\section{Perturbations and the action estimate}
\label{en:section}

An unperturbed anti-self-dual connection has nonnegative action. For a
perturbed equation the action may be negative, but its negative part can
be bounded in terms of the perturbation. We first prove a bound that is
uniform as the necks in our cobordisms become long. The important point
is to use the gradient-flow identity on a cylindrical region; estimating
the perturbation there by its four-dimensional $L^2$ norm alone would
introduce an unwanted factor of the neck length.

We must then choose perturbations that satisfy this bound and make all
the required moduli spaces regular. They must agree whenever two families
have a common boundary face, and they must preserve the chosen local
Floer complexes. The torus ends require the correlated marking data of
Section~\ref{loc:torus-geometry}. The construction of these simultaneous
choices occupies the rest of the section and is assembled in
Theorem~\ref{en:simultaneous-data}.

\subsection{A uniform estimate on stretched pieces}

We use the action normalization of \cite[Proposition~2.6]{KM-web}. On a compact
four-dimensional piece $Q$, with the orbifold integral and the corresponding
invariant inner product on $\mathfrak{so}(3)$, the Chern--Weil identity is
\begin{equation}
 \kappa_Q(A)=\frac{1}{32\pi^2}
       \bigl(\|F_A^-\|_{L^2(Q)}^2-\|F_A^+\|_{L^2(Q)}^2\bigr).
 \label{en:core-action}
\end{equation}
Consequently, a solution of $F_A^+=q(A)$ satisfies
\begin{equation}
 \kappa_Q(A)\geq-\frac{1}{32\pi^2}\|q(A)\|_{L^2(Q)}^2.
 \label{en:core-bound}
\end{equation}
The sign convention for $q$ is immaterial to this inequality.

Applying \eqref{en:core-bound} to an entire long cylinder would give an
unhelpful bound proportional to its length. The gradient structure gives a
better estimate. Normalize a lift $\mathcal C$ of the Chern--Simons functional
and the metric on configuration space so that its drop is the action on a
cylinder and its negative gradient equation is the unperturbed ASD equation.
A lift along a particular path suffices; no globally real-valued lift of
$\mathcal C$ is asserted. Perturbations $f_t$, in contrast, are globally defined
bounded gauge-invariant real-valued functions.

\begin{lemma}[Neck estimate]\label{en:neck}
Suppose $a(t)$ satisfies
\[
 \frac{d a}{dt}=-\operatorname{grad}(\mathcal C+f_t)(a(t)),
 \qquad t\in[t_-,t_+].
\]
Then
\begin{align}
 \kappa_{[t_-,t_+]}(a)
 &=\int_{t_-}^{t_+}\left\|\frac{da}{dt}\right\|^2dt
   +f_{t_+}(a(t_+))-f_{t_-}(a(t_-))
   -\int_{t_-}^{t_+}(\partial_t f_t)(a(t))\,dt,
 \label{en:neck-identity}\\
 \kappa_{[t_-,t_+]}(a)
 &\geq-\|f_{t_-}\|_\infty-\|f_{t_+}\|_\infty
          -\int_{t_-}^{t_+}\|\partial_t f_t\|_\infty\,dt.
 \label{en:neck-bound}
\end{align}
For a constant $f_t=f$, the last bound improves to $-\operatorname{osc}(f)$.
\end{lemma}
\begin{proof}
The chain rule gives
\[
 \frac{d}{dt}(\mathcal C(a(t))+f_t(a(t)))
 =-\left\|\frac{da}{dt}\right\|^2+(\partial_t f_t)(a(t)).
\]
Integration and the identification of the drop of $\mathcal C$ with the
Chern--Weil integral give \eqref{en:neck-identity}. Dropping the nonnegative
integral gives both inequalities. The calculation retains the chosen path
lift, including its action period. It makes no use of regularity or
nondegeneracy.
\end{proof}

Call a decomposition of a family into cylindrical regions and complementary
compact cores \emph{controlled} if its necks carry equations of this gradient
form and if there are constants $\eta,\epsilon\geq0$ such that
\begin{align}
 \sum_{\text{necks }j}\left(
   \|f_{j,t_-}\|_\infty+\|f_{j,t_+}\|_\infty
   +\int\|\partial_t f_{j,t}\|_\infty\,dt\right)&\leq\eta,
 \label{en:budget-neck}\\
 \sum_{\text{cores }Q}\|q_Q(A)\|_{L^2(Q)}^2&\leq\epsilon.
 \label{en:budget-core}
\end{align}
These bounds are required for every configuration in the relevant bundle
sectors and uniformly in the metric-family parameter, including arbitrary
neck lengths. The first expression may be replaced by its sharper telescoped
version when adjacent necks have the same end perturbation.

\begin{proposition}[Uniform action bound]\label{en:uniform-bound}
Every solution for controlled data satisfies
\begin{equation}
 \kappa_{\mathrm{total}}\geq
       -\eta-\frac{\epsilon}{32\pi^2}.
 \label{en:uniform-action}
\end{equation}
The same conclusion holds for endpoint-matched broken configurations if the
data on the matched ends agree and the displayed bounds apply to the
underlying finite collection of cobordism pieces.
\end{proposition}
\begin{proof}
The Chern--Weil integral is additive on the decomposition. Sum
\eqref{en:core-bound} over the cores and \eqref{en:neck-bound} over the necks.
For infinite ends, first truncate and then pass to the limit using convergence
to the prescribed critical points.

For a broken configuration the same argument applies to its pieces. At an
inserted Floer breaking, the endpoint values of $f$ in
\eqref{en:neck-identity} cancel, because the critical point and perturbation
are the same on the two sides. The nonnegative gradient energies add.
Thus subdividing a cylindrical trajectory into several levels does not
increase the perturbation error in the lower bound for action. This
cancellation is preferable to
assigning a separate oscillation bound to each level. The other compact
pieces and cutoff regions belong to the fixed finite construction and retain
their original bounds. This proves the assertion also for broken products.
\end{proof}

The required uniformity can be checked before making any regularity claim.
For example, put $f_t=\vartheta_t f$ on a neck, where $\vartheta_t$ is constant
on its long middle interval and has transitions in collars of fixed width.
Then
\[
 \int\|\partial_t f_t\|_\infty\,dt
 \leq\operatorname{TV}(\vartheta)\|f\|_\infty.
\]
The right-hand side does not depend on the length of the middle interval.
On a compact family of core metrics, a uniformly bounded holonomy
perturbation of size $O(\lambda)$ has squared $L^2$ norm $O(\lambda^2)$.
For finitely many such pieces, the total error in the action bound is therefore
$O(\lambda)+O(\lambda^2)$. This is an estimate on the admissible space of
perturbations. Scaling a particular regular perturbation need not preserve
regularity; regular data must be chosen inside the resulting small open
set.

\subsection{Transversality with the endpoint complexes fixed}
\label{en:collar-space}

The perfect local complexes of Section~\ref{lm:section} were obtained
by particular choices of perturbation. Those choices must remain fixed
on the infinite ends of each cobordism. There is nevertheless freedom
to perturb the equation on a finite cylindrical region inside the
cobordism. We use this freedom for transversality.

Fix a good endpoint functional $\mathcal C+f_\pi$, using the local and
terminal models of Proposition~\ref{lm:perfect} where required. This
fixes the critical points and the trajectory differential. On a finite
product subcollar $J\times Y$, add a time-dependent perturbation
of the form
\[
 h(t,a)=\sum_{i,j}c_{ij}\chi_i(t)f_j(a).
\]
Here $f_j$ are marked cylinder functions separating tangent vectors
of the configuration quotient, and the $\chi_i$ are dense among smooth
time bumps supported in $J$. We call $h$ a secondary perturbation,
to distinguish it from the perturbation already fixed on the end.
It is zero outside a finite cylindrical region. Within that region it
need not vanish near configurations that represent endpoint critical
points. Thus even a solution that is constant on this region can be
tested by all the cylinder-function directions, without changing the
Floer complex on the end.

Choose a weighted $\ell^1$ norm $\sum C_{ij}|c_{ij}|$, with weights
dominating the tame cylinder-function estimates and the time derivatives.
As in the usual diagonal construction of the perturbation space, the
weights can control every prescribed derivative order. In particular the
norm controls
\begin{equation}\label{en:collar-norm}
 \sup_t\|h_t\|_\infty+
 \int_J\|\partial_t h_t\|_\infty\,dt+
 \sup_{t,a}\|\operatorname{grad}h_t(a)\|_\infty.
\end{equation}
Every individual time bump times a cylinder function belongs to this
separable Banach space. Multiplying by parameter bumps gives the relative
family version. The marked cylinder functions are those of
\cite[Section 3.2]{KM-web}; their analytic estimates have the bounded-operator
form of \cite[Propositions 3.7 and 3.15]{KM-yaft}.

\begin{proposition}[Ordinary collar surjectivity]\label{en:collar}
Let $X$ be a completed bifold cobordism with fixed nondegenerate
asymptotic data. Suppose:
\begin{enumerate}
\item every connected component has an ordinary strongly marked end
      with a finite product subcollar admitting the secondaries above;
\item every nonlocal equation perturbation is a time-dependent gradient
      perturbation on a product cylindrical region; the complementary
      cores are unperturbed;
\item each finite gauge quotient used on a component has zero restriction
      in the marking quotient at its chosen ordinary end.
\end{enumerate}
Then the universal connection-and-secondary linearization is surjective
at every solution. The assertion holds with the endpoint complexes fixed,
in arbitrarily small open balls of secondary perturbations, and at
interior parameter points using variations vanishing on previously fixed
parameter-boundary collars.
\end{proposition}

\begin{proof}
Write the equation as $F_A^++q(A)=0$ and its linearization as
$L_A=d_A^++Dq_A$. On the usual unweighted Sobolev completions the
nondegenerate ends make the gauge-fixed operator
$D_A=(d_A^*,L_A)$ Fredholm. Let $(\phi,\eta)$ annihilate both its range
and every secondary direction. Thus
\[
 d_A\phi+L_A^*\eta=0.
\]
Gauge covariance at a solution gives $L_A d_A=0$. Testing against
compactly supported $d_A\xi$ yields $\Delta_A\phi=0$. Cutoff integration
by parts on the cylindrical ends gives $d_A\phi=0$. A nonzero parallel
section cannot be $L^2$ on an infinite end. Hence $\phi=0$, and the
remaining equation is
\begin{equation}\label{en:adjoint}
 L_A^*\eta=0.
\end{equation}

In temporal gauge on the control collar, write
$\eta=dt\wedge\zeta+*_Y\zeta$. The perturbation has no dependence on the
temporal component of $A$. The temporal component of
\eqref{en:adjoint} therefore says $d_a^*\zeta=0$. Orthogonality to the
variation $\chi_i(t)f_j(a)$ says
\[
 \int_J\chi_i(t)\,df_j|_{a(t)}[\zeta(t)]\,dt=0.
\]
Time localization implies that every $df_j$ annihilates $\zeta(t)$
at each $t$. Since $\zeta$ is horizontal and marked cylinder functions
separate tangent vectors, $\zeta=0$ on the entire subcollar.

On an unperturbed core, elliptic unique continuation propagates this
vanishing. On a nonlocal product collar use Lemma~\ref{en:temporal-uc} below,
rather than local unique continuation for a nonlocal equation. Alternating
these two arguments reaches the entire connected regular part of $X$;
orbifold smoothness extends the conclusion across its singular set.
Thus $\eta=0$ everywhere. The connection operator is Fredholm, so adding
the secondary directions to its range gives a closed subspace of finite
codimension. Its annihilator is zero, which proves surjectivity.

At an interior parameter point, multiply the secondary variation by a
parameter bump equal to one there and zero on the fixed boundary collars.
This proves the relative interior assertion.

The quotient restriction in the statement means restriction is zero in
the \emph{marked gauge quotient}, not merely that the quotient preserves
the endpoint critical set. All control-collar cylinder functions are
therefore already invariant. There is no averaging loss of directions.
If a quotient element fixes a connection class, its representative,
after composition with an allowed gauge transformation, is parallel.
At the strong end it is a marked automorphism and hence the identity;
parallel uniqueness gives the identity on the component. The effective
finite quotient acts freely, and the upstairs surjectivity descends.
\end{proof}

\begin{lemma}[Temporal continuation across a nonlocal collar]\label{en:temporal-uc}
Suppose a smooth Hilbert-space-valued section satisfies
$u'=B(t)u$, where $B(t)$ is self-adjoint on a common first-order
Sobolev domain and $B'(t)$ is bounded on $L^2$ on compact time intervals.
If $u$ vanishes on a nonempty time interval, it vanishes throughout
the connected interval of definition.
\end{lemma}

\begin{proof}
Where $h(t)=\|u(t)\|^2>0$, differentiation gives
\[
 h'=2\langle Bu,u\rangle,\qquad
 h''=2\langle B'u,u\rangle+4\|Bu\|^2,
\]
and consequently
\[
 (\log h)''\geq-2\|B'\|.
\]
If $\|B'\|\leq C$, the function $\log h+Ct^2$ is convex.
Its supporting tangent line at an interior point is a finite lower
bound at a finite endpoint. Thus $h$ cannot tend to zero at such an
endpoint of a positivity interval. This proves the assertion in
both time directions.
\end{proof}

For the adjoint equation above, put $u=(0,\zeta)$ in
$\Omega^0(Y;\operatorname{ad}E)\oplus\Omega^1(Y;\operatorname{ad}E)$.
The temporal constraint and spatial equation give such an evolution,
with extended Hessian, up to the fixed sign convention,
\[
 B(t)=
 \begin{pmatrix}
 0&d_a^*\\
 d_a&*_Yd_a+\operatorname{Hess}_a f_t
 \end{pmatrix}.
\]
The product metric is independent of physical time. Differentiating
$d_a$ contributes multiplication by $a'(t)$; differentiating the
perturbation Hessian gives a bounded $L^2$ operator, smoothly in $t$.
This verifies the hypotheses of the lemma. In particular, the proof
does not apply local unique continuation across a spatially nonlocal
holonomy term.

\subsubsection{Relative regularity and ideal configurations}

Sard--Smale gives regular secondary perturbations inside any sufficiently
small open ball of the permitted perturbation space. This is how we
obtain smallness and regularity at the same time; scaling a perturbation
after making it regular would not justify the assertion. The condition
that the endpoint differential vanish imposes no additional restriction
on these perturbations supported in finite collars. A uniform spectral
gap as the endpoint perturbation tends
to zero is also unnecessary: choose the endpoint perturbation first,
then use its fixed positive gap.

Relative boundary regularity still comes from the actual gluing theorem.
Surjectivity in a parameter interior cannot identify a torus-cut face
or prescribe its gauge quotient. These are addressed separately below.
All nonlocal perturbations can be kept on product collars, including
the collars inherited at a glued face. The compact cores can carry
the unperturbed ASD equation.

The freeness argument also applies to an ideal background retaining
the ordinary strong end. A sufficiently small neighborhood of that
background is disjoint from its nontrivial finite translates. Therefore
the quotient does not identify different branches over that same ideal
background. The four local tetrahedral bubble branches of
\cite[Proposition~2.9 and Section~3.4]{KM-web} retain their even multiplicity. This argument is local to an
open piece with a strong end; it asserts nothing about a closed quotient
without such an end.

\subsection{Compactification and gluing at the torus ends}
\label{en:torus-ends}

In the excision comparison, stretching the internal torus separates
an exterior cobordism from a cylinder or from two copies of
$T^2\times D^2$. We need to know that only the flat connections on
these torus pieces can occur at the ends of the zero- and
one-dimensional moduli spaces. We must also check the gluing
multiplicity, retaining the two boundary lifts together as in
Section~\ref{loc:torus-geometry}.

Keep the torus necks and caps unperturbed. Choose the secondary
perturbations on finite collars at the ordinary ends of the exterior,
and put all dots on the exterior factors. Proposition~\ref{en:collar}
then makes the exterior regular in every relevant relative bundle
class, with data invariant under the finite action used in
Lemma~\ref{loc:matching-orbit}. The next lemma provides the index gap
that excludes nonflat torus pieces.

\begin{lemma}[Flat cap inverse and index gap]\label{en:torus-gap}
Choose the cap metric as a flat $T^2$ times a disk completed with a
cylindrical end, with the product flat bundle fixed in Section~\ref{loc:torus-geometry}.
Its flat ASD deformation operator is invertible and has index zero.
Every nonflat finite-energy unperturbed ASD cap has index at least four.
Every nonconstant unperturbed ASD torus level has parameterized index
at least four.
\end{lemma}

\begin{proof}
Each of the three real characters in the adjoint local system is
nontrivial on a circle of $T^2$. The twisted torus Laplacian therefore
has a positive lower bound in every form degree. The product Laplacian
on the completed cap retains that bound. At a flat connection the
scaled ASD operator
\[
 D=d_A^*\oplus\sqrt2\,d_A^+
\]
has $D^*D$ equal to the twisted Hodge Laplacian on one-forms and $DD^*$
equal to its counterpart on zero-forms and self-dual two-forms.
The spectral bound gives a bounded inverse. This proves both kernel
and cokernel vanish, not just the vanishing of ordinary cohomology.
The same reasoning applies to the flat cylinder.

For the finite stabilizer, the three nonidentity elements of the
$V_4$ centralizer have lift obstructions $dx$, $dy$, and $dx+dy$.
They are excluded by the determinant-one marking, including its
extension by the angular class $ds$ alone. The latter exchanges
the two lifts and does not stabilize either. Thus no discrete
stabilizer factor has been concealed in the acyclicity statement.

Compare a cap solution with the oppositely oriented flat reference.
The double has underlying manifold $T^2\times S^2$. Writing $x,y$ for
its two degree-two generators, its bundle has
$w_2=x+\varepsilon y$, where $\varepsilon\in\{0,1\}$ records the
possible boundary lift. The Pontryagin square evaluates to
$2\varepsilon\bmod4$, so the relative action belongs to
$\tfrac12\mathbb Z$. The boundary adjoint operators of the two lifts
are conjugate; their spectral corrections agree. Relative to the
index-zero flat cap, the index is $8\kappa$.
Unperturbed ASD gives $\kappa\geq0$, with equality only for a flat
connection. Thus a nonflat cap has $\kappa\geq\tfrac12$ and index
at least four.

For a cylinder level, the corresponding mapping-torus comparison
is on $T^4$. The fixed $T^2$ class plus a class
$\lambda\smile[dt]$ again has even Pontryagin square, giving
half-integral action. Its relative index is $8\kappa$, with equal
adjoint spectral corrections at the two ends. A nonconstant ASD
level has positive action, proving the same bound. Equivalently,
the angular lift exchange has spectral-flow shift four.
\end{proof}

\begin{proposition}[Torus compactification and gluing]\label{en:torus-gluing}
For the composite and product torus-stretching families of
Section~\ref{loc:torus-geometry}, the torus ends of the zero- and one-dimensional
cut-down moduli spaces consist exactly of the regular matching
exterior configurations glued to the flat cylinder or to the two flat
caps. Both ends count $Q_{\mathfrak a}$ with multiplicity one. The assertion is
relative to the fixed ordinary boundary data and is compatible with
ordinary Floer breaking.
\end{proposition}

\begin{proof}
First exclude extra torus pieces. Fredholm indices add across the
acyclic torus ends. All cut-down constraints are on the exterior
$Z_{\mathrm{cut}}$. Let $D\leq1$ be the total cut-down family dimension,
let $r\geq1$ be the codimension of the parameter face, and let $p_\partial$
be its dimension. If $m$ is the cut-down degree, $j_h$ the cap indices
and $d_\ell$ the parameterized indices of genuine torus levels, the
exterior family has expected dimension
\begin{equation}\label{en:exterior-index}
 e_{\mathrm{ext}}=d_{Z_{\mathrm{cut}}}+p_\partial-m
 =D-r-\sum_h j_h-\sum_\ell d_\ell.
\end{equation}
There is no continuous torus matching space to subtract. A constant
flat cylinder is matching data and has no translation dimension to
subtract; a genuine nonconstant level contributes its parameterized
index in this formula. Any nonflat cap or nonconstant torus level
would make $e_{\mathrm{ext}}$ negative. The corresponding exterior moduli
space is empty by its independently achieved regularity. For a disconnected
exterior this means regularity of the
joint family, using independent collar directions on its components
and counting a shared parameter only once. At a torus breaking without
a parameter degeneration the same argument uses $r=0$. Additional
ordinary breakings only reduce the dimension available.
Smooth bubbles on the torus-only pieces also
force a negative-index exterior. No regularity assertion for
high-index unperturbed cap or torus moduli is needed here.

For a surviving flat broken configuration, the torus Hessian is
invertible with a fixed spectral gap. The exterior solution has
exponentially decaying tails. Choose a bounded right inverse to the
regular exterior operator on the complement of its finite-dimensional
kernel, including the linearized dot constraints. The flat local
piece has the inverse in Lemma~\ref{en:torus-gap}.

Splice the solutions and their inverses along necks of length $R$.
The equation error decays exponentially. The linear parametrix error
is the sum of exponentially small tail errors and cutoff commutators.
Allowing the \emph{parametrix} cutoff transitions to grow with $R$
makes the latter $O(R^{-1})$. These cutoffs are used only in solving
the linear equation; they do not lengthen any physical secondary
perturbation support. The absence of a torus zero mode prevents a
new small cokernel mode: a normalized approximate cokernel section
would limit either to a cokernel section on an exterior or cap, or
escape down the neck, where the spectral gap excludes it. Thus the
right-inverse estimate is uniform after the exterior kernel is split
off. For large $R$ the parametrix error has norm
less than one. A Neumann series gives a right inverse, and the
quadratic remainder estimate gives the unique nearby solution
modulo gauge by contraction mapping. The exterior kernel and the
neck length are the chart parameters.

These estimates are uniform on compact subsets of the regular broken
stratum. Uhlenbeck compactness and exponential convergence give the
converse: a sequence tending to the torus end approaches one of these
flat broken configurations, since the index argument has excluded
other torus or cap factors. Hence the gluing charts describe the full
end under consideration.

For the product family, there are two cap boundaries but only one
chosen stretching parameter. First allow independent lengths
$(R_1,R_2)$. Acyclic gluing gives the local chart
\[
 \mathcal M_{\mathrm{ext}}\times[R_0,\infty)^2,
\]
with submersive projection to the lengths. Restricting to
$R_1=R_2$ gives the required chart
$\mathcal M_{\mathrm{ext}}\times[R_0,\infty)$.
Thus this joint degeneration has codimension one, not two; the
diagonal restriction loses no regularity.

Construct the charts before the extra finite quotient. The matching
orbits on the composite are
$(\mathcal X_{00}\sqcup\mathcal X_{11})/G$, and the product selects
$\mathcal X_{00}/G_0$. Lemma~\ref{loc:matching-orbit} identifies these charts
and their counts once each. The tree and forest incidence graphs
there account for all central-lift multiplicities.

At an ordinary breaking use the fixed good ordinary endpoint data.
Splicing in disjoint cylindrical regions gives the ordinary gluing
parameters together with these torus parameters. In the dimensions
at issue an additional independent degeneration at a torus end has
codimension at least two and gives no new endpoint. This refers to
an additional breaking, not to the two prescribed cap lengths sharing
one parameter. Exterior tetrahedral bubbling retains the ordinary
four-branch local model and its even count, by the free-quotient
argument following Proposition~\ref{en:collar}.
\end{proof}

The underlying elliptic slice, Uhlenbeck compactness and cylindrical-end
nonlinear estimates are those of the bifold construction in
\cite[Section~3.4]{KM-web} and the broken-metric family framework in
\cite[Section~3.9]{KM-unknot}. The arguments above verify the additional
restricted perturbation directions, torus-only inverses, index gap,
diagonal length restriction and finite multiplicities needed for these
particular families.

\subsection{Compatible choices with a uniform action bound}
\label{en:choices}

We have established transversality for the permitted perturbations and
described gluing at the torus ends. We now choose the data for all the
cobordisms and families in Section~\ref{loc:section} at once. The
choices are made in an order that fixes each boundary face before the
interior of a family is treated. Their smallness will give a uniform
lower bound for the action of every configuration counted in the
three scalar expressions.

\subsubsection{Choose the local models in the normalization first}

Begin with the families of metrics used in the all-$L$ exact-triangle
argument of \cite{KM-tri}. These include the local sphere models and
the permitted metric choices on the special face. Take their
restrictions for the metrics on the ordinary all-$L$ cobordisms.
Thus we choose the map metrics as part of this family, rather than
prescribing them in advance and seeking an extension.

Fix the complete local $S^{2,1}$ and sphere models at this stage,
including their allowed generic metric families and their common
corner data. Subsequently, the secondary perturbations for the
counted all-$L$ operations will be supported on collars at ordinary
external ends, outside these local pieces. The local models can
therefore be used with the metric freedom allowed in the published
argument; no additional relative metric assertion for the special
face is required.

For clarity, the compactified pentagon has successive boundary faces
\[
 Y_2,\quad S_1,\quad S^{2,1},\quad S_2,\quad Y_1.
\]
These are the disjoint-cut adjacencies, not the cyclic intersection order
of the five hypersurfaces. The special face is adjacent to the two sphere
faces. The construction and permitted local metric freedom are those of
\cite[Sections 6--7]{KM-tri}. The two-parameter families used to prove the normalization do not
occur in the scalar expressions. We need not impose the smallness
bound for action on these auxiliary families, although their data
must agree with the selected data on the counted boundary faces.

Choose each remaining collared geometry before its equation perturbations.
A compact initial path from a prescribed metric to a neck-adapted metric
can be included before stretching. It is still a one-parameter family
on the same cobordism and changes neither its topological index nor
the number of parameters.

\subsubsection{A finite order of choices}

The order below is useful because it leaves no equation requiring a choice
to be invariant under its own subsequent compositions.
\begin{enumerate}
\item Choose the common endpoint data once for each labelled marked end.
      For a perfect local or terminal complex, choose the small model
      perturbation, then a good additional perturbation vanishing near
      its nondegenerate critical set.
\item Choose the ordinary maps, source and cap fillings, $p_j,O,L_j,F_j$,
      the excision maps, and the reduction maps. Repeated labels mean
      identical data.
\item Choose the common cut-open exterior with its correlated torus
      data, and finite reference composite and product cores.
\item Choose the interval families relative to their already chosen
      faces: composition, spectator translation, the two torus
      stretchings, product continuation, and the all-$L$ $J$ families.
\item Complete the all-$L$ normalization relative to its selected
      faces, within the families of metrics in the published construction.
\end{enumerate}
No interval interior is prescribed as another interval's endpoint.
The localization construction uses an algebraic sum and composition of
these homotopies. It does not assert that they are faces of a further
unconstructed higher-dimensional family.

The local relation needs no further choice constraint. If
$p_j:U\longrightarrow D$ are the relative maps into an actual perfect
local complex, then $d_U=d_D=0$. The homology relation
$\sum_j[p_j]=0$ implies the literal chain relation $\sum_jp_j=0$.
Interior changes of a filling map are chain homotopies and hence change
none of its entries with these endpoints. This is why the $p_j$ can be
made regular and small independently.

For relative extension, use collared identifications to view the permitted
perturbations as sections of the fixed Banach spaces. Pull back face data
over parameter collars and extend with a partition of unity in parameter
space. A parameter derivative is not a physical time derivative in the
energy identity. Finitely many extension charts have a finite total norm
loss. Choose successive generic corrections in smaller open balls,
vanishing on already fixed parameter collars. Proposition~\ref{en:collar}
supplies interior surjectivity; ordinary marked gluing and
Proposition~\ref{en:torus-gluing} supply regularity at the fixed
collars. The conditions for all relative bundle classes and endpoint
labels form a countable collection, so their residual regularity sets
can be intersected inside each small open ball.

Dot sections are chosen once on the ordinary factors and pulled back on
matched faces. Their relative extensions can be made transverse separately.
They change no equation perturbation and incur no action cost. Their
associated line bundles and the boundary identifications must be retained;
in this construction the dots lie on the ordinary source and cap pieces,
where the ordinary marking is fixed.

\subsubsection{The estimate on a whole capped constituent}

Apply the exact neck identity \eqref{en:neck-identity} and the core
bound \eqref{en:core-bound}. Fixed-width physical cutoffs give the
length-independent total-variation bound above. Translating such a
transition or lengthening the constant middle of a neck has the same
bound. The cap and torus equations remain exactly unperturbed.

Let $\rho$ bound the endpoint and secondary norms after the finite
extension losses. There are constants $C_1,C_2$, fixed by the chosen geometric
families and cutoff functions, so that each basic counted piece has a cylindrical
contribution to the action error at most $C_1\rho$ and a squared perturbation norm on the compact core
of at most $C_2\rho^2$. These are bounds on the whole configuration space in all
relevant sectors, not estimates for a selected solution.

Let $L$ bound the number of basic factors in a capped constituent.
The list of composites specified in Appendix~\ref{app:computation}
contains at most $18$ operators in any one term;
adding source and cap permits $L=20$, with fixed component multiplicities
absorbed in the constants. Thus every constituent satisfies
\begin{equation}\label{en:total-budget}
 \eta+\frac{\epsilon}{32\pi^2}
 \leq L C_1\rho+\frac{L C_2\rho^2}{32\pi^2}.
\end{equation}
For a prescribed $\delta>0$, choose $\rho$ so both summands are less than
$\delta/4$, leaving room for subsequent relative choices. Regularity is
obtained inside that open set. No lower bound on perturbation size is
imposed by the surjectivity argument.

At a Floer breaking, the two appearances of the same endpoint value of
$f$ in \eqref{en:neck-identity} cancel. Arbitrarily many trajectory levels therefore
do not create arbitrarily many oscillation penalties. The same applies at
ordinary matched ends of a composed word. There remain only the finitely
many prescribed transition regions and compact cores. The bound persists
on the broken faces.

The gluing threshold may grow as $\rho$ decreases and the local spectral
gaps shrink. Choose $\rho$ and the endpoint
data first; use their gluing threshold afterward. All the extra length is
assigned to gradient cylinders, where \eqref{en:neck-identity} is independent of
length, rather than to a volume estimate for an uncontrolled perturbation.

\begin{proposition}[Coherent regular data with quantitative control]\label{en:coherence}\label{en:relative-choice}
For the collared family geometries of Section~\ref{loc:section}, with the
ordinary marked gluing and the torus identifications proved above, the
fixed perfect endpoint complexes, ordinary regularity and agreement on
common faces can be imposed simultaneously with an arbitrarily small
uniform perturbation error in the action bound.
\end{proposition}

\begin{proof}
Use the finite order above, relative affine extension, and residual
regular choices in the open balls supplied by
Proposition~\ref{en:collar}. Use the flat models of Lemma~\ref{en:torus-gap}
on the isolated torus-only pieces and the published sphere and $S^{2,1}$ models on the
normalization faces. Formula \eqref{en:total-budget} applies to every word;
the endpoint cancellation proves the assertion on broken faces.
The local chain relations persist because their endpoints are perfect.
The auxiliary families proving normalization are completed relative to
the already chosen faces. They do not occur in the scalar expressions
and do not require this action bound.
\end{proof}


\subsection{Simultaneous geometric realization}

\begin{theorem}[Simultaneous geometric data]\label{en:simultaneous-data}
Use the actual local and terminal models of Proposition~\ref{lm:perfect},
the local fillings and framed arcs of Section~\ref{loc:section}, and the
product Hopf-core bundle and correlated torus boundary of
Section~\ref{loc:torus-geometry}. Choose the all-$L$ families of metrics and
their local sphere and $S^{2,1}$ models before the equation perturbations,
as above. Then for every $\delta>0$ the finite system of operations in
the three scalar expressions admits shared choices with the following
properties.
\begin{enumerate}
\item The local and terminal complexes are the asserted actual perfect
      models, with good ordinary endpoint perturbations. Every occurrence
      of a labelled marked end uses the same complex and perturbation.
\item The composition, spectator-naturality and inverse-excision families
      have regular zero- and one-dimensional cut-down moduli spaces,
      with common data on their faces. Their compactifications give
      precisely the identities \eqref{loc:composition},
      \eqref{loc:naturality} and \eqref{loc:foura}--\eqref{loc:fourd},
      with the common matching term $Q_{\mathfrak a}$.
      Ordinary Floer breakings are included, and tetrahedral bubble
      ends have even multiplicity over $\F$.
\item Every complete capped constituent of a scalar expression,
      including its ordinary source, cap and cobordism factors, has
      controlled data with
      $\eta+\epsilon/(32\pi^2)<\delta$.
      The bound is uniform through its stretched and broken faces.
\item The selected geometric all-$L$ operation extends to the
      normalization construction with its counted boundary data fixed.
      The auxiliary normalization families that do not occur in those
      expressions are not required to satisfy this action bound.
\end{enumerate}
\end{theorem}

\begin{proof}
Choose the common endpoint data after the collared families of metrics.
For the local and terminal ends, Proposition~\ref{lm:perfect} supplies
arbitrarily small perfect models. The additional good endpoint
perturbations may vanish near the nondegenerate critical set, by
\cite[Section~3.2]{KM-web}, so those generators are retained.

The explicit core and restriction calculations in
Section~\ref{loc:torus-geometry} fix the partial marking, bundle
sector and finite quotient. The local filling transitions and their
framed arcs remain in product spectator regions. The product
Hopf-tube markings on the ordinary factors are proper at the relevant
ends. Thus the geometric composition and translation families have
exactly the stated marked limiting pieces. This uses the underlying
constructions, rather than assuming the analytic family statement of
Proposition~\ref{loc:family-data}.

Every ordinary component and every torus-cut exterior component
retains an ordinary strong end. The additional marking quotients
restrict trivially there. Choose all nonlocal secondaries on product
collars, leaving the complementary cores unperturbed.
Proposition~\ref{en:collar} gives universal surjectivity in these
permitted spaces, including the relative parameter interiors.
The finite quotient loses neither collar directions nor the four
local tetrahedral branches at an ideal background.

For ordinary separating compositions and spectator translations,
use the ordinary marked family compactification with the proper
factors just described. For the torus faces,
Proposition~\ref{en:torus-gluing} gives the flat gluing charts and
excludes additional torus-only pieces. Lemma~\ref{loc:matching-orbit}
identifies the two surviving terms once each as $Q_{\mathfrak a}$.
Their sum with ordinary composition and continuation gives the
geometric inverse homotopy. These are statements about actual
regular parameter ends; no homology excision isomorphism is being
substituted for them.

Now follow the finite order of choices above. Boundary values agree
because the same labelled factors and the same cut exterior were
chosen first. Pull those values over parameter collars and extend
relatively. Sard--Smale in the small open balls gives interior
regularity; the ordinary and torus gluing charts give regularity near
already fixed faces. Extend the dot sections compatibly and make
them transverse separately. Proposition~\ref{en:coherence} and
\eqref{en:total-budget} give the prescribed bound on every capped
constituent and its broken limits. In the perfect local complexes,
the relative homology relations are literal chain relations throughout
these choices.

For normalization, the complete local sphere and $S^{2,1}$ models,
including their permitted generic metric family and corner data,
were fixed before choosing the secondary perturbations in the families
that define the counted $J$ operations. These perturbations lie on
external ordinary collars, outside the local pieces. Their sphere
endpoints consequently agree with the fixed local corner models. The
two ordinary-composition faces agree at
their common corner because they use the same ordinary map data.
The sphere-face exterior data extend between those corners; the
isolated sphere model remains the published one. The special face
likewise retains its preselected local model, and only its exterior
perturbations require relative extension. The ordinary collar
argument supplies that extension and the pentagon interior
regularity. This completes the construction of
\cite[Sections~6--7]{KM-tri} and its identity comparison with the
counted faces fixed. Those additional families need not satisfy the smallness
condition imposed on the scalar expressions. All four assertions follow.
\end{proof}

The theorem concerns realization of the specified operations. Scalar
vanishing is obtained below by applying the uniform action estimate
to their separately computed capped indices.

\subsection{The action required by the index formula}

We finish by applying the estimate to the scalar expressions. A term
in one of these expressions counts configurations on several cobordism
pieces, with matching critical points on their common ends. Gluing
their bundle data determines a closed bundle class. Its action is the
sum of the actions on the pieces, even though a connection obtained
by splicing the solutions need not itself be anti-self-dual. This
allows us to combine the bound just proved with the closed index
calculation in Section~\ref{geo:section}.

For a closed bifold $(X,\Sigma)$ write
\begin{equation}
 I_{\mathrm{top}}(X,\Sigma)
 =-\frac32\bigl(\chi(X)+\sigma(X)\bigr)
       +\chi(\Sigma)+\frac{\nu(\Sigma)}2-\frac{\tau(\Sigma)}2.
 \label{en:top-index}
\end{equation}
Here $\nu(\Sigma)=\Sigma\cdot\Sigma$ is the self-intersection of
\cite[Definition~2.5]{KM-web}, namely half the relative Euler number of the
squared normal bundle with its canonical boundary section; $\tau$ is the
number of tetrahedral points. The uncut deformation index is
$8\kappa+I_{\mathrm{top}}$ by
\cite[Proposition~2.6]{KM-web}. This formula is additive over disconnected
components. In particular one must not replace its ambient term by
$-3(1-b_1+b^+)$ while silently changing the number of components.

Suppose a matrix-product term is represented by endpoint-matched solutions
on its marked pieces. Use the matching isomorphisms to glue the bundle data.
On a truncation, splice connections with a product collar at each matched
end; exponential convergence lets the resulting action tend to the sum of
the piece actions. This constructs a closed bundle class and gives
additivity of Chern--Weil action. Fredholm index additivity across
nondegenerate ends gives the corresponding index equation. The actual
empty Hopf factor and acyclic torus flat factors have no continuous
stabilizer or critical-manifold dimension to add to that equation.
Their product closures have zero action and zero $I_{\mathrm{top}}$.
The matching-orbit identification of Lemma~\ref{loc:matching-orbit}
preserves the exterior action and index; the two flat torus fillings
add zero action.

The spliced connection need not solve the ASD equation. The action bound is
applied to the original solutions on the pieces. Likewise, no equality
between a closed point count and the algebraic trace is used. A trace-count
identity would require the appropriate self-gluing class in the finite gauge
subgroup; omitting it can change a covering multiplicity. It cannot change
the local index or Chern--Weil integral of a given glued bundle class.
Different bundle sectors may have different actions, but then the index
equation assigns them different dimensions.

For the prescribed geometries, Proposition~\ref{geo:allterms} gives
\[
 \chi(X)+\sigma(X)=0,\qquad
 \chi(\Sigma)+\nu(\Sigma)/2-\tau(\Sigma)/2=5/2.
\]
Every constituent has one metric-family parameter. A dot imposes one real
codimension, as in \cite[Section~3.5]{KM-web}. Thus a zero-dimensional
contribution with $k$ dots must satisfy
\begin{equation}
 0=8\kappa+\frac52+1-k.
 \label{en:required-action}
\end{equation}
There is no additional parameter for each fixed ordinary factor in the
product. The sole counted parameter belongs to the one homotopy factor.

\begin{proposition}[Scalar vanishing]\label{en:vanishing}
For the normalized scalar expressions of Proposition~\ref{op:formula},
with the capped geometries and dot counts of
Proposition~\ref{geo:allterms}, one has
\[
 \alpha=\beta=\chi=0.
\]
\end{proposition}
\begin{proof}
For three dots, \eqref{en:required-action} requires $\kappa=-1/16$; for two
dots it requires $\kappa=-3/16$. Choose the data in
Theorem~\ref{en:simultaneous-data} with
$\eta+\epsilon/(32\pi^2)<1/16$.
Proposition~\ref{en:uniform-bound} rules out every matched configuration that
could contribute to any constituent. Hence each constituent count is zero
and so are their finite sums. The cap summands for $\chi$ are used inside the
fixed full expression; the argument does not require each summand to define
an independent functional on the quotient containing the secondary
operation.
\end{proof}

The scalar conclusion uses the realization theorem, the identity-normalized
formula and the capped topological calculation together. The analytic
construction supplies the small regular representatives; the fixed index
equation supplies the negative action that excludes their contributions.