\section{The index calculation for the closed foams}\label{geo:section}

The three scalar expressions in the preceding section are counts of
configurations on composites of foam cobordisms. A source foam closes
the incoming web and a cap closes the outgoing web. After identifying
the remaining marked ends, each composite determines a closed bifold.
We shall compute the topological term in its ASD index formula. The
answer, together with the number of dots and the one metric parameter,
will force any contributing configuration to have negative action.
Section~\ref{en:section} will exclude that possibility by an estimate
for small perturbations.

For this calculation, the choice of chain homotopy matters. A relation
between maps on homology does not specify the topology of a family
realizing that relation. We use the particular foam cobordisms and
families described in Proposition~\ref{loc:family-data}, keeping their
normal framings and the auxiliary four-manifolds introduced by excision.
The topological calculation can be made before choosing regular
perturbations; the compatible analytic choices are supplied later by
Theorem~\ref{en:simultaneous-data}.

\subsection{Conventions for normal numbers and gluing}

For a foam $\Sigma$ in an oriented four-manifold, write $\nu(\Sigma)$ for the
self-intersection number of \cite[Definition~2.5]{KM-web}, and let $\tau(\Sigma)$
be its number of tetrahedral points.  Thus $\nu$ is \emph{half} the relative
Euler number of the squared normal bundle of the normalized facets, with the
canonical seam section.  Put
\begin{equation}\label{geo:index}
 Q(\Sigma)=\chi(\Sigma)+\tfrac12\nu(\Sigma)-\tfrac12\tau(\Sigma),\qquad
 I_{\rm top}(X,\Sigma)=-\tfrac32\bigl(\chi(X)+\sigma(X)\bigr)+Q(\Sigma).
\end{equation}
The closed deformation index is $8\kappa+I_{\rm top}$ by
\cite[Proposition~2.6]{KM-web}.  Formula~\eqref{geo:index} is additive over
connected components.  In particular, no fixed value of $b_0(X)$ is implicit
when an auxiliary component is introduced.

For relative comparisons all boundary collars and their normal sections are
fixed.  Gluing foams along a web $K$ subtracts $\chi(K)$ from their summed
Euler characteristics.  The normal obstructions add when the prescribed
sections are identified; this condition is part of our gluing convention.
We write $\Delta(\chi,\nu,\tau)$ for the difference between two fillings of
the same collared boundary.  Such a notation is used only after an actual
relative geometric comparison has been specified.

\begin{lemma}[A hypersurface normal section]\label{geo:normalzero}
Suppose that a closed foam $\Sigma$ is contained in an oriented embedded
three-manifold $M$ in an oriented four-manifold, and that its seam and
vertex models are the standard models in $M$.  Then $\nu(\Sigma)=0$.
\end{lemma}
\begin{proof}
Let $F$ be a normalized facet and $N_F$ its ambient normal bundle.  The line
bundle $N_MF\subset N_F$ gives a nowhere-zero section of $N_F^{[2]}$: the
images of the two unit vectors in a fiber coincide in the squared bundle.
This construction does not require $F$ to be orientable.  At its boundary,
the three-plane used to define the canonical seam section is $TM$.  The
section just constructed consequently restricts to that canonical section.
It has zero relative Euler obstruction.  The same statement holds after
rounding the standard tetrahedral corners in $M$, and summing over the
normalized facets proves the claim.
\end{proof}

\subsection{A reference closed foam}

We begin with a simpler composite, using the standard planar maps $s$
and $\gamma$. Its index can be calculated directly from the reduction
foams in Section~\ref{alg:section}. We will then compare each composite
that occurs in the scalar formula with this reference foam. All the
comparisons are made relative to fixed boundary collars, so that the
normal numbers can be compared as well as the Euler characteristics.

Use the spherical embedding of the generalized Petersen graph $G(10,2)$,
with edges
\[
 \{i,i+1\},\quad \{i,10+i\},\quad\{10+i,10+(i+2)\},
 \qquad i\in\mathbb Z/10.
\]
The integer additions in the first and third families are taken modulo ten
within their respective sets of ten vertices.  Remove vertices $0,1$ and
their incident edges.  The remaining exterior has four ends in cyclic order
$(9,10,2,11)$.  The webs $K_0,K_1,L_1$ used for $A,B,C$ are obtained by
inserting, respectively,
\begin{align*}
 K_0:&\quad (9,10),(2,11),\\
 K_1:&\quad (9,11),(10,2),\\
 L_1:&\quad (20,21),(20,9),(20,11),(21,10),(21,2).
\end{align*}
These insertions use one common disk and the inherited spherical rotation
system.  In particular, the two adjacent matchings are not identified by
an unspecified permutation of their external ends.

Recall how the basis foams were constructed. Reduce a web by bigon,
triangle and square moves supported in faces of its planar diagram,
and finish by filling the resulting theta web. Each step is a standard
foam cobordism. The resulting foam, or its reflection with the direction
of the cobordism reversed, supplies a cap or a source respectively. A square branch
specifies one of its two adjacent pairings.  At each stage use the standard
untwisted local foam in the indicated face disk times an interval and the
product on its complement.  The triangle and square models are those of
\cite[Figures~10--13]{KM-web}.  The
complete reduction sequences, including intermediate vertex names and dot
positions, are in \texttt{support/geometry/movie\_manifest.txt}; the
accompanying rotation data fix their facial supports.

For example, the common undotted source underlying $C_{14},C_{19}$ reduces
by the successive faces
\[
 (2,12,10,21),\ (3,4,14),\ (5,6,16,21),\ (7,15,17),\
 (8,9,19,21),\ (11,20),
\]
followed by the theta on vertices $13,18$.  The three squares use the
pairings $(01)(23),(01)(23),(03)(12)$ in the displayed cyclic orders.
The cap underlying $B_{33}$ uses
\[
 (2,10,12),\ (9,11,19),\ (3,4,14,20),\ (5,6,16),\
 (7,8,18,22),\ (13,21),
\]
and then the theta on $15,17$; its two squares use $(03)(12)$.
The dot on $C_{14}$ is on terminal theta edge $1$, and that on $C_{19}$ is
on the selected bigon facet.  The two dots on $B_{33}$ lie on its selected
bigon facet and terminal theta edge $0$.

The \emph{reference closure}, denoted $\Sigma_{\rm raw}$, for a source $P$
and cap $R$ is
\begin{equation}\label{geo:rawclosure}
                  P\ ;\ s\ ;\ \gamma\ ;\ R.
\end{equation}
Here $s$ is the standard planar unzip, and $\gamma$ the standard planar
saddle between the adjacent matchings.  The literal composite $ba$ is not
used in this definition.

Figure~\ref{fig:reference-middle} shows the local slices of this planar
middle. The cap at its right end is read in the direction displayed in
Figures~\ref{fig:B-half-foams} and~\ref{fig:B33-half-foam}; a source at
its left end is read in the reverse direction, as in
Figure~\ref{fig:C-half-foams}. For the two summands in the test for
$\alpha$, the corresponding reference closures use $P=C_{14},C_{19}$
and $R=\operatorname{cap}(B_{33})$. This picture specifies the reference
foam whose topology is computed below. The cobordisms and metric families
defining $h$ require the subsequent relative comparisons; the picture
does not identify $h$ with the composite $\gamma s$.

\begin{figure}[htbp]
\centering
\input{figures/reference_middle}
\caption{The local slices of the planar middle in
$\Sigma_{\rm raw}=P\,;\,s\,;\,\gamma\,;\,R$.
The labels give the four attachment vertices in the common exterior.
The first cobordism unzips the edge joining vertices $20,21$; the
second is the ordinary saddle between the two adjacent matchings.}
\label{fig:reference-middle}
\end{figure}

\begin{proposition}[Index of the reference closure]\label{geo:raw}
Every source--cap pair used in the three scalar tests has a reference closure
with $Q=5/2$ and $\nu=0$.  Its framed closure has
$\chi(X)+\sigma(X)=0$.
\end{proposition}
\begin{proof}
Construct each facial move in $S^2\times I$, using disjoint time intervals
and the fixed product collars.  The induced child rotation is precisely
the rotation obtained by the prescribed local replacement.  Thus the
construction continues inductively through the reduction sequence.  Cap
its terminal theta by the standard three disks meeting along an interval
in a ball.  The source, the planar middle of \eqref{geo:rawclosure}, and
the cap consequently lie in $S^2\times I$ with a three-ball at each end,
an oriented $S^3$.  Lemma~\ref{geo:normalzero} gives $\nu=0$.  This argument
places the whole foam in one three-manifold; planarity of its individual
slices alone would not give the required normal section.

A local bigon or triangle reduction contributes zero to the Euler
characteristic of a half-foam relative to its outgoing web; a square
contributes $-1$.  Indeed the local foam has Euler characteristic $1$,
whereas the outgoing local web has characteristic $1,1,2$, respectively.
The half-theta has characteristic $1$.  A triangle contributes one
tetrahedral point, and the other moves contribute none.  Hence
\begin{equation}\label{geo:halfdata}
 \chi(P)=1-\#\{\text{squares in }P\},\qquad
 \tau(P)=\#\{\text{triangles in }P\}.
\end{equation}
The finite list reduces to the following table.  Branch numbers here are
zero-based; basis subscripts remain one-based.
\begin{center}
\begin{tabular}{c|c|r|r|r|r}
Web & branches & squares & triangles & $\chi$ & $\tau$\\\hline
$C$ & $0,1,4,5$ & 3&2&$-2$&2\\
$C$ & $2,3$ &2&4&$-1$&4\\
$B$ & $0$ &1&5&0&5\\
$B$ & $1,2$ &2&3&$-1$&3
\end{tabular}
\end{center}
One has $\chi(C)=-10$, $\chi(A)=\chi(B)=-9$.  The global $s$ and $\gamma$
foams each have characteristic $-10$.  Subtracting the three gluing webs
therefore gives
\[
 \chi(\Sigma_{\rm raw})=\chi(P)+\chi(R)+8,\qquad
 \tau(\Sigma_{\rm raw})=\tau(P)+\tau(R).
\]
Each $C$ row has $\chi-\tau/2=-3$, and each $B$ row has
$\chi-\tau/2=-5/2$.  Thus $Q=-3-5/2+8=5/2$ for all eighteen possible
branch pairs, including all the required pairs.

The underlying framed scalar is a cobordism from the empty Hopf object
to itself.  Joining its ordinary ends adds the time-circle handle to the
closed $S^4$ picture, giving $S^4\#(S^1\times S^3)$, with characteristic
and signature zero.  The marking Hopf annuli close to product tori; their
Euler and normal numbers are zero.  They do not change $Q$.
\end{proof}

The supporting calculation checks that each prescribed reduction takes
place in a face of the planar web obtained at that stage. This verifies
that the entire sequence can be realized as just described. The reason
that the normal number is zero is the geometric one in the proof: the
whole foam lies in an oriented three-manifold, and the resulting section
of the squared normal bundle extends the prescribed seam section.

\subsection{Comparing the augmented cobordisms}

The scalar formula uses cobordisms obtained from the all-$L$ triangle
by the inclusions and projections in the local reduction formulas.
We must compare these augmented cobordisms with the standard maps
used above. The crossing and the position of the added edge are part
of the comparison: they determine a normal number. We therefore
specify the change from the pictures in \cite{KM-tri} to our four
labelled boundary points before making the index calculation.

To compare the all-$L$ construction with the standard mixed triangle, fix
the printed introduction/Section~8 convention of \cite{KM-tri}.  Its four
ports are $NW,NE,SE,SW$; $K_0$ pairs horizontally, $K_1$ vertically, and
$K_2$ joins $NW$ to $SE$ underneath the other diagonal.  Identify these
ports with $(9,10,2,11)$ only after the following cyclic transport.
Set
\[
 v_{NW}=(1,1,1),\quad v_{NE}=(1,-1,-1),\quad
 v_{SE}=(-1,1,-1),\quad v_{SW}=(-1,-1,1).
\]
The screen coordinates $(X,Y,Z)=(-z,x,-y)$ have determinant $+1$.
The $NW$--$SE$ segment has depth $-1$, and the $NE$--$SW$ segment has
depth $+1$.  Thus they realize the stated crossing, with positive depth
towards the viewer.  The rotation
\[
 R(x,y,z)=(y,z,x),\qquad \rho=(NE\ SW\ SE),
\]
fixes $NW$, preserves orientation, and fixes time.  It transports the
printed $(K_2,K_1,K_0)$ to the actual $(K_0,K_2,K_1)$.  The printed top
augmentation edge travels to the left edge between ports $9,11$.
The entire local comparison, including its collars and edge strip, is
transported.  No symmetry of the exterior web is required.

Write
\[
 Q_H=P_{X,A}\widehat H I_Z,\qquad
 Q_F=P_{Y,W}\widehat F I_{X,A}.
\]
These symbols in this subsection denote the actual composite foams.

\begin{proposition}[Three relative replacements]\label{geo:replacements}
For the standard embedded models with the transported collars just
specified, the following relative differences hold:
\begin{equation}\label{geo:shifts}
\begin{array}{c|rrr}
 \text{replacement}&\Delta\chi&\Delta\nu&\Delta\tau\\\hline
 ba-\gamma&-1&2&0\\
 Q_H-s&1&0&2\\
 Q_F-a&1&-1&1
\end{array}
\end{equation}
Each preserves $Q$.
\end{proposition}
\begin{proof}
The proof of \cite[Theorem~9.1]{KM-tri} identifies the literal $ba$ with
the saddle $\gamma$ summed in a facet with $\Psi_0$.  Since
$(\chi,\nu,\tau)(\Psi_0)=(1,2,0)$ and a facet sum subtracts two from
Euler characteristic, this gives the first row.

For $Q_H$, name the source vertices $20,21$ as above.  The triangle
inclusion expands $20$ to $(22,20,23)$, with the augmentation edge
$(22,23)$.  After $\widehat H$, the upper and lower square paths are
$9,22,24,10$ and $11,23,25,2$, with vertical edges $(22,23),(24,25)$.
The $A$ projection pairs $22$ with $24$, and $23$ with $25$.
The two tetrahedra are joined by the three seam paths through $20$,
through $22,24$, and through $23,25$.  The intervening facets are the
standard untwisted disks.  This is the collapsed-side triangle comparison
on a theta boundary: its two fillings have characteristics $2,1$ and
numbers of tetrahedra $2,0$.  Both lie in the same local three-dimensional
host, with the same collar framing, so their normal difference is zero.
This proves the second row.  This theta boundary must be distinguished
from the prism boundary for an inverse on the triangle side.

For the last row, use the embedded identity in the \emph{proof} of
\cite[Lemma~8.2]{KM-tri}, followed by its complete Figure~9 movie.
The middle band in that movie is the standard tetrahedral cobordism
summed with positive $\Psi_2$.  The surrounding square inclusion and
projection give
\begin{equation}\label{geo:embeddedidentity}
 Q_F\mathbin{\#}_{\rm tet}\Psi_2
       \cong a\mathbin{\#}_{\rm facet}\Psi.
\end{equation}
The source identifies the capping disk with $D_+$ of its
Proposition~4.7.  The identity therefore specifies both the foam and the
facet; equality of the resulting homology maps would not be sufficient.

From the definition $\Psi_n=\mathbb{RP}^2\cup D_1\cup\cdots\cup D_n$
in \cite[Section~4]{KM-tri}, one obtains
$\chi(\Psi_n)=1+n$, $\nu(\Psi_n)=2-n/2$, and
$\tau(\Psi_n)=n(n-1)/2$.  In particular positive $\Psi_2$ has tuple
$(3,1,1)$.  In a standard tetrahedral sum the two removed cones each
have characteristic $1$, and their common link $K_4$ has characteristic
$-2$.  The glued characteristic is therefore the sum minus four, and
the two tetrahedral points disappear.  The in-three-plane normal lines
on the removed facet sectors match under the specified cone-link gluing,
so $\nu$ adds.  Inserting $\Psi_2$ changes the tuple by $(-1,1,-1)$.
The source's description of $\Psi$ as
$\overline{\Psi}_2\#_{\rm tet}\Psi_2$ gives its tuple $(2,0,0)$, so a
facet sum with $\Psi$ changes none of the three numbers.
Solving \eqref{geo:embeddedidentity} now gives
$\Delta(Q_F-a)=(1,-1,1)$.  The normal sign is determined by this equation.
The external reflected cap is attached to both sides with the same collar;
it does not introduce a second reflection.  The final assertion follows
by substitution in \eqref{geo:index}.
\end{proof}

For the representative source and cap used to compute $\alpha$, the
reference foam has $(\chi,\nu,\tau)=(5,0,5)$.
The three replacements give successively
\[
 (5,0,5)\longmapsto(4,2,5)\longmapsto(5,2,7)
                 \longmapsto(6,1,8).
\]
There is a separate Euler check on the actual seven-piece augmented movie:
its pieces have characteristics $-2,-10,-11,-11,-10,-10,-1$, while the
six intermediate webs have characteristics $-10,-11,-11,-11,-9,-9$.
Thus $\chi=-55+61=6$; the tetrahedral counts sum to eight.  The augmented
foam need not lie in the three-sphere containing the reference foam.  Its normal number is supplied by
Proposition~\ref{geo:replacements}.

\subsection{Excision and the retained empty factors}

The construction in Section~\ref{loc:section} replaces a local foam
relation by a sum of one-parameter families on larger cobordisms.
We now check that the extra pieces introduced there do not change
$I_{\rm top}$ after the source and cap have been attached. By the
\emph{excess} of a piece we mean its contribution to the difference
between this closed index and the index of the corresponding original
cobordism. The comparison uses the punctures, framed arcs, product
spectator tubes and Hopf core specified in
Proposition~\ref{loc:family-data}. An arbitrary inverse of the excision
map on homology would not determine these data.

Keep the actual empty Hopf complex $U$ in the geometry, even when it is
suppressed from formulas for maps.  The reference comparison is with
$F_j\otimes1_U$.  Its closed $U$ product adds $S^1\times S^3$ with two
Hopf tori, and has zero ambient and singular contribution to
\eqref{geo:index}.

The standard four-ended Hopf core $W_H$ has relative handles of indices
$1,2,2,3$.  Its Kirby description gives
\[
 W_H\cong(I\times S^3)\#(I\times S^3)\#(S^2\times S^2),
 \qquad \chi(W_H)=\sigma(W_H)=0;
\]
see \cite[Section~4, Lemma~4.1 and Figures~6--8]{ISST}.  For completeness,
the zero normal contribution concerns the \emph{total} four annuli.
Label them $(i,\varepsilon)$ for $i=1,2$, $\varepsilon=+,-$.
The nonzero incoming linking entries pair equal $i$ and unequal
$\varepsilon$; the outgoing entries pair unequal $i$ and equal
$\varepsilon$.  Each entry is $+1$, and both ordered sums are four.
Their difference, the relative normal contribution, is zero.  The
reversed block has the same zero excess.  This uses the fixed boundary
framings, not merely the fact that the sheets are annuli.

A product spectator $K\times I$ has characteristic $\chi(K)$, which
must be retained.  If $P_j$ is a filling of $Q$ and $O$ its common
exterior, then for spectator $K$
\[
 \chi(L_j)=\chi(K)+\chi(P_j),\qquad
 \chi(\Psi_{\rm spectator})=\chi(K)+\chi(Q).
\]
Subtracting the intermediate web in either side of the naturality
comparison leaves $\chi(K)+\chi(P_j)$.  Attaching $O$ gives
\begin{equation}\label{geo:spectator}
           \chi(P_j)+\chi(O)-\chi(Q)=\chi(F_j).
\end{equation}
Thus the spectator has zero \emph{relative excess}.  The same product
collars identify the normal sections, and introduce no tetrahedra.

The local relative invariant punctures a four-ball away from the foam,
changing its ambient characteristic from one to zero.  The equivalent
boundary-connected insertion has a four-ball and a joining one-handle,
also with net ambient contribution zero.  No unpaired zero-handle is
lost in passing between these descriptions.

The inverse comparison uses the torus cut of
\cite[proof of Theorem~5.6, Figure~5]{KM-unknot}.  Its removed $T^3\times I$
and the two inserted $T^2\times D^2$ have characteristic and signature
zero and are disjoint from the foam.  The cuts are closed internal
three-manifolds, so ordinary signature additivity applies without a
corner correction.  Consequently composition comparison, either torus
stretching comparison, and product continuation have zero excess.
The matching-orbit identification of Section~\ref{loc:torus-geometry}
uses these same pieces with the same product bundle and cap framings.
It changes neither the topology nor the action of a matched
configuration; the torus-only reference factors are flat.
With the notation of the localization construction, every term
\[
 R_j,\quad ON_jV',\quad OL_jH_{\rm comp},\quad
 OL_jH_{\mathrm{cut},c},\quad OL_jH_{\mathrm{cut},p},\quad OL_jH_{\rm cont}
\]
has the same closed $I_{\rm top}$ as the corresponding capped $F_j$.
This is a calculation for sums of one-parameter families.  Two varying
families are never multiplied together.

\subsection{The index of every term in the scalar expressions}

It remains to combine these comparisons for the finite sum in
Proposition~\ref{op:formula}. There are two points to check: the
successive local reductions must preserve the topological term in the
closed index, and each term must still have just one family parameter.
The second point follows from the form of the sum: its homotopies are
added, and each composite contains only one varying factor.

\begin{proposition}[Index of the specified expansion]\label{geo:allterms}
For the localization families on the specified geometries, every
constituent of the capped three-scalar expression has
$\chi(X)+\sigma(X)=0$ and $Q=5/2$.  Each has one family parameter.  It
has three dots for $\alpha,\beta$ and two dots for either fixed cap
summand of $\chi$.
\end{proposition}
\begin{proof}
The two changes $ba-\gamma$ and $Q_F-a$ have sum $(0,1,1)$.
A previous triangle collapse followed by its reverse expansion adds
$(1,0,2)$; a square inclusion/projection branch has the same relative
triple as the square cylinder.  For a triangle-side inverse the local
boundary is $L_3$, of characteristic $-3$.  Its product filling has
$(\chi,\nu,\tau)=(0,0,0)$ and its collapse/expansion filling has
$(1,0,2)$.  In \eqref{geo:spectator}, one must subtract this prism
boundary, not a theta boundary.  The difference of the two resulting
closed triples is still $(1,0,2)$.

It follows that the change from the reference closure to any term in
the scalar expression is
\begin{equation}\label{geo:generalshift}
                  (\Delta\chi,\Delta\nu,\Delta\tau)=(n,1,2n+1),
\end{equation}
where the actual position of the local filling determines $n$:
\begin{center}
\begin{tabular}{l|c}
Constituent & $n$\\\hline
Main $\widehat J$, $T_X$, or $L_{H,B}$ &1\\
$L_{H,A}$, the $Q_H$ filling &1\\
$L_{H,A}$, the $s$ filling &0\\
A square term of $T_A$ &0\\
Triangle term at position $j$, product filling &$j-1$\\
Triangle term at position $j$, collapse/expansion filling &$j$
\end{tabular}
\end{center}
For the last two rows, $j-1$ counts the preceding paired triangle
collapses and expansions.  The channel lengths are $2,4,4,3$, so
$0\leq n\leq4$.  All terms in the local construction retain the value for
their chosen filling by the zero-excess calculation above.  Finally,
\[
 \Delta Q=n+\tfrac12-\tfrac12(2n+1)=0.
\]
Proposition~\ref{geo:raw} supplies the baseline.  Every homotopy is a
sum of one-parameter constituents, and all surrounding maps have zero
parameters.  The source has one dot, $B_{33}$ has two, and $B_2,B_7$
have one each; none of the constructions adds dots.
\end{proof}

The supporting term catalogue checks the sources and targets in every composite and assigns
\eqref{geo:generalshift} to each filling slot.  It verifies completeness
and arithmetic, but assumes the geometric identifications proved or
specified above.  Its entries do not constitute independent evidence for
an embedded isotopy or a normal framing.

There is a final distinction concerning closure.  A closed count equals
an algebraic trace only with the appropriate class dual to the glued
hypersurface included in the finite gauge subgroup;
\cite[Proposition~5.5 and its following remark]{KM-unknot} explains the
possible extra factor of two.  We do not use a trace-count equality.
Instead, a matched broken configuration provides bundle identifications
at its nondegenerate ends.  Splicing those data and closing the retained
empty factors gives closed bundle data to which the additive index
formula applies.  The spliced connection need not be anti-self-dual.
Finite gauge covers do not change its Chern--Weil action or local Fredholm
index; the Hopf restriction has zero stabilizer Lie algebra.

Thus, subject to the realization and regularity statements, a contribution
to a zero-dimensional, $k$-dotted, one-parameter constituent must satisfy
\begin{equation}\label{geo:requiredaction}
 0=8\kappa+\tfrac52-k+1,
 \qquad
 \kappa=\begin{cases}-1/16,&k=3,\\-3/16,&k=2.\end{cases}
\end{equation}
This is an index requirement on the specified contributions, not yet
an assertion that they vanish.  The uniform energy estimate is the
additional analytic input needed for that conclusion.