\section{Walker-exact covariance and actual-vertex phases}
\label{slide:sec:phase-walker-covariance}

The phase-zero calculation gives the geometric skeleton.  Here we restore the
integer weights and check that the same transported Lagrangian makes the
Walker corrections agree at every actual vertex and in both directions.

The notation in this half of the paper records a completed slide vertex.
Thus \(S_A^\epsilon\) is the complete phase word at the vertex \(A\); it is
not the fixed-site local cylinder \(S_{i\mid I}\) used in the first half.
The former varies over a Boolean family of completed slides, whereas the
latter records one local phase choice at a fixed handle site.

We now restore the Walker integers that were suppressed in the ordinary
covariance identities of \cref{slide:prop:ordinary-phase-zero-covariance}.  The
calculation uses the common Lagrangian proved in the preceding section:
for every \(A\subseteq[s]\) and every \(a\notin A\),
\begin{equation}
 \Lambda_A=\Lambda,
 \qquad
 (h_{a\mid A})_*\Lambda=\Lambda,
 \qquad
 (h_{a\mid A}^{-1})_*\Lambda=\Lambda.
 \label{slide:eq:walker-common-lagrangian-data}
\end{equation}
The edge cylinder remains
\(H_{a\mid A}=C(h_{a\mid A},0)\); its weight is part of its definition.
All products below are read right-factor first, and each Maslov triple is
written in its composition order before it is simplified.

The common Lagrangian first determines the two phase-zero seam weights.
For nonzero phases, marked conjugacy transports the words built from the
actual curves and support collars at each vertex.  Checking the forward and
reverse Maslov triples separately then gives phase-cylinder covariance,
which combines with the phase-zero identities to give completed-endpoint
covariance.  Thus the weights are computed from the transported marking
data, rather than from an old phase word assigned to a moved vertex.

\subsection{The two phase-zero seams}

\begin{lemma}[Seam Lagrangians and central weights]
\label{slide:lem:phase-zero-seam-data}
At a completed vertex \(A\), the forward compression body transports its
incoming declared Lagrangian to \(\Lambda_A=\Lambda\) on the high boundary,
whereas the reverse compression body inverse-transports its outgoing declared
Lagrangian to the same high-boundary subspace.  For a completely marked
cobordism \(M\) and a displayed boundary copy \(S\), write
\(\lambda_M(S)\) for the induced boundary Lagrangian transported into the
fixed group \(H_1(Y;\mathbb Q)\) through the retained parameterization; on an
oppositely oriented target copy, this uses inverse transport.  On the two
sides of the edge cylinder one has
\begin{equation}
 \lambda_{H_{a\mid A}}(Y)=(h_{a\mid A})_*\Lambda,
 \qquad
 \lambda_{H_{a\mid A}}(\overline Y)
 =(h_{a\mid A}^{-1})_*\Lambda.
 \label{slide:eq:edge-cylinder-seam-lagrangians}
\end{equation}
Consequently, for the totals in
\eqref{slide:eq:phase-zero-weight-totals},
\begin{equation}
 \operatorname{wt}\bigl(J_A^0(\mathbf c)\bigr)=c_\Sigma,
 \qquad
 \operatorname{wt}\bigl(R_A^0(\mathbf r)\bigr)=r_\Sigma.
 \label{slide:eq:phase-zero-central-weights}
\end{equation}
\end{lemma}

\begin{proof}
 First identify both high-boundary Lagrangians with the common meridian span.
Then read the two edge-cylinder seam subspaces from the chosen source and
target parameterizations.  In each composition order the seam has three equal
entries, so the correction vanishes and only the assigned integer total
remains.

The high-boundary kernel of the forward compression body is the old
disk-kernel together with the lines of all actual co-core meridians.  Its
transported Lagrangian is therefore
\[
 \lambda_g\oplus\bigoplus_{t=1}^k
 \langle[m_t^A]\rangle_{\mathbb Q}=\Lambda_A=\Lambda.
\]
Orientation reversal gives the same complete meridian span as the
inverse-trans\-ported high-boundary Lagrangian of the reverse compression body.
These are statements about the directly completed compression bodies.  The
two identities in \eqref{slide:eq:edge-cylinder-seam-lagrangians} follow from the
source parameterization \(\operatorname{id}_Y\) and target parameterization
\(h_{a\mid A}\) of the mapping cylinder.

For the forward central factor in
\eqref{slide:eq:central-phase-zero-traces}, the seam triple in actual product order
is
\[
 (\Lambda,\Lambda,\operatorname{id}_{Y*}^{-1}\Lambda)
 =(\Lambda,\Lambda,\Lambda).
\]
Thus its weight is \(0+c_\Sigma-0=c_\Sigma\).  For the independently typed reverse
central factor, the seam triple is
\[
 (\operatorname{id}_{Y*}\Lambda,\Lambda,\Lambda)
 =(\Lambda,\Lambda,\Lambda),
\]
and its weight is \(r_\Sigma+0-0=r_\Sigma\).  These computations see
\(\mathbf c\) and \(\mathbf r\) only through their totals; their coordinates
remain label-attached external data.
\end{proof}

\begin{proposition}[Weighted phase-zero edge covariance]
\label{slide:prop:phase-zero-walker-covariance}
For every \(A\subseteq[s]\), every \(a\notin A\), and all
\(\mathbf c,\mathbf r\in\mathbb Z^k\), the two ordered corrections are
\begin{equation}
 \boxed{\mu_Y(\Lambda,\Lambda,(h_{a\mid A}^{-1})_*\Lambda)=0}
 \label{slide:eq:forward-phase-zero-walker-correction}
\end{equation}
and
\begin{equation}
 \boxed{\mu_Y((h_{a\mid A})_*\Lambda,\Lambda,\Lambda)=0}.
 \label{slide:eq:reverse-phase-zero-walker-correction}
\end{equation}
They give equalities of completely marked Walker morphisms
\begin{equation}
 \boxed{
 H_{a\mid A}J_A^0(\mathbf c)
 =J_{A\cup\{a\}}^0(\mathbf c)}
 \label{slide:eq:forward-phase-zero-walker-covariance}
\end{equation}
and
\begin{equation}
 \boxed{
 R_{A\cup\{a\}}^0(\mathbf r)H_{a\mid A}
 =R_A^0(\mathbf r)}.
 \label{slide:eq:reverse-phase-zero-walker-covariance}
\end{equation}
The common weights in the two displays are \(c_\Sigma\) and \(r_\Sigma\), respectively.
At \(A=\varnothing\), these phase-zero traces agree with the fixed-site
traces established in \cref{fixed:thm-main-package}, with the same endpoint
markings and totals.  This comparison is made only at the old completed
vertex.
\end{proposition}

\paragraph{The phase-zero collar comparison.}
The phase-zero edge identity is obtained by comparing the exposed outgoing
row after composing with the edge cylinder.  The endpoint parameterization
square makes the ambient maps agree on the complete marking, while the common
Lagrangian data make both ordered Maslov triples vanish.  Forgetting the
central integer is therefore a geometric equality before the Walker weights
are restored.

\begin{proof}
In the forward composite
\(H_{a\mid A}J_A^0(\mathbf c)\), the right factor transports the incoming
Lagrangian to \(\Lambda\), the declared seam Lagrangian is \(\Lambda\), and
the left cylinder inverse-transports its target Lagrangian to
\((h_{a\mid A}^{-1})_*\Lambda\).  This is exactly the ordered triple in
\eqref{slide:eq:forward-phase-zero-walker-correction}.  After its order has been
fixed, \eqref{slide:eq:walker-common-lagrangian-data} identifies the third entry
with \(\Lambda\), so repeated-pair vanishing gives zero.  The composite
weight is therefore
\[
 c_\Sigma+0-\mu_Y(\Lambda,\Lambda,(h_{a\mid A}^{-1})_*\Lambda)=c_\Sigma.
\]
The forward ordinary equality
\eqref{slide:eq:forward-phase-zero-ordinary} retains both endpoint markings, and
the right side has weight \(c_\Sigma\) by
\cref{slide:lem:phase-zero-seam-data}; hence it upgrades to
\eqref{slide:eq:forward-phase-zero-walker-covariance}.

The reverse product is calculated separately.  In
\(R_{A\cup\{a\}}^0(\mathbf r)H_{a\mid A}\), the right cylinder transports
the source Lagrangian to \((h_{a\mid A})_*\Lambda\), the seam Lagrangian is
\(\Lambda\), and the reverse compression body inverse-transports its
low-boundary Lagrangian to \(\Lambda\).  Thus the ordered triple is the one
in \eqref{slide:eq:reverse-phase-zero-walker-correction}; it is not a permutation
of the forward triple.  Once again
\eqref{slide:eq:walker-common-lagrangian-data} and repeated-pair vanishing give
zero, and the weight is
\[
 0+r_\Sigma-\mu_Y((h_{a\mid A})_*\Lambda,\Lambda,\Lambda)=r_\Sigma.
\]
The independently typed ordinary equality
\eqref{slide:eq:reverse-phase-zero-ordinary}, together with the weight of its right
side, proves \eqref{slide:eq:reverse-phase-zero-walker-covariance}.  In both
directions the slide cylinder contributes exactly zero.

At \(A=\varnothing\), the direct completed chart, complete compression
body, endpoint rows, and endpoint parameterizations are those used in the
fixed-site construction.  Its phase-zero theorem identifies every labelled
order with that completed trace and gives weights \(c_\Sigma\) and \(r_\Sigma\).  Equality
of the underlying cobordisms with complete marking and of these integers yields
the stated comparison.  No phase word at a slid vertex is involved.
\end{proof}

\subsection{Phase words from completed markings}

For every completed vertex \(A\) and every label \(t\in[k]\), define the
supported three-twist word from the actual oriented product-framed row at
that vertex:
\begin{equation}
 \sigma_{t;A}:=T_{m_t^A}T_{\ell_t^A}T_{m_t^A},
 \qquad
 \Theta_{t;A}(u):=C(\sigma_{t;A},u).
 \label{slide:eq:actual-vertex-phase-factor}
\end{equation}
It is extended by the identity outside the actual one-holed-torus support
collar of label \(t\).  Thus \(\sigma_{t;A}\) is defined afresh from
\(\mathfrak C_A\), rather than transported from a phase word at a different
vertex.

For \(\epsilon\in\{0,1\}^k\), put
\(I_\epsilon=\{t:\epsilon_t=1\}\).  If
\(\rho=(\rho_1,\ldots,\rho_q)\) is an ordering of \(I_\epsilon\), set
\begin{equation}
 S_{A,\rho}^{\epsilon}(\mathbf u^+)
 :=\Theta_{\rho_q;A}(u_{\rho_q}^+)\cdots
   \Theta_{\rho_1;A}(u_{\rho_1}^+).
 \label{slide:eq:labelled-phase-order-word}
\end{equation}
The labelled pair \((t,u_t^+)\) moves as one unit under an order
replacement.  When \(I_\epsilon\) is empty, the right side means
\(C(\operatorname{id}_Y,0)\).

\begin{proposition}[Supported phase words and completed traces]
\label{slide:prop:endpoint-phase-words}
For every \(A\subseteq[s]\), every
\(\epsilon,\eta\in\{0,1\}^k\), and every
\(\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-\in\mathbb Z^k\), the word
in \eqref{slide:eq:labelled-phase-order-word} is independent of \(\rho\) as a
Walker morphism.  More precisely,
\begin{equation}
 S_{A,\rho}^{\epsilon}(\mathbf u^+)
 =S_A^\epsilon(\mathbf u^+)
 :=C\!\left(\sigma_A^\epsilon,U^+(\epsilon)\right),
 \label{slide:eq:endpoint-phase-cylinder}
\end{equation}
where
\begin{equation}
 \sigma_A^\epsilon:=\prod_{t\in I_\epsilon}\sigma_{t;A},
 \qquad
 U^+(\epsilon):=\sum_{t\in I_\epsilon}u_t^+,
 \label{slide:eq:endpoint-phase-total}
\end{equation}
and the mapping-class product denotes the common product.  The same
construction, with \(I_\eta:=\{t:\eta_t=1\}\), defines
\(S_A^\eta(\mathbf u^-)=C(\sigma_A^\eta,U^-(\eta))\), including the
empty word, where
\(U^-(\eta)=\sum_{t\in I_\eta}u_t^-\).

Define the completed-endpoint traces by
\begin{equation}
 \begin{aligned}
 J_A^\epsilon(\mathbf c,\mathbf u^+)
 &:=S_A^\epsilon(\mathbf u^+)J_A^0(\mathbf c),\\
 R_A^\eta(\mathbf r,\mathbf u^-)
 &:=R_A^0(\mathbf r)S_A^\eta(\mathbf u^-).
 \end{aligned}
 \label{slide:eq:completed-endpoint-phase-traces}
\end{equation}
Their weights are
\begin{equation}
 \operatorname{wt}(J_A^\epsilon)=c_\Sigma+U^+(\epsilon),
 \qquad
 \operatorname{wt}(R_A^\eta)=r_\Sigma+U^-(\eta).
 \label{slide:eq:completed-endpoint-phase-weights}
\end{equation}
Only at \(A=\varnothing\) do we invoke the assertion that these morphisms
agree with the order-independent full phase composites established in
\cref{fixed:thm-main-package}, with the same labelled parameters.
\end{proposition}

\paragraph{Vertexwise phase words and seam cancellation.}
Build the phase word at the completed vertex from the actual marked
three-twist neighborhoods, not from a word transported from another vertex.
Disjoint supports make adjacent phase factors commute, while the common
meridian Lagrangian makes each ordered Walker seam vanish.  The direct
compression trace can then be joined to the phase word with the same
label-attached weights; only at the old vertex is this construction compared
with the previously established fixed-site word.

\begin{proof}
The support collars of distinct labels are disjoint at every completed
vertex: this holds within a transported two-label sector, between distinct
protected sectors, and between either kind of sector and every unused site.
On the symplectic plane
\[
 P_t^A
 :=\operatorname{span}_{\mathbb Q}
   \{[m_t^A],[\ell_t^A]\}
 \subset H_1(Y;\mathbb Q),
\]
the positive
three-twist convention gives
\begin{equation}
 (\sigma_{t;A})_*[m_t^A]=-[\ell_t^A],
 \qquad
 (\sigma_{t;A})_*[\ell_t^A]=[m_t^A],
 \label{slide:eq:phase-plane-action}
\end{equation}
and both \(\sigma_{t;A}\) and its inverse are the identity on every other
direct summand.

Let \(i\neq j\).  Applying the Walker product law first to the ordered
product \(\Theta_{j;A}(v)\Theta_{i;A}(u)\) gives the correction
\begin{equation}
 \mu_Y((\sigma_{i;A})_*\Lambda,\Lambda,
       (\sigma_{j;A})_*^{-1}\Lambda).
 \label{slide:eq:forward-adjacent-phase-seam}
\end{equation}
On the \(i\)- and \(j\)-planes its triples of lines are, respectively,
\begin{equation}
 (\langle\ell_i^A\rangle,\langle m_i^A\rangle,
   \langle m_i^A\rangle),
 \qquad
 (\langle m_j^A\rangle,\langle m_j^A\rangle,
   \langle\ell_j^A\rangle).
 \label{slide:eq:forward-adjacent-phase-lines}
\end{equation}
Every remaining summand has three equal entries.  Direct-sum additivity and
repeated-pair vanishing make \eqref{slide:eq:forward-adjacent-phase-seam} zero.
The opposite ordered product
\(\Theta_{i;A}(u)\allowbreak\Theta_{j;A}(v)\) has the independently
ordered correction
\begin{equation}
 \mu_Y((\sigma_{j;A})_*\Lambda,\Lambda,
       (\sigma_{i;A})_*^{-1}\Lambda),
 \label{slide:eq:reverse-adjacent-phase-seam}
\end{equation}
whose \(i\)- and \(j\)-plane triples are
\[
 (\langle m_i^A\rangle,\langle m_i^A\rangle,
   \langle\ell_i^A\rangle),
 \qquad
 (\langle\ell_j^A\rangle,\langle m_j^A\rangle,
   \langle m_j^A\rangle).
\]
It vanishes separately.  Since the two marked mapping classes have disjoint
supports, they commute pointwise, and hence
\begin{equation}
 \Theta_{j;A}(v)\Theta_{i;A}(u)
 =C(\sigma_{j;A}\sigma_{i;A},u+v)
 =\Theta_{i;A}(u)\Theta_{j;A}(v).
 \label{slide:eq:adjacent-phase-swap}
\end{equation}

There is no additional correction in a longer word.  Let
\(\kappa=(\kappa_1,\ldots,\kappa_q)\) be an ordered prefix of distinct labels,
put
\[
 K_\kappa:=\{\kappa_1,\ldots,\kappa_q\},
 \qquad
 \sigma_{\kappa;A}
 :=\sigma_{\kappa_q;A}\cdots\sigma_{\kappa_1;A},
\]
and adjoin a new label \(j\notin K_\kappa\).  Its correction is
\[
 \mu_Y((\sigma_{\kappa;A})_*\Lambda,\Lambda,
       (\sigma_{j;A})_*^{-1}\Lambda).
\]
On a plane indexed by \(t\in K_\kappa\) the three lines are
\((\langle\ell_t^A\rangle,\langle m_t^A\rangle,
\langle m_t^A\rangle)\); on the \(j\)-plane they are
\((\langle m_j^A\rangle,\langle m_j^A\rangle,
\langle\ell_j^A\rangle)\); elsewhere they coincide.  It is therefore zero.
Induction followed by finite adjacent replacements proves
\eqref{slide:eq:endpoint-phase-cylinder}, with each weight kept attached to its
label and summed only after the order is fixed.  The same proof applies to
the minus-labelled parameters.

For the forward composite in
\eqref{slide:eq:completed-endpoint-phase-traces}, the ordered internal seam is
\begin{equation}
 \mu_Y(\Lambda,\Lambda,(\sigma_A^\epsilon)_*^{-1}\Lambda)=0,
 \label{slide:eq:forward-phase-trace-seam}
\end{equation}
while the independently typed reverse seam is
\begin{equation}
 \mu_Y((\sigma_A^\eta)_*\Lambda,\Lambda,\Lambda)=0.
 \label{slide:eq:reverse-phase-trace-seam}
\end{equation}
Their repeated pairs, together with
\cref{slide:lem:phase-zero-seam-data}, prove
\eqref{slide:eq:completed-endpoint-phase-weights}.

At the old vertex, each actual curve, framing, and support collar is the
corresponding fixed-site datum, so each
\(\sigma_{t;\varnothing}\) is literally the supported three-twist word used
in \cref{fixed:thm-main-package}.  The direct completed compression
body and its endpoint rows are also the same.  The fixed-site theorem assigns
weights
\(c_\Sigma+U^+(\epsilon)\) and \(r_\Sigma+U^-(\eta)\), exactly as above.  Equality of
the marked cobordisms and Walker integers proves the old-vertex comparison.
This comparison follows the construction of the actual-vertex words and
supplies no word, edge, or covariance statement at a slid vertex.
\end{proof}

\subsection{Complete conjugacy and all-phase covariance}

\begin{proposition}[Phase-cylinder commutation across an edge]
\label{slide:prop:phase-cylinder-commutation}
Let \(A\subseteq[s]\), \(a\notin A\), and \(B=A\cup\{a\}\).  Complete
marking transport gives, for every \(t\in[k]\),
\begin{equation}
 h_{a\mid A}\sigma_{t;A}h_{a\mid A}^{-1}=\sigma_{t;B}
 \label{slide:eq:complete-phase-conjugacy}
\end{equation}
as marked mapping classes with their support collars and product framings.
Consequently,
\begin{equation}
 \boxed{
 H_{a\mid A}S_A^\epsilon(\mathbf u^+)
 =S_B^\epsilon(\mathbf u^+)H_{a\mid A}}
 \label{slide:eq:forward-phase-cylinder-commutation}
\end{equation}
and, with the reverse-trace typing,
\begin{equation}
 \boxed{
 S_B^\eta(\mathbf u^-)H_{a\mid A}
 =H_{a\mid A}S_A^\eta(\mathbf u^-)}.
 \label{slide:eq:reverse-phase-cylinder-commutation}
\end{equation}
Every side of the first equality has weight \(U^+(\epsilon)\), and every
side of the second has weight \(U^-(\eta)\).
\end{proposition}

\paragraph{Naturality of the complete markings.}
Transport the complete meridian, longitude, framing, and support-collar data
through the edge map.  Naturality of the three twists gives the
conjugacy of the marked mapping classes; the two directions are then checked
separately by writing their ordered Maslov triples.  The common Lagrangian
forces the relevant repeated pair, so the edge transports the phase word
without adding a new central weight.

\begin{proof}
An orientation-preserving marked diffeomorphism conjugates a positive\linebreak[4] Dehn
twist about an oriented product-framed curve to the positive twist about the
transported curve.  For \(t=i_a,j_a\),
\cref{slide:prop:edge-homology-lagrangian} transports the actual embedded meridian
and longitude, their framings, and their complete support collars from the
row at \(A\) to the row at \(B\).  For every other label, the supported edge
fixes a neighborhood of the complete marking pointwise.  Applying twist
naturality to all three factors proves
\eqref{slide:eq:complete-phase-conjugacy}; this is framed marking conjugacy, not an
inference from homology.  Multiplication in any of the verified supported
orders gives
\begin{equation}
 h_{a\mid A}\sigma_A^\epsilon
 =\sigma_B^\epsilon h_{a\mid A},
 \qquad
 h_{a\mid A}\sigma_A^\eta
 =\sigma_B^\eta h_{a\mid A}.
 \label{slide:eq:product-phase-conjugacy}
\end{equation}

It remains to compare the Walker integers.  The two corrections in the
forward-typed equality, in the orders in which the factors occur, are
\begin{align}
 \mu_Y((\sigma_A^\epsilon)_*\Lambda,\Lambda,
       (h_{a\mid A}^{-1})_*\Lambda)&=0,
 \label{slide:eq:forward-phase-commutation-left-correction}\\
 \mu_Y((h_{a\mid A})_*\Lambda,\Lambda,
       (\sigma_B^\epsilon)_*^{-1}\Lambda)&=0.
 \label{slide:eq:forward-phase-commutation-right-correction}
\end{align}
The first vanishes because its last two entries agree by
\eqref{slide:eq:walker-common-lagrangian-data}; the second vanishes because its
first two entries agree.  Thus the two sides of
\eqref{slide:eq:forward-phase-cylinder-commutation} are
\begin{align*}
 C(h_{a\mid A},0)C(\sigma_A^\epsilon,U^+(\epsilon))
 &=C(h_{a\mid A}\sigma_A^\epsilon,U^+(\epsilon)),\\
 C(\sigma_B^\epsilon,U^+(\epsilon))C(h_{a\mid A},0)
 &=C(\sigma_B^\epsilon h_{a\mid A},U^+(\epsilon)),
\end{align*}
and \eqref{slide:eq:product-phase-conjugacy} identifies their marked mapping
classes.

For the reverse-trace typing, the two corrections are separately
\begin{align}
 \mu_Y((h_{a\mid A})_*\Lambda,\Lambda,
       (\sigma_B^\eta)_*^{-1}\Lambda)&=0,
 \label{slide:eq:reverse-phase-commutation-left-correction}\\
 \mu_Y((\sigma_A^\eta)_*\Lambda,\Lambda,
       (h_{a\mid A}^{-1})_*\Lambda)&=0.
 \label{slide:eq:reverse-phase-commutation-right-correction}
\end{align}
The first has equal first and second entries, while the second has equal
second and third entries.  These are the corrections for
\(C(\sigma_B^\eta,U^-(\eta))\allowbreak C(h_{a\mid A},0)\) and
\(C(h_{a\mid A},0)\allowbreak C(\sigma_A^\eta,U^-(\eta))\), respectively, in their
actual order.  They vanish independently, and the second equality in
\eqref{slide:eq:product-phase-conjugacy} identifies the marked maps.  This proves
\eqref{slide:eq:reverse-phase-cylinder-commutation} with unchanged phase weight and
zero slide weight.
\end{proof}

\begin{proposition}[All-phase completed-endpoint edge covariance]
\label{slide:prop:all-phase-edge-covariance}
For every \(A\subseteq[s]\), every \(a\notin A\), all
\(\epsilon,\eta\in\{0,1\}^k\), and all
\(\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-\in\mathbb Z^k\),
\begin{equation}
 \boxed{
 H_{a\mid A}J_A^\epsilon(\mathbf c,\mathbf u^+)
 =J_{A\cup\{a\}}^\epsilon(\mathbf c,\mathbf u^+)}
 \label{slide:eq:all-phase-forward-edge-covariance}
\end{equation}
and
\begin{equation}
 \boxed{
 R_{A\cup\{a\}}^\eta(\mathbf r,\mathbf u^-)
 H_{a\mid A}
 =R_A^\eta(\mathbf r,\mathbf u^-)}.
 \label{slide:eq:all-phase-reverse-edge-covariance}
\end{equation}
Their common weights are, respectively,
\(c_\Sigma+U^+(\epsilon)\) and \(r_\Sigma+U^-(\eta)\).
\end{proposition}

\begin{proof}
 At each vertex, disjoint support collars make the elementary phase maps
commute on the marked surface, so their labelled products depend only on the
active set and their total weight.  The edge carries those phase maps to the
next vertex, and the phase-zero covariance then supplies the completed trace
identity.  The argument keeps the forward and reverse composites separately
typed.

In the displayed calculation below, the fixed arguments
\((\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-)\) are suppressed from the
symbols \(J\), \(R\), and \(S\), but are unchanged at every equality.
Using the completed-endpoint definition, the forward phase-cylinder
commutation, and then phase-zero covariance gives the typed chain
\begin{align*}
 H_{a\mid A}J_A^\epsilon
 &=H_{a\mid A}S_A^\epsilon J_A^0\\
 &=S_{A\cup\{a\}}^\epsilon H_{a\mid A}J_A^0\\
 &=S_{A\cup\{a\}}^\epsilon J_{A\cup\{a\}}^0
 =J_{A\cup\{a\}}^\epsilon.
\end{align*}
Every step is an equality of Walker morphisms by
\cref{slide:prop:phase-cylinder-commutation,slide:prop:phase-zero-walker-covariance}, so
no correction is suppressed.  Its common weight is
\(c_\Sigma+U^+(\epsilon)\).

The reverse calculation is independently typed:
\begin{align*}
 R_{A\cup\{a\}}^\eta H_{a\mid A}
 &=R_{A\cup\{a\}}^0S_{A\cup\{a\}}^\eta H_{a\mid A}\\
 &=R_{A\cup\{a\}}^0H_{a\mid A}S_A^\eta\\
 &=R_A^0S_A^\eta
 =R_A^\eta,
\end{align*}
with common weight \(r_\Sigma+U^-(\eta)\).  Throughout both calculations the
parameters remain label-indexed.  The resulting complete morphisms retain
only the total base weight and the active phase-weight sum; they neither
transport nor recover an individual summand.
\end{proof}