\section{Completed vertices, homology, and phase-zero covariance}
\label{slide:sec:vertices-homology-phasezero}

The supported representatives of \cref{slide:sec-setup} must fit together
as maps of one global marked boundary.  The completed charts
\(\mathfrak C_A\), their outgoing parameterizations \(p_A\), and the edge
maps \(F_{a\mid A}\) have already been defined.  We now check that their
meridian and longitude rows glue with the complete marking intact.  This
geometric check makes the subsequent homology action meaningful and supplies
the common rational Lagrangian that controls the Walker seams.

First glue the selected local rows to the common complement, so that each
vertex is represented by one boundary with complete marking rather than by
independent sector choices.  Next compare the meridian spans of adjacent
vertices and compute the integral homology action of an edge, including its
dual-longitude correction.  The actual inverse is then distinguished from a
mere coordinate rewrite, after which the common Lagrangian and the
phase-zero Walker identities follow.  The order is essential: the homology
formulas record the already constructed marked maps; they do not replace the
gluing and marking argument.

\subsection{Compatible vertex rows and the common Lagrangian}

Recall that \(\bar m_t^A,\bar\ell_t^A\) are the geometric curves on the
outgoing boundary of \((W,\mathfrak C_A)\), whereas
\(m_t^A,\ell_t^A\subset Y\) are their counterparts in the fixed abstract
surface.

\begin{proposition}[Compatible completed-vertex rows]
\label{slide:prop:compatible-vertex-rows}
For every subset \(A\subseteq[s]\), the sector and complement rows defining
\(p_A\) glue smoothly and orientation-preservingly, and
\begin{equation}
 p_A(\bar m_t^A)=m_t^A,
 \qquad
 p_A(\bar\ell_t^A)=\ell_t^A
 \quad (1\leq t\leq k),
 \label{slide:eq:completed-vertex-marking-row}
\end{equation}
with labels, orientations, product framings, and support collars retained.
There is no independent postcomposition of an individual glued row.  Moreover,
the vertex Lagrangian from \eqref{slide:eq:vertex-lagrangian} satisfies
\begin{equation}
 \boxed{\Lambda_A=\Lambda}
 \qquad (A\subseteq[s])
 \label{slide:eq:vertex-lagrangian-equality}
\end{equation}
as a subspace of the one fixed group \(H_1(Y;\mathbb Q)\).
\end{proposition}

\begin{proof}
 Choose the old or slid complete row in each protected sector and glue it to
the unchanged complement through the common product collars.  Disjoint
collars make this a simultaneous global marking rather than a collection of
independent choices.  The homology calculation then compares the old and
slid meridian spans in each active handle, which proves equality of the
vertex Lagrangians in the one fixed homology group.

In each protected sector, choose the complete old row or the complete slid
row according as the sector label lies outside or inside \(A\).  The two
coordinate identifications in
\eqref{slide:eq:source-local-chart}--\eqref{slide:eq:target-local-chart} use the same
product coordinates on the entire interface collar.  Both local rows agree
there, as smooth product germs, with the common complement row.  Distinct
closed sector collars are disjoint, so no triple-overlap compatibility
condition occurs.  The selected local rows therefore glue to one smooth
global parameterization.  Every piece preserves orientation, and the pieces
agree on their overlaps; hence the glued map does as well.  The construction
retains every oriented framed support collar and gives
\eqref{slide:eq:completed-vertex-marking-row}.  Since this simultaneous gluing is
the definition of \(p_A\), it leaves no further postcomposition choice.

It remains to compare meridian spans.  If \(b\notin A\), the sector
meridians are the old classes \([m_{i_b}]\) and \([m_{j_b}]\).  If
\(b\in A\), the transported positive-slide row gives
\begin{equation}
 [m_{i_b}^A]=[m_{i_b}]+[m_{j_b}],
 \qquad
 [m_{j_b}^A]=[m_{j_b}].
 \label{slide:eq:slid-vertex-meridians}
\end{equation}
Thus the slid meridian plane is contained in the old plane.  The converse
inclusion follows from
\begin{equation}
 [m_{i_b}]=[m_{i_b}^A]-[m_{j_b}^A],
 \qquad
 [m_{j_b}]=[m_{j_b}^A].
 \label{slide:eq:old-meridians-in-slid-row}
\end{equation}
The active pairs use \(2s\) distinct labels, so these calculations lie in
disjoint handle summands.  The old summand \(\lambda_g\) and every unused
meridian line are unchanged.  Taking their direct sum proves both inclusions
in \eqref{slide:eq:vertex-lagrangian-equality}.  This also covers \(k=2s\), when
the sum over unused meridians is empty.
\end{proof}

\subsection{Complete edge marking and integral homology}

The edge transition \(h_{a\mid A}\) and its weight-zero cylinder
\(H_{a\mid A}\) were defined in \eqref{slide:eq:edge-transition-cylinder}.  Their
types include both endpoint parameterizations.

\begin{proposition}[Edge marking, homology, and Lagrangian transport]
\label{slide:prop:edge-homology-lagrangian}
For every \(A\subseteq[s]\) and \(a\notin A\), the boundary transition
transports the actual embedded active rows by
\begin{align}
 h_{a\mid A}(m_{i_a}^A)&=m_{i_a}^{A\cup\{a\}},&
 h_{a\mid A}(\ell_{i_a}^A)&=\ell_{i_a}^{A\cup\{a\}},\notag\\
 h_{a\mid A}(m_{j_a}^A)&=m_{j_a}^{A\cup\{a\}},&
 h_{a\mid A}(\ell_{j_a}^A)&=\ell_{j_a}^{A\cup\{a\}},
 \label{slide:eq:edge-embedded-marking}
\end{align}
with orientations, product framings, ribbons, detector, and support collars
retained.  It fixes the complete marked row of every other label in its
product coordinates.  On integral homology,
\begin{equation}
 \begin{aligned}
 (h_{a\mid A})_*[m_{i_a}^A]
  &=[m_{i_a}^{A\cup\{a\}}]
    =[m_{i_a}^A]+[m_{j_a}^A],&
 (h_{a\mid A})_*[\ell_{i_a}^A]
  &=[\ell_{i_a}^{A\cup\{a\}}]=[\ell_{i_a}^A],\\
 (h_{a\mid A})_*[m_{j_a}^A]
  &=[m_{j_a}^{A\cup\{a\}}]=[m_{j_a}^A],&
 (h_{a\mid A})_*[\ell_{j_a}^A]
  &=[\ell_{j_a}^{A\cup\{a\}}]
    =[\ell_{j_a}^A]-[\ell_{i_a}^A].
 \end{aligned}
 \label{slide:eq:edge-integral-homology}
\end{equation}
For every \(t\notin\{i_a,j_a\}\), including active labels in other sectors
and unused labels,
\begin{equation}
 \begin{aligned}
 (h_{a\mid A})_*[m_t^A]
   &=[m_t^{A\cup\{a\}}]=[m_t^A],\\
 (h_{a\mid A})_*[\ell_t^A]
   &=[\ell_t^{A\cup\{a\}}]=[\ell_t^A].
 \end{aligned}
 \label{slide:eq:edge-spectator-homology}
\end{equation}
The map also fixes the old summand \(H_1(X;\mathbb Z)\) pointwise.

As an actual map from the target marked basis to the source marked basis, its
inverse satisfies
\begin{equation}
 \begin{aligned}
 (h_{a\mid A}^{-1})_*[m_{i_a}^{A\cup\{a\}}]
   &=[m_{i_a}^A],&
 (h_{a\mid A}^{-1})_*[\ell_{i_a}^{A\cup\{a\}}]
   &=[\ell_{i_a}^A],\\
 (h_{a\mid A}^{-1})_*[m_{j_a}^{A\cup\{a\}}]
   &=[m_{j_a}^A],&
 (h_{a\mid A}^{-1})_*[\ell_{j_a}^{A\cup\{a\}}]
   &=[\ell_{j_a}^A].
 \end{aligned}
 \label{slide:eq:actual-edge-inverse}
\end{equation}
By contrast, merely rewriting the source classes in target-vertex coordinates
inside the fixed abstract group gives
\begin{equation}
 \begin{aligned}
 [m_{i_a}^A]
   &=[m_{i_a}^{A\cup\{a\}}]-[m_{j_a}^{A\cup\{a\}}],&
 [\ell_{i_a}^A]
   &=[\ell_{i_a}^{A\cup\{a\}}],\\
 [m_{j_a}^A]
   &=[m_{j_a}^{A\cup\{a\}}],&
 [\ell_{j_a}^A]
   &=[\ell_{j_a}^{A\cup\{a\}}]+[\ell_{i_a}^{A\cup\{a\}}].
 \end{aligned}
 \label{slide:eq:source-in-target-coordinates}
\end{equation}
Equation~\eqref{slide:eq:source-in-target-coordinates} is a coordinate rewrite,
not the action of \((h_{a\mid A}^{-1})_*\).

Finally, \((h_{a\mid A})_*\) is an integral symplectic automorphism, and
\begin{equation}
 \boxed{
 (h_{a\mid A})_*\Lambda_A=\Lambda_{A\cup\{a\}}=\Lambda,
 \qquad
 (h_{a\mid A}^{-1})_*\Lambda=\Lambda.}
 \label{slide:eq:edge-lagrangian-transport}
\end{equation}
\end{proposition}

\paragraph{Supported transport on homology.}
The supported map first identifies the active complete rows and fixes the
spectators.  Its local positive slide gives the meridian shear and the dual
longitude correction; the inverse relations distinguish actual transport
from a mere coordinate rewrite.  The resulting integral symplectic map carries
the common meridian span to the next vertex Lagrangian.

\begin{proof}
By \cref{slide:lem:local-model-identification}, the supported representative
transports the complete local marking in the \(a\)-sector and is the identity
on every other row.  Conjugation by \(p_A\) and \(p_{A\cup\{a\}}\) proves
\eqref{slide:eq:edge-embedded-marking}.  The compatibility of the complete
markings from
\cref{slide:sec-setup}, including the ribbon and detector tests,
ensures that these identities specify the selected relative class and are not
inferred from homology alone.

The positive local slide has the meridian plus sign and the compensating
dual-longitude minus sign.  Transport through the orientation-preserving
coordinates, with the ordered relabelling \(1\mapsto i_a\) and
\(2\mapsto j_a\), gives \eqref{slide:eq:edge-integral-homology}.  No permutation
or orientation reversal of the ordered active pair occurs.  The identity
exterior gives \eqref{slide:eq:edge-spectator-homology} and the pointwise identity
on the old summand.  Applying the actual inverse to the four target marked
classes gives \eqref{slide:eq:actual-edge-inverse}; solving the last equalities in
\eqref{slide:eq:edge-integral-homology} inside the common abstract group instead
gives \eqref{slide:eq:source-in-target-coordinates}.  The resulting change of basis
is unitriangular on the active four-dimensional summand and the identity
elsewhere, hence is an integral automorphism.

To check symplecticity without suppressing signs, abbreviate
\(m_i=m_{i_a}^A\), \(m_j=m_{j_a}^A\),
\(\ell_i=\ell_{i_a}^A\), and \(\ell_j=\ell_{j_a}^A\).  Since
\(a\notin A\), this is the old row in the active sector.  Using
\(m_r\mathbin\cdot\ell_t=\delta_{rt}\), the matching pairings are
\begin{equation}
 (m_i+m_j)\mathbin\cdot\ell_i=1,
 \qquad
 m_j\mathbin\cdot(\ell_j-\ell_i)=1,
 \label{slide:eq:matching-symplectic-pairings}
\end{equation}
and the only nontrivial cross calculation is
\begin{equation}
 (m_i+m_j)\mathbin\cdot(\ell_j-\ell_i)
 =0-1+1-0=0.
 \label{slide:eq:cross-symplectic-pairing}
\end{equation}
The other cross pairing is \(m_j\mathbin\cdot\ell_i=0\); transformed
meridians pair mutually to zero, as do transformed longitudes.  Skew symmetry
and identity on the orthogonal complementary summands complete the integral
symplecticity check.  In particular, the minus sign in the last formula of
\eqref{slide:eq:edge-integral-homology} is forced: a plus sign would make
\eqref{slide:eq:cross-symplectic-pairing} equal to \(2\).

For the forward Lagrangian direction, retaining every summand gives
\begin{align*}
 (h_{a\mid A})_*\Lambda_A
 &=\lambda_g\oplus
   \langle[m_i]+[m_j],[m_j]\rangle_{\mathbb Q}
   \oplus
   \bigoplus_{t\notin\{i_a,j_a\}}
      \langle[m_t^A]\rangle_{\mathbb Q}\\
 &=\lambda_g\oplus
   \langle[m_i],[m_j]\rangle_{\mathbb Q}
   \oplus
   \bigoplus_{t\notin\{i_a,j_a\}}
      \langle[m_t^A]\rangle_{\mathbb Q}
 =\Lambda.
\end{align*}
The transformed active span lies in the old active span, and the reverse
inclusion uses \([m_i]=([m_i]+[m_j])-[m_j]\).  Thus neither inclusion is
implicit.  Proposition~\ref{slide:prop:compatible-vertex-rows} identifies both
endpoint Lagrangians with \(\Lambda\).  For the reverse direction,
\eqref{slide:eq:actual-edge-inverse} maps the two target active meridians to the two
source active meridians and fixes every other meridian line.  This proves the
second equality in \eqref{slide:eq:edge-lagrangian-transport} directly.
\end{proof}

\subsection{Direct phase-zero traces}

For each \(A\subseteq[s]\), let
\(J_A^0(\mathbf 0)\colon X\to Y\) be the completed forward compression body
in the chart \(\mathfrak C_A\), with outgoing parameterization \(p_A\) and
Walker weight zero.  Let
\(R_A^0(\mathbf 0)\colon Y\to X\) be its reverse, again with the completed
endpoint marking and weight zero.  These are direct constructions at the
completed vertex.  They do not factor through partially slid charts or
elementary attachment arrows.

For arbitrary external vectors \(\mathbf c,\mathbf r\in\mathbb Z^k\), set
\begin{equation}
 c_\Sigma=\sum_{t=1}^k c_t,
 \qquad
 r_\Sigma=\sum_{t=1}^k r_t,
 \label{slide:eq:phase-zero-weight-totals}
\end{equation}
and define
\begin{equation}
 \begin{aligned}
 J_A^0(\mathbf c)
   &:=C(\operatorname{id}_Y,c_\Sigma)J_A^0(\mathbf 0),\\
 R_A^0(\mathbf r)
   &:=R_A^0(\mathbf 0)C(\operatorname{id}_Y,r_\Sigma).
 \end{aligned}
 \label{slide:eq:central-phase-zero-traces}
\end{equation}
The coordinates of \(\mathbf c\) and \(\mathbf r\) remain label-attached
external data.  The complete morphisms in \eqref{slide:eq:central-phase-zero-traces}
depend on them only through the totals \(c_\Sigma\) and \(r_\Sigma\); an individual
coordinate cannot be recovered from the resulting morphism.

At the old vertex \(A=\varnothing\), the ordinary cobordisms with complete
marking underlying \eqref{slide:eq:central-phase-zero-traces} agree with the
old-vertex traces established in \cref{fixed:thm-main-package}; the endpoint rows
and the assigned totals \(c_\Sigma\) and \(r_\Sigma\) agree as well.  This comparison uses
only the old completed chart and imports no slid-site word or edge covariance.
We do not use it here to identify a Walker composite before its ordered seam
has been computed.

\subsection{Ordinary forward and reverse covariance}

For a Walker morphism \(M\), write \(\underline M\) for the same completely
marked cobordism with only its integer weight forgotten.

\begin{proposition}[Ordinary phase-zero edge covariance]
\label{slide:prop:ordinary-phase-zero-covariance}
For every \(A\subseteq[s]\), every \(a\notin A\), and all
\(\mathbf c,\mathbf r\in\mathbb Z^k\), one has
\begin{equation}
 \boxed{
 \underline H_{a\mid A}\,\underline J_A^0(\mathbf c)
 =\underline J_{A\cup\{a\}}^0(\mathbf c)}
 \label{slide:eq:forward-phase-zero-ordinary}
\end{equation}
and
\begin{equation}
 \boxed{
 \underline R_{A\cup\{a\}}^0(\mathbf r)\,
 \underline H_{a\mid A}
 =\underline R_A^0(\mathbf r)}.
 \label{slide:eq:reverse-phase-zero-ordinary}
\end{equation}
Both formulas use the same source-to-target cylinder
\(H_{a\mid A}=C(h_{a\mid A},0)\), not a cylinder of its inverse.  These are
ordinary completely marked identities; the ordered Maslov corrections and
the corresponding Walker-weighted identities will be computed separately.
\end{proposition}

\paragraph{Collar collapse and completed vertices.}
Collapse the source-to-target cylinder onto the outgoing collar of the
forward trace and use the parameterization square to identify the resulting
marked cobordism with the directly completed next vertex.  The reverse
identity is checked independently with the inverse ambient map, while the
same source cylinder and its direction are retained.  Forgetting the central
integer isolates this geometric equality; the ordered Walker corrections are
computed only afterward.

\begin{proof}
Write \(f_{a\mid A,+}=F_{a\mid A}|_{\partial_+W}\).  The definition of the
boundary transition is equivalent to
\begin{equation}
 p_{A\cup\{a\}}\circ f_{a\mid A,+}
 =h_{a\mid A}\circ p_A.
 \label{slide:eq:forward-parameterization-square}
\end{equation}
Collapse \(\underline H_{a\mid A}\) onto the outgoing collar of the
compression body representing \(\underline J_A^0(\mathbf c)\).  The exposed
outgoing row is \(h_{a\mid A}\circ p_A\).  By
\eqref{slide:eq:forward-parameterization-square}, and because
\(F_{a\mid A}\) is the identity on \(X\) and its collar, that ambient map is
an isomorphism from the collapsed representative to the directly completed
compression body at \(A\cup\{a\}\).  It matches both endpoint
parameterizations and every component of the complete marking.  Forgetting
the central integer makes the same argument valid for every \(\mathbf c\),
which proves \eqref{slide:eq:forward-phase-zero-ordinary}.  The occurrence of
\(h_{a\mid A}\), rather than its inverse, is forced by the source-to-target
direction in \eqref{slide:eq:forward-parameterization-square}.

The reverse equality is typed independently.  Reverse the orientations and
the endpoint roles, and use the orientation-preserving map
\begin{equation}
 \overline{F_{a\mid A}^{-1}}\colon
 \overline{(W,\mathfrak C_{A\cup\{a\}})}
 \longrightarrow
 \overline{(W,\mathfrak C_A)}.
 \label{slide:eq:reversed-edge-inverse}
\end{equation}
It is the identity on the common boundary \(X\) and its collar.  Collapsing
\(\underline H_{a\mid A}\) against the incoming collar of the reverse
compression body at \(A\cup\{a\}\) exposes
\(h_{a\mid A}^{-1}\circ p_{A\cup\{a\}}\).  Rearranging
\eqref{slide:eq:forward-parameterization-square} gives
\begin{equation}
 p_A\circ f_{a\mid A,+}^{-1}
 =h_{a\mid A}^{-1}\circ p_{A\cup\{a\}}.
 \label{slide:eq:reverse-parameterization-square}
\end{equation}
Consequently \(\overline{F_{a\mid A}^{-1}}\) identifies the collapsed
composite with the reverse completed compression body at \(A\), retaining
both endpoint rows and the complete marking.  Again the underline removes
the central integer and nothing else, so the argument holds for every
\(\mathbf r\).  This proves \eqref{slide:eq:reverse-phase-zero-ordinary}.  The
inverse in \eqref{slide:eq:reverse-parameterization-square} comes from the incoming
collapse of the same \(h_{a\mid A}\)-cylinder; it does not change that
cylinder to \(C(h_{a\mid A}^{-1},0)\).
\end{proof}

The arguments include the single-direction case \(s=1\), retaining all
\(k-2\) spectators; when also \(g=1\) and \(k=2\), they reduce to the local
marking, homology, Lagrangian, and ordinary covariance identities.  For
\(s=2\) and \(s=3\), the same gluing and direct-sum proofs apply sector by
sector, with no extra compatibility choice.  The unused-label sum is empty
when \(k=2s\), while for \(k>2s\) every unused row is fixed pointwise.
Arbitrary positive, zero, and negative coordinates in \(\mathbf c\) and
\(\mathbf r\) are allowed.  Finally, no argument depends on whether active
labels are numerically adjacent: the pairwise-disjoint protected sectors and
the injectivity of \(a\mapsto(i_a,j_a)\) control all supports and direct
summands.