\section{Local relations and geometric chain homotopies}
\label{loc:section}

The local relations for foams have a familiar consequence. Suppose several
foam cobordisms agree outside a ball, and that the relative invariants of
the pieces inside the ball sum to zero. Then the maps induced by the full
cobordisms also sum to zero; this is \cite[Corollary~4.5]{KM-web}. We need a
chain-level version of this statement, together with a geometric description
of the chain homotopy.

The reason for keeping track of the homotopy is the argument to come.
After attaching specified foams at its incoming and outgoing ends, each
term will be a count of instantons on a closed foam, with the metric
varying in a one-parameter family. The topology of this closed foam enters
the index formula. An arbitrary algebraic chain homotopy would not give
us the four-manifold and foam needed for that calculation.

We first describe the cobordisms and the families of metrics that will
supply these chain homotopies. The marking data, including the framed
arcs used to define $\J$, are part of this description.
Proposition~\ref{loc:family-data} states that the data can be chosen
compatibly for all the families; its analytic proof is deferred to
Section~\ref{en:section}. We then combine the resulting identities and
apply them to the successive local reductions of $K_0$. This will give a
specified chain whose differential is $sc$, for each of the two source
cycles needed in the scalar calculation.

All complexes are over $\F$. If $f:E\to F$ is any linear map between complexes,
write $\delta f=d_Ff+fd_E$. Thus $\delta(fg)=(\delta f)g+f(\delta g)$,
and the maps associated to fixed cobordisms are $\delta$-closed. No absolute
integer grading on $\J$ is used in these identities.

\subsection{Local fillings, marked arcs and the empty factor}

Let $Q\subset S^3$ be the web obtained by intersecting a local foam with
the boundary of a four-ball. Suppose that $(P_j,\Sigma_j)$ is a finite
collection of fillings in four-balls, with a fixed identification of their
boundaries with $(S^3,Q)$ and a common boundary collar. Removing a small
ball disjoint from $\Sigma_j$ turns a filling into a cobordism from the
empty web to $Q$. Fix the puncture and a framed arc to the basepoint
on $S^3$, common to these fillings and disjoint from every foam. The resulting
cobordism defines a chain map
\[
 p_j:U\longrightarrow D,\qquad
 U=\CF(\varnothing),\quad D=\CF(Q).
\]
Here $U=\F u_\varnothing$, with $d_U=0$, is the geometric Floer complex
for the empty web constructed in Section~\ref{lm:section}. Although it
has only one generator, its boundary three-sphere still carries the Hopf
link and marking used in the definition of $\J$. We keep this sphere in
all the cobordisms below. The puncture and framed arc are precisely the
additional data in \cite[Definition~3.11]{KM-web}; we do not assert that
the construction is independent of the arc.

Suppose the fillings are inserted in cobordisms from a web $K$ to a web $K'$,
with a common exterior. Put
\[
 E=\CF(K),\quad F=\CF(K'),\quad
 S=\CF(K\sqcup Q),
\]
where the split union defining $S$ lies in one connected three-sphere.
The letters $D,E,F$ in this local construction are names for complexes;
in particular $E$ here is not the four-ended exterior of
Section~\ref{alg:section}.
Join the local ball to the incoming boundary by a thin tube in the common
exterior, disjoint from the foam and all dots. A regular neighborhood of the
incoming collar, tube and ball gives an insertion cobordism. Its map and
the map of the remaining exterior have types
\[
 L_j:E\longrightarrow S,\qquad O:S\longrightarrow F.
\]
Write $F_j:E\to F$ for the map of the original cobordism. The local
zero-handle and the one-handle given by the joining tube cancel. Thus the
insertion cobordism is the boundary connected sum of the cylinder on $K$
and the punctured filling, with their specified arcs. The cylinder on $K$
will be called the \emph{spectator cylinder}: the local replacement does
not change its foam. Stretching a neck between the insertion cobordism
and the remaining exterior gives a one-parameter family. Its two limits
are the original cobordism and the broken composite, so its chain
homotopy satisfies
\begin{equation}\label{loc:composition}
 \delta R_j=F_j+OL_j.
\end{equation}
If $F_j$ is a product representative, comparison with the strict chain
identity requires its product-continuation homotopy as well.

We will also use the excision cobordism that joins two Hopf-marked
three-spheres. It transfers $K$ and $Q$ from separate spheres to their
split union in one sphere, leaving an empty-web sphere as the other
component. This cobordism, which we call the Hopf-excision core, gives
chain maps
\begin{equation}\label{loc:types}
 \Psi_{K,Q}:E\otimes D\longrightarrow S\otimes U,
 \qquad V=\Psi_{K,\varnothing}:E\otimes U\longrightarrow E\otimes U.
\end{equation}
Let $V'$ denote the reverse core map. Both tensor factors in
\eqref{loc:types} represent connected boundary components, including the
empty-web components. They cannot be discarded when specifying the
four-manifold, markings or bundle sectors.

\subsection{Moving the local filling past the excision cobordism}

There are two ways to combine the filling $p_j$ with excision. We can
first create $Q$ in a separate sphere and then apply $\Psi_{K,Q}$.
Alternatively, we can first apply $V$ and then insert the filling into
the sphere containing $K$. Moving the filling from one side of the
excision region to the other gives the chain homotopy between these
composites. We denote it by $N_j$; the required identity is
\begin{equation}\label{loc:naturality}
 \delta N_j=\Psi_{K,Q}(1_E\otimes p_j)+(L_j\otimes1_U)V,
 \qquad N_j:E\otimes U\longrightarrow S\otimes U.
\end{equation}
To describe this family, observe that the Hopf surgery in the excision
cobordism is disjoint from product regions containing the webs. Choose
one such spectator region $\mathbb R\times B_Q$ connecting
the second incoming sphere to the outgoing sphere containing the split web.
Place the compact transition of the punctured filling there, with empty
incoming web and outgoing web $Q$, and with product structure near the
lateral boundary. The puncture and framed basepoint arc lie outside this
product ball. Translate the transition along the $\mathbb R$ coordinate.
At one end it occurs before excision, and at the other it occurs after
excision. These are exactly the two compositions in
\eqref{loc:naturality}. There is one relative-position parameter. At finite
parameter values an isotopy supported in the spectator tube identifies the
underlying marked cobordism; the family can be viewed as a family of pulled
back metrics and perturbations on that fixed cobordism.

The smooth-core version of this translation is described in
\cite[Section~4, equation~(20)]{ISST}. The partial marking is checked
below, and the web compactification is treated in Section~\ref{en:section}.
The singular local filling never crosses the Hopf surgery, and the
translation preserves its facets, dots and prescribed normal collars.

Fix the bundle and the gauge subgroup on the excision core before making
this translation. Within the spectator region retain the ordinary partial
marking of $p_j$. In particular, do not enlarge the marking across the
whole complement of $Q$ and then require its classes to extend over an
arbitrary $P_j$. That stronger extension condition is neither given nor
needed.

The ordinary filling factor at either limiting composition has product
Hopf-tube marking. On each connected component its marking domain minus
the singular set is the product of an interval with a Hopf-ball complement.
Restriction to either endpoint is consequently an isomorphism on $H^0$
and $H^1$ over $\F$. It is both left- and right-proper in the sense of
\cite[Definition~3.5]{KM-web}. These proper factors supply the
Mayer--Vietoris condition of \cite[Proposition~3.4]{KM-web}; one need not
assert properness for an isolated excision block with disconnected ends.
The additional finite core quotient remains part of the auxiliary map.

For clarity, an elementary endpoint calculation explains why the core
marking does not introduce new ordinary critical generators. Let $H$ be
the Hopf link in a ball $B_H$ and let $B_1,\ldots,B_m$ be spectator balls,
where $m=1$ or $2$. Set
\[
 M_m=S^3\setminus\bigl(H\cup\textstyle\bigcup_a\operatorname{int}B_a\bigr),
 \qquad U_H=B_H\setminus H.
\]
Attaching the simply connected region outside $B_H$ along its boundary
sphere shows that restriction gives an isomorphism
\begin{equation}\label{loc:endh1}
 H^1(M_m;\F)\xrightarrow{\ \cong\ }H^1(U_H;\F)=\F^2.
\end{equation}
Hence the marking enlargement has zero covering kernel, by
\cite[Lemma~2.4]{KM-web}. The ranks of $H^2$ are $m+1$ and $2$,
respectively. For $m=2$ the extra bundle choice is detected by the sphere
separating the two spectator balls. Fix its $w_2$ evaluation to be zero.
Every flat connection has this value, since a flat bundle on $S^2$ is
trivial. Compactness of the small-perturbation critical set excludes
critical points in the other sphere sector for sufficiently small
perturbations. This verifies the endpoint identification; it is not a
claim that all marking classes extend across all local fillings.

The subgroup for a disconnected composition must include its
Mayer--Vietoris classes. The explicit calculation in
Section~\ref{loc:torus-geometry} specifies this subgroup and its
restrictions at all ordinary and torus ends.

\subsection{A geometric inverse and the correlated torus ends}
\label{loc:torus-geometry}

Next we compare the composite of the excision cobordism and its reverse
with the product cobordism. At the chain level we need a homotopy
$H_{\mathrm{exc}}$ satisfying
\begin{equation}\label{loc:inverse}
 \delta H_{\mathrm{exc}}=VV'+1_{E\otimes U}.
\end{equation}
As with the preceding comparison, an inverse on homology is not enough:
we need to know the cobordisms on which the homotopy counts instantons.
The geometric comparison cuts the composite along a three-torus and
fills its two new boundary components with copies of $T^2\times D^2$.
There is a marking issue in making this comparison. Each torus end has
two critical lifts, but the allowable change of marking exchanges the
two lifts at both ends at once. We must therefore keep track of the
pair of lifts, rather than identify the two lifts independently at
each end. We give the marking calculation first.

\paragraph{The excision cobordism and its marking.}

Write $W_H,\overline W_H$ for the forward and reverse Hopf-excision cores,
$X_c$ for their composite, $N=T^2\times S^1$ for its internal torus,
and $X_p$ for the result of cutting there and inserting two
$T^2\times D^2$ caps. Let $Z_{\mathrm{cut}}$ be the common cut exterior.
The ordinary incoming and outgoing manifolds each have two
three-sphere components, including the retained empty-web factor.

The excision block is described by $T^2\times\mathcal U$, where
$\mathcal U$ is a disk with corners. The four solid-torus pieces attached to its sides contain
the Hopf annuli. After deleting those annuli, their complements are
torus collars extending the sides of $\mathcal U$. Deleting the spectator
$B^3\times I$ tubes does not change $H^1$, by Mayer--Vietoris with
intersection $S^2\times I$. Thus the marking-domain complement has the
same $H^1$ as $T^2\times\mathcal U$. Ordinary end restriction
identifies its two $H^1$ generators with the Hopf meridians.
The underlying disk and spectator construction are the excision model
of \cite[proof of Theorem 5.6]{KM-unknot} and \cite[Section 4]{ISST}.

Gluing $\mathcal U$ to its reverse along the two intermediate intervals produces
an annulus $\mathcal A$. Use inverse torus identifications on the reverse core;
there is no torus monodromy around $\mathcal A$. If $z$ denotes its angular class,
the composite marking complement, denoted $C_c$, has
\[
 H^1(C_c;\F)=H^1(T^2;\F)\oplus\F z.
\]
Cutting the central circle gives two annuli $\mathcal A_1,\mathcal A_2$ and sends
$z$ to $(z_1,z_2)$. Each $z_i$ restricts to $ds$ on its torus boundary
and to zero on every ordinary end. Filling the central boundary circle
by a disk kills $z_i$.

The cohomology of the marking complement is not the same as the
cohomology that specifies the additional finite quotient by changes of
marking. For the latter we use the ambient cobordism. The individual four-ended core
has $H^1(W_H;\F)=0$, as does its reverse. The two-component intermediate
boundary gives one Mayer--Vietoris class $c$ on the composite:
\[
 H^1(X_c;\F)=\F c,\qquad H^1(X_p;\F)=0.
\]
The class $c$ evaluates to one on the core loop crossing the two
different intermediate components, so it restricts to $z$ in the
marking complement. It is precisely the extra class needed for
composition across that disconnected boundary. Writing $\Gamma$ for this finite group on each of the indicated pieces,
we obtain
\begin{equation}\label{loc:torus-groups}
 \Gamma_{W_H}=\Gamma_{\overline W_H}=0,\quad
 \Gamma_c=\F c,\quad
 \Gamma_{\mathrm{cut}}=\F(z_1,z_2),\quad
 \Gamma_p=0.
\end{equation}
Their restrictions to ordinary external ends are zero in the ordinary
marking quotient. This follows also by restriction through
$H^1(S^3;\F)=0$.

More explicitly, in the complement basis $(x_i,y_i,z_i)$ the ordinary
restriction has matrix
$\left(\begin{smallmatrix}1&0&0\\0&1&0\end{smallmatrix}\right)$,
up to the fixed meridian interchange, and torus restriction is the
identity. After capping, only $(x_i,y_i)$ remain and ordinary restriction
is injective. In ambient cohomology, restriction across $N$ is
$(a,b)\mapsto(a+b)ds$, whose kernel is the diagonal. These matrices
also show that no further endpoint-trivial partial-marking kernel
survives on the product side. The fixed product coordinates are used:
the meridian coefficient alone would not rule out extra $dx,dy$ terms
after an arbitrary change of torus framing.

The capped manifold $X_p$ has two product components. A connected cap
attached along one connected boundary cannot join two components of $Z_{\mathrm{cut}}$.
It follows that $Z_{\mathrm{cut}}=Z_1\sqcup Z_2$, with each $Z_i$ carrying the ordinary
strong ends of its product component. In particular $N$ separates $X_c$.
Its absolute dual in $H^1(X_c;\F)$ is zero. The nonzero class $c$ above
is the disconnected-composition class, not the dual of this separating
cut. Consequently the general trace hypothesis involving the class dual
to the cut imposes no further condition in this topology.

Fix the flat bundle $P_T$ on $T^2$ with orthogonal half-turn holonomies.
Use its product pullback over $\mathcal U$, its reverse, and both caps. The gluing
identifications are fixed to agree with this one reference. On $N$ this
gives
\[
 w_2(P_N)=\operatorname{pr}_{T^2}^*w_2(P_T);
\]
both mixed two-torus evaluations are zero. The third holonomy of the
flat connection is the identity. This specification is needed: merely
requiring nonzero $w_2[T^2]$ would leave other sectors.

The spectator region is excluded from this marking. Thus this core
construction preserves the ordinary marking, proper factors and framed
arcs of the translation in \eqref{loc:naturality}.

We also need to realize the class $z$ by a change of the marking
isomorphism, rather than merely record it in cohomology. This change is
made on the Hopf core; it need not extend as a gauge transformation
across the spectator foam. For existence, let $L_z$ be the real line
bundle with $w_1(L_z)=z$. A unit section of
$L_z\otimes(\mathbb R\oplus E_\mu)$ supplies local unit quaternions
whose conjugations patch to a marking automorphism with lift
obstruction $z$. The first obstruction is in $H^4(C_c;\mathbb Z)$,
which vanishes for this core with boundary, and there are no higher
obstructions in dimension four. Its square is liftable. This realizes
the indicated order-two deck action without imposing a new condition
on the spectator foam.

\paragraph{Retaining the two torus lifts jointly.}

In the specified bundle sector on $N$, the flat critical point has two
lifts when we take the quotient by determinant-one gauge transformations.
Denote them by $t_0,t_1$. The angular gauge class $ds$ exchanges them.
Each is acyclic: the cohomology with coefficients in its adjoint local
system is zero. Indeed, that local system is a sum of three real
characters, each nontrivial on $T^2$. The same argument proves
acyclicity on $T^2\times D^2$.

On the pair of oppositely oriented torus boundaries, the restriction
of \eqref{loc:torus-groups} acts by
\[
 (t_i,t_j)\longmapsto(t_{i+1},t_{j+1}).
\]
Thus the four ordered pairs of critical lifts give two orbits. The
complex for the pair of torus boundaries, with this quotient, is
\begin{equation}\label{loc:joint-torus}
 \mathcal T_\Delta=\F\mathfrak a\oplus\F\mathfrak b,\qquad
 \mathfrak a=[00]=[11],\quad \mathfrak b=[01]=[10].
\end{equation}
It has zero differential: the two orbits differ in grading by four
and admit no index-one trajectory. One may equivalently begin with
the two perfect determinant-one torus complexes and take their
diagonal quotient. Quotienting them separately would also identify
$\mathfrak a$ with $\mathfrak b$, which the geometry does not permit.

\begin{lemma}[The matching torus limits]\label{loc:matching-orbit}
Use the same exterior metric, equation, dot data and relative sectors on
$Z_{\mathrm{cut}}$, with perturbations and cuts invariant under the
indicated marking quotient. Let $\mathcal X_{ij}$ be its moduli problem with determinant-one
torus limits $(t_i,t_j)$, before the extra finite quotient. Let $G$ be
that extra quotient and $G_0$ the kernel of its boundary-lift action.
If its image is the diagonal exchange, then
\begin{equation}\label{loc:toruscontract}
 (\mathcal X_{00}\sqcup\mathcal X_{11})/G
 \;\cong\;\mathcal X_{00}/G_0.
\end{equation}
For \eqref{loc:torus-groups}, $G=\Gamma_{\mathrm{cut}}$ and $G_0=0$.
With the fixed product cap framings, the composite cylinder and the
two caps select the left and right sides of \eqref{loc:toruscontract},
respectively.
\end{lemma}

\begin{proof}
The generator is an actual marking-deck transformation induced by
$c|_{Z_{\mathrm{cut}}}$. It exchanges $\mathcal X_{00}$ and $\mathcal X_{11}$ and
preserves the equation and cuts. Each matching orbit has exactly one
representative in $\mathcal X_{00}$, modulo $G_0$. This proves the
bijection before counting. It is not division of a mod-two count by two.

The flat cylinder has matching lifts $00$ and $11$, interchanged by
its extending angular class. It selects the matching orbit $\mathfrak a$ once.
The product flat cap pair has the single lift $00$ in the chosen
boundary framings. The class $ds$ cannot extend across a disk cap.
Consequently the group extending across both caps is exactly $G_0$,
and the pair selects the right side of \eqref{loc:toruscontract}.
Both cap framings have been aligned with the same $P_T$ reference;
changing only one of them would select the other orbit.

There is no further determinant-central multiplicity. The incidence
graph for the original $W_H,\overline W_H$ composition has two vertices
and two edges, and its one-dimensional $H^0$ Mayer--Vietoris cokernel
is exactly the included class $c$. The torus-cylinder decomposition
has graph $Z_1$--cylinder--$Z_2$, a tree. The cap decompositions have
two disjoint single-edge graphs. Their corresponding cokernels are
zero. This is the central-lift issue in the composition convention of
\cite[Propositions 5.4--5.5]{KM-unknot}.
\end{proof}

The bijection is induced by marking-deck transformations on a fixed
exterior; these cover the same underlying connection problem.
It preserves curvature action, index and cut-down data. Both local
fillings are flat and have zero action. No trajectory switching $t_0$
and $t_1$ is inserted.

\paragraph{The common term in the two neck-stretching limits.}
Let
\[
 Q_{\mathfrak a},Q_{\mathfrak b}:E\otimes U\longrightarrow E\otimes U
\]
be the operations obtained by counting on the cut exterior with the
torus boundary condition $\mathfrak a$ or $\mathfrak b$, respectively.
The preceding lemma says that the flat cylinder and the two flat caps
both give the term $Q_{\mathfrak a}$. Let $W_c$ and $W_p$ be the maps
of the composite and product cobordisms before stretching the torus
necks. The four comparison homotopies then satisfy
\begin{align}
 \delta H_{\mathrm{comp}}&=VV'+W_c,\label{loc:foura}\\
 \delta H_{\mathrm{cut},c}&=W_c+Q_{\mathfrak a},\label{loc:fourb}\\
 \delta H_{\mathrm{cut},p}&=W_p+Q_{\mathfrak a},\label{loc:fourc}\\
 \delta H_{\mathrm{cont}}&=W_p+1.\label{loc:fourd}
\end{align}
Each line has one parameter on a fixed cobordism: ordinary composition,
neck stretching on the composite, simultaneous neck stretching at the
two product caps, and product continuation. The two middle broken terms
agree by Lemma~\ref{loc:matching-orbit}. Proposition~\ref{en:torus-gluing}
proves that they are the full torus ends in the dimensions being counted,
including the one-parameter diagonal of the two cap neck lengths.

Their sum is $H_{\mathrm{exc}}$ and satisfies \eqref{loc:inverse}.
The operation $Q_{\mathfrak b}$ is retained in the joint boundary model;
neither flat filling selects it, and no vanishing assertion about it is
needed. The replacement pieces $T^3\times I$ and $T^2\times D^2$ are
smooth and disjoint from the foam. This comparison changes no dot,
normal framing or parameter count. Its topological excess is computed
in Proposition~\ref{geo:allterms}.

\begin{proposition}[Geometric comparison families]\label{loc:family-data}
For the finite collection of local relations below, the identities
\eqref{loc:composition}, \eqref{loc:naturality} and
\eqref{loc:foura}--\eqref{loc:fourd} can be realized by the indicated marked
cobordisms and one-parameter families with shared endpoint data. The
local complexes are the actual perfect complexes used for the relative
chains $p_j$. On each common broken face, the metric, perturbation,
bundle restriction, marking quotient and dot data agree.

The zero- and one-dimensional cut-down spaces are regular. Their ends
are the stated parameter limits, ordinary Floer breakings and the even
local tetrahedral bubble branches of the bifold construction. The torus
limits use the matching coefficient $Q_{\mathfrak a}$ with multiplicity
one. Moreover, for any $\varepsilon>0$, the data can be chosen so that the
total perturbation error in the action estimate for each complete
capped term is less than $\varepsilon$, as in
Theorem~\ref{en:simultaneous-data}.
\end{proposition}

\begin{proof}
The constructions above specify the collared cobordisms, the actual
subgroups and their restrictions, the bundle sectors and the flat
matching terms. Ordinary fillings are proper at the required cuts.
Every ordinary comparison component, including each component of
$Z_{\mathrm{cut}}$, retains a strongly marked ordinary end.
The only additional isolated pieces are the torus cylinder and caps.

The analytic proof is supplied in Section~\ref{en:section}:
Proposition~\ref{en:collar} gives universal surjectivity by varying
additional time-dependent perturbations on ordinary end collars, with
the endpoint complexes fixed;
Proposition~\ref{en:torus-gluing} gives the torus compactification and
its gluing charts. The relative choice argument and quantitative bounds
in Theorem~\ref{en:simultaneous-data} then select one shared collection
of data. That argument uses the explicit geometries above and their
common boundary faces, not the localization identity or the value of
any scalar coefficient. Thus it introduces no circular dependence on the algebraic
consequences proved next.
\end{proof}

\subsection{The chain homotopy for a local relation}

We can now combine the preceding comparisons. The formula below has
three terms for each filling: the composition homotopy, the homotopy
that moves the filling past excision, and the homotopy that cancels
excision with its reverse. Keeping the one-generator complex $U$ in
the formula makes the domains and codomains explicit.

\begin{proposition}[The localization chain homotopy]
\label{loc:constructor}
Use the geometric comparison maps of
Proposition~\ref{loc:family-data}. Define a map on
$E\otimes U$ by
\begin{equation}\label{loc:K}
 \widetilde K=\sum_j\left(
 R_j\otimes1_U+(O\otimes1_U)N_jV'
 +(OL_j\otimes1_U)H_{\mathrm{exc}}\right).
\end{equation}
Then
\begin{equation}\label{loc:residual}
 \delta\widetilde K=
 \sum_jF_j\otimes1_U+
 (O\otimes1_U)\Psi_{K,Q}
       \left(1_E\otimes\sum_jp_j\right)V'.
\end{equation}
In particular, if $\sum_jp_j=0$ as actual chain maps $U\to D$, the
identification $U=\F u_\varnothing$ gives a homotopy $K_{\mathrm{loc}}$
with $\delta K_{\mathrm{loc}}=\sum_jF_j$. Every constituent of
\eqref{loc:K} has exactly one family parameter.
\end{proposition}

\begin{proof}
All fixed cobordism maps are $\delta$-closed. Apply the Leibniz rule and the
three identities to obtain, for each $j$,
\begin{align*}
 \delta\bigl(R_j\otimes1_U\bigr)
   &=F_j\otimes1_U+OL_j\otimes1_U,\\
 \delta\bigl((O\otimes1_U)N_jV'\bigr)
   &=(O\otimes1_U)\Psi_{K,Q}(1_E\otimes p_j)V'
       +(OL_j\otimes1_U)VV',\\
 \delta\bigl((OL_j\otimes1_U)H_{\mathrm{exc}}\bigr)
   &=(OL_j\otimes1_U)VV'+OL_j\otimes1_U.
\end{align*}
The repeated terms cancel over $\F$. Summing proves
\eqref{loc:residual}. The identification with $K_{\mathrm{loc}}$ is an
identification of linear map spaces; it is not a claim that each
underlying cobordism is a product with an empty factor. Only $R_j$, $N_j$
or one summand of $H_{\mathrm{exc}}$ varies in each constituent. All other
factors are fixed, so no second parameter is introduced.
\end{proof}

If $d_D=0$, equality of the relative invariants $\sum_j[p_j]=0$ implies
$\sum_jp_j=0$ literally. This is why the local complex must be a genuine
zero-differential Floer complex. If instead we replaced $D$ by its homology using an arbitrary chain
equivalence, the comparison would introduce further homotopies for which
we had not specified any cobordisms.
The relation is used over $\F$; no equality of action-weighted relative
invariants is asserted.

\subsection{The local relations used below}

We apply this construction to four kinds of local relation. To find the
boundary web $Q$, enclose the local foam in a four-ball and inspect its
intersection with the boundary sphere. In the present cases this amounts
to doubling the corresponding tangle. The results are:
\begin{center}
\begin{tabular}{lll}
\hline
Relation & Local boundary & Occurrences\\
\hline
Square decomposition & Cube & $4$\\
Triangle-side inverse & Triangular prism $L_3$ & $13$\\
Source comparison, $A$ component & Theta & $1$\\
Source comparison, $B$ component & Handcuff & $1$\\
\hline
\end{tabular}
\end{center}

For the triangle relation, let $T$ be the triangle tangle with three
external legs and $Y$ the three-legged star. Doubling $Y$ gives theta;
doubling $T$ gives $L_3$. The star-side composite and the triangle-side
composite therefore require different local complexes. The two relative
relations are the fillings in \cite[Figures~10--11 and
Proposition~6.6]{KM-web}. A rank-six identification alone cannot exchange
these boundaries.

For the source comparison, the local incoming tree has two vertices joined
by one edge and exterior pairs $(9,11)$ and $(10,2)$. The $A$ smoothing
pairs $(9,10)$ and $(2,11)$. Both smoothing arcs join the two vertices in
the doubled boundary, yielding theta. The $B$ smoothing pairs $(9,11)$
and $(10,2)$; each arc gives a loop at one vertex, leaving a connecting
bridge. Its boundary is the handcuff. The latter has no flat connection:
a sphere meeting the bridge once makes the bridge meridian trivial, while
bifold holonomy requires a nontrivial half-turn. Thus it has an empty
complex for sufficiently small perturbations.

The theta source filling has the collapsed-side triangle comparison.
The three seam paths between its two tetrahedra are, in the labelled local
movie,
\[
 t_1\!-!20\!-!t_2,\qquad
 t_1\!-!22\!-!24\!-!t_2,\qquad
 t_1\!-!23\!-!25\!-!t_2.
\]
The intervening facets are the prescribed planar disks, so this is the
relative filling of Figure~10, with its specified collars. Its equality
with the ordinary source filling is a relative theta relation. In an
actual perfect theta complex it is a literal chain equality. The
handcuff relative chain is zero in the empty complex. Application of
Proposition~\ref{loc:constructor} therefore gives the two components of
\[
 \delta L_H=P_X\widehat H I_Z+(s,0),
\]
with the common geometric data of Proposition~\ref{loc:family-data}.
The fourth square relation gives
$\delta T_X=1+I_XP_X$. These are the transfer homotopies used by the
normalized scalar formula.

\subsection{Combining the reductions of $K_0$}

Recall that the two cycles $c_1,c_3$ used in the finite calculation have
$[sc_1]=[sc_3]=0$. Thus each $sc_i$ is a boundary. We now construct a
single operator $T_A$ such that $T_Asc_i$ is a primitive for this boundary:
$d(T_Asc_i)=sc_i$. The point of using the reductions of $K_0$ is to
express $T_A$ as a sum of the geometric homotopies already described.

Let $A=\CF(K_0)$, with the two facial square reductions at
$(2,3,13,11)$ and $(8,9,10,18)$ in the labelled web. The second reduction
is made on both branches of the first. Write
\[
 P_i:A\to A_i,\quad I_i:A_i\to A,\qquad
 Q_{ij}:A_i\to A_{ij},\quad J_{ij}:A_{ij}\to A_i
 \quad (i,j\in\{0,1\}).
\]
The four subsequent routes contain $2,4,4,3$ triangle collapses,
respectively. Denote their composite collapses and reverse maps by
\[
 R_{ij}:A_{ij}\to D_{ij},\qquad S_{ij}:D_{ij}\to A_{ij}.
\]
The terminal webs are a double-bigon theta, theta, theta and $L_2$, with
actual zero-differential complexes of dimensions $24,6,6,12$, respectively,
as constructed in Proposition~\ref{lm:perfect}.

Let $K$, $L_i$ and $T_{ij}$ be the geometric homotopies supplied by
Proposition~\ref{loc:constructor}, so that
\begin{equation}\label{loc:routeidentities}
 \delta K=1+\sum_iI_iP_i,\qquad
 \delta L_i=1+\sum_jJ_{ij}Q_{ij},\qquad
 \delta T_{ij}=1+S_{ij}R_{ij}.
\end{equation}
The last homotopy is obtained by combining the successive triangle
relations. If $r_k:A_{k-1}\to A_k$ and
$s_k:A_k\to A_{k-1}$ are successive triangle maps and
$\delta h_k=1+s_kr_k$, then
\[
 T=h_1+s_1h_2r_1+\cdots+
 s_1\cdots s_{m-1}h_mr_{m-1}\cdots r_1
\]
satisfies $\delta T=1+(s_1\cdots s_m)(r_m\cdots r_1)$.
Indeed, the terminal term of each summand cancels the initial term of the
next. Every summand still has one parameter.

\begin{proposition}[Common source primitive]\label{loc:primitive}
With the actual maps and homotopies in \eqref{loc:routeidentities}, set
\begin{align*}
 T_A&=K+\sum_i I_iL_iP_i+
       \sum_{i,j}I_iJ_{ij}T_{ij}Q_{ij}P_i,\\
 \Pi&=\sum_{i,j}I_iJ_{ij}S_{ij}R_{ij}Q_{ij}P_i.
\end{align*}
Then $\delta T_A=1+\Pi$. If $z\in A$ is a boundary and every $D_{ij}$ has
zero differential, then $\Pi z=0$ and $d(T_Az)=z$.
\end{proposition}

\begin{proof}
The differential of the first summand contributes $1+\sum_iI_iP_i$.
The second contributes
$\sum_iI_iP_i+\sum_{i,j}I_iJ_{ij}Q_{ij}P_i$.
The third contributes
$\sum_{i,j}I_iJ_{ij}Q_{ij}P_i+\Pi$.
Cancellation gives the operator identity. Write $z=da$.
Since $R_{ij}Q_{ij}P_i$ is a chain map, its value on $z$ is
$d_{D_{ij}}(R_{ij}Q_{ij}P_i a)=0$. Hence $\Pi z=0$.
Finally $dz=0$ gives $dT_Az=(1+\Pi)z=z$.
\end{proof}

Apply this proposition to $z=sc_1$ and $z=sc_3$. The finite computation
says that the full cycles $c_1,c_3$ lie in $\ker H(s)$, so these full images
are boundaries. Individual half-foam summands need not be in that kernel.
The same $T_A$ works for both sources, and the construction asserts neither
$\Pi=0$ nor a contraction of the entire complex $A$.

For later use, the primitive on the augmented all-$L$ complex is
\begin{equation}\label{loc:augprimitive}
 \widehat x=T_X\widehat H I_Zc+
 I_X\bigl(L_Hc+(T_Asc,0)\bigr).
\end{equation}
Differentiation, before using the terminal vanishing, gives
\[
 d\widehat x=\widehat H I_Zc+I_{X,A}\Pi sc.
\]
Proposition~\ref{loc:primitive} removes the residual term. Thus
$d\widehat x=\widehat H I_Zc$. This is the primitive used in the normalized secondary operation of
the next section. Its construction adds no dots. Each expanded term contains
one of the comparison families described here, surrounded by fixed maps;
its capped topology is the subject of Proposition~\ref{geo:allterms}.