\section{Floer complexes for the local webs}\label{lm:section}

The bigon, triangle and square relations in \cite{KM-web} are relations
between maps on instanton homology. Our argument will also need their
chain-level representatives. For example, if two foams with the same
boundary define the same homology class, their relative invariants differ
by a boundary in the Floer complex of that boundary web. We shall arrange
that, for the particular local webs needed here, the differential is zero.
The two relative invariants will then be equal as chains.

We call such a choice of Floer complex \emph{perfect}. This means that the
complex is defined by nondegenerate critical points and counts of
trajectories, with a choice of perturbation for which its differential
vanishes. Simply replacing a Floer complex by its homology would not
provide the geometric representatives needed later. The task in this
section is to construct perfect complexes by choosing the perturbations.
Section~\ref{loc:section} will explain how to use the resulting local
chain relations inside a larger foam cobordism.

Write $\mathcal F=SO(3)/V_4$ for the space of ordered triples of mutually
orthogonal unoriented lines in $\mathbb R^3$. We refer to its points as
orthogonal flags. In this section $L_n$ denotes the planar $n$-sided prism,
so that $L_4$ is the cube. This prism notation is local to this section;
it should not be confused with the six webs in the octahedral diagram.
Let $N$ be the double-bigon theta web. More explicitly, its vertices are $p,q,r,s,t,u$, and its edges are
\[
 pq,\ pr,\ ps,\ qt,\ qu,\ ru,\ ru,\ ts,\ ts.
\]
The doubled edges bound the two specified bigons; collapsing both gives
the theta web $\Theta$. These webs are drawn in
Figure~\ref{fig:local-webs}; the marking Hopf link is omitted.

\input{figures/local_webs}

\begin{proposition}\label{lm:perfect}
For each of the following planar webs, there are arbitrarily small good
holonomy perturbations for which the geometric complex $\CF(Q)$ has the
indicated number of generators and differential zero:
\[
\begin{array}{c|rrrrrr}
Q&\varnothing&\Theta&L_2&L_3&L_4&N\\ \hline
\dim\CF(Q)&1&6&12&6&24&24.
\end{array}
\]
The generators may all be chosen to be flat connections.  For each fixed
$Q$, their unperturbed Chern--Simons lifts can be chosen equal.  The planar
handcuff web has an empty complex for all sufficiently small perturbations.
\end{proposition}

As in the definition of $\J$, the Hopf link used for marking is present
in every occurrence of $\CF$. Thus even for the empty web there is a
three-dimensional gauge-theory problem: $\CF(\varnothing)$ is the
one-generator complex of the marked Hopf link. This observation matters
when the empty-web factor is retained in a cobordism or an excision.

We use the holonomy perturbations and the notion of a good perturbation
from \cite[Section 3.2]{KM-web}. The proof has two parts. First we find
small perturbations with exactly the numbers of critical points listed
above. We then compare those numbers with known lower bounds for the
homology dimensions. Equality forces the differentials to vanish. Most
of the flat loci are Morse--Bott; the cube requires a separate analysis
at the intersection of three branches.

\subsection{Analytic reduction and trace perturbations}

The marking allows us to describe the flat connections by holonomy
without taking a quotient by overall conjugation. More precisely,
\cite[Lemma 2.10]{KM-web} identifies $\mathcal R^\sharp(Q)$ with the
representations of the web complement for which every meridian has
order two. For a planar web, choose the usual meridians associated to
its edges. The holonomy of an edge meridian is a half-turn $R_v$ about
an unoriented line $v$. At a vertex, the three lines are mutually
orthogonal. The representation varieties below can therefore be
calculated in terms of these lines.  The identity
\begin{equation}\label{lm:trace}
 \operatorname{tr}(R_vR_w)=4(v\mathbin{\cdot}w)^2-1
\end{equation}
holds for unit representatives.  Words in based meridians define the
allowed cylinder functions after the usual thickening and integration
over a transverse disk.  We may include the finitely many functions used
here in the defining collection of cylinder functions.  This enlargement
preserves the analytic estimates and the separation property in the
construction of the perturbation space.

Two elementary properties will be useful.  The function
$(3-\operatorname{tr}g)/4$ is nonnegative on $SO(3)$ and vanishes at the
identity.  Also, the differential of trace is zero at the identity and
at every half-turn: writing a tangent vector as $gX$, with $X$ skew
symmetric, gives $\operatorname{tr}(gX)=0$ whenever $g$ is symmetric.
Consequently a trace word whose holonomy is an identity or a half-turn
has zero differential on the full configuration space.  This remains
true after thickening, since flat holonomies of the parallel loops are
conjugate.  Thus the flat critical points constructed below persist
exactly, rather than merely up to an unspecified displacement.

Here is the analytic reduction used in the smooth cases.  Suppose the
entire flat locus $M$ is compact and smooth, and the kernel of the
gauge-fixed Chern--Simons Hessian is $TM$.  In a Coulomb slice, its
self-adjoint elliptic Hessian is invertible on the normal complement.
Compactness of $M$ gives a uniform inverse on finitely many charts.
The implicit function theorem therefore solves the normal equation for
$\operatorname{CS}+\epsilon f$, and its reduced function on $M$ is
\begin{equation}\label{lm:mb-reduced}
 \operatorname{CS}|_M+\epsilon f|_M+O(\epsilon^2)
\end{equation}
in $C^2$, uniformly on those charts.  If $f|_M$ is Morse, there is
precisely one critical point near each of its critical points and none
elsewhere near $M$.  If those specified flat points are already critical
for the full $f$, they are the unique resulting critical points.

We also use uniform compactness as the perturbation tends to zero, not
merely compactness for each fixed perturbation.  For the fixed finite
collection of cylinder functions, the perturbed critical equation bounds
curvature by a constant tending to zero with the coefficients.  Local
orbifold gauge fixing on a finite cover, followed by the elliptic
estimates for that equation, gives convergence modulo gauge to a flat
connection for any sequence of critical points with perturbations
tending to zero.  The same argument applies in the tame perturbation
norm of \cite[Section 3.2]{KM-web}.  Thus a complete description of the
flat locus excludes additional small-perturbation critical points away
from its neighborhoods.  Strong marking supplies the gauge slices used
here.  All these statements concern the ordinary three-dimensional
critical-point problem; they make no assertion about regularity for a
new four-dimensional excision family.

Once finitely many nondegenerate critical points have been obtained,
\cite[Section 3.2]{KM-web} permits a further arbitrarily small good
perturbation that vanishes near their neighborhoods.  It preserves the
critical points and makes the trajectory moduli spaces regular.  The
absence of other critical points is stable under a sufficiently small
such perturbation, by the same compactness argument.

Let $h_1,h_2$ be the two marked Hopf meridians and put $h_3=h_1h_2$.
Their flat axes give the reference orthogonal flag.  For a vertex with
meridians $m_1,m_2,m_3$, set
\begin{equation}\label{lm:flag-height}
 f_{\mathcal F}=
 \sum_{i,j=1}^3\lambda_i\mu_j
 \frac{\operatorname{tr}\operatorname{Hol}(m_ih_j)+1}{4},
 \qquad \lambda=(0,1,3),\quad\mu=(0,1,4).
\end{equation}
On a flat flag this is $\operatorname{tr}(D_\lambda R^tD_\mu R)$.
Its critical equation is
$[D_\lambda,R^tD_\mu R]=0$.  Since both diagonal matrices have simple
spectra, there are exactly six critical flags.  In an $ik$ rotation
direction the Hessian is a nonzero multiple of
$(\lambda_i-\lambda_k)(\mu_{\sigma(i)}-\mu_{\sigma(k)})$.
Every critical point is therefore nondegenerate.  At each of these flags
every word in \eqref{lm:flag-height} is an identity or a half-turn, so
the full differential vanishes there.

\subsection{The smooth local and terminal models}

The theta flat space is $\mathcal F$.  Its Hessian kernel is its tangent
space, and \eqref{lm:flag-height} gives six flat generators by
\eqref{lm:mb-reduced}.  Its homology rank is six by
\cite[Proposition 5.7]{KM-web}.

For $L_3$, name the top vertices $t_0,t_1,t_2$ and the corresponding bottom
vertices $b_0,b_1,b_2$.  The top triangle's edge axes are pairwise
orthogonal, so write
\[
 t_0t_1=z,\qquad t_1t_2=x,\qquad t_2t_0=y.
\]
The vertical axes are forced to be $x,y,z$, respectively, and the bottom
axes are forced to agree with their corresponding top axes.  These
assignments satisfy all the bottom vertex relations.  They describe the
whole representation space, so $\mathcal R^\sharp(L_3)=\mathcal F$.
The description is also valid infinitesimally: each forced axis is a
nonzero cross product of two transverse axes, so it is obtained by an
invertible local elimination in projective coordinates.  No additional
infinitesimal variables remain.  Equivalently, the orthogonality
Jacobian has rank fifteen on the eighteen axis coordinates, with kernel
the three simultaneous rotations.  Holonomy identifies this kernel with
the kernel of the gauge-fixed Chern--Simons Hessian.  Thus this is a
Morse--Bott flat manifold, and \eqref{lm:flag-height} gives six flat
generators.

The two-sided prism has a different unperturbed flat space.  Write its
two bigon edge pairs as $a,b$ and $u,v$, and its other two edges as $c,d$.
The planar meridian presentation gives
\[
 ab=c=d=uv,\qquad a^2=b^2=u^2=v^2=c^2=d^2=1.
\]
The first vertex determines an orthogonal flag $(x,y,z)$, with
$a=R_x$, $b=R_y$, and $c=d=R_z$.  The remaining axes are
\begin{equation}\label{lm:l2-axes}
 u=R_{\cos\theta\,x+\sin\theta\,y},\qquad
 v=R_{-\sin\theta\,x+\cos\theta\,y},
 \qquad \theta\in\mathbb R/\pi\mathbb Z.
\end{equation}
This exhausts the representations.  The sign changes of an oriented
frame act on $\theta$ by reflection, and hence
\begin{equation}\label{lm:l2-space}
 \mathcal R^\sharp(L_2)
 =SO(3)\mathbin{\times_{V_4}}\mathbb{RP}^1
 =\mathbb P(L_x\oplus L_y)\longrightarrow\mathcal F,
\end{equation}
where the $L_i$ are the tautological lines.  This is a connected smooth
four-manifold.  The action on the $SO(3)$ factor is free, including at
the reflection-fixed fiber points.  The elimination leading to
\eqref{lm:l2-axes} solves all remaining axes smoothly and over the dual
numbers; it leaves exactly one fiber tangent direction.  It therefore
also proves the Morse--Bott assertion.

The cylinder function
\[
 q=\frac{3-\operatorname{tr}\operatorname{Hol}(m_am_u)}4
\]
restricts to $\sin^2\theta$.  For any fixed $\delta>0$, the restriction of
$f_{\mathcal F}+\delta q$ has exactly twelve Morse critical points,
the six coordinate flags with $\theta=0,\pi/2$.  In local flag and fiber
coordinates its Hessian is block diagonal, with fiber entry
$\pm2\delta$.  All twelve points are full critical points, by the trace
observation above.  Multiplying this function by a sufficiently small
positive number gives precisely twelve nondegenerate flat generators.

\begin{remark}\label{lm:l2-remark}
The literal unperturbed description of $L_2$ as two flag manifolds in
the proof of \cite[Lemma 5.12]{KM-web} is not used here.
Equation \eqref{lm:l2-axes} exhibits the intervening family explicitly:
$\operatorname{tr}(R_xR_{\cos\theta x+\sin\theta y})
=4\cos^2\theta-1$ varies.  Two flag critical manifolds are obtained
\emph{after} perturbing by a small multiple of $q$.
\end{remark}

For $N$, fix the axes at $p$ by
$pq=x$, $pr=y$, $ps=z$.  The first bigon forces $qu=y$ and the second
forces $qt=z$.  The two internal ordered axis pairs can be written
\[
\begin{split}
 ru &: (\cos\theta\,x+\sin\theta\,z,
             -\sin\theta\,x+\cos\theta\,z),\\
 ts &: (\cos\phi\,x+\sin\phi\,y,
             -\sin\phi\,x+\cos\phi\,y).
\end{split}
\]
There is no equation coupling $\theta$ and $\phi$.  Thus
\[
 \mathcal R^\sharp(N)=SO(3)\mathbin{\times_{V_4}}
    (\mathbb{RP}^1\times\mathbb{RP}^1)
 =\mathbb P(L_x\oplus L_z)\mathbin{\times_{\mathcal F}}
   \mathbb P(L_x\oplus L_y).
\]
The same invertible axis eliminations give a smooth five-dimensional
Morse--Bott flat manifold.  Pairing the first edge of each bigon with
$pq$ in the trace formula gives $q_1=\sin^2\theta$ and
$q_2=\sin^2\phi$.  The function
$f_{\mathcal F}+\delta(q_1+q_2)$ has exactly $6\cdot2\cdot2=24$
nondegenerate restricted critical points, all full flat critical points.
The reduction and compactness arguments give precisely these twenty-four
generators.

The empty web has the single nondegenerate flat point of the marked
Hopf model.  For the planar handcuff, a sphere meeting its bridge once
shows that the bridge meridian is null-homotopic in the complement:
the meridian contracts on the once-punctured sphere.  It cannot have
order-two holonomy.  There are no flat points, and uniform
small-perturbation compactness gives an empty critical set.

\subsection{A finite-dimensional lemma for the singular cube model}

For the cube, fixing one orthogonal flag leaves three one-dimensional
branches of flat connections, meeting at a point. Near that point the
normal part of the Chern--Simons function will have the form
$\lambda xyz$, with $\lambda\ne0$. We first give two elementary lemmas
for this singularity. The first removes higher-order terms; the second
shows how a small quadratic perturbation of mixed signs leaves only
one critical point near the intersection. The estimate in the second
lemma is needed to exclude critical points at distances larger than
the perturbation scale but still tending to the intersection.

\begin{lemma}\label{lm:normal-form}
Let $f$ be a real analytic germ on $\mathbb R^3$ whose full differential
vanishes on the three coordinate axes, with $f(0)=0$, zero quadratic
part, and cubic part $\lambda xyz$, where $\lambda\ne0$.  There is an
analytic change of variables tangent to the identity taking $f$ to
$\lambda xyz$.  The construction is uniform in compact parameter
families for which these hypotheses hold and $|\lambda|$ is bounded
away from zero.
\end{lemma}

\begin{proof}
The ideal of analytic germs vanishing on the coordinate axes is
$I=(yz,xz,xy)$.  This follows by grouping the convergent monomial series
according to which two variables occur.  Write $f=\lambda xyz+h$.
Then $h$ has order at least four, and every component of $\nabla h$
belongs to $I$.  With $\mathbf i=(yz,xz,xy)^t$, choose analytic
$R=O(|(x,y,z)|)$ and $b=O(|(x,y,z)|^2)$ such that
$\nabla h=R\mathbf i$ and $h=b^t\mathbf i$.  These choices follow by
the same monomial grouping, with convergence on a smaller fixed
neighborhood.  For $f_t=\lambda xyz+th$, put
$M_t=\lambda\mathbf 1+tR$.  Shrinking the neighborhood makes $M_t$
invertible for every $t\in[0,1]$.  The analytic vector field
$V_t=-M_t^{-t}b$ has order at least two and satisfies
$df_t(V_t)=-h$.  Its time-dependent local flow $\varphi_t$ obeys
\[
 \frac d{dt}f_t(\varphi_t)=h(\varphi_t)+df_t(V_t)(\varphi_t)=0.
\]
Thus $f\circ\varphi_1=\lambda xyz$.  Uniform bounds give the parameter
statement.  For parameters $r$ the flow fixes $r$ and solves the normal
equation fiberwise; no group action on nonflat configurations is needed.
\end{proof}

\begin{lemma}\label{lm:mixed-sign}
Suppose
\[
 f_\epsilon(x,y,z)=\lambda xyz+\epsilon q_\epsilon(x,y,z),
 \quad dq_\epsilon(0)=0,
 \quad \operatorname{Hess}q_\epsilon(0)
       \longrightarrow\operatorname{diag}(a,b,c),
\]
where $abc\ne0$, $\lambda\ne0$, and the $q_\epsilon$ are uniformly
$C^3$ on a fixed neighborhood.  If $a,b,c$ have mixed signs, then for
all sufficiently small positive $\epsilon$ the origin is the only
critical point in a fixed neighborhood, and it is nondegenerate.
The conclusion is uniform with compact additional parameters and
uniform nonzero bounds on the displayed coefficients.
\end{lemma}

\begin{proof}
Let $e_\epsilon$ denote the norm of the difference between the Hessian
and its diagonal limit.  Taylor expansion gives
\[
 \nabla q_\epsilon=(ax,by,cz)
       +O(e_\epsilon r+r^2),\qquad
 r=\max(|x|,|y|,|z|).
\]
We claim that every nonzero critical point has $r\le M\epsilon$ for
one fixed $M$.  Suppose, after relabeling, that $|x|=r$, and put
$s=\max(|y|,|z|)$.  The $y,z$ critical equations give
\[
 rs\le C\epsilon(s+e_\epsilon r+r^2).
\]
For $r\ge M\epsilon$ with $M$ large enough, absorption yields
$s\le C'\epsilon(e_\epsilon+r)$.  On a sufficiently small fixed
neighborhood and for small $e_\epsilon$, the $x$ equation then gives
\[
 \tfrac12\epsilon|a|r\le|\lambda yz|
       \le C''\epsilon^2(e_\epsilon+r)^2,
 \qquad r\le C'''\epsilon(e_\epsilon+r)^2.
\]
Shrinking that neighborhood and then taking $\epsilon$ small contradicts
$r\ge M\epsilon$, proving the claim.

Rescale $(x,y,z)=\epsilon(u,v,w)$ and divide the gradient equations by
$\epsilon^2$.  On the resulting fixed compact set, they converge in
$C^1$ to
\[
 \lambda vw+au=0,\qquad
 \lambda uw+bv=0,\qquad
 \lambda uv+cw=0.
\]
A nonzero solution must have all three coordinates nonzero, and hence
$u^2=bc/\lambda^2$, $v^2=ac/\lambda^2$, and
$w^2=ab/\lambda^2$.  These equations are impossible for mixed signs.
The origin has invertible derivative $\operatorname{diag}(a,b,c)$.
Uniform local invertibility excludes additional solutions near it;
compactness excludes solutions on the remaining bounded set.  The
original origin is exactly critical and has nondegenerate Hessian
$\epsilon\operatorname{Hess}q_\epsilon(0)$.
\end{proof}

Diagonality in the branch coordinates is essential in this lemma.
An arbitrary indefinite quadratic form is not covered.  Likewise,
rescaling alone would not exclude solutions whose distance is much
larger than $\epsilon$; that is the role of the preceding estimate.

\subsection{The Chern--Simons function near the cube's flat connections}

We now calculate the flat connections on the cube and relate that
calculation to the two preceding lemmas. It is necessary to keep track
of infinitesimal deformations as well as the set of flat connections:
the latter alone would not determine the first nonzero term of the
Chern--Simons function.

Number the top square $0,1,2,3$ and the bottom square $4,5,6,7$,
with spokes $i(i+4)$.  Fix the axes at vertex $0$ as
$v_{01}=e_1$, $v_{03}=e_2$, $v_{04}=e_3$.  Every remaining
representation is described by homogeneous coordinates
$[A_0:A_1],[B_0:B_1],[C_0:C_1]\in\mathbb{RP}^1$ with
\[
\begin{array}{lll}
v_{12}=(0,A_0,A_1),&v_{15}=(0,-A_1,A_0),\\
v_{23}=(B_0,0,B_1),&v_{37}=(-B_1,0,B_0),\\
v_{45}=(C_0,C_1,0),&v_{47}=(-C_1,C_0,0).
\end{array}
\]
The relations at vertices $2,5,7$ are exactly
\begin{equation}\label{lm:cube-homogeneous}
 A_1B_1=A_1C_1=B_1C_1=0.
\end{equation}
The axes $v_{26},v_{56},v_{67}$ are forced by cross products; they
are nonzero on this solution set, and their vertex-$6$ equations follow
from \eqref{lm:cube-homogeneous}.  Thus the fixed-frame flat variety
is three projective lines meeting at their common origin.

In the affine chart at that origin set $a=A_1/A_0$, $b=B_1/B_0$,
$c=C_1/C_0$.  The forced remaining vectors are
\[
 v_{26}=(b,a,-1),\qquad
 v_{56}=(-c,1,a),\qquad
 v_{67}=(-1,-c,-b).
\]
The three vertex-$6$ pairings are
\begin{equation}\label{lm:vertex6}
 v_{26}\cdot v_{56}=-bc,\qquad
 v_{26}\cdot v_{67}=-ac,\qquad
 v_{56}\cdot v_{67}=-ab.
\end{equation}
Thus the local ideal is exactly $(ab,ac,bc)$.  All eliminations used
invert nonzero coordinate functions and take nonzero cross products,
so this is a scheme-level local calculation, not merely a description
of the real zero set.  Away from the common point the fixed-frame
variety is smooth of dimension one.  Restoring the flag gives a
six-dimensional Hessian kernel at the common flag and a
four-dimensional kernel on the smooth branches.

For clarity we explain why the cubic in the reduced Chern--Simons
function is nonzero.  The local holonomy correspondence used here
also identifies infinitesimal flat deformations, including their
second-order obstructions.  Indeed parallel transport of a flat
connection with coefficients in $\mathbb R[t]/(t^3)$ gives a
representation over that ring; conversely the representation gives
locally constant transition functions and a flat bundle.  Gauge
equivalence is the corresponding change of trivializations.  The
finite local isotropy and the fixed marked Hopf factor impose their
same relations in both descriptions; the latter is rigid and fixes
the local framing.  This is the infinitesimal version of
\cite[Lemma 2.10]{KM-web}, applied in a gauge slice.

Elliptic Lyapunov--Schmidt reduction at the common flag gives a real
analytic reduced potential $g(r,a,b,c)$, where $r$ is a flag chart and
$a,b,c$ are the three normal kernel directions.  The three smooth
flat sheets can simultaneously be straightened onto the three normal
coordinate axes: their normal tangent lines are independent, so the
sum of their graph parametrizations has invertible differential.
All these axes consist of critical points and have the same
Chern--Simons value $C_0$.  The normal quadratic part vanishes, and
the only possible cubic is
\[
 g(r,a,b,c)=C_0+\lambda(r)abc+O(|(a,b,c)|^4).
\]
Every other cubic monomial has a derivative that is nonzero on some
coordinate axis.

If $\lambda(r)$ vanished, the full reduced gradient at that flag
would have no quadratic obstruction in the three normal directions.
The flag derivatives also have no such term, because the normal
Hessian vanishes along the entire flag.  Every first-order normal
vector would then extend through second order.  But the vector
$(a,b,c)=(t,t,0)$ cannot extend to a solution modulo $t^3$:
the coefficient of $t^2$ in $ab$ is one, independently of all
second-order corrections.  The infinitesimal holonomy correspondence
contradicts the alleged vanishing.  Thus $\lambda(r)\ne0$, uniformly
away from zero on finitely many charts of the compact flag.
Lemma~\ref{lm:normal-form} supplies the exact fiberwise form
$C_0+\lambda(r)abc$.

Define the three trace functions
\[
 q_a=\frac{3-\operatorname{tr}\operatorname{Hol}(m_{12}m_{03})}4,
 \quad q_b=\frac{3-\operatorname{tr}\operatorname{Hol}(m_{23}m_{01})}4,
 \quad q_c=\frac{3-\operatorname{tr}\operatorname{Hol}(m_{45}m_{01})}4.
\]
On the three flat branches their formulas are respectively
$a^2/(1+a^2)$, $b^2/(1+b^2)$, and $c^2/(1+c^2)$.
Their normal Hessians at the common flag are diagonal in these branch coordinates.
For example $q_a$ is nonnegative on the full configuration space and
vanishes at that flag, so its Hessian is positive semidefinite.  It
vanishes identically along the other two flat branches and the flag;
their tangent directions therefore lie in the Hessian kernel,
including all mixed entries.  Its second derivative along the
$a$-branch is two.  The same argument applies to $q_b,q_c$.

Perturb first by $\epsilon(q_a+q_b-q_c)$.  The four Tait flag orbits
are exactly critical for this full function.  After solving the
invertible complement, the limiting normal Hessian at the common
flag is $\operatorname{diag}(2,2,-2)$; the elimination changes it only
by terms tending to zero.  The flat flag remains exactly critical,
so the reduced perturbation has zero normal gradient there.
Lemma~\ref{lm:mixed-sign} excludes all other nearby normal critical
points, uniformly over the flag, and makes the normal Hessian
invertible.

At the three outer Tait flags the fiber Hessians are respectively
$-2\epsilon,-2\epsilon,2\epsilon$.  On compact portions of the
open branches the restricted derivative is nonzero.  Ordinary
Morse--Bott reduction there, together with the preceding central
analysis and uniform small-perturbation compactness, proves that the
entire perturbed critical set is precisely four Morse--Bott copies of
$\mathcal F$.  Add $\delta f_{\mathcal F}$, with $\delta$ chosen
sufficiently small \emph{after} $\epsilon$ is fixed.  Each flag then
has its six coordinate critical points, all still flat.  The normal
inverse may grow as $\epsilon$ tends to zero; the stated order of
choices accounts for this.  There are exactly twenty-four
nondegenerate generators.  A further good perturbation supported
away from them completes the geometric Floer complex.

\subsection{Zero differentials, actions and the local chain relations}

We have constructed the critical points. To show that the differentials
vanish, we use lower bounds for the dimensions of the corresponding
homology groups. Each bound comes from a part of the local relations
in \cite{KM-web} that does not require the chain homotopies we are
trying to construct.

For a bigon, the initial bubble-bursting calculation in the proof of
\cite[Proposition 6.5]{KM-web} gives maps with
\[
 ca=db=1,\qquad cb=da=0.
\]
They inject two copies of the collapsed web's homology into the
expanded web's homology.  Starting from the theta rank six, this
gives lower bounds twelve for $L_2$ and twenty-four for $N$.
For $L_3$, the collapsed-side triangle identity $ca=1$, proved by
theta pairings in \cite[Proposition 6.6, Figure 10]{KM-web}, injects
the tetrahedron homology, of rank six by
\cite[Proposition 5.10]{KM-web}.  It does not use the triangle-side
inverse relation.  Finally the initial neck-cutting and
bubble-bursting identities in the proof of
\cite[Proposition 6.8]{KM-web} inject the direct sum of the two
twelve-dimensional square resolutions into the cube.  They give
the lower bound twenty-four before the cube-side inverse relation
is established.

Each lower bound equals the number of geometric generators just
constructed.  For a finite complex over $\F$,
$\dim H(C,d)=\dim C-2\rk d$, so $d=0$ in each case.  The marked
Hopf unit has the same conclusion immediately.  This proves the
generator and differential assertions of Proposition~\ref{lm:perfect}.

Each of the flat spaces above is connected, including the cube's
three branches with their common flag.  Along a lifted path of flat
spatial connections, the four-dimensional curvature in temporal gauge
is $dt\wedge\dot A$, so its Chern--Weil action is zero.  The same
calculation applies to loops closing by an allowed marked gauge
transformation.  The action cover is therefore trivial along the
flat locus, giving the common unperturbed lifts asserted in
Proposition~\ref{lm:perfect}.  The perturbed critical values generally
differ; they are not being identified.

For fixed critical endpoints and Fredholm index, the loop identity
$\operatorname{SF}=8\kappa$ of \cite[equation (7)]{KM-web} fixes the
topological action of all trajectory classes contributing to one
differential entry.  Thus its zero mod-two count remains zero if
that entry is weighted by its topological action.  This is an
entrywise cancellation statement, not an assertion that individual
trajectories are absent.  Cancellation among different filling
cobordisms requires their own index and dot comparisons.

We can now return to the purpose of this section. A foam filling of a
local web defines a relative invariant in its Floer complex. Once one
of the perfect complexes above is fixed, such cycles
$p_j$ satisfying $\sum_j[p_j]=0$ satisfy the literal chain equality
$\sum_jp_j=0$.  This applies to the local filling relations used
in the localization construction, with their actual boundary
identifications.  In particular the two triangle composites have
different local boundaries.  Doubling the three-legged star gives
$\Theta$, whereas doubling the triangle tangle gives $L_3$.
For the triangle-side relation one must use the $L_3$ complex and
the two fillings in \cite[Figure 11]{KM-web}.  The collapsed-side
identity and the rank-six calculation above imply that the
expansion--collapse projection on $\J(L_3)$ is the identity;
applying it to the standard filling gives the equality of those
two classes, and hence of their relative chains in this complex.
This argument supplies the local chain equality.  Its transport
through a marked exterior, with coherent regular data and the
claimed action control, remains the separate localization task.