three-sphere. The associated map
$\widehat J:\widehat Z\to\widehat Y$ satisfies
$\delta\widehat J=\widehat F\widehat H$. For the next pair of
cobordisms in cyclic order there is a corresponding map $J_{GF}$ with
$\delta J_{GF}=\widehat G\widehat F$.

The two-parameter family over the pentagon compares these
nullhomotopies. Together with the calculation on its special face, it
gives a further chain homotopy $R$ satisfying
\begin{equation}\label{op:normalization}
 \delta R=\widehat G\widehat J+J_{GF}\widehat H+1_{\widehat Z}.
\end{equation}
Indeed, the pentagon boundary formula in
\cite[Section~6, equation~(12)]{KM-tri} has two ordinary composition
faces, two sphere faces and the $S^{2,1}$ face. The sphere terms vanish
by the stated connected-sum calculation, and Section~7 identifies the
special-face operation up to chain homotopy with the identity. Combining
that homotopy with the pentagon gives \eqref{op:normalization}.

\begin{proposition}[Normalized cycle formula]\label{op:formula}
Assume that the actual geometric families satisfy
\eqref{op:normalization}, that \eqref{op:homology-comparison} holds, and
that $\widehat x$ is as in Lemma~\ref{op:primitive-lemma}. Let $\ell$ be
an actual cap cocycle on $B$ whose homology functional annihilates
$\im\gamma$. Then
\begin{equation}\label{op:scalar}
 \ell h([c])=
 \widehat q\bigl(\widehat J\widehat c+\widehat F\widehat x\bigr),
 \qquad
 \widehat q=\ell b\operatorname{pr}_W P_Y.
\end{equation}
\end{proposition}
\begin{proof}
Set $\widehat w=\widehat J\widehat c+\widehat F\widehat x$.
The identity $\delta\widehat J=\widehat F\widehat H$ and
$d\widehat x=\widehat H\widehat c$ imply $d\widehat w=0$.
Applying the normalization identity \eqref{op:normalization} to this
cycle gives
\begin{equation}\label{op:actual-boundary}
 \widehat G\widehat w
 =\widehat c+d\bigl(R\widehat c+J_{GF}\widehat x\bigr).
\end{equation}
To check it, apply \eqref{op:normalization} to $\widehat c$, and apply
$\delta J_{GF}=\widehat G\widehat F$ to $\widehat x$; the two
$J_{GF}\widehat H\widehat c$ terms cancel.

Let $u$ be the $W$ component of $H(P_Y)[\widehat w]$. Applying $H(P_Z)$
to \eqref{op:actual-boundary} and using
\eqref{op:projection-comparison} yields $r(u)=[c]$.
By the definition of the secondary map,
$h([c])=b(u)\bmod\im\gamma$. Evaluating with $\ell$ proves
\eqref{op:scalar}. Its independence of the lift $u$ follows from
$ba=\gamma$ on homology, exactly as in the definition of $h$.
\end{proof}

\begin{remark}\label{op:no-arbitrary-homotopy}
The identity term in \eqref{op:normalization} is what ensures that the
class $u$ maps to $[c]$, rather than to its image under an unspecified
automorphism. Merely knowing that a cone comparison is a
quasi-isomorphism would leave this ambiguity. There is a related
ambiguity in choosing $\widehat J$: two solutions of
$\delta\widehat J=\widehat F\widehat H$ differ by a cycle in the
mapping complex, and its contribution to \eqref{op:scalar} need not
vanish. Finally, $\widehat x$ need not be a cycle. If we replace
$\widehat F$ by a chain-homotopic map when evaluating it on
$\widehat x$, we must include the correction from that homotopy.
Formula~\eqref{op:scalar} avoids these ambiguities by using the specified
normalized family and the original chain map $\widehat F$.
\end{remark}

\subsection{Which families enter the action estimate}
Substitution of \eqref{op:primitive} into \eqref{op:scalar} gives four
types of term:
\begin{equation}\label{op:four-types}
\begin{split}
 &\widehat q\widehat J I_Zc,\qquad
 \widehat q\widehat F T_X\widehat H I_Zc,\\
 &\widehat q\widehat F I_X L_Hc,\qquad
 \widehat q\widehat F I_X(T_As c,0)^t.
\end{split}
\end{equation}
Substitute into $T_A,T_X,L_H$ the expressions in
Section~\ref{loc:section}. Each resulting summand is a count on a
composition of fixed cobordisms with just one cobordism carrying a
one-parameter family. Attach the source filling and the cap specified
by $\ell$ to obtain the corresponding closed foam. The dots on these
fillings remain in place, and the construction introduces no new dots.
The two evaluations using $\ell_2$ have three dots in total. Each of
the two fixed cap summands of the evaluation using $\ell_1$ has two.
Although the latter summands can be counted separately within this
chosen expansion, only their sum defines a functional on
$\coker\gamma$.

The estimate in Section~\ref{en:section} will be applied to the
complete capped terms in \eqref{op:four-types}. It bounds how far the
perturbations can lower their topological action. The map $R$, the
interior of the pentagon, and the homotopy identifying the $S^{2,1}$
operation with the identity do not occur in these terms. They establish
the normalization, so they must be compatible with the counted families,
but they do not themselves need to satisfy this small bound on the action
error. Theorem~\ref{en:simultaneous-data} preserves this distinction.

We must make these choices in the right order. First fix the families
of metrics used in the published construction, including the local
sphere models, the allowed family on $S^{2,1}$, and their common
restrictions at the corners. Only then choose the additional
perturbations for the family $\widehat J$ that enters the count.
These perturbations are supported on collars of the ordinary external
ends, away from the isolated local pieces. The two faces corresponding
to ordinary composition therefore use the same maps at their common
corners, and their sphere limits agree with the local models already
chosen.

Theorem~\ref{en:simultaneous-data} extends the choices over the
remaining families while preserving these restrictions. The available
generic metrics are those of \cite[Proposition~7.1]{KM-tri}; we do not
assert such an extension for an arbitrary local model chosen in advance.
This order of choices keeps the normalization required in
Proposition~\ref{op:formula}.